1 Introduction

The nature of confinement in a Yang–Mills theory is an important and complex question that is not settled yet. In fact, there is no consensus on what mechanism leads to confinement. Various candidates include center vortices, magnetic monopoles, Abelian Higgs mechanism, the Gribov mechanism, or the creation of an Abrikosov-type string via dual Meissner effect; see [1] for a review. Even the phenomenology of the strong interaction’s strings/flux tubes is challenged as a manifestation of quark confinement [2]. What is clear is that confinement stems from field self-interactions in the non-perturbative, i.e., the strong coupling, regime.

In this paper, we numerically study two self-interacting scalar field theories which may correspond, in the high-temperature limit, to Quantum Chromodynamics (QCD) and General Relativity (GR). QCD is the non-Abelian – and thus self-interacting – theory of the strong interaction and the prototypical confining theory. Its phenomenology has been – and continue to be – experimentally scrutinized. Many methods have been developed to study it. They have reached a high degree of sophistication. Thus, our primary purpose here is not to study QCD with a simplified approach but rather to use the strong interaction as a testing ground, once our model is checked with simpler free-field theories. Nevertheless, we hope that the present simple model can yield interesting insight in the confinement mechanism. Gravity’s GR is also a self-interacting theory which ensuing non-linearities make its resolution notoriously difficult. Studying it is not as developed as for QCD and our model provides a relatively simple way to approach its phenomenology. The verification of our approach in the free-field case and its relevance to QCD phenomenology, indicate that its predictions for GR should also be pertinent.

The paper is organized as follows. We first introduce the Lagrangians of two scalar self-interacting field theories. Next, the corresponding instantaneous potentials are expressed in terms of path integrals on a lattice, and calculated with a standard numerical Monte-Carlo technique. We then interpret the results obtained in the high-temperature regime. Finally, we discuss how the scalar theories may approximate the non-zero spin field theories of QCD (spin 1, i.e. vector-field) and GR (spin 2, i.e. rank-2 tensor-field). In particular, we recall how the Lagrangian of GR can be expressed in a polynomial form which, when expressed in the static limit, is identical to one of the scalar Lagrangians previously introduced in the high-temperature limit. A first consequence of the high-temperature limit is that numerical calculations become extremely fast and tractable. Furthermore, the high-temperature limit takes the innate quantum nature of a path-integral formalism to the classical limit. While this is a limitation for modeling the inherently quantum strong interaction, it is advantageous for approximating GR since it alleviates the notorious difficulties associated with quantum gravity. The strong interaction being intrinsically non-classical, there is no reason to expect that the high-temperature potential corresponds to the known quark–quark phenomenological static potential. It is in fact not the case: the high-temperature (i.e. classical) potential derived with the method discussed in this article has approximately a Yukawa form, which is very different from the phenomenological quark–quark linear potential. Although the classical potential cannot be compared to the phenomenological one, it is still a valuable result since it can be compared to results from other approaches obtained in the classical limit of QCD. An agreement would then validate the method used in this article. Furthermore, the short distance quantum effects can be introduced had-oc by allowing the theory’s coupling to run. Checking whether such running transforms the Yukawa potential into the expected linear potential would then validate further the approach and makes it relevant to the study of confinement. As we already made clear, the method discussed here in the context of the strong interaction is not intended to compete with the more mature and sophisticated approaches that are already used to study the strong interaction, such as Lattice QCD, the Schwinger–Dyson approach or the Anti-de Sitter/Conformal Field Theory (AdS/CFT) duality. Rather, the primary purpose of studying the strong interaction is to check the technique so that it can be applied with confidence to the more complex – but classical – case of GR. A secondary benefit is that new approaches may provide fresh views on mechanisms important to the confinement problem. As a specific example, it is interesting to see in the approach discussed here how a mass scale emerges out of a conformal Lagrangian. Such a phenomenon is at the heart of recent advances in our understanding of QCD in its non-perturbative regime [3]. At the end of this article, after having validated our approach using the free-field and QCD cases, we discuss the predictions of our model in the GR case. In particular, the model naturally explains many cosmological observations without need for non-baryonic dark matter. To show how these observations can be naturally and accurately explained by a phenomenology well-known in the QCD context is the main goal of this article. These observations include the rotation curves of disk galaxies, the fact that they are approximately flat, The correlation between the rotation speeds and baryonic masses of disk galaxies [4, 5], the correlation between the dark mass of elliptical galaxies and their ellipticities [6], the dynamics of galaxy clusters and the Bullet cluster observations [7].

We have grouped the technical aspects of this work in several appendices: In Appendix A we recall the numerical technique used to obtain the potentials. In Appendix B we give step-by-step instructions for programming a basic calculation. In Appendix C, we discuss the question of the choice of boundary conditions and show the independence of our results on such choices. In Appendix D, we recover several analytically known potentials to validate the technique developed in this paper. In Appendix E, we verify that the results are independent of the choice of lattice parameters, such as mesh spacing, decorrelation coefficient or lattice size. In Appendix F we recover the post-Newtonian formalism from the scalar Lagrangian, thereby further verifying that it correctly approximates GR in the classical limit.

This approach is simple and fast. We give in Appendix B the time necessary for a standard computer to perform a calculation to good precision. Extending this figure to state-of-art lattice machines implies that a one minute calculation would reach a relative precision of \(2 \times 10^{-7}\), comparable to the intrinsic precision of a 32-bit machine. It should be remarked that the method and algorithms are not specially optimized, since the speed is adequate for the problems discussed in this paper.

2 Scalar Lagrangians

Confinement stems from field self-interactions in large coupling or strong field regimes. Lagrangian densities for self-interacting theories can be constructed from the graphs shown in Fig. 1. We discuss here the simple case of a scalar (spin 0) field \(\phi \). We consider only massless fields, like those describing the QCD and GR forces.

Fig. 1
figure 1

Graphs used to construct the Lagrangians. The graph on the left corresponds to the free part of the theory yielding a 1 / r potential. The two other graphs create field self-interactions that distort this potential. In case (1), the forms of these terms are imposed by choosing a dimensionless coupling (no consistent cubic term can be formed). In case (2) they are chosen for a coupling of dimension [energy\(]^{-2}\)

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The form of the Lagrangian density in case (1) is

$$\begin{aligned} \mathcal {L}=\partial _{\mu }\phi \partial ^{\mu }\phi -g^2\phi ^{4} , \end{aligned}$$
(1)

with g the coupling, dimensionless as for QCD. For case (2) the Lagrangian density is

$$\begin{aligned} \mathcal {L}=\partial _{\mu }\phi \partial ^{\mu }\phi +g\phi \partial _{\mu }\phi \partial ^{\mu }\phi +g^{2}\phi ^{2}\partial _{\mu }\phi \partial ^{\mu }\phi , \end{aligned}$$
(2)

with g a coupling of dimension [energy\(]^{-2}\), as for GR.

There is no cubic term in Eq. (1) since it would not be Lorentz invariant. One could insist on having such a term by adding a unit 4-vector \(e^\mu \) to make the Lorentz structure of Eq. (1) internally consistent when a cubic term is included. However, such a \(g\phi ^2\partial _{\mu }\phi e^{\mu }\) term would have no influence since it disappears from the Euler–Lagrange equation:

$$\begin{aligned} \frac{\partial \mathcal {L}}{\partial \phi }-\partial _\mu \frac{\partial \mathcal {L}}{\partial \partial ^\mu \phi }= -2g^2 \phi ^3-\partial ^2 \phi , \end{aligned}$$
(3)

as we see from the absence of a term linear in g. Only the terms stemming from the quartic component (proportional to \(g^2)\) and from the quadratic one remain.

Although the scalar field Lagrangians in Eqs. (1) and ( 2) stem from the same diagrams, they display different structures. There is no cubic term in Eq. (1) and the quartic term contains no partial derivatives. Hence, it does not contribute to the propagation of the field and acts similarly to a mass term. In fact, the analytical solution for the equation of motion is known, with the field obeying a massive dispersion relation [8], whose resulting potential can be well fit by a Yukawa potential, as we show in Sect. 4.1. In Eq. ( 2), the cubic term is preserved. All terms contain partial derivatives and thus contribute to the field propagation.

3 Computation of the potential between two static sources

The potential at a point \(x_{2}\) from a point-like source at \(x_{1}\) can be obtained, in the high-temperature limit, by the two-point Green function \(G_{2p}(x_{1}-x_{2})\). In the Feynman path-integral formalism \(G_{2p}\) is

$$\begin{aligned} G_{2p}(x_{1}-x_{2})=\frac{1}{Z}\intop \mathrm {D}\varphi \,\varphi (x_{1})\varphi (x_{2})\mathrm {e}^{-\mathrm {i}\, S_{\mathrm {s}}}, \end{aligned}$$
(4)

where \(S_{\mathrm {s}}\equiv {\int {\mathrm {d}^{4}x\,\mathcal {L}}}\) is the action, \(\intop {\mathrm {D}\varphi }\) sums over all possible field configurations, and \(Z\equiv \intop {\mathrm {D}\varphi \,\mathrm {e}^{-\mathrm {i}\, S_{\mathrm {s}}}}\). On an Euclidean spacetime lattice simulation, the temperature corresponds to inverse of the lattice spacing in the time direction. Hence, in the high-temperature limit, the sum \(\intop {\mathrm {D}\varphi }\) is reduced to configurations in position space. The suppression of time allows us to identify the potential to \(G_{2p}\).

Such expression of the potential is unconventional and restrictive since it applies only to stationary systems. However, its simplicity allows for fast calculations, which in turn may allow the implementation of forces too complex, e.g. gravity with tensor fields, to be efficiently implemented on the lattice in the standard way. Since such a way of calculating potentials is unusual, we will specifically verify its validity for instantaneous potentials, the only cases studied here. This will be demonstrated by correctly recovering potentials in known cases of free-field theories (theories without self-interacting fields). Although adding terms to the Lagrangian does not affect the above definition of the potential, one might speculate that self-interacting terms may void its validity. However, we will also recover the known expected potential for the self-interacting \(\phi ^4\) theory as well as the strong interaction phenomenological static potential once short distance quantum effects are accounted for. Based on this we can assume with some confidence that this method of calculating potentials remains valid for the self-interacting field case.

3.1 Quantum versus classical calculations

(In this section we keep \(\hbar \) in expressions to keep track of quantum effects.) In principle, the path-integral formalism provides intrinsically quantum calculations. However, the results obtained here will be classical since we employ a lattice covering position space only (high-temperature limit) [9, 10]. Briefly, the system being time-independent, we have \(S_{\mathrm {s}}\equiv {\int {\mathrm {d}^{4}x\,\mathcal {L}}=\tau S}\), with \(\tau =\intop _{t_{0}}^{\infty }{\mathrm {d}t\rightarrow \infty }\) and \(S\equiv {\int {\mathrm {d}^{3}x\,\mathcal {L}}}\). The exponential term in Eq. (4) becomes \(\mathrm {e}^{\mathrm {-i}S_{\mathrm {s}}/\hbar }=\mathrm {e}^{\mathrm {-i}\tau S/\hbar }\). Compared to the usual \(\mathrm {e}^{\mathrm {-i}S/\hbar }\) exponential in path-integral quantum field theory, quantum effects are suppressed as \(\hbar /\tau \). Since \(\tau \rightarrow \infty \), this is equivalent to the classical \(\hbar \rightarrow 0\) limit.

We remark that our definition of the potential allows one to compute only instantaneous potentials, which are suited for stationary systems and the purpose of the present article. Consequently, computing potentials from the time-propagation of the system, as for example in the Feynman–Kac formula [11, 12] or using Wilson loops [13], is not applicable.

In the rest of the paper, we will not write anymore the factor \(\tau \) since, as a constant rescaling factor, it is irrelevant to classical calculations (a rescaled extremal action remains extremal or, equivalently, rescaling S amounts to rescaling \(\mathcal {L}\), whose rescaling factor cancels out in the Euler–Lagrange Equation). However, \(\tau \) must be considered when e.g. discussing the dimensions of the fields and couplings. In other words, the action S has a dimension \([\hbar \times \)energy] rather than \([ \hbar ]\).

3.2 Goal, formalism and verifications

The goal is to compute the potential between two approximately static (\(v\ll c\)) sources in the non-perturbative regime. The lattice technique, based on a standard Monte-Carlo method exploiting the Metropolis algorithm, is described in Appendix A. A practical example is given in Appendix B. In Appendix C, we discuss the delicate point of the choice of boundary conditions and show that the results are independent of such choices. In Appendix D, the method is verified by recovering analytically known free-field potentials in two or three spatial dimensions, as well as the \(g^2 \phi ^4\) theory potential in three dimensions. In Appendix E, we verify the independence of the results on the choice of lattice parameters.

4 Results

4.1 \(g^2 \phi ^4\) theory

The \(g^2 \phi ^4\) theory can be solved analytically [8]. The analytical solution of the equation of motion is known, with the field obeying a massive dispersion relation. Indeed, the resulting potentials are well fit by a Yukawa potential over the whole range of coupling values we used (\(0\le g^2 \le 4\)); see Fig. 2 for an example of potential between two sources located at \(x_1=21\) and \(x_2=35\), and for \(g^2 = 0.5\). Figure 2 also demonstrates the independence of the result with the choice of boundary conditions. The expected function describing \(\phi \) is the Jacobi elliptic function \(\text{ sn }(x,-1)\) [8]. Its autocorrelation \(\text{ sn } \star \text{ sn }\), which provides the potential \(G_{2p}\), fits well our numerical results; see Fig. 3, but only for coupling values \(g^2 > 1\).

Fig. 2
figure 2

Potential \(G_{2p}\) for the \(g^2 \phi ^4\) theory, plotted as a function of distance. The simulations used a lattice size \(N=55\), a coupling \(g^2=0.5\), a decorrelation coefficient (see Sect. B.1) \(N_{\mathrm {cor}}=20\) and \(N_{\mathrm {s}}=5\times 10^{4}\) configurations. The results are independent of the choice of boundary conditions. They are well fit by two Yukawa potentials centered on each two sources at \(d=\pm 7\) from the lattice center and with a mass parameter \(m_{\text{ eff }}=0.64\) (continuous line)

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The effect of the quartic term \(g^2 \phi ^4\) can be embodied using a field effective mass \(m_{\text{ eff }}\). It can be obtained by fitting \(G_{2p}\) with a Yukawa potential \(e^{-m_{\text{ eff }}~x}/x\). Such effective mass is shown as a function of \(g^2\) in Fig. 4. A polynomial fit to these data yields \(m_{\text{ eff }} = (0.76\pm 0.02)( g^2)^{0.39\pm 0.02}+0.05\pm 0.02\), and a logarithmic fit yields \(m_{\text{ eff }} = (1.13\pm 0.03)\text{ ln }\big ((g^2)^{0.51\pm 0.02}+1\big )+0.06\pm 0.02\) with a marginally better \(\chi ^2\) (10% smaller). As already mentioned, the effectively massive behavior of the \(g^2 \phi ^4\) theory can be understood from the lack of partial derivatives in the quartic term. It thus does not contribute to the field propagation and is akin to a mass term. Evidently, this field effective mass depends on the value of the coupling \(g^2\) (Fig. 4).

Fig. 3
figure 3

Comparison between the potential computed numerically using Eq. (1) (squares) and the analytical expectation \(\text{ sn }(x,-1)\star \text{ sn }(x,-1)\) (line). The simulation used a coupling \(g^2 =3\), a lattice size \(N=55\), \(N_{\mathrm {s}}=5\times 10^{4}\) configurations, a decorrelation coefficient \(N_{\mathrm {cor}}=15\) and random field boundary conditions

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Fig. 4
figure 4

The \(g^2\)-dependence of the effective mass generated by the quartic term \(g^2 \phi ^4\). The continuous line is a polynomial fit to the data and the dashed line is a logarithmic fit. The simulation used a lattice size \(N=55\), \(N_{\mathrm {s}}=5\times 10^{4}\) configurations, a decorrelation coefficient \(N_{\mathrm {cor}}=15\) and random field boundary conditions

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4.2 (\(g \phi \partial _\mu \phi \partial ^\mu \phi + g^2 \phi ^2 \partial _\mu \phi \partial ^\mu \phi \)) theory

The (\(g \phi \partial _\mu \phi \partial ^\mu \phi + g^2 \phi ^2 \partial _\mu \phi \partial ^\mu \phi \)) theory in its non-perturbative regime yields the static potential shown in Fig. 5. It varies approximately linearly with distance (except at mid-distance between the two sources where the potential flattens out, as expected for a solely attractive force). Also shown is the \(g=0\) case which yields the expected free massless field potential in 1 / x, with x the distance to one of the source. In Fig. 6, results are shown with the 1 / x contribution subtracted in order to isolate the linear part stemming from the (\(g \phi \partial _\mu \phi \partial ^\mu \phi + g^2 \phi ^2 \partial _\mu \phi \partial ^\mu \phi \)) term.

Fig. 5
figure 5

The potential for the (\(g \phi \partial _\mu \phi \partial ^\mu \phi + g^2 \phi ^2 \partial _\mu \phi \partial ^\mu \phi \)) theory (triangles). The circles are the free-field case (\(g=0\)). The two sources are located on the x-axis at \(d=\pm 7\) from the lattice center \(x_{\mathrm {center}}=28\), \(y=0\) and \(z=0\). The coupling is \(g=7.5\times 10^{-3}\), the lattice size is \(N=85\), the decorrelation parameter is \(N_{\mathrm {cor}}=20\) and \(N_{\mathrm {s}}=3.5\times 10^{4}\) paths were used. To conveniently compare the results, a constant offset is added to the free-field case so that the potentials at the position of the first source, \(x=21\), match

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Fig. 6
figure 6

Same as Fig. 5 but with the 1 / x – free-field – contribution subtracted. The lines illustrate the nearly linear trend of the potential

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The linear x-dependence of the potential can be interpreted in a similar manner as done in the QCD static case; see e.g. [1]. In QCD, the linear potential is often pictured as originating from the collapse of the force’s field lines into a flux tube linking the two sources. This collapse is caused by strong non-perturbative field self-interactions: the gluons strongly attract each other toward the region of their highest density i.e. the segment joining the sources. If the sources are static and since the field has collapsed into a segment, the two dimensions transverse to the segment become irrelevant. The strong non-perturbative effects have constrained the field flux from three spatial degrees of freedom to one. The resulting force is constant, as easily calculated or visualized with Faraday’s flux picture.

A similar phenomenon seems to occur with the theory using (\(g \phi \partial _\mu \phi \partial ^\mu \phi + g^2 \phi ^2 \partial _\mu \phi \partial ^\mu \phi \)) theory. The strong field self-interaction effectively reduces the three-dimensional system to a one-dimensional system, yielding a linear potential. The linear potential can also be understood from field correlations. The potential is given by the two-point Green function, i.e. the autocorrelation function of the field; see Eq. (4). It is a general property that the autocorrelation of the sum of two uncorrelated functions is the sum of the autocorrelations of each function separately: if \(f=g+h\), then \(f\star f=g\star g+h\star h\), i.e. two uncorrelated fields g and h (i.e. g and h do not influence each others) from two sources imply the superposition principle. The total potential \(f\star f\) is the sum of the potentials \(g\star g\) and \(h\star h\). If the fields mutually influence each others, correlations appear. The superposition principle is thus broken. When fields are interacting strongly enough, they become strictly correlated. The total field (observed sufficiently away from the sources so that the three-dimensional symmetry around a point-like source is broken) is then constant on a segment linking the two sources. It must quickly go to zero outside the segment due to the boundary conditions at infinity. The field is then approximately a one-dimensional rectangular function. The autocorrelation of a rectangular function being a triangle function, the potential between the two sources has a linear x-dependence, as seen in Fig. 6. In all, the linear potential arises due to the long field autocorrelation length – stemming from the partial derivatives present in each term of the Lagrangian – that couples the field at different locations in space. This explanation is consistent with the naive dimensional argument inspired from QCD.

With this interpretation, we can immediately generalize the present two-source linear potential result to a uniform and homogeneous disk of sources. The three-dimensional system is then reduced to two dimensions. This yields a logarithmic potential (see Fig 12). For a uniform and homogeneous spherical distribution of sources, the system stays three-dimensional and retains a static 1 / x potential.

5 Connections to QCD and GR

In this section, we explore the possible connections of the Lagrangians in Eqs. (1) and (2) with QCD and GR, respectively. We first discuss the benefits that such connections would provide.

Numerical studies of QCD using lattice gauge theory are sophisticated and well checked. They have provided invaluable information on this theory. As stated in the introduction, the present work is not meant to compete with these more exact and more mature approaches. The primary goal of making the connection to QCD is that, if the QCD phenomenology is recovered, it would support the validity of the method beyond the checks already discussed. The method can then be used with confidence to address more complicated cases, e.g. GR whose tensorial field makes it currently impractical to put on the lattice. This primary goal being clarified, there are still advantages of such approach within the sole context of the strong interaction: (1) a simple approach showing the onset of confinement may isolate its essential aspects and make them easier to study, including analytically. Furthermore, (2) if the confinement mechanism is manifest in a simplified approach, it may provide guidances for an analytical resolution of the problem. It could (3) also provide an alternate approach to a complex problem. In fact, lattice QCD has greatly benefited from other non-perturbative approaches that can be fully solved only numerically, e.g. the Schwinger–Dyson equations approach. (4) It makes calculations fast compared to typical lattice running time – a few hours of running on a standard personal computer are sufficient. (5) It provides a pedagogical model of confinement.

Efforts to solve GR in its non-perturbative regime by numerical methods also constitute an active field of study [14]. Its object is strongly interacting systems for which gravitational backreaction cannot be ignored. As in QCD, the equations governing these systems are non-linear and can be non-perturbative. Except in rare cases, they can only be solved numerically. A non-perturbative component to the potential would not be identified using the usual perturbative (post-Newtonian) approximation, and a non-perturbative linear contribution to the potential is in general compatible with the field equations of bound systems [15, 16]. In fact, it is emphasized in Refs. [15, 16] that bound states can be understood semi-classically as divergences of the perturbative expansion arising from ladder-type Feynman diagrams. Consequently, even more than for QCD, possessing alternate methods to approach the complex problem of bound systems in GR is beneficial.

In the rest of this section, we discuss the possibility of such method, based on the fact that the Lagrangians in Eqs. (1) and (2) may be related to the QCD and GR Lagrangians. Such relation has already been put forward in the case of QCD [17, 18] where it was asserted that Eq. (1) is equivalent to QCD in either the classical limit or the low energy limit. We will show that similarly, Eq. (2) may describe GR in the classical limit. We first start by recalling the Lagrangians of QCD and GR.

5.1 QCD Lagrangian

The QCD field Lagrangian is

$$\begin{aligned} \mathcal {L}^{\mathrm{field}}_{\mathrm{QCD}}= & {} -\frac{1}{4}F_{\mu \nu }^a F^{\mu \nu }_a \nonumber \\= & {} \frac{1}{4}(\partial _{\nu }A_\mu ^a - \partial _{\mu }A^a_\nu ) (\partial ^{\mu }A^{\nu a} - \partial ^{\nu }A^{\mu a} ) \nonumber \\&+\sqrt{\pi \alpha _s}f^{abc}\big (\partial _{\nu }A_\mu ^a - \partial _{\mu }A_\nu ^a\big )A^{\mu b}A^{\nu c} \nonumber \\&-\pi \alpha _s f^{abe}f^{cde}A^a_\mu A^b_\nu A^{\mu c} A^{\nu d}, \end{aligned}$$
(5)

with \(\alpha _s\) the QCD coupling constant, \(A_\mu ^a\) the gluon field and with the SU(3) index \(a=1,\ldots , 8\). We do not include Fadeev–Popov ghosts in the Lagrangian. Their necessity depends on the choice of gauge, e.g. there are no ghosts in the axial gauge \(e^\mu A^a_\mu \) or in the light-cone gauge \(A^+=0\). Furthermore, Fadeev–Popov ghosts are needed in QCD essentially to force the gluon propagator to remain transverse when loop diagrams are included and since we are discussing here limits using spinless fields, there is no such need. Fermions are also not included in Eq. (5) since it is sufficient to work in the pure field sector for a first study of confinement in the static case.

5.2 General relativity

Einstein’s field equations of GR are generated by the Einstein–Hilbert Lagrangian density:

$$\begin{aligned} \mathcal {L}_{\mathrm {GR}}=\frac{1}{16\pi G}\sqrt{\mathrm {det}(g_{\mu \nu })}\, g_{\mu \nu }R^{\mu \nu }, \end{aligned}$$
(6)

where G is the Newton constant, \(g_{\mu \nu }\) the metric and \(R_{\mu \nu }\) the Ricci tensor. The gravity field \(\varphi _{\mu \nu }\) is defined as the difference between \(g_{\mu \nu }\) and a constant reference metric \(\eta _{\mu \nu }\): \(\varphi _{\mu \nu }\equiv (g_{\mu \nu }-\eta _{\mu \nu })/\sqrt{16\pi G}\). Here, \(\eta _{\mu \nu }\) will be the flat metric. Expanding \(\mathcal {L}_{\mathrm {GR}}\) in terms of \(\varphi _{\mu \nu }\) yields (see e.g. [19] or [20]):

$$\begin{aligned}&\mathcal {L}_{\mathrm {GR}}={[\partial \varphi \partial \varphi ]+}\sqrt{16\pi G}[\varphi \partial \varphi \partial \varphi ]+16\pi G[\varphi ^{2}\partial \varphi \partial \varphi ] \nonumber \\&\quad +\cdots -\sqrt{16\pi G}\,\varphi _{\mu \nu }T^{\mu \nu }-\frac{16\pi G}{2}\varphi _{\mu \nu }\varphi _{\sigma \lambda }T^{\mu \nu }\eta ^{\sigma \lambda }+\cdots ,\nonumber \\ \end{aligned}$$
(7)

where \(T^{\mu \nu }\) is the energy-momentum tensor and \([\varphi ^{n}\partial \varphi \partial \varphi ]\) is a shorthand notation for a sum over the possible Lorentz invariant terms of the form \(\varphi ^{n}\partial \varphi \partial \varphi \). For example, \([\partial \varphi \partial \varphi ]\) is explicitly given by the Fierz–Pauli Lagrangian [21], the first order approximation of GR that leads to Newton’s gravity:

$$\begin{aligned}{}[\partial \varphi \partial \varphi ]=\frac{1}{2}\partial ^{\lambda }\varphi _{\mu \nu }\partial _{\lambda }\varphi ^{\mu \nu }-\frac{1}{2}\partial ^{\lambda }\varphi _{\mu }^{\mu }\partial _{\lambda }\varphi _{\nu }^{\nu } \nonumber \\ -\partial ^{\lambda }\varphi _{\lambda \nu }\partial _{\mu }\varphi ^{\mu \nu }+\partial ^{\nu }\varphi _{\lambda }^{\lambda }\partial ^{\mu }\varphi _{\mu \nu }. \end{aligned}$$
(8)

Equation (7) is expanded in term of the dimensionless quantity \(\sqrt{16\pi G}\varphi \). It is justified to truncate Eq. (7), even for strong gravity fields, since in general \(\sqrt{16\pi G}\ll 1/\mu \) with \(\mu \) the typical mass scale or inverse distance scale of the system considered. A pertinent reference metric \(\eta _{\mu \nu }\) is chosen according to the system being considered. It can be e.g. the Minkowski metric for systems inducing weak curvatures, or the Schwarzschild metric for a system involving a black hole. Hence, Eq. (7) is applicable to both the weak-field approximations of GR and non-perturbative gravity.

5.3 Lagrangians in the static case

Equation (7) can be simplified. Consider first the \([\partial \varphi \partial \varphi ]\) term given by Eq. (8). The Euler–Lagrange equation obtained by varying \(\varphi _{\mu \nu }\) in \(\int {\mathrm {d}^{4}x\big (\big [\partial \varphi \partial \varphi \big ]-}\) \(\sqrt{16\pi G}\,\varphi _{\mu \nu }T^{\mu \nu }\big )\) leads to \(\partial ^{2}\varphi ^{\mu \nu }=-16\pi G\big (T^{\mu \nu }-\frac{1}{2}\eta ^{\mu \nu }\text{ Tr }~T\big )\). Since \(T^{00}\) dominates over the other components of \(T^{\mu \nu }\) in the static limit, \(\partial ^{2}\varphi ^{\mu \mu }\gg \partial ^{2}\varphi ^{\mu \nu }\) for \(\mu \ne \nu \) and \(\partial ^{2}\varphi ^{00}\simeq \partial ^{2}\varphi ^{ii}\). Thus, \(\varphi ^{00}=\varphi ^{ii}+a_{i}x+b_{i}\). Since \(\varphi ^{\mu \nu }\rightarrow 0\) for \(x\rightarrow \infty \), \(a_{i}=0\), \(b_{i}=0\) and \(\varphi ^{00}=\varphi ^{ii}\). Replacing the \(\varphi ^{ii}\) terms in Eq. (8) by \(\varphi ^{00}\) and applying the harmonic gauge condition \(\partial ^{\mu }\varphi _{\mu \nu }=\frac{1}{2}\partial _{\nu }\varphi _{\kappa }^{\kappa }\), leads to \([\partial \varphi \partial \varphi ]=\partial _{\lambda }\varphi ^{00}\partial ^{\lambda }\varphi _{00}\). Consider now the next order term, \([\varphi \partial \varphi \partial \varphi ]\). The \(\varphi ^{\mu \nu }\) \((\mu \ne \nu )\) terms that were neglected in the lower order operations leading to \([\partial \varphi \partial \varphi ]=\partial _{\lambda }\varphi ^{00}\partial ^{\lambda }\varphi _{00}\) can be ignored with respect to the \({\varphi ^{00}}\partial _{\lambda }\varphi ^{00}\partial ^{\lambda }\varphi ^{00}\) term: the terms \([\partial \varphi \partial \varphi ]\) lead to linear field equations, so they do not contribute to the non-linearity due to the \([\varphi ^{n}\partial \varphi \partial \varphi ]\) terms, while they remain negligible compared to the linear terms \(\partial _{\lambda }\varphi ^{00}\partial ^{\lambda }\varphi ^{00}\). The \(\partial _{\lambda }\varphi ^{00}\partial ^{\lambda }\varphi _{00}\) structure dominates all other components of the form \([\partial \varphi \partial \varphi ]\). Thus, if the influence of the \({\varphi ^{00}}\partial _{\lambda }\varphi ^{00}\partial ^{\lambda }\varphi ^{00}\) term is not negligible compared to the effect of the \(\partial _{\lambda }\varphi ^{00}\partial ^{\lambda }\varphi ^{00}\) term (i.e. there are strong field non-linearities) then it is justified to neglect the components \(\varphi ^{\mu \nu }\) \((\mu \ne \nu )\) of \([\partial \varphi \partial \varphi ]\). The corrections coming from the modification of the Euler–Lagrange equation enter only at the level of the terms \(\varphi ^{n+1}\partial ^{2}\varphi \), with \(n\ge 2\), and can likewise be neglected. Consequently, in the static case – which identifies to the high-temperature limit in Euclidean space – , the GR Lagrangian may be simplified to

$$\begin{aligned} \mathcal {L}_{\mathrm {GR,stat.}}=\sum _{n=0}^{\infty }\left( 16\pi G\right) ^{n/2}\varphi ^{n}[a_{n}\partial \varphi \partial \varphi -T_{00}], \end{aligned}$$
(9)

where the subscript \(\mathrm {{stat.}}\) reminds us that \(\mathcal {L}{}_{\mathrm {GR}}\) is expressed in the static limit. We define the scalar field \(\varphi \equiv \varphi _{00}\). We found above that \(a_0 =1\) and confirm it in Appendix F by deriving the weak-field potential from Eq. (9), and comparing it to the Einstein–Infeld–Hoffmann equations [22]. The values of the other \(a_{n}\) coefficients are not important here since Eq. (9) will be used only at first orders. We set \(a_{n}=1\) for all n in the rest of the paper. For two static point sources located at \(x_{1}\) and \(x{}_{2}\), \(\mathcal {L}_{\mathrm {GR,stat.}}\) becomes

$$\begin{aligned}&\mathcal {L}_{\mathrm {GR,stat.}}=\sum _{n=0}^{\infty }\left( 16\pi G\right) ^{n/2} [\varphi ^{n}\partial \varphi \partial \varphi \nonumber \\&\quad -\left( 16\pi G\right) ^{1/2}\,\varphi ^{n+1}(\delta ^{(4)}(x-x_{1})+\delta ^{(4)}(x-x_{2}))^{n+1}].\nonumber \\ \end{aligned}$$
(10)

Gravitation is a spin-2 theory and, like any spin-even theory, is always attractive. Satisfactorily, the scalar, spin-0, approximation is also spin-even and thus also always attractive. In all, we see that the pure field part of \(\mathcal {L}_{\mathrm {GR,stat.}}\) is identical to Eq. (2), thus supporting that GR can be adequately described in its static limit with the Lagrangian in Eq. (2).

Likewise in QCD, the chromoelectric component \(A_0^a \equiv A\) of the gluon field dominates in the case of two static sources and the QCD Lagrangian may be approximated by

$$\begin{aligned} \mathcal {L}=\partial _{\mu }A \partial ^{\mu }A -\pi \alpha _s A ^{4}. \end{aligned}$$
(11)

Contrary to Eq. (5), Eq. (11) has no cubic term. In the strong coupling regime where non-perturbative effects are dominant, either \(\pi \alpha _s A^4 \gg \partial A \partial A\) if only the quartic term dominates in Eq. (5), or \(\sqrt{\pi \alpha _s} A^2 \gg \partial A\) if the cubic term dominates over the quadratic one. The latter inequality also implies \(\sqrt{\pi \alpha _s} A^4 \gg A^2 \partial A\) so in any case, the quartic term dominates over the other terms. Thus, the lack of a cubic term in Eq. (11) may not preclude that this equation describes QCD in a classical limit. In fact, it has been advanced that Eq. (11) describes QCD in either the classical limit when quantum fluctuations are neglected, or in the low energy, strong coupling, limit  [17, 18].

In QCD, a two-source system corresponds to a meson which, to be colorless and have zero baryon number, must be made of constituent quark and antiquark of opposite colors. QCD being a spin-1 theory, the force either attracts or repulses. However, the force between the two static sources is solely attractive since they must carry opposite colors. Hence, as for gravity, the always attractive spin-0 field is also satisfactory in the case of strong interaction. This also applies to baryons when viewed as quark–diquark structures.

In all, the QCD and GR Lagrangians, Eqs. (5) and (7), respectively, may be approximated in the classical and static limits by the scalar Lagrangians of Eqs. (11) and (9), respectively. We remark that the invariance principles underlying the theories, e.g. gauge invariance under SU(3)\(_{c}\) for QCD, cannot be considered in scalar theories. However, they are manifest in the sense that the form of the scalar Lagrangians stem from the ones of the full theories whose Lagrangian forms themselves are imposed by invariance principles. Likewise, the non-abelian nature of the theories cannot be formalized with a scalar theory but is still manifest in the non-linear terms of the Lagrangian. Related features that the full theories possess, e.g. different colors charges, must be accounted for ad hoc, e.g. by inserting the appropriate Casimir factor in the static potential.

5.4 Onset of the strong regime for QCD and GR

The QCD coupling at long distance is large [23]. There, effects from the higher order cubic and quartic terms of Eq. (5) become dominant and quarks are confined. In the gravitational force case, what can lead to a non-perturbative regime is either interactions at very short distances (presumably described by quantum gravity, which we do not consider here), or a large system mass M, which is what concerns us here. The gravitational force is a function of its total field squared, \(\varphi ^2\) and is also proportional to M. Normalizing \(\varphi \) to \(\varphi ^{2}=M\phi ^{2}\) makes explicit the importance of the higher order terms \(\left( 16\pi G\right) ^{n/2}M^{n/2+1}\varphi ^{n}\partial \varphi \partial \varphi \) in Eq. (10) for large values of M. Rescaling \(\mathcal {L}\) by \(1/M\), the three-dimensional action based on Eq. (10) is

$$\begin{aligned} S= & {} \int \mathrm {d}^{3}x[\partial _{i}\phi \partial ^{i}\phi +g\phi \partial _{i}\phi \partial ^{i}\phi +g^{2}\phi ^{2}\partial _{i}\phi \partial ^{i}\phi +\cdots \nonumber \\&-\,g\phi (\delta ^{(3)}(x-x_{1})+\delta ^{(3)}(x-x_{2}))-\cdots ],\nonumber \\ \end{aligned}$$
(12)

with \(g=\sqrt{16\pi MG}\). (The integration of the Lagrangian density being here three-dimensional; see Sect. 3.1, we summed over the spatial index i rather than the Lorentz index \(\mu \).) We note that rescaling thus the Lagrangian emphasizes further the quantum effect suppression discussed in Sect. 3.1. In all, quantum effects for the theory given by Eq. (10) are suppressed by \(\hbar /M\tau \), with M a large mass in our regime of interest.

In QCD, the non-linear regime arises for distances greater than \(2 \times 10^{-16}\) m when \(\alpha _s\) becomes large: \(\alpha _s \simeq 0.5\) in the \(\overline{MS}\) scheme [23]. Likewise, such a regime should arise in GR when \(g=\sqrt{16 \pi G M}\) is large, with M the system mass, and the system size is comparable to the field correlation length L. For a galaxy, typically \(\sqrt{16 \pi G M/L} \simeq 10^{-2}\), with \(L=10\) kpc taken to be a third of a typical galaxy size, and \(M=3 \times 10^{11} M_\odot \), a typical galaxy mass. \(g \simeq 10^{-2}\) is large enough to trigger the non-linear regime onset; see Fig. 5.

5.5 Applying the scalar model to QCD and GR in the high-temperature limit

5.5.1 QCD

While GR is a classical theory, and thus relevant for comparison with results obtained using the Lagrangian in Eq. (10), there is a priori no well-known classical limit pertinent to QCD, although semi-classical approximations do exist [15, 16, 24]. Certainly, an approximate Yukawa potential such as the one obtained in the high-temperature limit for the \(g^2 \phi ^4\) theory; see Fig. 2, does not resemble the linear potential expected in the QCD static case. This indicates that, provided Eq. (1) indeed describes the classical aspect of QCD, quantum effects are essential for obtaining a linear potential. Whereas, in the pure field sector of QCD, quantum effects such as quark condensates are irrelevant, others such as the running of the coupling cannot be disregarded when comparing meaningfully with phenomenology. Accounting for it can be done by mapping the \( g^2\)-dependence of \(m_{\text{ eff }}\) into a 4-momentum transfer \(Q^2\)-dependence assuming that \(g^2/\pi \) runs as the QCD coupling \(\alpha _s(Q^2)\) [23]. (Equations (1) and (5) allow us to identify \(g^2/\pi \) to \(\alpha _s\).) By augmenting the \(g^2 \phi ^4\) theory with the running of the coupling, we introduce short-distance quantum effects in an otherwise classical setting. We used the AdS/QCD results on \(\alpha _s(Q^2)\) [25,26,27] for \(Q^2 \lesssim 1\) GeV\(^2\), supplemented by the perturbative QCD expression of \(\alpha _s(Q^2)\) at fourth order for larger \(Q^2\). Such \(\alpha _s(Q^2)\) agrees well with the phenomenology [28, 29] for all \(Q^2\). Specifically, we used \(\alpha _s(Q^2)\) in the conventional \(\overline{MS}\) renormalization scheme. The result is shown in Fig. 7.

Fig. 7
figure 7

The running effective mass \(m_{\text{ eff }}(Q^2)\) as a function of the 4-momentum transfer \(Q^2\) corresponding to the \(g^2\) values shown in Fig. 4

Full size image

Using these results, the potential of the \(g^2(x) \phi ^4\) theory becomes

$$\begin{aligned} G_{2p}(x) = -C_f \frac{g^2(x)}{\pi } \frac{\mathrm{e}^{-m_{\text{ eff }}(x)x}}{x}, \end{aligned}$$
(13)

where we included the appropriate color factor \(C_f =4/3\) for a more pertinent comparison with QCD, and where \(g^2(Q^2)\) and \(m_{\text{ eff }}(Q^2)\) were Fourier-transformed to position space. The result is shown in Fig. 8 and compared with the well-established Cornell potential [30,31,32,33,34] that successfully describes hadron spectroscopy and that is consistent with quenched SU(3) QCD lattice calculations [35]. \(G_{2p}(x)\) displays the expected linear behavior in the relevant x range and agrees well with the Cornell potential. (For this comparison, x in Eq. (13) must be rescaled by \(1/C_f\), since not having color factors in Eq. (11) underestimates the magnitude of the field coupling by \(C_f\) compared to the SU(3) QCD theory.) The string tension, fit from the results between 0.35 and 0.6 fm in Fig. 8, is \(\sigma =0.157 \pm 0.012\) GeV\(^2\). This yields a confinement parameter \(\kappa = \sqrt{ \sigma \pi /2} = 0.496 \pm 0.038\) GeV that agrees with the value \(\kappa = 0.523 \pm 0.024\) GeV found in [3].

Fig. 8
figure 8

High-temperature potential from the \(g(x)^2 \phi ^4\) theory as a function of distance after adding the runnings of \(g^2\) and of \(m_{\text{ eff }}\). The strong interaction’s Cornell potential is shown by the dashed line

Full size image

The appearance of a mass scale in the \(g^2 \phi ^4\) theory, despite its conformal Lagrangian, is a valuable feature. Understanding the emergence of a GeV-size mass scale is one of the outstanding problems of strong interaction; see e.g. [24]. In the present work, such emergence can be understood from the absence of derivatives in the quartic term \(g^2 \phi ^4\), which thus does not contribute to field propagation but acts as a field mass. QCD also possesses a non-propagating \(A^a_\mu A^b_\nu A^{\mu c} A^{\nu d}\) term in its Lagrangian. This suggests that the mass scale emergence in strong interaction is a classical effect due to the specific form of the quartic term of the QCD Lagrangian. The cubic term may contribute similarly, or may instead generate a long correlation length as it does in the GR case. Although the good agreement of the \(g^2(x) \phi ^4\) theory with strong interaction suggests a minor role of the cubic term, its actual effect cannot be explored in the present model.

In all, the running effective mass and the potential obtained from the Lagrangian in Eq. (11) are consistent with what is expected from QCD. If this consistency underlines the pertinence of the \(g(x)^2 \phi ^4\) theory to model QCD, then it indicates that quantum effects, responsible for the running of the strong coupling, play a role in shaping the potential. Augmenting the \(g^2 \phi ^4\) theory at high-temperature with the running of \(g^2\) is necessary to map the values of the field’s effective mass into a \(Q^2\)-running mass, and to change the Yukawa potential into a linear potential. (An equivalent solution leading to a linear potential without introducing a running of the coupling would be to express the dressed gluon propagator as a sum of massive field propagators as done in Refs. [17, 18, 36].) A consistency between the \(g^2(x) \phi ^4\) theory and QCD also encourages the application of the present approach to the more complex case of GR, a fortiori since GR is already a classical theory and would not need to be supplemented ad hoc by quantum effects.

5.5.2 General relativity

The (\(g \phi \partial _\mu \phi \partial ^\mu \phi + g^2 \phi ^2 \partial _\mu \phi \partial ^\mu \phi \)) theory calculations in the high-temperature limit yield a potential which varies approximately linearly with distance; see Figs. 5 and 6.

This can be pictured as a collapse of the three-dimensional system into one dimension. As discussed in Sect. 5.4 and shown numerically, typical galaxy masses are enough to trigger the onset of the strong regime for GR.

Hence, for two massive bodies, such as two galaxies or two galaxy clusters, this would result in a string containing a large gravity field that links the two bodies – as suggested by the map of the large structures of the universe. That this yields quantitatively the observed dark mass of galaxy clusters and naturally explains the Bullet Cluster observation [7] was discussed in [37].

For a homogeneous disk, the potential becomes logarithmic. Furthermore, if the disk density falls exponentially with the radius, as it is the case for disk galaxies, it is easy to show that a logarithmic potential yields flat rotation curves: a body subjected to such a potential and following a circular orbit (as stars do in disk galaxies to good approximation) follows the equilibrium equation:

$$\begin{aligned} v(r)=\sqrt{G'M(r)}, \end{aligned}$$
(14)

with v the tangential speed and M(r) the disk mass integrated up to r, the orbit radius. \(G'\) is an effective coupling constant of dimension GeV\(^{-1}\) similar to the effective coupling \(\sigma \) (string tension) in QCD. Disk galaxies density profiles typically fall exponentially: \(\rho (r)=M_{0}\mathrm{e}^{-r/r_{0}}/(2\pi r_{0}^{2})\), where \(M_{0}\) is the total galactic mass and \(r_{0}\) is a characteristic length particular to a galaxy. Such a profile leads to, after integrating \(\rho \) up to r:

$$\begin{aligned} v(r)=\sqrt{G'M_{0}\left( 1-(r/r_{0}+1)\mathrm{e}^{-r/r_{0}}\right) }, \end{aligned}$$
(15)

At small r, the speed rises as \(v(r)\simeq \sqrt{G'M_{0}}r/r_{0}\) and flatten at large r: \(v\simeq \sqrt{G'M_{0}}\). This is what is observed for disk galaxies and the present approach yields rotation curves agreeing quantitatively with observations [37]. Flat rotation curves are an essential feature of galaxy dynamic that is natural in the present approach. In contrast, they are implemented in an ad hoc way in the standard WIMP/axion approach to dark matter, the profile of the dark matter halo being chosen specifically to yield flat rotation curves.

A problematic observation for non-baryonic dark matter is the correlation between the rotation speeds and baryonic masses of disk galaxies, a phenomenon known as the Tully–Fisher relation [4, 5]. The relation is puzzling since it is a dynamical relation that does not involve the galaxy dark mass while this one dominates the total galaxy mass. We showed in [37] how this relation is also naturally explained from the logarithmic potential arising in a disk galaxy. This mass–rotation relation of GR can be paralleled to the mass–angular-momentum relation of QCD at the origin of the Regge trajectories [1, 38]. The two relations are strikingly similar.

For a uniform and homogeneous spherical distribution of matter, the system remains three-dimensional and the static potential stays proportional to 1 / r, in contrast to the linear or logarithmic potentials of the one- and two-dimensional cases, respectfully. The dependence of the potential with the system’s symmetry suggested a search for a correlation between the shape of galaxies and their dark masses [37]. Evidence for such correlation has been found [6].

6 Summary and conclusion

We have numerically studied non-linearities in two particular scalar field theories. In the high-temperature limit these may provide a possible description of the strong interaction and of General Relativity (GR) that would simplify their study in their non-perturbative regime, and allow for fast numerical calculations. The overall validity of the method is verified by recovering analytically known potentials. We further verified the validity of the simplifications in the case of GR by recovering the post-Newtonian formalism.

Several approaches to QCD in its non-perturbative regime already exist, such as e.g. Lattice Gauge Theory, Dyson–Schwinger Equations, Functional Renormalization Group, Stochastic Quantization, or the AdS/CFT correspondence to name a few. They are more advanced and more exact that the simplified method discussed here. What primarily justifies the development of this simpler approximation to QCD is that it provides further checks of the method, which we can then be applied with confidence to the GR case. A subsidiary benefit of testing the method with QCD is that it may help to isolate important ingredients leading to confinement. In that context, this method is able to provide: (1) the expected field function, (2) a mechanism – straightforwardly applicable to QCD – for the emergence of a mass scale out of a conformal Lagrangian, (3) a running of the field effective mass, and (4) a potential agreeing with the phenomenological Cornell linear potential up to 0.8 fm, the relevant range for hadronic physics. Quantum effects, which cause couplings to run, are necessary for producing the linear potential and must be supplemented to the approach. A non-running coupling, even with a large value, would only yield a short range Yukawa potential. That the method recovers the essential features of QCD supports its application to GR. Since GR is a classical theory, no running coupling needs to be supplemented. The main goal of the paper is to provides a simple but reliable method to study GR in its non-perturbative regime since compared to QCD, non-perturbative methods for GR are not as developed. It is interesting to develop a method for GR that relates to QCD’s formalism since QCD and GR share similar field Lagrangians, with cubic and quartic field self-interactions terms, and since – contrary to GR – QCD’s non-perturbative phenomenology is well studied both experimentally and theoretically. QCD’s phenomenology may then serve as a guide to suggest, identify and understand non-perturbative effects in GR.

In fact, the similarity between QCD and GR field Lagrangians may explain intriguing parallels between on the one hand observations interpreted with dark matter and on the other hand the phenomenology of the strong interaction. For examples, the total masses of galaxies and galaxy clusters are much larger than the sum of the masses of their known components, just like for hadrons; galaxies and hadrons share similar mass-rotation correlations (the Tully–Fisher relation and Regge trajectories, respectively); the large-scale balancing of gravitational attraction by dark energy – known as the cosmic coincidence problem – may be paralleled with the large-distance suppression of the strong interaction into a much weaker Yukawa force; the universe’s large scale stringy structure observed by weak gravitational lensing is evocative of massive bodies, such as galaxies – or at larger scale two galaxy clusters – linked by a QCD-like string/flux tube.

These parallels and the similar structure of GR and QCD’s Lagrangians suggest that massive galactic objects such as galaxies or cluster of galaxies have triggered the non-perturbative regime of GR and that dark matter and dark energy are manifestations of it. It is clear that the self-interactions terms in GR have the right effects to explain dark matter and dark energy, the question being whether galaxies are massive enough to trigger these effects. This is the question we addressed in the second part of this paper. In QCD, the non-perturbative regime arises for distances greater than \(2 \times 10^{-16}\) m, while we find that for GR it arises at galactic scales. The non-perturbative effects then explain quantitatively the dynamics of spiral galaxies and clusters [37]. In particular, in the case spiral galaxies, the logarithmic potential resulting from the strong field non-linearities trivially yields flat rotation curves. Finally, for a uniform and homogeneous spherical distribution of matter, the non-linearity effects should balance out [37]. Evidences for the consequent correlation expected between galactic ellipticity and galactic dark mass have been found [6]. We thus have a natural explanation for dark matter and dark energy, which according to the method developed and tested in this article, agrees with galactic and cluster dynamics. With the most probable phase-spaces for WIMPS and axions dark matter candidates now ruled out, it is important to consider the present explanation for the nature of Dark Matter and Dark Energy.