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Hindawi Publishing Corporation Advances in High Energy Volume 2015, Article ID 902396, 7 pages http://dx.doi.org/10.1155/2015/902396

Research Article Weyl-Invariant Extension of the Metric-Affine Gravity

R. Vazirian,1 M. R. Tanhayi,2 and Z. A. Motahar3

1 Plasma Physics Research Center, Islamic Azad University, Science and Research Branch, Tehran 1477893855, Iran 2Department of Physics, Islamic Azad University, Central Tehran Branch, Tehran 8683114676, Iran 3Department of Physics, University of Malaya, 50603 Kuala Lumpur, Malaysia

Correspondence should be addressed to M. R. Tanhayi; [email protected]

Received 30 September 2014; Accepted 28 November 2014

Academic Editor: Anastasios Petkou

Copyright © 2015 R. Vazirian et al. 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 SCOAP3.

Metric-affine geometry provides a nontrivial extension of the where the metric and are treated as the two independent fundamental quantities in constructing the spacetime (with nonvanishing torsion and nonmetricity). In this paper,westudythegenericformofactioninthisformalismandthenconstructtheWeyl-invariantversionofthistheory.Itisshown that, in Weitzenbock¨ space, the obtained Weyl-invariant action can cover the conformally invariant teleparallel action. Finally, the related field equations are obtained in the general case.

1. Introduction with the metric and torsion-free condition is relaxed; thus, in addition to , the would Extended theories of gravity have become a field of interest contain an antisymmetric part and nonmetric terms as well. in recent years due to the lack of a theory which could The transition from Riemann to metric-affine geometry has fully describe gravity as one of the fundamental interactions led to metric-affine gravity (MAG), an extended theory of together with other ones. Shortcoming of the current gravita- gravity which has been extensively studied during recent tion theory in cosmological scales, despite all the successes in decades from different viewpoints, and a variety of non- solar system tests, is another reason for developing the idea of Riemannian cosmology models are proposed based on it extending it and has confirmed the need to surpass Einstein’s (e.g., see [6–11]). generaltheoryofrelativity(GR)[1–3]. The extension can be The underlying idea of GR that explains gravity in terms made in different ways such as geometrical, dynamical, and of geometric properties of spacetime remains unaffected in higher or even a combination of them [4, 5]. In MAGandjustnewconceptsareaddedtothispictureby this paper, we will focus on geometric extension to gravity introducing the torsion and nonmetricity in descrip- under the notion of metric-affine formalism. tion of gravity. Almost all of the spacetime geometries such as Our current description of gravity is based, via theory of Riemann-Cartan, Weyl, and Minkowski geometries can be GR, on in which metric is the only obtained by constraining the three aforementioned tensors geometrical object needed to determine the spacetime struc- [12, 13]; this is one of the peculiarities of metric-affine ture and connection is considered to be metric connection, formalism. the well-known Christoffel symbols. In such a context, as a In this paper, we study the conformally invariant metric- priori, the connection is symmetric, that is, torsion-free con- affine theory. The conformal theories of gravity are of great dition, and is also assumed to be compatible with the metric importance; for example, in the framework of quantum grav- (∇𝜆𝑔𝜇] =0).Toenlargethisscheme,onecansetthetwo ity it is proved that such theories are better to renormalize [14] presumptions aside and think of metric-affine geometry in and also, from the phenomenological point of view, it is which metric and connection are independent geometrical shown that the solutions of conformal invariant theories (e.g., quantities. Therefore, connection is no longer compatible Weyl gravity) can explain the extraordinary speed of rotation 2 Advances in High Energy Physics curves of galaxies and also they can address the cosmological from which it is implied that inner product of vectors and so constant problem [15, 16]. Thus, considering the conformal their length and the angle between them are not conserved invariance within the context of MAG seems to be worth- when parallel transported along a curve in spacetime [4, 26]. while, which has been studied in some papers [17, 18]. (There Another difference that arose in this setup is the increase are other choices of the conformal transformation of the of dynamical degrees of freedom. Noting that the affine con- torsion which are studied in [19–22].) We first consider nection is independent of the metric and is not constrained in the action in its general format in MAG and then fix the general, it thus contains 64 independent components in four- degrees of freedom by imposing the conformal invariance. dimensional spacetime. As it is obvious by definitions given The organization of the paper is as follows. The conven- above, torsion and nonmetricity are rank-three tensors being tions and general aspects of metric-affine geometry are briefly antisymmetric in first and symmetric in last pair of indices, reviewed in Section 2.InSection 3, the most general second respectively. Due to these symmetry properties, the two order action in metric-affine formalism is presented in terms quantities will constitute the 64 independent components of of free parameters. Weyl-invariant version of the action is the affine connection (24 components of and investigated in Section 4 by determining the free parameters 40 components of nonmetricity tensor). Thereby, the overall and the subject will be concluded in the fifth section. The field number of the independent components will become 74 by equations of the general action are presented in the Appendix. taking into account the 10 independent components of the metric, a symmetric second rank tensor, together with the 2. Metric-Affine Geometry connection. It is straightforward to show that the affine connection As mentioned, the main idea in MAG is the independence can be expressed in the following form: of metric and connection, both of which are being the fun- 𝜆 𝜆 𝜆 𝜆 damental quantities indicating the spacetime structure and Γ 𝜇] = { 𝜇]} +𝐾 𝜇] +𝐿 𝜇], (6) carry their own dynamics in contrast with GR where metric 𝜆 is the only independent dynamical variable [23, 24]. Note where { 𝜇]} is the usual Christoffel symbol and the rest isa that, in Palatini formalism of GR, such an assumption holds, combination of torsion and nonmetricity tensors defined as but it brings nothing novel to the theory. In presence of the follows: gravitational field, spacetime can be curved or twisted and 𝜆 𝜎𝜆 thesefeaturesofspacetimearecharacterizedbycurvatureand 𝐾 𝜇] =𝑔 (𝑆𝜇𝜎] +𝑆]𝜎𝜇 +𝑆𝜇]𝜎), torsion tensors defined in terms of the affine connection and 1 (7) its derivatives as 𝐿𝜆 = 𝑔𝜎𝜆 (−𝑄 −𝑄 +𝑄 ). 𝜇] 2 𝜇]𝜎 ]𝜇𝜎 𝜎𝜇] 𝜆 𝜆 𝜆 𝜆 𝛼 𝜆 𝛼 R =−𝜕Γ +𝜕 Γ +Γ Γ −Γ Γ (1) 𝛽𝜇] ] 𝛽𝜇 𝜇 𝛽] 𝛼𝜇 𝛽] 𝛼] 𝛽𝜇 Accordingly, the affine curvature tensor will take the form of and (square brackets and parentheses in relations show R𝜆 =𝑅𝜆 ({}) −∇ 𝐿𝜆 +∇ 𝐿𝜆 antisymmetrization and symmetrization over indices, respec- 𝛽𝜇] 𝛽𝜇] {]} 𝛽𝜇 {𝜇} 𝛽] tively) 𝜆 𝛼 𝜆 𝛼 𝜆 𝜆 +𝐿 𝛼𝜇𝐿 𝛽] −𝐿 𝛼]𝐿 𝛽𝜇 −∇{]}𝐾 𝛽𝜇 +∇{𝜇}𝐾 𝛽] Γ𝛼 −Γ𝛼 𝛼 𝛼 𝜇] ]𝜇 (8) 𝑆 ≡Γ = . (2) +𝐾𝜆 𝐾𝛼 −𝐾𝜆 𝐾𝛼 +𝐿𝜆 𝐾𝛼 𝜇] [𝜇]] 2 𝛼𝜇 𝛽] 𝛼] 𝛽𝜇 𝛼𝜇 𝛽] 𝛼 𝜆 𝛼 𝜆 𝜆 𝛼 These two definitions are obtained from the covariant deriva- +𝐿 𝛽]𝐾 𝛼𝜇 −𝐿 𝛽𝜇𝐾 𝛼] −𝐿 𝛼]𝐾 𝛽𝜇, tives commutator acting on a vector field, 𝜆 where 𝑅 𝛽𝜇]({}) indicates and [∇ ,∇] 𝑉𝜆 =𝑅𝜆 𝑉𝛽 +2𝑆 𝛼∇ 𝑉𝜆, 𝜇 ] 𝛽𝜇] 𝜇] 𝛼 (3) bracesareusedtoshowcovariantderivativeinRiemann geometry. Contraction of (8) will result in two different rank- ∇ where denotes associated with the two tensors [4, 23]; one of them is the affine Ricci tensor affine connection and for a is defined as obtained by contracting the first index with the last pair of 𝜆 𝜆 𝜆 𝜆 𝛽 𝛽 𝜆 indices (R𝛽] ≡ R 𝛽𝜆]) and the other one is the so-called ∇𝜇𝐴 ] =𝜕𝜇𝐴 ] +Γ 𝛽𝜇𝐴 ] −Γ ]𝜇𝐴 𝛽. (4) homothetic curvature tensor which resulted from contrac- (R̂ ≡ R̂𝜆 ) It must be emphasized that the third index of the connection tion of the first two indices 𝜇] 𝜆𝜇] . Generation of the is conventionally chosen to be in the direction along which two second rank curvature tensors is due to the fact that the differentiation is done (the order of indices must be carefully only symmetry of the affine curvature tensor is the antisym- considered since the connection coefficients are supposed not metric property of the last pair of indices. However, there to be completely symmetric with nonvanishing torsion ten- is just one independent affine Ricci scalar and the contraction sor) [23, 25]. of homothetic curvature tensor with metric gives rise to Nonmetricity tensor is the other geometrical object vanishing scalar because of its antisymmetric nature [4, 23]. defined as Having more complete description of geometry, in the fol- lowing section, we will focus on constructing the generic ∇𝜆𝑔𝜇] ≡𝑄𝜆𝜇] (5) gravitational action based on metric-affine geometry. Advances in High Energy Physics 3

𝜇] 𝛼𝛽 𝜆 𝛾 3. Metric-Affine Second Order Action +𝑎11𝑔 𝑔 𝑔𝜆𝛾𝑄 𝜇]𝑄 𝛼𝛽

𝜇] 𝜆 𝜇] 𝜆 With independence of the metric and connection in MAG, +𝑎12𝑔 𝑄 𝜇]𝑆𝜆 +𝑎13𝑔 𝑄 𝜇𝜆𝑆] the general gravitational action is expressed in terms of met- 𝜇𝛼 𝜆 ] ric and connection and their derivatives. Also the coupling +𝑎14𝑔 𝑄 𝜇]𝑆𝜆𝛼 ) of matter action with connection in addition to metric and matter fields is allowed in the case of MAG which forms (9) the dissimilarity between this approach and the Palatini 𝑆 ≡𝑆 𝛽 𝑎 method [23, 27]. In this paper, we limit our discussion to the in which 𝜇 𝜇𝛽 and 𝑖’s denote different coupling con- stants, 𝑔 stands for the determinant of metric, and 𝜅 contains geometrical part of the action, but what the form of the action 𝜅 = 16𝜋𝑙2 is. Obviously, the first choice is to replace the Einstein-Hilbert the Planck length, for example, in four dimensions 𝑝. (EH) action with its counterpart in metric-affine formalism, The relevant field equations can be obtained by varying the but this is not the only possibility and extension of EH action action with respect to metric and connection independently, into a more general one through the metric-affine formalism and then one obtains after a trivial but rather lengthy calcu- can be done by considering torsion and nonmetricity tensors lation the field equations which are given in the Appendix. In as well as curvature tensor in construction of the action. (In thefollowingsection,wearegoingtofixthefreeparameters our study, only the geometrical part of the action is dealt whichappearintheactionduetoitsgeneralformof with, so the inconsistency that is mentioned in [28, 29]does definition. not appear in our consideration.) In order to construct the gravitational Lagrangian, we follow the approach of [23, 30] 4. Conformal Invariance of the Action which is an effective field theory approach and appropriate scalar terms are constructed at each order by applying power The action in the form of (9) has free parameters that must counting analysis. To start with and in natural units 𝑐=ℎ=1, be fixed. This can be done by imposing some constraints by choosing [𝑑𝑥] = [𝑑𝑡],allofthegeometricalquan- =[𝑙] or initial conditions where we use the conformal invariance tities, which are needed in construction of the action, are condition. The first extension of Einstein’s gravity was done by expressed in terms of length and, consequently, Weyl in 1919 and then developed by Cartan and Dirac (for to make the action dimensionless, the coupling constant is review see [2, 3, 32, 33]). In [19], conformal invariance was related to Planck length 𝑙𝑝. The highest power of the length considered in Riemann-Cartan geometry where the torsion dimensioninthescalarsshowstheorderofactionsothefirst plays the role of an effective Weyl gauge field. Conformal term of the generalized action, which is the Ricci scalar that torsion gravity and conformal symmetry in teleparallelism isreplacedbytheaffineone,isofsecondorder. werestudiedin[34]. Restricting our discussion to second order action and Under the conformal transformation of metric of the considering symmetries of torsion and nonmetricity tensors, form there will be four scalars written in terms of torsion tensor 𝑔 (𝑥) =Ω2𝑔 (𝑥) , and its derivative, eight terms made up of the nonmetricity 𝜇] 𝜇] (10) tensor and its derivative, and three terms constructed from the nonmetricity, which is associated with ∇,transformsas the contraction of torsion and nonmetricity tensors. (Terms of higher orders have been presented in [31]and,foraless 𝜆 𝜆 𝜆 𝑄𝜇] =2𝑔𝜇]∇ ln Ω+𝑄 𝜇], (11) general case, in [23].) Accordingly, the generic second order gravitational action in 𝑛 dimensions takes the following where Ω is a scalar function of 𝑥.However,theconformal form: (a similar action has been studied in [31] with different transformation of the torsion tensor is somehow ambiguous approach in the context of scalar, vector, tensor theory. Note and different forms of transformation are considered, namely, also that it is proved that, due to the symmetry properties, the weak conformal transformation under which the torsion other possible forms of nonvanishing scalars can be rewritten tensor remains unchanged and strong conformal transforma- usingthetermsofthisaction) tion in which the torsion tensor transforms similar to (10); 𝜆 𝑆 =Ω2𝑆 𝜆 I namely, one has 𝜇] 𝜇] [17]. Being interested in MAG conformal invariance of the action, we choose the weak form 1 which preserves the antisymmetric part of the affine connec- = ∫ 𝑑𝑛𝑥√−𝑔 (𝑎 R +𝑎∇𝜇𝑆 𝜅 0 1 𝜇 tion. In accordance with all what is mentioned above, one can easily show that the affine connection is not changed under 𝜇 𝜇 𝜎 𝜆 𝜇]𝜆 +𝑎2𝑆 𝑆𝜇 +𝑎3𝑆 𝜆 𝑆𝜇𝜎 +𝑎4𝑆 𝑆𝜇]𝜆 the conformal transformation of the form (10).

𝜇 𝜎 ]𝜎 𝜇 It is straightforward to show that, under conformal +𝑎5∇ 𝑄 𝜇𝜎 +𝑎6𝑔 ∇𝜇𝑄 ]𝜎 transformation, the action (9) transforms in the following way: +𝑎𝑔𝜇]𝑄𝜆 𝑄𝜎 +𝑎𝑔𝜇𝜆𝑄] 𝑄𝜎 7 𝜇𝜆 ]𝜎 8 𝜇𝜆 ]𝜎 1 I =Ω𝑛−2I + 𝜇] 𝜎 𝜆 MAG MAG 𝜅 +𝑎9𝑔 𝑄 𝜇𝜆𝑄 ]𝜎 𝑛 𝑛−2 𝜇𝛼 ]𝛽 𝜆 𝛾 × ∫ 𝑑 𝑥Ω +𝑎10𝑔 𝑔 𝑔𝜆𝛾𝑄 𝜇]𝑄 𝛼𝛽 4 Advances in High Energy Physics

× √−𝑔 ([2𝑎 +2𝑛𝑎]∇2 Ω 𝜇𝛼 ]𝛽 𝜆 𝛾 5 6 ln +𝑎10 [𝑔 𝑔 𝑔𝜆𝛾𝑄 𝜇]𝑄 𝛼𝛽 +[4𝑎 +4𝑛𝑎 +4𝑎 +4𝑛𝑎 7 8 9 10 1 𝜇] 𝛼𝛽 𝜆 𝛾 − 𝑔 𝑔 𝑔𝜆𝛾𝑄 𝜇]𝑄 𝛼𝛽] 2 𝜆 𝑛 +4𝑛 𝑎11](∇ ln Ω) (∇𝜆 ln Ω) 𝜇] 𝜆 +𝑎12 [𝑔 𝑄 𝜇]𝑆𝜆 + [−𝑎5 − (𝑛−2) 𝑎6 +2𝑎8 +4𝑎10 𝜇𝛼 𝜆 ] 𝜇] 𝜆 −𝑛𝑔 𝑄 𝜇]𝑆𝜆𝛼 ] +4𝑛𝑎11]𝑔 𝑄 𝜇]∇𝜆 ln Ω 𝜇] 𝜆 +𝑎13 [𝑔 𝑄 𝜇𝜆𝑆] +[2𝑎5 +4𝑎7 +2𝑛𝑎8 𝜇 𝜆 −𝑔𝜇𝛼𝑄𝜆 𝑆 ]]). +4𝑎9]𝑄 𝜇𝜆∇ ln Ω 𝜇] 𝜆𝛼 (14) +[4𝑎5 +4𝑛𝑎6 +2𝑛𝑎12 +2𝑎13

𝜆 ThisisthegeneralformoftheWeyl-invariantmetric-affine +2𝑎14]𝑆𝜆∇ ln Ω) gravity (WMAG), noting that a compensating Weyl scalar is 𝑛−2 (12) needed to cancel out the Ω factor which appears in (12). More simplification can be done if one is interested in the 2 {𝜆} in which ∇ ln Ω=∇{𝜆}∇ ln Ω. The conformal invariance reduced form of metric-affine space; for example, in Einstein- of the action results in vanishing of the additional terms at Weyl-Cartan space, one has ∇𝜆𝑔𝜇] =−2𝐴𝜆𝑔𝜇],where𝐴𝜆 is a the above relation, which leads one to write vector field [22]. By applying this condition to (14),itturnsto 𝑎 the following simple form: 𝑎 =− 5 , 6 1 𝑛 I = 𝑎 𝑛𝑎 WMAG 𝜅 𝑎 =− 5 −𝑎 − 8 , 9 2 7 2 (13) × ∫ 𝑑𝑛𝑥√−𝑔 (𝑎 R +𝑎∇𝜇𝑆 +𝑎𝑆 𝑆𝜇 𝑎 𝑎 𝑎 0 1 𝜇 2 𝜇 𝑎 = 5 − 8 − 10 , 11 2 2𝑛 2𝑛 𝑛 𝜇] 𝜎 𝜆 𝜇]𝜆 +𝑎3𝑔 𝑆𝜇𝜆 𝑆]𝜎 +𝑎4𝑆𝜇]𝜆𝑆 ). 𝑎14 =−𝑛𝑎12 −𝑎13. (15) Inserting (13) in (9) leads to Wewouldliketomentionthat(15) is expressible in a special form when the affine curvature scalar and covariant 1 I = derivative are redefined in terms of the Riemannian part plus WMAG 𝜅 a particular combination of torsion and nonmetricity square-

𝑛 𝜇 𝜇 terms by using (6), (7),and(8). Therefore, the affine curvature × ∫ 𝑑 𝑥√−𝑔 0(𝑎 R +𝑎1∇ 𝑆𝜇 +𝑎2𝑆𝜇𝑆 scalar takes the form of 𝜇] 𝜇 𝜇] 𝜎 𝜆 𝜇] 𝜎 𝜆 𝜇]𝜆 R =𝑅({}) −4𝑔 ∇{𝜇}𝑆] −4𝑆𝜇𝑆 +2𝑔 𝑆𝜇𝜆 𝑆]𝜎 +𝑎3𝑔 𝑆𝜇𝜆 𝑆]𝜎 +𝑎4𝑆𝜇]𝜆𝑆 𝜇]𝜆 ]𝜎 𝜇 𝜇] 𝜎 1 +𝑆𝜇]𝜆𝑆 +𝑔 ∇{𝜇}𝑄 ]𝜎 −𝑔 ∇{𝜇}𝑄 ]𝜎 +𝑎 [𝑔𝜇]∇ 𝑄𝜎 − 𝑔]𝜎∇ 𝑄𝜇 5 𝜇 ]𝜎 𝑛 𝜇 ]𝜎 1 1 + 𝑔𝜇𝜆𝑄] 𝑄𝜎 − 𝑔𝜇]𝑄𝜎 𝑄𝜆 𝜇𝜆 ]𝜎 𝜇𝜆 ]𝜎 (16) 1 𝜇] 𝜎 𝜆 2 2 − 𝑔 𝑄 𝜇𝜆𝑄 ]𝜎 2 1 𝜇𝛼 ]𝛽 𝜆 𝛾 1 𝜇] 𝛼𝛽 𝜆 𝛾 + 𝑔 𝑔 𝑔𝜆𝛾𝑄 𝜇]𝑄 𝛼𝛽 − 𝑔 𝑔 𝑔𝜆𝛾𝑄 𝜇]𝑄 𝛼𝛽 1 𝜇] 𝛼𝛽 𝜆 𝛾 4 4 + 𝑔 𝑔 𝑔𝜆𝛾𝑄 𝜇]𝑄 𝛼𝛽] 2𝑛2 𝜇] 𝜆 𝜇] 𝜆 𝜇𝛼 𝜆 ] +2𝑔 𝑄 𝜇]𝑆𝜆 −2𝑔 𝑄 𝜇𝜆𝑆] −2𝑔 𝑄 𝜇]𝑆𝜆𝛼 . +𝑎 [𝑔𝜇]𝑄𝜆 𝑄𝜎 7 𝜇𝜆 ]𝜎 And, in Einstein-Weyl-Cartan space, it becomes −𝑔𝜇]𝑄𝜎 𝑄𝜆 ] 𝜎 2 𝜇𝜆 ]𝜎 R =𝑅({}) −2(𝑛−1) ∇{𝜎}𝐴 − (𝑛−1)(𝑛−2) 𝐴

𝜇𝜆 ] 𝜎 −4(𝑛−2) 𝐴𝜇𝑆 −4𝑔𝜇]∇ 𝑆 −4𝑆 𝑆𝜇 +𝑎8 [𝑔 𝑄 𝜇𝜆𝑄 ]𝜎 𝜇 {𝜇} ] 𝜇 (17) 𝑛 +2𝑔𝜇]𝑆 𝜎𝑆 𝜆 +𝑆 𝑆𝜇]𝜆. − 𝑔𝜇]𝑄𝜎 𝑄𝜆 𝜇𝜆 ]𝜎 𝜇]𝜆 2 𝜇𝜆 ]𝜎 With the same procedure, one obtains 1 − 𝑔𝜇]𝑔𝛼𝛽𝑔 𝑄𝜆 𝑄𝛾 ] 𝜇 𝜇] 𝜇 𝜇 2𝑛 𝜆𝛾 𝜇] 𝛼𝛽 ∇ 𝑆𝜇 =𝑔 ∇{𝜇}𝑆] +2𝑆 𝑆𝜇 + (𝑛−2) 𝐴 𝑆𝜇 (18) Advances in High Energy Physics 5 and substituting (17) and (18) in (15) results in de Sitter backgrounds should contain such mixed symmetry tensor field of rank-3 [39–41]. 1 I𝑠𝑝. = Our obtained action (9) and its Weyl-invariant exten- WMAG 𝜅 sion cover the theories made up from Riemann curvature equipped with the torsion and nonmetricity, albeit studying × ∫ 𝑑𝑛𝑥√−𝑔 (𝑎 𝑅 ({}) −2(𝑛−1) 𝑎 ∇ 𝐴𝜎 0 0 {𝜎} the Weyl invariance of such theories needs some modifi- cations; however, as discussed, in our method, the Weyl 2 − (𝑛−1)(𝑛−2) 𝑎0𝐴 invariance can only be obtained by fixing the coefficients, 𝜇] which we have considered as a special case. −(4𝑎0 −𝑎1)𝑔 ∇{𝜇}𝑆]

𝜇 −(4𝑎0 −2𝑎1 −𝑎2)𝑆𝜇𝑆 Appendix 𝜇] 𝜎 𝜆 Field Equations of the General Action +(2𝑎0 +𝑎3)𝑔 𝑆𝜇𝜆 𝑆]𝜎 Inthisappendix,weobtaintherelatedfieldequationsof(9) +(𝑎 +𝑎)𝑆 𝑆𝜇]𝜆 0 4 𝜇]𝜆 in 4 dimensions. At first, let us variate the action with respect to the metric, which leads to −(4(𝑛−2) 𝑎 − (𝑛−2) 𝑎 )𝐴𝜇𝑆 ). 0 1 𝜇 1 1 (19) 𝑎 {R − 𝑔 R}+𝑎 {∇ 𝑆 − 𝑔 ∇𝛽𝑆 } 0 (𝜇]) 2 𝜇] 1 (𝜇 ]) 2 𝜇] 𝛽 Now by imposing the torsion-free condition on (19) and set- 1 𝛼 +𝑎2 {− 𝑔𝜇]𝑆𝛼𝑆 +𝑆𝜇𝑆]} ting 𝑎0 =1,uptoacompensatingWeylscalar,itbecomessim- 2 ilar to what is studied as a conformally invariant extension of 1 +𝑎 {− 𝑔 𝑆 𝑆𝛼𝜎𝜆 +𝑆 𝜎𝑆 𝜆} Einstein-Hilbert action [35]. 3 2 𝜇] 𝛼𝜆𝜎 𝜇𝜆 ]𝜎 It is worth noting that when the action in (14) is trans- 1 𝜌𝜎𝜆 𝛼 𝜆 𝜌𝜎 ferred to Weitzenbockspacebysettingthecurvatureand¨ +𝑎4 {− 𝑔𝜇]𝑆𝜌𝜎𝜆𝑆 +2𝑆 ]𝜆𝑆𝛼𝜇 −𝑆𝜌𝜎𝜇𝑆 ]} nonmetricity to zero [13], it reduces to 2

1 +𝑎5 {−∇]∇𝜇 ln √−𝑔−(∇] ln √−𝑔) (∇𝜇 ln √−𝑔) I = WMAG 𝜅 𝛼 𝛼𝜅 +2𝑄 𝛼(𝜇∇]) ln √−𝑔+4𝑆(𝜇∇]) ln √−𝑔+𝑔𝜇𝜅∇]𝑄𝛼 × ∫ 𝑑𝑛𝑥√−𝑔 (𝑎 ∇𝜇𝑆 +𝑎𝑆 𝑆𝜇 1 1 𝜇 2 𝜇 +∇ 𝑄𝜎 − 𝑔 ∇𝛽𝑄𝜎 +2∇𝑆 +𝑄 𝜌𝛾𝑄 𝜇 𝜎] 2 𝜇] 𝜎𝛽 ] 𝜇 𝜌 ]𝛾𝜇 𝜇] 𝜎 𝜆 𝜇]𝜆 +𝑎3𝑔 𝑆𝜇𝜆 𝑆]𝜎 +𝑎4𝑆𝜇]𝜆𝑆 ) 𝜎 𝜅 𝜎 −𝑄 𝜎]𝑄 𝜅𝜇 −4𝑆𝜇𝑆] −4𝑆(𝜇𝑄 𝜎|])} (20) 𝛽 𝛽 which is similar to the one used in teleparallel theories of +𝑎6 {−𝑔𝜇]∇𝛽∇ ln √−𝑔−𝑔𝜇] (∇𝛽 ln √−𝑔) (∇ ln √−𝑔) gravity [29, 36]. For example, in [37], from the tetrad and 𝜎 𝛼 scalar field analysis of torsion, it is shown that such an action −𝑄]𝜎 ∇𝜇 ln √−𝑔+2𝑄 𝜇]∇𝛼 ln √−𝑔 with specific coefficients of 𝑎1 =0, 𝑎2 =−1/3, 𝑎3 =1/2,and 𝛽 𝛼 +4𝑔𝜇]𝑆 ∇𝛽 ln √−𝑔+2∇𝛼𝑄 𝜇] 𝑎4 =1/4can indeed be a conformally invariant teleparallel 1 action. − 𝑔 𝑔𝜎𝛽 ∇ 𝑄𝛼 +2𝑔 ∇ 𝑆𝛽 +𝑄 𝜎𝜌 𝑄 2 𝜇] 𝛼 𝜎𝛽 𝜇] 𝛽 𝜇 ]𝜎𝜌 5. Conclusion 𝛼𝜆 𝛼 𝜎 𝛼 −2𝑄 ]𝑄𝛼𝜆𝜇 −4𝑔𝜇]𝑆 𝑆𝛼 +2𝑆𝜇𝑄]𝜎 −4𝑆𝛼𝑄 𝜇]} In this paper, we have first studied the general form of second 1 +𝑎 {2𝑄𝛽 ∇ √−𝑔 + 2∇ 𝑄𝛽 − 𝑔 𝑄𝜆 𝑄 𝜎𝛼 order metric-affine action which is constructed from all the 7 𝛽𝜇 ] ln ] 𝛽𝜇 2 𝜇] 𝜆𝛼 𝜎 possible forms of 15 scalar terms made up of affine curva- 𝛽 𝜆 𝛽 ture, torsion, and nonmetricity tensors. The Weyl-invariant −𝑄 𝛽𝜇𝑄 𝜆] −4𝑆]𝑄 𝛽𝜇} extension is obtained by imposing conformal invariance as a condition on the action. The resultant action reduces to +𝑎 {𝑄 𝛽∇ √−𝑔 + 𝑔 𝑄 𝛽𝜎∇ √−𝑔+𝑔 ∇𝜅𝑄𝛽 the conformally invariant teleparallel action in transition to 8 𝜇𝛽 ] ln 𝜇] 𝛽 𝜎 ln 𝜇] 𝛽𝜅

Weitzenbock¨ space [37]. Studying conformally invariant tor- 𝛽𝜆 𝜎 𝛽 𝛽𝜆 sion theories is important because, aside from the conformal +𝑔 ∇]𝑄𝜇𝛽𝜆 +2𝑄 𝜎[𝜇𝑄]]𝛽 −𝑄] 𝑄𝜇𝛽𝜆 invariance property, they can address some important issues 1 of theoretical physics (e.g., see [38]). Torsion theories may be −𝑔 𝑄 𝜎𝜅𝑄𝛽 − 𝑔 𝑄𝛽𝜆 𝑄𝜎 𝜇] 𝜎 𝛽𝜅 2 𝜇] 𝜆 𝜎𝛽 viewed as a rank-3 mixed symmetry tensor field. From the spacetime symmetry and group theoretical point of view, it −2𝑆 𝑄 𝜎 −2𝑔 𝑆𝜌𝑄𝛽 } is proved that linear conformal quantum gravity in flat and ] 𝜇𝜎 𝜇] 𝛽𝜌 6 Advances in High Energy Physics

𝜇 𝜇 𝜎 𝛽𝜎 𝜎 +𝑎 {−4𝑄]𝛽 𝛿 }+𝑎 {𝛿[]𝑄𝜇]𝛽 −2𝛿 𝑆]} +𝑎9 {2𝑄]𝜇𝜎∇ ln √−𝑔−2𝑄]𝜇𝜎𝑄𝛽 +2∇ 𝑄]𝜇𝜎 11 𝛽 𝜆 12 𝜆 𝛽 𝜆 +𝑎 {𝑄 𝛼[𝜇𝛿]] −𝑔𝜇]𝑆 −𝑆𝜇𝛿] } 1 𝜎𝛽𝜆 𝜆 𝜎 𝜎 13 𝛼 𝜆 𝜆 𝜆 − 𝑔𝜇]𝑄 𝑄𝜆𝛽𝜎 −𝑄 𝜇𝜎𝑄 𝜆] −4𝑆 𝑄]𝜇𝜎} 2 𝜅 𝛿I +𝑎 {𝑄[𝜇]] +𝑆𝜇] +𝑆 ]𝜇}=− 𝑀 . 1 14 𝜆 𝜆 𝜆 𝜆 𝜎 𝜎 𝜆𝛼𝛽 √−𝑔 𝛿Γ 𝜇] +𝑎10 {2𝑄 𝜇]∇𝜎 ln √−𝑔 + 2∇𝜎𝑄 𝜇] − 𝑔𝜇]𝑄 𝑄𝜆𝛼𝛽 2 (A.2) +𝑄 𝛼𝛽𝑄 −2𝑄𝜎𝛼 𝑄 −4𝑆 𝑄𝜎 } ] 𝜇𝛼𝛽 ] 𝜎𝛼𝜇 𝜎 𝜇] Conflict of Interests 𝛽 𝜎 𝛼𝛽 𝜎 +𝑎11 {2𝑔𝜇]𝑄𝜎𝛽 ∇ ln √−𝑔 + 2𝑔𝜇]𝑔 ∇𝜎𝑄 𝛼𝛽 The authors declare that there is no conflict of interests regarding the publication of this paper. 1 𝜆𝜎 𝛼 𝛼 𝜌 − 𝑔𝜇]𝑄 𝜎𝑄𝜆𝛼 +𝑄𝜇𝛼 𝑄]𝜌 2 Acknowledgments 𝜎𝛼𝛽 𝜎 𝛽 −2𝑔𝜇]𝑄 𝑄𝜎𝛼𝛽 −4𝑔𝜇]𝑆 𝑄𝜎𝛽 } The authors would like to thank M. V. Takook for his helpful guidelines. This work was supported in part by Islamic Azad 𝛽 𝛽 𝜆 University, Central Tehran Branch. +𝑎12 {𝑔𝜇]𝑆 ∇𝛽 ln √−𝑔 +𝜇 𝑔 ]∇𝛽𝑆 −2𝑔𝜇]𝑆 𝑆𝜆 1 References − 𝑔 𝑆𝜆𝑄 𝛽 +𝑆 𝑄 𝛼} 2 𝜇] 𝜆𝛽 𝜇 ]𝛼 [1] S. Capozziello and V. Faraoni, Beyond Einstein Gravity: A Sur- vey of Gravitational Theories for Cosmology and Astrophysics, +𝑎 {𝑆 ∇ √−𝑔+∇𝑆 −2𝑆 𝑆 13 𝜇 ] ln ] 𝜇 ] 𝜇 Springer,NewYork,NY,USA,2011. [2] S. Nojiri and S. D. Odintsov, “Introduction to modified gravity 1 𝛼 𝜆 𝜆 − 𝑔𝜇]𝑆 𝑄 𝜆𝛼 +2𝑄 𝜆[𝜇𝑆]]} and gravitational alternative for dark energy,” International 2 Journal of Geometric Methods in Modern Physics,vol.4,no.1, pp. 115–145, 2007. 𝜆 𝜆 𝜌 𝜆 +𝑎14 {𝑆𝜆𝜇]∇ ln √−𝑔+𝑔𝜌]∇ 𝑆𝜆𝜇 −2𝑆 𝑆𝜆𝜇] [3] S. Capozziello and M. De Laurentis, “Extended theories of grav- ity,” Physics Reports,vol.509,no.4-5,p.167,2011. 1 𝜆𝛽𝜌 𝜆𝛼 𝛽𝜌 [4] H. F. M. Goenner, “On the history of unified field theories,” − 𝑔𝜇]𝑆 𝑄𝜆𝛽𝜌 −𝑆𝜆𝛼]𝑄 𝜇 +𝑆𝜇 𝑄]𝛽𝜌 2 Living Reviews in Relativity,vol.7,article2,2004. [5] S. Nojiri and S. D. Odintsov, “Unified cosmic history in mod- 𝜎𝜆 𝛽 𝜆 𝜅 𝛿I𝑀 −𝑆 𝑄 +𝑆 𝑄 }=− , ified gravity: from 𝐹(𝑅) theory to Lorentz non-invariant mod- 𝜆𝜇] 𝜎 𝜆] 𝜇𝛽 −𝑔 𝛿𝑔𝜇] √ els,” Physics Reports, vol. 505, no. 2–4, pp. 59–144, 2011. (A.1) [6]F.W.Hehl,J.D.McCrea,E.W.Mielke,andY.Ne’eman, “Metric-affine gauge theory of gravity: field equations, Noether where I𝑀 stands for matter action. identities, world , and breaking of dilation invariance,” Variation of the action with respect to the connection Physics Reports,vol.258,no.1-2,pp.1–171,1995. yields the following equation: [7] D. Puetzfeld, “Prospects of non-Riemannian cosmology,” in 𝜇] ] 𝜇 𝜇] 𝑎0 {−𝑔 ∇𝜆 ln √−𝑔 + 𝛿 𝜆∇ ln √−𝑔 +𝜆 𝑄 Proceedings of 22nd Texas Symposium on Relativistic Astro- physics at Stanford University, Stanford, Calif, USA, December −𝑄 𝜎𝜇 𝛿] −2𝑆 𝜇] +2𝑔𝜇]𝑆 −2𝑆𝜇𝛿] } 𝜎 𝜆 𝜆 𝜆 𝜆 2004. [𝜇 ]] 𝛼[𝜇 ]] 𝜇] [𝜇 ]] [8] D. Puetzfeld, “Status of non-riemannian cosmology,” New +𝑎1 {𝛿 𝜆∇ ln √−𝑔+𝑄𝛼 𝛿 𝜆 −𝑔 𝑆𝜆 +2𝑆 𝛿 𝜆} Astronomy Reviews,vol.49,no.2,pp.59–64,2005. [𝜇 ]] []𝜇] [𝜇]] +𝑎2 {2𝑆 𝛿 𝜆}+𝑎3 {2𝑆𝜆 }+𝑎4 {2𝑆 𝜆} [9] A. V. Minkevich and A. S. Garkun, “Isotropic cosmology in metric-affine gauge theory of gravity,” http://arxiv.org/abs/gr- 𝜇] ] 𝜇 𝜇] 𝛾 qc/9805007. +𝑎5 {𝑔 ∇𝜆 ln √−𝑔+𝛿 ∇ ln √−𝑔 − 2𝑔 𝑄 𝛾𝜆 𝜆 [10] N. J. Poplawski, “Acceleration of the universe in the Einstein 𝑓(𝑅) 𝜌𝜇 ] 𝜇] ] 𝜇 frame of a metric-affine gravity,” Classical and Quantum −𝑄𝜌 𝛿𝜆 −2𝑔 𝑆𝜆 −2𝛿𝜆𝑆 } Gravity,vol.23,no.6,pp.2011–2020,2006.

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