# Predicting Planck Scale and Newtonian Constant from a Yang-Mills Gauge Theory: 1- and 2-Loop Estimates

Authors: Rodrigo F. Sobreiro; Anderson A. Tomaz
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

Recently, a model for an emergent gravity based onSO(5) Yang-Mills action in Euclidian 4-dimensional spacetime was proposed. In this work we provide some 1- and 2-loop computations and show that the model can accommodate suitable predicting values for the Newtonian constant. Moreover, it is shown that the typical scale of the expected transition between the quantum and the geometrodynamical theory is consistent with Planck scale. We also provide a discussion on the cosmological constant problem.


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## Body

## 1. Introduction

Quantization of the gravitational field is one of most import problems in physics since the beginning of the 20th century. The long pursuit of a theory of quantum gravity has generated a variety of theoretical proposals to describe the quantum sector of gravity; see, for instance, Loop Quantum Gravity [1, 2], Higher Derivatives Quantum Gravity [3, 4], Causal Sets [5], Causal Dynamical Triangulations [6], String Theory [7, 8], Asymptotic Safety [9, 10], Emergent Gravities [4, 11], Hořava-Lifshitz Gravity [12], the Nojiri-Odintsov-Hořava-Lifshitz Instability Free Gravity [13, 14], Topological Gauge Theories [15], and so on. Each one of these theories carries its own set of advantages and disadvantages. On the other hand, gauge theories are relentless in describing the high energy regime of particle physics [16–18]. Hence, one can question if gravity could also be described by a gauge theory in its high energy sector. In fact, since the seminal papers [19–21] about gauge theoretical descriptions of gravity, it is known that gravity can be, at least, dressed as a gauge theory for the local isometries of spacetime. See also [22, 23]. Although consistent with general relativity, these models also have problems with its quantization.In [24] it was proposed an induced gravity model from a pure Yang-Mills theory based on de Sitter-type groups. In this model, gravity emerges as an effective phenomenon originated by a genuine Yang-Mills action in flat space. The transition between the quantum gauge sector and the classical geometrical sector is mediated by a mass parameter, identified with the Gribov parameter [25–43]. The combination of the running of this parameter and the running of the coupling parameter would provide a good scenario for an Inonu-Wigner contraction [44] of the gauge group, deforming it to a Poincaré-type group. Because the original action is not invariant under the resulting group, the model actually suffers a dynamical symmetry breaking to Lorentz-type groups. At this point, with the help of the mass parameter, the gauge degrees of freedom are identified with geometrical objects, namely, the vielbein and spin connection. At the same time the mass parameter and the coupling parameter are combined to generate the gravity Newtonian constant and an emergent gravity is realized. Moreover, a cosmological constant inherent to the model is also generated. See also [45, 46] for details.The aim of the present work is to provide estimates for the emergent parameters of the model above discussed by applying the usual apparatus of quantum field theory (QFT). We concentrate our efforts on 1- and 2-loop computations. In particular, from the explicit expressions of the running coupling and the Gribov parameters, we are able to fit the actual value of Newtonian constant and to obtain a renormalization group cut-off very close to the Planck scale, as expected to be the transition scale from quantum to classical gravity.The cosmological constant is an essential point, which can be related to the accelerated expansion of the Universe. Observational data and quantum field theory prediction for the cosmological constant strongly disagree in numerical values [18, 47]. Following [48, 49], we can expect that the cosmological parameter generated by the model should combine with the value found in theoretical calculations by quantum field theory in a way that the effective final value fits the observational data. It is worth mentioning that the cosmological parameter generated by the model is related to the Gribov parameter [24]. In [50], a preliminary estimative for these running parameters at 1-loop approximation was scratched. From this reasonable starting, we develop here a refinement on early predictions and we show a numerical improvement at 1-loop calculations. Further, we improve the techniques up to 2-loop estimates. Hence, we present a best estimative for the Gribov parameter in order to fit the model with a suitable emergent gravity.This work is organized as follows. In Section2 we resume some concepts and ideas about our effective gravity model. In Section 3, we present the first results that we obtained for 1-loop estimates. In Section 4, we present the main calculations results for running parameters at 2 loops are performed. In Section 5, we discuss shortly our results and some perspectives will be cast.


## 2. Effective Gravity from a Gauge Theory

In [24], a quantum gravity theory was constructed based on an analogy with quantum chromodynamics; see also [45, 46]. In this section we will briefly discuss the main ideas, definitions, and conventions behind this model1.The starting action is the Yang-Mills action:(1)SYM=12∫FAB∗FBA,where FAB are the field strength 2-form and F=dY+κYY, where d is the exterior derivative, κ is the coupling parameter, and Y is the gauge connection 1-form, that is, the fundamental field in the adjoint representation. The Hodge dual operator in 4-dimensional Euclidian spacetime is denoted by ∗. The action equation (1) is invariant under SO(5) gauge transformations, Y↦U-11/κd+YU, with U∈SO(5). The infinitesimal version of the gauge transformation is(2)Y⟼Y+∇α,where ∇=d+κY is the full covariant derivative and α is the infinitesimal gauge parameter.It is possible to decompose the gauge group according toSO(5)=SO(4)⊗S(4), where SO(4) is the stability group S(4) is the symmetric coset space. Thus, defining J5a≡Ja, the gauge field is also decomposed:(3)Y=YABJAB=AabJab+θaJa,where capital Latin indices A,B,… run as {5,0,1,2,3} and the small Latin indices a,b,… vary as {0,1,2,3}. The decomposed field strength reads(4)F=FABJAB=Ωab-κ4θaθbJab+KaJa,where Ωab=dAab+κAacAcb   and  Ka=dθa+κAabθb. Thus, it is a simple task to find that the Yang-Mills action equation (1) can be rewritten as(5)SYM=12∫Ωab∗Ωab+12Ka∗Ka-κ2Ωab∗θaθb+κ216θaθb∗θaθb.Before we advance to the next stage of the model, let us quickly point out some important aspects of Yang-Mills theories and their analogy with a possible quantum gravity model. To start with, Yang-Mills theories present two very important properties, namely, renormalizability and asymptotic freedom [16]. The Yang-Mills action is, in fact, renormalizable, at least to all orders in perturbation theory [51] which means that it is stable at quantum level. In this context, the so-called BRST symmetry has a fundamental hole. Asymptotic freedom [52, 53], on the other hand, means that, at high energies, the coupling parameter is very small and we can use perturbation theory in our favor. However, as the energy decreases, the coupling parameter increases and the theory becomes highly nonperturbative. In this regime, the so-called Gribov ambiguities problem takes place [25, 26]. Essentially, the gauge fixing2 is not strong enough to eliminate all spurious degrees of freedom from the Faddeev-Popov path integral; a residual gauge symmetry survives the Faddeev-Popov procedure. The elimination of the Gribov ambiguities is not entirely understood; however, it is known that a mass parameter is required and a soft BRST symmetry breaking associated with this parameter appears; see, for instance, [30–32, 34, 37, 54, 55]. This parameter is known as Gribov parameter γ, and it is fixed through minimization of the quantum action, δΣ/δγ2=0, the so-called gap equation. The action that describes the improved theory (free of infinitesimal ambiguities) is known as Gribov-Zwanziger action [54] and has a more refined version [27, 34] by taking into account few-dimension two operators and their condensation effects.It is clear that the fieldθ has the same degrees of freedom that a soldering forms in spacetime manifold (the vielbein). However, the field θ carries UV dimension 1 while the vielbein is dimensionless. The presence of a mass scale is then very important to identify the field θ with an effective soldering form. We will show in the next sections that the Gribov parameter is a very good candidate for this purpose. The next step is to perform the rescalings(6)A⟶1κA,θ⟶γκθ,at the action equation (5), achieving(7)S=12κ2∫Ω¯ab∗Ω¯ab+γ22K¯a∗K¯a-γ22Ω¯ab∗θaθb+γ416θaθb∗θaθb,where Ω¯ab=dAab+AacAcb, K¯a=Dθa, and the covariant derivative is now D=d+A.The transition from the action (7) to a gravity action is performed by studying the running behaviour of the quantity γ/κ. It is expected [24] that this quantity vanishes for a specific energy scale. This property induces an Inonu-Wigner contraction SO(5)↦ISO(4) [44]. However, since the action (7) is not invariant under ISO(4) gauge transformations, the theory actually suffers a symmetry breaking to the stability group SO(4). The broken theory is ready to be rewritten as a gravity theory. The map (see [24]) is a simple identification of the gauge fields with geometric effective entities according to δaaδbbAab=ωab and δaaθa=ea. Where the indices {a,b,c,…} belong to the tangent space of the effective deformed spacetime, ω is the spin connection 1-form and e is the vielbein 1-form. Thus, with the extra parametric identifications(8)γ2=κ24πG=4Λ23,where G is the Newtonian constant and Λ2 is the renormalized cosmological constant3. Hence, the action equation (7) generates the following effective gravity action:(9)SGrav=116πG∫32Λ2Rab⋆Rab-12ϵabcdRabeced+Ta⋆Ta+Λ212ϵabcdeaebeced,where Rab=dωab+ωacωcb and Ta=dea+ωabeb are, respectively, the curvature and torsion 2-forms. The symbol ⋆ stands for the Hodge dual operator in M4 (the deformed spacetime).


## 3. Running Parameters and 1-Loop Estimates

Now4, we return to the original action (1) and we take only its quadratic part5, considering also the Gribov-Zwanziger quadratic term6 [27],(10)Squad=∫d4x14∂μYνA-∂νYμA2+12α∂μYμA2+φ¯μAB∂2φμAB+-λ2κfABCYμAφμBC+fABCYμAφ¯μBC-λ4dNN-12,where φμAB,φ¯μAB is a pair of complex conjugate bosonic fields and λ is, essentially, the Gribov parameter. Here we will use N=5 since we are building a Yang-Mills for the SO(5) group. In the future we will employ this value, but, for now, we continue using N in general. The parameter α is the gauge parameter associated with the gauge fixing. Herein, the limit α→0 must be employed in order to enforce the Landau gauge condition; that is, ∂μYμA=0. The choice of the gauge is, in principle, arbitrary as long as the gauge is renormalizable. However, most of the developments in the Gribov problem were made in the Landau gauge [25, 27, 30, 34]. Moreover, the Landau gauge is a very simple gauge to work with. Nevertheless, there are recent evidences that the Gribov parameter is a gauge invariant parameter [40, 41, 43, 56, 57], a very welcome feature for the model.At 1-loop, the effective action7 is defined through(11)e-Γ1=∫DΦe-Squad,which, in d dimensions, yields(12)Γ1=-λ4dNN-12+d-12NN-12∫ddp2πdln⁡p4+2Nκ2λ4.To control the divergences of the quantum action we employ the MS¯ renormalization scheme to obtain(13)Γr1=-λ4dNN-12-d-132π2NN-12Nκ2λ4ln⁡2Nκ2λ4μ4-83,where(14)γ4≡2κ2λ4is a more convenient mass parameter. At first sight, this choice is a mere algebraicansatz to simplify future computations with this parameter. In Appendix B we demonstrate why this choice is actually better than λ for our purposes. Thus, for d=4, (13) turns to(15)Γr1=-γ42k24NN-12-332π2NN-12Nγ42lnNγ4μ4-83.Following the Gribov-Zwanziger prescription [28], the Gribov parameter can be determined by minimizing the quantum action; that is, ∂Γr(1)/∂γ2=0. The result is(16)Nκ216π258-38lnNγ4μ4=1.Or, equivalently,(17)γ2=e5/6Nμ2e-4/316π2/Nκ2.Moreover, the 1-loop coupling parameter is found to be [52](18)Nκ216π2=111/3ln⁡μ2/Λ¯2,where Λ¯ is the renormalization group cut-off. By inserting (18) into (17) for N=5 we find(19)γ2=e5/65Λ¯2μ2Λ¯2-35/9.Thus, the higher the energy scale is, the smaller the Gribov parameter would be. This behaviour is plotted in Figure 1.Figure 1
The running of the Gribov parameter as function of the energy scale squared. The energy squaredμ2 is in units of Λ¯2 and the Gribov parameter is normalized in units of (e5/6/5)Λ¯2.As we have mentioned in Section2, the ratio between the two quantum parameters, after we combine (18) and (19), namely,(20)γ2κ2=αΛ¯2μ2Λ¯2-35/9ln⁡μ2Λ¯2,with α=55e5/6/48π25, is crucial for the present model. The behaviour of this ratio is illustrated in Figure 2. It is clear that the expected behaviour γ2/κ2→0 is attained at μ2=Λ¯2.Figure 2
The running of the ratioγ2/κ2 as function of energy scale squared. The energy scale μ2 is in units of Λ¯2 and the ratio is in units of αΛ¯2.The simple inversion of (20) gives(21)κ2γ2=1αΛ¯2μ2Λ¯235/91ln⁡μ2/Λ¯2,which shows the running behaviour of the ratio κ2/γ2 which is displayed in Figure 3. We remark that there is a discontinuity (Landau pole) at μ=Λ¯. We interpret this discontinuity as an indication of the transition between the quantum and classical regimes of the model. For μ<Λ¯ we expect a geometrical regime while for μ>Λ¯, the theory is at the quantum region.Figure 3
The ratioκ2/γ2 as a function of the energy scale squared. The energy scale is in units of Λ¯2 and the ratio is in units of 1/(αΛ¯2).To estimate the Newtonian constant and the renormalization group cut-off we emphasize that we are not assigning any running behaviour to the Newtonian constant. According to (8), our aim consists in identifying the ratio γ2/κ2 with G only after an energy scale is chosen. So the Newtonian constant is fixed as an effective quantity.It is also important to realize that the deep infrared behaviour of Figures1, 2, and 3 does not reproduce the expected behaviour at zero momenta, as known by QCD lattice simulations [58–63]. In the deep infrared regime the coupling parameter κ goes to a finite value, that is, an infrared fixed point. However, this extreme behaviour is not relevant for the purposes of the present work.


### 3.1. Numerical Estimates at 1-Loop

We first follow the procedure performed in [50] where the strategy was not to solve the gap equation by fixing Λ¯ and μ, which is the traditional way, but to fix the Newtonian constant and find if this is a consistent solution. Nevertheless, for the sake of consistency, we need a coupling constant as small as possible. Furthermore, we must have μ2>Λ¯2. Accordingly, we possess a certain range to work with; namely,(22)0<Nκ216π2<1,0<lnμ2Λ¯2<1.Let us take, for instance, μ2=2Λ¯2 in (21), which provides ln⁡(μ2/Λ¯2)=0.6931, satisfying Ineq. (22). One way to obtain the scale Λ¯ is setting the Newtonian constant to its experimental value; that is, G=6.707×10-33TeV-2 in (21), providing(23)Λ¯2≈2.122×1033TeV2.This result allows us to estimate the renormalized cosmological constant. Combining (8), (19), and (23) we obtain(24)Λ2≈1.106×1032TeV2.We notice that the cut-off value (23) is just right above the Planck scale, given by Ep2=1.491×1032TeV2.


#### 3.1.1. Methods of Enhancement at 1 Loop

The main goal here is to calculate the best values forNκ2/16π2 and ln⁡(μ2/Λ¯2) in accordance with Ineq. (22). To handle this task we apply three methods, labelled by M1, M2, and M3, as follows.M 1 : Taylor Series Method. Let us rewrite (18) as(25)1a=113ln⁡b,where(26)a=Nκ216π2,b=μ2Λ¯2,for mere simplification. Next, we expand the right hand side of (25) as a Taylor series at the critical point μ=Λ¯, that is, b=1, as follows:(27)ln⁡b=∑n=1∞1n-1n-1b-1n,with 0<ln⁡b<1 as stated by Ineq. (22). We investigate the series (27) under two perspectives as follows.Perspective (i): The Endpoint Extremum. The series expansion of ln⁡(b) has radius of convergence equal to 1. Precisely, the alternating series test ensures that the series does not converge at b=0 and converges at b=2. Hence, the series is convergent for 0<b⩽2. Therefore, the endpoint extremum occurs at b=2 which happens also to be a global maximum. A curious fact about that series occurs when we truncate the series expansion at even nth order: any of these truncations have a maximum at b=2. Again such maximum is a global one and it happens at the endpoint. From (25), it is clear that, by fixing a global maximum for ln⁡(b), a minimum value for Nκ2/16π2 is set. Thus, with b=2, we obtain ln⁡(μ2/Λ¯2)=0.6931 and Nκ2/16π2=0.3935, which are both in accordance with the intervals described in Ineq. (22).Perspective (ii): A Bound on the Taylor Series for ln ⁡ ( b ). At this point, we are looking for a certain bound for the series expansion (27). Hence, since a<1, we have(28)lnb>311⟺∑n=1∞1n-1n-1b-1n>311.In this range we solve the Ineq. (28) for several n values. We notice that the choices of b values are restricted to 1.314<b<bsup while n is even and where bsup values decrease while even n values increase. The bsup values are displayed in Table 1.Table 1
The superior bound forb range using only even values for n.
n b s u p 2 2.674 4 2.476 8 2.305 10 2.261 20 2.158 50 2.079 100 2.046 1000 2.007 5000 2.002 10000 2.000Still looking at Table1, for instance, n=8⇒bsup≈2.305, n=10⇒bsup≈2.261, and—as actually we would expect—n→∞⇒bsup→2.000. However, if n is odd we obtain for all intervals b>1.313, of course, and no upper bound. Hence, we have a confirmation of our first choice for ln⁡(μ2/Λ¯2) given by μ2=2Λ¯2.Besides the best choice that we can perform, we still have freedom to choose any value consistent with Ineq. (22). If we pick, for instance, a=0.4300, we have b=1.886⇒ln⁡(b)≡ln⁡(μ2Λ¯2)≈0.6342, which provides, from (21) and (8),(29)Λ¯2≈1.845×1033TeV2,(30)Λ2≈1.208×1032TeV2.These results can be interpreted as a numerical verification of the values equations (23) and (24) due to the fact that their order of magnitude are maintained. In this sense we confirm the first insight presented in [50].M 2 : Equilibrium Value Method. Now we use a simple method of enhancement which we calledequilibrium value between two functions at certain point. First, in order to simplify, we take (18) as(31)fκ2hμ2,Λ¯2=311,where(32)fκ2=Nκ216π2,hμ2,Λ¯2=ln⁡μ2Λ¯2.To obtain small values for h(μ2,Λ¯2) and f(κ2), we made an equilibrium choice; that is, h(μ2,Λ¯2)=f(κ2). Consequently, it provides(33)hμ2,Λ¯2=3111/2⟹ln⁡μ2Λ¯2≈0.5222⟹μ2Λ¯2≈1.686.Thus, from (21) and (8), we find(34)Λ¯2≈1.449×1033TeV2,(35)Λ2≈1.468×1032TeV2.We conclude that (34) and (35) do not show any significant improvement with respect to (23) and (24).M 3 : Method by Geometrical Series. Here we employ a geometrical series to treat the logarithm that will be used. Due to Ineq. (22), we can treat the logarithm in (18) as a geometrical series. First, we define(36)r=1-ln⁡μ2Λ¯2.Hence, we can use8(37)11-r=∑n=0∞rnin (18), providing(38)Nκ216π2=31111-r=311∑n=0∞rn.Second, we use Ineq. (22) and (38) to write(39)∑n=0∞rn<113.Now, we test several truncations of expression equation (39) to deal with an nth-degree polynomial inequality. Such procedure permits us to find r∈(0,0.7273) as an optimum valid range. To clarify this point, for instance, we mount Table 2 displaying the evolution of this range, which directly determines the value of the logarithm.Table 2
The superior boundrsup for the range of values for r as a function of the of nth-degree polynomial.
n r s u p 5 0.7974 8 0.7470 10 0.7367 20 0.7276 30 0.7273 40 0.7273 100 0.7273 1000 0.7273We notice thatn>30 does not bring any significant improvement for the superior bound of r. In this way, we choose r≈0.7273 as an optimal extreme valid value, which implies that ln⁡(μ2/Λ¯2)≈0.2727 and Nκ2/16π2≈0.3803. With these values and using (21) and (8) we find the following results:(40)Λ¯2≈1.052×1032TeV2,Λ2≈2.810×1032TeV2.Then, Λ¯2 decreases by one order of magnitude when compared to (23), (29), and (34). It is straightforward to see how these values can be obtained directly from (38). We stress out that the superior bound for n<30 leaves us with an invalid range for Nκ2/16π2.For the other extreme we chooser=1.000×10-4, providing ln⁡(μ2/Λ¯2)≈0.9999 and Nκ2/16π2≈0.2728. We use these values to find(41)Λ¯2≈4.851×1033TeV2,(42)Λ2≈7.666×1031TeV2.In this case a better value for the renormalized cosmological constant is found. However, the renormalization group cut-off is the worst found until this point. To summarize all results that we found in each method we built Table 3.Table 3
The cut-off and renormalized cosmological constant obtained in each method. The columnI lists our initial estimates. The other columns M1, M2, M3a, and M3b are related to the values obtained by Taylor series, equilibrium value, and geometric series, respectively. The column Pr exhibits the physical predictions, that is, the Planck energy squared and the absolute value predicted by the quantum field theory for the cosmological constant [47].
I M 1 M 2 M 3 a M 3 b Pr Λ ¯ 2 (TeV2) 2.122 × 1 0 33 1.845 × 1 0 33 1.449 × 1 0 33 1.052 × 1 0 32 4.851 × 1 0 33 1.491 × 1 0 32 Λ 2  (TeV2) 1.106 × 1 0 32 1.208 × 1 0 32 1.468 × 1 0 32 2.810 × 1 0 32 7.666 × 1 0 31 3.710 × 1 0 28Comparing the numerical values for the cut-off and the renormalized cosmological constant which were obtained through the three methodsM1, M2, and M3 and listed in Table 3, we observe that the orders of magnitude of those results are almost unchanged. A unique exception occurs to the cut-off in the column M3b, which is caused by the extreme high value to the logarithm ln⁡(μ2/Λ¯2).


#### 3.1.2. FixingΛ¯ as the Planck Energy

We introduce here a different path to find values for the Newtonian constant and the renormalized cosmological constant. In this manner we made all slightly different since we fit the cut-offΛ¯2 equal to energy Planck; that is, Λ¯2=Ep2=1.491×1016TeV. Previously, in Section 3.1, we found an optimum logarithm to fix the cut-off and the renormalized cosmological constant. With the help of fixed logarithms and (21) we compute the Newtonian constant Gp for each method as displayed in Table 4.Table 4
Newtonian constant and cosmological constant values based on the logarithm computed in each method in Section3.1 with the cut-off equal to the Planck energy.
I M 1 M 2 M 3 a M 3 b G p  (TeV-2) 9.551 × 1 0 - 32 8.301 × 1 0 - 32 6.521 × 1 0 - 32 5.254 × 1 0 - 32 2.183 × 1 0 - 31 Λ p  (TeV) 7.766 × 1 0 30 9.765 × 1 0 30 1.510 × 1 0 31 6.271 × 1 0 31 2.355 × 1 0 30We observe that all values forGp are in 1 order of magnitude above G. After confrontation with the values presented in Table 2 we notice a better estimate using method M3b, that is, while we apply the logarithm obtained with the geometrical series for κ2. The method M3a ensures the best value for G=6.707×10-33TeV2. As a consequence, the price to pay is a renormalized cosmological constant with a higher value than one encountered in the method M3b. However, the order of magnitude is the same while we compare the values for Λ2 found through the methods M3a and M3b.


## 3.1. Numerical Estimates at 1-Loop

We first follow the procedure performed in [50] where the strategy was not to solve the gap equation by fixing Λ¯ and μ, which is the traditional way, but to fix the Newtonian constant and find if this is a consistent solution. Nevertheless, for the sake of consistency, we need a coupling constant as small as possible. Furthermore, we must have μ2>Λ¯2. Accordingly, we possess a certain range to work with; namely,(22)0<Nκ216π2<1,0<lnμ2Λ¯2<1.Let us take, for instance, μ2=2Λ¯2 in (21), which provides ln⁡(μ2/Λ¯2)=0.6931, satisfying Ineq. (22). One way to obtain the scale Λ¯ is setting the Newtonian constant to its experimental value; that is, G=6.707×10-33TeV-2 in (21), providing(23)Λ¯2≈2.122×1033TeV2.This result allows us to estimate the renormalized cosmological constant. Combining (8), (19), and (23) we obtain(24)Λ2≈1.106×1032TeV2.We notice that the cut-off value (23) is just right above the Planck scale, given by Ep2=1.491×1032TeV2.


### 3.1.1. Methods of Enhancement at 1 Loop

The main goal here is to calculate the best values forNκ2/16π2 and ln⁡(μ2/Λ¯2) in accordance with Ineq. (22). To handle this task we apply three methods, labelled by M1, M2, and M3, as follows.M 1 : Taylor Series Method. Let us rewrite (18) as(25)1a=113ln⁡b,where(26)a=Nκ216π2,b=μ2Λ¯2,for mere simplification. Next, we expand the right hand side of (25) as a Taylor series at the critical point μ=Λ¯, that is, b=1, as follows:(27)ln⁡b=∑n=1∞1n-1n-1b-1n,with 0<ln⁡b<1 as stated by Ineq. (22). We investigate the series (27) under two perspectives as follows.Perspective (i): The Endpoint Extremum. The series expansion of ln⁡(b) has radius of convergence equal to 1. Precisely, the alternating series test ensures that the series does not converge at b=0 and converges at b=2. Hence, the series is convergent for 0<b⩽2. Therefore, the endpoint extremum occurs at b=2 which happens also to be a global maximum. A curious fact about that series occurs when we truncate the series expansion at even nth order: any of these truncations have a maximum at b=2. Again such maximum is a global one and it happens at the endpoint. From (25), it is clear that, by fixing a global maximum for ln⁡(b), a minimum value for Nκ2/16π2 is set. Thus, with b=2, we obtain ln⁡(μ2/Λ¯2)=0.6931 and Nκ2/16π2=0.3935, which are both in accordance with the intervals described in Ineq. (22).Perspective (ii): A Bound on the Taylor Series for ln ⁡ ( b ). At this point, we are looking for a certain bound for the series expansion (27). Hence, since a<1, we have(28)lnb>311⟺∑n=1∞1n-1n-1b-1n>311.In this range we solve the Ineq. (28) for several n values. We notice that the choices of b values are restricted to 1.314<b<bsup while n is even and where bsup values decrease while even n values increase. The bsup values are displayed in Table 1.Table 1
The superior bound forb range using only even values for n.
n b s u p 2 2.674 4 2.476 8 2.305 10 2.261 20 2.158 50 2.079 100 2.046 1000 2.007 5000 2.002 10000 2.000Still looking at Table1, for instance, n=8⇒bsup≈2.305, n=10⇒bsup≈2.261, and—as actually we would expect—n→∞⇒bsup→2.000. However, if n is odd we obtain for all intervals b>1.313, of course, and no upper bound. Hence, we have a confirmation of our first choice for ln⁡(μ2/Λ¯2) given by μ2=2Λ¯2.Besides the best choice that we can perform, we still have freedom to choose any value consistent with Ineq. (22). If we pick, for instance, a=0.4300, we have b=1.886⇒ln⁡(b)≡ln⁡(μ2Λ¯2)≈0.6342, which provides, from (21) and (8),(29)Λ¯2≈1.845×1033TeV2,(30)Λ2≈1.208×1032TeV2.These results can be interpreted as a numerical verification of the values equations (23) and (24) due to the fact that their order of magnitude are maintained. In this sense we confirm the first insight presented in [50].M 2 : Equilibrium Value Method. Now we use a simple method of enhancement which we calledequilibrium value between two functions at certain point. First, in order to simplify, we take (18) as(31)fκ2hμ2,Λ¯2=311,where(32)fκ2=Nκ216π2,hμ2,Λ¯2=ln⁡μ2Λ¯2.To obtain small values for h(μ2,Λ¯2) and f(κ2), we made an equilibrium choice; that is, h(μ2,Λ¯2)=f(κ2). Consequently, it provides(33)hμ2,Λ¯2=3111/2⟹ln⁡μ2Λ¯2≈0.5222⟹μ2Λ¯2≈1.686.Thus, from (21) and (8), we find(34)Λ¯2≈1.449×1033TeV2,(35)Λ2≈1.468×1032TeV2.We conclude that (34) and (35) do not show any significant improvement with respect to (23) and (24).M 3 : Method by Geometrical Series. Here we employ a geometrical series to treat the logarithm that will be used. Due to Ineq. (22), we can treat the logarithm in (18) as a geometrical series. First, we define(36)r=1-ln⁡μ2Λ¯2.Hence, we can use8(37)11-r=∑n=0∞rnin (18), providing(38)Nκ216π2=31111-r=311∑n=0∞rn.Second, we use Ineq. (22) and (38) to write(39)∑n=0∞rn<113.Now, we test several truncations of expression equation (39) to deal with an nth-degree polynomial inequality. Such procedure permits us to find r∈(0,0.7273) as an optimum valid range. To clarify this point, for instance, we mount Table 2 displaying the evolution of this range, which directly determines the value of the logarithm.Table 2
The superior boundrsup for the range of values for r as a function of the of nth-degree polynomial.
n r s u p 5 0.7974 8 0.7470 10 0.7367 20 0.7276 30 0.7273 40 0.7273 100 0.7273 1000 0.7273We notice thatn>30 does not bring any significant improvement for the superior bound of r. In this way, we choose r≈0.7273 as an optimal extreme valid value, which implies that ln⁡(μ2/Λ¯2)≈0.2727 and Nκ2/16π2≈0.3803. With these values and using (21) and (8) we find the following results:(40)Λ¯2≈1.052×1032TeV2,Λ2≈2.810×1032TeV2.Then, Λ¯2 decreases by one order of magnitude when compared to (23), (29), and (34). It is straightforward to see how these values can be obtained directly from (38). We stress out that the superior bound for n<30 leaves us with an invalid range for Nκ2/16π2.For the other extreme we chooser=1.000×10-4, providing ln⁡(μ2/Λ¯2)≈0.9999 and Nκ2/16π2≈0.2728. We use these values to find(41)Λ¯2≈4.851×1033TeV2,(42)Λ2≈7.666×1031TeV2.In this case a better value for the renormalized cosmological constant is found. However, the renormalization group cut-off is the worst found until this point. To summarize all results that we found in each method we built Table 3.Table 3
The cut-off and renormalized cosmological constant obtained in each method. The columnI lists our initial estimates. The other columns M1, M2, M3a, and M3b are related to the values obtained by Taylor series, equilibrium value, and geometric series, respectively. The column Pr exhibits the physical predictions, that is, the Planck energy squared and the absolute value predicted by the quantum field theory for the cosmological constant [47].
I M 1 M 2 M 3 a M 3 b Pr Λ ¯ 2 (TeV2) 2.122 × 1 0 33 1.845 × 1 0 33 1.449 × 1 0 33 1.052 × 1 0 32 4.851 × 1 0 33 1.491 × 1 0 32 Λ 2  (TeV2) 1.106 × 1 0 32 1.208 × 1 0 32 1.468 × 1 0 32 2.810 × 1 0 32 7.666 × 1 0 31 3.710 × 1 0 28Comparing the numerical values for the cut-off and the renormalized cosmological constant which were obtained through the three methodsM1, M2, and M3 and listed in Table 3, we observe that the orders of magnitude of those results are almost unchanged. A unique exception occurs to the cut-off in the column M3b, which is caused by the extreme high value to the logarithm ln⁡(μ2/Λ¯2).


### 3.1.2. FixingΛ¯ as the Planck Energy

We introduce here a different path to find values for the Newtonian constant and the renormalized cosmological constant. In this manner we made all slightly different since we fit the cut-offΛ¯2 equal to energy Planck; that is, Λ¯2=Ep2=1.491×1016TeV. Previously, in Section 3.1, we found an optimum logarithm to fix the cut-off and the renormalized cosmological constant. With the help of fixed logarithms and (21) we compute the Newtonian constant Gp for each method as displayed in Table 4.Table 4
Newtonian constant and cosmological constant values based on the logarithm computed in each method in Section3.1 with the cut-off equal to the Planck energy.
I M 1 M 2 M 3 a M 3 b G p  (TeV-2) 9.551 × 1 0 - 32 8.301 × 1 0 - 32 6.521 × 1 0 - 32 5.254 × 1 0 - 32 2.183 × 1 0 - 31 Λ p  (TeV) 7.766 × 1 0 30 9.765 × 1 0 30 1.510 × 1 0 31 6.271 × 1 0 31 2.355 × 1 0 30We observe that all values forGp are in 1 order of magnitude above G. After confrontation with the values presented in Table 2 we notice a better estimate using method M3b, that is, while we apply the logarithm obtained with the geometrical series for κ2. The method M3a ensures the best value for G=6.707×10-33TeV2. As a consequence, the price to pay is a renormalized cosmological constant with a higher value than one encountered in the method M3b. However, the order of magnitude is the same while we compare the values for Λ2 found through the methods M3a and M3b.


## 3.1.1. Methods of Enhancement at 1 Loop

The main goal here is to calculate the best values forNκ2/16π2 and ln⁡(μ2/Λ¯2) in accordance with Ineq. (22). To handle this task we apply three methods, labelled by M1, M2, and M3, as follows.M 1 : Taylor Series Method. Let us rewrite (18) as(25)1a=113ln⁡b,where(26)a=Nκ216π2,b=μ2Λ¯2,for mere simplification. Next, we expand the right hand side of (25) as a Taylor series at the critical point μ=Λ¯, that is, b=1, as follows:(27)ln⁡b=∑n=1∞1n-1n-1b-1n,with 0<ln⁡b<1 as stated by Ineq. (22). We investigate the series (27) under two perspectives as follows.Perspective (i): The Endpoint Extremum. The series expansion of ln⁡(b) has radius of convergence equal to 1. Precisely, the alternating series test ensures that the series does not converge at b=0 and converges at b=2. Hence, the series is convergent for 0<b⩽2. Therefore, the endpoint extremum occurs at b=2 which happens also to be a global maximum. A curious fact about that series occurs when we truncate the series expansion at even nth order: any of these truncations have a maximum at b=2. Again such maximum is a global one and it happens at the endpoint. From (25), it is clear that, by fixing a global maximum for ln⁡(b), a minimum value for Nκ2/16π2 is set. Thus, with b=2, we obtain ln⁡(μ2/Λ¯2)=0.6931 and Nκ2/16π2=0.3935, which are both in accordance with the intervals described in Ineq. (22).Perspective (ii): A Bound on the Taylor Series for ln ⁡ ( b ). At this point, we are looking for a certain bound for the series expansion (27). Hence, since a<1, we have(28)lnb>311⟺∑n=1∞1n-1n-1b-1n>311.In this range we solve the Ineq. (28) for several n values. We notice that the choices of b values are restricted to 1.314<b<bsup while n is even and where bsup values decrease while even n values increase. The bsup values are displayed in Table 1.Table 1
The superior bound forb range using only even values for n.
n b s u p 2 2.674 4 2.476 8 2.305 10 2.261 20 2.158 50 2.079 100 2.046 1000 2.007 5000 2.002 10000 2.000Still looking at Table1, for instance, n=8⇒bsup≈2.305, n=10⇒bsup≈2.261, and—as actually we would expect—n→∞⇒bsup→2.000. However, if n is odd we obtain for all intervals b>1.313, of course, and no upper bound. Hence, we have a confirmation of our first choice for ln⁡(μ2/Λ¯2) given by μ2=2Λ¯2.Besides the best choice that we can perform, we still have freedom to choose any value consistent with Ineq. (22). If we pick, for instance, a=0.4300, we have b=1.886⇒ln⁡(b)≡ln⁡(μ2Λ¯2)≈0.6342, which provides, from (21) and (8),(29)Λ¯2≈1.845×1033TeV2,(30)Λ2≈1.208×1032TeV2.These results can be interpreted as a numerical verification of the values equations (23) and (24) due to the fact that their order of magnitude are maintained. In this sense we confirm the first insight presented in [50].M 2 : Equilibrium Value Method. Now we use a simple method of enhancement which we calledequilibrium value between two functions at certain point. First, in order to simplify, we take (18) as(31)fκ2hμ2,Λ¯2=311,where(32)fκ2=Nκ216π2,hμ2,Λ¯2=ln⁡μ2Λ¯2.To obtain small values for h(μ2,Λ¯2) and f(κ2), we made an equilibrium choice; that is, h(μ2,Λ¯2)=f(κ2). Consequently, it provides(33)hμ2,Λ¯2=3111/2⟹ln⁡μ2Λ¯2≈0.5222⟹μ2Λ¯2≈1.686.Thus, from (21) and (8), we find(34)Λ¯2≈1.449×1033TeV2,(35)Λ2≈1.468×1032TeV2.We conclude that (34) and (35) do not show any significant improvement with respect to (23) and (24).M 3 : Method by Geometrical Series. Here we employ a geometrical series to treat the logarithm that will be used. Due to Ineq. (22), we can treat the logarithm in (18) as a geometrical series. First, we define(36)r=1-ln⁡μ2Λ¯2.Hence, we can use8(37)11-r=∑n=0∞rnin (18), providing(38)Nκ216π2=31111-r=311∑n=0∞rn.Second, we use Ineq. (22) and (38) to write(39)∑n=0∞rn<113.Now, we test several truncations of expression equation (39) to deal with an nth-degree polynomial inequality. Such procedure permits us to find r∈(0,0.7273) as an optimum valid range. To clarify this point, for instance, we mount Table 2 displaying the evolution of this range, which directly determines the value of the logarithm.Table 2
The superior boundrsup for the range of values for r as a function of the of nth-degree polynomial.
n r s u p 5 0.7974 8 0.7470 10 0.7367 20 0.7276 30 0.7273 40 0.7273 100 0.7273 1000 0.7273We notice thatn>30 does not bring any significant improvement for the superior bound of r. In this way, we choose r≈0.7273 as an optimal extreme valid value, which implies that ln⁡(μ2/Λ¯2)≈0.2727 and Nκ2/16π2≈0.3803. With these values and using (21) and (8) we find the following results:(40)Λ¯2≈1.052×1032TeV2,Λ2≈2.810×1032TeV2.Then, Λ¯2 decreases by one order of magnitude when compared to (23), (29), and (34). It is straightforward to see how these values can be obtained directly from (38). We stress out that the superior bound for n<30 leaves us with an invalid range for Nκ2/16π2.For the other extreme we chooser=1.000×10-4, providing ln⁡(μ2/Λ¯2)≈0.9999 and Nκ2/16π2≈0.2728. We use these values to find(41)Λ¯2≈4.851×1033TeV2,(42)Λ2≈7.666×1031TeV2.In this case a better value for the renormalized cosmological constant is found. However, the renormalization group cut-off is the worst found until this point. To summarize all results that we found in each method we built Table 3.Table 3
The cut-off and renormalized cosmological constant obtained in each method. The columnI lists our initial estimates. The other columns M1, M2, M3a, and M3b are related to the values obtained by Taylor series, equilibrium value, and geometric series, respectively. The column Pr exhibits the physical predictions, that is, the Planck energy squared and the absolute value predicted by the quantum field theory for the cosmological constant [47].
I M 1 M 2 M 3 a M 3 b Pr Λ ¯ 2 (TeV2) 2.122 × 1 0 33 1.845 × 1 0 33 1.449 × 1 0 33 1.052 × 1 0 32 4.851 × 1 0 33 1.491 × 1 0 32 Λ 2  (TeV2) 1.106 × 1 0 32 1.208 × 1 0 32 1.468 × 1 0 32 2.810 × 1 0 32 7.666 × 1 0 31 3.710 × 1 0 28Comparing the numerical values for the cut-off and the renormalized cosmological constant which were obtained through the three methodsM1, M2, and M3 and listed in Table 3, we observe that the orders of magnitude of those results are almost unchanged. A unique exception occurs to the cut-off in the column M3b, which is caused by the extreme high value to the logarithm ln⁡(μ2/Λ¯2).


## 3.1.2. FixingΛ¯ as the Planck Energy

We introduce here a different path to find values for the Newtonian constant and the renormalized cosmological constant. In this manner we made all slightly different since we fit the cut-offΛ¯2 equal to energy Planck; that is, Λ¯2=Ep2=1.491×1016TeV. Previously, in Section 3.1, we found an optimum logarithm to fix the cut-off and the renormalized cosmological constant. With the help of fixed logarithms and (21) we compute the Newtonian constant Gp for each method as displayed in Table 4.Table 4
Newtonian constant and cosmological constant values based on the logarithm computed in each method in Section3.1 with the cut-off equal to the Planck energy.
I M 1 M 2 M 3 a M 3 b G p  (TeV-2) 9.551 × 1 0 - 32 8.301 × 1 0 - 32 6.521 × 1 0 - 32 5.254 × 1 0 - 32 2.183 × 1 0 - 31 Λ p  (TeV) 7.766 × 1 0 30 9.765 × 1 0 30 1.510 × 1 0 31 6.271 × 1 0 31 2.355 × 1 0 30We observe that all values forGp are in 1 order of magnitude above G. After confrontation with the values presented in Table 2 we notice a better estimate using method M3b, that is, while we apply the logarithm obtained with the geometrical series for κ2. The method M3a ensures the best value for G=6.707×10-33TeV2. As a consequence, the price to pay is a renormalized cosmological constant with a higher value than one encountered in the method M3b. However, the order of magnitude is the same while we compare the values for Λ2 found through the methods M3a and M3b.


## 4. Numerical Estimates at 2 Loops

For 2 loops, an explicit analytical computation is a virtually impossible task. To work out this equation, sophisticated algebraic programs were built. For instance,FORM andQGraph programs are frequently used as well as the recent developments about such computational packages [64–69]. In this section we borrow the main results of 2-loop computations from [65].


### 4.1. 2-Loopβ-Function

First, recalling that theβ-function at 2 loops [65, 70] is given by(43)βκ2=-11N3κ216π22-343N2κ216π23,the 2-loop running coupling constant is(44)Nκ216π2=111/3ln⁡μ2/Λ¯2-102121lnln⁡μ2/Λ¯2ln⁡μ2/Λ¯22,where Λ¯ is the cut-off of the energy scale. The evolution of the coupling κ, related to energy scale μ, is displayed in Figure 4.Figure 4
The behaviour of the coupling parameter related to the energy scale.


### 4.2. 2-Loop Gap Equation

From [65], the main result is the 2-loop Gribov gap equation in the MS¯ with massive quarks. Here, we are dealing with a theory without fermions. Hence, from [65], the 2-loop gap equation reduces to the simpler form:(45)1=Nκ216π258-38ln⁡Nγ4μ4+Nκ216π2238931536+82540963π2+29768π2-6548ln⁡Nγ4μ4+35128ln2Nγ4μ4+13720485π2-13174096π2.First of all, we compute the system formed by (44) and (45) to analyze the behaviour of the Gribov parameter related to energy scale. Such procedure gives us the following two functions, where we have now the mass parameter related to logarithm:(46)γm2=15μ2hμ-HμeWmμ,γp2=15μ2hμ-HμeWpμ,where(47)hμ=ln⁡μ2Λ¯2,Hμ=1496105PμQμ,Pμ=hμ33hμ+65,Qμ=11hμ-34ln⁡hμ2,Wmμ=11680QμSμ-2Tμ,Wpμ=11680QμSμ+2Tμ,Sμ=a1h3μ+a2h2μ+a3ln2hμ,Tμ=Qμb1h4μ+b2h3μ-b3h2μ-b4ln2hμ+b5ln⁡hμhμ-b6ln⁡hμh2μ,where a1=255,552, a2=251,680, a3=2,404,480, b1=2,368,796,672, b2=173,775,360, b3=605d0, b4=5780d0, b5=3740d0, b6=537,123,840, d0=221,384+b0, and b0=21-3,487+2,4753+8225π2.The behaviour ofγm2 and γp2 in (46) can be clearly seen in Figures 5 and 6, respectively. The behaviour of γp2 and γm2 indicates uniquely γm2 as the one that has the expected typical running behaviour of a mass parameter in the Gribov-Zwanziger scenario. Thus, we necessarily keep γm2 for the next computations.Figure 5
Gribov parameterγm2 as a function of the energy scale μ2. Both γm2 and μ2 are in units of Λ¯2.Figure 6
Gribov parameterγp2 as a function of the energy scale μ2. Both γp2 and μ2 are in units of Λ¯2.


### 4.3. Methods of Enhancement at 2 Loops

Following similar steps that we have made in Section3.1.1, we are looking for the best logarithm for the sake of better estimates of the renormalized cosmological constant Λ2 and the energy cut-off Λ~2. Before we advance in applying these methods, we refer to the definitions (32). Moreover, we skip the initial choice, as made at 1 loop, μ2/Λ¯2=2, because it provides Nκ2/16π2=1.036, which is outside the acceptable range for the coupling parameter; see Ineq. (22). Hence, we proceed with the methods of enhancement as we have done in Section 3.1.1.M 1 : Taylor Series Method. We are looking for small logarithms through a Taylor expansion of (44), which can be written as(48)fκ=311-135121hμ-1+288121hμ-12-475121hμ-13+12522hμ-14-4602605hμ-15+Ohμ-16,where, for simplicity, the expansion above is displayed up to fifth order. However, we must keep in mind that we can truncate such expansion at any arbitrary order. If we truncate the above expansion of f(κ) at fourth order and consider 0<f(κ)<1, then we find h(μ)>0.6938. Therefore, we obtain 0.6938<h(μ)<1.000. All truncations beyond the fourth order do not imply significant improvements in the inferior limit hinf, in the interval hinf<h(μ)<1, of the intervals for h(μ), since each order of truncation modifies such limit (see Table 5).Table 5
The inferior limithinf for the range of values for h(μ) according to nt, which is the order of truncation of the expansion for f(κ).
n t h i n f 2 0.6340 3 0.6806 4 0.6938 5 0.6983 6 0.6998 7 0.7004 8 0.7006 10 0.7007 50 0.7007 100 0.7007 500 0.7007 1000 0.7007Since we are dealing with a perturbation expansion, we need to get to small values forh(μ). Nevertheless, because of the second term in (44), which is resulting from the contribution at 2 loops in the computation, a choice for the logarithm close to any hinf results in high numerical values of the renormalized cosmological constant Λ2. Hence, the best choice for the logarithm is ln⁡(μ2/Λ¯2)=0.9999. Employing this logarithm value and combining (44), (45), and (8), we obtain(49)Λ¯2≈2.269×1032TeV2,(50)Λ2≈7.665×1031TeV2.The value (49) is very close to the order of magnitude of the Planck energy Ep2. The result (41) is almost better than that we found in Section 3.1.1 at 1-loop approximation. The result (50) certifies the 1-loop result for Λ2.M 2 : Equilibrium Value Method. If we apply f(κ)=h(μ) in (44), we obtain the result h(μ)=0.7599. However, this value, even though it obeys Ineq. (22), it does not provide a real value for the Gribov parameter γm2, according to (45).M 3 : Method by Geometric Series. In this case, we will treat the logarithm as a geometric series. For such aim, we employ once again (36) and (37) by treating r as the ratio of a geometric series. In this way, we combine (36), (37), and (44) to find(51)fκ=311∑n=0ntrn+102121∑n=0ntrn2ln⁡∑n=0ntrn,where nt is the truncation order. Now, we must solve the inequality 0<f(κ)<1 to find the range for valid logarithms. Each nt results in an inequality in the form 0<r<rsup. To clarify this point, the superior limit rsup as a function of nt is displayed in Table 6.Table 6
The superior limitrsup according to nt, which is the order of truncation of the expansion for f(κ).
n t r s u p 3 0.3054 4 0.3011 5 0.2998 6 0.2995 7 0.2994 8 0.2994 9 0.2993 10 0.2993The truncation at fourth order works just as good as the 1-loop case because the values for the superior limit forr do not reveal any significant changes. At first sight, we could work with the interval 0<r<0.2993; however there is another constraint due to the Gribov parameter function γm2, according to (46) and (47). The square root in γm2 only has a real solution if 0.7882<h(μ)<1 or, equivalently, 0<r<0.2117. This interval is actually more restrictive than the 1-loop treatment. Therefore, we are satisfied with the value r=0.2117. Hence, using h(μ)=0.7883 in the equation system formed by (44), (45), and (8), we get(52)Λ¯2≈4.593×1031TeV2,Λ2≈1.879×1032TeV2.On the other hand, for the other extremes, we chooser=1.000×10-4 and the same equation system (see (44), (45), and (8)), we find(53)Λ¯2≈2.269×1032TeV2,Λ2≈7.665×1031TeV2.The values (53) match the first ones that we found using M1.


### 4.4. The Logarithmic Elimination Option

A straight and simple manner to simplify the gap equation (45) consists in setting up all logarithms to zero. For this purpose we can set(54)γ2μ2=15.Thus, (45) can easily be solved, providing Nκ2/16π2≈0.4013 and ln⁡(μ2/Λ¯2)≈0.9067. These results leave us with the following 2-loop values for the energy cut-off and the renormalized cosmological constant:(55)Λ¯2≈1.066×1030TeV2,Λ2≈3.589×1031TeV2.We remark that, at 1 loop, this strategy is not permitted because the value for the coupling parameter is found to be higher than 1; namely,Nκ2/16π2=1.600.


## 4.1. 2-Loopβ-Function

First, recalling that theβ-function at 2 loops [65, 70] is given by(43)βκ2=-11N3κ216π22-343N2κ216π23,the 2-loop running coupling constant is(44)Nκ216π2=111/3ln⁡μ2/Λ¯2-102121lnln⁡μ2/Λ¯2ln⁡μ2/Λ¯22,where Λ¯ is the cut-off of the energy scale. The evolution of the coupling κ, related to energy scale μ, is displayed in Figure 4.Figure 4
The behaviour of the coupling parameter related to the energy scale.


## 4.2. 2-Loop Gap Equation

From [65], the main result is the 2-loop Gribov gap equation in the MS¯ with massive quarks. Here, we are dealing with a theory without fermions. Hence, from [65], the 2-loop gap equation reduces to the simpler form:(45)1=Nκ216π258-38ln⁡Nγ4μ4+Nκ216π2238931536+82540963π2+29768π2-6548ln⁡Nγ4μ4+35128ln2Nγ4μ4+13720485π2-13174096π2.First of all, we compute the system formed by (44) and (45) to analyze the behaviour of the Gribov parameter related to energy scale. Such procedure gives us the following two functions, where we have now the mass parameter related to logarithm:(46)γm2=15μ2hμ-HμeWmμ,γp2=15μ2hμ-HμeWpμ,where(47)hμ=ln⁡μ2Λ¯2,Hμ=1496105PμQμ,Pμ=hμ33hμ+65,Qμ=11hμ-34ln⁡hμ2,Wmμ=11680QμSμ-2Tμ,Wpμ=11680QμSμ+2Tμ,Sμ=a1h3μ+a2h2μ+a3ln2hμ,Tμ=Qμb1h4μ+b2h3μ-b3h2μ-b4ln2hμ+b5ln⁡hμhμ-b6ln⁡hμh2μ,where a1=255,552, a2=251,680, a3=2,404,480, b1=2,368,796,672, b2=173,775,360, b3=605d0, b4=5780d0, b5=3740d0, b6=537,123,840, d0=221,384+b0, and b0=21-3,487+2,4753+8225π2.The behaviour ofγm2 and γp2 in (46) can be clearly seen in Figures 5 and 6, respectively. The behaviour of γp2 and γm2 indicates uniquely γm2 as the one that has the expected typical running behaviour of a mass parameter in the Gribov-Zwanziger scenario. Thus, we necessarily keep γm2 for the next computations.Figure 5
Gribov parameterγm2 as a function of the energy scale μ2. Both γm2 and μ2 are in units of Λ¯2.Figure 6
Gribov parameterγp2 as a function of the energy scale μ2. Both γp2 and μ2 are in units of Λ¯2.


## 4.3. Methods of Enhancement at 2 Loops

Following similar steps that we have made in Section3.1.1, we are looking for the best logarithm for the sake of better estimates of the renormalized cosmological constant Λ2 and the energy cut-off Λ~2. Before we advance in applying these methods, we refer to the definitions (32). Moreover, we skip the initial choice, as made at 1 loop, μ2/Λ¯2=2, because it provides Nκ2/16π2=1.036, which is outside the acceptable range for the coupling parameter; see Ineq. (22). Hence, we proceed with the methods of enhancement as we have done in Section 3.1.1.M 1 : Taylor Series Method. We are looking for small logarithms through a Taylor expansion of (44), which can be written as(48)fκ=311-135121hμ-1+288121hμ-12-475121hμ-13+12522hμ-14-4602605hμ-15+Ohμ-16,where, for simplicity, the expansion above is displayed up to fifth order. However, we must keep in mind that we can truncate such expansion at any arbitrary order. If we truncate the above expansion of f(κ) at fourth order and consider 0<f(κ)<1, then we find h(μ)>0.6938. Therefore, we obtain 0.6938<h(μ)<1.000. All truncations beyond the fourth order do not imply significant improvements in the inferior limit hinf, in the interval hinf<h(μ)<1, of the intervals for h(μ), since each order of truncation modifies such limit (see Table 5).Table 5
The inferior limithinf for the range of values for h(μ) according to nt, which is the order of truncation of the expansion for f(κ).
n t h i n f 2 0.6340 3 0.6806 4 0.6938 5 0.6983 6 0.6998 7 0.7004 8 0.7006 10 0.7007 50 0.7007 100 0.7007 500 0.7007 1000 0.7007Since we are dealing with a perturbation expansion, we need to get to small values forh(μ). Nevertheless, because of the second term in (44), which is resulting from the contribution at 2 loops in the computation, a choice for the logarithm close to any hinf results in high numerical values of the renormalized cosmological constant Λ2. Hence, the best choice for the logarithm is ln⁡(μ2/Λ¯2)=0.9999. Employing this logarithm value and combining (44), (45), and (8), we obtain(49)Λ¯2≈2.269×1032TeV2,(50)Λ2≈7.665×1031TeV2.The value (49) is very close to the order of magnitude of the Planck energy Ep2. The result (41) is almost better than that we found in Section 3.1.1 at 1-loop approximation. The result (50) certifies the 1-loop result for Λ2.M 2 : Equilibrium Value Method. If we apply f(κ)=h(μ) in (44), we obtain the result h(μ)=0.7599. However, this value, even though it obeys Ineq. (22), it does not provide a real value for the Gribov parameter γm2, according to (45).M 3 : Method by Geometric Series. In this case, we will treat the logarithm as a geometric series. For such aim, we employ once again (36) and (37) by treating r as the ratio of a geometric series. In this way, we combine (36), (37), and (44) to find(51)fκ=311∑n=0ntrn+102121∑n=0ntrn2ln⁡∑n=0ntrn,where nt is the truncation order. Now, we must solve the inequality 0<f(κ)<1 to find the range for valid logarithms. Each nt results in an inequality in the form 0<r<rsup. To clarify this point, the superior limit rsup as a function of nt is displayed in Table 6.Table 6
The superior limitrsup according to nt, which is the order of truncation of the expansion for f(κ).
n t r s u p 3 0.3054 4 0.3011 5 0.2998 6 0.2995 7 0.2994 8 0.2994 9 0.2993 10 0.2993The truncation at fourth order works just as good as the 1-loop case because the values for the superior limit forr do not reveal any significant changes. At first sight, we could work with the interval 0<r<0.2993; however there is another constraint due to the Gribov parameter function γm2, according to (46) and (47). The square root in γm2 only has a real solution if 0.7882<h(μ)<1 or, equivalently, 0<r<0.2117. This interval is actually more restrictive than the 1-loop treatment. Therefore, we are satisfied with the value r=0.2117. Hence, using h(μ)=0.7883 in the equation system formed by (44), (45), and (8), we get(52)Λ¯2≈4.593×1031TeV2,Λ2≈1.879×1032TeV2.On the other hand, for the other extremes, we chooser=1.000×10-4 and the same equation system (see (44), (45), and (8)), we find(53)Λ¯2≈2.269×1032TeV2,Λ2≈7.665×1031TeV2.The values (53) match the first ones that we found using M1.


## 4.4. The Logarithmic Elimination Option

A straight and simple manner to simplify the gap equation (45) consists in setting up all logarithms to zero. For this purpose we can set(54)γ2μ2=15.Thus, (45) can easily be solved, providing Nκ2/16π2≈0.4013 and ln⁡(μ2/Λ¯2)≈0.9067. These results leave us with the following 2-loop values for the energy cut-off and the renormalized cosmological constant:(55)Λ¯2≈1.066×1030TeV2,Λ2≈3.589×1031TeV2.We remark that, at 1 loop, this strategy is not permitted because the value for the coupling parameter is found to be higher than 1; namely,Nκ2/16π2=1.600.


## 5. Conclusions

The Gribov mass parameter in our theory is the central point in the development of the induced gravity discussed in [24]. It is responsible for addressing the deformation of the Yang-Mills theory in the infrared regime to a geometrical theory of gravity. In this model, the Gribov parameter and the Yang-Mills coupling constant combine in order to provide the value of Newtonian constant. In the present work we have developed 1- and 2-loop estimates to accommodate reliable values for the prediction of Newtonian constant. Moreover, the renormalization group scale could also be determined. Furthermore, a discussion about the cosmological constant was performed.Our results show that the experimental value of the Newtonian constant is a solution of the theory and that the renormalization group scale always lie around Planck scale, a good feature for a quantum gravity model candidate. We have also improved the estimates in order to attain the best values for these constants, at 1 and 2 loops.Concerning the cosmological constant problem, our model provides an inherent gravitational cosmological constant. Following [48, 49], it would be nice that the sum of such constant with the one predicted by QFT for the Standard Model vacuum [47] would compensate each other and provide a very small value for the observational cosmological constant. However, although very high, our cosmological constant differs from the QFT prediction by a few orders in magnitude.Of course, much has to be investigated. For instance, since the emergence of gravity relies on the Gribov parameter and the soft BRST breaking [31, 32, 36], the Gribov-Zwanziger refinement [34, 42, 43] should also be considered with all the extra mass parameters. These extra masses should also refine the values we have computed in the present work. One interesting feature that should be mentioned of the theory is about the renormalized cosmological constant: first of all, at the quantum level, it is related to the Gribov parameter, so it is a running mass parameter; second, at classical level, its value does not run anymore. Its huge fixed value takes place at the classical sector of the theory. This is very important because it suppresses the quadratic curvature term in (9), at classical level, ensuring the general relativity limit of the theory, at least for a torsionless regime.



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