What if string theory has a de Sitter excited state?

Joydeep Chakravarty and

Keshav Dasgupta

Department of Physics, McGill University,

3600 Rue University, Montreal, H3A 2T8, QC Canada.

We propose precise effective field theory criteria to obtain a four-dimensional de Sitter space within M-theory. To this effect, starting with the state space described by the action of metric perturbations, fluxes etc over the supersymmetric Minkowski vacuum in eleven-dimensions,  we discuss the most general low energy effective action in terms of the eleven-dimensional fields including non-perturbative and non-local terms. Given this, our criteria to obtain a valid four-dimensional de Sitter solution at far IR involve satisfying the Schwinger-Dyson equations of the associated path integral, as well as obeying positivity constraints on the dual IIA string coupling and its time derivative. For excited states, the Schwinger-Dyson equations imply an effective emergent potential different from the original potential. We show that while vacuum solutions and arbitrary coherent states fail to satisfy these criteria, a specific class of excited states called the Glauber-Sudarshan states obey them. Using the resurgent structure of observables computed using the path integral over the Glauber-Sudarshan states, four-dimensional de Sitter in the flat slicing can be constructed using a Glauber-Sudarshan state in M-theory.

Among other novel results, we discuss the smallness of the positive cosmological constant, including the curious case where the cosmological constant is very slowly varying with time. We also discuss the resolution of identity with the Glauber-Sudarshan states, generation and the convergence properties of the non-perturbative and the non-local effects, the problems with the static patch and other related topics. We analyze briefly the issues related to the compatibility of the Wilsonian effective action with Borel resummations and discuss how they influence the effective field theory description in a four-dimensional de Sitter space.

§ INTRODUCTION

An important question in theoretical physics is to understand whether our four dimensional observable universe can be described using a UV complete description. Over the last few decades string theory/M-theory has provided penetrating insights into important questions in quantum gravity. The theory is consistent in ten/eleven dimensions, and we obtain a four dimensional universe upon compactification. In this light, we can ask whether upon string compactification, is it possible to obtain a four dimensional universe with a positive cosmological constant or not.

The Gibbons-Maldacena-Nunez no-go theorem [1] has played a guiding light in searching for such solutions. By circumventing the no-go theorem, various works have provided a seemingly positive answer to the above question by constructing meta-stable de Sitter like vacua in string theory which involve lifting from a scale-separated AdS space [2]. However it has been pointed out that there can be various issues with such meta-stable realization of de Sitter vacua [3, 4, 5, 6].

In our work, we attempt to address the topic by focussing on a rather simple question: can the issues associated with the construction of a vacuum de Sitter solution be circumvented if we search for a de Sitter excited state instead? Regarding this, a very conservative criterion would be to reproduce the quantum analogue of the equations of motion such as the Einstein equations, i.e. the Schwinger-Dyson equations, for the on-shell degrees of freedom describing four-dimensional de Sitter spacetime.

The goal of our present work is two-fold. The first objective is to lay down precise effective field theory criteria to obtain four dimensional de Sitter spacetime in type IIB string theory starting with M-theory, and compactifying over an internal eight-manifold. We lay down our criteria in terms of the Schwinger-Dyson equations from the associated path integral description. As we show, the criteria are readily satisfied by a certain class of excited states called Glauber-Sudarshan states, but not by the vacuum or by arbitrary excited states. The second objective is to precisely understand the implications of the observables computed over the Glauber-Sudarshan states. In order to overcome the Gibbons-Maldacena-Nunez no-go theorem we need to take non-perturbative contributions to such observables, which requires an analysis of the resurgent structure of the associated path integral.

A reading guide: For the readers' convenience, the key results of the paper are summarized in <ref> as well as in Table <ref>. Following that a partial list of new results derived in this paper is listed in <ref>. The outline of the paper appears in section <ref> with appropriate links.
We pedagogically discuss the Glauber-Sudarshan state in M-theory in <ref> and <ref>. The rest of the work deals with understanding the consequences, building upto the EFT analysis in <ref>.
In the conclusion section <ref>, we have a slightly non-technical longer discussion in a question and answer format which summarizes the results from this paper exhaustively without going into technical details.

§.§ Brief summary of the key results

Four-dimensional de Sitter in flat slicing

within the framework of M-theory

Metric     $ds^2 = g_s^{-8/3}\left(-dt^2 + dx_1^2 + dx_2^2\right) + g_s^{-2/3}{\rm H}^2(y) g_{mn}(y) dy^m dy^n + g_s^{4/3} \delta_{ab} dw^a dw^b$,

(M-theory)     on ${\bf R}^{2,1}\times {\cal M}_6 \times \dfrac{\mathbb{T}^2}{{\cal G}}$ (see (<ref>))

$g_s$ is the Type IIA string coupling.

Compactification to 4d dS: Take $g_s \to 0$, and T-duality along $x_3$.

Path Integral    Total action $\hat{\bf S}_{\rm tot}\lc {\bf g}_{\rm AB}, {\bf C}_{\rm ABD} \rc$ includes all possible perturbative terms,

with specific non-perturbative and non-local contributions, as given in (<ref>).

Path integral description utilizes the action $\hat{\bf S}_{\rm tot}\lc {\bf g}_{\rm AB}, {\bf C}_{\rm ABD} \rc$ from (<ref>)

with the supersymmetric vacuum $\ket{\Omega}$ defined over ${\bf R}^{2,1}\times {\cal M}_6 \times \dfrac{\mathbb{T}^2}{{\cal G}}$ (see (<ref>)).

EFT Criteria     1. Restriction of time dependence of Type IIA string coupling:

on action      $g_s(t) \ll 1, \, \, \, \, $ and $ \, \, \,\, \dfrac{\partial g_s(t)}{ \partial t} \propto g_s^{k} \, \, (k>0)$, (see (<ref>))

for four-dim dS     2. Satisfying the Schwinger Dyson equation for 11-dim fields:

$\dfrac{\delta \hat{\bf S}_{\rm tot}\lc \langle{ {\bf g}_{\rm AB}}\rangle_\sigma, \langle{ {\bf C}_{\rm ABD}}\rangle_\sigma \rc}{ \delta\langle{ {\bf g}_{\rm AB}}\rangle_\sigma}  =   \dfrac{\delta \hat{\bf S}_{\rm tot}\lc \langle{ {\bf g}_{\rm AB}}\rangle_\sigma, \langle{ {\bf C}_{\rm ABD}}\rangle_\sigma \rc}{ \delta\langle{{\bf C}_{\rm ABD}}\rangle_\sigma} =0$, (see (<ref>))

Solutions     The vacuum $\ket{\Omega}$ itself does not satisfy EFT criteria.

Glauber-Sudarshan states satisfy the above EFT criteria, defined as

$\vert\sigma\rangle = \mathbb{D}\lc \sigma, {\bf g}_{\rm AB}, {\bf C}_{\rm ABD}\rc \ket{\Omega} $

where $\mathbb{D}(\sigma, {\bf g}_{\rm AB}, {\bf C}_{\rm ABD})$ is a displacement operator for the GS state given in (<ref>).

The metric  $\langle{\bf g}_{\mu \nu}\rangle_\sigma={\langle \sigma |{\bf g}_{\mu \nu}|\sigma \rangle/ \langle\sigma\vert\sigma\rangle}$ is given by $ds^2 = \lc\Lambda t^2 \rc^{-8/3} (-dt^2 + d{\bf x}^2)$.

An overview of the EFT criteria. Here, in our EFT criteria, for representation purposes we have only included the bosonic sector.

To pose the effective field theory (EFT) criteria, we work with M-theory on a supersymmetric background with the topology ${\bf R}^{2,1}\times {\cal M}_6 \times {\mathbb{T}^2\over {\cal G}}$, where ${\cal M}_6 $ denotes a six-dimensional  internal manifold while ${\cal G}$ is an orbifold action without a fixed point. Here as conventional, one direction of $\mathbb{T}^2$ denotes the M-theory circle, while a T-duality along the other direction combined with ${\bf R}^{2,1}$ gives a four dimensional universe upon compactification in type IIB theory. Alternatively, using this setup, we can simply attempt to lift a four-dimensional de Sitter space in type IIB in the flat slicing to a eleven-dimensional configuration. To systematically do this, we need to carefully specify the dependence of the eleven-dimensional metric on the dual type IIA string coupling $g_s$, which is a function of the flat-slicing (conformal) temporal coordinate as does the volume of ${\cal M}_6 $.

We write down the most general perturbative action ${\bf Q}_{\rm pert}({\bf \Xi})$ in (<ref>) at far IR using the set of on-shell fields $\{{\bf \Xi}\}$, where the set goes over the G-flux ${\bf G}_{\rm ABCD}$ and the metric ${\bf g}_{\rm AB}$ components with $({\rm A, B}) \in {\bf R}^{2,1}\times {\cal M}_6 \times {\mathbb{T}^2\over {\cal G}}$. (We will leave the fermionic degrees of freedom for later work.) Here ${\bf Q}_{\rm pert}({\bf \Xi})$ incorporates all possible higher derivative terms corresponding to the bosonic fields. The coefficients of the higher derivative terms
systematically carry information about the full UV theory in the sense of an exact renormalization group (ERG) approach.

However, from the Gibbons-Maldacena-Nunez theorem, a simple scaling analysis tells us that accounting solely from perturbative corrections does not allow us to create a de Sitter solution. Consequently, we must take into account the full action $\hat{\bf S}_{\rm tot} ({\bf \Xi})$, in (<ref>) which incorporates the perturbative series  ${\bf Q}_{\rm pert}({\bf \Xi})$ from (<ref>) in the full non-perturbative and non-local interactions. An example of a non-perturbative contribution is given by wrapping five-brane instantons on the manifold $\mathcal{M}_6$.

The generic interacting theory on the supersymmetric backgound on ${\bf R}^{2,1}\times {\cal M}_6 \times {\mathbb{T}^2\over {\cal G}}$ provides us with an interacting vacuum state $\ket{\Omega}$. A state in our theory, called the Glauber-Sudarshan state and denoted by $\ket{\sigma}$, is obtained by acting with operator insertions over the interacting vacuuum. More precisely, $\ket{\sigma}= {\mathbb D}_{\sigma}({\bf \Xi})\ket{\Omega}$, where ${\mathbb D}_{\sigma}({\bf \Xi})$ denotes the displacement operator with a specific insertions of the on-shell fields from the set $\{{\bf \Xi}\}$ as in (<ref>).

With all this in place, we now outline the key results related to the first objective, which are also summarized in Table <ref>. In order to obtain a consistent four-dimensional description, the EFT criteria involve satisfying the Schwinger-Dyson equations for the on-shell fields, along with imposing  positivity conditions on Type IIA string coupling $g_s$. The EFT criteria are not satisfied by the vacuum state or by a generic coherent state. However we show that a very specific class of excited states labelled by Glauber-Sudarshan (GS) states satisfy the aforementioned criteria.

We now describe some key results obtained within our second objective, i.e. implications of the Glauber-Sudarshan states. Over a Glauber-Sudarshan state, we show that the series expansion of perturbative action leads to an asymptotic series with a specific factorial growth. In literature, this specific form of the asymptotic series is known as the Gevrey series. A Borel resummation of the Gevrey series leads to a resurgent trans-series from where one could predict the non-perturbative terms in the effective action. For the Glauber-Sudarshan states, the non-perturbative term in $\hat{\bf S}_{\rm tot} ({\bf \Xi})$ can be thus be recovered this way leading us to the final form as given in (<ref>). This allows us to determine the complete non-perturbative, as well as the non-local, contributions to the Schwinger-Dyson equations as shown in (<ref>). The non-local terms, which appear from integrating out the massless off-shell degrees of freedom are also shown to take a trans-series form.
Lastly, the Borel resummation of the Gevrey series can be used to understand the smallness of the positive cosmological constant, including the curious case of a very slowly varying cosmological constant. The latter also leads to a consistent Glauber-Sudarshan state which we illustrate here in some details.

§.§ What are the new results in this paper?

Apart from the key results discussed above, we now provide a quick glance at the other results in our work. These are important towards concretely establishing the central results in Table <ref> as well as opening up avenues for further research.

$\bullet$ A new form for the displacement operator which is taken here to be complex instead of real.

$\bullet$ Resolution of identity with the Glauber-Sudarshan states.

$\bullet$  The factorial growths of the nodal diagrams and the corresponding amplitudes.

$\bullet$ Possible reasons for the Gevrey growth, and subsequent Borel resummation.

$\bullet$ Generation of non-local quantum terms by integrating out the massless off-shell states.

$\bullet$  Generation of the non-perturbative quantum effects by summing over the instanton saddles.

$\bullet$ Convergence properties of all the quantum terms in the Borel resummed effective action.

$\bullet$ Elucidating the differences between the Glauber-Sudarshan states and the coherent states.

$\bullet$ Reason for the smallness of the positive cosmological constant.

$\bullet$ Constructing a Glauber-Sudarshan state associated to a slowly varying cosmological constant.

$\bullet$ Proof and the derivation of the equations of motion as Schwinger-Dyson equations.

$\bullet$ Computing the emergent potential and satisfying the Effective Field Theory (EFT) criteria.

$\bullet$ The problems and the deceptive dynamics with the usage of the de Sitter static patch.

$\bullet$ Studying the compatibility of the Wilsonian effective action with Borel resummed action.

§.§ Organization of the paper and summary

The paper is organized in the following way. In <ref> we analyze the reasons why would a four-dimensional de Sitter spacetime cannot exist as a vacuum solution in string theory. Our analysis is in type IIB but the problems are generic. In <ref> we study this from a M-theory uplift, with special emphasis on the vacuum EOMs in <ref> and including the quantum corrections in <ref>. In <ref> we discuss how narrowly a de Sitter spacetime misses being a vacuum solution. This section partially constitutes a review of the failure of a  de Sitter vacua in string theory as a follow-up to [3], but put in a different light.

Having discussed the failure, our aim in <ref> is to realize this as a Glauber-Sudarshan state in string theory. We start in section <ref> by pointing out issues in the static patch, where one might have thought of realizing a de Sitter spacetime, by including the problems and the subtleties coming from the Wilsonian integrating-out procedure itself. The way out is to realize de Sitter spacetime as an excited states, and in <ref> we point out why we couldn't realize it as a coherent state, but only as a Glauber-Sudarshan state. In <ref> we quantify the displacement operator and using this show how the de Sitter metric appears from a path-integral formulation as an emergent quantity.

The path-integral formulation has its own set of subtleties which we point out in <ref>. In <ref> we argue that the path-integral (<ref>) can be rewritten as a binomial expansion, using which we show in <ref> how one may determine the nested structure and how one may actually compute it. The binomial expansion form also helps us to see the factorial growth much more clearly which we discuss in <ref>. In fact using this we can compute the one-point functions much more efficiently. The special asymptotic behavior appearing from the path-integral leads to the so-called Gevrey series on which we can do the Borel resummation. This is detailed in <ref>, wherein we also discuss the origin of the non-perturbative effects.

The Borel resummation actually leads, for a certain well-defined toy set-up with scalar degrees of freedom, to an actual determination of the cosmological constant. This and other details are the subject of <ref>. In <ref> we combine the ideas from the binomial expansion and the Borel resummation to provide a closed form expression for the four-dimensional cosmological constant in
(<ref>). This was already argued in [7] to be positive definite, but it wasn't clear from there that it could be made small. In <ref> we discuss procedures how to make the four-dimensional cosmological constant small. Our procedures include introducing new computational tools called the Borel Boxes, as well as finding ways to allow for convergent summing over the Borel Boxes, which we describe in some details in <ref>. The final expression after summing over a subset of Borel Boxes is given by (<ref>), which does show considerable decrease if we carefully take the degeneracies into account. However the analysis is technically challenging and is not clear whether the decrease could be as small as
$10^{-120} {\rm M}_p^2$. We leave a detailed study on this for an upcoming work [8]. Finally in <ref> we study the curious case of the cosmological constant slowly varying with time.
In the limit where the variation is very slow, we argue that this may be easily accommodated in our set-up by constructing a new Glauber-Sudarshan state. It is possible that this might have some connection to the recent DESI BAO result [10], but we do not make any detailed elaboration here and leave this for future work.

The path-integral formulation in <ref> provides an emergent metric configuration, but doesn't provide a detailed connection to other emergent degrees of freedom like fluxes and fermionic condensates. Such a connection appears from the Schwinger-Dyson equations which may also be derived from the path-integral. Our aim in <ref> is the derivation of the Schwinger-Dyson equations and study the consequences. These equations are specifically designed to provide the dynamics of the emergent degrees of freedom, but are in general harder to implement in real practice. This improves drastically once we determine the resolution of identity for the Glauber-Sudarshan states as we show in <ref>. The resolution of identity allows us to rewrite the Schwinger-Dyson equations in a more efficient way which, in turn, leads us to determine the exact reason for the failure of the vacuum solution from <ref>. In addition to that, it provides a way to incorporate the non-local and the non-perturbative effects much more clearly. We discuss this in <ref> wherein we show two things: one, how the Borel resummation of the Gevrey series introduces the non-perturbative effects and two, how by integrating out the massless off-shell states introduces the non-local effects. For the latter, to get a convergent form, one needs a more detailed mathematical analysis which we elaborate in <ref>. The final convergent forms for both the non-perturbative and the non-local effects are useful but raises a different question related to the compatibility of the Wilsonian effective action with Borel resummation. We discuss this briefly in <ref>, but leave a more detailed study for future work. Finally in <ref> we show that the Schwinger-Dyson equations lead to two set of equations in the presence of Faddeev-Popov ghosts and gauge fixing terms. One set of equation, given by the first equation in (<ref>), is somewhat similar to the EOMs that we studied in <ref> in the sense that they are almost the same EOMs except the fields are replaced by their expectation values. However the second set of equations are new. These set of equations, coming from the second equation in (<ref>) and the two other equations from (<ref>), signify the deviations from the vacuum configuration.

One consequence of our study of path-integrals, Borel resummations and the Schwinger-Dyson equations is the construction of the two EFT criteria given by (<ref>), that we discuss in <ref>. Another consequence of our analysis, and especially from the study of the Schwinger-Dyson equations, is the construction of an emergent potential which is different from the vacuum potential. We study this in <ref> and show that, using this emergent potential, the EOMs for the expectation values of the on-shell degrees of freedom take a simpler form as in (<ref>). Moreover, since the expectation values are never at the minima of the potential $-$ plus the potential being an emergent one $-$ they are not subjected to the EFT criteria of [6]. The first EFT criterion of (<ref>) then restricts the dynamics to lie within the temporal domain of (<ref>), and in <ref> we argue that this criterion is violated in the static patch leading to non-convergence of the instanton series. The second criterion of (<ref>) is deeply rooted in the quantum series expressed using  (<ref>), and is determined from actual computations of wrapped five-brane instantons. <ref> is dedicated to the study of these instantonic effects. We specifically take only five-brane instantons to illustrate this, dedicating <ref> and <ref> to the BBS instanons [11] and <ref> for the KKLT instantons [2]. The study of the BBS instantons in <ref> however reveals the incompleteness of the non-local interactions that we had derived in <ref>, which we rectify in <ref>. Once all corrections are made, the effective low energy action takes the form as in (<ref>), and the corresponding Schwinger-Dyson equations become (<ref>).
In <ref> we study the KKLT instantons and discuss the special case of a temporally varying cosmological constant $-$ that we studied in <ref> $-$ and show how our Schwinger-Dyson equations may easily accommodate such a scenario.

We provide our conclusions in <ref> where in <ref> we take the opportunity to give a detailed summary of our work in a non-technical question-and-answer format. There is a dual motive for such a format: not only is our aim to answer questions justifying the validity of the Glauber-Sudarshan states in providing a positive cosmological constant background, but to also  summarize some of the earlier related works that are sometimes difficult to access due to their overly technical natures. We dedicate <ref> in pointing out some of the future directions and open questions. In Appendix <ref> we list various coordinates patches that one puts to study four-dimensional de Sitter spacetime.

§ DE SITTER SPACE AS A VACUUM SOLUTION IN STRING THEORY?

To see why we would like to view four-dimensional de Sitter space-time as a coherent state, or more appropriately as a Glauber-Sudarshan state[The distinction between coherent states and Glauber-Sudarshan states will be spelled out in section <ref>.], we need to take a step back and start by asking some basic questions. With this in mind, let us take the following four-dimensional action:

S = ∫d^4x √(-g_4)[1/16 πG_N(R_4 - Λ/2) + 14g_YM^2 F ∧∗F + .... ], where $G_{\rm N}$ is the four-dimensional Newton's constant, $g_{\rm YM}$ is the gauge coupling for gauge field ${\bf F}$, $\Lambda$ is the positive cosmological constant, and the dotted terms denote higher order interactions.

If (<ref>) is considered, then there is nothing to show! The theory admits, in four-dimension, a positive cosmological constant solution which we can identify with the de Sitter space-time. The issue here is the middle term that is proportional to $\sqrt{-{\bf g}_4} \Lambda$: string theory or M-theory does not come equipped with a term like this! The question then is the following:
Can we show that this term comes out from dimensional reduction of a ten or eleven dimensional action? In other words, can the following action in (say) eleven-dimensions:

S = M_p^9∫d^11x √(-g_11)(R_11 + G_4 ∧∗G_4) + M_p^9∫C_3 ∧G_4 ∧G_4 + M_p^3∫C_3 ∧𝕏_8 + ..., reproduce the action (<ref>) by some dimensional reduction to four-dimensions? Here we denote the three-form flux as ${\bf C}_3$ with ${\bf G}_4 = d{\bf C}_3 + ...$, and $\mathbb{X}_8$ is a fourth order curvature polynomial. Note that (<ref>) has no scale other than ${\rm M}_p$ and the size of the internal manifold; and therefore both $G_{\rm N}$ and $g^2_{\rm YM}$ in (<ref>) should come only from the two aforementioned scales.

The challenge is to reproduce the four-dimensional cosmological constant term. Clearly since $\Lambda$ does not explicitly show up in (<ref>), it must appear from the flux and the dotted terms in (<ref>) which we, for our purpose, package them as the energy-momentum tensor ${\bf T}_{\rm MN}$. The condition to get a positive cosmological constant solution is simply the statement that [1]:

𝕋^μ_μ  >   45 𝕋^m_m, where $x^\mu$ is the coordinate of four-dimensional space-time and $y^m$ is the coordinate of the internal seven-dimensional compact manifold. If (<ref>) is satisfied then we can get a four-dimensional de Sitter space-time. Unfortunately, as shown in [1], this simple condition cannot be satisfied by any combination of fluxes, branes, anti-branes, and orientifold planes. Somewhat surprisingly, even perturbative quantum correction (to any order) fails to satisfy (<ref>) [1]. Why is this the case? Where do all the energies from fluxes, branes, planes etc. go, and why do they fail to give positive energy to the space-time? A more puzzling question is regarding the perturbative corrections. Why aren't they effective in lifting the energy of space-time to some positive value? However even if perturbative corrections were effective, why is the cosmological constant, which measures the amount of positive energy, so small?

We have already raised too many questions, so in the following we will try to answer at least a subset of them. The first one, related to the energies of the classical sources like fluxes, branes and planes, is relatively easy. The answer lies in the strange property of gravity itself: the back-reactions of the classical sources not only curve the background, but also create negative potentials. Thus all the positive energies from the classical sources are basically nullified by the negative potentials created by the back-reactions of them on space-time. This is also the reason why negative cosmological constant solutions are so ubiquitous in string theory: it doesn't require much effort to create an effective negative potential using classical sources.

This means, to create an effective positive energy we require some kind of sources that do not back-react so strongly on the space-time and yet still provide enough positive energies to make the net effect positive. Clearly since classical sources cannot do this, the sole burden now falls on quantum corrections. Herein, and as alluded to earlier, lies a surprise: the perturbative quantum corrections fail to do this. Why is this the case?
Could non-perturbative corrections save the day? If yes, does this mean that de Sitter space can exist as a vacuum solution in the presence of non-perturbative corrections?

To start answering the aforementioned questions, let us first quantify some details. Instead of finding a four-dimensional de Sitter space in M-theory, we will look for such a space-time in type IIB. One specific ansätze for the metric configuration is the following:

ds^2 = 1H^2(y)Λ|t|^2(-dt^2 + dx_1^2 + dx_2^2 + dx_3^2) + H^2(y) g_mn(y) dy^m dy^n, which would give us a four-dimensional de Sitter space-time in a flat-slicing with the conformal time $t$ ranging as $-\infty < t < 0$ as shown in figure <ref>; and
where ${\rm H}(y)$ is the warp-factor with $g_{mn}(y)$ being the metric on the six-dimensional internal space that is generically a non-Kähler manifold (which may or may not be complex). One could however raise a question on the form of the metric (<ref>) itself: could we even express the background metric in the above form? The answer is more clear from a M-theory uplift, where we will show that a description like (<ref>) is only valid in a limited temporal domain.

§.§ The M-theory uplift of the IIB background

As mentioned above, somewhat surprisingly such a configuration is dual to M-theory on a compact eight-manifold[And not a compact seven-manifold!], which is a torus fibered over a six-dimensional base, implying that we can continue using the action (<ref>) [12]. The step to reach M-theory is to first T-dualize along the spatial $x_3$ direction of (<ref>), and then uplift the resulting configuration to eleven-dimensions. T-dualizing along $x_3$ leads to a time-dependent type IIA coupling $g_s \equiv {\rm H}(y) \sqrt{\Lambda}\vert t \vert$. The coupling ${g_s\over {\rm H}(y)}$ remains smaller than 1, i.e. ${g_s \over {\rm H}(y)} < 1$, in the interval:

-1√(Λ)   <    t   <   0, which is surprisingly the trans-Planckian bound advocated in [13]! Thus the de Sitter space-time is automatically defined within the allowed temporal domain governed by the so-called TCC (see also figure <ref>). One might however ask why is this the case. For example, lifting to M-theory means we are automatically going to strong type IIA coupling, so shouldn't this be an issue here?

(5,5) – (5,0) – (0,5) – (5,5);
[blue] (0,5) .. controls (2.7,3.7) .. (5,3.3);
[dashed] (0,5) .. controls (3.5,2.5) .. (5,2);
[fill] (3,5)  node[above]  $t =0$ ;
[fill] (3,2.9) node [rotate=-25,above] $t_c = -\frac{1}{\sqrt{\Lambda}}$ ;
[fill] (3.4,3.7) node [rotate=-10,above] $t = $ constant;
[fill] (2.5,2.5) node [rotate=-45,below] $t = -\infty$;

Flat slicing, with a constant Poincare time slice denoted in blue and $t$ being the conformal time. Here $t_c$ denotes the TCC cutoff.

The answer is actually no because of the contributions from the non-perturbative and the non-local counter-terms. In fact, and as mentioned earlier, it is the non-perturbative and the non-local quantum terms that are actually responsible in generating the kind of background we want. (The perturbative terms are just red-herrings in the problem.) Both these quantum terms go as ${\rm exp}\left(-{{\rm M}_p^2\over g_s^2}\right)$ and therefore if we keep $g_s$ strong then it will be difficult to allow for a convergent series of these quantum effects. This means, while M-theory is defined at strong type IIA coupling, we want to be in the opposite limit, i.e. in the limit when ${g_s\over {\rm H}(y)} < 1$.
(A more elaborate analysis of the aforementioned statement has appeared in [14, 15, 16] which the readers may look up for more details.)

Before moving further, let us quickly clarify one question that may arise at this stage: why go to M-theory? Can we not do the analysis directly in the type IIB set-up? The answer is unfortunately no because the type IIB background with the metric (<ref>) is actually at the constant coupling limit of F-theory [17] and therefore the IIB coupling $g_s^{(b)} = 1$ where even S-duality does not help. Going away from this point would switch on time-dependent dilaton and axion (because of the dynamical seven-branes) making the analysis even harder to track. (Additionally, we really do not have a well defined Lagrangian description for the type IIB case.) In fact we will soon see that keeping the internal manifold and the fluxes time-independent will become problematic. There will also more severe problems associated with the very existence of the vacuum solution (<ref>) itself that we will encounter soon. Such a conclusion will be borne out from a careful analysis of the system, but for the time-being we shall suffice with the  metric ansätze (<ref>).

Assuming this to be the case, we can now lift[The lifting from type IIB involves at least one T-duality, so one needs to take into account all the $\alpha'$ corrections to the Buscher's T-duality rules. This is surprisingly tractable as shown in the upcoming work [9], but we will avoid these subtle nuances here. ] the metric (<ref>) to eleven dimensions by using the type IIA string coupling $g_s \equiv {\rm H}(y) \sqrt{\Lambda} \vert t \vert$ in the following way:

ds^2 = g_s^-8/3(-dt^2 + dx_1^2 + dx_2^2) + g_s^-2/3H^2(y) g_mn(y) dy^m dy^n + g_s^4/3 δ_ab dw^a dw^b, where $(w^a, w^b) = (x_3, x_{11}) \in {\mathbb{T}^2\over {\cal G}}$ with ${\cal G}$ being an orbifold action without a fixed point; and $(y^m, y^n) \in {\cal M}_6$ such that the internal eight-manifold is locally ${\cal M}_6 \times {\mathbb{T}^2\over {\cal G}}$. The warp-factor ${\rm H}(y)$ is assumed to be smooth over the six-manifold ${\cal M}_6$. Question is whether (<ref>), or (<ref>), solves the background EOMs?

When we talk of EOMs in supergravity we assume that we are at low energies where all massive stringy and KK degrees of freedom have been integrated away, and the theory is completely specified by a small subset of known low energy degrees of freedom. In other words we want the EOMs of a low energy Wilsonian effective action whose solution is (<ref>). Note that we have made at least two crucial assumptions:

$\bullet$ Existence of a well-defined Wilsonian effective action for the finite set of low energy degrees of freedom, and

$\bullet$ Existence of a supergravity limit where we can consistently truncate any quantum series to a finite subset.

§.§ Analysis of the EOMs and consistency conditions

We shall see below that both the above assumptions are problematic, and in particular it looks unlikely for the second assumption to be true. The reasoning is simple: even if we are at weak type IIA coupling, the curvature tensors can become very strong in the temporal domain (<ref>) and specifically when we approach late time where $g_s \to 0$. There is fortunately a resolution to the above conundrum of strong curvature [9], but even then the first problem could actually rule out our whole procedure itself! Unfortunately however these are not the only issues. There are other difficulties that we will face soon when we start writing the EOMs themselves, assuming of course that it is allowed to do so.

We will then start with a narrow minded approach wherein we will assume that the aforementioned two points are true: namely, there exists a Wilsonian effective action that allows for a well-defined truncated supergravity limit. Can (<ref>), or (<ref>), be a solution to such a system?

To see this we will have to write all possible EOMs allowed by the system from say eleven-dimensional point of view. This is a formidable exercise, but can be done with some efforts. We have already proposed a metric ansätze in (<ref>) with dominant $g_s$ behavior, and we take the following ansätze for the G-flux components:

G_MNPQ(x, y; g_s) = ∑_k = 0^∞G^(k)_MNPQ(x, y) (g_sH(y))^2k3, where $k \in {\mathbb{Z}\over 2}$, $({\rm M, N, P, Q}) \in {\bf R}^{2, 1} \times {\cal M}_6 \times {\mathbb{T}^2\over {\cal G}}$, ${\bf x} \in {\bf R}^2, y \in {\cal M}_6$ and $t = t(g_s)$ is the conformal time. Notice that, according to (<ref>), $k = 0$ constitute the time-independent components of the G-flux, and $k > 0$ are the sub-dominant contributions that control the temporal behavior of the G-flux components[The actual ansätze that works for all string theories descending from M-theory is a bit more involved as discussed in [9]. But for the present exposition, these technicalities are not needed and therefore may be avoided. Interested readers may look up our upcoming paper [9].]. Plugging (<ref>) and (<ref>) in the EOMs lead to a complicated set of coupled equations as shown in
[14, 15, 16] which can nevertheless be combined together to extract the following consistency condition:

12Λ∫d^6y √(g_6(y))  H^8(y) + 58 ∫d^6 y √(g_6(y)) G^(3/2)_mnab(y) G^(3/2)mnab(y)

+      2κ^2 T_2(n_b + n_b)
+ 12 ∫d^6 y
√(g_6(y))  H^4(y) ([ℂ^a_a(y)]^(0) +
[ℂ^m_m(y)]^(0) - 2H^4(y)[ℂ^i_i(y)]^(0))= 0,
where $\Lambda$ is the four-dimensional cosmological constant, ${\rm T}_2$ is the tension of the 2-brane and the anti-2-brane (their number being $n_b$ and $\overline{n}_b$ respectively), $\kappa$ is a constant, ${\rm H}(y)$ is the smooth warp-factor appearing in (<ref>),
$\left[\mathbb{C}^{\rm M}_{\rm M}(y)\right]^{(k)}$ is the trace of the quantum term[The notation differs a bit from [14] because we will define the quantum series in a slightly different way to avoid repeating details from [14].] (see (<ref>)), and the indices are raised or lowered by the unwarped metric components (i.e. the $g_s$ independent parts of the metric components).

The first term in (<ref>) is a consequence of the backreactions from fluxes, branes and O-planes and is positive definite. The second and the third terms, which are also positive-definite, are from the G-fluxes and from branes and anti-branes (both integer and fractional) respectively, and the last term is from the quantum effects. Some of the results mentioned above can now be easily quantified from (<ref>).

$\bullet$ In the absence of quantum terms, there is no solution in the system for $\Lambda > 0$. If $\Lambda < 0$, solutions can easily exist. This shows that branes, anti-branes, O-planes and fluxes are simply red-herrings in the problem, i.e. they cannot give rise to positive $\Lambda$.

$\bullet$ The time-independent part of the G-flux, namely ${\cal G}^{(0)}_{\rm MNPQ}$, does not enter (<ref>). In fact one can easily argue that the time-independent part of the G-flux components cannot solve the background EOMs. This issue does not go away if the use static-patch because of other factors plaguing the physics in a
static-patch as we shall discuss later.

$\bullet$ Solution with $\Lambda > 0$ would appear to exist only in the presence of quantum terms and in particular only if the quantum terms satisfy the following inequality:

∫d^6y √(g_6(y))  H^8(y) [ℂ^i_i(y)]^(0)
>   12 ∫d^6y √(g_6(y))  H^4(y)([ℂ^a_a(y)]^(0) + [ℂ^m_m(y)]^(0)), where $x^i \in {\bf R}^2, w^a \in {\mathbb{T}^2\over {\cal G}}$ and
$y^m \in {\cal M}_6$. The condition (<ref>), while encouragingly reminiscent of the bound (<ref>) (but now completely in the language of quantum terms), actually creates more puzzles than resolving them. The first is the issue of hierarchy, which amounts to saying as to how many quantum terms should we take to justify (<ref>); and second is the unclear nature of (<ref>), which amounts to asking whether only perturbative, non-perturbative or non-local quantum terms, or a combination of all of them, would be required to maintain the inequality in the temporal domain $-{1\over \sqrt{\Lambda}} < t < 0$.

The last few points raised above are important and therefore let us clarify them in the following. From the discussion it is clear that only the quantum terms can possibly allow a $\Lambda > 0$ solution to exist in the system, but this immediately raises the following subtlety in
(<ref>): the first three set of terms therein are classical, whereas the last set of terms are quantum. Thus somehow we are balancing the classical terms with the quantum terms, so we have to carefully specify what (<ref>) entails. The quantum terms $\mathbb{Q}_{\rm T}$ are typically classified  as a series in
${g_s^a\over {\rm M}_p^b}$ perturbatively, or as a series in
${\rm exp}\left(-{{\rm M}_p^b\over g_s^a}\right)$ non-perturbatively,
so the contributions that actually enter (<ref>) are the traces of the corresponding energy-momentum tensors i.e.

ℂ^P_M(y, g_s) = g^PN(y) ℚ_T(y, [g_s])g^MN(y, g_s) ≡∑_k = 0^∞∑__{l_i},{n_j}[ℂ^P_M(y)]^(k, {l_i}, {n_j})(g_sH(y))^θ({l_i},{n_j}) - a̅^P_M + 2k3,
where ${\bf g}_{\rm MN}(y, g_s)$ is the warped and $g_{\rm MN}(y)$ is the un-warped (i.e. $g_s$ independent part of the) metric components from (<ref>); $\theta(\{l_i\}, \{n_j\})$ is the quantum scaling to be defined momentarily, and $\bar{a}^{\rm P}_{\rm M} \in {\mathbb{Z}\over 3}$ such that:

(a̅^0_0,  a̅^i_i,  a̅^m_m,  a̅^a_a) = (83,  83,  23,  -43), which are clearly related to the $g_s$ exponents in (<ref>). This is not surprising because the definition (<ref>) involves derivative with respect to the warped metric. Putting everything together, this means that the only way (<ref>) would make sense is if the
$g_s$ and ${\rm M}_p$ scalings of all terms match with each other. The $g_s$ and ${\rm M}_p$ scalings of the classical terms are easy to compute
using (<ref>) and (<ref>), so the whole burden now lies in matching them to the $g_s$ and ${\rm M}_p$ scalings of the quantum pieces. The question now is following. How do we express the quantum series $\mathbb{Q}_{\rm T}$ succinctly to accomplish the aforementioned job? In the following sub-section we will analyze this in some details, but due to the technical nature of the content the readers, who would like to see the final answers, could skip it on their first reading.

§.§ Analysis of the quantum series and the Exact RG constraints

The quantum series $\mathbb{Q}_{\rm T}$ that we want to write should be expressed in terms of the on-shell degrees of freedom so as to appear in the EOMs, and therefore consequently in the constraint (<ref>). This means all the massless off-shell degrees of freedom will have to be integrated out giving rise to the non-local quantum terms. These are exactly the non-local counter-terms discussed in details in [14, 15, 16]. (For a simpler derivation of this the readers may refer to section <ref> and our upcoming paper [9].) In fact, when we talk of the on-shell degrees of freedom, we have inherently assumed that we have some knowledge of the allowed degrees of freedom (DOFs) in M-theory. Since the UV behavior of M-theory is completely unknown, all we can say with certainty are the DOFs at the energy scale much below the KK scale $\hat\mu$. The various energy scales are shown in figure <ref>.

[thick,->] (0,-1) – + (0,5.5) ;
(-0.25,-0.9) –  (0.25,-0.9);
(x) at (-0.25,-0.9) [label=left:$k_{\text{IR}}$] ;
(x) at (0.25, -0.9) [label=right: IR cutoff] ;
(-0.25,0) –  (0.25,0);
(x) at (-0.25,0) [label=left:$\mu$] ;
(x) at (0.25,0) [label=right:LHC cutoff] ;
(-0.25,2) –  (0.25,2);
(x) at (-0.25,2) [label=left:$\hat{\mu}$] ;
(x) at (0.25,2) [label=right:KK scale] ;
(-0.25,3) –  (0.25,3);
(x) at (-0.25,3) [label=left:${\rm M}_s$] ;
(x) at (0.25,3) [label=right:String scale] ;
(-0.25,4) –  (0.25,4);
(x) at (-0.25,4) [label=left:${\rm M}_p$] ;
(x) at (0.25,4) [label=right:Planck scale] ;

[pattern=north east lines,pattern color=ashgrey]
(-0.20,-0.89) rectangle +(0.4,0.88);

Energy scales in the problem.

This means, as long as $k_{\rm IR} < k < \mu << \hat\mu$, where
$k_{\rm IR}$ is the IR cut-off, we can confidently ascertain the DOFs. Such a scenario calls for a Wilsonian effective action (or more appropriately an action constructed from an Exact Renormalization Group procedure), wherein all the massive, massless off-shell, as well as the KK DOFs have been integrated away, including the massive, as well as the massless off-shell, DOFs on the branes. To provide a finer reiteration of the aforementioned claims: our analysis with the quantum series relies on two very important conditions.

1. Knowledge of the precise set of degrees of freedom in the energy scale $k$ where $k_{\rm IR} < k < \mu << \hat\mu$ with $\mu$ being the typical energy scale in a scattering experiment and $\hat\mu$ being the KK scale, and

2. Existence of an Exact Renormalization Group effective action constructed from integrating out all the massive, as well as the massless off-shell, degrees of freedom.

Once the aforementioned two conditions are satisfied we can easily quantify the quantum series $\mathbb{Q}_{\rm T}$. The Exact Renormalization Group procedure now instructs us to keep all possible powers of the curvatures and the G-flux components, including all possible derivatives, to construct the perturbative series. The non-perturbative and the non-local terms may then be constructed by exponentiating these terms in appropriate ways as shown in [14, 15, 16]. Putting everything together it is not too hard to see that generically there can at most be 60 on-shell curvature tensors and at most 40 on-shell G-flux components. (The fermionic extension can be easily inserted in, but we will not do so here. Interested readers may look up [9, 18].) This would mean that a typical quantum series may be presented succinctly using these on-shell tensors as the following series:

ℚ_T(x, y, w, [g_s]) = ∑_{l_i}, {n_j} C_{l_i}, {n_j} M_p^σ_nl [g^-1] ∏_j = 0^3 [∂]^n_j ∏_k = 1^60 (R_A_k B_k C_k D_k)^l_k ∏_p = 61^100 (G_A_p B_p C_p D_p)^l_p + O(exp(-M_p^σ')),
where ${\cal C}_{\{l_i\}, \{n_j\}}$ are constants for a given values in the set $(\{l_i\}, \{n_j\})$; $(l_i, n_j) \in (+\mathbb{Z}, +\mathbb{Z})$;
$(n_0, n_1, n_2, n_3)$ are the number of derivatives along temporal, ${\bf R}^2, {\cal M}_6$ and ${\mathbb{T}^2\over {\cal G}}$ respectively; and
$\sigma_{nl} \equiv \sum\limits_{j = 0}^3 n_j + 2\sum\limits_{k = 1}^{60} l_k + \sum\limits_{p = 61}^{100} l_p$ with $\sigma'$ being the scaling of a similar series [14, 15, 16]. All the curvature and the flux tensors, including the inverse metric components are expressed in terms of their warped form, and therefore include the $g_s$ factors. The presence of
$[g_s]$ in $\mathbb{Q}_{\rm T}({\bf x}, y, w, [g_s])$ signifies this, including $({\bf x}, y, w) \in ({\bf R}^2, {\cal M}_6, {\mathbb{T}^2\over {\cal G}})$ dependence generically.

Despite the complicated nature of (<ref>) one can, with some effort, work out the precise $g_s$ scalings at any orders in $(\{l_i\}, \{n_j\})$. This have been demonstrated in [14] for simpler cases and in [15, 16] for more complicated cases. We will not repeat any of the details here $-$ the readers may look up [14, 15, 16] $-$ and only present the form of the $g_s$ scalings of (<ref>) in the following way:

ℚ_T(x, y, w, [g_s]) = ∑_k = 0^∞∑_{l_i}, {n_j} C_{l_i}, {n_j}M_p^σ_nl
(g_sH(y)H_o(x))^θ({l_i}, {n_j}) + 2k3 + O(exp(-M_p^σ')), where ${\rm H}_o({\bf x})$ is a new warp-factor that only appears if we demand a dependence on the spatial coordinate ${\bf x} \in {\bf R}^2$. The $g_s$ scaling of the non-perturbative and the non-local terms, shown here simply as ${\cal O}\left({\rm exp}(-{\rm M}_p^{\sigma'})\right)$ is quite non-trivial and has been worked out in [15, 16]. Taking  derivatives of this term with respect to the warped metric components, and then making it traceless using the unwarped metric components, introduce the factors $\bar{a}^{\rm P}_{\rm M}$ from (<ref>) in (<ref>). The EOM constraint then tells us that at the lowest order, i.e. for $k = 0$ in (<ref>), the following identification may be done:

θ({l_i}, {n_j}) - a̅^M_M = ζ^M_M,     σ_nl = σ_classical,     [ℂ^M_M(y)]^(0) ≡∑__{l_i},{n_j}[ℂ^P_M(y)]^(0, {l_i}, {n_j}), where the repeated indices are not summed over and $\sigma_{\rm classical}$ is the ${\rm M}_p$ scaling of the classical pieces (we are ignoring $\sigma'$ to maintain the simplicity of the ensuing analysis). The set $(\{l_i\}, \{n_j\})$ is then the number of integer solutions to
$\theta(\{l_i\}, \{n_j\}) - \bar{a}^{\rm M}_{\rm M} = \zeta^{\rm M}_{\rm M}$, with $\zeta^{\rm M}_{\rm M}$ taking the following values:

ζ^M_M ≡(ζ^0_0,   ζ^i_i,   ζ^m_m,   ζ^a_a) = (-2,  -2,  0,   +2), which reproduces the following two choices for $\theta(\{l_i\}m \{n_j\})$, namely, $\theta(\{l_i\}, \{n_j\}) = {2\over 3}$ perturbatvely and $\theta(\{l_i\}, \{n_j\}) = {8\over 3}$ non-perturbatively. One may then combine (<ref>) with (<ref>) to see whether they may provide possible ways to realize the consistency conditions in (<ref>) and (<ref>).

§.§ An almost possible way of attaining a de Sitter vacuum

The readers who have skipped section <ref> may note that
the identifications (<ref>) and (<ref>) are the only results we need to resolve the classical-quantum issue that we raised in the paragraph above (<ref>). The classical terms, which are the Einstein tensors ${\bf G}_{\mu\nu} \equiv {\bf R}_{\mu\nu} - {1\over 2} {\bf g}_{\mu\nu} {\bf R}$ for $(\mu, \nu) \in {\bf R}^{2, 1}$, scale in the following way:

(g_sH(y) H_o(x))^-2 ×1M_p^σ_classical, and therefore, from (<ref>) and (<ref>), we look for $\theta(\{l_i\}, \{n_j\})$ in the quantum terms that would scale in exactly the same way. To facilitate the discussion, let us take two case: one, with time-independent G-fluxes and the other with time-dependent G-fluxes. With the first case, the $g_s$ scalings of various terms entering the EOMs along ${\bf R}^{2, 1}$ directions are given in the second column of Table <ref>. We see that all other ingredients would scale as (<ref>) except the fluxes. Unfortunately however the three set of quantum terms: perturbative, non-perturbative and the non-local ones, the scaling behavior imply:

θ_1 - θ_2 = 23,      and     θ_1 - θ_2 = 83where with $\theta_i$, for $i = 1, 2$, the former denote the $g_s$ scalings of the perturbative quantum series in (<ref>) and the latter for the non-perturbative and the non-local quantum series (which we don't discuss here. See [15] and [16] for details). The precise values of $\theta_i$, while can be seen from [14, 15, 16] or easily worked out directly from (<ref>), are not important. What is important is the relative minus sign, which implies an infinite number of solutions with no $g_s$ hierarchy. Due to the subtlety with localized fluxes, the ${\rm M}_p$ hierarchy is also removed (see [14]), meaning that to order ${g_s^{-2}\over {\rm M}_p^{\sigma_{\rm classical}}}$, there are an infinite number of quantum terms with no apparent $g_s$ or ${\rm M}_p$ hierarchies, signalling a breakdown of EFT. Similar breakdown happens for both the non-perturbative and the non-local terms, the latter appearing from integrating out the massless off-shell states.

This is a serious problem, but
what if we try to ignore the quantum terms altogether and solve the space-time EOMs using only the branes and the anti-branes? From the second column, since these states scale exactly as (<ref>), it appears that we should be able to make use of this. Plugging it in the EOMs, the result we get is the following:

Λ= -16V_6 T_2 κ^2(n_b + n̅_b), where $(\kappa^2, {\rm T}_2)$ are the data related to the branes; ${\rm V}_6$ is the volume of the six-dimensional base of the internal eight-manifold; and $(n_b, \bar{n}_b)$ are the number of the $-$ integer and fractional $-$ M2 and the $\overline{\rm M2}$ branes respectively. The cosmological constant $\Lambda$ from (<ref>) is negative! This means in the dual type IIB side, we are dealing with the following configuration:

1Λt^2(-dt^2 + dx_1^2 + dx_2^2 + dx_3^2)    1|Λ|z^2(-dη^2 + dx_1^2 + dx_2^2 + dz^2), which would generically be a non-supersymmetric ${\bf AdS}_4$ space-time in the Poincare patch. Including the internal space, we expect the configuration to be of the form ${\bf AdS}_4 \times {\bf M}_6$, with ${\rm V}_6$ giving the volume of ${\bf M}_6$. Because of this volume factor, it is a scale-separated AdS space. Unfortunately due to the ill-behaved quantum terms, such a configuration is most likely in the swampland. (See [19] for more details on the scale-separated AdS spaces using the language of (<ref>).) This means branes or the anti-branes, including the IIB O-planes[The information of O-planes is encoded in the geometry of the eight-manifold in M-theory [12].], can at best give a non-supersymmetric ${\bf AdS}_4$ spacetime but never a de Sitter space.

The story appears to improve rather dramatically once we take time-dependent G-fluxes and slowly moving branes. (The latter is not important, at least for solving the EOMs, but is crucial for flux quantizations and anomaly cancellations. We will however avoid these discussion for the time being.) The $g_s$ scalings of various contributing factors are shown in the third column of Table <ref>. The crucial change from (<ref>) is that $\theta_1 \to \theta_1'$ and $\theta_2 \to -\theta_2'$, such that:

θ_1' + θ_2' ≡θ({l_i}, {n_j}) = 23,      and      θ_1' + θ_2' ≡θ({l_i}, {n_j}) = 83, for the perturbative and the non-perturbative (as well as the non-local) quantum series respectively; and $\theta(\{l_i\}, \{n_j\})$ is defined earlier in section <ref>.
Unfortunately however the perturbative solution with $\theta(\{l_i\}, \{n_j\}) = {2\over 3}$ does not satisfy either (<ref>) or (<ref>), which amounts to saying that the perturbative quantum terms, just like the classical terms, are red herrings in the problem. This however doesn't mean that the perturbative terms are completely useless. They do contribute at higher orders in $g_s$, a fact that may be verified by working out the EOMs in the presence of (<ref>). At the lowest order in $g_s$, it is only the non-perturbative and non-local quantum terms that can save the day. These terms come from the BBS-like instanton effects [11], which are basically instantonic five-branes wrapped around the six-manifold ${\cal M}_6$. These non-perturbative (and non-local) effects are slightly more complicated to compute because of the temporal behavior of the six-manifold ${\cal M}_6$ as seen from (<ref>), but they can be done (see details in [15, 16]). The result precisely gives $\theta(\{l_i\}, \{n_j\}) = {8\over 3}$, and plugging the values of $(l_i, n_j)$ in (<ref>) provides us with quantum terms that are quartic in curvature tensors (amongst other higher order mixed terms). The back-reaction of these quartic terms shows that (<ref>) solves the EOMs provided we modify the internal metric $g_s^{-2/3} g_{mn}(y)$ of ${\cal M}_6$ by adding sub-dominant contributions as $g_s^{-2/3} g_{mn}(y) \to \sum\limits_{k = 0}^\infty g_{mn}^{(k)}(y) g_s^{-2/3 + 2k/3}$ with $k \in {\mathbb{Z}\over 2}$, which in-turn will make the internal metric (<ref>) in the IIB side
time-dependent[To keep the four-dimensional Newton's constant time-independent in type IIB, a simple way would be to split ${\cal M}_6$ to ${\cal M}_4 \times {\cal M}_2$ and define the sub-dominant temporal dependence as multiplicative factors, say for example ${\rm F}_1(t)$ and ${\rm F}_2(t)$ respectively. Making ${\rm F}_1^2(t) F_2(t) = 1$ would keep the four-dimensional Newton's constant time-independent. Happily, such a choice also solves the EOMs as demonstrated in [14].].  The inclusion of quartic corrections  was anticipated for sometime (see [4]), and here we see why this may be the case. Four further encouraging results appear simultaneously:

1. The non-locality factors in the  non-local quantum terms  are integrated away to give rise to local interactions which contribute to
$\theta(\{l_i\}, \{n_j\}) = {8\over 3}$, thus behaving very similar to the BBS instantons.

2. The fermionic terms, coming from the Rarita-Schwinger fermions in M-theory, contribute only as polynomial powers of the condensates of lower dimensional Dirac fermions thus creating the anticipated non-perturbative effects.

Contributions to the EOMs     2c|$\hat{g}_s \equiv {g_s\over {\rm H}(y){\rm H}_o({\bf x})}$ scaling

${\bf G}_{\mu\nu} = {1\over 4\pi{\rm G}_{\rm N}}\sum\limits_{i = 1}^6\mathbb{T}^{[i]}_{\mu\nu}$    Time-independent fluxes     Time-dependent fluxes

${\bf G}_{\mu\nu} \equiv {\bf R}_{\mu\nu} - {1\over 2}{\bf g}_{\mu\nu} {\bf R}$     ${1\over \hat{g}_s^2}$     ${1\over \hat{g}_s^2}$

$\mathbb{T}_{\mu\nu}^{[1]}= \mathbb{T}_{\mu\nu}^{[{\rm fluxes}]}$     ${1\over \hat{g}_s^4}$     ${1\over \hat{g}_s^2}$

$\mathbb{T}_{\mu\nu}^{[2]}= \mathbb{T}_{\mu\nu}^{[{\rm M2}_{({\rm i+f})}]}$     ${1\over \hat{g}_s^2}$     ${1\over \hat{g}_s^2}$

$\mathbb{T}_{\mu\nu}^{[3]}= \mathbb{T}_{\mu\nu}^{[\overline{\rm M2}_{({\rm i+f})}]}$     ${1\over \hat{g}_s^2}$     ${1\over \hat{g}_s^2}$

$\mathbb{T}_{\mu\nu}^{[4]}= \mathbb{T}_{\mu\nu}^{[{\rm perturbative}]}$     $\hat{g}_s^{\theta_1-\theta_2 - {8\over 3}}$     $\hat{g}_s^{\theta'_1 + \theta'_2 - {8\over 3}}$

$\mathbb{T}_{\mu\nu}^{[5]}= \mathbb{T}_{\mu\nu}^{[{\rm non-perturbative}]}$     ${\hat{g}_s^{\theta_1-\theta_2 - {8\over 3}}\over \hat{g}_s^2}~{\rm exp}\left(-{\hat{g}_s^{\theta_1 - \theta_2}\over \hat{g}_s^2}\right)$     ${\hat{g}_s^{\theta'_1+\theta'_2 - {8\over 3}}\over \hat{g}_s^2}~{\rm exp}\left(-{\hat{g}_s^{\theta'_1 + \theta'_2}\over \hat{g}_s^2}\right)$

$\mathbb{T}_{\mu\nu}^{[6]}= \mathbb{T}_{\mu\nu}^{[{\rm non-local}]}$     ${\hat{g}_s^{\theta_1-\theta_2 - {8\over 3}}\over \hat{g}_s^2}~{\rm exp}\left(-{\hat{g}_s^{\theta_1 - \theta_2}\over \hat{g}_s^2}\right)$     ${\hat{g}_s^{\theta'_1+\theta'_2 - {8\over 3}}\over \hat{g}_s^2}~{\rm exp}\left(-{\hat{g}_s^{\theta'_1 + \theta'_2}\over \hat{g}_s^2}\right)$

[]The $g_s$ scalings of various terms appearing the EOMs along the $2+1$ space-time directions. The second column depicts the scalings with time-independent fluxes and the third column depicts the same with time-dependent fluxes. ${\rm M2}_{({\rm i+f})}$ denotes both integer and fractional M2 branes. $\theta_i$ and $\theta'_i$, for $i = 1, 2$, denote the $g_s$ scaling for the quantum series (<ref>) for the time-independent and time-dependent fluxes respectively. Details about what these $g_s$ scalings imply are described in the text.

${\bf 3.}$ The expression for the four-dimensional cosmological constant changes from being purely negative in (<ref>), to now taking the following form:

Λ= 112 V_6  |𝕋|^[NP, NL] - 16V_6 T_2κ^2(n_b + n̅_b) - 5384V_6 |G|^2, where $\vert\mathbb{T}\vert^{[{\rm NP, NL}]}$ involve integrals of the traces of the unwarped, i.e. $g_s$ independent, non-perturbative and non-local energy-momentum tensors along the ${\bf R}^{2, 1}$ directions. $\vert{\rm G}\vert^2$ is also defined in a similar vein. For both cases, the integrals are carried over the unwarped six-dimensional base (meaning that they involve the unwarped six-dimensional metric components).

${\bf 4.}$ The order by order study in $g_s$ is important. To the lowest order in $g_s$ the expression for cosmological constant is
(<ref>). At the next order in $g_s$, new flux components (recall the ${2k\over 3}$ factor in (<ref>)), as well as new quantum terms et cetera appear. They are all balanced against each other at every order in $g_s$ in such a way that the lowest order result, for example (<ref>), does not change.

The last two points confirm what we have been saying all along: the branes, anti-branes, O-planes and the perturbative quantum corrections are all red-herrings in the problem. Positive cosmological constant can only come from the non-perturbative and the non-local quantum terms, and only if they have the requisite $g_s$ and ${\rm M}_p$ hierarchies. However (<ref>) is more delicate: it appears to balance the quantum terms, i.e. the non-perturbative and the non-local quantum terms, against the classical terms, i.e. the branes, fluxes and the O-planes. This would have been impossible if the quantum terms were vanishing for small $g_s$. Fortunately, while the usual instanton like effects go as ${\rm exp}\left(-{1\over g_s^2}\right)$, thus vanishing for $g_s \to 0$, here the effects go as  ${\rm exp}\left(-{{g}_s^{\theta'_1 + \theta'_2}\over {g}_s^2}\right) = {\rm exp}\left(-g_s^{2/3}\right)$, which do not vanish for $g_s \to 0$. This may be easily confirmed by plugging in the second relation from (<ref>) in the non-perturbative and the non-local results from the third column of Table <ref>. Interestingly, using the first relation from (<ref>), the instanton like effects would have vanished for small $g_s$, thus confirming once again that the first relation in (<ref>) cannot help in constructing a positive cosmological constant solution in the type IIB side.

Despite this, the temporal $-$ albeit sub-dominant $-$ dependence of the internal metric in the type IIB side is a slight matter of concern now. Questions can be raised regarding Bianchi identities, flux quantizations and anomaly cancellation. However it has been shown in [14, 15, 16] that all the three criteria can be satisfied order-by-order in $g_s$. For example, taking derivatives of (<ref>) with respect to the ${\bf C}_{\rm MNP}$ fields provide corrections to the EOMs of the three-form fields in the presence of all order quantum terms. Once we restrict the flux components along the internal directions and integrate the equations over the eight-manifold, we get the anomaly cancellation formula to any order in $g_s$. Similarly taking the derivatives of (<ref>) with respect to the dual six-form fields (not field strengths!), provide the Bianchi identities to any order in $g_s$. Restricting over four-cycles (considered as boundaries of  five-cycles), provide the necessary flux quantization conditions, again to any order in $g_s$. These have been discussed in [14, 15, 16] for type IIB theory and to some extent in the upcoming work [9] for the $SO(32)$ and the ${\rm E}_8 \times {\rm E}_8$ theories suggesting the possibility of almost attaining a de Sitter vacuum solution. What goes wrong, or more appropriately, where is the error in the above analysis?

§ GLAUBER-SUDARSHAN STATES AND THE FAILURE OF VACUUM SOLUTIONS

Our above analysis may provide a false hope to the readers that a four-dimensional de Sitter space-time can exist as a vacuum solution in string theory. Unfortunately this is not the case, despite all the apparent  successes mentioned in the previous section. Where did we go wrong? The mistake lies in the two assumptions that we made earlier, namely the existence of an Exact Renormalization Group effective action, and the existence of a supergravity description at the energy scale $k_{\rm IR} < k < \mu << \hat\mu$. Let us clarify this by first recalling some, often misunderstood, facts about supergravity and string theory.

$\bullet$ String theory is not supergravity, not even approximately. When a string is quantized over a flat time-independent
background, the spectrum consists of massless states and a tower of massive states appearing from the scale ${\rm M}_s$ and above (as depicted in figure <ref>).

$\bullet$ Supergravity generically[But not always! Type IIB string theory is an example.] provides a Lagrangian description of these massless states and are represented by some consistent truncation procedure that keeps the marginal and relevant interactions between these states. Most irrelevant interactions (with respect to the string or the Planck scale) take us away from the supergravity description.

$\bullet$ The energy scale is very important, as we have no idea how string theory behaves in the far UV (i.e. at very short distances) and what degrees of freedom control the far UV dynamics, although it is presumed that the UV behavior is well-defined without any pathologies. The point is that, no matter what degrees of freedom exist in the far UV, an Exact Renormalization Group (ERG) procedure will tell us that once we integrate from $\Lambda_{\rm UV}$ till $\mu$ (see figure <ref>) all the information of the UV can be encoded in the coefficients of the infinite towers of marginal, relevant and irrelevant interactions of the massless states!

$\bullet$ Over a curved but still time-independent background, although we do not know how to quantize a string, it is believed that the supergravity interactions involving the same massless states continue to provide the dynamics of the theory. Unfortunately the irrelevant interactions are not well known and are difficult to derive in general. Even going to the energy scale $\hat\mu$ that introduces the KK states (see figure <ref>) becomes hard to deal within the supergravity approximation. Thus the safest energy scale is the one where $k_{\rm IR} < k < \mu << \hat\mu$ wherein we could take all order perturbative interactions involving only the massless states and venture beyond the supergravity approximation consistently.

$\bullet$ String perturbation theory is not convergent but is rather asymptotic, which is good because it tells us how to introduce non-perturbative effects, like D-branes, instantons, world-sheet instantons et cetera in a natural and consistent way. The original discovery of the D-branes successfully used this asymptotic behavior to argue for the existence of states whose mass go as inverse the string coupling instead of inverse square of the string coupling [20]. These states themselves carry towers of massless and massive degrees of freedom on their world-volumes and within the energy range $k_{\rm IR} < k < \mu << \hat\mu$ one could express the complete UV dynamics from an infinite towers of interactions involving the world-volume massless states following ERG. This perturbative series is again not convergent but is asymptotic which tells us what the non-perturbative contributions are in the aforementioned energy range.

$\bullet$ Supersymmetry is also important. The world-sheet GSO projection typically eliminates the tachyon and preserves certain amount of
space-time supersymmetry. However there is a misconception that breaking supersymmetry, spontaneously or explicitly, always guarantees a positive energy. This is definitely untrue. For example anti-brane breaks supersymmetry spontaneously, but does not change the cosmological constant of the theory. Similarly non self-dual four-form flux in M-theory breaks supersymmetry but keeps the cosmological constant unchanged. In fact no classical sources can break suspersymmetry and create positive cosmological constant without violating the no-go theorems [1]. The connection between supersymmetry breaking and the sign of the cosmological constant is more subtle. For example zero and negative cosmological constant spacetimes may or may not be supersymmetric, but a positive cosmological constant spacetime is always non-supersymmetric[As an example taking ${\rm AdS}_4$ in the Poincare patch and changing the coordinates $z \to it$ and $t \to iz$, where $t$ is the conformal time, would appear to give a supersymmetric ${\rm dS}_4$. (This is sometime also termed as a dS slicing of an AdS space.) However a careful study of the fluxes supporting ${\rm AdS}_4$ will tell us that the fluxes now become imaginary, thus ruling out this oft-used trick to generate ${\rm dS}_4$.]. Our problem here is with the latter: can we generate a four-dimensional positive cosmological constant, and therefore non-supersymmetric, space-time as a vacuum solution from string theory? The answer will turn out to be no.

§.§ Why doesn't de Sitter spacetime exist as a vacuum solution?

Our little discussion above concluded with the claim that four-dimensional de Sitter spacetime cannot exist as a vacuum solution in string theory.
Why is this the case? The reason is that all of the above facts change when we have a time-dependent background: an example being four-dimensional de Sitter space-time in the flat-slicing as shown in figure <ref>. The reasoning is simple, and can be illustrated directly in four-dimensional QFT. Consider a QFT defined over a four-dimensional de Sitter background in a flat-slicing. (The slicing will not be important in the ensuing discussion and we will make this clear soon. Meanwhile we will continue with the flat-slicing because of its technical simplicity.) From our modern understanding of QFT, we know that there is no distinction between renormalizable or non-renormalizable theories: “non-renormalizable$"$ theory simply means that, beyond some scale, the theory would need new degrees of freedom for it to make sense. An example is Einstein gravity itself. Traditionally it was classified as a “non-renormalizable$"$ theory, but now we know that beyond the string scale, the theory gets a tower of increasingly massive degrees of freedom which would render all UV amplitudes finite. Alternatively, a theory at a given scale may be defined by integrating out exactly those massive degrees of freedom. Once we do that, we should not think of extending the theory beyond that specific scale. Failing to maintain this would result in all kind of wrong conclusions and nomenclatures[Despite the simplicity of the above argument, it has unfortunately been constantly misused and mis-interpreted in the literature. We want to affirm the readers early on that we will always be in the energy range $k_{\rm IR} < k < \mu << \hat\mu$ where no massive degrees of freedom will enter and the dynamics will be completely governed by the massless degrees of freedom, albeit beyond the supergravity description.].

The above conclusion relies on an important fact: our ability to integrate out the UV degrees of freedom. What if we could not do this? Such a scenario could arise if there is no well-defined UV description, or if the frequencies of the fluctuations are changing with respect to time. The former could be ruled out because, despite our ignorance of the UV degrees of freedom and the UV dynamics, the short-distance behavior in string theory is well-defined with no pathologies whatsoever as mentioned earlier. This means, the only way a Wilsonian integrating out procedure would fail if the frequencies are themselves changing with respect to time. Unfortunately this is exactly what happens when we study fluctuations over a de Sitter background: the frequencies, and therefore the corresponding energies, are constantly red-shifted. This immediately implies a failure of the integrating-out procedure because if the energies of the modes are changing with respect to time, there is no meaning of integrating out the high energy modes. A common misconception at this point is to view the de Sitter space-time in a static patch and declare it as a way out of this problem. Unfortunately this argument doesn't work. Part of the reason being the highly misleading dynamics in a static patch. To see this, consider the two scenarios presented in the middle and right of figure <ref>, the first one corresponds to the behavior of a mode inside a static patch,
and the second one corresponds to the behavior of the same mode when seen outside the static patch.

[thick] (0,8.0) – (0,0)        (8,0) – (8,8.0) (0,8.0) – (8,0) (0,0) –(8,8) (0,8) – (8,8) (0,0) –(8,0) ;
[fill] (6,4)  node[right]   R ;
[fill] (2,4)  node[left]   L ;
[fill] (4,6)  node[above]   F ;
[fill] (4,2)  node[below]   P ;

[thick] (0,6.5) – (0,-0.5)        (3,-0.5) – (3,6.5)  ;
[blue] (0,1) .. controls (0.5,2) .. (1,1) .. controls (1.5,0) .. (2.0,1) .. controls  (2.5, 2) .. (3,1) ;
[blue] (0,3) .. controls (0.5,4) .. (1,3) .. controls (1.5,2) .. (2.0,3) .. controls  (2.5, 4) .. (3,3) ;
[blue] (0,5) .. controls (0.5,6) .. (1,5) .. controls (1.5,4) .. (2.0,5) .. controls  (2.5, 6) .. (3,5) ;

[fill] (3,-0.5)  node[right]   IR ;
[fill] (3,6.5)  node[right]   UV ;

[thick] (0,6.5) – (0,-0.5)        (3,-0.5) – (3,6.5)  ;
[blue] (0,1) .. controls (0.5,2) .. (1,1) .. controls (1.5,0) .. (2.0,1) .. controls  (2.5, 2) .. (3,1) ;
[blue] (0,3) .. controls (0.5,4) .. (1,3) .. controls (1.5,2) .. (2.0,3) .. controls  (2.5, 4) .. (3,3) ;
[blue] (0,5) .. controls (0.5,6) .. (1,5) .. controls (1.5,4) .. (2.0,5) .. controls  (2.5, 6) .. (3,5) ;
[blue] (3,5) .. controls (3.5, 4) .. (4,5) .. controls (4.5, 6) .. (5,5);
[blue, dashed] (5,5) .. controls (5.5, 4) .. (6,5) ;
[blue] (0,5) .. controls (-0.5, 4) .. (-1,5) .. controls (-1.5, 6) .. (-2,5);
[blue, dashed] (-2,5) .. controls (-2.5, 4) .. (-3,5) ;
[purple] (3,3) .. controls (3.75, 2) .. (4.5,3) .. controls (5.25, 4) .. (6,3);
[purple, dashed] (6,3) .. controls (6.75, 2) .. (7.5,3) ;
[purple] (0,3) .. controls (-0.75, 2) .. (-1.5,3) .. controls (-2.25, 4) .. (-3.0,3);
[purple, dashed] (-3.0,3) .. controls (-3.75, 2) .. (-4.5,3) ;
[red] (3,1) .. controls (4, 0) .. (5,1) .. controls (6, 2) .. (7,1);
[red, dashed] (7,1) .. controls (8, 0) .. (9,1) ;
[red] (0,1) .. controls (-1, 0) .. (-2,1) .. controls (-3, 2) .. (-4,1);
[red, dashed] (-4,1) .. controls (-5, 0) .. (-6,1) ;
[fill] (3,-0.5)  node[right]   IR ;
[fill] (3,6.5)  node[right]   UV ;

Left: Penrose diagram for a static patch of a four-dimensional de Sitter space-time. Middle: The region inside the black lines indicate the static patch, where a mode can possess the same frequency. Right: Similar configuration as the middle where, inside a static patch, a mode can possess the same frequency, while it changes outside the static patch. The colour scheme outside the static patch indicates slow red-shifting of the modes as we go from the UV to the IR. The underlying temporal coordinate used here is the one from the flat-slicing.

The regions inside the black lines in both the figures correspond to the static patch. An observer inside the static patch would see the UV frequencies (drawn in blue). From UV to IR, the frequencies would appear to not change with respect to time $-$ using the temporal coordinate of the flat-slicing $-$ and therefore the observer would attempt to integrate them out to write an EFT in the IR. However from figure <ref> we see that this would be a mistake because the frequencies do change outside the static patch, so at any given energy scale the frequencies are actually constantly changing even inside the static patch! Going to other coordinate patches related to the static patch would fail to alleviate the
problem. This and other related issues fall under the so-called trans-Planckian problems in de Sitter space-time [13].

Our above argument should convince the readers that allowing a de Sitter as a vacuum solution in string theory spoils the two essential properties that we need to allow for a low energy description, namely the integrating out mechanism following ERG, and the existence of a supergravity description. In addition to that, since the fluctuations are related to the zero modes of the closed string, the quantization of the string becomes a problem and it is not clear how the spectra of the string would look like. This issue would extend to the open string quantization also, leading to the same issue with the world-volume spectra on branes. Non-existence of supersymmetry is another problem: the bulk zero point energies do not cancel, and therefore lead to the cosmological constant problem that we barely avoided using supergravity descriptions with time-independent backgrounds. Question is, are there ways out of this conundrum? In the following we tabulate three possible ways.

${\bf 1}.$ Discard completely the possibility of de Sitter as a vacuum solution and replace it by quintessence, or string gas cosmology or other alternatives.

${\bf 2.}$ Allow the existence of a de Sitter vacuum solution but replace Wilsonian Effective Field Theories by Open Quantum Field Theories.

${\bf 3.}$ Discard completely a de Sitter vacuum solution and replace it by an excited state in the theory over a supersymmetric Minkowski vacuum.

The first one is an interesting alternative. While quintessence has it's own share of problems (see for example [21]), string gas cosmology [22] can serve as a viable alternative: it is based on string theory and there are recent developments connecting it to M(atrix) models [23] which appear to be very promising. It will be interesting to see how ERG ideas play out in this setting and how EFTs are constructed. We will have more to say on this a little later.

The second one is more subtle. Open QFTs [24] are traditionally described by isolating relevant degrees of freedom from an “environment$"$ which allows them to gain or lose energies to the environment. Since neither energy is conserved, nor an EFT description exists, the dynamics of the theory typically follow some Markovian process which may be quantified in certain settings (see for example [25] and references therein). The problem with this picture appears when we try to use it in string theory: in a UV complete theory, like string theory or M-theory, there appears to be no clear demarcation between the relevant degrees of freedom and the environment, and therefore a concrete realization over a temporally varying background is equally hard. So far there has not been much progress implementing this idea in string theory, although some recent works in field theories have shed some interesting light on the de Sitter problem [26].

This brings us to the third one: realizing a de Sitter space-time as an excited state over a supersymmetric Minkowski background. Our bet is on this. Three immediate encouraging  results are as follows.

${\bf 1.}$ Being supersymmetric Minkowski, or more appropriately being a warped supersymmetric Minkowski with a compact non-Kähler internal space, the zero point energies automatically cancel, and so does the Dine-Seiberg runaway [27] problem. The latter being due to the stabilized moduli.

${\bf 2.}$ Being time-independent background, a Wilsonian EFT can be constructed. The fluctuations over such a vacuum are time-independent and, although our knowledge of the short-distance physics is negligible, ERG techniques can be implemented leading to a well-defined EFT in the energy scale $k_{\rm IR} < k < \mu << \hat\mu$, where $\hat\mu$ is quantified by the KK modes from the non-Kähler internal space.

${\bf 3.}$ Being an excited state over a supersymmetric Minkowski, the supersymmetry breaking required for a de Sitter space can be realized spontaneously, meaning that the state breaks supersymmetry whereas the vacuum doesn't. Since the zero point energies continue to cancel, the positive cosmological constant would come from those quantum corrections responsible for violating the no-go theorems [1].

§.§ Coherent states versus the Glauber-Sudarshan states

With the aforementioned encouragements, we now ask what kind of excited state over a Minkowski minimum[By this we mean local minima unless mentioned otherwise.] should we look for to reproduce a de Sitter space-time once we take an expectation value of the metric operator over this state. Since the space-time is almost “classical$"$, as a first trial we should look for a quantum state that is closest to classical trajectories[Such a line of thought will eventually lead us to a dead-end, but we will push on.].  An immediate answer would be a coherent state which is the closest analog to a classical trajectory from quantum theory. Coherent states are usually described as displacement operators acting on free Gaussian vacua, which may alternatively be represented by specific linear combinations of the number operators associated with the quantization of harmonic oscillators. These harmonic oscillators clearly arise from quantizing a free field theory over a Minkowski minima. Once we have multiple minima (with identical values), as shown in figure <ref>, the coherent states are not as simple as the one known for a single minimum and are typically represented in quantum mechanics by the so-called Gazeau-Klauder coherent states [28]:

|σ⟩≡|J, γ⟩= 1√(N(J)) ∑_n ≥0 J^n/2√(ρ_n)  e^-iE_n γ |n ⟩, where ${\rm E}_n$ are the eigenvalues of a Hamiltonian ${\rm H}$, such that
${\rm H} \vert n \rangle = \omega {\rm E}_n \vert n \rangle$, $\gamma \in \mathbb{R}, 0 \le {\rm J} \in \mathbb{R}, \rho_n$ is an arbitrary function of $n$ with $\rho_0 = 1$, such that when ${\rm E}_n = n$ and $\rho_n = n!$, we reproduce the well-known coherent states for the case with a single minimum. Additionally demanding $\langle\sigma\vert\sigma\rangle = 1$ implies that ${\rm N(J)} \equiv \sum\limits_{n \ge 0}{{\rm J}^n\over \rho_n}$ be a convergent series within an appropriate radius of convergence.
Within this radius of convergence if the moment problem allows a positive
solution for the moment, then there exists a resolution of the identity (see [28] for details).

(0,5) .. controls (0.3,3) .. (1,1) .. controls (1.5,0) .. (2.0,1) .. controls  (2.5, 2) .. (3,1) .. controls (3.5,0) ..(4.0,1) .. controls (4.5,2.5).. (5,5) ;
[right] at (8.95,2.65) $ \longrightarrow$;

(0,5) – (0,0)  (2,5) – (2,0) (0,0) – (2,0) (0,0.1) – (2,0.1);
(1, 0.5) circle(1pt) (1,1.0) circle(1pt) (1, 1.5) circle(1pt) ;

(0,5) .. controls (0.3,3) .. (1,1) .. controls (1.5,0) .. (2.0,1) .. controls  (2.5, 2) .. (3,1) .. controls (3.5,0) ..(4.0,1) .. controls (4.5,2).. (5,1) ;
[black] (5.2,1) circle (1pt)   (5.6,1) circle (1pt)   (5.9,1) circle (1pt);
(9,5) .. controls (8.7,3) .. (8,1) .. controls (7.5,0) .. (7.0,1) .. controls  (6.5, 2) .. (6,1);
[right] at (10.25,2.65) $ \longrightarrow \qquad $;

(0,5) – (0,0)  (2,5) – (2,0) (0,0) – (2,0) (0,1) – (2,1) (0,2.5) – (2,2.5) (0,3.5) – (2,3.5);
[ashgrey2] (1.98,2.3) rectangle (0.0,3.51);
[ashgrey] (1.98,0.99) rectangle (0.0,0.01);
(1, 4.5) circle(1pt) (1,4.0) circle(1pt) (1, 5.0) circle(1pt) ;

Top diagrams indicate the lowest two states for the quartic potential, i.e. $\ket{0}_S$ and $\ket{0}_A$, with the true vacuum being the symmetric wavefunction $\ket{0}_S$. Bottom diagrams indicate that for a multi-valley potential the energy levels split into different bands, with the bottom of the brown energy band being the true vacuum $\ket{0}_S$ corresponding to the most symmetric wavefunction.

The coherent states for the multiple minima case, or more appropriately $-$ since they have the same values $-$ multiple wells' case, are not necessarily the mimimum uncertainity states although their temporal evolutions follow quantum approximations to classical motions. As an example consider the double well case as depicted by the first figure in figure <ref>, or by a more elaborate version in figure <ref>. The symmetric wave-function will have the lowest energy, with the energy of the anti-symmmetric one being slightly more than the symmetric one. One can construct a coherent state following (<ref>) and its temporal evolution follows the sequence of graphs plotted in figure 6 of [29]. From the figure one may see the appearance of Schrödinger cat-like states. For more complicated wells, the system forms energy bands with the lowest energy states at the bottom of the grey band as depicted in figure <ref>. Coherent states are difficult to construct for such cases, although some numerical works have been done.

Left: excitations over the left potential well in the example a quartic potential. Right: The symmetric and the antisymmetric wavefunctions corresponding to the true ground state and first excitation of the double well problem.

Few issues prohibit us from using the aforementioned coherent states as representations of the excited states over a Minkowski vacuum in string theory. First, the coherent state construction in (<ref>) is mostly in quantum mechanics and its extension to quantum field theory is non-trivial. Secondly, interacting field theories are an infinite collection of interacting harmonic oscillators which would make the construction of coherent states pretty hard. One might think of alleviating the second problem by going to weakly coupled scenarios where typically we would expect almost decoupled oscillators. Unfortunately, this is exactly what we cannot do. Thus the three main issues that actually prohibit us from using the standard notion of the coherent states in the low energy effective field theory description of string or M-theory are as follows.

${\bf 1.}$ Type IIB theory in which we want to realize a coherent state over a supersymmetric Minkowski vacuum is at a constant coupling point of F-theory [17]. This means that the IIB coupling $g^{(b)}_s \equiv 1$, with the moduli stabilized at vanishing axio-dilaton. Zero type IIB coupling would correspond to infinitely negative value of the background dilaton. We would however like to maintain $g^{(b)}_s \equiv 1$ even for the emergent de Sitter background to avoid dynamical seven-branes et cetera (the latter also being states in the theory [30]).

${\bf 2.}$ Once we have a non-trivial coupling, there are no free vacua $\{\vert 0 \rangle\}$ and one has to replace them by an interacting vacuum $\vert\Omega\rangle$. However now we have no idea how to construct the equivalent of higher states $\{\vert n \rangle\}$ in the interacting system! In QFT we avoid this delicate problem by using an $i\epsilon$ prescription and by computing the correlation functions by tilting the temporal intervals along the slightly imaginary direction, i.e. using ${\rm T} \to \pm\infty(1 - i\epsilon)$

${\bf 3.}$ Coherent states are supposed to mimic classical trajectories or classical configurations. However de Sitter space-time is not actually a classical configuration as non-perturbative effects are needed to stabilize it within the temporal domain (<ref>). Thus whatever excited states we construct should have the relevant set of quantum terms somehow embedded in them.

The second and the third problems raised above are fatal: they clearly tell us that we cannot use the standard notion of coherent state from say (<ref>). What can we use, now that we only have the interacting vacuum $\vert\Omega\rangle$ to our disposal? Our proposal
is to use the Glauber-Sudarshan state $-$ named after [31] $-$ which is constructed by displacing the interacting vacuum in the following way [14, 15, 16]:

|σ⟩= 𝔻(σ) |Ω⟩∝𝔻(σ) lim_T→∞(1-iϵ) exp(-i H T) |0⟩_min, where $\mathbb{D}(\sigma)$ is the displacement operator that is not unitary; $\vert 0 \rangle_{\rm min}$ is the free vacuum associated to any chosen Minkowski minimum[To see the second equality, express $\vert 0 \rangle_{\rm min}$ in terms of the linear combination of the eigenstates of the total Hamiltonian ${\bf H}$, in the following way:
|0 ⟩_min = c_o |Ω⟩+ ∑_n = 1^∞c_n|n ⟩, where $c_o = \langle\Omega\vert 0\rangle_{\rm min}$. Acting with
$e^{-i{\bf H}{\rm T}}$ on $\vert 0\rangle_{\rm min}$, with the assumption that there is an energy gap between the interacting vacuum $\vert\Omega\rangle$ and the higher states $\vert n\rangle$, and then taking the limit of ${\rm T} \to \infty(1-i\epsilon)$, decouples all the higher states $\vert n\rangle$ and gives us:
|Ω⟩= lim_T →∞(1-iϵ) e^-iHT|0⟩_min⟨Ω|0⟩_min, implying that the proportionality constant in (<ref>) is
${e^{-i{\rm E}_o {\rm T}}\over \langle\Omega\vert 0\rangle_{\rm min}}$. Plugging the second equality in the definition of $\langle\varphi\rangle_\sigma = {\langle\sigma\vert \varphi\vert \sigma\rangle\over \langle\sigma\vert \sigma\rangle}$ then restricts the path integral over the Minkowski minimum. ]; and ${\bf H}$ is the total Hamiltonian that contains the full non-perturbative and non-local contributions. The latter will be described in details in section <ref>. By construction it is not a coherent state per se, but mimics somewhat the construction of a coherent state for a single well which is a displacement of the free vacuum. (As an example, a coherent state $\vert \sigma\rangle_{\rm c}$ can be described as $\vert\sigma\rangle_{\rm c} = \mathbb{D}_{\rm c}(\sigma) \vert 0 \rangle_{\rm min}$, where $\mathbb{D}_{\rm c}(\sigma)$ is the displacement operator, or more appropriately as in (<ref>).) However the definition (<ref>) remains unchanged even if we have multiple minima: the first equality in (<ref>) is universal, whereas the second equality depends on which Minkowski minimum we choose. As an example consider the two potentials depicted in figure <ref>. The Glauber-Sudarshan state is the displacement of the interacting vacua associated with the minimum energy state for the full interacting potential. For computational purpose we could, using (<ref>), describe it as localized around a Minkowski minimum in each of the two configurations respectively by choosing $\vert 0 \rangle_{\rm min}$ appropriately. For the potential on the left of figure <ref> the states could be localized around any one of the two minima. For the potential on the right of figure <ref>, the second minima associated with the positive potential should be removed by quantum corrections in the theory, and therefore only the Minkowski (and AdS) minima remain. Glauber-Sudarshan states are then defined, using (<ref>), over the Minkowski $-$ and not the AdS $-$ minima. Such states clearly answer all the objections raised above.

${\bf 1.}$ By construction (<ref>) is not classical as $\vert\Omega\rangle$ is far from classical. Because of this non-classical nature, $\vert\sigma\rangle$ somehow contains all order quantum corrections which are in fact necessary to overcome the no-go theorems.

${\bf 2.}$ Since the Glauber-Sudarshan states can be localized around the Minkowski minimum of the potential, but are not related to free vacuum associated with the potential, they cannot be generated by simple Bogoliubov transformations. The corresponding EOMs satisfied by these states will consequently be more involved, albeit tractable quantitatively.

${\bf 3.}$ The definition of the Glauber-Sudarshan state is universal and works for all types of potentials irrespective of their natures. All we need are the interacting vacua expressed using the Minkowski minima (as in (<ref>)) for each cases, and by appropriately displacing them we can construct the required states. Even if the potential has, in addition, a lowest energy AdS minimum (or a set of AdS minima), the Glauber-Sudarshan states are still constructed only over the Minkowski minimum[This raises two questions. One, what if the potential only has AdS minima? And two, how far the AdS minima should be from the Minkowski minima in a given potential? The answer to the first question, albeit an unlikely scenario in string theory, is that for such cases construction of the Glauber-Sudarshan states would be difficult. These states are not constructed over the AdS minimum as the boundary condition would not match. One possibility from our earlier discussion would be that the quantum corrections uplift some of the AdS minima to Minkowski minima (and nothing beyond that to avoid the aforementioned EFT issues!). Glauber-Sudarshan states may then be constructed over any one of such Minkowski vacuum using (<ref>). However if the potential also has Minkowski minima this uplift might be harder because the Minkowski minima cannot be uplifted further. For the second question, we expect the AdS minima to be far away in the field space, or be separated by a large potential wall, from the Minkowski minima to reduce the tunnelling effects.].

(0,5) .. controls (0.5,2.5) .. (1,1) .. controls (1.5,0) .. (1.8,1) .. controls  (2.5, 4) .. (3.2,1) .. controls (3.5,0) ..(4.0,1) .. controls (4.5,2.5).. (5,5) ;
[magenta] (0,1) ..controls (0.7,1) .. (0.9,1.3) .. controls (1.3,2.0) .. (1.9,1.3) ..controls (2.2,1.0).. (2.5,0.95) ..controls (2.8,1.0) .. (3.1,1.3) .. controls (3.7,2.0) .. (4.1,1.3) ..controls (4.3,1.0).. (5,1.0);
[blue] (0,1.5) ..controls (0.7,1.5) .. (0.9,1.8) .. controls (1.3,2.5) .. (1.8,1.9) ..controls (2.2,1.5).. (2.5,1.5) ..controls (2.8,1.5) .. (3.05,1.2) .. controls (3.5,0.6) .. (4.0,1.2) ..controls (4.3,1.5).. (5,1.5);

(0,5) .. controls (0.5,2.5) .. (1,1) .. controls (1.5,0) .. (1.8,1) .. controls  (2.5, 4) .. (3.0,3) .. controls (3.5,2) ..(4.0,3) .. controls (4.7,4.5).. (4.9,5) ;
[blue] (0,1) ..controls (0.7,1) .. (0.9,1.3) .. controls (1.3,2.0) .. (1.9,1.3) ..controls (2.2,1.0).. (2.5,0.95) –(5,1.0);

Left: Out of the two wave-functions for the double-well potential  the symmetric one (drawn in red) represents the lowest energy state $\vert 0 \rangle$. The Glauber-Sudarshan state will be related to the interacting vacuum for the full potential and not the lowest energy state $\vert 0 \rangle$, i.e.
$\vert 0 \rangle \rightarrow \vert\Omega\rangle$. However using (<ref>) we could express it around any one of the (Minkowski) minima in the full double well potential. Right: For more general potential with unequal minima,  the Glauber-Sudarshan state will again be related to the interacting vacuum $\vert\Omega\rangle$ associated with the full potential. Again for computational purpose using (<ref>) one could use the localized wave-function near the lowest energy, i.e. the Minkowski minimum, to realize the state. However the second minimum, associated with positive energy, should be removed by quantum corrections so that only the Minkowski minimum survives with no potential conflict with EFT considerations.

The above arguments suggest that the Glauber-Sudarshan states, and not the coherent states, are responsible in creating a four-dimension de Sitter space-time with positive cosmological constant (see also [32]). However one could raise the following questions. How do we quantify the displacement operator $\mathbb{D}(\sigma)$, and especially the value of $\sigma$? How do the EOMs show up in this new viewpoint? How is the positive energy realized, or how are the no-go theorems avoided using the Glauber-Sudarshan states? How are the other constraints like anomaly cancellation, flux quantizations et cetera in M-theory are taken care of now? How do these states satisfy the Effective Field Theory criteria? In the following sections we will answer most of these questions but leave the details of the EFT criteria, and how these states satisfy them, for section <ref>.

§.§ Quantifying the displacement operator and the emergent metric

Our first attempt would be to quantify the displacement operator $\mathbb{D}(\sigma)$. For the standard coherent state, the displacement operator is unitary and may be expressed by exponentiating the creation operator $a_k^\dagger$ associated with the momentum $k$ (typically in $3+1$ dimensions, although this definition could be extended to higher dimensions). Going beyond the single minumum to the multiple minima case, the coherent state takes the form (<ref>), which may not have a simple representation using a displacement operator. Additionally, and as emphasized earlier, the low energy effective field theory in type IIB theory is at order 1 coupling, so clearly the theory is interacting. For such a case, as mentioned earlier, we talk about Glauber-Sudarshan state (<ref>) (instead of the coherent state) where the displacement operator isn't necessarily unitary, nor it may be represented by exponentiating a creation operator because defining a creation operator itself is complicated in an interacting theory. Nevertheless one could define an effective annihilation operator $a_{\rm eff}(k)$, constructed from mixing the creation and the annihilation operators for all $k$ as described in [15], that would annihilate these states:

a_eff(k) |σ(k)⟩= 0, where $\vert\sigma(k)\rangle$ is the Glauber-Sudarshan state for a given mode $k$ over the supersymmetric Minkowski background. Unfortunately this definition is not very illuminating, partly because of the non $c$-number nature of the commutator of $a_{\rm eff}(k)$ and
$a_{\rm eff}^\dagger(k)$, and partly because of the absence of a simple description of $\mathbb{D}(\sigma)$, where $\sigma = \{\sigma(k)\} ~\forall~k$, in terms of $a_{\rm eff}(k)$ and
$a_{\rm eff}^\dagger(k)$. How do we then quantify $\mathbb{D}(\sigma)$?

The answer actually appears from a slightly different consideration. To see this let us restrict ourselves to only a set of scalar fields $\varphi_1, \varphi_2, \varphi_3$ and $\varphi_4$ which form the representative samples of the spacetime metric in ${\bf R}^{2, 1}$, the internal metric in
${\cal M}_4 \times {\cal M}_2 \times {\mathbb{T}^2\over {\cal G}}$, the three-form flux components and the fermionic condensates respectively. Such a choice has a hidden agenda: we can avoid inserting ghosts in the effective action, although we will show a way to insert them in the generic case. One might however get worried about representing tensor and scalar components by purely scalar fields $\varphi_i$. Notwithstanding the fact that this is just a toy model, in string field theory we do use something like a scalar field $\Phi$ to represent the level of the string field (see for example the recent work [33]). We are not using string field theory, so the choice of scalar fields should be viewed as an exercise in simplicity[Our work, as discussed in [7, 8] has other parallels to string field theory, but we will not dwell on this aspect here anymore.]. The quantity $\sigma$, that enters in the definition of the displacement operator $\mathbb{D}(\sigma)$, then may be quantified as:

σ= {σ_i} = (α_1μν, α_2mn, β_ABC, γ)   →  (α_1, α_2, β, γ), where $(\mu, \nu) \in {\bf R}^{2, 1}, ~(m, n) \in {\cal M}_4 \times {\cal M}_2 \times {\mathbb{T}^2\over {\cal G}}, ~({\rm A, B})\in {\bf R}^{2, 1} \times{\cal M}_4 \times {\cal M}_2 \times {\mathbb{T}^2\over {\cal G}}$ with $\gamma$ related to the condensate $\overline\Psi^{\rm A} \Psi_{\rm A}$. Since we are taking scalar degrees of freedom, the tensor indices are not important and therefore $\sigma_i$ may be related directly to $(\alpha_1, \alpha_2, \beta, \gamma)$ as shown on the right of (<ref>). A possible choice of the displacement operator then takes the following form:

log 𝔻(σ) = ∫_k_IR^μ∏_j = 1^4 d^11k_j ∑_{n_i},{m_i}, {p_i} z_n_1m_1p_1...n_4m_4p_4 ∏_i = 1^4 k_i^2n_iσ_i^(m_i, p_i)(k_i) φ_i^∗m_i(k_i), which is expressed completely in the language of the Fourier components of the fields, and $z_{n_1m_1p_1...n_4m_4p_4}$ are dimensionful[All quantities are measured with respect to ${\rm M}_p$, although we could also choose the KK scale $\hat\mu$. For the latter there would be additional suppression factors proportional to powers of ${\hat\mu\over {\rm M}_p}$. See [7] for more details.] constants with $n_i \ge 0, m_i \ge 1, p_i \ge 1$. The eleven-dimension integral is henceforth done over
the Minkowski spacetime, unless mentioned otherwise, with 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 form of the displacement operator (<ref>) differs from the one we considered in [7] in the sense that it is complex and more elaborate, although one could make it even more generic by viewing $\widetilde\varphi_i^m(k_i)$ to be the first term in the binomial expansion of $\varphi_i^m(x, y)$ in Fourier modes. However no matter how generic the displacement operator is, it is clear that we need to impose the following constraints on the $z_{n_1m_1p_1...n_4m_4p_4}$ coefficients:

z_n_12p_10..0   =   z_0..n_22p_20..0   =   z_0..n_32p_3000   =   z_0..n_41p_4   =   0, to avoid generating massive states in the theory[There could be potential ghosts for $n_i > 0$ and $m_i = 2, ~\forall p_i \ge 1$, which will require more careful analysis. The constraint (<ref>) eliminates them here, but generically ghosts could in principle appear once we take the actual tensors into account. We will discuss this later.]. The other quantity defined in (<ref>) is $\sigma_i^{(m_i, p_i)}$ such that $\sigma_i^{(1, 1)}(k_i) \equiv \sigma_i(k_i)$. Without loss of generality we can keep $z_{0110..0} = z_{0000110..0} = .. = {1\over 2}$. Note that for $m_i > 2$, (<ref>) introduces polynomial interactions in the theory whereas for $(m_1, m_2, m_3, m_4)$ taking values $(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0)$ and $(0, 0, 0, 1)$, (<ref>) simply shifts the kinetic terms once we insert $\mathbb{D}(\sigma)$ inside a path-integral. To see how this works out precisely, let us consider an example where we want to compute the expectation value of $\varphi_1$ over the Glauber-Sudarshan state $\vert\sigma\rangle$ in the following way:

= ∫∏_i = 1^4 Dφ_i  e^-S_tot φ_1(x, y)  𝔻^†(σ) 𝔻(σ) ∫∏_i = 1^4 Dφ_i  e^-S_tot 𝔻^†(σ) 𝔻(σ), where $\hat\varphi_1$ is the operator associated to the field $\varphi_1$, and ${\bf S}_{\rm tot}$ is the total action that involves the kinetic terms of the four fields as well as their all-order perturbative interactions. Other possible ingredients in ${\bf S}_{\rm tot}$, as well as the reason for choosing only the perturbative interactions, will be elaborated soon[The $i\epsilon$ prescription appears in the usual way from the path integral (<ref>) over the Minkowski background. ]. Note that the integrations are done over the “on-shell” degrees of freedom, which means that we have given small masses to the “off-shell” degrees of freedom to avoid generating non-local interactions. As such they do not appear in the energy range $k_{\rm IR} < k < \mu$. A more careful analysis will be presented in section <ref>, wherein we will analyze the consequence of the non-local interactions. Coming back, the form of $\mathbb{D}(\sigma)$ in (<ref>) implies that it is complex, but $\mathbb{D}^\dagger(\sigma) \ne \mathbb{D}^{-1}(\sigma)$, and therefore inserting this in the path integral (<ref>) and imposing (<ref>), gives us the following three results.

${\bf 1.}$ The kinetic terms of the scalar fields, namely
$k_i^2 \vert\widetilde\varphi_i(k_i)\vert^2$, shift by the $\sigma_i(k_i)$ factors from the $m_i = p_i = 1, n_i = 0$ terms of (<ref>) in the following way:

k_i^2 |φ_i(k_i)|^2   →  k_i^2 |φ_i(k_i) - σ_i(k_i)k_i^2|^2, which is exactly what we require to keep the one-point functions in (<ref>) non-zero. This is then the meaning of the displacement in the displacement operator $\mathbb{D}(\sigma)$. For $m_i = 1$ but $p_i > 1$ and $n_i > 0$, there will be additional contribution to
(<ref>) which may be related to the $\alpha'$ corrections to the T-duality rules that we alluded to in footnote <ref> (see the upcoming paper [9] for more details on this).

2. The displacement operator presented in (<ref>) is complex, which implies that the any other Glauber-Sudarshan state $\vert\sigma'\rangle$ is not orthogonal[Unfortunately even for a real choice of $\mathbb{D}(\sigma)$, as in [7], orthogonality cannot be guaranteed.] to $\vert\sigma\rangle$, i.e.
$\langle\sigma'\vert\sigma\rangle$ is
non-zero even for large difference between $\sigma'$ and $\sigma$. Because of this the resolution of identity is harder to find here. Nevertheless let us define the following operator:

O = ∑_σ, σ^∗  k⊗ |σ(k)⟩⟨σ(k)|∏_k ⟨σ(k)|σ(k)⟩, 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 $\prod\limits_k$ is the usual product over $k$. For any arbitrary state $\vert\sigma'\rangle$, one can see that ${\cal O}\vert\sigma'\rangle \propto \vert\sigma'\rangle$, because
when ${\rm Re}~\sigma \ne {\rm Re}~\sigma'$, the exponential factors in $\mathbb{D}(\sigma)$ make sure that the corresponding path-integral is suppressed by:

exp(-|σ(k) -σ'(k)|^24k^2 + |σ(k) + σ'(k)|^24k^2 - |σ(k)|^2k^2 + σ^∗(k)σ'(k) - σ(k)σ^'∗(k)2k^2),
where for both ${\rm Re}~\sigma(k) > {\rm Re}~\sigma'(k)$ and ${\rm Re}~\sigma(k) < {\rm Re}~\sigma'(k)$ this becomes arbitrarily small provided ${\rm Im}~\sigma > {\rm Im}~\sigma'$; and for $\sigma(k) = \sigma'(k)$ this becomes identity[Alternatively one could think of transforming $\sigma(k) \to - \sigma(k) \equiv \hat\sigma(k)$, Then $\forall ~{\rm Re}~\sigma'(k) > 0$ and ${\rm Re}~\sigma'(k) \ne {\rm Re}~\hat\sigma(k)$ orthogonality can be ensured. Unfortunately, when $\sigma'(k) = + \hat\sigma(k)$, (<ref>) is non-zero but not an identity anymore.]. Thus
it is the closest we can come to decoupling these states despite the absence of a clear notion of orthogonality. Therefore ${\cal O}$ does almost behave like an identity operator for all practical purposes.
(We will provide a better construction in section <ref> where we will see that we need to add another operator to (<ref>) to convert it into an identity.)

${\bf 3.}$ The remaining terms in the displacement operator (<ref>) for $m_i \ge 3$ only changes the coefficients of the interaction terms in the total action ${\bf S}_{\rm tot} \equiv {\bf S}_{\rm kin} + {\bf S}_{\rm int}$. If we denote the interaction term
${\bf S}_{\rm int}$ as a function of the four fields $\varphi_i$ and the coefficients $\{g_l\}$ of the irrelevant perturbative interactions, then in the presence of (<ref>), we expect:

S_int(φ_1,..,φ_4; {g_l})   →
S_int(φ_1,..,φ_4; {g_l} + {z}), where $\{z\}$ denotes the set of the coefficients $z_{n_1m_1...m_4p_4}$,
$\forall (n_i, m_i, p_i)$. Thus in the presence of (<ref>), the action that appears in the path-integral (<ref>) could become slightly different from the expected Wilsonian effective action, or more appropriately, the action after the ERG operation. Note that we have only considered perturbative pieces in both ${\bf S}_{\rm int}$ and the displacement operator $\mathbb{D}(\sigma)$ from (<ref>). An alternative choice of the displacement operator that shifts $\varphi_i$ in the interaction term ${\bf S}_{\rm int}$ as:

S_int(φ_1,...,φ_4; {g_l})   →
S_int(φ_1 - σ_1, ..., φ_4 - σ_4; {g_l}), i.e. not as in (<ref>) but as global change like (<ref>), does not help because taking an expectation value (<ref>) only gives the tree-level result that reduces $\sigma_i$ to 0. This is a self-consistency test otherwise the result would have been in tension with the no-go theorems [1].

With the above inputs we are ready to interpret (<ref>) as a way to extract the form of the background using path-integrals. The background data, like metric and flux components as well as the fermionic condensates, may now be viewed as expectation values or as emergent quantities. As an example, the space-time metric (albeit from M-theory point of view) now takes the form:

⟨ĝ_μν⟩_σ ≡⟨σ|ĝ_μν|σ⟩⟨σ|σ⟩ = ∫{Dg_AB}{ DC_ABD}{ DΨ_A}  { DΨ_A} e^-S_tot  𝔻^†(σ)
g_μν(x, y)
𝔻(σ) ∫{Dg_AB}{ DC_ABD}{ DΨ_A} { DΨ_A} e^-S_tot  𝔻^†(σ)
where $\mathbb{D}(\sigma)$ now differs from (<ref>) by the replacement of the scalar fields with the tensor fields related to metric and flux components and with the fermionic condensates (the latter is to avoid Grassmanian integrals). The total action is also different because now it involves, in addition to ${\bf S}_{\rm kin}$ and ${\bf S}_{\rm int}$, the topological interactions ${\bf S}_{\rm top}$, gauge fixing terms ${\bf S}_{\rm gf}$, as well as ghost interactions ${\bf S}_{\rm ghost}$. Two immediate questions arise:

${\bf 1.}$ What equations of motion do the expectation values satisfy? Can they be derived in the generic set-up?

${\bf 2.}$ Is it possible to compute explicitly the path-integrals in say (<ref>) or (<ref>) even if we don't know the exact coefficients of the irrelevant operators?

In the following sections we will argue that both the above questions can be answered satisfactorily at least in the toy model with scalar fields. With tensors and fermions, complications as well as subtleties arise due to the formidable nature of ${\bf S}_{\rm tot}$ from the presence of ghosts and gauge-fixing terms. In this paper we will spare the readers, as well as ourselves, from these challenges and only consider the theory with four scalar fields. This is because, once we know how to tackle the path integral in (<ref>), the generic case (<ref>) will not be very hard. Therefore, in the following, we will start by answering the second question first and then derive the corresponding EOMs in section <ref>.

§ PATH INTEGRAL, FACTORIAL GROWTH AND BOREL RESUMMING A GEVREY SERIES

The path integral in (<ref>) basically computes the one-point function of the scalar field $\varphi_1$. In the absence of the displacement operator $\mathbb{D}(\sigma)$ from (<ref>), one point functions would typically vanish, and the only non-zero answer from (<ref>) would be the background value (which is the warped Minkowski background here). Thus to get a non-zero answer we need (<ref>), but also the full perturbative interactions ${\bf S}_{\rm int}$. Why is that, and what about the non-perturbative interactions? These and other questions will be discussed in this section, and in particular we will be able to quantify precisely how the path-integral computation leads to a Gevrey growth. Previously such a claim was presented without a formal proof in [7]. Here we will fill in the gaps. In the process we will also show that the path integral computation follows a simple binomial expansion.

§.§ Organizing the path integral (<ref>) using binomial expansions

Let us start with the perturbative interactions.
The presence of these interactions is easy to see: it appears directly from the presence of the interacting vacua $\vert\Omega\rangle$ in the definition of the Glauber-Sudarshan state (<ref>). The shift in (<ref>) appears from the non-unitary nature of the displacement operator when we define $\mathbb{D}(\sigma) \mathbb{D}^\dagger(\sigma)$, or more appropriately $\mathbb{D}(\sigma, \varphi_i(x, y)) \mathbb{D}^\dagger(\sigma, \varphi_i(x, y))$, at a coincident point. Clearly this is very different from what we would have expected using coherent states, and therefore this gives yet another motivation to use the Glauber-Sudarshan states instead of the coherent states[One could however ask the following question: what would happen if we take a unitary $\mathbb{D}(\sigma, \varphi_i)$ but define the product $\mathbb{D}(\sigma, \varphi_i(x_1, y_1)) \mathbb{D}^\dagger(\sigma, \varphi_i(x_2, y_2))$ at two different points? The answer could be provided in multiple ways, but the simplest one is as follows. Because of the presence of a free vacuum in the definition of the displacement operator for a coherent state, the path integral structure, if any, will unfortunately only have the tree-level pieces but no interaction terms. This will clash with the no-go theorems [1], plus when we make $x_1 \to x_2$, the system will have no solution other than the Minkowski one.]. Using the Glauber-Sudarshan states, somewhat surprisingly, the non-perturbative effects are not that hard to see because they depend on factorial growths of the amplitudes at a given order. The factorial growths are of the Gevrey kind [34, 35, 36, 37, 38, 7], so what we want here is to Borel resum a Gevrey series. The net result of such a summation is the appearance of a condensed form for the expression of $\langle\varphi_i\rangle_\sigma$. From here the non-perturbative effects could be discerned.

The aforementioned story is non-trivial, so we need to carefully
address the path integral in (<ref>). Fortunately this has already been done in great details in [7], so we will present and elaborate on some salient features that were not discussed in [7]. One of the key question is of course the reason behind the Gevrey growth, and how to perform the Borel resummation of a Gevrey series. To answer this, we need to address an even more basic question: how to compute the path integral in (<ref>) efficiently?

Before starting the path integral computation we should note that, because of the shift in the kinetic term from (<ref>), one-point functions would be non-zero and therefore this cannot be the usual analysis that we perform for the Feynman path integral. Two key differences are:

${\bf 1.}$ The path integral in (<ref>) can be represented by the so-called nodal diagrams that form a bigger class of diagrams than the Feynman diagrams.

${\bf 2.}$ Feynman diagrams require the presence of complex fields otherwise certain non-trivial (and necessary) cancellations cannot be performed. Nodal diagrams work for both real and complex fields. With the latter, certain restrictions appear, but they do not alter much of the physics.

Both the above points have been described in [7], which the readers may look up for details. Here we want to emphasize the fact that the easiest way to do the path integral in (<ref>) is to break up the momenta from $k_{\rm IR} < k < \mu$ into small and equal pieces so as to convert any integrals into summation via the standard formula:

∫_a^b dx f(x) = lim_N →∞ b-aN ∑_j = 0^N f(a + j(b-aN)). Since our integrals will be over eleven-dimensional momenta, that factor of ${b-a\over {\rm N}}$ may be converted to the inverse volume, i.e. ${1\over {\rm V}}$, of the eleven-dimensional space. This is related to the IR cut-off $k_{\rm IR}$ that we continue to impose here. With all these in places, the path integral structure in (<ref>) may be represented by eq. (3.4) in [7]. The path integral has the following structural breakdown.

${\bf 1.}$ The measures of all the fields in terms of their real and complex Fourier components. In [7] we took only three scalar fields, but here we take four.

${\bf 2.}$ The displacement operator $\mathbb{D}^\dagger(\sigma) \mathbb{D}(\sigma)$. In [7] we took a real displacement operator. Here we take a complex one (<ref>) although
$\mathbb{D}^\dagger(\sigma) \mathbb{D}(\sigma)$ doesn't change from what we had in [7].

${\bf 3.}$ The kinetic terms of all the fields written in terms of their Fourier components. In [7] we took a Lorentzian formalism, here we take an Euclidean approach.

${\bf 4.}$ The interactions terms in terms of powers and derivatives of the four fields. Here we take one specific set of interactions, and we will discuss the generic case a bit later.

${\bf 5.}$ Momentum conservation directly imposed on the Fourier components of the fields. As in [7], we can impose momentum conservation on one Fourier component of the $\varphi_4$ field.

The important parts in the above breakdown of the path integral structure are in points 2, 3 and 4. Combining 2 and 3, gives us (<ref>), which is what we term as the shift in the vacua. This shift, as mentioned earlier, is one of the important aspects of our construction as it keeps the one-point functions non-zero. Point 3 is the interaction term, which may be given the following binomial expansion:

∂^n φ^p_1(x, y)   →  Z_φ_1 ≡1V^p p! (∑_i = 1^p k'_i)^n _k'_i→k_i(φ_1(k_1) + φ_1(k_2) + φ_1(k_3) + ..... +
φ_1(k_N_μ))^p, with $n > 0$ because we chose to integrate out the off-shell degrees of freedom by giving them a small mass (see discussion after (<ref>)).
The operation $\underset{k'_i \to k_i}{\bigodot}$ chooses the appropriate set of (i.e. $p$ number of) $k'_i$ momenta that would tally with the $p$ product of the Fourier components; and ${\rm N}_\mu$ is the number of division of the momenta $k$ between $k_{\rm IR} \equiv k_1$ and $\mu << \hat\mu$.
This is most clearly presented as products of the Fourier components, as we did in eq. (3.4) of [7], but the binomial form is easier to handle. The most dominant terms in the binomial expansion go as products of $p$ number of Fourier components with all momenta unequal. They all scale as $p!$, which are cancelled by the ${1\over p!}$ factor in (<ref>). If we denote the dominant contribution as ${\cal A}(k_i, k_{i+1},.., k_{i+p})$
via an ordered set of $p$ momenta, then (<ref>) may be rewritten as:

V^p Z_φ_1=      A(k_1, k_2, .., k_p) (∑_i = 1^p k_i)^n + A(k_p+1, k_p+2, .., k_2p) (∑_i = p+1^2p k_i)^n + .....

+      1(p-1)! A(k”_1, k”_1, .., k”_1, k”_2) ((p-1)k”_1+k”_2)^n  + 1p!
A(k'_1, k'_1,.., k'_1) (pk'_1)^n,

where we see that the sub-dominant contributions are being suppressed by
$(p-n)!$ with $n$ being the order of the binomial expansion. The total number of terms  would depend on ${\rm N}_\mu$, i.e. on how many finer pieces are we dividing the momenta $k_{\rm IR} < k < \mu$. For the present case it should be ${\rm N}_\mu^p$, and clearly ${\rm N}_\mu >> 1$. This means that the number of interaction terms from the $\varphi_1$ sector may be simply be arranged as the following series:

Z_φ_1= 1V^p (A_1 + A_2 + A_3 + A_4 + ..... + A_N_μ^p-1 + A_N_μ^p), where ${\cal A}_1 = {\cal A}(k_1,..,k_p)(k_1+..+ k_p)^n = \widetilde\varphi_1(k_1)..\widetilde\varphi_1(k_p)(k_1+..+k_p)^n$, and so on in the same order as in (<ref>). The above is of course only the $\varphi_1$ sector as mentioned, so we will have similar binomial series for $\varphi_2, \varphi_3$ and $\varphi_4$. However momentum conservation will put some restrictions in the $\varphi_4$ sector[The analysis is independent of how we assign the momentum conservation rule, as is shown in [7] and [8]. This is also quite easy to see from our above analysis, so will leave it for our diligent readers as an exercise.] because (a) not all terms of the Fourier components for $\varphi_4$ are independent, and (b) the total momentum expansion for the $\varphi_4$ sector would now simply be a linear combination of the momenta for the other three sectors. These subtleties need to be carefully taken care of, which we did in [7], but here we will not discuss this further. Instead, we just rearrange the path integral in (<ref>), using the aforementioned information, in the following suggestive way:

Num ⟨φ̂_1⟩_σ= ∫∏_i = 1^4 [Dφ_i] e^-S_kin + log[𝔻^†(σ)𝔻(σ)] (1 +g
[1pt][7pt]Z_φ_1Z_φ_2 Z_φ_3 Ẑ_φ_4 + .....) ·1V∑_j= 1^N_μφ_1(k̂_j)ψ_k̂_j(z) where Num denotes the numerator of the path integral (<ref>), the over-bracket denotes the contraction of the $\widetilde\varphi_1(\hat{k}_j)$ field with one of the field in ${\cal Z}_{\varphi_1}$, and $\psi_{\hat{k}_j}(z)$ for $z \equiv (x, y)$ denotes the wave-function over the Minkowski background. The momentum conservation condition is imposed on the $\varphi_4$ field components and it is reflected by $\hat{\cal Z}_{\varphi_4}$ in (<ref>). $\mathbb{D}(\sigma)$ is the displacement operator from (<ref>), $g$ is the coupling constant expressed in inverse powers of ${\rm M}_p$, and the dotted terms are the higher order interactions. Two questions arise:

${\bf 1.}$ Can we compute the over-bracket contraction explicitly?

${\bf 2.}$ Can we go to all orders in the perturbative expansion and compute the over-bracket contraction explicitly?

The answer to both the questions are in the affirmative, but we will have to develop the story a little bit more before we can get to the final answer. This is what we turn to next.

§.§ Computing the path integral (<ref>) and nested integrals

The numerator of the path integral (<ref>) is as shown in (<ref>), and the denominator will be similar but without the source term. The shift in the vacuum is denoted by the $\log\left[\mathbb{D}^\dagger(\sigma) \mathbb{D}(\sigma)\right]$ term, and this results to at least (<ref>), but could in-principle also shift the coupling constant $g$ (see earlier discussion). The computation of the over-bracket contraction is quite a challenging exercise, as evident from the diagramatic study in [7], so we will not pursue this here. Instead we will give a toy computation to motivate the actual analysis. Let us then start with the following interacting part of the path integral:

1p!V^p∫dϕ_1 dϕ_2 dϕ_3 .... e^-b_1(ϕ_1 - β_1b_1)^2 e^-b_2(ϕ_2 - β_2b_2)^2 e^-b_3(ϕ_3 - β_3b_3)^2...
(k”_1 + k”_2 +..+k”_p)^n _k”→k'(ϕ_1' + ϕ_2' + ....)^p,

where $\phi_i = \phi(k_i)$, $\phi_{i'} = \phi(k'_i), b_i = b(k_i), \beta_i = \beta(k_i)$ and the operation $\underset{k''_i \to k'_i}{\bigodot}$, as in (<ref>), chooses the appropriate set (i.e. $p$ number) of $k''_i$ momenta that would tally with the $p$ product of the field components $\phi(k'_i)$. In the end all set of momenta, $k_i, k'_i$ and $k''_i$, needs to be matched properly to extract the value of the path integral. For example if we consider $p-1$ fields to have the same momenta and one field with a different moments, we are looking at the following interaction in the path integral (<ref>):

1(p-1)!V^p  ϕ^p-1_i' ϕ_j” ((p-1)k'_i + k”_j)^n, for all values of $i$ and $j$. Looking at (<ref>) one might get worried about a term linear in $\phi$, but due to the shifted vacua in (<ref>), such a term is non-zero. A simple way to then perform the path integral is to first fix a value of $i$, and then sum over all values of $j$. After which we repeat the process with a different value of $i$, and then sum over all the corresponding $j$, and so on. The result may be summarized in the following way:

W^(1)_ϕ=     1 V^p∑_r, l A^(1)_rl (β_1b_1)^p-1-r k_1^l(β_2b_2 k_2^n-l + β_3b_3 k_3^n-l +
β_4b_4 k_4^n-l + β_5b_5 k_5^n-l +
β_6b_6 k_6^n-l + ... +β_N_μb_N_μ k_N_μ^n-l)

+     1 V^p∑_r, l A^(2)_rl (β_2b_2)^p-1-r k_2^l(β_1b_1 k_1^n-l + β_3b_3 k_3^n-l +
β_4b_4 k_4^n-l + β_5b_5 k_5^n-l +
β_6b_6 k_6^n-l + ...+β_N_μb_N_μ k_N_μ^n-l)

+     1 V^p∑_r, l A^(3)_rl (β_3b_3)^p-1-r k_3^l(β_1b_1 k_1^n-l + β_2b_2 k_2^n-l +
β_4b_4 k_4^n-l + β_5b_5 k_5^n-l +
β_6b_6 k_6^n-l + ...+β_N_μb_N_μ k_N_μ^n-l)

+      1 V^p∑_r, l A^(4)_rl (β_4b_4)^p-1-r k_4^l(β_1b_1 k_1^n-l + β_2b_2 k_2^n-l +
β_3b_3 k_3^n-l + β_5b_5 k_5^n-l +
β_6b_6 k_6^n-l + ...+β_N_μb_N_μ k_N_μ^n-l)

.........                   .........

+      1 V^p∑_r, l A^(m)_rl (β_mb_m)^p-1-r k_m^l(β_1b_1 k_1^n-l + β_2b_2 k_2^n-l +
β_3b_3 k_3^n-l + β_5b_5 k_5^n-l +
β_6b_6 k_6^n-l + ...+β_qb_q k_q^n-l)

where $m \equiv {\rm N}_\mu$ and $q \equiv {\rm N}_\mu - 1$, which are decided from the way we divided the momentum $k$, lying between $k_{\rm IR}$ and $\mu$, into ${\rm N}_\mu$ equal pieces. The other quantities, namely ${\bf A}^{(i)}_{rl} = {\bf A}^{(i)}_{rl}[b_i]$, are defined in the following way:

A^(s)_rl[b_s] =  (p-1)^l-1(p-2)!( p-1 r) ( n l) ∫_0^∞dy  e^-b_s y^2 y^r, where the $s$ dependence comes from $b_s$ in the Gaussian integrals, and we have integrated the fields from 0 to $\infty$, instead of $-\infty$ to $+\infty$. Thus depending on even or odd $r$, the coefficients will be slightly different, although powers of $b_i$ will remain unaltered. The set of terms in (<ref>) has a hidden nested structure which is easy to see. Moving diagonally and summing over the terms in the upper triangle leads to one nested integral structure. In the lower triangle, if we move from bottom to top we can see another nested integral structure. These may be combined to give us:

W^(1)_ϕ=      1V^p-2 ∑_r, l∫_k_IR^μd^11k  (β(k)b(k))^p-1-r k^l ∫_k^μd^11k' A_rl[b(k), b(k')] k^'(n-l)β(k')b(k')

+      1V^p-2 ∑_r, l∫_k_IR^μd^11k  k^(n-l)β(k)b(k)
∫_k^μd^11k' A_rl[b(k), b(k')] (β(k')b(k'))^p-1-r k'^l,

showing that the amplitude coming from an interaction like (<ref>) is suppressed by ${\rm V}^{p-2}$. But there is more. Since two triangles can form a square, the two nested integrals may be combined under one roof to give us the final amplitude:

W^(1)_ϕ≡      1/V^p ∑_i, j   [thick, baseline=(0, 0cm),
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(a) – node[midway, above right, pos=0.2] $k$ (1);
(1) – node[midway, above right, pos=0.01] $(p-1)k'$ (2);
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(2) – node[midway, right, pos=0.7] $k'$ (6);
(2) – node[midway, right, pos=0.7] $k'$ (7);

=       1V^p-2 ∑_r, l∫_k_IR^μd^11k  k^(n-l)β(k)b(k)
∫_k_IR^μd^11k' A_rl[b(k), b(k')] (β(k')b(k'))^p-1-r k'^l,

which is a general result and continues to hold even if we have multiple distribution of the momenta. The diagrammatic representation that we present in (<ref>) is an example of a nodal diagram, and is a very convenient computational tool for the one-point functions in a path integral like (<ref>). The suppression factor of ${\rm V}^{p-2}$ tells us that the most dominant diagram is for the case when all the $p$ field components in (<ref>) have different momenta. The amplitude for this case will simply be products of integrals with integrands proportional to ${\beta(k)\over b(k)}$ along with powers of momenta coming from the $n$-th power of the momentum sum in (<ref>). All integrals are computed from $k_{\rm IR}$ to $\mu$ since all nested pieces add up to this.

The readers may have easily guessed that the above computation is related to the $\varphi_2$ or $\varphi_3$ fields. For $\varphi_4$ additional complications come from momentum conservation which we will not discuss here. (The factor $b(k)$ appearing here and in (<ref>) is precisely the massless propagator with $\beta$ being the shift for the $\varphi_3$ field in (<ref>).) For the $\varphi_1$ fields, as evident from (<ref>), the analysis is slightly different due to the presence of a source. The most dominant diagram with the source is clearly given by the following contraction:

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), which is suppressed by ${\rm V}$. The completely un-suppressed diagram will be when the source itself has a different momentum compared to the $p$ fields. Unfortunately such a diagram is projected out by the
denominator of the path integral (<ref>). The story is similar to the vacuum bubble case in field theory, which we shall leave it for our diligent readers to work out. The amplitude for the contraction in (<ref>) is now easy to work out and it takes the form:

A_p ≡∑_{r_i}  ∏_j = 1^p-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_p-1r_p[{a(k_j)}, a(k)]
k^r_p(α^2(k)a^2(k) + 12a(k))ψ_k(x, y),
where ${\bf A}_{r_1r_2...r_{p-1}r_p}[\{a(k_j)\}, a(k)]$ is similar to the
structure in (<ref>) but involve multiple Gaussian integrals and more complicated permutation factors. The explicit form is not important, so we will not work it out here. The integrands are defined in terms of the propagators $a(k_j), a(k)$ as well as the shift parameters $\alpha(k_j), \alpha(k)$ which may be related to $\alpha_1$ or $\alpha_2$ factors from (<ref>).
Note also the presence of the volume suppression factor for the integral involving source wave-function $\psi_k(x, y)$. This is a consequence of the contraction in (<ref>) with the source term. Since both $\alpha(k)$ and $a(k)$ have hidden volume factors, the ${\alpha^2(k)\over a^2(k)}$ term becomes sub-dominant compared to the ${1\over 2a(k)}$ term. The latter is related to the Feynman propagator (but since we are taking real fields, it is not exactly a two-point function).

All these that we said above is interesting, but is only the result to first order in $g$. Our aim is to extend it to all orders in $g$ and see how the diagrams grow. We plan to answer the following two questions.

${\bf 1.}$ Can we consistently argue that the nodal diagrams show factorial growths even if we don't know the exact coefficients of the higher order terms?

${\bf 2.}$ Is it possible to sum up all the dominant nodal diagrams to all perturbative orders in $g$ in a self-consistent way? Will such summation reveal the non-perturbative structure underlying the system?

In the following section we will answer the first question, and leave the details of the second question for the subsequent section.

§.§ Factorial growths of the nodal diagrams and 1-point functions

To study the growth of the dominant nodal diagrams we go back to the product structure in the path integral (<ref>). There are four key factors given by ${\cal Z}_{\varphi_i}$ with $i = 1,.., 4$. Let us start with ${\cal Z}_{\varphi_1}$ associated with the $\varphi_1$ field components. This appears from the binomial expansion in (<ref>), and may be arranged as a series in (<ref>). The most dominant terms, which are products of $p$ fields, appear when all the $p$ momenta are different. Such terms scale as $p!$, but since we have a ${1\over p!}$, they scale as 1. When we raise the series in (<ref>) to ${\rm N}$-th power, the dominant contributions, which are again terms with unequal momenta, would naturally scale as ${\rm N}!$. Due to the magic of the binomial expansion, the dominant scaling is always ${\rm N}!$ irrespective of the number of terms in the expansion[For example, if we have $(a + b + c + d)^4$, or $(a + b + c + d + e)^4$, or $(a + b + c + d + e + ...)^4$, the dominant terms, which are here with different alphabets, always scale as $4!$.]. This should partially explain the ${\rm N}!$ growths at order ${\rm N}$, but the story is a bit more subtle because there are many $-$ depending on what value we assign to ${\rm N}_\mu$ $-$ terms with the dominant contributions. Does this effect the growth of the dominant nodal diagrams as we are eventually adding up these contributions?

The answer actually appears from the nested structure that we discussed earlier. Here however, since all fields have different momenta, the nested structure is pretty involved, but a simple diagrammatic approach can shed some light on it. Recall that now we are looking at contractions of the following form: