Higher-Order Corrections to the Effective Field Theory of Low-Energy Axions

Abstract

Dark matter (DM) can be composed of a collection of axions, or axion-like particles (ALPs), whose existence is due to the spontaneous breaking of the Peccei–Quinn $U\left(1\right)$ symmetry, which is the most compelling solution of the strong $CP$-problem of quantum chromodynamics (QCD). Axions must be spin-0 particles with very small masses and extremely weak interactions with themselves as well as with the particles that constitute the Standard Model. In general, the physics of axions is detailed by a quantum field theory of a real scalar field, $\varphi$. Nevertheless, it is more convenient to implement a nonrelativistic effective field theory with a complex scalar field, $\psi$, to characterize the mentioned axions in the low-energy regime. A possible application of this equivalent description is for studying the collapse of cold dark matter into more complex structures. There have been a few derivations of effective Lagrangians for the complex field $\psi$, which were all equivalent after a nonlocal space transformation between $\varphi$ and $\psi$ was found, and some other corrections were introduced. Our contribution herein is to further provide higher-order corrections; in particular, we compute the effective field theory Lagrangian up to order ${\left({\psi }^{*}\psi \right)}^5$, also incorporating the fast-oscillating field fluctuations into the dominant slowly varying nonrelativistic field.

1. Introduction

Despite the broad consistency of the Standard Model (SM) of particle physics and the success of general relativity (GR) to describe, respectively, the short and large scales of the universe, there are still unsolved problems in modern physics. In particular, the nature of dark matter is one of the most intriguing enigmas as it is linked to both frameworks. Experimental observations indicate that DM should be abundant throughout the universe, contributing about five times more to the universe’s energy density than ordinary (baryonic) matter. Additionally, the vast majority of DM should be cold and collisionless [1]. In order to fully resolve the puzzle of dark matter, one needs, on one hand, to identify a plausible dark matter candidate within realistic models of nuclear and particle physics and, on the other hand, to develop an accurate, systematically improvable, theoretical approach suitable for low-energy phenomena associated with cold DM.

Axions, and axion-like particles, are some of the most strongly supported possibilities for the constituents that make up the dark matter of the universe [2]. The hypothetical particle was originally introduced as the pseudo-Goldstone boson associated with the spontaneously broken Peccei–Quinn $U\left(1\right)$ symmetry in order to address the strong CP problem of QCD [3,4,5]. Axions can be produced in the early universe with an abundance that is compatible with the observed dark matter density via a combination of cosmic string decay [6,7] and the vacuum misalignment mechanisms [8,9,10]. The axions from the cosmic string decay mechanism are incoherent, while the ones from the vacuum misalignment mechanism are coherent. An important feature is that both produce axions that are extremely nonrelativistic. Finally, Refs. [11,12,13,14,15,16] conclude that gravitational self-interactions may bring the axions in the early universe into thermal equilibrium, producing Bose–Einstein condensates.

A Bose–Einstein condensate of spin-0 bosons that fulfill certain properties of axion-like particles can be described by quantum field theory for a real scalar field $\varphi \left(x\right)$, whose interactions are depicted by a potential $V\left(\varphi \right)$ that is a periodic function of $\varphi$. With nonrelativistic applications in mind, one can introduce a convenient field redefinition that relates $\varphi$ to a complex scalar field, $\psi$, which is more suitable for describing the underlying theory in its low-energy limit. The resulting effective field theory (EFT) for a complex field $\psi$ was first constructed in Refs. [17,18]. However, it is worth mentioning that there have been other articles whose goal was to propose a formalism to describe low-energy axions. The authors of Ref. [19], by integrating out relativistic fluctuations of $\varphi$, developed a nonrelativistic effective Lagrangian for $\psi$ whose expression is completely different than that given in Ref. [17]. An exact relationship between the real scalar field $\varphi$ and the complex scalar field $\psi$ is given in Ref. [20], which, correcting some errors in both [17] and [19], demonstrates that the two proposed nonrelativistic effective Lagrangians are equivalent. And, lastly, Ref. [21] developed a method that gives a sequence of improvements for equations for a complex field $\psi$ with harmonic time dependence.

Having at hand what we could call a unique effective field theory for low-energy axions, the main purpose of this study was the computation of higher-order corrections, up to ${\left({\psi }^{*}\psi \right)}^5$. This was motivated by the fact that axion dynamics are primarily governed by attractive gravitational interaction and scalar–scalar contact forces. Therefore, a Bose–Einstein condensate can be formed but, instead of being a state with cosmologically long-range correlations and located in the high-density/occupancy regime, the mentioned interactions may lead to localized Bose clumps that only exhibit short-range correlations with either scarce occupancy or low coherence. Under these conditions, the system may be poorly approximated using classical field theory since other behavior would be possible, and higher-order corrections would be important. Another relevant aspect to consider when constructing EFTs for low-energy axions is the field fluctuations that oscillate rapidly in time scales $m_a^{-1}$ [20]. Typically, in the nonrelativistic limit, these fluctuations would be averaged to 0 on time scales much larger than the inverse of the axion mass. However, as we shall see later, nonlinear self-couplings of axions can induce a back-reaction of the fast-oscillating terms on the dominant slowly varying nonrelativistic field; thus, they must be treated with caution.

We would like also to emphasize that, from our point of view, since the nature of dark matter is a complete mystery, with no clue about the kind of particles that constitute it and how they interact, effective field theory approaches (EFTs) may be needed to study dark matter in a quantitative and systematic way. Particularly, higher-order corrections of nonrelativistic effective field theory of a weakly interacting scalar particle are needed because (i) since the kind of interactions among dark matter particles and also with standard model ones are unknown, radiative corrections need to be calculated at high orders to account for quantum effects; (ii) weak interactions can contain patterns or valuable information that are only revealed when considering higher orders; (iii) high-order corrections are necessary to understand how the theory behaves in some asymptotic limits; and (iv) as in other examples of EFTs, such as chiral perturbation theory or (potential) nonrelativistic QCD, computing higher orders is usually necessary years after their establishment to ensure that they are in line with the experimental data.

Moreover, within the context of axion physics, the collapse of large axion clouds to constitute dark Mmatter has not been addressed. While we know that large clouds of ordinary matter show Jeans instabilities and collapse, we do not know how this process happens (if it does) for dark matter. If dark matter is in a proportion 4:1 to ordinary matter, it is expected that compact objects of dark matter exist in our universe, and this is an issue that needs to be addressed. The time and length for which the collapse happens determine the size, temperature, and final mass of the compact object. Note that many times in the literature, Bosonic stars have been studied without having an estimate of size or temperature for these objects coming from the characteristics of dark matter [22,23,24,25]. Therefore, extending the original calculus of Jeans to axion dark matter is of interest, but we need the thermodynamics of an axion gas in order to do so. However, the computation of partition function from finite-temperature quantum field theory (QFT) results in boundary conditions for path integrals that cannot be resolved. Fortunately, axion dark matter is cold and nonrelativistic, and thus a quantum-mechanical description given by nonrelativistic EFTs provides a way to overcome the difficulties of finite-temperature QFT: a one-particle potential can be defined, and, from it, we can find the macroscopic description of an axion gas. In a large cloud, the extremely nonlinear characteristic of the axion’s self-interaction potential, together with the weak coupling to photons, results in a reheating through any collapse, and higher-order corrections to the nonrelativistic dynamics of axions may be necessary to obtain a realistic model, specifically if computational physics is applied to resolve the collapse.

The manuscript is arranged as follows: Section 2 simply defines the relativistic field theory of a real scalar field. Afterwards, we present in Section 3 the general procedure through which the relativistic Lagrangian, described by the real scalar field $\varphi$, transforms into the nonrelativistic Lagrangian, described by the complex scalar field $\psi$. Next, Section 4 is devoted to presenting our computation of the ${\left({\psi }^{*}\psi \right)}^5$ higher-order corrections taking into account the effects of fast-oscillating fields into the dominant slowly varying nonrelativistic one due to the axion’s nonlinear self-interactions. Finally, some conclusions and possible future steps are given in Section 5.

2. Axion Relativistic Field Theory

For momentum scales much smaller than the axion’s decay constant, $f_a$, the axion field can be represented using an elementary quantum field $\varphi \left(x\right)$ that is a real Lorentz scalar. Since $\varphi$ ultimately represents an angle, $\varphi$ must obey the shift symmetry $\varphi \to \varphi +2\pi f_a$. At even smaller momenta, i.e., below the QCD confinement scale (≈$1\mathrm{GeV}$), the self-interactions of axions can be described using a relativistic potential $V\left(\varphi \right)$, known as the axion potential. This EFT is often regarded as relativistic axion field theory (see Ref. [26] for a review). The Lagrangian density of the field $\varphi$ can be written as follows: (Note here that some authors define $V\left(\varphi \right)$ without including the mass term $\frac{1}{2}{m}_a^2{\varphi }^2$ and, thus, it is directly depicted in the Lagrangian).

$$\mathcal{L}=\frac{1}{2}{\partial }_u\varphi {\partial }^u\varphi -V\left(\varphi \right),$$

(1)

where $V\left(\varphi \right)$ is a periodic function of $\varphi$ with period $2\pi f_a$ as a result of the aforementioned shift symmetry. It is worth mentioning that $\varphi$ also has $Z_2$ symmetry $\varphi \left(x\right)\to -\varphi \left(x\right)$. Consequently, $V\left(\varphi \right)$ is an even function of $\varphi$; for $\varphi /f_a\ll 1$, it can be expanded in powers of ${\varphi }^2$:

$$V\left(\varphi \right)=\frac{1}{2}{m}_a^2{\varphi }^2+m_a^2{f}_a^2\sum _{n=2}^{\infty }\frac{{\lambda }_{2n}}{\left(2n\right)!}{\left(\frac{{\varphi }^2}{f_a^2}\right)}^n,$$

(2)

where ${\lambda }_{2n}$ are dimensionless coupling constants. In reasonable axion models, the constants ${\lambda }_{2n}$ have natural values of order 1. The Feynman rule for the ($2n$)-axion vertex is $-i{\lambda }_{2n}{m}_a^2/f_a^{2n-2}$, while the quantum-loop suppression factor for the QCD axion in the postinflation scenario is approximately $m_a^2/f_a^2\simeq {10}^{-48\pm 4}$. (± does not symbolize uncertainty but the possible range of magnitude of the quantum-loop suppression factor as a direct consequence of the allowed regions of $m_a$ and $f_a$. In the postinflationary scenario ${10}^9\mathrm{GeV}\lesssim f_a\lesssim {10}^{12}\mathrm{GeV}$ and ${10}^{-6}\mathrm{eV}\lesssim m_a\lesssim {10}^{-3}\mathrm{eV}$ [11]). Consequently, for most purposes, the effects of quantum loops can be omitted, i.e., relativistic axion field theory can be treated as classical field theory. It is important to mention that in fundamental quantum field theory, the series of the effective potential must end at power ${\varphi }^4$ so the system can be renormalizable [27]. However, this requirement is not necessary for either classical field theory or effective field theory [26].

In the literature, one can find two alternatives for the relativistic axion potential. The first one is known as the instanton potential, and it is given by the following expression:

$$V\left(\varphi \right)=m_a^2{f}_a^2\left[1-cos\left(\frac{\varphi }{f_a}\right)\right].$$

(3)

This potential was first derived by Peccei and Quinn [3,28] using a dilute instanton gas approximation; thus, there is no known way to improve it. Therefore, the instanton potential should be considered only as a good phenomenological approximation, subsequently implemented in most phenomenological studies regarding axion physics [29,30]. If one compares the Taylor series expansion of Equation (3) with (2), the dimensionless coupling constants ${\lambda }_{2n}$ for axion self-interactions are defined by ${\lambda }_{2n}={(-1)}^{n+1}$. It is clear that ${\lambda }_4=-1$, which implies that axion pair interactions are attractive.

The second option for the axion’s relativistic potential is known as chiral potential because it is derived frothe leading-order chiral effective field theory considering the light pseudoscalar mesons of QCD and the axion field:

$$V\left(\varphi \right)=m_{\pi }^2{f}_{\pi }^2\left[1-\sqrt{1-\frac{4z}{{(1+z)}^2}{sin}^2\left(\frac{\varphi }{2f_a}\right)}\right],$$

(4)

where $m_{\pi }$ is the pion’s mass, $f_{\pi }$ is the pion’s decay constant, and $z=m_u/m_d$ is the ratio of the up and down quark masses. This potential was first derived by Di Vecchia and Veneziano [31], continued in diverse studies such as those in Refs. [26,30,32]. Note also that the instanton potential can be derived from the chiral potential in Equation (4). To do so, one should apply the binomial expansion to the square root, then truncate it after the ${sin}^2(\varphi /2f_a)$ term, and finally use a trigonometric identity.

According to the chiral potential, the dimensionless coupling constant for the 4-axion vertex is

$${\lambda }_4=-\frac{1-z+z^2}{(1+z^2)}.$$

(5)

Since $z>0$, it is easy to see that ${\lambda }_4<0$; consequently, the axion pair interactions are, once again, attractive [17,26].

3. Nonrelativistic Effective Field Theory

The predominant method used to build a nonrelativistic effective field theory for axions is to transform the real scalar field $\varphi$ into a complex scalar field $\psi$. This may seem inconsistent because the complex scalar field is formed by two real scalar fields. However, the aim of the procedure is to obtain a nonrelativistic Lagrangian from the original relativistic one while maintaining the number of degrees of freedom. The relativistic Lagrangian is quadratic in time derivatives of $\varphi$, while the nonrelativistic one has only linear terms in time derivatives of $\psi$. Furthermore, the number of propagating degrees of freedom in a Lagrangian containing up to quadratic terms in time derivatives is equal to the number of real fields in it. On the other hand, the number of propagating degrees of freedom for a Lagrangian containing only first derivatives in time is equal to half the number of real fields in it. Therefore, the number of degrees of freedom in both Lagrangians remains equal.

Although both Refs. [20,33] implemented different methods in order to construct an EFT for nonrelativistic axions, the final effective Lagrangian was demonstrated to be the same. Therefore, it becomes compelling to take the results of these investigations as a basis for future developments and applications. Specifically, the procedure first detailed in Ref. [20] is implemented herein to latercompute the higher-order corrections.

3.1. Nonrelativistic EFT Lagrangian Density

The analysis in Ref. [20] begins from the relativistic Lagrangian of $\varphi$, where only one self-interaction term is utilized in the derivation. In this work, four self-interaction terms are taken into account,

$$\begin{array}{cc}\hfill \mathcal{L}=& \frac{1}{2}{\partial }_u\varphi {\partial }^u\varphi -\frac{1}{2}{m}_a^2{\varphi }^2-m_a^2{f}_a^2\sum _{n=2}^5\frac{{\lambda }_{2n}}{\left(2n\right)!}\frac{{\varphi }^{2n}}{f_a^{2n}},\hfill \end{array}$$

(6)

where the constants $m_a^2{\lambda }_{2n}/f_a^{2n-2}$ indicate the strength of the self-interaction terms. To simplify this notation, it is convenient to define

$$\begin{array}{cc}\hfill {\overline{\lambda }}_{2n}& =\frac{m_a^2}{f_a^{2n-2}}{\lambda }_{2n},\hfill \end{array}$$

(7)

and thus the relativistic Lagrangian becomes

$$\begin{array}{cc}\hfill \mathcal{L}=& \frac{1}{2}{\partial }_u\varphi {\partial }^u\varphi -\frac{1}{2}{m}_a^2{\varphi }^2-\sum _{n=2}^5{\overline{\lambda }}_{2n}\frac{{\varphi }^{2n}}{\left(2n\right)!}.\hfill \end{array}$$

(8)

Consequently, the Hamiltonian density of the system is

$$\begin{array}{cc}\hfill \mathcal{H}=& \pi (t,x)\dot{\varphi }(t,x)-\mathcal{L}\hfill \\ \hfill =& \frac{1}{2}{\pi }^2-\frac{1}{2}{(▽\varphi )}^2+\frac{1}{2}{m}_a^2{\varphi }^2+\sum _{n=2}^5{\overline{\lambda }}_{2n}\frac{{\varphi }^{2n}}{\left(2n\right)!},\hfill \end{array}$$

(9)

where $\pi =\partial \mathcal{L}/\partial \dot{\prec }=\dot{\prec }$ is the canonical momentum field of $\varphi$. Finally, the system’s equations of motion are

$$\begin{array}{cc}\hfill \dot{\varphi }=+\frac{\delta \mathcal{H}}{\delta \pi }=& \pi ,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill \dot{\pi }=-\frac{\delta \mathcal{H}}{\delta \varphi }=& ({▽}^2-m_a^2)\varphi -\sum _{n=2}^5{\overline{\lambda }}_{2n}\frac{{\varphi }^{2n-1}}{(2n-1)!}.\hfill \end{array}$$

(11)

To obtain the nonrelativistic EFT for low-energy axions, one should typically relate $\varphi (t,x)$ and $\pi (t,x)$ with $\psi (t,x)$ as follows:

$$\begin{array}{cc}\hfill \varphi & =\frac{1}{\sqrt{2m_a}}\left(e^{-i{m}_{a}t}\psi +e^{i{m}_{a}t}{\psi }^{*}\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \pi & =-i\sqrt{\frac{m_a}{2}}\left(e^{-i{m}_{a}t}\psi -e^{i{m}_{a}t}{\psi }^{*}\right).\hfill \end{array}$$

(13)

However, in order to obtain relativistic corrections in the EFT, the authors in Ref. [20] propose the following nonlocal canonical field transformation:

$$\begin{array}{cc}\hfill \varphi & =\frac{1}{\sqrt{2m_a}}{\mathcal{P}}^{-\frac{1}{2}}\left(e^{-i{m}_{a}t}\psi +e^{i{m}_{a}t}{\psi }^{*}\right),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \pi & =-i\sqrt{\frac{m_a}{2}}{\mathcal{P}}^{\frac{1}{2}}\left(e^{-i{m}_{a}t}\psi -e^{i{m}_{a}t}{\psi }^{*}\right),\hfill \end{array}$$

(15)

where

$$\mathcal{P}\equiv \sqrt{1-\frac{{▽}^2}{m_a^2}},$$

(16)

is an operator that will be later expanded in powers of ${▽}^2/m_a^2$. The operator $\mathcal{P}$ is defined so that the equation of motion of $\psi$ does not contain rapidly oscillating terms in free field theory. It is also important to mention that $m_a\mathcal{P}$ corresponds to the total energy of a free relativistic particle.

The transformations in Equations (14) and (15) uniquely define both $\psi$ and ${\psi }^{*}$; that is, an expression of $\psi$ in terms of $\varphi$ and $\pi$ can be derived

$$\psi =\sqrt{\frac{m_a}{2}}{e}^{i{m}_{a}t}{\mathcal{P}}^{\frac{1}{2}}\left[\varphi +\frac{i}{m_a}{\mathcal{P}}^{-1}\pi \right].$$

(17)

Additionally, it is possible to obtain the equation of motion of $\psi$. First, we derive Equation (17) with respect to time and obtain

$$\begin{array}{c}\hfill \dot{\psi }=i{m}_a\sqrt{\frac{m_a}{2}}{e}^{i{m}_{a}t}{\mathcal{P}}^{\frac{1}{2}}\left[\varphi +\frac{i}{m_a}{\mathcal{P}}^{-1}\pi \right]+\sqrt{\frac{m_a}{2}}{e}^{i{m}_{a}t}{\mathcal{P}}^{\frac{1}{2}}\left[\dot{\varphi }+\frac{i}{m_a}{\mathcal{P}}^{-1}\dot{\pi }\right].\end{array}$$

(18)

Afterwards, by replacing Equation (17) in (18) and then multiplying by i, the following expression is obtained:

$$i\dot{\psi }=-m_a\psi +\sqrt{\frac{m_a}{2}}{e}^{i{m}_{a}t}{\mathcal{P}}^{\frac{1}{2}}\left[i\dot{\varphi }-\frac{1}{m_a}{\mathcal{P}}^{-1}\dot{\pi }\right],$$

(19)

On the other hand, from Equation (16), we have:

$${▽}^2-m_a^2=-m_a^2{\mathcal{P}}^2,$$

(20)

and therefore Equation (11) can be written as

$$\dot{\pi }=-m_a^2{\mathcal{P}}^2\varphi -\sum _{n=2}^5{\overline{\lambda }}_{2n}\frac{{\varphi }^{2n-1}}{(2n-1)!},$$

(21)

in such a way that Equation (19) is given by

$$\begin{array}{cc}\hfill i\dot{\psi }=& -m_a\psi +m_a\mathcal{P}\left[\sqrt{\frac{m_a}{2}}{e}^{i{m}_{a}t}{\mathcal{P}}^{\frac{1}{2}}\left(\varphi +\frac{i}{m_a}{\mathcal{P}}^{-1}\pi \right)\right]\hfill \\ \hfill & +\frac{1}{\sqrt{2m_a}}\sum _{n=2}^5{e}^{i{m}_{a}t}{\mathcal{P}}^{-\frac{1}{2}}\left({\overline{\lambda }}_{2n}\frac{{\varphi }^{2n-1}}{(2n-1)!}\right),\hfill \end{array}$$

(22)

when using Equations (10) and (21). Lastly, by replacing Equations (14) and (17), one obtains the equation of motion for the field $\psi$:

$$\begin{array}{c}\hfill i\dot{\psi }=m_a(\mathcal{P}-1)\psi +\sum _{n=2}^5\frac{n{\overline{\lambda }}_{2n}}{2^{n-1}\left(2n\right)!m_a^n}{e}^{i{m}_{a}t}{\mathcal{P}}^{-\frac{1}{2}}{\mathcal{A}}^{2n-1},\end{array}$$

(23)

where

$$\mathcal{A}=e^{-i{m}_{a}t}{\mathcal{P}}^{-\frac{1}{2}}\psi +e^{i{m}_{a}t}{\mathcal{P}}^{-\frac{1}{2}}{\psi }^{*}.$$

(24)

As a side note, the Lagrangian that describes the dynamics of $\psi$ can be found by requiring that its equation of motion matches the one above:

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathrm{eff}}=\frac{i}{2}(\dot{\psi }{\psi }^{*}-\psi {\dot{\psi }}^{*})-m_a{\psi }^{*}(\mathcal{P}-1)\psi -\sum _{n=2}^5{\overline{\lambda }}_{2n}{\left(\frac{1}{2m_a}\right)}^n\frac{{\mathcal{A}}^{2n}}{\left(2n\right)!}.\end{array}$$

(25)

3.2. Importance of Rapid Field Fluctuations

Once the equation of motion of $\psi$ is derived, it becomes tempting to expand $\mathcal{P}$ in powers of ${▽}^2/m_a^2$ to obtain an expression that describes the behavior of low-energy axions. However, this would not incorporate how the fast-oscillation terms inside $\psi$ affect the behavior of the slowly varying field. In the original work in Ref. [20], to show the importance of this effect, all expressions of $\mathcal{P}$ were approximated as

$$\mathcal{P}-1\approx -\frac{{▽}^2}{2m_a^2},{\mathcal{P}}^{-\frac{1}{2}}\approx 1.$$

(26)

This is also replicated in this paper but only with two self-interaction terms in the effective potential so as not to overload the text with more mathematical expressions than are necessary. By inputting Equation (26) into (23), one obtains

$$\begin{array}{cc}\hfill i\dot{\psi }\approx & -\frac{1}{2m_a}{▽}^2\psi +\frac{{\overline{\lambda }}_4}{8m_a^2}{\left|\psi \right|}^2\psi +\frac{{\overline{\lambda }}_6}{4·4!m_a^3}{\left|\psi \right|}^4\psi +\frac{{\overline{\lambda }}_4}{4!m_a^2}(e^{-2i{m}_{a}t}{\psi }^3+3e^{2i{m}_{a}t}{\left|\psi \right|}^2{\psi }^{*}\hfill \\ \hfill & +e^{4i{m}_{a}t}{{\psi }^{*}}^3)+\frac{{\overline{\lambda }}_6}{8·5!m_a^3}(e^{-4i{m}_{a}t}{\psi }^5+5e^{-2i{m}_{a}t}{\left|\psi \right|}^2{\psi }^3+10e^{2i{m}_{a}t}{\left|\psi \right|}^4{\psi }^{*}\hfill \\ & +5e^{4i{m}_{a}t}{\left|\psi \right|}^2{{\psi }^{*}}^3+e^{6i{m}_{a}t}{{\psi }^{*}}^5).\hfill \end{array}$$

(27)

All terms inside parenthesis are fast-oscillating factors. Typically, they would be approximated to 0 in the limit of large $m_a$, i.e., when averaging over time scales $\Delta t\gg m_a^{-1}$. Nevertheless, to analyze how the equation behaves, let us expand $\psi$ as the following ansätze

$$\psi ={\psi }_s+e^{2i{m}_{a}t}\delta \psi ,$$

(28)

where ${\psi }_s$ is slow-varying, and $e^{2i{m}_{a}t}\delta \psi$ is clearly not. Then, by inserting Equation (28) into (27) and only keeping terms up to linear order in $\delta \psi$, which also vary in time more slowly than $e^{\pm i{m}_{a}t}$, one obtains

$$\begin{array}{cc}\hfill i{\dot{\psi }}_s\approx & -\frac{1}{2m_a}{▽}^2{\psi }_s+\frac{{\overline{\lambda }}_4}{8m_a^2}|{\psi }_s{|}^2{\psi }_s+\frac{{\overline{\lambda }}_6}{4·4!m_a^3}{\left|{\psi }_s\right|}^4{\psi }_s\hfill \\ \hfill & +\frac{{\overline{\lambda }}_4}{8m_a^2}\left({\psi }_s^2\delta \psi +2{\psi }_s^2\delta {\psi }^{*}\right)\hfill \\ \hfill & +\frac{{\overline{\lambda }}_6}{4·4!m_a^3}\left(2|{\psi }_s{|}^2{\psi }_s^2\delta \psi +3{\left|{\psi }_s\right|}^4\delta {\psi }^{*}\right).\hfill \end{array}$$

(29)

The last two terms on the right-hand side (RHS) of Equation (29) show that the dynamics of the slow-varying field are affected by fast-oscillating factors in nontrivial ways for each self-interaction included in the calculation.

3.3. Defining the Iterative Process

To study the effect of fast-oscillating factors, $\psi$ must be expanded as an infinite series of harmonics

$$\psi (t,x)=\sum _{\nu =-\infty }^{+\infty }{\psi }_{\nu }(t,x)e^{i\nu m_{a}t},$$

(30)

where each ${\psi }_{\nu }$ varies slowly on a time scale $m_a^{-1}$. Additionally, the mode with $\nu =0$ is designated as the lowest slowly varying contribution of the field, and it is denoted as ${\psi }_{\nu =0}\equiv {\psi }_s$. In the nonrelativistic limit, it is assumed that $|{\psi }_{\nu }-{\psi }_s|\ll |{\psi }_s|$, i.e., ${\psi }_s$ is the dominant term of $\psi$. Next, it is convenient to define

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_{\nu }& \equiv {\mathcal{P}}^{-\frac{1}{2}}{\psi }_{\nu },\hfill \end{array}$$

(31)

and

$$\begin{array}{cc}\hfill \tilde{G}& \equiv e^{i{m}_{a}t}{\mathcal{P}}^{-\frac{1}{2}}{\mathcal{A}}^3,\tilde{F}\equiv e^{i{m}_{a}t}{\mathcal{P}}^{-\frac{1}{2}}{\mathcal{A}}^5,\hfill \\ \hfill \tilde{D}& \equiv e^{i{m}_{a}t}{\mathcal{P}}^{-\frac{1}{2}}{\mathcal{A}}^7,\tilde{R}\equiv e^{i{m}_{a}t}{\mathcal{P}}^{-\frac{1}{2}}{\mathcal{A}}^9,\hfill \end{array}$$

(32)

in such a way that, by placing Equations (30) and (31) into Equations (32), it is possible to expand $\tilde{G}$, $\tilde{F}$, $\tilde{D}$, and $\tilde{R}$ as an infinite series of harmonics:

$$\begin{array}{cc}\hfill \tilde{G}(t,x)& =\sum _{\nu =-\infty }^{+\infty }{\tilde{G}}_{\nu }(t,x)e^{\nu i{m}_{a}t},\tilde{F}(t,x)=\sum _{\nu =-\infty }^{+\infty }{\tilde{F}}_{\nu }(t,x)e^{\nu i{m}_{a}t},\hfill \\ \hfill \tilde{D}(t,x)& =\sum _{\nu =-\infty }^{+\infty }{\tilde{D}}_{\nu }(t,x)e^{\nu i{m}_{a}t},\tilde{R}(t,x)=\sum _{\nu =-\infty }^{+\infty }{\tilde{R}}_{\nu }(t,x)e^{\nu i{m}_{a}t},\hfill \end{array}$$

(33)

where, for brevity, the mode functions ${\tilde{G}}_{\nu }$, ${\tilde{F}}_{\nu }$, ${\tilde{D}}_{\nu }$, and ${\tilde{R}}_{\nu }$ can be found in Appendix A. Inputting Equation (31) and Equations (A1) to (A4) into Equation (23) results in the following equation of motion for each mode $\nu$:

$$\begin{array}{cc}\hfill i{\dot{\psi }}_{\nu }-\nu m_a{\psi }_{\nu }=& m_a(\mathcal{P}-1){\psi }_{\nu }+\frac{{\overline{\lambda }}_4}{4!m_a^2}{\tilde{G}}_{\nu }\hfill \\ \hfill & +\frac{3{\overline{\lambda }}_6}{4·6!m_a^3}{\tilde{F}}_{\nu }+\frac{{\overline{\lambda }}_8}{2·8!m_a^4}{\tilde{D}}_{\nu }\hfill \\ \hfill & +\frac{5{\overline{\lambda }}_{10}}{16·10!m_a^5}{\tilde{R}}_{\nu }.\hfill \end{array}$$

(34)

Then, by multiplying Equation (34) by ${\mathcal{P}}^{-\frac{1}{2}}$ and rearranging terms, it is possible to obtain

$${\mathsf{\Psi }}_{\nu }=-\frac{i}{m_a}{\mathsf{\Gamma }}_{\nu }{\dot{\mathsf{\Psi }}}_{\nu }+{\overline{\lambda }}_4{G}_{\nu }+{\overline{\lambda }}_6{F}_{\nu }+{\overline{\lambda }}_8{D}_{\nu }+{\overline{\lambda }}_{10}{R}_{\nu },$$

(35)

where

$${\mathsf{\Gamma }}_{\nu }={(1-\nu -\mathcal{P})}^{-1},$$

(36)

and

$$\begin{array}{cc}\hfill G_{\nu }& =\frac{{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-\frac{1}{2}}}{4!m_a^3}{\tilde{G}}_{\nu },F_{\nu }=\frac{3{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-\frac{1}{2}}}{4·6!m_a^4}{\tilde{F}}_{\nu },\hfill \\ \hfill D_{\nu }& =\frac{{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-\frac{1}{2}}}{2·8!m_a^5}{\tilde{D}}_{\nu },R_{\nu }=\frac{5{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-\frac{1}{2}}}{16·10!m_a^6}{\tilde{R}}_{\nu }.\hfill \end{array}$$

(37)

Now, given a physical object $\mathcal{F}(t,x)$, the spatial and temporal variations are compared with $m_a^{-1}$, and the self-interactions are mediated by the constants ${\overline{\lambda }}_{2n}$. It is useful to parameterize the magnitude of spatial and temporal variations as

$$\frac{{\nabla }^2\mathcal{F}}{m_a^2}\sim {ϵ}_x\mathcal{F},\frac{\dot{\mathcal{F}}}{m_a}\sim {ϵ}_t\mathcal{F}.$$

(38)

It is appropriate to assume ${ϵ}_x$, ${ϵ}_t$, ${\overline{\lambda }}_{2n}\ll 1$ for weakly interacting systems in the nonrelativistic limit. Then, Equation (35) holds for all orders of ${ϵ}_x,{ϵ}_t$, and ${\overline{\lambda }}_{2n}$. Additionally, the first term on the RHS in Equation (35) is suppressed relative to ${\mathsf{\Psi }}_{\nu }$ by ${ϵ}_t$, the second term is suppressed relative to $G_{\nu }$ by ${\overline{\lambda }}_4$, the third term is suppressed relative to $F_{\nu }$ by ${\overline{\lambda }}_6$, the fourth term is suppressed relative to $D_{\nu }$ by ${\overline{\lambda }}_8$, and the fifth term is suppressed relative to $R_{\nu }$ by ${\overline{\lambda }}_{10}$. Therefore, the RHS of (35) can be interpreted as a perturbative source for ${\mathsf{\Psi }}_{\nu }$.

As long as functions ${\psi }_{\nu }$ are constructed so that each mode satisfies Equation (34), the full series satisfies the equation of motion (23). Taking this into account, the authors of Ref. [20] proposed the following iterative process to find $\psi$. The $0th$-order approximation of ${\mathsf{\Psi }}_{\nu }$ is

$${\mathsf{\Psi }}_{\nu }^{\left(0\right)}(t,x)=\left\{\begin{array}{cc}{\mathsf{\Psi }}_s(t,x)\hfill & \mathrm{if}\nu =0,\hfill \\ 0\hfill & \mathrm{if}\nu \ne 0,\hfill \end{array}\right.$$

(39)

For $\nu \ne 0$, the $nth$ correction to ${\mathsf{\Psi }}_{\nu }$ at the $nth$ iteration is denoted as ${\mathsf{\Psi }}_{\nu }^{\left(n\right)}$. Mathematically, this results in ${\mathsf{\Psi }}_{\nu }$ being expressed as

$${\mathsf{\Psi }}_{\nu }(t,x)=\sum _{n=0}^{\infty }{\mathsf{\Psi }}_{\nu }^{\left(n\right)}(t,x),$$

(40)

where ${\mathsf{\Psi }}_{\nu }^{\left(0\right)}=0$ because the way of defining the $0th$-order approximation. Additionally, ${\mathsf{\Psi }}_0^{\left(n\right)}$ is set to 0 for $n>0$. As mentioned before, the perturbative source of ${\mathsf{\Psi }}_{\nu }$ is Equation (35), i.e., ${\mathsf{\Psi }}_{\nu }^{\left(n\right)}$ is always proportional to the n powers of ${ϵ}_t$ and ${\overline{\lambda }}_{2n}$. Additionally, for all $\nu$, $G_{\nu }$, $F_{\nu }$, $D_{\nu }$, and $R_{\nu }$ are expanded as

$$\begin{array}{cc}\hfill & G_{\nu }(t,x)=\sum _{n=0}^{\infty }{G}_{\nu }^{\left(n\right)}(t,x),F_{\nu }(t,x)=\sum _{n=0}^{\infty }{F}_{\nu }^{\left(n\right)}(t,x),\hfill \\ \hfill & D_{\nu }(t,x)=\sum _{n=0}^{\infty }{D}_{\nu }^{\left(n\right)}(t,x),R_{\nu }(t,x)=\sum _{n=0}^{\infty }{R}_{\nu }^{\left(n\right)}(t,x),\hfill \end{array}$$

(41)

where $G_{\nu }^{\left(n\right)}$, $F_{\nu }^{\left(n\right)}$, $D_{\nu }^{\left(n\right)}$, and $R_{\nu }^{\left(n\right)}$ contain all the terms proportional to the nth power of ${ϵ}_t$ and ${\overline{\lambda }}_{2n}$ in Equation (37), respectively. Next, by substituting Equation (40) and Equation (41) into Equation (35), and collecting terms with the same power of ${ϵ}_t$ and ${\overline{\lambda }}_{2n}$, one obtains the following expression:

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_{\nu }^{\left(n\right)}=& -\frac{i}{m_a}{\mathsf{\Gamma }}_{\nu }{\dot{\mathsf{\Psi }}}_{\nu }^{(n-1)}+{\overline{\lambda }}_4{G}_{\nu }^{(n-1)}+{\overline{\lambda }}_6{F}_{\nu }^{(n-1)}+{\overline{\lambda }}_8{D}_{\nu }^{(n-1)}+{\overline{\lambda }}_{10}{R}_{\nu }^{(n-1)}.\hfill \end{array}$$

(42)

It is important to note that the above equation only works for $\nu \ne 0$. By substituting Equation (37) into (34) for $\nu =0$, it is possible to derive the following:

$$\begin{array}{cc}\hfill i{\dot{\psi }}_s=& m_a(\mathcal{P}-1){\psi }_s+m_a{\mathsf{\Gamma }}_0^{-1}{\mathcal{P}}^{\frac{1}{2}}\left[{\overline{\lambda }}_4{G}_0+{\overline{\lambda }}_6{F}_0\right.+\left.{\overline{\lambda }}_8{D}_0+{\overline{\lambda }}_{10}{R}_0\right].\hfill \end{array}$$

(43)

Therefore, one comes across the following way of proceeding. First, obtain an expression for $G_0$, $F_0$, $D_0$, and $R_0$ in terms of ${\mathsf{\Psi }}_s$ up to the nth power of ${ϵ}_t$ and ${\overline{\lambda }}_{2n}$. This results in the last term in the RHS in Equation (43) only containing ${\mathsf{\Psi }}_s$ terms proportional up to the $(n+1)$th power of ${ϵ}_t$ and ${\overline{\lambda }}_{2n}$. Next, expand all expressions of $\mathcal{P}$ in Equation (43) up to the $(n+1)$th power of ${ϵ}_x$. However, only the terms proportional up to the $(n+1)$th power of ${ϵ}_x$, ${ϵ}_t$ and ${\overline{\lambda }}_{2n}$ are kept.

One apparent problem with this process is that new degrees of freedom are added from the first term on the RHS in Equation (42). How exactly this dilemma is solved is more effectively shown in the next section. Nevertheless, the basic idea is to represent all expressions of ${\dot{\mathsf{\Psi }}}_{\nu }^{\left(m\right)}$, where $m\le n$, in terms of ${\mathsf{\Psi }}_s$ and powers of ${ϵ}_t$. These terms are maintained as such throughout the whole process up until the effective Lagrangian is calculated. A final step is added at this point: all terms that contain ${ϵ}_t$, other than $\frac{i}{2}({\dot{\psi }}_s{\psi }_s^{*}-{\psi }_s{\dot{\psi }}_s^{*})$, are expressed in terms of ${\psi }_s$ by replacing the $0th$-order approximation equation of motion. The final expression is the $nth$ approximation of the nonrelativistic effective Lagrangian of $\psi$ when ${\psi }_s$ is the dominant term.

The above procedure is iterative because $G_{\nu }^{\left(n\right)}$, $F_{\nu }^{\left(n\right)}$, $D_{\nu }^{\left(n\right)}$, and $R_{\nu }^{\left(n\right)}$ are expressed in terms of ${\mathsf{\Psi }}_{\nu }^{\left(m\right)}$, where $m\le n$, and, from Equation (42), these terms are expressed in terms of $G_{\nu }^{(m-1)}$, $F_{\nu }^{(m-1)}$, $D_{\nu }^{(m-1)}$, and $R_{\nu }^{(m-1)}$. In other words, in order to obtain $G_{\nu }^{\left(n\right)}$, $F_{\nu }^{\left(n\right)}$, $D_{\nu }^{\left(n\right)}$, and $R_{\nu }^{\left(n\right)}$, it is necessary to already know an expression for all $G_{\nu }^{\left(m\right)}$, $F_{\nu }^{\left(m\right)}$, $D_{\nu }^{\left(m\right)}$, and $R_{\nu }^{\left(m\right)}$, where $m\le n$.

4. Higher-Order Corrections

Before implementing the iterative process just described in the previous section, some other considerations need to be applied for practical purposes. First, every new iteration does not take into account the highest order of ${\overline{\lambda }}_{2n}$ of the previous iteration. Consequently, the 0th-order iteration takes into account all four interaction terms; meanwhile, the 1st-order iteration only takes into account the three lowest-order interaction terms and so on. This is applied in order to reduce the number of expressions. However, as a result of this methodology, the $\mathcal{P}$ operator must be expanded with care. Secondly, the lengths of $G_{\nu }^{\left(n\right)}$, $F_{\nu }^{\left(n\right)}$, $D_{\nu }^{\left(n\right)}$, and $R_{\nu }^{\left(n\right)}$ increase drastically with each iteration. As a result, it is necessary to introduce new notation to divide these expressions into pieces. How exactly this is achieved is better shown in the subsequent subsections. Finally, to reduce the numerical constants associated with some expressions, it is convenient to define

$$\begin{array}{c}\hfill \alpha \left(n\right)=2^n{(n!)}^2.\end{array}$$

(44)

4.1. Zeroth-Order Iteration

Given that the $0th$-order approximation of ${\mathsf{\Psi }}_{\nu }$ is Equation (39), Equations (45)–(48) can be derived from the expressions in Equation (37):

$$\begin{array}{cc}\hfill G_{\nu }^{\left(0\right)}=& \frac{{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{4!m_a^3}\left[{\mathsf{\Psi }}_s^3{\delta }_{\nu ,-2}+3{\mathsf{\Psi }}_s|{\mathsf{\Psi }}_s{|}^2{\delta }_{\nu ,0}+3{\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2{\delta }_{\nu ,2}+{{\mathsf{\Psi }}_s^{*}}^3{\delta }_{\nu ,4}\right],\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill F_{\nu }^{\left(0\right)}=\frac{3{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{4·6!m_a^4}[& {\mathsf{\Psi }}_s^5{\delta }_{\nu ,-4}+5{\mathsf{\Psi }}_s^3|{\mathsf{\Psi }}_s{|}^2{\delta }_{\nu ,-2}+10{\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^4{\delta }_{\nu ,0}\hfill \\ \hfill & +10{\mathsf{\Psi }}_s^{*}|{\mathsf{\Psi }}_s{|}^4{\delta }_{\nu ,2}+5{{\mathsf{\Psi }}_s^{*}}^3{\left|{\mathsf{\Psi }}_s\right|}^2{\delta }_{\nu ,4}+{{\mathsf{\Psi }}_s^{*}}^5{\delta }_{\nu ,6}],\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill D_{\nu }^{\left(0\right)}=\frac{{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{2·8!m_a^5}[& {\mathsf{\Psi }}_s^7{\delta }_{\nu ,-6}+7{\mathsf{\Psi }}_s^5|{\mathsf{\Psi }}_s{|}^2{\delta }_{\nu ,-4}+21{\mathsf{\Psi }}_s^3{\left|{\mathsf{\Psi }}_s\right|}^4{\delta }_{\nu ,-2}\hfill \\ \hfill & +35{\mathsf{\Psi }}_s|{\mathsf{\Psi }}_s{|}^6{\delta }_{\nu ,0}+35{\mathsf{\Psi }}_s^{*}|{\mathsf{\Psi }}_s{|}^6{\delta }_{\nu ,2}+21{{\mathsf{\Psi }}_s^{*}}^3{\left|{\mathsf{\Psi }}_s\right|}^4{\delta }_{\nu ,4}\hfill \\ \hfill & +7{{\mathsf{\Psi }}_s^{*}}^5{\left|{\mathsf{\Psi }}_s\right|}^2{\delta }_{\nu ,6}+{{\mathsf{\Psi }}_s^{*}}^7{\delta }_{\nu ,8}],\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill R_{\nu }^{\left(0\right)}=\frac{5{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{16·10!m_a^6}[& {\mathsf{\Psi }}_s^9{\delta }_{\nu ,-8}+9{\mathsf{\Psi }}_s^7|{\mathsf{\Psi }}_s{|}^2{\delta }_{\nu ,-6}+36{\mathsf{\Psi }}_s^5|{\mathsf{\Psi }}_s{|}^4{\delta }_{\nu ,-4}+84{\mathsf{\Psi }}_s^3{\left|{\mathsf{\Psi }}_s\right|}^6{\delta }_{\nu ,-2}\hfill \\ \hfill & +126{\mathsf{\Psi }}_s|{\mathsf{\Psi }}_s{|}^8{\delta }_{\nu ,0}+126{\mathsf{\Psi }}_s^{*}|{\mathsf{\Psi }}_s{|}^8{\delta }_{\nu ,2}+84{{\mathsf{\Psi }}_s^{*}}^3{\left|{\mathsf{\Psi }}_s\right|}^6{\delta }_{\nu ,4}\hfill \\ \hfill & +36{{\mathsf{\Psi }}_s^{*}}^5|{\mathsf{\Psi }}_s{|}^4{\delta }_{\nu ,6}+9{{\mathsf{\Psi }}_s^{*}}^7{\left|{\mathsf{\Psi }}_s\right|}^2{\delta }_{\nu ,8}+{{\mathsf{\Psi }}_s^{*}}^9{\delta }_{\nu ,10}],\hfill \end{array}$$

(48)

For $\nu =0$, these expressions take the following form

$$\begin{array}{cc}\hfill G_0^{\left(0\right)}=& \frac{2{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{\alpha \left(2\right)m_a^3}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^2\right),F_0^{\left(0\right)}=\frac{3{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{\alpha \left(3\right)m_a^4}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^4\right),\hfill \\ \hfill D_0^{\left(0\right)}=& \frac{4{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{\alpha \left(4\right)m_a^5}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^6\right),R_0^{\left(0\right)}=\frac{5{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{\alpha \left(5\right)m_a^6}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^8\right).\hfill \end{array}$$

(49)

4.2. First-Order Iteration

In this iteration, ${\mathsf{\Psi }}_{\nu }={\mathsf{\Psi }}_{\nu }^{\left(1\right)}$. With this in mind, Equations (50)–(52) can be derived again from the expressions in Equation (37)

$$\begin{array}{cc}\hfill G_{\nu }^{\left(1\right)}=\frac{3{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{4!m_a^3}[& {\mathsf{\Psi }}_s^2\left({\mathsf{\Psi }}_{2+\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{-\nu }^{\left(1\right)*}\right)+2{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{2-\nu }^{\left(1\right)*}\right)\hfill \\ \hfill & +{{\mathsf{\Psi }}_s^{*}}^2\left({\mathsf{\Psi }}_{-2+\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{4-\nu }^{\left(1\right)*}\right)],\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill F_{\nu }^{\left(1\right)}=\frac{15{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{4·6!m_a^4}[& {\mathsf{\Psi }}_s^4\left({\mathsf{\Psi }}_{4+\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{-2-\nu }^{\left(1\right)*}\right)+4{\mathsf{\Psi }}_s^2{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{2+\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{-\nu }^{\left(1\right)*}\right)\hfill \\ \hfill & +6|{\mathsf{\Psi }}_s{|}^4\left({\mathsf{\Psi }}_{\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{2-\nu }^{\left(1\right)*}\right)+4{{\mathsf{\Psi }}_s^{*}}^2{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{-2+\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{4-\nu }^{\left(1\right)*}\right)\hfill \\ \hfill & +{{\mathsf{\Psi }}_s^{*}}^4\left({\mathsf{\Psi }}_{-4+\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{6-\nu }^{\left(1\right)*}\right)],\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill D_{\nu }^{\left(1\right)}=\frac{7{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{2·8!m_a^5}[& {\mathsf{\Psi }}_s^6\left({\mathsf{\Psi }}_{6+\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{-4-\nu }^{\left(1\right)*}\right)+6{\mathsf{\Psi }}_s^4{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{4+\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{-2-\nu }^{\left(1\right)*}\right)\hfill \\ \hfill & +15{\mathsf{\Psi }}_s^2|{\mathsf{\Psi }}_s{|}^4\left({\mathsf{\Psi }}_{2+\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{-\nu }^{\left(1\right)*}\right)+20{\left|{\mathsf{\Psi }}_s\right|}^6\left({\mathsf{\Psi }}_{\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{2-\nu }^{\left(1\right)*}\right)\hfill \\ \hfill & +15{{\mathsf{\Psi }}_s^{*}}^2|{\mathsf{\Psi }}_s{|}^4\left({\mathsf{\Psi }}_{-2+\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{4-\nu }^{\left(1\right)*}\right)+6{{\mathsf{\Psi }}_s^{*}}^4{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{-4+\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{6-\nu }^{\left(1\right)*}\right)\hfill \\ \hfill & +{{\mathsf{\Psi }}_s^{*}}^6\left({\mathsf{\Psi }}_{-6+\nu }^{\left(1\right)}+{\mathsf{\Psi }}_{8-\nu }^{\left(1\right)*}\right)].\hfill \end{array}$$

(52)

For $\nu =0$, these expressions take the following form:

$$\begin{array}{cc}\hfill G_0^{\left(1\right)}=& \frac{3{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{4!m_a^3}\left[{\mathsf{\Psi }}_s^2{\mathsf{\Psi }}_2^{\left(1\right)}+2{\left|{\mathsf{\Psi }}_s\right|}^2{\mathsf{\Psi }}_2^{\left(1\right)*}+{{\mathsf{\Psi }}_s^{*}}^2\left({\mathsf{\Psi }}_4^{\left(1\right)*}+{\mathsf{\Psi }}_{-2}^{\left(1\right)}\right)\right],\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill F_0^{\left(1\right)}=\frac{15{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{4·6!m_a^4}[& {\mathsf{\Psi }}_s^4\left({\mathsf{\Psi }}_4^{\left(1\right)}+{\mathsf{\Psi }}_{-2}^{\left(1\right)*}\right)+4{\mathsf{\Psi }}_s^2|{\mathsf{\Psi }}_s{|}^2{\mathsf{\Psi }}_2^{\left(1\right)}+6{\left|{\mathsf{\Psi }}_s\right|}^4{\mathsf{\Psi }}_2^{\left(1\right)*}\hfill \\ \hfill & +4{{\mathsf{\Psi }}_s^{*}}^2{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_4^{\left(1\right)*}+{\mathsf{\Psi }}_{-2}^{\left(1\right)}\right)+{{\mathsf{\Psi }}_s^{*}}^4\left({\mathsf{\Psi }}_6^{\left(1\right)*}+{\mathsf{\Psi }}_{-4}^{\left(1\right)}\right)],\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill D_0^{\left(1\right)}=\frac{7{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{2·8!m_a^5}[& {\mathsf{\Psi }}_s^6\left({\mathsf{\Psi }}_6^{\left(1\right)}+{\mathsf{\Psi }}_{-4}^{\left(1\right)*}\right)+6{\mathsf{\Psi }}_s^4|{\mathsf{\Psi }}_s{|}^2\left({\mathsf{\Psi }}_4^{\left(1\right)}+{\mathsf{\Psi }}_{-2}^{\left(1\right)*}\right)+15{\mathsf{\Psi }}_s^2{\left|{\mathsf{\Psi }}_s\right|}^4{\mathsf{\Psi }}_2^{\left(1\right)}\hfill \\ \hfill & +20|{\mathsf{\Psi }}_s{|}^6{\mathsf{\Psi }}_2^{\left(1\right)*}+15{{\mathsf{\Psi }}_s^{*}}^2{\left|{\mathsf{\Psi }}_s\right|}^4\left({\mathsf{\Psi }}_4^{\left(1\right)*}+{\mathsf{\Psi }}_{-2}^{\left(1\right)}\right)\hfill \\ \hfill & +6{{\mathsf{\Psi }}_s^{*}}^4{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_6^{\left(1\right)*}+{\mathsf{\Psi }}_{-4}^{\left(1\right)}\right)+{{\mathsf{\Psi }}_s^{*}}^6\left({\mathsf{\Psi }}_8^{\left(1\right)*}+{\mathsf{\Psi }}_{-6}^{\left(1\right)}\right)],\hfill \end{array}$$

(55)

Afterwards, it is necessary to express all ${\mathsf{\Psi }}_{\nu }^{\left(1\right)}$ in terms of ${\mathsf{\Psi }}_s$. This is computed by taking ${\overline{\lambda }}_{10}=0$ and $n=1$ in Equation (42),

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_{\nu }^{\left(1\right)}=& {\overline{\lambda }}_4{G}_{\nu }^{\left(0\right)}+{\overline{\lambda }}_6{F}_{\nu }^{\left(0\right)}+{\overline{\lambda }}_8{D}_{\nu }^{\left(0\right)},\hfill \end{array}$$

(56)

where the first term on the RHS in Equation (42) becomes zero because ${\mathsf{\Psi }}_{\nu }^{\left(0\right)}=0$. By inspecting Equations (45)–(47), it becomes clear that the only nonzero modes are $\nu =-6$, $-4$, $-2$, 2, 4, 6, and 8 when utilizing only the first three self-interaction terms. For example, ${\mathsf{\Psi }}_2^{\left(1\right)}$ equals

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_2^{\left(1\right)}=& {\overline{\lambda }}_4\frac{3{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}}{4!m_a^3}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)+{\overline{\lambda }}_6\frac{15{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}}{2·6!m_a^4}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^4\right)+{\overline{\lambda }}_8\frac{35{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}}{2·8!m_a^5}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^6\right).\hfill \end{array}$$

(57)

Replacing all the required modes in Equations (53)–(55) results in

$$\begin{array}{cc}\hfill G_0^{\left(1\right)}=& {\overline{\lambda }}_4\frac{3{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(3\right)m_a^6}\mathrm{G}{\left(4\right)}_{0,\left(1\right)}^{\left(1\right)}+{\overline{\lambda }}_6\frac{6{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^7}\mathrm{G}{\left(6\right)}_{0,\left(1\right)}^{\left(1\right)}+{\overline{\lambda }}_8\frac{5{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(5\right)m_a^8}\mathrm{G}{\left(8\right)}_{0,\left(1\right)}^{\left(1\right)},\hfill \\ \hfill F_0^{\left(1\right)}=& {\overline{\lambda }}_4\frac{2{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^7}\mathrm{F}{\left(4\right)}_{0,\left(1\right)}^{\left(1\right)}+{\overline{\lambda }}_6\frac{5{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(5\right)m_a^8}\mathrm{F}{\left(6\right)}_{0,\left(1\right)}^{\left(1\right)}+{\overline{\lambda }}_8\frac{15{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(6\right)m_a^9}\mathrm{F}{\left(8\right)}_{0,\left(1\right)}^{\left(1\right)},\hfill \\ \hfill D_0^{\left(1\right)}=& {\overline{\lambda }}_4\frac{5{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(5\right)m_a^8}\mathrm{D}{\left(4\right)}_{0,\left(1\right)}^{\left(1\right)}+{\overline{\lambda }}_6\frac{3{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(6\right)m_a^9}\mathrm{D}{\left(6\right)}_{0,\left(1\right)}^{\left(1\right)}+{\overline{\lambda }}_8\frac{7{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(7\right)m_a^{10}}\mathrm{D}{\left(8\right)}_{0,\left(1\right)}^{\left(1\right)},\hfill \end{array}$$

(58)

where a new notation is introduced. For example, $\mathrm{G}{\left(4\right)}_{0,\left(1\right)}^{\left(1\right)}$ corresponds to the term related to ${\overline{\lambda }}_4$ inside $G_0^{\left(1\right)}$. The extra subindex $\left(1\right)$ indicates that this term arises from the part of $G_0^{\left(1\right)}$ that only includes ${\mathsf{\Psi }}_{\nu }^{\left(1\right)}$. This seems redundant as $G_0^{\left(1\right)}$ is completely expressed in terms of ${\mathsf{\Psi }}_{\nu }^{\left(1\right)}$; however, this detail becomes important in the next iterations. The exact form of all the terms in Equation (58) is presented in Appendix B.

4.3. Second-Order Iteration

In this iteration, $G_{\nu }^{\left(2\right)}$ and $F_{\nu }^{\left(2\right)}$ need to be computed. These terms are divided into two in order to reduce the length of their expressions:

$$\begin{array}{cc}\hfill G_{\nu }^{\left(2\right)}=& G_{\nu ,\left(1\right)}^{\left(2\right)}+G_{\nu ,\left(2\right)}^{\left(2\right)},F_{\nu }^{\left(2\right)}=F_{\nu ,\left(1\right)}^{\left(2\right)}+F_{\nu ,\left(2\right)}^{\left(2\right)},\hfill \end{array}$$

(59)

where $G_{\nu ,\left(1\right)}^{\left(2\right)}$ and $F_{\nu ,\left(1\right)}^{\left(2\right)}$ contain only terms proportional to ${\mathsf{\Psi }}_{\mu }^{\left(1\right)}{\mathsf{\Psi }}_{\sigma }^{\left(1\right)}$. On the other hand, $G_{\nu ,\left(2\right)}^{\left(2\right)}$ and $F_{\nu ,\left(2\right)}^{\left(2\right)}$ contain only terms proportional to ${\mathsf{\Psi }}_{\mu }^{\left(2\right)}$. The four terms shown in Equation (59) can be derived from the expressions in Equation (37):

$$\begin{array}{cc}\hfill G_{\nu ,\left(1\right)}^{\left(2\right)}=& \frac{3{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{4!m_a^3}\sum _{{\nu }_1}[{\mathsf{\Psi }}_s\left({\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{2+\nu -{\nu }_1}^{\left(1\right)}+2{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{-\nu +{\nu }_1}^{\left(1\right)*}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{2-\nu -{\nu }_1}^{\left(1\right)*}\right)\hfill \\ \hfill & +{\mathsf{\Psi }}_s^{*}\left({\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{\nu -{\nu }_1}^{\left(1\right)}+2{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{-2+\nu +{\nu }_1}^{\left(1\right)}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{4-\nu -{\nu }_1}^{\left(1\right)*}\right)],\hfill \end{array}$$

(60)

$$\begin{array}{cc}\hfill F_{\nu ,\left(1\right)}^{\left(2\right)}=& \frac{15{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{2·6!m_a^4}\sum _{{\nu }_1}[{\mathsf{\Psi }}_s^3\left({\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{4+\nu -{\nu }_1}^{\left(1\right)}+2{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{-2-\nu +{\nu }_1}^{\left(1\right)*}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{-\nu -{\nu }_1}^{\left(1\right)*}\right)\hfill \\ \hfill & +3{\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{2+\nu -{\nu }_1}^{\left(1\right)}+2{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{-\nu +{\nu }_1}^{\left(1\right)*}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{2-\nu -{\nu }_1}^{\left(1\right)*}\right)\hfill \\ \hfill & +3{\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{\nu -{\nu }_1}^{\left(1\right)}+2{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{-2+\nu +{\nu }_1}^{\left(1\right)}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{4-\nu -{\nu }_1}^{\left(1\right)*}\right)\hfill \\ \hfill & +{{\mathsf{\Psi }}_s^{*}}^3\left({\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{-2+\nu -{\nu }_1}^{\left(1\right)}+2{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{-4+\nu +{\nu }_1}^{\left(1\right)}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{6-\nu -{\nu }_1}^{\left(1\right)*}\right)].\hfill \end{array}$$

(61)

Its important to note that the sums in Equations (60) and (61) are finite since ${\mathsf{\Psi }}_{\nu }^{\left(1\right)}$ only exists for $\nu =-4$, $-2$, 2, 4, and 6, when only the first two self-interaction terms are considered. For $\nu =0$, these expressions take the following form:

$$\begin{array}{cc}\hfill G_{0,\left(1\right)}^{\left(2\right)}=& \frac{3{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{4!m_a^3}\sum _{{\nu }_1}[{\mathsf{\Psi }}_s\left({\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{2-{\nu }_1}^{\left(1\right)}+2{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{2-{\nu }_1}^{\left(1\right)*}\right)\hfill \\ \hfill & +{\mathsf{\Psi }}_s^{*}\left({\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{-{\nu }_1}^{\left(1\right)}+2{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{-2+{\nu }_1}^{\left(1\right)}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{4-{\nu }_1}^{\left(1\right)*}\right)],\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill F_{0,\left(1\right)}^{\left(2\right)}=& \frac{15{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{2·6!m_a^4}\sum _{{\nu }_1}[{\mathsf{\Psi }}_s^3\left({\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{4-{\nu }_1}^{\left(1\right)}+2{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{-2+{\nu }_1}^{\left(1\right)*}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{-{\nu }_1}^{\left(1\right)*}\right)\hfill \\ \hfill & +3{\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{2-{\nu }_1}^{\left(1\right)}+2{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{2-{\nu }_1}^{\left(1\right)*}\right)\hfill \\ \hfill & +3{\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{-{\nu }_1}^{\left(1\right)}+2{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{-2+{\nu }_1}^{\left(1\right)}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{4-{\nu }_1}^{\left(1\right)*}\right)\hfill \\ \hfill & +{{\mathsf{\Psi }}_s^{*}}^3\left({\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{-2-{\nu }_1}^{\left(1\right)}+2{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{-4+{\nu }_1}^{\left(1\right)}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)*}{\mathsf{\Psi }}_{6-{\nu }_1}^{\left(1\right)*}\right)].\hfill \end{array}$$

(63)

Meanwhile,

$$\begin{array}{cc}\hfill G_{\nu ,\left(2\right)}^{\left(2\right)}=\frac{3{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{4!m_a^3}[& {\mathsf{\Psi }}_s^2\left({\mathsf{\Psi }}_{2+\nu }^{\left(2\right)}+{\mathsf{\Psi }}_{-\nu }^{\left(2\right)*}\right)+2{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{\nu }^{\left(2\right)}+{\mathsf{\Psi }}_{2-\nu }^{\left(2\right)*}\right)\hfill \\ & +{{\mathsf{\Psi }}_s^{*}}^2\left({\mathsf{\Psi }}_{-2+\nu }^{\left(2\right)}+{\mathsf{\Psi }}_{4-\nu }^{\left(2\right)*}\right)],\hfill \end{array}$$

(64)

$$\begin{array}{cc}\hfill F_{\nu ,\left(2\right)}^{\left(2\right)}=\frac{15}{4}\frac{{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{6!m_a^4}[& {\mathsf{\Psi }}_s^4\left({\mathsf{\Psi }}_{4+\nu }^{\left(2\right)}+{\mathsf{\Psi }}_{-2-\nu }^{\left(2\right)*}\right)+4{\mathsf{\Psi }}_s^2{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{2+\nu }^{\left(2\right)}+{\mathsf{\Psi }}_{-\nu }^{\left(2\right)*}\right)\hfill \\ \hfill & +6|{\mathsf{\Psi }}_s{|}^4\left({\mathsf{\Psi }}_{\nu }^{\left(2\right)}+{\mathsf{\Psi }}_{2-\nu }^{\left(2\right)*}\right)+4{{\mathsf{\Psi }}_s^{*}}^2{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{-2+\nu }^{\left(2\right)}+{\mathsf{\Psi }}_{4-\nu }^{\left(2\right)*}\right)\hfill \\ \hfill & +{{\mathsf{\Psi }}_s^{*}}^4\left({\mathsf{\Psi }}_{-4+\nu }^{\left(2\right)}+{\mathsf{\Psi }}_{6-\nu }^{\left(2\right)*}\right)].\hfill \end{array}$$

(65)

where, for $\nu =0$, they become

$$\begin{array}{cc}\hfill G_{0,\left(2\right)}^{\left(2\right)}& =\frac{3{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{4!m_a^3}\left[{\mathsf{\Psi }}_s^2{\mathsf{\Psi }}_2^{\left(2\right)}+2{\left|{\mathsf{\Psi }}_s\right|}^2{\mathsf{\Psi }}_2^{\left(2\right)*}+{{\mathsf{\Psi }}_s^{*}}^2\left({\mathsf{\Psi }}_4^{\left(2\right)*}+{\mathsf{\Psi }}_{-2}^{\left(2\right)}\right)\right],\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill F_{0,\left(2\right)}^{\left(2\right)}=\frac{15}{4}\frac{{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{6!m_a^4}[& {\mathsf{\Psi }}_s^4\left({\mathsf{\Psi }}_4^{\left(2\right)}+{\mathsf{\Psi }}_{-2}^{\left(2\right)*}\right)+4{\mathsf{\Psi }}_s^2|{\mathsf{\Psi }}_s{|}^2{\mathsf{\Psi }}_2^{\left(2\right)}+6{\left|{\mathsf{\Psi }}_s\right|}^4{\mathsf{\Psi }}_2^{\left(2\right)*}\hfill \\ \hfill & +4{{\mathsf{\Psi }}_s^{*}}^2{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_4^{\left(2\right)*}+{\mathsf{\Psi }}_{-2}^{\left(2\right)}\right)+{{\mathsf{\Psi }}_s^{*}}^4\left({\mathsf{\Psi }}_6^{\left(2\right)*}+{\mathsf{\Psi }}_{-4}^{\left(2\right)}\right)].\hfill \end{array}$$

(67)

Next, all ${\mathsf{\Psi }}_{\nu }^{\left(1\right)}$ in Equations (62) and (63) are expressed in terms of ${\mathsf{\Psi }}_s$ by implementing Equation (56) but taking into account that ${\overline{\lambda }}_8={\overline{\lambda }}_{10}=0$ in this iteration. This results in

$$\begin{array}{cc}\hfill G_{0,\left(1\right)}^{\left(2\right)}=& {\overline{\lambda }}_4^2\frac{2{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^9}\mathrm{G}{(4,4)}_{0,\left(1\right)}^{\left(2\right)}+{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{5{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(5\right)m_a^{10}}\mathrm{G}{(4,6)}_{0,\left(1\right)}^{\left(2\right)}+{\overline{\lambda }}_6^2\frac{9{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(6\right)m_a^{11}}\mathrm{G}{(6,6)}_{0,\left(1\right)}^{\left(2\right)},\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill F_{0,\left(1\right)}^{\left(2\right)}=& {\overline{\lambda }}_4^2\frac{25{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(5\right)m_a^{10}}\mathrm{F}{(4,4)}_{0,\left(1\right)}^{\left(2\right)}+{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{30{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(6\right)m_a^{11}}\mathrm{F}{(4,6)}_{0,\left(1\right)}^{\left(2\right)}+{\overline{\lambda }}_6^2\frac{147{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(7\right)m_a^{12}}\mathrm{F}{(6,6)}_{0,\left(1\right)}^{\left(2\right)}.\hfill \end{array}$$

(69)

Similarly, it is necessary to express all ${\mathsf{\Psi }}_{\nu }^{\left(2\right)}$ in Equations (66) and (67) in terms of ${\mathsf{\Psi }}_s$. These expressions are computed by taking $n=2$ in Equation (42):

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_{\nu }^{\left(2\right)}=& -\frac{i}{m_a}{\mathsf{\Gamma }}_{\nu }{\dot{\mathsf{\Psi }}}_{\nu }^{\left(1\right)}+{\overline{\lambda }}_4{G}_{\nu }^{\left(1\right)}+{\overline{\lambda }}_6{F}_{\nu }^{\left(1\right)},\hfill \end{array}$$

(70)

where ${\dot{\mathsf{\Psi }}}_{\nu }^{\left(1\right)}$ is calculated by deriving Equation (56), e.g., for $\nu =2$, this results in

$$\begin{array}{cc}\hfill -\frac{i}{m_a}{\mathsf{\Gamma }}_2{\dot{\mathsf{\Psi }}}_2^{\left(1\right)}=& -i{\overline{\lambda }}_4\frac{6{\left({\mathsf{\Gamma }}_2\right)}^2{\mathcal{P}}^{-1}}{4!m_a^4}\left(|{\mathsf{\Psi }}_s{|}^2{\dot{\mathsf{\Psi }}}_s^{*}\right)-i{\overline{\lambda }}_4\frac{3{\left({\mathsf{\Gamma }}_2\right)}^2{\mathcal{P}}^{-1}}{4!m_a^4}\left({{\mathsf{\Psi }}_s^{*}}^2{\dot{\mathsf{\Psi }}}_s\right)\hfill \\ \hfill & -i{\overline{\lambda }}_6\frac{15{\left({\mathsf{\Gamma }}_2\right)}^2{\mathcal{P}}^{-1}}{4·6!m_a^5}\left(6|{\mathsf{\Psi }}_s{|}^4{\dot{\mathsf{\Psi }}}_s^{*}\right.+\left.4{{\mathsf{\Psi }}_s^{*}}^2{\left|{\mathsf{\Psi }}_s\right|}^2{\dot{\mathsf{\Psi }}}_s\right).\hfill \end{array}$$

(71)

As mentioned before, the only nonzero modes are $\nu =-6$, $-4$, $-2$, 2, 4, 6, and 8 when only the first two self-interaction terms are utilized. As an example of calculation, ${\mathsf{\Psi }}_2^{\left(2\right)}$ equals

$$\begin{array}{cc}\hfill & {\mathsf{\Psi }}_2^{\left(2\right)}=-i{\overline{\lambda }}_4\frac{3{\left({\mathsf{\Gamma }}_2\right)}^2{\mathcal{P}}^{-1}}{4!m_a^4}\left(2|{\mathsf{\Psi }}_s{|}^2{\dot{\mathsf{\Psi }}}_s^{*}+{{\mathsf{\Psi }}_s^{*}}^2{\dot{\mathsf{\Psi }}}_s\right)-i{\overline{\lambda }}_6\frac{15{\left({\mathsf{\Gamma }}_2\right)}^2{\mathcal{P}}^{-1}}{4·6!m_a^5}\left(6|{\mathsf{\Psi }}_s{|}^4{\dot{\mathsf{\Psi }}}_s^{*}+4{{\mathsf{\Psi }}_s^{*}}^2{\left|{\mathsf{\Psi }}_s\right|}^2{\dot{\mathsf{\Psi }}}_s\right)\hfill \\ \hfill & +{\overline{\lambda }}_4^2\frac{3{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}}{{(4!)}^2{m}_a^6}\left[{\mathsf{\Psi }}_s^2({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^3\right)+6{\left|{\mathsf{\Psi }}_s\right|}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)+3{{\mathsf{\Psi }}_s^{*}}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^2\right)\right]\hfill \\ \hfill & +{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}}{40·{(4!)}^2{m}_a^7}[15{\mathsf{\Psi }}_s^2({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^3{\left|{\mathsf{\Psi }}_s\right|}^2\right)+60{\left|{\mathsf{\Psi }}_s\right|}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^4\right)\hfill \\ \hfill & +30{{\mathsf{\Psi }}_s^{*}}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^4\right)+20{\mathsf{\Psi }}_s^2|{\mathsf{\Psi }}_s{|}^2({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^3\right)+90{\left|{\mathsf{\Psi }}_s\right|}^4{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)\hfill \\ \hfill & +60{{\mathsf{\Psi }}_s^{*}}^2{\left|{\mathsf{\Psi }}_s\right|}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^2\right)+5{{\mathsf{\Psi }}_s^{*}}^4({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^3\right)]\hfill \\ \hfill & +{\overline{\lambda }}_6^2\frac{45{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}}{16·{(6!)}^2{m}_a^8}[{\mathsf{\Psi }}_s^4({\mathsf{\Gamma }}_6+{\mathsf{\Gamma }}_{-4}){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^5\right)+20{\mathsf{\Psi }}_s^2{\left|{\mathsf{\Psi }}_s\right|}^2({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^3{\left|{\mathsf{\Psi }}_s\right|}^2\right)\hfill \\ \hfill & +60|{\mathsf{\Psi }}_s{|}^4{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^4\right)+40{{\mathsf{\Psi }}_s^{*}}^2{\left|{\mathsf{\Psi }}_s\right|}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^4\right)\hfill \\ \hfill & +5{{\mathsf{\Psi }}_s^{*}}^4({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^3{\left|{\mathsf{\Psi }}_s\right|}^2\right)].\hfill \end{array}$$

(72)

Replacing all the required modes in Equations (66) and (67) results in

$$\begin{array}{cc}\hfill G_{0,\left(2\right)}^{\left(2\right)}=& i{\overline{\lambda }}_4\frac{9{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(3\right)m_a^7}\mathrm{G}{(4,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}+i{\overline{\lambda }}_6\frac{6{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^8}\mathrm{G}{(6,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}+{\overline{\lambda }}_4^2\frac{6{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^9}\mathrm{G}{(4,4)}_{0,\left(2\right)}^{\left(2\right)}\hfill \\ \hfill & +{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{5{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(5\right)m_a^{10}}\mathrm{G}{(4,6)}_{0,\left(2\right)}^{\left(2\right)}+{\overline{\lambda }}_6^2\frac{45{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(6\right)m_a^{11}}\mathrm{G}{(6,6)}_{0,\left(2\right)}^{\left(2\right)},\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill F_{0,\left(2\right)}^{\left(2\right)}=& i{\overline{\lambda }}_4\frac{6{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^8}\mathrm{F}{(4,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}+i{\overline{\lambda }}_6\frac{25{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(5\right)m_a^9}\mathrm{F}{(6,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}+{\overline{\lambda }}_4^2\frac{25{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(5\right)m_a^{10}}\mathrm{F}{(4,4)}_{0,\left(2\right)}^{\left(2\right)}\hfill \\ \hfill & +{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{15{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(6\right)m_a^{11}}\mathrm{F}{(4,6)}_{0,\left(2\right)}^{\left(2\right)}+{\overline{\lambda }}_6^2\frac{735{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{4\alpha \left(7\right)m_a^{12}}\mathrm{F}{(6,6)}_{0,\left(2\right)}^{\left(2\right)},\hfill \end{array}$$

(74)

where a new feature is added to the notation. The roman numerals inside a term indicate that its complete expression is proportional to ${ϵ}_t$. Furthermore, the number depicted in roman numerals is equal to the order of ${ϵ}_t$, e.g., $\mathrm{G}{(4,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}$ is proportional to ${\dot{\mathsf{\Psi }}}_s$ and its complex conjugate.

Substituting Equations (68), (69), (73), and (74) into (59) results in

$$\begin{array}{cc}\hfill G_0^{\left(2\right)}=& i{\overline{\lambda }}_4\frac{9{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(3\right)m_a^7}\mathrm{G}{(4,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}+i{\overline{\lambda }}_6\frac{6{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^8}\mathrm{G}{(6,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}\hfill \\ \hfill & +{\overline{\lambda }}_4^2\frac{2{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^9}\left(\mathrm{G}{(4,4)}_{0,\left(1\right)}^{\left(2\right)}+3\mathrm{G}{(4,4)}_{0,\left(2\right)}^{\left(2\right)}\right)\hfill \\ \hfill & +{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{5{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(5\right)m_a^{10}}\left(2\mathrm{G}{(4,6)}_{0,\left(1\right)}^{\left(2\right)}+\mathrm{G}{(4,6)}_{0,\left(2\right)}^{\left(2\right)}\right)\hfill \\ \hfill & +{\overline{\lambda }}_6^2\frac{9{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(6\right)m_a^{11}}\left(2\mathrm{G}{(6,6)}_{0,\left(1\right)}^{\left(2\right)}+5\mathrm{G}{(6,6)}_{0,\left(2\right)}^{\left(2\right)}\right),\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill F_0^{\left(2\right)}=& i{\overline{\lambda }}_4\frac{6{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^8}\mathrm{F}{(4,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}+i{\overline{\lambda }}_6\frac{25{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(5\right)m_a^9}\mathrm{F}{(6,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}\hfill \\ \hfill & +{\overline{\lambda }}_4^2\frac{25{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(5\right)m_a^{10}}\left(2\mathrm{F}{(4,4)}_{0,\left(1\right)}^{\left(2\right)}+\mathrm{F}{(4,4)}_{0,\left(2\right)}^{\left(2\right)}\right)\hfill \\ \hfill & +{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{15{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(6\right)m_a^{11}}\left(4\mathrm{F}{(4,6)}_{0,\left(1\right)}^{\left(2\right)}+\mathrm{F}{(4,6)}_{0,\left(2\right)}^{\left(2\right)}\right)\hfill \\ \hfill & +{\overline{\lambda }}_6^2\frac{147{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{4\alpha \left(7\right)m_a^{12}}\left(2\mathrm{F}{(6,6)}_{0,\left(1\right)}^{\left(2\right)}+5\mathrm{F}{(6,6)}_{0,\left(2\right)}^{\left(2\right)}\right).\hfill \end{array}$$

(76)

The detailed expressions of all the terms in Equations (75) and (76) are presented in Appendix C.

4.4. Third-Order Iteration

Similar to the previous iteration, $G_{\nu }^{\left(3\right)}$ is divided intp three terms in order to reduce its length and improve its readability:

$$\begin{array}{cc}\hfill G_{\nu }^{\left(3\right)}=& G_{\nu ,\left(1\right)}^{\left(3\right)}+G_{\nu ,\left(1\right)\left(2\right)}^{\left(3\right)}+G_{\nu ,\left(3\right)}^{\left(3\right)},\hfill \end{array}$$

(77)

where $G_{\nu ,\left(1\right)}^{\left(3\right)}$ contains only terms proportional to ${\mathsf{\Psi }}_{\mu }^{\left(1\right)}{\mathsf{\Psi }}_{\sigma }^{\left(1\right)}{\mathsf{\Psi }}_{\rho }^{\left(1\right)}$. On the other hand, $G_{\nu ,\left(1\right)\left(2\right)}^{\left(3\right)}$ contains only terms proportional to ${\mathsf{\Psi }}_{\mu }^{\left(1\right)}{\mathsf{\Psi }}_{\sigma }^{\left(2\right)}$. Lastly, $G_{\nu ,\left(3\right)}^{\left(3\right)}$ contains only terms proportional to ${\mathsf{\Psi }}_{\mu }^{\left(3\right)}$. The three terms depicted in Equation (77) can be derived from G’s expression in Equation (37):

$$\begin{array}{cc}\hfill G_{\nu ,\left(1\right)}^{\left(3\right)}=\frac{{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{4!m_a^3}\sum _{{\nu }_1,{\nu }_2}[& {\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{{\nu }_2}^{\left(1\right)}\left({\mathsf{\Psi }}_{2+\nu -{\nu }_1-{\nu }_2}^{\left(1\right)}+3{\mathsf{\Psi }}_{-\nu +{\nu }_1+{\nu }_2}^{\left(1\right)\ast }\right)\hfill \\ \hfill & +{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)\ast }{\mathsf{\Psi }}_{{\nu }_2}^{\left(1\right)\ast }\left(3{\mathsf{\Psi }}_{-2+\nu +{\nu }_1+{\nu }_2}^{\left(1\right)}+{\mathsf{\Psi }}_{4-\nu -{\nu }_1-{\nu }_2}^{\left(1\right)\ast }\right)],\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill G_{\nu ,\left(1\right)\left(2\right)}^{\left(3\right)}=\frac{6{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{4!m_a^3}\sum _{{\nu }_1}[& {\mathsf{\Psi }}_s\left({\mathsf{\Psi }}_{2+\nu -{\nu }_1}^{\left(2\right)}{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}+{\mathsf{\Psi }}_{-\nu +{\nu }_1}^{\left(2\right)\ast }{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(2\right)}{\mathsf{\Psi }}_{-\nu +{\nu }_1}^{\left(1\right)\ast }+{\mathsf{\Psi }}_{2-\nu -{\nu }_1}^{\left(2\right)\ast }{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)\ast }\right)\hfill \\ \hfill & +{\mathsf{\Psi }}_s^{*}({\mathsf{\Psi }}_{\nu -{\nu }_1}^{\left(2\right)}{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}+{\mathsf{\Psi }}_{-2+\nu +{\nu }_1}^{\left(2\right)}{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)\ast }+{\mathsf{\Psi }}_{{\nu }_1}^{\left(2\right)\ast }{\mathsf{\Psi }}_{-2+\nu +{\nu }_1}^{\left(1\right)}\hfill \\ \hfill & +{\mathsf{\Psi }}_{4-\nu -{\nu }_1}^{\left(2\right)\ast }{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)\ast }\left)\right],\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill G_{\nu ,\left(3\right)}^{\left(3\right)}=\frac{3{\mathsf{\Gamma }}_{\nu }{\mathcal{P}}^{-1}}{4!m_a^3}[& {\mathsf{\Psi }}_s^2\left({\mathsf{\Psi }}_{2+\nu }^{\left(3\right)}+{\mathsf{\Psi }}_{-\nu }^{\left(3\right)\ast }\right)+2{\left|{\mathsf{\Psi }}_s\right|}^2\left({\mathsf{\Psi }}_{\nu }^{\left(3\right)}+{\mathsf{\Psi }}_{2-\nu }^{\left(3\right)\ast }\right)\hfill \\ \hfill & +{{\mathsf{\Psi }}_s^{*}}^2\left({\mathsf{\Psi }}_{-2+\nu }^{\left(3\right)}+{\mathsf{\Psi }}_{4-\nu }^{\left(3\right)\ast }\right)].\hfill \end{array}$$

(80)

Once again, it is noteworthy to mention that the sums in Equations (78) and (79) are finite since ${\mathsf{\Psi }}_{\nu }^{\left(1\right)}$a and ${\mathsf{\Psi }}_{\nu }^{\left(2\right)}$ only exists for $\nu =-2$, 2, and 4 when considering only the first self-interaction term. For $\nu =0$, these expressions take the following form

$$\begin{array}{cc}\hfill G_{0,\left(1\right)}^{\left(3\right)}=\frac{{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{4!m_a^3}\sum _{{\nu }_1,{\nu }_2}[& {\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}{\mathsf{\Psi }}_{{\nu }_2}^{\left(1\right)}\left({\mathsf{\Psi }}_{2-{\nu }_1-{\nu }_2}^{\left(1\right)}+3{\mathsf{\Psi }}_{{\nu }_1+{\nu }_2}^{\left(1\right)\ast }\right)\hfill \\ \hfill & +{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)\ast }{\mathsf{\Psi }}_{{\nu }_2}^{\left(1\right)\ast }\left(3{\mathsf{\Psi }}_{-2+{\nu }_1+{\nu }_2}^{\left(1\right)}+{\mathsf{\Psi }}_{4-{\nu }_1-{\nu }_2}^{\left(1\right)\ast }\right)],\hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill G_{0,\left(1\right)\left(2\right)}^{\left(3\right)}=\frac{6{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{4!m_a^3}\sum _{{\nu }_1}[& {\mathsf{\Psi }}_s\left({\mathsf{\Psi }}_{2-{\nu }_1}^{\left(2\right)}{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(2\right)\ast }{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}+{\mathsf{\Psi }}_{{\nu }_1}^{\left(2\right)}{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)\ast }+{\mathsf{\Psi }}_{2-{\nu }_1}^{\left(2\right)\ast }{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)\ast }\right)\hfill \\ \hfill & +{\mathsf{\Psi }}_s^{*}({\mathsf{\Psi }}_{-{\nu }_1}^{\left(2\right)}{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)}+{\mathsf{\Psi }}_{-2+{\nu }_1}^{\left(2\right)}{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)\ast }+{\mathsf{\Psi }}_{{\nu }_1}^{\left(2\right)\ast }{\mathsf{\Psi }}_{-2+{\nu }_1}^{\left(1\right)}\hfill \\ \hfill & +{\mathsf{\Psi }}_{4-{\nu }_1}^{\left(2\right)\ast }{\mathsf{\Psi }}_{{\nu }_1}^{\left(1\right)\ast }\left)\right],\hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill G_{0,\left(3\right)}^{\left(3\right)}=& \frac{3{\mathsf{\Gamma }}_0{\mathcal{P}}^{-1}}{4!m_a^3}\left[{\mathsf{\Psi }}_s^2{\mathsf{\Psi }}_2^{\left(3\right)}+2{\left|{\mathsf{\Psi }}_s\right|}^2{\mathsf{\Psi }}_2^{\left(3\right)\ast }+{{\mathsf{\Psi }}_s^{*}}^2\left({\mathsf{\Psi }}_4^{\left(3\right)\ast }+{\mathsf{\Psi }}_{-2}^{\left(3\right)}\right)\right].\hfill \end{array}$$

(83)

Next, all ${\mathsf{\Psi }}_{\nu }^{\left(1\right)}$ and ${\mathsf{\Psi }}_{\nu }^{\left(2\right)}$ in Equations (81) and (82) are expressed in terms of ${\mathsf{\Psi }}_s$ by implementing Equations (56) and (70) but taking into account that ${\overline{\lambda }}_6={\overline{\lambda }}_8={\overline{\lambda }}_{10}=0$ in this iteration. This results in

$$\begin{array}{cc}\hfill G_{0,\left(1\right)}^{\left(3\right)}=& {\overline{\lambda }}_4^3\frac{25{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(5\right)m_a^{12}}\mathrm{G}{(4,4,4)}_{0,\left(1\right)}^{\left(3\right)},\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill G_{0,\left(1\right)\left(2\right)}^{\left(3\right)}=& i{\overline{\lambda }}_4^2\frac{12{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^{10}}\mathrm{G}{(4,4,\mathbf{I})}_{0,\left(1\right)\left(2\right)}^{\left(3\right)}+{\overline{\lambda }}_4^3\frac{25{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(5\right)m_a^{12}}\mathrm{G}{(4,4,4)}_{0,\left(1\right)\left(2\right)}^{\left(3\right)}.\hfill \end{array}$$

(85)

Afterwards, it is necessary to express all ${\mathsf{\Psi }}_{\nu }^{\left(3\right)}$ in Equation (83) in terms of ${\mathsf{\Psi }}_s$. This is achieved by taking $n=3$ in Equation (42):

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_{\nu }^{\left(3\right)}=& -\frac{i}{m_a}{\mathsf{\Gamma }}_{\nu }{\dot{\mathsf{\Psi }}}_{\nu }^{\left(2\right)}+{\overline{\lambda }}_4{G}_{\nu }^{\left(2\right)},\hfill \end{array}$$

(86)

where ${\dot{\mathsf{\Psi }}}_{\nu }^{\left(2\right)}$ is calculated by deriving Equation (70) with respect to time; e.g., for $\nu =2$, this results in

$$\begin{array}{cc}\hfill & -\frac{i}{m_a}{\mathsf{\Gamma }}_2{\dot{\mathsf{\Psi }}}_2^{\left(2\right)}=-{\overline{\lambda }}_4\frac{3{\left({\mathsf{\Gamma }}_2\right)}^3{\mathcal{P}}^{-1}}{4!m_a^5}\left(2|{\mathsf{\Psi }}_s{|}^2{\ddot{\mathsf{\Psi }}}_s^{*}+{{\mathsf{\Psi }}_s^{*}}^2{\ddot{\mathsf{\Psi }}}_s\right)-{\overline{\lambda }}_4\frac{6{\left({\mathsf{\Gamma }}_2\right)}^3{\mathcal{P}}^{-1}}{4!m_a^5}\left({\mathsf{\Psi }}_s{\left({\dot{\mathsf{\Psi }}}_s^{*}\right)}^2+2{\mathsf{\Psi }}_s^{*}{\left|{\dot{\mathsf{\Psi }}}_s\right|}^2\right)\hfill \\ \hfill & -i{\overline{\lambda }}_4^2\frac{3{\left({\mathsf{\Gamma }}_2\right)}^2{\mathcal{P}}^{-1}}{{(4!)}^2{m}_a^7}[2{\mathsf{\Psi }}_s{\dot{\mathsf{\Psi }}}_s({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^3\right)+3{\mathsf{\Psi }}_s^2({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^2{\dot{\mathsf{\Psi }}}_s^{*}\right)\hfill \\ \hfill & +6{\mathsf{\Psi }}_s{\dot{\mathsf{\Psi }}}_s^{*}{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)+6{\mathsf{\Psi }}_s^{*}{\dot{\mathsf{\Psi }}}_s{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)+12{\left|{\mathsf{\Psi }}_s\right|}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\dot{\mathsf{\Psi }}}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)\hfill \\ \hfill & +6|{\mathsf{\Psi }}_s{|}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^2{\dot{\mathsf{\Psi }}}_s\right)+6{\mathsf{\Psi }}_s^{*}{\dot{\mathsf{\Psi }}}_s^{*}{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^2\right)+6{{\mathsf{\Psi }}_s^{*}}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left(|{\mathsf{\Psi }}_s{|}^2{\dot{\mathsf{\Psi }}}_s\right)\hfill \\ \hfill & +3{{\mathsf{\Psi }}_s^{*}}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^2{\dot{\mathsf{\Psi }}}_s^{*}\right)].\hfill \end{array}$$

(87)

As mentioned before, the only nonzero modes are $\nu =-2$, 2, and 4 when only the first two self-interaction terms are utilized. As an example of calculation, ${\mathsf{\Psi }}_3^{\left(2\right)}$ equals

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_2^{\left(3\right)}=& -{\overline{\lambda }}_4\frac{3{\left({\mathsf{\Gamma }}_2\right)}^3{\mathcal{P}}^{-1}}{4!m_a^5}\left(2|{\mathsf{\Psi }}_s{|}^2{\ddot{\mathsf{\Psi }}}_s^{*}+{{\mathsf{\Psi }}_s^{*}}^2{\ddot{\mathsf{\Psi }}}_s\right)-{\overline{\lambda }}_4\frac{6{\left({\mathsf{\Gamma }}_2\right)}^3{\mathcal{P}}^{-1}}{4!m_a^5}\left({\mathsf{\Psi }}_s{\left({\dot{\mathsf{\Psi }}}_s^{*}\right)}^2+2{\mathsf{\Psi }}_s^{*}{\left|{\dot{\mathsf{\Psi }}}_s\right|}^2\right)\hfill \\ \hfill & -i{\overline{\lambda }}_4^2\frac{3}{{(4!)}^2{m}_a^7}\left[{\left({\mathsf{\Gamma }}_2\right)}^2{\mathcal{P}}^{-1}\right(2{\mathsf{\Psi }}_s{\dot{\mathsf{\Psi }}}_s({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^3\right)\hfill \\ \hfill & +3{\mathsf{\Psi }}_s^2({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^2{\dot{\mathsf{\Psi }}}_s^{*}\right)+6{\mathsf{\Psi }}_s{\dot{\mathsf{\Psi }}}_s^{*}{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)+6{\mathsf{\Psi }}_s^{*}{\dot{\mathsf{\Psi }}}_s{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)\hfill \\ \hfill & +12|{\mathsf{\Psi }}_s{|}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\dot{\mathsf{\Psi }}}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)+6{\left|{\mathsf{\Psi }}_s\right|}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^2{\dot{\mathsf{\Psi }}}_s\right)+6{\mathsf{\Psi }}_s^{*}{\dot{\mathsf{\Psi }}}_s^{*}{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^2\right)\hfill \\ \hfill & +6{{\mathsf{\Psi }}_s^{*}}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left(|{\mathsf{\Psi }}_s{|}^2{\dot{\mathsf{\Psi }}}_s\right)+3{{\mathsf{\Psi }}_s^{*}}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^2{\dot{\mathsf{\Psi }}}_s^{*}\right))\hfill \\ \hfill & +3{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}({\mathsf{\Psi }}_s^2({\left({\mathsf{\Gamma }}_4\right)}^2-{\left({\mathsf{\Gamma }}_{-2}\right)}^2){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^2{\dot{\mathsf{\Psi }}}_s^{*}\right)\hfill \\ \hfill & +2|{\mathsf{\Psi }}_s{|}^2{\left({\mathsf{\Gamma }}_2\right)}^2{\mathcal{P}}^{-1}\left(2|{\mathsf{\Psi }}_s{|}^2{\dot{\mathsf{\Psi }}}_s^{*}+{{\mathsf{\Psi }}_s^{*}}^2{\dot{\mathsf{\Psi }}}_s\right)-{{\mathsf{\Psi }}_s^{*}}^2{\left({\mathsf{\Gamma }}_2\right)}^2{\mathcal{P}}^{-1}\left(2|{\mathsf{\Psi }}_s{|}^2{\dot{\mathsf{\Psi }}}_s+{\mathsf{\Psi }}_s^2{\dot{\mathsf{\Psi }}}_s^{*}\right)\left)\right]\hfill \\ \hfill & +{\overline{\lambda }}_4^3\frac{3{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}}{{(4!)}^3{m}_a^9}\left[3{\mathsf{\Psi }}_s\right(3{\left({\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)\right)}^2\hfill \\ \hfill & +2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^2\right)({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^3\right))\hfill \\ \hfill & +2{\mathsf{\Psi }}_s^{*}\left(9{\left|{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^2\right)\right|}^2+{\left|({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^3\right)\right|}^2\right)\hfill \\ \hfill & +3{\mathsf{\Psi }}_s^2({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left(2|{\mathsf{\Psi }}_s{|}^2({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^3\right)+3{{\mathsf{\Psi }}_s^{*}}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)\right)\hfill \\ \hfill & +6|{\mathsf{\Psi }}_s{|}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}({\mathsf{\Psi }}_s^2({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({{\mathsf{\Psi }}_s^{*}}^3\right)+6{\left|{\mathsf{\Psi }}_s\right|}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)\hfill \\ \hfill & +3{{\mathsf{\Psi }}_s^{*}}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^2\right))+3{{\mathsf{\Psi }}_s^{*}}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}({{\mathsf{\Psi }}_s^{*}}^2({\mathsf{\Gamma }}_4+{\mathsf{\Gamma }}_{-2}){\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^3\right)\hfill \\ \hfill & +6|{\mathsf{\Psi }}_s{|}^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^2\right)+3{\mathsf{\Psi }}_s^2{\mathsf{\Gamma }}_2{\mathcal{P}}^{-1}\left({\mathsf{\Psi }}_s^{*}{\left|{\mathsf{\Psi }}_s\right|}^2\right)\left)\right].\hfill \end{array}$$

(88)

Replacing all the required modes in Equation (83) results in

$$\begin{array}{cc}\hfill G_{0,\left(3\right)}^{\left(3\right)}=& -{\overline{\lambda }}_4\frac{9{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(3\right)m_a^8}\mathrm{G}{(4,\mathbf{I}\mathbf{I})}_{0,\left(3\right)}^{\left(3\right)}-{\overline{\lambda }}_4\frac{18{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(3\right)m_a^8}\mathrm{G}{(4,\mathbf{I},\mathbf{I})}_{0,\left(3\right)}^{\left(3\right)}\hfill \\ \hfill & +i{\overline{\lambda }}_4^2\frac{6{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^{10}}\mathrm{G}{(4,4,\mathbf{I})}_{0,\left(3\right)}^{\left(3\right)}+{\overline{\lambda }}_4^3\frac{25{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(5\right)m_a^{12}}\mathrm{G}{(4,4,4)}_{0,\left(3\right)}^{\left(3\right)},\hfill \end{array}$$

(89)

and substituting Equations (84), (85), and (89) into (77) terminates in

$$\begin{array}{cc}\hfill G_0^{\left(3\right)}=& -{\overline{\lambda }}_4\frac{9{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(3\right)m_a^8}\left(\mathrm{G}{(4,\mathbf{I}\mathbf{I})}_{0,\left(3\right)}^{\left(3\right)}+2\mathrm{G}{(4,\mathbf{I},\mathbf{I})}_{0,\left(3\right)}^{\left(3\right)}\right)\hfill \\ \hfill & +i{\overline{\lambda }}_4^2\frac{6{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^{10}}\left(2\mathrm{G}{(4,4,\mathbf{I})}_{0,\left(1\right)\left(2\right)}^{\left(3\right)}+\mathrm{G}{(4,4,\mathbf{I})}_{0,\left(3\right)}^{\left(3\right)}\right)\hfill \\ \hfill & +{\overline{\lambda }}_4^3\frac{25{\mathsf{\Gamma }}_0{\mathcal{P}}^{-\frac{1}{2}}}{2\alpha \left(5\right)m_a^{12}}\left(\mathrm{G}{(4,4,4)}_{0,\left(1\right)}^{\left(3\right)}+2\mathrm{G}{(4,4,4)}_{0,\left(1\right)\left(2\right)}^{\left(3\right)}+\mathrm{G}{(4,4,4)}_{0,\left(3\right)}^{\left(3\right)}\right).\hfill \end{array}$$

(90)

The complete expressions for all the terms in Equation (90) are presented in Appendix D.

4.5. Expansion of the $\mathcal{P}$ Operator

Substituting Equations (49), (58), (75), (76), and (90) into Equation (43) results in

$$\begin{array}{cc}\hfill i{\dot{\psi }}_s=& m_a(\mathcal{P}-1){\psi }_s+{\overline{\lambda }}_4\frac{2{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(2\right)m_a^2}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^2\right)+{\overline{\lambda }}_6\frac{3{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(3\right)m_a^3}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^4\right)\hfill \\ \hfill & +{\overline{\lambda }}_4^2\frac{3}{2\alpha \left(3\right)m_a^5}\mathrm{G}{\left(4\right)}_{0,\left(1\right)}^{\left(1\right)}+i{\overline{\lambda }}_4^2\frac{9}{2\alpha \left(3\right)m_a^6}\mathrm{G}{(4,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}\hfill \\ \hfill & -{\overline{\lambda }}_4^2\frac{9}{2\alpha \left(3\right)m_a^7}\left(\mathrm{G}{(4,\mathbf{I}\mathbf{I})}_{0,\left(3\right)}^{\left(3\right)}+2\mathrm{G}{(4,\mathbf{I},\mathbf{I})}_{0,\left(3\right)}^{\left(3\right)}\right)+{\overline{\lambda }}_8\frac{4{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^4}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^6\right)\hfill \\ \hfill & +{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{2}{\alpha \left(4\right)m_a^6}\left(3\mathrm{G}{\left(6\right)}_{0,\left(1\right)}^{\left(1\right)}+\mathrm{F}{\left(4\right)}_{0,\left(1\right)}^{\left(1\right)}\right)+i{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{6}{\alpha \left(4\right)m_a^7}\left(\mathrm{G}{(6,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}+\mathrm{F}{(4,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}\right)\hfill \\ \hfill & +{\overline{\lambda }}_4^3\frac{2}{\alpha \left(4\right)m_a^8}\left(\mathrm{G}{(4,4)}_{0,\left(1\right)}^{\left(2\right)}+3\mathrm{G}{(4,4)}_{0,\left(2\right)}^{\left(2\right)}\right)+i{\overline{\lambda }}_4^3\frac{6}{\alpha \left(4\right)m_a^9}\left(2\mathrm{G}{(4,4,\mathbf{I})}_{0,\left(1\right)\left(2\right)}^{\left(3\right)}+\mathrm{G}{(4,4,\mathbf{I})}_{0,\left(3\right)}^{\left(3\right)}\right)\hfill \\ \hfill & +{\overline{\lambda }}_{10}\frac{5{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(5\right)m_a^5}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^8\right)+{\overline{\lambda }}_4{\overline{\lambda }}_8\frac{5}{\alpha \left(5\right)m_a^7}\left(\mathrm{G}{\left(8\right)}_{0,\left(1\right)}^{\left(1\right)}+\mathrm{D}{\left(4\right)}_{0,\left(1\right)}^{\left(1\right)}\right)\hfill \\ \hfill & +{\overline{\lambda }}_6^2\frac{5}{2\alpha \left(5\right)m_a^7}\mathrm{F}{\left(6\right)}_{0,\left(1\right)}^{\left(1\right)}+i{\overline{\lambda }}_6^2\frac{25}{2\alpha \left(5\right)m_a^8}\mathrm{F}{(6,\mathbf{I})}_{0,\left(2\right)}^{\left(2\right)}\hfill \\ \hfill & +{\overline{\lambda }}_4^2{\overline{\lambda }}_6\frac{5}{2\alpha \left(5\right)m_a^9}\left(2\mathrm{G}{(4,6)}_{0,\left(1\right)}^{\left(2\right)}+\mathrm{G}{(4,6)}_{0,\left(2\right)}^{\left(2\right)}+10\mathrm{F}{(4,4)}_{0,\left(1\right)}^{\left(2\right)}+5\mathrm{F}{(4,4)}_{0,\left(2\right)}^{\left(2\right)}\right)\hfill \\ \hfill & +{\overline{\lambda }}_4^4\frac{25}{2\alpha \left(5\right)m_a^{11}}\left(\mathrm{G}{(4,4,4)}_{0,\left(1\right)}^{\left(3\right)}+2\mathrm{G}{(4,4,4)}_{0,\left(1\right)\left(2\right)}^{\left(3\right)}+\mathrm{G}{(4,4,4)}_{0,\left(3\right)}^{\left(3\right)}\right)\hfill \\ \hfill & +{\overline{\lambda }}_6{\overline{\lambda }}_8\frac{3}{\alpha \left(6\right)m_a^8}\left(5\mathrm{F}{\left(8\right)}_{0,\left(1\right)}^{\left(1\right)}+\mathrm{D}{\left(6\right)}_{0,\left(1\right)}^{\left(1\right)}\right)\hfill \\ \hfill & +{\overline{\lambda }}_4{\overline{\lambda }}_6^2\frac{3}{2\alpha \left(6\right)m_a^{10}}\left(6\mathrm{G}{(6,6)}_{0,\left(1\right)}^{\left(2\right)}+15\mathrm{G}{(6,6)}_{0,\left(2\right)}^{\left(2\right)}+20\mathrm{F}{(4,6)}_{0,\left(1\right)}^{\left(2\right)}+5\mathrm{F}{(4,6)}_{0,\left(2\right)}^{\left(2\right)}\right)\hfill \\ \hfill & +{\overline{\lambda }}_8^2\frac{7}{2\alpha \left(7\right)m_a^9}\mathrm{D}{\left(8\right)}_{0,\left(1\right)}^{\left(1\right)}\hfill \\ \hfill & +{\overline{\lambda }}_6^3\frac{147}{4\alpha \left(7\right)m_a^{11}}\left(2\mathrm{F}{(6,6)}_{0,\left(1\right)}^{\left(2\right)}+5\mathrm{F}{(6,6)}_{0,\left(2\right)}^{\left(2\right)}\right).\hfill \end{array}$$

(91)

All instances of $\mathcal{P}$ in Equation (91) need to be expressed in terms of ${ϵ}_x$. However, the order of this expansion is not clear since, with each new iteration, an additional self-interaction term is neglected. Therefore, according to ${\overline{\lambda }}_6$, the iterative process ends at the 2nd iteration, but according to ${\overline{\lambda }}_8$, the iterative process concludes at the 1st iteration. Consequently, each term inside Equation (91) is treated differently based on the constant that multiplies it. First, all terms proportional to ${\overline{\lambda }}_{10}$ are expanded up to the 1st order of ${ϵ}_x$, ${ϵ}_t$ and ${\overline{\lambda }}_{2n}$. Next, from the remaining terms, all the terms proportional to ${\overline{\lambda }}_8$ are expanded up to the 2nd order of ${ϵ}_x$, ${ϵ}_t$ and ${\overline{\lambda }}_{2n}$. Afterwards, all the unused expressions proportional to ${\overline{\lambda }}_6$ are expanded up to the 3rd order of ${ϵ}_x$, ${ϵ}_t$ and ${\overline{\lambda }}_{2n}$. Finally, all leftover terms are expanded up to the 4th order of ${ϵ}_x$, ${ϵ}_t$ and ${\overline{\lambda }}_{2n}$. For example,

$$\begin{array}{cc}\hfill {\overline{\lambda }}_8\frac{4{\mathcal{P}}^{-\frac{1}{2}}}{\alpha \left(4\right)m_a^4}\left({\mathsf{\Psi }}_s{\left|{\mathsf{\Psi }}_s\right|}^6\right)\simeq & {\overline{\lambda }}_8\frac{4}{\alpha \left(4\right)m_a^4}\left({\psi }_s{\left|{\psi }_s\right|}^6\right)+{\overline{\lambda }}_8\frac{1}{\alpha \left(4\right)m_a^6}\hfill \\ \hfill & \times \left(3{\psi }_s^2|{\psi }_s{|}^4{▽}^2{\psi }_s^{*}+4{\left|{\psi }_s\right|}^6{▽}^2{\psi }_s+{▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^6\right)\right).\hfill \end{array}$$

(92)

Once $\mathcal{P}$ is expanded, the equation of motion takes the following form:

$$\begin{array}{cc}\hfill 0=& i{\dot{\psi }}_s+\frac{1}{2m_a}{▽}^2{\psi }_s+\frac{1}{8m_a^3}{▽}^4{\psi }_s+\frac{1}{16m_a^5}{▽}^6{\psi }_s+\frac{5}{128m_a^7}{▽}^8{\psi }_s\hfill \\ \hfill & -{\overline{\lambda }}_4\frac{2}{\alpha \left(2\right)m_a^2}\left({\psi }_s{\left|{\psi }_s\right|}^2\right)-{\overline{\lambda }}_4\frac{1}{2\alpha \left(2\right)m_a^4}\left({\psi }_s^2{▽}^2{\psi }_s^{*}+2{\left|{\psi }_s\right|}^2{▽}^2{\psi }_s+{▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^2\right)\right)\hfill \\ \hfill & -{\overline{\lambda }}_4\frac{1}{16\alpha \left(2\right)m_a^6}[2{\psi }_s^{*}{\left({▽}^2{\psi }_s\right)}^2+4{\psi }_s|{▽}^2{\psi }_s{|}^2+5{\psi }_s^2{▽}^4{\psi }_s^{*}+10{\left|{\psi }_s\right|}^2{▽}^4{\psi }_s\hfill \\ \hfill & +5{▽}^4\left({\psi }_s{\left|{\psi }_s\right|}^2\right)+2{▽}^2\left({\psi }_s^2{▽}^2{\psi }_s^{*}+2{\left|{\psi }_s\right|}^2{▽}^2{\psi }_s\right)]\hfill \\ \hfill & -{\overline{\lambda }}_4\frac{1}{64\alpha \left(2\right)m_a^8}[2{▽}^2{\psi }_s{\left|{▽}^2{\psi }_s\right|}^2+10{\psi }_s^{*}{▽}^2{\psi }_s{▽}^4{\psi }_s+10{\psi }_s{▽}^2{\psi }_s^{*}{▽}^4{\psi }_s\hfill \\ \hfill & +10{\psi }_s{▽}^2{\psi }_s{▽}^4{\psi }_s^{*}+15{\psi }_s^2{▽}^6{\psi }_s^{*}+30{\left|{\psi }_s\right|}^2{▽}^6{\psi }_s+15{▽}^6\left({\psi }_s{\left|{\psi }_s\right|}^2\right)\hfill \\ \hfill & +5{▽}^4\left({\psi }_s^2{▽}^2{\psi }_s^{*}+2{\left|{\psi }_s\right|}^2{▽}^2{\psi }_s\right)+{▽}^2\left(2{\psi }_s^{*}{\left({▽}^2{\psi }_s\right)}^2+4{\psi }_s{\left|{▽}^2{\psi }_s\right|}^2+5{\psi }_s^2{▽}^4{\psi }_s^{*}\right.\hfill \\ \hfill & \left.+10|{\psi }_s{|}^2{▽}^4{\psi }_s\right)]-{\overline{\lambda }}_6\frac{3}{\alpha \left(3\right)m_a^3}\left({\psi }_s{\left|{\psi }_s\right|}^4\right)\hfill \\ \hfill & -{\overline{\lambda }}_6\frac{3}{4\alpha \left(3\right)m_a^5}\left(2{\psi }_s^2|{\psi }_s{|}^2{▽}^2{\psi }_s^{*}+3{\left|{\psi }_s\right|}^4{▽}^2{\psi }_s+{▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^4\right)\right)\hfill \\ \hfill & -{\overline{\lambda }}_6\frac{3}{16\alpha \left(3\right)m_a^7}{▽}^2\left(2{\psi }_s^2|{\psi }_s{|}^2{▽}^2{\psi }_s^{*}+3{\left|{\psi }_s\right|}^4{▽}^2{\psi }_s\right)\hfill \\ \hfill & -{\overline{\lambda }}_6\frac{3}{32\alpha \left(3\right)m_a^7}\left(2{\psi }_s^3{\left({▽}^2{\psi }_s^{*}\right)}^2+10{\psi }_s^2{\left|{\psi }_s\right|}^2{▽}^4{\psi }_s^{*}\right.\hfill \\ \hfill & +\left.12{\psi }_s|{\psi }_s{|}^2|{▽}^2{\psi }_s{|}^2+6{\psi }_s^{*}|{\psi }_s{|}^2{\left({▽}^2{\psi }_s\right)}^2+15{\left|{\psi }_s\right|}^4{▽}^4{\psi }_s+5{▽}^4\left({\psi }_s{\left|{\psi }_s\right|}^4\right)\right)\hfill \\ & +{\overline{\lambda }}_4^2\frac{51}{8\alpha \left(3\right)m_a^5}\left({\psi }_s{\left|{\psi }_s\right|}^4\right)\hfill \\ \hfill & +{\overline{\lambda }}_4^2\frac{51}{32\alpha \left(3\right)m_a^7}{▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^4\right)+{\overline{\lambda }}_4^2\frac{51}{32\alpha \left(3\right)m_a^7}\left(2{\psi }_s^2|{\psi }_s{|}^2{▽}^2{\psi }_s^{*}+3{\left|{\psi }_s\right|}^4{▽}^2{\psi }_s\right)\hfill \\ \hfill & +{\overline{\lambda }}_4^2\frac{3}{64\alpha \left(3\right)m_a^7}\left[36{\psi }_s^2{▽}^2\left({\psi }_s^{*}{\left|{\psi }_s\right|}^2\right)+72{\left|{\psi }_s\right|}^2{▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^2\right)+{{\psi }_s^{*}}^2{▽}^2\left({\psi }_s^3\right)\right]\hfill \\ \hfill & +{\overline{\lambda }}_4^2\frac{3}{512\alpha \left(3\right)m_a^9}[170{▽}^4\left({\psi }_s{\left|{\psi }_s\right|}^4\right)+{▽}^2(72{\psi }_s^2{▽}^2\left({\psi }_s^{*}{\left|{\psi }_s\right|}^2\right)+144{\left|{\psi }_s\right|}^2{▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^2\right)\hfill \\ \hfill & +2{{\psi }_s^{*}}^2{▽}^2\left({\psi }_s^3\right))+240{\psi }_s^2{▽}^4\left({\psi }_s^{*}{\left|{\psi }_s\right|}^2\right)+480{\left|{\psi }_s\right|}^2{▽}^4\left({\psi }_s{\left|{\psi }_s\right|}^2\right)-{{\psi }_s^{*}}^2{▽}^4\left({\psi }_s^3\right)\hfill \\ \hfill & +68{\psi }_s^3{\left({▽}^2{\psi }_s^{*}\right)}^2+340{\psi }_s^2|{\psi }_s{|}^2{▽}^4{\psi }_s^{*}+408{\psi }_s|{\psi }_s{|}^2|{▽}^2{\psi }_s{|}^2+204{\psi }_s^{*}{\left|{\psi }_s\right|}^2{\left({▽}^2{\psi }_s\right)}^2\hfill \\ \hfill & +510|{\psi }_s{|}^4{▽}^4{\psi }_s+{▽}^2\left(136{\psi }_s^2|{\psi }_s{|}^2{▽}^2{\psi }_s^{*}+204{\left|{\psi }_s\right|}^4{▽}^2{\psi }_s\right)\hfill \\ \hfill & +144{\psi }_s{▽}^2{\psi }_s{▽}^2({\psi }_s^{*}|{\psi }_s{|}^2)+{\psi }_s^2{▽}^2\left(72{{\psi }_s^{*}}^2{▽}^2{\psi }_s+144{\left|{\psi }_s\right|}^2{▽}^2{\psi }_s^{*}\right)\hfill \\ \hfill & +144{\psi }_s{▽}^2{\psi }_s^{*}{▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^2\right)+144{\psi }_s^{*}{▽}^2{\psi }_s{▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^2\right)+{\left|{\psi }_s\right|}^2{▽}^2\left(144{\psi }_s^2{▽}^2{\psi }_s^{*}\right.\hfill \\ \hfill & \left.+288|{\psi }_s{|}^2{▽}^2{\psi }_s\right)+4{\psi }_s^{*}{▽}^2{\psi }_s^{*}{▽}^2{\psi }_s^3+6{{\psi }_s^{*}}^2{▽}^2\left({\psi }_s^2{▽}^2{\psi }_s\right)]-{\overline{\lambda }}_8\frac{4}{\alpha \left(4\right)m_a^4}\left({\psi }_s{\left|{\psi }_s\right|}^6\right)\hfill \\ \hfill & -{\overline{\lambda }}_8\frac{1}{\alpha \left(4\right)m_a^6}\left(3{\psi }_s^2|{\psi }_s{|}^4{▽}^2{\psi }_s^{*}+4{\left|{\psi }_s\right|}^6{▽}^2{\psi }_s+{▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^6\right)\right)+{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{44}{\alpha \left(4\right)m_a^6}({\psi }_s|{\psi }_s{|}^6)\hfill \\ \hfill & +{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{1}{8\alpha \left(4\right)m_a^8}|{\psi }_s{|}^2(1035{\psi }_s^2|{\psi }_s{|}^2{▽}^2{\psi }_s^{*}+1292{\left|{\psi }_s\right|}^4{▽}^2{\psi }_s+1035{\psi }_s^3{\left(▽{\psi }_s^{*}\right)}^2\hfill \end{array}$$

(93)

$$\begin{array}{cc}\hfill & +4140{\psi }_s|{\psi }_s{|}^2{\left|▽{\psi }_s\right|}^2+1806{\psi }_s^{*}{\left|{\psi }_s\right|}^2{\left(▽{\psi }_s\right)}^2)-{\overline{\lambda }}_4^3\frac{49}{\alpha \left(4\right)m_a^8}\left({\psi }_s{\left|{\psi }_s\right|}^6\right)\hfill \\ \hfill & -{\overline{\lambda }}_4^3\frac{1}{16\alpha \left(4\right)m_a^{10}}(588{\psi }_s^2|{\psi }_s{|}^4{▽}^2{\psi }_s^{*}+784{\left|{\psi }_s\right|}^6{▽}^2{\psi }_s+196{▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^6\right)\hfill \\ \hfill & +153{\psi }_s^2{▽}^2\left({\psi }_s^{*}{\left|{\psi }_s\right|}^4\right)+306{\left|{\psi }_s\right|}^2{▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^4\right)+3{{\psi }_s^{*}}^2{▽}^2\left({\psi }_s^3{\left|{\psi }_s\right|}^2\right)+{\psi }_s^4{▽}^2\left({{\psi }_s^{*}}^3\right)\hfill \\ & +306{\psi }_s^2|{\psi }_s{|}^2{▽}^2\left({\psi }_s^{*}{\left|{\psi }_s\right|}^2\right)+459|{\psi }_s{|}^4{▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^2\right)+4{{\psi }_s^{*}}^2{\left|{\psi }_s\right|}^2{▽}^2\left({\psi }_s^3\right))\hfill \\ \hfill & -{\overline{\lambda }}_{10}\frac{5}{\alpha \left(5\right)m_a^5}\left({\psi }_s{\left|{\psi }_s\right|}^8\right)+{\overline{\lambda }}_4{\overline{\lambda }}_8\frac{225}{2\alpha \left(5\right)m_a^7}({\psi }_s|{\psi }_s{|}^8)+{\overline{\lambda }}_6^2\frac{655}{6\alpha \left(5\right)m_a^7}({\psi }_s|{\psi }_s{|}^8)\hfill \\ \hfill & +{\overline{\lambda }}_6^2\frac{5}{288\alpha \left(5\right)m_a^9}|{\psi }_s{|}^4(25716{\psi }_s^2|{\psi }_s{|}^2{▽}^2{\psi }_s^{*}+30145{\left|{\psi }_s\right|}^4{▽}^2{\psi }_s+38574{\psi }_s^3{\left(▽{\psi }_s^{*}\right)}^2\hfill \\ \hfill & +128580{\psi }_s|{\psi }_s{|}^2{\left|▽{\psi }_s\right|}^2+56290{\psi }_s^{*}|{\psi }_s{|}^2{\left(▽{\psi }_s\right)}^2)-{\overline{\lambda }}_4^2{\overline{\lambda }}_6\frac{21575}{24\alpha \left(5\right)m_a^9}({\psi }_s|{\psi }_s{|}^8)\hfill \\ \hfill & +{\overline{\lambda }}_4^4\frac{43125}{64\alpha \left(5\right)m_a^{11}}({\psi }_s|{\psi }_s{|}^8)+{\mathrm{ETT}}_{\mathrm{eq}.\mathrm{m}.}+\mathcal{O}\left(5\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\mathrm{ETT}}_{\mathrm{eq}.\mathrm{m}.}=& -i{\overline{\lambda }}_4^2\frac{81}{32\alpha \left(3\right)m_a^6}({\dot{\psi }}_s|{\psi }_s{|}^4)-i{\overline{\lambda }}_4^2\frac{9}{128\alpha \left(3\right)m_a^8}[18{\dot{\psi }}_s\left({\psi }_s{\left|{\psi }_s\right|}^2{▽}^2{\psi }_s^{*}\right.\hfill \\ \hfill & \left.+{\psi }_s^{*}{\left|{\psi }_s\right|}^2{▽}^2{\psi }_s\right)+9{\left|{\psi }_s\right|}^4{▽}^2{\dot{\psi }}_s-16{\psi }_s^2{▽}^2\left(2|{\psi }_s{|}^2{\dot{\psi }}_s^{*}+{{\psi }_s^{*}}^2{\dot{\psi }}_s\right)\hfill \\ \hfill & +32|{\psi }_s{|}^2{▽}^2\left(2|{\psi }_s{|}^2{\dot{\psi }}_s+{\psi }_s^2{\dot{\psi }}_s^{*}\right)+3{{\psi }_s^{*}}^2{▽}^2\left({\psi }_s^2{\dot{\psi }}_s\right)+9{▽}^2\left(|{\psi }_s{|}^4{\dot{\psi }}_s\right)]\hfill \\ \hfill & -{\overline{\lambda }}_4^2\frac{9}{128\alpha \left(3\right)m_a^7}(32{\psi }_s^2|{\psi }_s{|}^2{\ddot{\psi }}_s^{*}+33{\left|{\psi }_s\right|}^4{\ddot{\psi }}_s+16{\psi }_s^3{\left({\dot{\psi }}_s^{*}\right)}^2\hfill \\ \hfill & +96{\psi }_s|{\psi }_s{|}^2|{\dot{\psi }}_s{|}^2+18{\psi }_s^{*}|{\psi }_s{|}^2{\left({\dot{\psi }}_s\right)}^2)-i{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{15}{\alpha \left(4\right)m_a^7}({\dot{\psi }}_s|{\psi }_s{|}^6)\hfill \\ \hfill & +i{\overline{\lambda }}_4^3\frac{42}{\alpha \left(4\right)m_a^9}({\dot{\psi }}_s|{\psi }_s{|}^6)-i{\overline{\lambda }}_6^2\frac{3875}{144\alpha \left(5\right)m_a^8}({\dot{\psi }}_s|{\psi }_s{|}^8),\hfill \end{array}$$

(94)

are terms that contain temporal derivatives of ${\psi }_s$ that completely arise from the iterative process. Specifically, terms proportional to ${ϵ}_t{\psi }_s$ and ${ϵ}_t^2{\psi }_s$ appear from the 2nd and 3rd iteration, respectively. These extra temporal terms must be somehow treated in order to not introduce new degrees of freedom. How exactly this is achieved is shown in the next subsection. Meanwhile, these terms are treated as if no problem arises from their existence. Finally,

$$\begin{array}{cc}\hfill \mathcal{O}\left(5\right)=& {\overline{\lambda }}_6{\overline{\lambda }}_8\frac{762}{\alpha \left(6\right)m_a^8}({\psi }_s|{\psi }_s{|}^{10})-{\overline{\lambda }}_4{\overline{\lambda }}_6^2\frac{58293}{8\alpha \left(6\right)m_a^{10}}({\psi }_s|{\psi }_s{|}^{10})\hfill \\ \hfill & +{\overline{\lambda }}_8^2\frac{27895}{16\alpha \left(7\right)m_a^9}\left({\psi }_s{\left|{\psi }_s\right|}^{12}\right)-{\overline{\lambda }}_6^3\frac{810509}{32\alpha \left(7\right)m_a^{11}}({\psi }_s|{\psi }_s{|}^{12}),\hfill \end{array}$$

(95)

are terms of 6th or higher order, which contribute beyond the amount of iterations and ${\overline{\lambda }}_{2n}$ considered in this work. As a result, these expressions are not considered from now on.

By applying the Euler–Lagrange equation, it is possible to find the following hermitian Lagrangian that describes Equation (93):

$$\begin{array}{cc}\hfill \mathcal{L}=& \frac{i}{2}\left({\dot{\psi }}_s{\psi }_s^{*}-{\psi }_s{\dot{\psi }}_s^{*}\right)-\frac{1}{2m_a}▽{\psi }_s·▽{\psi }_s^{*}+\frac{1}{8m_a^3}{▽}^2{\psi }_s{▽}^2{\psi }_s^{*}\hfill \\ \hfill & -\frac{1}{16m_a^5}▽\left({▽}^2{\psi }_s\right)·▽\left({▽}^2{\psi }_s^{*}\right)+\frac{5}{128m_a^7}{▽}^4{\psi }_s{▽}^4{\psi }_s^{*}-{\overline{\lambda }}_4\frac{1}{\alpha \left(2\right)m_a^2}{\left|{\psi }_s\right|}^4\hfill \\ \hfill & -{\overline{\lambda }}_4\frac{1}{2\alpha \left(2\right)m_a^4}{\left|{\psi }_s\right|}^2\left({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*}\right)\hfill \\ \hfill & -{\overline{\lambda }}_4\frac{5}{16\alpha \left(2\right)m_a^6}{\left|{\psi }_s\right|}^2\left({\psi }_s^{*}{▽}^4{\psi }_s+{\psi }_s{▽}^4{\psi }_s^{*}\right)-{\overline{\lambda }}_4\frac{1}{16\alpha \left(2\right)m_a^6}({\psi }_s^2{\left({▽}^2{\psi }_s^{*}\right)}^2\hfill \\ \hfill & +4|{\psi }_s{|}^2|{▽}^2{\psi }_s{|}^2+{{\psi }_s^{*}}^2{\left({▽}^2{\psi }_s\right)}^2)-{\overline{\lambda }}_4\frac{1}{64\alpha \left(2\right)m_a^8}[15{\left|{\psi }_s\right|}^2\left({\psi }_s^{*}{▽}^6{\psi }_s+\right.\hfill \\ \hfill & \left.{\psi }_s{▽}^6{\psi }_s^{*}\right)+5({\psi }_s^2{▽}^2{\psi }_s^{*}{▽}^4{\psi }_s^{*}+{{\psi }_s^{*}}^2{▽}^2{\psi }_s{▽}^4{\psi }_s)+10|{\psi }_s{|}^2({▽}^2{\psi }_s{▽}^4{\psi }_s^{*}\hfill \\ \hfill & +{{\psi }_s^{*}}^2{▽}^2{\psi }_s{▽}^4{\psi }_s)+10|{\psi }_s{|}^2({▽}^2{\psi }_s{▽}^4{\psi }_s^{*}+{▽}^2{\psi }_s^{*}{▽}^4{\psi }_s)+2|{▽}^2{\psi }_s{|}^2({\psi }_s^{*}{▽}^2{\psi }_s\hfill \\ \hfill & +{\psi }_s{▽}^2{\psi }_s^{*}\left)\right]-{\overline{\lambda }}_6\frac{1}{\alpha \left(3\right)m_a^5}|{\psi }_s{|}^6-{\overline{\lambda }}_6\frac{3}{4\alpha \left(3\right)m_a^5}{\left|{\psi }_s\right|}^4\left({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*}\right)\hfill \\ \hfill & -{\overline{\lambda }}_6\frac{15}{32\alpha \left(3\right)m_a^7}|{\psi }_s{|}^4\left({\psi }_s^{*}{▽}^4{\psi }_s+{\psi }_s{▽}^4{\psi }_s^{*}\right)-{\overline{\lambda }}_6\frac{3}{16\alpha \left(3\right)m_a^7}{\left|{\psi }_s\right|}^2({\psi }_s^2{\left({▽}^2{\psi }_s^{*}\right)}^2\hfill \\ & +3|{\psi }_s{|}^2|{▽}^2{\psi }_s{|}^2+{{\psi }_s^{*}}^2{\left({▽}^2{\psi }_s\right)}^2)+{\overline{\lambda }}_4^2\frac{17}{8\alpha \left(3\right)m_a^5}{\left|{\psi }_s\right|}^6\hfill \\ \hfill & +{\overline{\lambda }}_4^2\frac{3}{64\alpha \left(3\right)m_a^7}|{\psi }_s{|}^2[\left(6{\mathrm{C}}_{4,4}^{\left[\mathbf{1}\right]}-387\right)|{\psi }_s{|}^2{|▽{\psi }_s|}^2+\left(2{\mathrm{C}}_{4,4}^{\left[\mathbf{1}\right]}-140\right)\left({{\psi }_s^{*}}^2{\left(▽{\psi }_s\right)}^2\right.\hfill \\ \hfill & \left.+{\psi }_s^2{\left(▽{\psi }_s^{*}\right)}^2\right)+{\mathrm{C}}_{4,4}^{\left[\mathbf{1}\right]}\left({\psi }_s^{*}|{\psi }_s{|}^2{▽}^2{\psi }_s+{\psi }_s{\left|{\psi }_s\right|}^2{▽}^2{\psi }_s^{*}\right)]\hfill \\ \hfill & +{\overline{\lambda }}_4^2\frac{1}{2048\alpha \left(3\right)m_a^9}[\left(6306+4{\mathrm{C}}_{4,4}^{\left[\mathbf{2}\right]}-2{\mathrm{C}}_{4,4}^{\left[\mathbf{3}\right]}\right){\left|{\psi }_s\right|}^4({\psi }_s^{*}{▽}^4{\psi }_s+{\psi }_s{▽}^4{\psi }_s^{*})\hfill \\ \hfill & +12{\mathrm{C}}_{4,4}^{\left[\mathbf{3}\right]}|{\psi }_s{|}^4{\left|{▽}^2{\psi }_s\right|}^2+\left(5436+{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}+{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}-4{\mathrm{C}}_{4,4}^{\left[\mathbf{7}\right]}+12{\mathrm{C}}_{4,4}^{\left[\mathbf{8}\right]}\right){\left|{\psi }_s\right|}^2\left({\psi }_s^2{\left({▽}^2{\psi }_s^{*}\right)}^2\right.\hfill \\ \hfill & \left.+{{\psi }_s^{*}}^2{\left({▽}^2{\psi }_s\right)}^2\right)+12{\mathrm{C}}_{4,4}^{\left[\mathbf{4}\right]}{\left|{\psi }_s\right|}^4{▽}^2\left(|▽{\psi }_s{|}^2\right)+\left(3936+8{\mathrm{C}}_{4,4}^{\left[\mathbf{2}\right]}-4{\mathrm{C}}_{4,4}^{\left[\mathbf{3}\right]}+{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}\right.\hfill \\ \hfill & \left.+{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}-4{\mathrm{C}}_{4,4}^{\left[\mathbf{7}\right]}+12{\mathrm{C}}_{4,4}^{\left[\mathbf{8}\right]}\right){\left|{\psi }_s\right|}^2({\psi }_s^2▽{\psi }_s^{*}·▽\left({▽}^2{\psi }_s^{*}\right)+{{\psi }_s^{*}}^2▽{\psi }_s·▽\left({▽}^2{\psi }_s\right))\hfill \\ \hfill & +\left(-13392+12{\mathrm{C}}_{4,4}^{\left[\mathbf{3}\right]}+3{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}\right){\left|{\psi }_s\right|}^2\left({\psi }_s{▽}^2{\psi }_s{(▽{\psi }_s^{*})}^2+{\psi }_s^{*}{▽}^2{\psi }_s^{*}{(▽{\psi }_s)}^2\right)\hfill \\ \hfill & +\left(2892+{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}+{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}-4{\mathrm{C}}_{4,4}^{\left[\mathbf{7}\right]}+12{\mathrm{C}}_{4,4}^{\left[\mathbf{9}\right]}\right){\left|{\psi }_s\right|}^2\left({\psi }_s^2{▽}^2\left({(▽{\psi }_s^{*})}^2\right)\right.\hfill \\ \hfill & \left.+{{\psi }_s^{*}}^2{▽}^2\left({(▽{\psi }_s)}^2\right)\right)+\left(-5976+12{\mathrm{C}}_{4,4}^{\left[\mathbf{3}\right]}+6{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}+3{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}-12{\mathrm{C}}_{4,4}^{\left[\mathbf{7}\right]}\right.\hfill \\ \hfill & \left.+36{\mathrm{C}}_{4,4}^{\left[\mathbf{8}\right]}\right)|{\psi }_s{|}^2{|▽{\psi }_s|}^2\left({\psi }_s{▽}^2{\psi }_s^{*}+{\psi }_s^{*}{▽}^2{\psi }_s\right)+\left(5832+3{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}+6{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}\right.\hfill \\ \hfill & \left.-12{\mathrm{C}}_{4,4}^{\left[\mathbf{7}\right]}+36{\mathrm{C}}_{4,4}^{\left[\mathbf{9}\right]}\right){\left|{\psi }_s\right|}^2\left({\psi }_s▽{\psi }_s·▽\left({(▽{\psi }_s^{*})}^2\right)+{\psi }_s^{*}▽{\psi }_s^{*}·▽\left({(▽{\psi }_s)}^2\right)\right)\hfill \\ \hfill & +\left(-11520+24{\mathrm{C}}_{4,4}^{\left[\mathbf{4}\right]}+3{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}\right){\left|{\psi }_s\right|}^2\left({\psi }_s▽{\psi }_s^{*}·▽\left(|▽{\psi }_s{|}^2\right)\right.\hfill \\ \hfill & \left.+{\psi }_s^{*}▽{\psi }_s·▽\left(|▽{\psi }_s{|}^2\right)\right)+12{\mathrm{C}}_{4,4}^{\left[\mathbf{2}\right]}{\left|{\psi }_s\right|}^4\left(▽{\psi }_s·▽\left({▽}^2{\psi }_s^{*}\right)+▽{\psi }_s^{*}·▽\left({▽}^2{\psi }_s\right)\right)\hfill \\ \hfill & +12{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}|{\psi }_s{|}^2|▽{\psi }_s{|}^4+12{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}{\left|{\psi }_s\right|}^2{(▽{\psi }_s)}^2{(▽{\psi }_s^{*})}^2\hfill \\ \hfill & +12{\mathrm{C}}_{4,4}^{\left[\mathbf{7}\right]}{|▽{\psi }_s|}^2\left({\psi }_s^2{(▽{\psi }_s^{*})}^2+{{\psi }_s^{*}}^2{(▽{\psi }_s)}^2\right)+12{\mathrm{C}}_{4,4}^{\left[\mathbf{9}\right]}\left({\psi }_s^3▽{\psi }_s^{*}·▽\left({(▽{\psi }_s^{*})}^2\right)\right.\hfill \\ \hfill & \left.+{{\psi }_s^{*}}^3▽{\psi }_s·▽\left({(▽{\psi }_s)}^2\right)\right)+12{\mathrm{C}}_{4,4}^{\left[\mathbf{8}\right]}\left({\psi }_s^3{▽}^2{\psi }_s^{*}{(▽{\psi }_s^{*})}^2+{{\psi }_s^{*}}^3{▽}^2{\psi }_s{(▽{\psi }_s)}^2\right)]\hfill \\ \hfill & -{\overline{\lambda }}_8\frac{1}{\alpha \left(4\right)m_a^4}|{\psi }_s{|}^8+{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{11}{\alpha \left(4\right)m_a^6}|{\psi }_s{|}^8-{\overline{\lambda }}_8\frac{1}{\alpha \left(4\right)m_a^6}{\left|{\psi }_s\right|}^6\left({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*}\right)\hfill \\ \hfill & +{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{1}{16\alpha \left(4\right)m_a^8}|{\psi }_s{|}^4[\left(8{\mathrm{C}}_{4,6}-2584\right)|{\psi }_s{|}^2{|▽{\psi }_s|}^2\hfill \\ \hfill & +{\mathrm{C}}_{4,6}{\left|{\psi }_s\right|}^2({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*})+\left(3{\mathrm{C}}_{4,6}-1035\right)\left({{\psi }_s^{*}}^2{\left(▽{\psi }_s\right)}^2+{\psi }_s^2{\left(▽{\psi }_s^{*}\right)}^2\right)]\hfill \\ \hfill & -{\overline{\lambda }}_4^3\frac{49}{4\alpha \left(4\right)m_a^8}|{\psi }_s{|}^8-{\overline{\lambda }}_4^3\frac{1}{8\alpha \left(4\right)m_a^{10}}|{\psi }_s{|}^4[\left(8{\mathrm{C}}_{4,4,4}-2020\right)|{\psi }_s{|}^2{|▽{\psi }_s|}^2\hfill \\ \hfill & +{\mathrm{C}}_{4,4,4}{\left|{\psi }_s\right|}^2({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*})\hfill \\ \hfill & +\left(3{\mathrm{C}}_{4,4,4}-831\right)\left({{\psi }_s^{*}}^2{\left(▽{\psi }_s\right)}^2+{\psi }_s^2{\left(▽{\psi }_s^{*}\right)}^2\right)]-{\overline{\lambda }}_{10}\frac{1}{\alpha \left(5\right)m_a^5}|{\psi }_s{|}^{10}+{\overline{\lambda }}_4{\overline{\lambda }}_8\frac{45}{2\alpha \left(5\right)m_a^7}{\left|{\psi }_s\right|}^{10}\hfill \\ \hfill & +{\overline{\lambda }}_6^2\frac{131}{6\alpha \left(5\right)m_a^7}|{\psi }_s{|}^{10}+{\overline{\lambda }}_6^2\frac{5}{288\alpha \left(5\right)m_a^9}{\left|{\psi }_s\right|}^6[(4{\mathrm{C}}_{6,6}-12858)({{\psi }_s^{*}}^2{\left(▽{\psi }_s\right)}^2+{\psi }_s^2{\left(▽{\psi }_s^{*}\right)}^2)\hfill \\ \hfill & +{\mathrm{C}}_{6,6}|{\psi }_s{|}^2({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*})+(10{\mathrm{C}}_{6,6}-30145)|{\psi }_s{|}^2{|▽{\psi }_s|}^2]\hfill \\ \hfill & -{\overline{\lambda }}_4^2{\overline{\lambda }}_6\frac{4315}{24\alpha \left(5\right)m_a^9}|{\psi }_s{|}^{10}+{\overline{\lambda }}_4^4\frac{8625}{64\alpha \left(5\right)m_a^{11}}{\left|{\psi }_s\right|}^{10}+{\mathrm{ETT}}_{\mathcal{L}},\hfill \end{array}$$

(96)

where

$$\begin{array}{cc}\hfill {\mathrm{ETT}}_{\mathcal{L}}=& -i{\overline{\lambda }}_4^2\frac{27}{64\alpha \left(3\right)m_a^6}{\left|{\psi }_s\right|}^4\left({\dot{\psi }}_s{\psi }_s^{*}-{\psi }_s{\dot{\psi }}_s^{*}\right)\hfill \\ \hfill & -i{\overline{\lambda }}_4^2\frac{9}{256\alpha \left(3\right)m_a^8}[2{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}|{\psi }_s{|}^2{|▽{\psi }_s|}^2({\dot{\psi }}_s{\psi }_s^{*}-{\psi }_s{\dot{\psi }}_s^{*})\hfill \\ \hfill & +\left(-32+{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}-2{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\right){\left|{\psi }_s\right|}^4({\psi }_s^{*}{▽}^2{\dot{\psi }}_s-{\psi }_s{▽}^2{\dot{\psi }}_s^{*})\hfill \\ \hfill & +2{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}{\left|{\psi }_s\right|}^2({{\psi }_s^{*}}^2{\dot{\psi }}_s{▽}^2{\psi }_s-{\psi }_s^2{\dot{\psi }}_s^{*}{▽}^2{\psi }_s^{*})\hfill \\ \hfill & +\left(-128+{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}-6{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\right){\left|{\psi }_s\right|}^4(▽{\psi }_s^{*}·▽{\dot{\psi }}_s-▽{\psi }_s·▽{\dot{\psi }}_s^{*})\hfill \\ \hfill & +2{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\left({{\psi }_s^{*}}^3{\dot{\psi }}_s{(▽{\psi }_s)}^2-{\psi }_s^3{\dot{\psi }}_s^{*}{(▽{\psi }_s^{*})}^2\right)\hfill \\ \hfill & +\left(37+{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}-3{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}\right){\left|{\psi }_s\right|}^4({\dot{\psi }}_s{▽}^2{\psi }_s^{*}-{\dot{\psi }}_s^{*}{▽}^2{\psi }_s)\hfill \\ \hfill & +\left(-90+2{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}-6{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\right)|{\psi }_s{|}^2({\psi }_s{\dot{\psi }}_s{(▽{\psi }_s^{*})}^2\hfill \\ \hfill & -{\psi }_s^{*}{\dot{\psi }}_s^{*}{(▽{\psi }_s)}^2\left)\right]-{\overline{\lambda }}_4^2\frac{9}{128\alpha \left(3\right)m_a^7}[{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}|{\psi }_s{|}^4({\ddot{\psi }}_s{\psi }_s^{*}+{\psi }_s{\ddot{\psi }}_s^{*})+\left(6{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}-33\right)|{\psi }_s{|}^4{\left|{\dot{\psi }}_s\right|}^2\hfill \\ \hfill & +\left(2{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}-16\right)|{\psi }_s{|}^2\left({\left({\dot{\psi }}_s\right)}^2{{\psi }_s^{*}}^2+{\psi }_s^2{\left({\dot{\psi }}_s^{*}\right)}^2\right)]-i{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{15}{8\alpha \left(4\right)m_a^7}{\left|{\psi }_s\right|}^6\left({\dot{\psi }}_s{\psi }_s^{*}-{\psi }_s{\dot{\psi }}_s^{*}\right)\hfill \\ \hfill & +i{\overline{\lambda }}_4^3\frac{21}{4\alpha \left(4\right)m_a^9}|{\psi }_s{|}^6\left({\dot{\psi }}_s{\psi }_s^{*}-{\psi }_s{\dot{\psi }}_s^{*}\right)-i{\overline{\lambda }}_6^2\frac{775}{288\alpha \left(5\right)m_a^8}{\left|{\psi }_s\right|}^8\left({\dot{\psi }}_s{\psi }_s^{*}-{\psi }_s{\dot{\psi }}_s^{*}\right),\hfill \end{array}$$

(97)

is the part of the Lagrangian corresponding to Equation (94). Additionally, ${\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{i}\right]}$, ${\mathrm{C}}_{4,4,6}$, ${\mathrm{C}}_{4,4,4}$, ${\mathrm{C}}_{6,6}$, ${\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{j}\right]}$, and ${\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}$, where $i=1,2,\cdots ,9$ and $j=1,2,3$ are all constants.

4.6. Treatment of the Extra Temporal Terms

In order to write the additional temporal terms as expressions depending only on ${\psi }_s$, ${ϵ}_x$, and ${\overline{\lambda }}_{2n}$, the equation of motion from a previous iteration is replaced in Equation (97). However, this must be performed carefully in order to not introduce incomplete terms. In this work, the following expressions are needed

$$\begin{array}{cc}\hfill i{\dot{\psi }}_s=& -\frac{1}{2m_a}{▽}^2{\psi }_s+{\overline{\lambda }}_4\frac{2}{\alpha \left(2\right)m_a^2}\left({\psi }_s{\left|{\psi }_s\right|}^2\right),\hfill \end{array}$$

(98)

$$\begin{array}{cc}\hfill i{\dot{\psi }}_s=& -\frac{1}{2m_a}{▽}^2{\psi }_s-\frac{1}{8m_a^3}{▽}^4{\psi }_s+{\overline{\lambda }}_4\frac{2}{\alpha \left(2\right)m_a^2}\left({\psi }_s{\left|{\psi }_s\right|}^2\right)\hfill \\ \hfill & +{\overline{\lambda }}_4\frac{1}{2\alpha \left(2\right)m_a^4}\left({\psi }_s^2{▽}^2{\psi }_s^{*}+2{\left|{\psi }_s\right|}^2{▽}^2{\psi }_s\right)+{\overline{\lambda }}_4\frac{1}{2\alpha \left(2\right)m_a^4}\left({▽}^2\left({\psi }_s{\left|{\psi }_s\right|}^2\right)\right)\hfill \\ \hfill & +{\overline{\lambda }}_6\frac{3}{\alpha \left(3\right)m_a^3}\left({\psi }_s{\left|{\psi }_s\right|}^4\right)-{\overline{\lambda }}_4^2\frac{51}{8\alpha \left(3\right)m_a^5}\left({\psi }_s{\left|{\psi }_s\right|}^4\right).\hfill \end{array}$$

(99)

where Equations (98) and (99) are the equations of motion when the iterative process, as shown in this section, ends at the $0th$ and $1st$ iteration, respectively. Additionally, it is necessary to compute ${\ddot{\psi }}_s$. This is achieved by deriving Equation (98) with respect to time and then implementing Equation (98) itself to replace the resulting ${\dot{\psi }}_s$. Doing so results in

$$\begin{array}{cc}\hfill {\ddot{\psi }}_s=& -\frac{1}{4m_a^2}{▽}^4{\psi }_s-{\overline{\lambda }}_4^2\frac{9}{2\alpha \left(3\right)m_a^4}\left({\psi }_s{\left|{\psi }_s\right|}^4\right)\hfill \\ \hfill & +{\overline{\lambda }}_4\frac{4}{\alpha \left(2\right)m_a^3}\left(|{\psi }_s{|}^2{▽}^2{\psi }_s+{\psi }_s{|▽{\psi }_s|}^2\right)\hfill \\ \hfill & +{\overline{\lambda }}_4\frac{2}{\alpha \left(2\right)m_a^3}\left({\psi }_s^{*}{\left({▽}^2{\psi }_s\right)}^2\right)\hfill \end{array}$$

(100)

If Equation (99) is substituted for the last term in Equation (97), then a term proportional to ${\overline{\lambda }}_4^2{\overline{\lambda }}_6^2$ would arise. However, this new term is beyond the 5th order. Consequently, even though this contribution would not be wrong, it would be fruitless to add it to the final Lagrangian as it is not complete. Taking this into account, by substituting Equations (98), (99), and (100) into Equation (97), one arrives at

$$\begin{array}{cc}\hfill {\mathrm{ETT}}_{\mathcal{L}}=& {\overline{\lambda }}_4^2\frac{27}{128\alpha \left(3\right)m_a^7}{\left|{\psi }_s\right|}^2\left({\psi }_s^{*}|{\psi }_s{|}^2{▽}^2{\psi }_s+{\psi }_s{\left|{\psi }_s\right|}^2{▽}^2{\psi }_s^{*}\right)\hfill \\ \hfill & +{\overline{\lambda }}_4^2\frac{9}{512\alpha \left(3\right)m_a^9}[\left(-29+{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}+{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}-2{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\right){\left|{\psi }_s\right|}^4({\psi }_s^{*}{▽}^4{\psi }_s+{\psi }_s{▽}^4{\psi }_s^{*})\hfill \\ \hfill & +\left(107-6{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}+2{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}-6{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}\right){\left|{\psi }_s\right|}^4{\left|{▽}^2{\psi }_s\right|}^2\hfill \\ \hfill & +\left(-16+2{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}+2{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}\right){\left|{\psi }_s\right|}^2\left({\psi }_s^2{\left({▽}^2{\psi }_s^{*}\right)}^2+{{\psi }_s^{*}}^2{\left({▽}^2{\psi }_s\right)}^2\right)\hfill \\ \hfill & +\left(-90+2{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}-6{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\right){\left|{\psi }_s\right|}^2\left({\psi }_s{▽}^2{\psi }_s{(▽{\psi }_s^{*})}^2+{\psi }_s^{*}{▽}^2{\psi }_s^{*}{(▽{\psi }_s)}^2\right)\hfill \\ \hfill & +2{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}|{\psi }_s{|}^2{|▽{\psi }_s|}^2\left({\psi }_s{▽}^2{\psi }_s^{*}+{\psi }_s^{*}{▽}^2{\psi }_s\right)\hfill \\ \hfill & +\left(-128+{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}-6{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\right){\left|{\psi }_s\right|}^4\left(▽{\psi }_s·▽\left({▽}^2{\psi }_s^{*}\right)+▽{\psi }_s^{*}·▽\left({▽}^2{\psi }_s\right)\right)\hfill \\ \hfill & +2{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\left({\psi }_s^3{▽}^2{\psi }_s^{*}{(▽{\psi }_s^{*})}^2+{{\psi }_s^{*}}^3{▽}^2{\psi }_s{(▽{\psi }_s)}^2\right)]\hfill \\ \hfill & +{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{15}{16\alpha \left(4\right)m_a^8}|{\psi }_s{|}^4\left({\psi }_s^{*}|{\psi }_s{|}^2{▽}^2{\psi }_s+{\psi }_s{\left|{\psi }_s\right|}^2{▽}^2{\psi }_s^{*}\right)-{\overline{\lambda }}_4^3\frac{27}{8\alpha \left(4\right)m_a^8}{\left|{\psi }_s\right|}^8\hfill \\ \hfill & -{\overline{\lambda }}_4^3\frac{3}{64\alpha \left(4\right)m_a^{10}}[\left(-2232+24{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}+24{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}+24{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}-120{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\right)|{\psi }_s{|}^2{|▽{\psi }_s|}^2\hfill \\ \hfill & +\left(-64+6{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}+3{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}+6{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}-18{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\right){\left|{\psi }_s\right|}^2\left({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*}\right)\hfill \\ \hfill & +\left(-828+6{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}+9{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}+6{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}-24{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\right)\left({\psi }_s^2{(▽{\psi }_s^{*})}^2+{{\psi }_s^{*}}^2{(▽{\psi }_s)}^2\right)]\hfill \\ \hfill & +{\overline{\lambda }}_6^2\frac{775}{576\alpha \left(5\right)m_a^9}|{\psi }_s{|}^6\left({\psi }_s^{*}|{\psi }_s{|}^2{▽}^2{\psi }_s+{\psi }_s{\left|{\psi }_s\right|}^2{▽}^2{\psi }_s^{*}\right)-{\overline{\lambda }}_4^2{\overline{\lambda }}_6\frac{75}{2\alpha \left(5\right)m_a^9}{\left|{\psi }_s\right|}^{10}\hfill \\ \hfill & +{\overline{\lambda }}_4^4\frac{12225}{128\alpha \left(5\right)m_a^{11}}{\left|{\psi }_s\right|}^{10}.\hfill \end{array}$$

(101)

Substituting Equation (101) in Equation (96) results in

$$\begin{array}{cc}\hfill \mathcal{L}=& \frac{i}{2}\left({\dot{\psi }}_s{\psi }_s^{*}-{\psi }_s{\dot{\psi }}_s^{*}\right)-{\mathcal{H}}_{\mathrm{eff}},\hfill \end{array}$$

(102)

where

$${\mathcal{H}}_{\mathrm{eff}}={\mathcal{T}}_{\mathrm{eff}}+V_{\mathrm{eff}}+W_{\mathrm{eff}},$$

(103)

and

$$\begin{array}{cc}\hfill {\mathcal{T}}_{\mathrm{eff}}=& \frac{1}{2m_a}▽{\psi }_s·▽{\psi }_s^{*}-\frac{1}{8m_a^3}{▽}^2{\psi }_s{▽}^2{\psi }_s^{*}+\frac{1}{16m_a^5}▽\left({▽}^2{\psi }_s\right)·▽\left({▽}^2{\psi }_s^{*}\right)\hfill \\ \hfill & -\frac{5}{128m_a^7}{▽}^4{\psi }_s{▽}^4{\psi }_s^{*},\hfill \end{array}$$

(104)

$$\begin{array}{cc}\hfill V_{\mathrm{eff}}=& {\overline{\lambda }}_4\frac{1}{\alpha \left(2\right)m_a^2}|{\psi }_s{|}^4+{\overline{\lambda }}_6\frac{1}{\alpha \left(3\right)m_a^5}|{\psi }_s{|}^6-{\overline{\lambda }}_4^2\frac{17}{8\alpha \left(3\right)m_a^5}|{\psi }_s{|}^6+{\overline{\lambda }}_8\frac{1}{\alpha \left(4\right)m_a^4}{\left|{\psi }_s\right|}^8\hfill \\ \hfill & -{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{11}{\alpha \left(4\right)m_a^6}|{\psi }_s{|}^8+{\overline{\lambda }}_4^3\frac{125}{8\alpha \left(4\right)m_a^8}|{\psi }_s{|}^8+{\overline{\lambda }}_{10}\frac{1}{\alpha \left(5\right)m_a^5}{\left|{\psi }_s\right|}^{10}\hfill \\ \hfill & -{\overline{\lambda }}_4{\overline{\lambda }}_8\frac{45}{2\alpha \left(5\right)m_a^7}|{\psi }_s{|}^{10}-{\overline{\lambda }}_6^2\frac{131}{6\alpha \left(5\right)m_a^7}{\left|{\psi }_s\right|}^{10}\hfill \\ \hfill & +{\overline{\lambda }}_4^2{\overline{\lambda }}_6\frac{5215}{24\alpha \left(5\right)m_a^9}|{\psi }_s{|}^{10}-{\overline{\lambda }}_4^4\frac{29475}{128\alpha \left(5\right)m_a^{11}}{\left|{\psi }_s\right|}^{10},\hfill \end{array}$$

(105)

$$\begin{array}{cc}\hfill W_{\mathrm{eff}}=& {\overline{\lambda }}_4\frac{1}{2\alpha \left(2\right)m_a^4}|{\psi }_s{|}^2\left({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*}\right)+{\overline{\lambda }}_4\frac{5}{16\alpha \left(2\right)m_a^6}{\left|{\psi }_s\right|}^2\left({\psi }_s^{*}{▽}^4{\psi }_s\right.\hfill \\ \hfill & \left.+{\psi }_s{▽}^4{\psi }_s^{*}\right)+{\overline{\lambda }}_4\frac{1}{16\alpha \left(2\right)m_a^6}\left({\psi }_s^2{\left({▽}^2{\psi }_s^{*}\right)}^2+4|{\psi }_s{|}^2{\left|{▽}^2{\psi }_s\right|}^2+{{\psi }_s^{*}}^2{\left({▽}^2{\psi }_s\right)}^2\right)\hfill \\ \hfill & +{\overline{\lambda }}_4\frac{1}{64\alpha \left(2\right)m_a^8}\left[15\right|{\psi }_s{|}^2\left({\psi }_s^{*}{▽}^6{\psi }_s+{\psi }_s{▽}^6{\psi }_s^{*}\right)+5({\psi }_s^2{▽}^2{\psi }_s^{*}{▽}^4{\psi }_s^{*}\hfill \\ \hfill & +{{\psi }_s^{*}}^2{▽}^2{\psi }_s{▽}^4{\psi }_s)({\psi }_s^2{▽}^2{\psi }_s^{*}{▽}^4{\psi }_s^{*}+{{\psi }_s^{*}}^2{▽}^2{\psi }_s{▽}^4{\psi }_s)+10|{\psi }_s{|}^2({▽}^2{\psi }_s{▽}^4{\psi }_s^{*}\hfill \\ \hfill & +{▽}^2{\psi }_s^{*}{▽}^4{\psi }_s)+2|{▽}^2{\psi }_s{|}^2({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*})]\hfill \\ \hfill & +{\overline{\lambda }}_6\frac{3}{4\alpha \left(3\right)m_a^5}{\left|{\psi }_s\right|}^4\left({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*}\right)\hfill \\ \hfill & +{\overline{\lambda }}_6\frac{15}{32\alpha \left(3\right)m_a^7}{\left|{\psi }_s\right|}^4\left({\psi }_s^{*}{▽}^4{\psi }_s+{\psi }_s{▽}^4{\psi }_s^{*}\right)\hfill \\ \hfill & +{\overline{\lambda }}_6\frac{3}{16\alpha \left(3\right)m_a^7}{\left|{\psi }_s\right|}^2\left({\psi }_s^2{\left({▽}^2{\psi }_s^{*}\right)}^2+3|{\psi }_s{|}^2{\left|{▽}^2{\psi }_s\right|}^2+{{\psi }_s^{*}}^2{\left({▽}^2{\psi }_s\right)}^2\right)\hfill \\ \hfill & -{\overline{\lambda }}_4^2\frac{3}{128\alpha \left(3\right)m_a^7}|{\psi }_s{|}^2[\left(12{\mathrm{C}}_{4,4}^{\left[\mathbf{1}\right]}-774\right)|{\psi }_s{|}^2{|▽{\psi }_s|}^2\hfill \\ \hfill & +\left(4{\mathrm{C}}_{4,4}^{\left[\mathbf{1}\right]}-280\right)\left({{\psi }_s^{*}}^2{\left(▽{\psi }_s\right)}^2+{\psi }_s^2{\left(▽{\psi }_s^{*}\right)}^2\right)\hfill \\ \hfill & +\left(2{\mathrm{C}}_{4,4}^{\left[\mathbf{1}\right]}+9\right)\left({\psi }_s^{*}|{\psi }_s{|}^2{▽}^2{\psi }_s+{\psi }_s{\left|{\psi }_s\right|}^2{▽}^2{\psi }_s^{*}\right)]\hfill \\ \hfill & -{\overline{\lambda }}_4^2\frac{1}{2048\alpha \left(3\right)m_a^9}[\left(5262+36{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}+36{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}-72{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}+4{\mathrm{C}}_{4,4}^{\left[\mathbf{2}\right]}-2{\mathrm{C}}_{4,4}^{\left[\mathbf{3}\right]}\right){\left|{\psi }_s\right|}^4\hfill \end{array}$$

(106)

$$\begin{array}{cc}\hfill & \times ({\psi }_s^{*}{▽}^4{\psi }_s+{\psi }_s{▽}^4{\psi }_s^{*})\hfill \\ \hfill & +\left(3852-216{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}+72{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}-216{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}+12{\mathrm{C}}_{4,4}^{\left[\mathbf{3}\right]}\right){\left|{\psi }_s\right|}^4{\left|{▽}^2{\psi }_s\right|}^2\hfill \\ \hfill & +\left(4860+72{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}+72{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}+{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}+{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}-4{\mathrm{C}}_{4,4}^{\left[\mathbf{7}\right]}+12{\mathrm{C}}_{4,4}^{\left[\mathbf{8}\right]}\right)\hfill \\ \hfill & \times |{\psi }_s{|}^2\left({\psi }_s^2{\left({▽}^2{\psi }_s^{*}\right)}^2+{{\psi }_s^{*}}^2{\left({▽}^2{\psi }_s\right)}^2\right)\hfill \\ \hfill & +\left(3936+8{\mathrm{C}}_{4,4}^{\left[\mathbf{2}\right]}-4{\mathrm{C}}_{4,4}^{\left[\mathbf{3}\right]}+{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}+{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}-4{\mathrm{C}}_{4,4}^{\left[\mathbf{7}\right]}+12{\mathrm{C}}_{4,4}^{\left[\mathbf{8}\right]}\right)|{\psi }_s{|}^2({\psi }_s^2▽{\psi }_s^{*}·▽\left({▽}^2{\psi }_s^{*}\right)\hfill \\ \hfill & +{{\psi }_s^{*}}^2▽{\psi }_s·▽\left({▽}^2{\psi }_s\right))+12{\mathrm{C}}_{4,4}^{\left[\mathbf{4}\right]}|{\psi }_s{|}^4{▽}^2\left(|▽{\psi }_s{|}^2\right)+12{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}|{\psi }_s{|}^2{|▽{\psi }_s|}^4\hfill \\ \hfill & +12{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}{\left|{\psi }_s\right|}^2{(▽{\psi }_s)}^2{(▽{\psi }_s^{*})}^2+\left(-16632+72{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}-216{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}+12{\mathrm{C}}_{4,4}^{\left[\mathbf{3}\right]}+3{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}\right)\hfill \\ \hfill & \times |{\psi }_s{|}^2\left({\psi }_s{▽}^2{\psi }_s{(▽{\psi }_s^{*})}^2+{\psi }_s^{*}{▽}^2{\psi }_s^{*}{(▽{\psi }_s)}^2\right)\hfill \\ \hfill & +\left(2892+{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}+{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}-4{\mathrm{C}}_{4,4}^{\left[\mathbf{7}\right]}+12{\mathrm{C}}_{4,4}^{\left[\mathbf{9}\right]}\right){\left|{\psi }_s\right|}^2\left({\psi }_s^2{▽}^2\left({(▽{\psi }_s^{*})}^2\right)+{{\psi }_s^{*}}^2{▽}^2\left({(▽{\psi }_s)}^2\right)\right)\hfill \\ \hfill & +\left(-5976+72{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}+12{\mathrm{C}}_{4,4}^{\left[\mathbf{3}\right]}+6{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}+3{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}-12{\mathrm{C}}_{4,4}^{\left[\mathbf{7}\right]}+36{\mathrm{C}}_{4,4}^{\left[\mathbf{8}\right]}\right)\hfill \\ \hfill & \times |{\psi }_s{|}^2{|▽{\psi }_s|}^2\left({\psi }_s{▽}^2{\psi }_s^{*}+{\psi }_s^{*}{▽}^2{\psi }_s\right)\hfill \\ \hfill & +\left(5832+3{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}+6{\mathrm{C}}_{4,4}^{\left[\mathbf{6}\right]}-12{\mathrm{C}}_{4,4}^{\left[\mathbf{7}\right]}+36{\mathrm{C}}_{4,4}^{\left[\mathbf{9}\right]}\right){\left|{\psi }_s\right|}^2\left({\psi }_s▽{\psi }_s·▽\left({(▽{\psi }_s^{*})}^2\right)\right.\hfill \\ \hfill & \left.+{\psi }_s^{*}▽{\psi }_s^{*}·▽\left({(▽{\psi }_s)}^2\right)\right)+\left(-11520+24{\mathrm{C}}_{4,4}^{\left[\mathbf{4}\right]}+3{\mathrm{C}}_{4,4}^{\left[\mathbf{5}\right]}\right){\left|{\psi }_s\right|}^2\left({\psi }_s▽{\psi }_s^{*}·▽\left(|▽{\psi }_s{|}^2\right)\right.\hfill \\ \hfill & \left.+{\psi }_s^{*}▽{\psi }_s·▽\left(|▽{\psi }_s{|}^2\right)\right)+\left(-4608+36{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}-216{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}+12{\mathrm{C}}_{4,4}^{\left[\mathbf{2}\right]}\right){\left|{\psi }_s\right|}^4\hfill \\ \hfill & \times \left(▽{\psi }_s·▽\left({▽}^2{\psi }_s^{*}\right)+▽{\psi }_s^{*}·▽\left({▽}^2{\psi }_s\right)\right)\hfill \\ \hfill & +12{\mathrm{C}}_{4,4}^{\left[\mathbf{7}\right]}{|▽{\psi }_s|}^2\left({\psi }_s^2{(▽{\psi }_s^{*})}^2+{{\psi }_s^{*}}^2{(▽{\psi }_s)}^2\right)+12{\mathrm{C}}_{4,4}^{\left[\mathbf{9}\right]}\left({\psi }_s^3▽{\psi }_s^{*}·▽\left({(▽{\psi }_s^{*})}^2\right)\right.\hfill \\ \hfill & \left.+{{\psi }_s^{*}}^3▽{\psi }_s·▽\left({(▽{\psi }_s)}^2\right)\right)+\left(72{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}+12{\mathrm{C}}_{4,4}^{\left[\mathbf{8}\right]}\right)\left({\psi }_s^3{▽}^2{\psi }_s^{*}{(▽{\psi }_s^{*})}^2\right.\hfill \\ \hfill & \left.+{{\psi }_s^{*}}^3{▽}^2{\psi }_s{(▽{\psi }_s)}^2\right)]+{\overline{\lambda }}_8\frac{1}{\alpha \left(4\right)m_a^6}{\left|{\psi }_s\right|}^6\left({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*}\right)\hfill \\ \hfill & -{\overline{\lambda }}_4{\overline{\lambda }}_6\frac{1}{16\alpha \left(4\right)m_a^8}|{\psi }_s{|}^4[\left(8{\mathrm{C}}_{4,6}-2584\right)|{\psi }_s{|}^2{|▽{\psi }_s|}^2\hfill \\ \hfill & +{\mathrm{C}}_{4,6}{\left|{\psi }_s\right|}^2({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*})+\left(3{\mathrm{C}}_{4,6}-1035\right)\left({{\psi }_s^{*}}^2{\left(▽{\psi }_s\right)}^2+{\psi }_s^2{\left(▽{\psi }_s^{*}\right)}^2\right)]\hfill \\ \hfill & +{\overline{\lambda }}_4^3\frac{1}{64\alpha \left(4\right)m_a^{10}}{\left|{\psi }_s\right|}^4[\left(-22856+64{\mathrm{C}}_{4,4,4}+72{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}+72{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}+72{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}\right.\hfill \\ \hfill & \left.-360{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\right)|{\psi }_s{|}^2{|▽{\psi }_s|}^2+\left(-192+8{\mathrm{C}}_{4,4,4}+18{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}+9{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}+18{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}\right.\hfill \\ \hfill & \left.-54{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\right){\left|{\psi }_s\right|}^2({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*})\hfill \\ \hfill & +\left(-9132+24{\mathrm{C}}_{4,4,4}+18{\mathrm{C}}_{4,4,\mathbf{I}\mathbf{I}}+27{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{1}\right]}+18{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{2}\right]}-72{\mathrm{C}}_{4,4,\mathbf{I}}^{\left[\mathbf{3}\right]}\right)\left({{\psi }_s^{*}}^2{\left(▽{\psi }_s\right)}^2\right.\hfill \\ \hfill & \left.+{\psi }_s^2{\left(▽{\psi }_s^{*}\right)}^2\right)]-{\overline{\lambda }}_6^2\frac{5}{576\alpha \left(5\right)m_a^9}{\left|{\psi }_s\right|}^6[(8{\mathrm{C}}_{6,6}-25716)({{\psi }_s^{*}}^2{\left(▽{\psi }_s\right)}^2+{\psi }_s^2{\left(▽{\psi }_s^{*}\right)}^2)\hfill \\ \hfill & +\left(2{\mathrm{C}}_{6,6}+155\right)|{\psi }_s{|}^2({\psi }_s^{*}{▽}^2{\psi }_s+{\psi }_s{▽}^2{\psi }_s^{*})+(20{\mathrm{C}}_{6,6}-60290)|{\psi }_s{|}^2{|▽{\psi }_s|}^2].\hfill \end{array}$$

5. Summary

Axions are one of the most strongly supported candidates for making up the dark matter in the universe. They were originally introduced as the pseudo-Goldstone bosons associated with spontaneously broken Peccei–Quinn symmetry, which address the strong $CP$-problem in QCD. Therefore, they could link, from their weaknesses, two of the most successful theories of particle physics: the Standard Model and general relativity.

Axions can be produced in the early universe in an abundance compatible with the observed dark matter density via appealing to some physical mechanisms that produce them at extremely low-energy conditions. Additionally, it has been argued that nonrelativistic axions that weakly interact among themselves can produce Bose–Einstein condensates that exhibit short-range correlations with either scarce occupancy or low coherence. Having at hand a derived effective field theory for low-energy axions that seems to be unique, since the predicted Bose–Eintein condensates demands higher-order corrections, we computed them up to ${\left({\psi }^{*}\psi \right)}^5$, also considering the effects of fast-oscillating fluctuations that influence the dominant slowly varying nonrelativistic field.

The unique nonrelativistic EFT for low-energy axions was reviewed and meticulously derived up to fifth-order corrections. We also showed a systematic, improvable, and iterative way of computing even higher-order terms, considering the back-reaction of fast-oscillating fields. Going forward in the calculation presented herein was beyond the scope of this study, which already collected all the information on the subject disseminated in various studies by different authors; their calculations were also completed and corrected up to following highest-order correction, i.e., up to ${\left({\psi }^{*}\psi \right)}^5$.

EFTs may be needed in the near future to study dark matter in a quantitative and systematic way. Particularly, higher-order corrections of nonrelativistic effective field theory of a weakly interacting scalar particle are needed because (i) radiative corrections may be needed at high orders to account for quantum effects; (ii) weak interactions may contain patterns or valuable information that are only revealed when considering higher orders; (iii) high-order corrections may be necessary to understand how the theory behaves in some asymptotic limits; and. (iv) as in other examples of EFTs such as chiral perturbation theory or (potential) nonrelativistic QCD, computing higher orders is usually necessary years after their establishment to ensure that they are in line with the experimental data.

Now, it remains to consider the possible future steps to follow in this line of research, which include the exploration of axion coupling with gravity and the development of computational algorithms that allow us to treat possible many-body axion bound-state systems. Both are under consideration already, and we expect to communicate results soon. The axions’ interactions among themselves may need to be determiend up to the precision level developed in this manuscript since, on one hand, gravity corresponds to a weak interaction and, on the other hand, many-body physics of weakly interacting particles could produce emergent phenomena encoded only in high-order terms.

Finally, unifying both the EFT perspective and the axion dark matter description, since dark matter is cold and nonrelativistic, a quantum-mechanical description given using EFTs provides a way to overcome the difficulties of finite-temperature QFT: a one-particle potential can be defined; from it, we can find the macroscopic description of an axion gas. In a large cloud, the extremely nonlinear characteristic of the axion’s self-interaction potential, together with the weak coupling to photons, would result in a reheating through any collapse, and higher-order corrections to the nonrelativistic dynamics of axions may be necessary to obtain a realistic model, esspecially if computational physics is applied to resolve the collapse.