Advances in High Energy Physics

Volume 2018, Article ID 2767410, 8 pages

https://doi.org/10.1155/2018/2767410

## Inhomogeneous Extra Space as a Tool for the Top-Down Approach

Correspondence should be addressed to Sergey G. Rubin; ur.tsil@niburiegres

Received 28 December 2017; Accepted 6 February 2018; Published 12 March 2018

Academic Editor: Elias C. Vagenas

Copyright © 2018 Sergey G. Rubin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The top-down approach for the 6-dimensional space has been elaborated. The connection between the cosmological constant and the extra space metric has been obtained. The metric can be found with the necessary accuracy. It is shown that descent from high energies to the low ones leads to the quantum corrections which influence weakly the metric of extra space.

#### 1. Introduction

Nowadays it becomes more or less clear that the physical laws are formed at high energies where we may only guess about the Lagrangian structure [1, 2]. It is assumed that the values of observable parameters are the result of the evolution of our Universe started at high energies. Observed low-energy physics depend on parameters and initial conditions which have been formed at high energies [3, 4].

The natural values of the physical parameters are assumed to be quantities of the order of the Planck scale. At the same time, the observed parameter is determined at low energies, and their values are concentrated around electroweak scales and below. The ratio of these two scales is a small parameter, which creates difficulties in constructing the primary theory at high energy. Many attempts have been made to reconcile these two contradictory positions, but the skepticism of the scientific community remains. Quantum corrections only aggravate the situation.

In this article two ideas are attracted to soften the problem. Firstly, the connection between the primary physical parameters and the observable ones is achieved by suitable choice of an extra space metric. Secondly, we determine parameters of Lagrangian at a high-energy scale in the spirit of the effective field theory. In this case, quantum corrections being applied to primary parameters do not spoil the result.

One of the aims of the fundamental physics is to postulate a Lagrangian depending on primary parameters and find them using their connection with observational values. Suppose that one managed to obtain a set of relationshipsbetween primary parameters at an energy scale and the observational parameters (the particle masses, coupling constants, etc.) at low energies. Solving these equations with appropriate precision, one could determine primary parameters at a chosen scale . The implementation of this plan in its entirety is a matter for the future. Nevertheless, an activity in this direction is observed. In the paper [5], warped geometry is used for the solution of the small cosmological constant problem. The hybrid inflation [6] has been developed to avoid the smallness of the inflaton mass. The electron-to-proton mass ratio is discussed in [7]. The seesaw mechanism is usually applied to explain the smallness of neutrino-to-electron mass ratio [8].

The aim of this paper is to establish and analyze only one connectionthat ought to be considered as small but necessary part of a future theory. It has been proved earlier [9, 10] that there exists a set of primary parameters that are responsible for the observable value of the cosmological constant (CC). Here much attention is paid to the problem of quantum corrections.

It is shown that the idea on an extra space existence facilitates connection of high-energy Lagrangian structure and the low-energy one. The observational smallness of the CC is used to find the extra space metric.

As a mathematical tool, we use the effective field theory technique, a well-known method for theoretical investigation of the energy dependence of physical parameters [11]. In this approach, parameters of the Wilson action are fixed at a high-energy scale and the renormalization flow is used to descend to low energies (the top-down approach) [12–15]. As is usually stated, the parameters of the Wilson action already contain quantum corrections caused by field fluctuations with energies between the chosen scale and maximal energy scale, the -dimensional Planck mass in our case. Therefore the natural value of these parameters is that are usually many orders of magnitude greater than the electroweak scale GeV.

The research is based on the multidimensional gravity. The interest in theories is motivated by inflationary scenarios starting with the work of Starobinsky [16]. The guiding principle underlying general relativity is the local invariance under coordinate transformations. We may use any invariant combination of quantities invariant under the general coordinate transformations keeping in mind two issues. Firstly, a theory must restore the Einstein-Hilbert action at low energies. Secondly, any gravitational action including the Einstein-Hilbert one is nonrenormalizable and should be considered as an effective theory.

The simplest extension of the gravitation theory is the one containing a function of the Ricci scalar . In the framework of such extension, many interesting results have been obtained. Some viable models in 4-dim space that satisfies the observable constraints are proposed in [17–19]. Stabilization of extra space as the pure gravitational effect has been studied in [20, 21]. It has been shown recently [9] that the model with the deformed nonuniform extra space is able to reproduce the 4-dim Minkowski metric.

The extra dimensions have now become a widespread tool to obtain new theoretical results [22–25]. The idea of inhomogeneous extra space has been developed in [9, 26, 27] and plays one of the central roles in this research. It influences low-energy physics together with physical parameters of a Lagrangian. At the same time an accidental formation of manifolds with various metrics and topologies may be considered as a source of different universes whose variety is connected with a continuous set of extra space metrics. Entropic mechanism of a metric stabilization is considered in [28]. Stationary extra space metric is the final result of a metric evolution governed by the classical equation of motion, and hence the final stationary metric depends on initial configuration. One could keep in mind an analogy with the black hole mass where the Schwarzschild metric depends on an initial matter distribution. In the framework of the scalar-tensor theory, Weinberg [29] has proved that the firm fine-tuning of initial parameters of a Lagrangian is necessary if metric and scalar fields are constant in space-time. The latter means that the solution of the problem should be sought in the class of nonuniform configurations of metrics and fields. Metrics of the deformed extra space discussed in this paper belong to this class.

The plan of the paper consists of three steps. In Section 2, we consider a scalar field as the source of quantum corrections to the Lambda term. It will be shown that they are small relative to primary parameter value at high energies where physical parameters are fixed initially. The appropriate metric of inhomogeneous extra space is discussed in Section 3. In Section 4, the scalar field quantum corrections on the inhomogeneous background are analyzed.

#### 2. Quantum Corrections Caused by the Scalar Field: Minkowski Space

The general goal of the top-down approach is to fix primary parameters by comparison with the experimental data at low energies. According to the effective field theory, quantum fluctuations with energies in the interval , had been involved in the parameters of action. It means that their natural values are of the order of the Planck scale. Primary parameters are assumed to be formed at high energies, in our case.

Descending to the electroweak scale or lower where all physical parameters are measured is the necessary step. This process is accompanied by an alternation of physical parameters due to quantum fluctuations. The aim of this section is to demonstrate that quantum corrections are small. The toy model for the scalar field is considered to study the quantum corrections which are the result of integrating out the quick modes in the energy interval . The scalar field action is written in the standard formacting in the ordinary 4-dimensional Minkowski space. In this section, we study quantum corrections to the parameter which may be considered as the primary cosmological constant at the scale .

The generating functional for action (3) at the scale plays the central role in the effective field theory approach. Here and in the following a subscript and superscript indicate an interval of momentum in the Euclidean space that are taken into account. Thus, functional (5) is the result of integrating out quick modes . The -dimensional Planck mass is considered as the maximal energy scale in the rest of the article.

Let us integrate out modes with Euclidean momentum in the interval in generating functional (5) and shift down to the electroweak scale GeV. To this end, one should decompose the scalar field as follows:Here quick and slow modes in 4-dim Euclidean space arecorrespondingly.

Substituting (6) into (5) gives the generating functional in the formwhereHere, we have taken into account orthogonality of and .

The way to integrate out the field from (8), provided that the coupling constant is small, is well known (see, e.g., textbook [11]). Consider generating functionalas a functional of an external current . Then the result of integrating out quick modes is as follows:After the Wick rotation, quantum correction acquires the formand can be easily calculated. As a result, the contribution (14) to the bare cosmological constant is small because of the inequality .

The integral in (14) is usually estimated keeping in mind that a cutoff scale is much greater than the Lagrangian parameters; that is, the inequality holds; see recent discussion in [30]. First estimation has been presented by Zeldovich in 1967 [31], where the proton mass was used as the maximum energy scale. In our case, the situation is different. Indeed, the scale is chosen such that while a natural value of the effective parameter ; see (4). In both cases, the corrections are proportional to the fourth power of the energy scale . This is not surprising, since the chosen scale is still much larger than the electroweak scale .

Estimation (15) for the quantum corrections at the scale GeV givesThis value is negligibly small as compared to the primary (bare) value of the termand is huge as compared to the observational value. The latter is not a great problem if our intention is to find values of the physical parameters at a high-energy scale. Indeed, quantum corrections must be compared to primary, physical parameters rather than the observational ones.

It is interesting to check for future studies that correction to the mass also contains the small parameter and hence is small. To verify this, let us estimate quantum corrections produced by terms proportional to . The latter can be extracted from (9) and (10) and has the formReceipt (11) with instead of leads to the quantum correction to the potential in the first multiplier in expression (8). Herefor . This means that the quantum correction to is small due to the last inequality in (4) and the choice of energy scale . Hence, the quantum correction to the mass is also small.

We can conclude that the quantum corrections are small in comparison with the primary physical parameter. The mass of the scalar particles remains on the order of the Planck scale, which only means that they cannot be created at low energies. A much more serious defect lies in the fact that it is not possible to neutralize the difference between the primary and observational values of CC. We must complicate the model to solve this problem.

To this end, one may draw on the method developed in articles [9, 10]. As has been shown in [9], the problem can be strongly facilitated on the classical level by the 6-dim scalar-tensor gravity with higher derivatives. Moreover, the way of explanation of the CC smallness in the framework of pure gravity without scalar fields was studied in [10]. The latter is shortly discussed in the next section and the Appendix for clarity.

#### 3. Inclusion of Inhomogeneous Extra Space

In this section, we shortly consider the connection of the CC value and the form of extra space. The discussion is performed on the classical level while the quantum corrections are considered in the next section. Following the ideas developed in [9, 10], consider gravity with the actionacting in a 6-dim space. The constant that was written explicitly in (3) is involved now in the definition of the function .

The metric is assumed to be the direct product of the 4-dim space and 2-dim compact space Here, and are metrics of the manifolds and , respectively. and are the coordinates of the subspaces and . We will refer to 4-dim space and -dim compact space as the main space and an extra space, respectively. The metric has the signature (), and the Greek indexes refer to 4-dimensional coordinates. Latin indexes run over .

There are three scales of energy: the 6-dim Planck mass , the characteristic size of the compact extra space , and the low-energy scale GeV. It is assumed that the scales , and satisfy the inequalities

The first inequality in (25) means that a characteristic energy scale of extra space is large (the experimental limit is GeV) and its geometry is stabilized shortly after the Universe creation [21, 22, 28, 32–34]. On the other side, quantum behavior dominates at the scale and if one intends to describe a metric of extra space classically, the second inequality must take place. In the following, everything is measured in the units.

As will be shown later, conditionfor the energy scale is an appropriate choice. Indeed, the inequality permits us to consider the masses of particles to be zero. At the same time, excitations of compact extra space geometry are known to form the Kaluza-Klein tower with energies . If we start from the energy scale , the excitations are suppressed and extra space metric represents a stationary configuration described by part of the classical equations of motionwhere .

Let us assume that the metric of our 4-dim space is the Minkowski metric, . The compact 2-dim manifold is supposed to be parameterized by the two spherical angles and . The choice of the extra space metric is as follows:There is continuous set of extra space metrics, solutions to the differential equations (27), characterized by additional conditions. Maximally symmetrical extra spaces that are used in great majority of literature represent a small subset of this set. As additional conditions, we fix the metric at the point The system of equations (27) together with these conditions completely determines the form of extra space metric.

Numerical solutions to (27) with additional conditions (29) are discussed in [9, 26]. It has been found that, due to high nonlinearity of the equation, the gravity can trap itself in a small region around even without matter contribution.

The next step consists of finding an appropriate 2-dim extra metric with the help of the observable value of CC. General connection is represented in the Appendix, formula (A.4). It should be stressed that our aim is not to calculate CC with extremal accuracy but to find physical parameters at high-energy scale . In this case, the left hand side of (A.4) can be safely substituted by zero and we arrive at the following connection between the physical parameters:

To be more specific, suppose that the primary parameters of the Lagrangian and the extra space size dictated by the parameter are known:Then, numerical solution to (30) with respect to the Ricci scalar can be obtained:Relative accuracy of this result can be improved if necessary. Thus, we have found the extra space metric (see Figure 1) of our toy model with appropriate precision.