Braneworld in Fًtق Gravity Theory with Noncanonical Scalar Matter Field

PHYSICAL REVIEW D 98, 084046 (2018)

Braneworld in fðTÞ gravity theory with noncanonical scalar matter field

† ‡ Jian Wang,1,2, Wen-Di Guo,1,2, Zi-Chao Lin,1,2,§ and Yu-Xiao Liu1,2,* 1Research Center of Gravitation & Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China 2Key Laboratory for Magnetism and Magnetic of the Ministry of Education, Lanzhou University, Lanzhou 730000, China

(Received 15 August 2018; published 26 October 2018)

In this paper, we investigate the braneworld scenario in fðTÞ gravity with a K-field as the background field. We consider various different specific forms of fðTÞ gravity and K-field and find a general way to construct the braneworld model. Based on our solutions, the split of branes is investigated. In addition, the stability of the braneworld is studied by investigating the tensor perturbation of the vielbein.

DOI: 10.1103/PhysRevD.98.084046

I. INTRODUCTION theory came up as an alternative to dark energy for the explanation of the acceleration of the Universe [50],itwas One of the most well-known and earliest extra-dimension widely investigated [51–55]. Braneworld models in fðTÞ theories was first proposed by T. Kaluza [1] and O. Klein [2] theories were studied in Refs. [56–58]. In the previous to unify Einstein’s general relativity and Maxwell’s electro- works [56,57], the solutions of braneworld scenarios in magnetism in the 1920s. The extra-dimension theories fðTÞ gravity with the form of fðTÞ¼T þ αTn were drew wide attention with the work of N. Arkani-Hamed, investigated by the first-order formalism, i.e., the super- S. Dimopoulos, and G. R. Dvali [3] and the works of L. potential way. In addition, the split of the brane in fðTÞ Randall and R. Sundrum [4,5] in the end of the 20th century. gravity was given in Ref. [56]. Furthermore, the tensor Later, various braneworld scenarios were developed such as perturbation of the braneworld was also studied in Ref. [58] the Gregory-Rubakov-Sibiryakov (GRS) model [6],the and it was shown that the solutions to the fðTÞ braneworld Dvali-Gabadadze-Porrati (DGP) model [7], the thick brane were stable. The localization of matter fields was also model [8–17], and the universal extra-dimension model [18], investigated in Ref. [56]. etc. Among these theories, one important model is the thick On the other hand, there are also braneworld scenarios in brane theory originated from the domain wall model pro- which the kinetic terms of background scalar fields are of posed by V.A. Rubakov and M. E. Shaposhnikov [19] in noncanonical form, i.e., the K-fields. The K-field theory was 1983. In the thick brane model, the brane could be generated first proposed as a new mechanism of inflation in cosmology by scalar fields [11,20–26], as well as vector fields and [59–61]. The lagrangian of a K-field can be written as spinor fields [12,27,28]. In addition, there are also brane- L ¼ FðX; ϕÞ − VðϕÞ, where X ¼ − 1 gMN∂ ϕ∂ ϕ is the world models without matter fields [14,29,30]. The standard 2 M N Fð ϕÞ model fields in the bulk can be localized near the brane kinetic term of the scalar field and X; is an arbitrary ϕ [6,11,31–34]. function of X and . The K-field theory was then applied to – The braneworld was also studied in different modified the thick braneworld models [62 67]. The analytical sol- gravity theories, for example, the scalar-tensor gravity utions of the thick K-brane models with two specific cases L ¼ − α 2 − ðϕÞ L ¼ − 2 2 − ðϕÞ theory [35–44], the metric fðRÞ gravity theory [45–47], X X V and X = V were given and the Palatini fðRÞ theory [48,49]. As the fðTÞ gravity by the first-order formalism in Ref. [65] with the perturbative procedure and in Ref. [68] with the nonperturbative procedure. Furthermore, the stability of thick K-brane was * Corresponding author. discussed in Ref. [69] and the trapping of bulk fermions [email protected] † [email protected] was also discussed in Ref. [67]. ‡ [email protected] In addition to being a mechanism of inflation, the K-fields §[email protected] also can be a dynamical dark energy model [70–72].Asa scalar field pervading the Universe, from the effective Published by the American Physical Society under the terms of field theory view point, the K-fields must interact with the the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to matter that is present [73]. One of the interesting cases is the the author(s) and the published article’s title, journal citation, interaction with the neutrino, which undergoes flavor oscil- and DOI. Funded by SCOAP3. lations when traveling through the K-field background.

2470-0010=2018=98(8)=084046(11) 084046-1 Published by the American Physical Society WANG, GUO, LIN, and LIU PHYS. REV. D 98, 084046 (2018)

ΓP ≡ ΓP − ΓP ¼ Pð∂ B − ∂ B Þ ð Þ By introducing some certain types of CPT violating terms in MN MN MN eB NeM MeN : 2:3 the neutrino action, it is possible to provide a unified explanation for all the existing neutrino data [73–75].One The difference between the Weitzenböck connection and ˜ P of theways that may give a CPT violating term is considering the Levi-Civita connection Γ MN widely used in general the effect of torsion [73]. Additionally, there is also work relativity is defined as the contorsion tensor about torsion-induced neutrino oscillations [76]. So torsion in the K-fields is one interesting possibility to the interaction P P ˜ P K MN ≡ Γ MN − Γ MN between dark matter and neutrino. We are especially inter- 1 ested in the higher-dimensional aspects of this model and ¼ ð P þ P − P Þ ð Þ 2 TM N TN M T MN : 2:4 corresponding braneworld scenario. In this paper, we will consider the braneworld model of From the torsion and contorsion tensors, the tensor SPMN fðTÞ K fðTÞ gravity with -fields. The forms of will be more can be defined as general. Instead of adopting the usual superpotential method, we develop a new way to give the solutions SPMN ≡ KMNP − gPNTQM þ gPMTQN ; ð2:5Þ and we get some specific solutions for different forms of Q Q fðTÞ and K-fields. Based on these solutions, we can then and then the torsion scalar T is given by study the energy density of the background fields and analyse the structure of the branes. Moreover, we find that 1 ð Þ ≡ PMN ð Þ by adding terms in the polynomial form of f T the number T 2 S TPMN: 2:6 of split sub-branes will increase, and this gives us a mechanism to generate sub-branes. Thus, one can write the Lagrangian of the teleparallel In the next section, we will give a review of the gravity as braneworld scenario in fðTÞ gravity briefly. And we will introduce the new method to construct the braneworld with c4e ð Þ L ¼ − T; ð2:7Þ K-fields in f T gravity. In Sec. III, we get the solutions for T 16πG different forms of fðTÞ and K-field. In particular, the form ð Þ M of f T is taken as a polynomial of T with arbitrary terms where e is the determinant of the vielbein eAðx Þ.Asto and the Lagrangian of background fields also contains fðTÞ gravity in the braneworld scenario, the action is given arbitrary terms of polynomials of the canonical kinetic by replacing T with fðTÞ, which is an arbitrary function of term. Based on these solutions, in Sec. IV, we study the T. Considering the matter field, the total action is energy density of matter fields along the extra dimension, Z Z which gives us the split of the brane. In Sec. V, we study the 1 5 5 S ¼ − d xefðTÞþ d xLM; ð : Þ tensor perturbation of our model and recover the four- 4 2 8 dimensional effective gravity. Finally, a brief summary is given in Sec. VI. c4 ¼ 1 L where we have taken 4πG for convenience, and M is the Lagrangian of background matter fields. Instead of II. BACKGROUND AND fðTÞ BRANEWORLD adopting a canonical form of matter fields, in our work, the kinetic term of a scalar field is replaced with an arbitrary In teleparallel gravity, the dynamical fields are vielbein function PðXÞ of the canonical kinetic term of the scalar ð MÞ fields eA x associated with each point in the manifold field X, i.e., the Lagrangian of the background matter is of M with spacetime coordinates x . In the case of the brane- the form world scenario considered here, the spacetime is a five- dimensional manifold. We use the indices A; B; C; and LM ¼ eðPðXÞ − VðϕÞÞ; ð2:9Þ M; N; P; which run from 0 to 4 to label the coordinates of tangent space and spacetime, respectively. Furthermore, where the canonical kinetic term is of the form X ¼ the relation of the metric and vielbein is given by 1 MN − 2 g ∂Mϕ∂Nϕ. Varying the action (2.8) with respect to the vielbein, we ¼ A B η ð Þ gMN eMeN AB; 2:1 get the field equation where ηAB ¼ diagð−1; 1; 1; 1; 1Þ. And the Weitzenböck 1 1 ΓP g fðTÞþ f ½g S RQeA ∂ eP connection MN is 4 MN 2 T RM P N Q A þ e−1g ∂ ðS RQeÞ − g TP S QR ΓP ≡ eP∂ eB ¼ −eB ∂ eP: ð2:2Þ RM Q N RM QN P MN B N M M N B 1 þ g f S RQ∂ T ¼ T ; ð2:10Þ The torsion tensor is defined as 2 RM TT N Q MN

084046-2 BRANEWORLD IN fðTÞ GRAVITY THEORY WITH … PHYS. REV. D 98, 084046 (2018) where fT and fTT represent, respectively, the derivative and considered. So it is natural to consider more general forms the second-order derivative of fðTÞ with respect to T: fT ≡ of fðTÞ and PðXÞ. To this end, we need to consider other 2 2 df=dT and fTT ≡ d f=dT , and T MN is the energy- method. Instead of using a superpotential approach to solve momentum tensor defined as these equations as usual, in this paper, we develop a new approach to solve the equations by which we could get δS −1 M some new and interesting solutions easily. T ¼ e g eA MN RM N δ A eR Considering that only two of the three equations are independent, we will focus on Eqs. (2.14) and (2.15). ¼ g ½PðXÞ − VðϕÞ þ P ∂ ϕ∂ ϕ; ð2:11Þ MN X M N Subtracting Eq. (2.15) from Eq. (2.14), we get where SM is the action corresponding to the Lagrangian 3 L ð Þ f A00 − 36f ðA0Þ2A00 ¼ −P ðϕ0Þ2; ð2:17Þ M, and PX is the derivative of P X with respect to X. 2 T TT X Next we consider the static flat braneworld scenario, for which the metric is which can also be written as

2 2AðyÞ μ ν 2 3 0 0 2 ds ¼ e ημνdx dx þ dy ; ð2:12Þ ∂ ð Þ¼− ðϕ Þ ð Þ 2 y fTA PX : 2:18 η ¼ where the Greek indices run from 0 to 3 and μν This equation gives a prescription to solve the system: ð−1 1 1 1Þ diag ; ; ; is the four-dimensional Minkowski met- noting that AðyÞ and T are functions of y only, we can see 2AðyÞ ric. In addition, e is the warped factor and y is the that the left-hand side of Eq. (2.18) only depends on y. coordinate of the extra dimension. We can choose the As for the right-hand side of the equation, it is a function A ¼ ð A A A A 1Þ 0 vielbein in the form eM diag e ;e ;e ;e ; , since it of ϕ only, because PðXÞ is an arbitrary function of imposes no constraints on the function fðTÞ and the torsion 1 MN 1 02 0 X ¼ − 2 g ∂Mϕ∂Nϕ ¼ − 2 ϕ . Regarding ϕ as a new scalar T [77,78]. Then the torsion scalar can be calculated variable, we can get a polynomial equation or transcen- ¼ −12 02 0 as T A . The prime denotes the derivative with dental equation of ϕ and y, which is easier to solve by respect to y through the whole paper. In addition, we also some skills than a differential equation of ϕ. After solving ϕ require the scalar field to depend on y only. Then the this polynomial equation or transcendental equation, we equation of motion of the background field can be given by can get the expression of ϕ0. Then, by integrating ϕ0 we at ϕ varying with respect to : last get the solution of ϕ. In the next section, we will give some specific solutions of the system with different ∂ ð ϕ0Þ¼ ð Þ y ePX eVϕ: 2:13 functions fðTÞ and PðXÞ. Using the metric assumption above, it is easy to write the equations of motion as follows: III. SOME SPECIFIC SOLUTIONS 1 1 In our work, we only consider the following warp factor þ ð3 00 þ 12 02Þ − 36 02 00 ¼ − solution, 4 f 2 fT A A fTTA A P V; ð2:14Þ AðyÞ¼−m ln ðcoshðkyÞÞ; ð3:1Þ

1 where k is an arbitrary constant with mass dimension one. It þ 6 02 ¼ − þ ϕ02 ð Þ 4 f fTA P V PX ; 2:15 should be noted that the bulk spacetime is asymptotically AdS5 at jyj → ∞ and hence the braneworld considered in 0 0 0 4A ϕ PX þ ∂yðPXϕ Þ¼Vϕ; ð2:16Þ this paper is embedded in an AdS5 spacetime. For the requirement that the warp factor is not divergent at the where Eqs. (2.14) and (2.15) are derived from the μν boundary of the extra dimension, we set m>0. components and 44 component of Eq. (2.10), respectively, In the following part, we will consider two specific forms and Eq. (2.16) could be given by simplifying Eq. (2.13). of fðTÞ: These are second-order differential equations and it is T not easy to get analytic solutions in general case. The most fðTÞ¼T0ðeT0 − 1Þð3:2Þ popular method to give braneworld solutions is the superpotential way. And the superpotential way was used in and braneworld scenario of fðTÞ gravity with canonical kinetic XN α term in Ref. [56]. In addition, for the specific form of the n þ1 fðTÞ¼ Tn þ C; ð3:3Þ noncanonical kinetic PðXÞ¼X þ α½ð1 þ bXÞn − 1, the þ 1 n¼0 n solution was also given by the superpotential way [57]. In both cases, only the form of fðTÞ¼T þ αTn was for which we have

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T T0 More complicatedly, if the right-hand side of Eq. (2.18) fTðTÞ¼e ð3:4Þ could be written as the same form of the exponential and function of X times a polynomial of X, we are able to read X off. Once again, if the forms of PðXÞ are pretty good, we XN could get the analytical solutions. So next, we will give n ð Þ fTðTÞ¼ αnT ; ð3:5Þ some examples of P X as we considered above, ¼0 n pffiffiffiffiffi −1 c −X PX ¼ −X0X e ; ð3:11Þ respectively. Note that in the second case, αn’s are arbitrary −2 constants with mass dimension n to ensure that fT is 2 −1− X mk 2 2 ϕ2 ¼ ð þ ϕ Þ 0 ð Þ dimensionless and N could be any positive integer. So the P 2 X 0 e ; 3:12 ϕ0 second case can represent a wide class fT. Next, we will give the solutions for both cases with different forms sffiffiffiffiffiffiffiffiffi ! qffiffiffiffiffi −X −1 of PðXÞ. −2X 2ϕ2 2 ¼ −3 2 − 2 0 ð Þ P mk 2 e ; 3:13 ϕ0 A. f T = expðT=T0Þ 2 2 ϕ For simplicity, we set T0 ¼ 24k m throughout the where X0, 0 and c are constants. The corresponding paper. The left-hand side of Eq. (2.18) gives solutions are given as follows.pffiffiffiffiffiffiffi For PX ¼ −ðX0=XÞ expðc −XÞ, the solution reads 3 0 3 −1 2ð Þ 2 −4 2tanh ky 2 ∂yðfTA Þ¼− e k mcosh ðkyÞ: ð3:6Þ 1 pffiffiffi 3 2 2 k m −2ky ϕðyÞ¼ 2 ln ½Li2ð−e Þ ck 2X0 This expression can be regarded as the product of an − kyð−ky − 2 þ 2 ln 2Þ exponential function of tanh2ðkyÞ and a polynomial of −1 tanh2ðkyÞ because cosh−2ðkyÞ¼1 − tanh2ðkyÞ. First, if the þ 2 tanhðkyÞðlnðcosh ðkyÞÞ − 1Þ; ð3:14Þ right-hand side of Eq. (2.18) can be written as some power n 2 2 −1 2ð Þ functions of X, for instance, X , we can get X as the nth VðyÞ¼−6m k ðe 2tanh ky − 1Þ root of the right-hand side of Eq. (3.6). And if the form of 3 2 ð Þ k m −1 P X we adopt is pretty good, it could be easy to get the − 2X0Ei ln þ 4 lnðcosh ðkyÞÞ 0 2X0 analytical solution of ϕðyÞ by integrating ϕ . In the ’ ð Þ¼ 1 2 following, we llp giveffiffiffiffiffiffiffi solutions in the case of P X X 2 2 − tanh ðkyÞ =c and PðXÞ¼C1 −X, where C1 is a constant. 2 For PðXÞ¼X, we can easily get the solution 3 2 −1 2ð Þ −4 þ 2tanh ky ð Þ 2 mk e cosh ky 1 2 2 −1 2ð Þ 2 ϕð Þ¼ ð Þ ð Þ − 6 2tanh ky ð Þ ð Þ y v1erf 2 tanh ky ; 3:7 m k e tanh ky ; 3:15

3 where Li2ðxÞ is the polylogarithm function of x and EiðxÞ ðϕÞ¼ 2 −2F 2ðϕÞ½ð1 − 4F 2ðϕÞÞ2 V 4 k me is the exponential integral function of x. For PðXÞ¼ðmk2=ϕ2ÞðX þ ϕ2Þ expð−1=2 − X=ϕ2Þ,we 2F 2ðϕÞ 2 0 0 0 þ 8mðe − 4F ðϕÞ − 1Þ; ð3:8Þ get the following ϕðyÞ and VðyÞ: qffiffiffiffiffiffiffi ¼ 3πm FðϕÞ¼ −1ðϕ Þ −1ð Þ 2ϕ0 ky where v1 2 , erf =v with erf x the ϕðyÞ¼ tan−1 tanh ; ð3:16Þ k 2 inverse of the errorpffiffiffiffiffiffiffi function erfðxÞ. For PðXÞ¼C1 −X,wehave ϕ 1 2 −1sin2ðk Þ 2 kϕ VðϕÞ¼ mk e 2 ϕ0 ð1 − 12mÞ 1 þ sin −1tanh2ðkyÞ 2 ϕ0 ϕðyÞ¼−v2e 2 tanhðkyÞ; ð3:9Þ ϕ 1sin2ðk Þ 4 kϕ þ 12 2 ϕ0 þ 3 ð Þ 2 2 1 ð−ϕ2 2Þ −1 ð−ϕ2 2Þ me cos : 3:17 VðϕÞ¼6k m e2ProductLog =v2 ½e 2ProductLog =v2 ϕ0 2 2 qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi þ ProductLogð−ϕ =v2Þ − 1; ð3:10Þ 2 2X X 1 For PðXÞ¼−3mk − 2 −2 exp − 2 −2 , we get ϕ0 2ϕ0 p3ffiffikm where v2 ¼ and ProductLogðxÞ gives the principal ϕ0 2C1 ϕðyÞ¼ tanhðkyÞ; ð3:18Þ solution for w in x ¼ wew. k

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ϕ ϕ these potential could be solved in our way. As for other 3 2 −1ðk Þ2 1ðk Þ2 ðϕÞ¼ 2 ϕ0 4 þ 4 ð 2 ϕ0 − 2Þ V 2 mk e m e forms of potential, whether they could be solved needs further discussion. 2 2 2 kϕ kϕ The solutions and potentials are obtained in classical þð4m − 2Þ 1 − þ 1 − : ϕ0 ϕ0 circumstances. Then another question arises naturally: if we ð3:19Þ consider quantum effects, is there any symmetry protecting these potentials from quantum fluctuations? The potential is P usually assumed to emerge from the microscopic physics N n B. f T = αnT P n =0 which leads to the braneworld model in a low energy limit ¼ N α n For fT n¼0 nT , the left-hand side of Eq. (2.18) is [8]. For the usual super potential way, the potential is the one given by in five-dimensional gauged supergravity with a USpð8Þ invariance [8,79].Butit’s an interesting and open question if 3 XN α one can construct a supergravity theory that the supersym- ∂ 0 ¼ − n 31þn4n ð1 þ 2 Þ y 2 fTA 2 m n metry conditions could give any potential desired [8].Weare n¼0 not able to answer this question and we focus on the classical × ð−k2m2ð1 − cosh−2ðkyÞÞÞncosh−2ðkyÞ: aspects of the braneworld in this paper. It is still unknown if all our potentials possess some symmetries that could protect ð3:20Þ them all from quantum fluctuation. Following the sprit of the last subsection, we consider the right-hand side of this equation as a polynomial of IV. THE SPLIT OF THE BRANE −2ð Þ cosh ky again. Then comparing with the right-hand side In this section, we investigate the split of the brane. This of Eq. (2.18), which is a polynomial of X, we can set phenomenon can be generated by a real scalar field [80], ∝ −2ð Þ ϕ0 ¼ ϕ −1ð Þ ϕ X cosh ky , i.e., 0cosh ky , where 0 is an two real scalar fields [81–84] or a complex scalar field [85]. ð Þ arbitrary constant, and read off the corresponding P X . In Ref. [56], the authors pointed out that the effect of The solution is given by torsion will influence the distribution of the energy density. As a consequence, the brane will have an incomplete split. 2ϕ0 ky ϕ ¼ tan−1 tanh ; ð3:21Þ While the authors of Ref. [56] only considered fðTÞ¼ k 2 T þ αTn, we can see more complicate phenomena caused by the arbitrary function of T in our work. From Eq. (2.10), XN 2 2 þ1 k m n we get the energy density ρðyÞ: P ¼ −α βðnÞ − ð2X þ ϕ2Þ ; ð : Þ n ϕ2 0 3 22 n¼0 0 ρðyÞ¼T UMUN MN XN kϕ 1þn ∂P V ¼ −α βðnÞ −k2m2sin2 ¼½P − Vg UMUN þ ∂ ϕ∂ ϕUMUN n ϕ MN ∂X M N n¼0 0 ¼ − ð Þþ ðϕÞ 3 ϕ n ϕ P X V þ α 2ð2 þ 1Þ −12 2 2 2 k 2 k nmk n k m sin cos 1 3 2 ϕ0 ϕ0 02 0 ¼ − fðTÞ − 6fTA − ∂y fTA : ð4:1Þ ϕ nþ1 4 2 n 2 2 2 k − 612 αn −k m sin ϕ0 It is easy to see that ρ only depends on fðTÞ and AðyÞ. Then 1 XN α ϕ n fðTÞ − n − 2 2 2 k ð Þ we will consider two different forms of from the 4 k m sin ϕ ; 3:23 previous sections and get the corresponding energy den- ¼1 n 0 n sities ρðyÞ to investigate the split of brane. T 3nþ14n−1ð2 þ1Þ For f ¼ eT0,wehave βðnÞ ¼ n T where mðnþ1Þ . We should note that the solutions have a large depend- 3 2 −1 2ð Þ ρ ¼ 2tanh ky ence on the forms of potentials. In other words, our method 2 mk e works only for some particular potential. This could be 1 2ð Þ 2 4 seen from our procedure. Our method is mainly based on × 4m e2tanh ky − tanh ðkyÞ − 1 þ sech ðkyÞ : solving the polynomial equation or transcendental equation gotten from Eq. (2.18). If we fix the forms of fðTÞ and ð4:2Þ PðXÞ, the polynomial equation or transcendental equation gives us some special forms of ϕ with corresponding Considering that there could be an arbitrary constant, or potential V as the solution. Thus only the system with so called cosmology constant in the potential V, we can

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T T0 ¼ α þ α FIG. 1. Plot of the energy density (4.2) for fT ¼ e with FIG. 2. Plot of the energy density (4.4) for fT 0 1T with k ¼ m ¼ 1. α0 ¼ −12, α1 ¼ −1. separate it from V by demanding V to be zero at y → ∞. Then we get the shape of the energy density (see Fig. 1). From Fig. 1, we can easily see that the brane is not split and, instead, it is localized near the zero point of the extra dimension. Next, we willP consider fðTÞ satisfying the differential ¼ N α n equation fT n¼0 nT which will result in a different energy density distribution along the extra dimension. We ð Þ get the expressionP for f T from the differential equation, þ1 α ð Þ¼ N n−1 n þ i.e., f T n¼1 n T C. Thus, the energy density is FIG. 3. Plot of the enrgy density (4.5) for fT ¼ α0 þ α1T þ 2 XN α2T with α0 ¼ 95.5, α1 ¼ 40, α2 ¼ 2.8. n 02n ρ ¼ αnð−12Þ A ðyÞ n¼0 6 6 6 ρ ¼ −720α2m k tanh ðkyÞ 3 02 3 00 × − 6 A ðyÞ − ð2n þ 1ÞA ðyÞ : ð4:3Þ 5 6 4 2 n þ 1 2 þ 1080α2m k tanh ðkyÞsech ðkyÞ þ 54α 4 4 4ð Þ To illustrate that the value of N (or the number of arbitrary 1m k tanh ky α ’ ð Þ 3 4 2 2 constants n s in the expression of f T ) could give us the − 54α1m k tanh ðkyÞsech ðkyÞ split of brane, we now need to fix N as some specific 3 − 3α 2 2 2ð Þþ α 2 2ð Þ ð Þ numbers and then give the calculations. First, let us set 0m k tanh ky 2 0mk sech ky : 4:5 N ¼ 1, i.e., f ¼ α0 þ α1T, then we get T The shape is shown in Fig. 3. We can see that the brane is 4 4 4 split into three sub-branes while α0 α1 α2 ρ ¼ 54α1m k tanh ðkyÞ , , are taken as 3 4 2 2 95.5, 40, 2.8, respectively. − 54α1m k tanh ðkyÞsech ðkyÞ At last, we consider N ¼ 3, i.e., fT ¼ α0 þ α1T þ 3 α 2 þ α 3 − 3α 2 2 2ð Þþ α 2 2ð Þ ð Þ 2T 3T , then 0m k tanh ky 2 0mk sech ky : 4:4 8 8 8 ρ ¼ 9072α3k m tanh ðkyÞ For simplicity, we will take k ¼ 1, m ¼ 1 in this section − 18144α 8 7 6ð Þ 2ð Þ and the shape of the energy density for N ¼ 1 is shown in 3k m tanh ky sech ky 6 6 6 Fig. 2. One can see that the brane is split into two sub- − 720α2k m tanh ðkyÞ branes which correspond to the two peaks in the energy 6 5 4 2 þ 1080α2k m tanh ðkyÞsech ðkyÞ density. Note that while α0, α1 are taken as some other 4 4 4 combinations, there could be still one peak in the energy þ 54α1k m tanh ðkyÞ density which means that the brane is not split. So in the 4 3 2 2 − 54α1k m tanh ðkyÞsech ðkyÞ following part, we will only display the graphs in which the 3 brane have the maximum numbers of sub-branes. Next, − 3α 2 2 2ð Þþ α 2 2ð Þ ð Þ 0k m tanh ky 2 0k msech ky : 4:6 we’ll show that the brane could split into more sub-branes with larger N. It is shown in Fig. 4. As expected, there areP four sub-brans. 2 þ1 α ¼ 2 ¼ α þ α þ α ð Þ¼ N n−1 n þ Secondly, for N , i.e., fT 0 1T 2T , the For the general case of f T n¼1 n T C, energy density is we could see that the maximum number of sub-branes

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ϕ ¼ ϕ¯ þ ϕ˜ ; ð5:5Þ

where ϕ¯ is the background field and ϕ˜ ¼ ϕ˜ ðxμ;yÞ is the perturbed field, we could get the perturbations of the μν components of energy-momentum tensor ∂V δ ¼ ð−ϕ¯ 0ϕ˜ 0Þ − ϕ˜ 2Aη þð − Þ 2Aγ Tμν PX ∂ϕ e μν P V e μν: ð5:6Þ

FIG. 4. Plot of the energy density (4.6) for fT ¼ α0 2 3 þα1T þ α2T þ α3T with α0 ¼ −116.4, α1 ¼ −47.2, α2 ¼ From Ref. [58] the perturbation of the field equations (2.10) −7.48, α3 ¼ −0.371. is given as

δ ¼ −1 δ ∂ ð PQÞþ −1 ∂ ð PQÞ increases with the number of terms in the polynomial TMN e fT gNP Q eSM e fTgNP Q eSM ð Þ Q ˜ Q P expression of f T . This conclusion could be roughly seen þ fTTδSMN ∂QT − fTδΓ PMSQN from Eq. (4.3). Substituting Eq. (3.1) into Eq. (4.3), the 1 − Γ˜ Q δ P þ δ ð Þ ð Þ right-hand side of Eq. (4.3) becomes a polynomial of fT PM SQN 4 gMNf T : 5:7 tanh2ðkyÞ whose highest order is N þ 1. So the distribution 2 of energy density will be similar to a polynomial of y . The μν components give þ 1 Thus, we can get N peaks by adjusting the values of α ’s. In other words, the more terms we adopt, the more 1 3 n 2Aγ þ 2A 6 02γ þ 00γ sub-branes we could get. 4 e μν fTe A μν 2 A μν 1 1 − 0γ0 − γ00 − −2A∂ ∂ργ V. TENSOR PERTURBATION AND EFFECTIVE A μν 4 μν 4 e ρ μν POTENTIAL ALONG THE EXTRA 2A 02 00 0 00 0 − f e ð36A A γμν − 6A A γμνÞ DIMENSION TT ∂ ∂ P ¯ ˜ V ˜ 2A In this section, we will investigate the linear tensor ¼ ð−∂yϕ∂yϕÞ − ϕ e ημν perturbation of the braneworld which satisfies the transverse- ∂X ∂ϕ 2A traceless condition. The perturbation of the vielbein þðP − VÞe γμν: ð5:8Þ fields is [58] Subtracting the background equation (2.14) from Eq. (5.8), AðyÞ a a e ðδ μ þ h μÞ 0 A ¼ ð Þ we get eM ; 5:1 01 1 1 0 0 00 −2A ρ 0 00 0 − f A γμν þ γμν þ e ∂ρ∂ γμν þ 6f A A γμν and the corresponding metric reads T 4 4 TT ∂ 2AðyÞ ¯ 0 ˜ 0 V ˜ e ðημν þ γμνÞ 0 ¼ P ð−ϕ ϕ Þ − ϕ ημν: ð5:9Þ g ¼ ; ð5:2Þ X ∂ϕ MN 01

Contracting this equation with ημν and considering the where transverse-traceless conditions (5.4),weget

a b b a γμν ¼ðδ μh ν þ δ νh μÞηab: ð5:3Þ ∂ ð−ϕ¯ 0ϕ˜ 0Þ − V ϕ˜ ¼ 0 ð Þ PX ∂ϕ ; 5:10 The transverse-traceless conditions are

μν μν and ∂μγ ¼ 0 ¼ η γμν: ð5:4Þ 1 1 0 00 0 0 0 00 −2A ρ The perturbation of the torsion tensor, contortion tensor and 6fTTA A γμν − fT A γμν þ γμν þ e ∂ρ∂ γμν ¼ 0; ρ ρ ρ 4 4 e.t.c (T μν, K μν, S μν) are given in Ref. [58]. The energy- momentum tensor is given as Eq. (2.11). With the perturba- ð5:11Þ tion of the vielbein fields (5.1) and the perturbation of the scalar field which is the same as Eq. (39) in Ref. [58].

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As in Ref. [58], we could write the tensor perturbation Then we will consider the localization of the zero mode equation as of graviton. As in Ref. [58], the zero mode of graviton is 2 μν R ð∂ þ 2H∂ þ η ∂μ∂νÞγμν ¼ 0; ð5:12Þ z z 3A− KðzÞdz Ψ0 ¼ N0e2 ; ð5:21Þ where 3 f where N0 is the normalization coefficient. The localization ¼ ∂ þ 12 −2Aðð∂ Þ3 − ∂2 ∂ Þ TT ð Þ H zA e zA zA zA ; 5:13 of the zero mode requires that 2 fT Z Z R and z is the conformal flat coordinate which is transformed Ψ2 ¼ 2 3A −2 Kdz ∞ ð Þ from y as 0dz N0e e dz < : 5:22 dz ¼ e−Ady: ð5:14Þ Substituting the expression of KðzÞ (5.16) into By introducing the Kaluza-Klein (KK) decomposition Eq. (5.22), we get

ρ ρ Z Z γμνðx ;zÞ¼ϵμνðx ÞFðzÞΨðzÞ; ð5:15Þ 2 2 −2m R Ψ0dz ¼ N0cosh ðkyÞjfTjdy: ð5:23Þ −3 ð Þþ ð Þ where FðzÞ¼e 2A z K z dz with

f In this paper, the forms of fT are considered to be a ð Þ¼12 −2Að∂2 ∂ − ð∂ Þ3Þ TT ð Þ K z e z A zA zA ; 5:16 polynomial of T or an exponential function of T, so the fT integrand is not divergent. Thus, to check whether the we get two equations from Eq. (5.12). One is the equation requirement (5.22) is satisfied, we only need to consider for the four-dimensional KK gravitons ϵμν, the asymptotic behavior of the integral when y approaches −2mð Þ −2mjkyj ðηρσ∂ ∂ þ ˜ 2Þϵ ð ρÞ¼0 ð Þ to infinity. Thus, we can replace cosh ky with e ρ σ mn μν x ; 5:17 and get the integral at infinity and the other is the Schrödinger-like equation for the extra- Z Z dimensional profile, 2 2 −2mjkyj Ψ0dz ¼ N0e jfTjdy: ð5:24Þ 2 2 ∞ ∞ ð−∂z þ UðzÞÞΨ ¼ m˜ nΨ; ð5:18Þ −12 2 2 2ð Þ where m˜ is the mass of the KK graviton and UðzÞ is the Note that, T is of the form k m tanh ky and fT is a n 2ð Þ effective potential polynomial of tanh ky or an exponential function of 2 tanh ðkyÞ in this paper. For both cases, jfTj is finite and we 2 UðzÞ¼∂zH þ H : ð5:19Þ can see that if and only if m>0 could the integrand be integrable at infinity, which is in coincidence with the The schrodinger-like equation (5.18) can be factorized as requirement of an asymptotically AdS5 spacetime. So we ð∂ þ HÞð−∂ þ HÞΨ ¼ m˜ 2Ψ; ð5:20Þ conclude that the zero mode of graviton could be localized. z z n Next, we will give the effective potential and zero mode 2 ¼ α þ α 1 þ α 2 which means that m˜ n > 0, i.e., any brane solution of of graviton for two specific cases fT 0 1T 2T T=T0 fðTÞ gravity theory with noncanonical scalar fields of and fT ¼ e . 2 the form PðXÞ is stable under the transverse-traceless For fT ¼ α0 þ α1T þ α2T , the effective potential can tensor perturbation. be read as

−2ð2þmÞ 2 2 2 2 2 4 U ¼ cosh ðkyÞ½−3k mα0ð8α0cosh ðkyÞþ32k mα1ð2 − 2ð2 þ 7mÞsinh ðkyÞþ15m sinh ðkyÞÞ 2 6 4 2 2 2 2 2 − 5mα0sinh ð2kyÞÞ þ 576k m ð24α0α2 þðα1 þ 2α0α2Þsinh ðkyÞð−8 − 22m þ 15m sinh ðkyÞÞÞtanh ðkyÞ 8 6 2 2 4 4 10 8 2 2 − 41472k m α1α2ð2 − 2ð2 þ 5mÞsinh ðkyÞþ5m sinh ðkyÞÞtanh ðkyÞþ82944k m α2ð8 − 2ð8 þ 19mÞsinh ðkyÞ 2 4 6 2 2 2 4 4 4 2 þ 15m sinh ðkyÞÞtanh ðkyÞ=½16ðα0 − 12k m α1tanh ðkyÞþ144k m α2tanh ðkyÞÞ ; ð5:25Þ and the zero mode of graviton is

−3m 2 2 2 4 4 4 Ψ0ðyÞ¼N0cosh 2 ðkyÞjα0 − 12k m α1tanh ðkyÞþ144k m α2tanh ðkyÞj: ð5:26Þ

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FIG. 6. Plots of the effective potential and the zero mode of T=T0 graviton for fT ¼ e .

VI. CONCLUSION In this paper, we developed a new method for finding solutions of the braneworld scenario in fðTÞ gravity theory with K-fields. Following our method, we found several FIG. 5. Plots of the effective potential, zero mode of graviton solutions to the cases that fðTÞ takes the forms of fðTÞ¼ and energy density of the background field for f ¼ P T T þ1 α α þ α 1 þ α 2 α ¼ 5 α ¼ 2 6 α ¼ 0 54 ð T0 − 1Þ ð Þ¼ N n−1 n þ 0 1T 2T with 0 , 1 . , 3 . . T0 e and f T n¼1 n T C. Then based on our solutions, we studied the distribution of the corresponding energy density along the extra dimension. The The plots of the effective potential and zero mode are shown results shows that the polynomial form of the fðTÞ will in Figs. 5(a) and 5(b), where we have taken k ¼ m ¼ 1. These two figures show that the effective potential has three cause split of the brane. Next, we considered the stability of wells so the corresponding zero mode has three peaks. our solutions by investigating the linear tensor perturbation Compared with the energy density of the background field of the vielbein. And we concluded that our solutions are [see Fig. 5(c)], we can see that the distribution of the zero stable. Finally, we demonstrated that the zero mode of mode of graviton is similar to that of the energy density, graviton could be localized for both forms of fT.In which implies that the zero mode could be localized near the addition, we calculated the effective potential of the KK brane. So we conclude that the split of the brane will cause modes of graviton along the extra dimension and gave the the split of the zero mode of graviton. zero mode of graviton. For f ¼ eT=T0, we get The brane we studied in this paper is flat, however, the T cosmology constant on the brane can also be nonvanishing. 1 Whether the method we developed in this paper is still valid ¼ 2 2mþ2ð Þð 2ð Þ 2ð Þ U 4 k sech ky sech ky tanh ky for dS thick brane or AdS thick brane can be studied further. We will investigate this in the future. þð8m − 6Þsech2ðkyÞþ15m2 − 6m þ 4Þ; ð5:27Þ the corresponding zero mode of graviton is ACKNOWLEDGMENTS

−1tanh2ðkyÞ −3m=2 We thank the referee for the comments that improved the Ψ0ðyÞ¼N0e 4 cosh ðkyÞ: ð5:28Þ paper. This work was supported by the National Natural Science Foundation of China (Grants No. 11875151, The corresponding figure for k ¼ m ¼ 1 is shown in Fig. 6. No. 11522541, and No. 11705070) and the Fundamental We can see that the effective potential is volcano-like and Research Funds for the Central Universities (Grant the zero mode is localized near the brane, this is similar to No. lzujbky-2018-k11). that of general relativity.

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