Localization of scalar matter with nonminimal derivative coupling on braneworlds

The localization of scalar matter with nonminimal derivative coupling (NMDC) in the braneworld model is studied. Two types of brane models are considered: a thin brane (the Modified Randall-Sundrum) and a thick brane generated by the bulk scalar field. The 5-dimensional theory of scalar field with NMDC is reduced to the effective 4-dimensional theory by imposing the fulfillment of localization conditions. In this article, the localization properties of field with NMDC are examined in both models. We found that the massless and massive scalar field with NMDC can be localized on the MRS thin brane. In the thick brane, the massless scalar field is localized.


Introduction
A braneworld is a concept that emerges from string theory.It offers a solution to the hierarchy problem encountered in the realm of high-energy physics.It proposes the existence of additional spatial dimension(s) beyond the familiar 3+1 dimensions.Its connection to our familiar 1+3dimensions universe is governed by an exponential factor named a warp factor.Within this framework, any weak-scale fields or matter, characterized at the TeV scale, are assumed to be confined on a hypersurface known as a brane.This brane is located within a higher-dimensional spacetime referred to as the bulk.Various brane models have been proposed, with some of the most well-known ones being the Randall-Sundrum (RS) model [1,2] and its alternative model, the Modified Randall-Sundrum (MRS) [3].Apart from the RS and RS-like models, which represent brane scenarios without thickness, there have been developments in brane models featuring thickness.The thick brane model extends the application of the theory in general relativity, offering a more versatile framework.Unlike thin branes, the thick brane concept does not require junction conditions to achieve continuous Einstein field equations.The solution of a thick brane is smooth, distinguishing it from thin branes.A review paper on thick branes is available in [4].
From the definition of the braneworld theory, weak-scale matter fields are ideally localized on a brane.For this reason, the localization of the weak energy field is one of the important issues that needs to be examined in the brane model.In the MRS model, it is shown that the extra coordinate transformation, dz = e −A dy, gives significant result in the field localization test.The MRS model has better localization properties of the scalar, vector, and spinor fields than the RS model [3].Several cases of localization of interaction field systems have also been discussed in the MRS model [5,6,7,8].
The modified theory of gravity is currently being widely applied to several prominent topics, including inflation, expansion of the late-time universe, gravitational waves, and astrophysical objects.One well-known modified theory of gravity is the nonminimal derivative coupling model (NMDC) of a scalar field interacting with gravity through a curvature tensor, as introduced by Amendola [9].This model is interesting to study as a candidate for various contexts, such as inflation [10,11], expansion of the universe [12,13,14,15], gravitational waves [16], black holes [17,18], wormholes [19,20], and neutron stars [21].In the 5-dimensional braneworld theory, NMDC has been discussed in terms of cosmology with the universal extra dimension model [22] and scalar field localization with the MRS thin brane model [23].
In this article, we discuss the localization of a scalar field with an NMDC term of the form ξG M N ∂ M Φ∂ N Φ, where G M N is the 5-dimensional Einstein tensor.The formulation of the 5-dimensional matter field equation is reduced to the standard 4-dimensional Klein-Gordon equation by providing the localization conditions that must be fulfilled.Here, we will review two braneworld models.For a better brane model regarding field localization in thin brane scenarios, we will examine the MRS brane.It also as a comment on Reference [23].For a comparative analysis in the context of thick branes, considering a relatively simple model that can represent thick branes, we will also examine the scalar thick brane model [24,25].In Section 2, we discuss the reduction of the 5-dimensional to 4-dimensional field equations by obtaining the localization conditions of the scalar field with NMDC.We then discuss the scalar field localization on the MRS brane model, Section 3, and on the thick brane model, Section 4. The conclusion is given in Section 5.

Field equation of 5D scalar field with nonminimal derivative coupling
In this section, we will derive the field equations, explain the localization mechanism, and derive the localization conditions that must be satisfied.

Field equation
Consider an action of 5-dimension scalar field Φ(x M ) with a nonminimal coupling, where the second term is a nonminimal derivative coupling that contains Einstein tensor G M N and ξ is a coupling constant.A braneworld metric with z extra coordinate is given by the following line element where e 2A(z) is a warp factor, which relates the brane to the extra dimension.The M, N indices represent 5-dimension bulk coordinate, and the µ, ν indices represent 4-dimension brane coordinate.In this article, two cases of braneworld will be reviewed, namely thin and thick branes.The thin brane case considered is the MRS model with the warp function A(z) = −k|z|.
In the case of thick brane, the smooth warp factor e A(z) is used as the gravitational solution for braneworld system.From metric (2), some geometrical quantities can be derived, including the Ricci tensor components where prime is a derivative with respect to z, the Ricci scalar 10th Asian Physics Symposium (APS 2023) Journal of Physics: Conference Series 2734 (2024) 012071 and the Einstein tensor components, By varying the action of scalar field with NMDC in (1) with respect to Φ(x M ), the equation of motion for scalar field Φ(x M ) can be derived, that is The 5-dimensional scalar field can be decomposed into Φ(x M ) = ϕ(x µ )χ(z).By considering a spacetime with the metric (2) and the field decomposition, the above field equation can be written into The following is a comment to Taufani et al [23] regarding the separation of 4-dimension and extra-dimension field equations.In [23], the 4-dimensional Klein-Gordon equation still depends on extra coordinate in a term α µν ≡ √ g(g µν + ξG µν ).By considering the conformally flat metric with a static warp factor, exp(A(z)), the Einstein tensor depends on extra coordinate z only.So, the standard 4-dimensional Klein-Gordon equation is a proper form of reducing the 5-dimensional equation to 4-dimension.If the metric is supposed to contain a dynamic warp factor, exp(A(t, z)), then the Einstein tensor depends on time and extra coordinates.So, it is possible to obtain a reduced NMDC term in the brane.In case of static warp factor, to satisfy the 4-dimensional Klein-Gordon equation, ∂ µ ∂ µ ϕ + m 2 ϕ = 0, the 5-dimensional field equation (7) reduces to an extra dimensional field equation By explicitly providing the Einstein tensor in equation ( 5), the extra dimensional field equation (8 For any thin brane model represented by the warp function A(z), the field solution will be found from the above equation of motion.

Localization conditions
From metric (2), action (1) can be written as follows where The dependence of Einstein tensor G M N on z coordinate only, enables reduction of the 5dimensional action (1) to the 4-dimensional scalar field action with the following localization conditions It turns out that the coupling constant ξ plays the role in shifting the integrands in the localization conditions.If the solution of the scalar field χ(z) in ( 9) satisfies the localization conditions in ( 12) and ( 13), then the field Φ is said to be localized on a brane characterized by the warp function A(z).

Localization on Modified Randall-Sundrum
As stated earlier, the MRS model demonstrates better field localization compared to the RS model.Therefore, for the field localization test in thin branes, we opt for the MRS model [3].

Field equation
The MRS model is a thin brane model with the warp function where k is a constant.The scalar field equation with NMDC in equation ( 9) becomes Since in the MRS thin brane model is Z 2 -symmetric, the integration limits in the localization condition equations can be chosen from zero to infinity [3].So the equations ( 12) and ( 13) can be written as For a massless scalar, the mass term should be m 2 = 0.

Field localization
The localization of the field on the MRS brane for both massless and massive modes is provided as follows.
3.2.1.Massless mode For the case of a massless scalar field, m = 0, the field equation ( 15) can be written into It gives a general solution where erfi is the imaginary error function.Then, an examination is conducted to determine whether this solution satisfies the normalization ( 16) and mass equation (17).If the solution satisfies the conditions N = 1 and m 2 = 0, then a massless scalar field is localized on the MRS thin brane.The simplest case is the case when c 1 = 0.The corresponding solution is which is in accordance with the mass equation ( 17), χ 0 gives m 2 = 0.

Massive mode
For the case of a scalar field with m = 0, the solution of equation ( 15) is where the constants is the Kummer confluent hypergeometric function, which is defined by the following series where (a) n = a(a + 1)(a + 2) • • • (a + n − 1).It is convergent for all finite z.The simplest solution corresponds to choosing a = 0 and thus m 2 = −(6k + 4).The function χ(z) = Be −2z , where for k > − 4 3 .

Localization on a scalar thick brane
To compare the localization of the scalar field with NMDC in thick and thin brane scenarios, we will investigate a thick brane generated from a bulk scalar.

Scalar thick brane
Consider a thick brane generated by a scalar bulk φ(x M ).The action is given by where V (φ) is a scalar bulk potential.The braneworld metric is defined by the following line element, ds 2 = e 2A(y) η µν dx µ dx ν − dy 2 , ( where A(y) is the warp function, that depends on the extra coordinate y.In this case, the bulk scalar is assumed to be a function of y only, φ = φ(y).By varying the action with respect to the scalar and metric tensor, the equation of motion for bulk scalar field and