Advances in High Energy Physics

Phenomenological Aspects of Quantum Gravity and Modified Theories of Gravity

Guest Editors: Ahmed Farag Ali, Giulia Gubitosi, Mir Faizal, and Barun Majumder Phenomenological Aspects of Quantum Gravity and Modified Theories of Gravity Advances in High Energy Physics

Phenomenological Aspects of Quantum Gravity and Modified Theories of Gravity

Guest Editors: Ahmed Farag Ali, Giulia Gubitosi, Mir Faizal, and Barun Majumder Copyright © 2017 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Advances in High Energy Physics.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board

Luis A. Anchordoqui, USA Frank Filthaut, Netherlands Anastasios Petkou, Greece T. Asselmeyer-Maluga, Germany Chao-Qiang Geng, Taiwan Alexey A. Petrov, USA M. Battaglia, Switzerland Philippe Gras, France Thomas Rössler, Sweden Botio Betev, Switzerland Xiaochun He, USA J. José Sanz-Cillero, Spain Lorenzo Bianchini, Switzerland Filipe R. Joaquim, Portugal Edward Sarkisyan-Grinbaum, USA Burak Bilki, USA KyungK.Joo,RepublicofKorea Sally Seidel, USA Adrian Buzatu, UK Aurelio Juste, Spain George Siopsis, USA Rong-Gen Cai, China Michal Kreps, UK Luca Stanco, Italy Duncan L. Carlsmith, USA Ming Liu, USA Satyendra Thoudam, Netherlands Ashot Chilingarian, Armenia Enrico Lunghi, USA Smarajit Triambak, South Africa Anna Cimmino, Belgium Piero Nicolini, Germany Elias C. Vagenas, Kuwait Andrea Coccaro, Switzerland Seog H. Oh, USA Nikos Varelas, USA Shi-Hai Dong, Mexico Sergio Palomares-Ruiz, Spain YauW.Wah,USA Edmond C. Dukes, USA Giovanni Pauletta, Italy Amir H. Fatollahi, Iran Yvonne Peters, UK Contents

Phenomenological Aspects of Quantum Gravity and Modified Theories of Gravity Ahmed Farag Ali, Giulia Gubitosi, Mir Faizal, and Barun Majumder Volume 2017, Article ID 1274326, 2 pages

Investigation of Free Particle Propagator with Generalized Uncertainty Problem F. Ghobakhloo and H. Hassanabadi Volume 2016, Article ID 2045313, 4 pages

Fermions Tunneling from Higher-Dimensional Reissner-Nordström Black Hole: Semiclassical and Beyond Semiclassical Approximation ShuZheng Yang, Dan Wen, and Kai Lin Volume 2016, Article ID 4647069, 6 pages

Matter Localization on Brane-Worlds Generated by Deformed Defects Alex E. Bernardini and Roldão da Rocha Volume 2016, Article ID 3650632, 10 pages

The Effects of Minimal Length, Maximal Momentum, and Minimal Momentum in Entropic Force Zhong-Wen Feng, Shu-Zheng Yang, Hui-Ling Li, and Xiao-Tao Zu Volume 2016, Article ID 2341879, 9 pages

Spontaneous Symmetry Breaking in 5D Conformally Invariant Gravity Taeyoon Moon and Phillial Oh Volume 2016, Article ID 5353267, 7 pages

On the UV Dimensions of Loop Quantum Gravity Michele Ronco Volume 2016, Article ID 9897051, 7 pages

Charged Massive Particle’s Tunneling from Charged Nonrotating Microblack Hole M. J. Soleimani, N. Abbasvandi, Shahidan Radiman, and W. A. T. Wan Abdullah Volume 2016, Article ID 1632469, 8 pages

Lorentz Invariance Violation and Modified Hawking Fermions Tunneling Radiation Shu-Zheng Yang, Kai Lin, Jin Li, and Qing-Quan Jiang Volume 2016, Article ID 7058764, 7 pages

Thermodynamical Study of FRW Universe in Quasi-Topological Theory H. Moradpour and R. Dehghani Volume 2016, Article ID 7248520, 10 pages

The Minimal Length and the Shannon Entropic Uncertainty Relation Pouria Pedram Volume 2016, Article ID 5101389, 8 pages

An Attempt for an Emergent Scenario with Modified Chaplygin Gas Sourav Dutta, Sudeshna Mukerji, and Subenoy Chakraborty Volume 2016, Article ID 7404218, 5 pages Hindawi Publishing Corporation Advances in High Energy Physics Volume 2017, Article ID 1274326, 2 pages http://dx.doi.org/10.1155/2017/1274326

Editorial Phenomenological Aspects of Quantum Gravity and Modified Theories of Gravity

Ahmed Farag Ali,1,2 Giulia Gubitosi,3 Mir Faizal,4 and Barun Majumder5

1 Benha University, Benha, Egypt 2Netherlands Institute for Advanced Study, Korte Spinhuissteeg 3, 1012 CG Amsterdam, Netherlands 3Imperial College London, London, UK 4British Columbia University, Vancouver, BC, Canada 5Indian Institute of Technology-Gandhinagar, Gujarat, India

Correspondence should be addressed to Ahmed Farag Ali; [email protected]

Received 12 December 2016; Accepted 20 December 2016; Published 15 February 2017

Copyright © 2017 Ahmed Farag Ali 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.

The phenomenological aspects of various approaches to horizon entropy in quasi-topological gravity. In the paper by quantum gravity and modified theories of gravity are getting S.-Z. Yang et al., the authors studied the Hawking radiation a lot of attention and cover many different topics such as in the context of Lorentz invariance violation which has been theories with minimal length, noncommutative geometry, predicted in different approaches to quantum gravity. In the theories with generalized uncertainty principle, violations paperbyM.J.Soleimanietal,theauthorsinvestigatetunnel- of Lorentz symmetries, and loop quantum cosmology. The ing of charged massive particles in charged TeV-scale black recent developments in these approaches with their phe- hole with the help of a generalized uncertainty relation that nomenological implications in particle physics, cosmology, has a minimal length and maximal momentum. In the paper and astrophysics are the main theme of this special issue. by M. Ronco, the author tackled Planck-scale dynamical In this special issue on quantum gravity phenomenology, we dimensional reduction in which he observed that the number have invited papers that address such issues. of UV dimensions can be used to constrain the ambiguities in In the paper by S. Dutta et al., the authors present a study the choice of these loop quantum gravity-based modifications on emergent universe scenario with modified Chaplygin gas of the Dirac space-time algebra. In the paper by T. Moon and and its implications with the initial big bang singularity. In the P. Oh, the authors studied spontaneous symmetry breaking paper by P. Pedram, the author uses the modified commuta- in 5D conformally invariant gravity and found that the tion relation, which has quadratic order in the momentum dimensional reduction via ADM decomposition gives rise to as deformation, to study the Beckner, Bialynicki-Birula, and the4DMinkowskivacuum.InthepaperbyZ.-W.Fenget Mycielski (BBM) inequality. With a particular choice of al., the authors studied the implications of minimal length self-adjoint representation, the author has shown that the theories on the entropic force realization of gravity and BBM inequality remains valid for various choices of the they derived, in this context, the modified Einstein field deformation parameter. In the paper by H. Moradpour equation and modified Friedman equations which may have and R. Dehghani, the authors tackled the unified first law various phenomenological implications. In the paper by A. E. of thermodynamics and its genuine connection with the Bernardini and R. Rocha, the authors have studied a family of apparent horizon of FRW universe in which they found that asymmetric thick brane configurations which are formed by whenever there is no energy exchange between the various defects. A left-right asymmetric chiral localization of spin 1/2 parts of cosmos, one could get an expression for the apparent particles is produced in the localization of fermion fields of 2 Advances in High Energy Physics the thick branes. In the paper by S.-Z. Yang et al., the authors investigated the origin of Hawking radiation in the context of fermions tunneling from higher-dimensional Reissner- Nordstromblackholeandcalculatedtheentropyoftheblack¨ hole. In the paper by F. Ghobakhloo and H. Hassanabadi, theauthorsinvestigatedthefreeparticlepropagatorinthe context of minimal length theories and discussed possible implications of this modified propagator. Ahmed Farag Ali Giulia Gubitosi Mir Faizal Barun Majumder Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 2045313, 4 pages http://dx.doi.org/10.1155/2016/2045313

Research Article Investigation of Free Particle Propagator with Generalized Uncertainty Problem

F. Ghobakhloo and H. Hassanabadi

Physics Department, Shahrood University of Technology, P.O. Box 316, Shahrood 3619995161, Iran

Correspondence should be addressed to F. Ghobakhloo; [email protected]

Received 1 March 2016; Revised 11 April 2016; Accepted 24 April 2016

Academic Editor: Mir Faizal

Copyright © 2016 F. Ghobakhloo and H. Hassanabadi. 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.

We consider the Schrodinger¨ equation with a generalized uncertainty principle for a free particle. We then transform the problem into a second-order ordinary differential equation and thereby obtain the corresponding propagator. The result of ordinary quantum mechanics is recovered for vanishing minimal length parameter.

1. Introduction Inourpaper,wearegoingtocombinethesetwosubjects. Namely,westudythefreeparticlepropagatorinSchrodinger¨ The generalization of classical action principle into quantum framework in minimal length formulation. In Section 2, theory appears in path integral formulation. Instead of a we review the essential concepts of GUP and write the single classical path, the quantum version considers a sum, or generalized Hamiltonian for free particle. In Section 3, we better, say, integral, of infinite possible paths [1, 2]. Although obtain the propagator for this system in which the details of the main idea of path integral approach was released by calculations are brought. N. Wiener, in an attempt to solve diffusion and Brownian problems, it was introduced in Lagrangian formulation of quantum mechanics of P.A. M. Dirac [1, 2]. Nevertheless, the 2. GUP-Corrected Hamiltonian present comprehensive formulation is named after Feynman An immediate consequence of the ML is the GUP and extracted from his Ph.D. thesis supervised by J. A. Wheeler [1, 2]. Feynman’s formulation is now an essen- ℏ Δ𝑝 Δ𝑥 ≥ +𝛼𝑙2 , tial ingredient in many fundamental theories of theoretical Δ𝑝 𝑝 ℏ (1) physics including quantum field theory, quantum gravity, and high energy physics [1–3]. where the GUP parameter 𝛼 is determined from a funda- Ontheotherhand,wearenowalmostsurefromfunda- mental theory. At low energies, that is, energies much smaller mental theories such as string theory and quantum gravity than the Planck mass, the second term on the right hand side that the ordinary quantum mechanics ought to be reformu- of (1) vanishes and we recover the well-known Heisenberg lated. In more precise words, a generalization of Heisenberg uncertainty principle. The GUP of (1) corresponds to the uncertainty principle, called generalized uncertainty princi- generalized commutation relation ple (GUP), should be considered at energies of order Planck 2 scale [4–7]. This generalization corresponds to a generaliza- [𝑥𝑜𝑝 ,𝑝𝑜𝑝 ]=𝑖ℏ(1+𝛽𝑝)0≤𝛽≤1, (2) tion of wave equation of quantum mechanics. Till now, var- 2 ious wave equations of quantum mechanics, different inter- where 𝑥𝑜𝑝 =𝑥, 𝑝𝑜𝑝 = 𝑝[1 + 𝛽(𝑝) ] and 0≤𝛽≤1.Thelimits actions,andotherrelatedmathematicalaspectsandphysical 𝛽→0and 𝛽→1correspond to the standard quantum concepts have been considered in this framework [8–18]. mechanics and extreme quantum gravity, respectively. 2 Advances in High Energy Physics

Equation (2) gives the minimal length in this case as in which the Lagrangian is given by [20] (Δ𝑥)min =2𝑙𝑝√𝛼.Itshouldbenotedthat,inthedeformed Schrodinger¨ equation, the Hamiltonian does not have any 󸀠󸀠 󸀠 2 (𝑥 −𝑥) 𝑝 2 explicit time dependence [19]: 𝐿 (𝑡) =𝑝⋅ − −4𝛽𝑚(𝐸(0)) . (11) (𝑡󸀠󸀠 −𝑡󸀠) 2𝑚 𝑛 2 𝑝𝑜𝑝 ( +𝑉(𝑥))𝜓𝑛 (𝑥) =𝐸𝑛𝜓𝑛 (𝑥) . (3) 2𝑚 Therefore, the propagator appears as

This deformed momentum operator modifies the original 1 𝐾(𝑥󸀠󸀠,𝑡󸀠󸀠;𝑥󸀠,𝑡󸀠)= Hamiltonian as 2𝜋ℏ 𝑝2 𝐻= +𝑉 (𝑥) , (4) 󸀠󸀠 󸀠 2 𝑓𝑓 𝑖 (𝑥 −𝑥) 𝑚 2 2𝑚 ⋅ ( −4𝛽𝑚(𝐸(0)) Δ𝑡) exp ℏ 2Δ𝑡 𝑛 where (12) 4 𝑝 󸀠󸀠 󸀠 2 𝑉 (𝑥) =𝛽 +𝑉(𝑥) . (5) 𝑖Δ𝑡 (𝑥 −𝑥)𝑚 𝑓𝑓 𝑚 ⋅ ∫ 𝑑𝑝 ⋅ (− (𝑝 − ) ) exp 2𝑚ℏ Δ𝑡 The problem becomes much simpler if we consider [16] 2 (0) 𝑝 =2𝑚(𝐸𝑛 −𝑉(𝑥)), or

2 (6) 4 2 (0) 󸀠󸀠 󸀠󸀠 󸀠 󸀠 1 𝑝 =4𝑚 (𝐸𝑛 −𝑉(𝑥)) . 𝐾(𝑥 ,𝑡 ;𝑥 ,𝑡 )= 2𝜋ℏ

Inthefreeparticlecase,wethereforehave 󸀠󸀠 󸀠 2 (𝑥 −𝑥) 𝑚 2 2 𝑖 (0) 𝑝 2 ⋅ exp ( −4𝛽𝑚(𝐸 ) Δ𝑡) ∫ 𝑑𝑈 (13) 𝐻= +4𝛽𝑚(𝐸(0)) . (7) ℏ 2Δ𝑡 𝑛 2𝑚 𝑛

Wenowcalculatethesinglefreeparticlepropagator 𝑖Δ𝑡 2 ⋅ exp (− 𝑈 ) corresponding to this deformed Hamiltonian in Section 3. 2𝑚ℏ

3. Perspicuous Form of Propagator with 󸀠 󸀠 If the wave function 𝜓(𝑥, 𝑡 ) is known at a time 𝑡 ,wecan (𝑥󸀠󸀠 −𝑥󸀠)𝑚 󸀠󸀠 󸀠󸀠 𝜓(𝑥, 𝑡 ) 𝑡 𝑈=(𝑝− ). (14) explicitly write the wave function at a later time Δ𝑡 using the propagation relation as [20] 𝑖𝐻󸀠󸀠 (𝑡 −𝑡󸀠) 󸀠󸀠 󸀠 In a more explicit form, the propagator for free particle under 𝜓(𝑥,𝑡 )=exp (− )𝜓(𝑥,𝑡 ). (8) ℏ minimal length is

󸀠󸀠 󸀠 For a small time interval 𝑡 −𝑡 =Δ𝑡,wehave 𝑚 𝐾(𝑥󸀠󸀠,𝑡󸀠󸀠;𝑥󸀠,𝑡󸀠)=(√ ) 󵄨 𝑖𝐻Δ𝑡 󵄨 1 2𝑖𝜋ℏΔ𝑡 󵄨 󵄨 ⟨𝑥 󵄨exp (− )󵄨 𝑥⟩ = ∫ 𝑑𝑝 󵄨 ℏ 󵄨 2𝜋ℏ 2 (15) (𝑥󸀠󸀠 −𝑥󸀠) 𝑚 󵄨 󵄨 𝑖 (0) 2 󸀠󸀠 󸀠 󵄨 𝑖 𝑝2 󵄨 ⋅ ( −4𝛽𝑚(𝐸 ) (𝑡 −𝑡)) . 󵄨 󵄨 exp ℏ 2(𝑡󸀠󸀠 −𝑡󸀠) 𝑛 ⋅(⟨𝑥󵄨exp − ( Δ𝑡)󵄨 𝑝⟩ 󵄨 ℏ 2𝑚 󵄨 󵄨 󵄨 (9) 󵄨 𝑖 (0) 2 󵄨 𝑑𝑝 Now, if we assume 𝛽=0, then the one-dimensional free ⋅⟨𝑝󵄨exp − (4𝛽𝑚 (𝐸 ) Δ𝑡)󵄨 𝑥⟩) = ∫ 󵄨 ℏ 𝑛 󵄨 2𝜋ℏ particle propagator is given by

2 𝑖 𝑝 2 ⋅ − ( +4𝛽𝑚(𝐸(0)) Δ𝑡) . 𝐾(𝑥󸀠󸀠,𝑡󸀠󸀠;𝑥󸀠,𝑡󸀠) exp ℏ 2𝑚 𝑛 2 󸀠󸀠 󸀠 (16) Therefore, the quantum mechanical propagator for small 𝑚 𝑖 (𝑥 −𝑥) 𝑚 󸀠 =(√ ) ( ). time interval Δ𝑡 = 𝑡 −𝑡 , corresponding to this nonlocal 2𝑖𝜋ℏΔ𝑡 exp ℏ 2(𝑡󸀠󸀠 −𝑡󸀠) Hamiltonian,canbewrittenas 𝑡󸀠+Δ𝑡 󸀠󸀠 󸀠󸀠 󸀠 󸀠 𝑖 𝑑𝑝 In order to obtain free particle propagator for a finite time 𝐾(𝑥 ,𝑡 ;𝑥 ,𝑡 )=∫ exp ( ∫ 𝐿 (𝑡) 𝑑𝑡) , (10) 󸀠󸀠 󸀠 ℏ 𝑡󸀠 2𝜋ℏ interval (𝑡 −𝑡 ) we divide the interval into 𝑁 subintervals of Advances in High Energy Physics 3

󸀠󸀠 󸀠 equal length Δ𝑡 such that (𝑡 −𝑡 )=𝑁Δ𝑡.Now,thepropagator In the limit 𝛽→0, the final form of propagator is given by of a finite time interval is written as 𝐾(𝑥󸀠󸀠,𝑡󸀠󸀠;𝑥󸀠,𝑡󸀠) 𝑚 𝑁 𝐾(𝑥󸀠󸀠,𝑡󸀠󸀠;𝑥󸀠,𝑡󸀠)=(√ ) 󸀠󸀠 󸀠 2 2𝑖𝜋ℏΔ𝑡 𝑚 𝑖𝑚 (𝑥 −𝑥) (22) =(√ ) ( ) 𝑖𝑚 2𝑖𝜋ℏ󸀠󸀠 (𝑡 −𝑡󸀠) exp 2ℏ (𝑡󸀠󸀠 −𝑡󸀠) ⋅ ∫ 𝑑𝑥 𝑑𝑥 𝑑𝑥 ⋅⋅⋅𝑑𝑥 1 2 3 𝑁−1 exp 2ℏΔ𝑡 (17) 2 2 2 which is the result in ordinary quantum mechanics. ⋅((𝑥1 −𝑥0) +(𝑥2 −𝑥1) +⋅⋅⋅+(𝑥𝑁 −𝑥𝑁−1) )

−𝑖4𝛽𝑚 2 4. Conclusion ⋅ ( (𝐸(0)) (𝑡󸀠󸀠 −𝑡󸀠)) . exp ℏ 𝑛 We considered the nonrelativistic free particle propagation The integral in (17) can be calculated as [20] probleminananalyticalmannerinminimallengthfor- malism. We first transformed arising differential equation ∫ 𝑑𝑥 𝑑𝑥 𝑑𝑥 ⋅⋅⋅𝑑𝑥 𝑖𝜆 into a second-order differential equation which included a 1 2 3 𝑁−1 exp modifiedeffectivepotential.Wenextcalculatedthepropa- gator. Apart from the application of the study, the work is of ⋅((𝑥 −𝑥 )2 +(𝑥 −𝑥 )2 +⋅⋅⋅+(𝑥 −𝑥 )2) 1 0 2 1 𝑁 𝑁−1 (18) pedagogical interest in graduate physics. (𝑁−1)/2 2 1 𝑖𝜋 𝑖𝜆𝑁 (𝑥 −𝑥0) = ( ) exp ( ). Competing Interests √𝑁 𝜆 𝑁 The authors declare that they have no competing interests. Substituting (18) into (17), the propagator is obtained as

𝑚 𝑁 1 𝑖𝜋 (𝑁−1)/2 References 𝐾(𝑥󸀠󸀠,𝑡󸀠󸀠;𝑥󸀠,𝑡󸀠)=(√ ) ( ) 2𝑖𝜋ℏΔ𝑡 √𝑁 𝜆 [1] M. Chaichian and A. Demichev, Path Integrals in Physics, Institute of Physics, Bristol, UK, 2001. 2 𝑖𝜆 (𝑥 −𝑥 ) [2] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path ⋅ ( 𝑁 0 ) (19) exp 𝑁 Integrals,McGrawhill,NewYork,NY,USA,1965. [3]J.J.Sakurai,Modern Quantum Mechanics, Addison-Wesley, −𝑖4𝛽𝑚 2 Reading, Mass, USA, 1994. ⋅ ( (𝐸(0)) Δ𝑡) , exp ℏ 𝑛 [4] A. F. Ali, M. M. Khalil, and E. C. Vagenas, “Minimal length in quantum gravity and gravitational measurements,” Europhysics 󸀠󸀠 󸀠 where 𝜆 = 𝑚/2ℏΔ𝑡.Replacing𝑥𝑁 and 𝑥0 by 𝑥 and 𝑥 , Letters, vol. 112, no. 2, Article ID 20005, 2015. 󸀠󸀠 󸀠 respectively, and using (𝑡 −𝑡)=𝑁Δ𝑡,weobtainthefinal [5] S. Das, E. C. Vagenas, and A. F. Ali, “Discreteness of space from expression as GUP II: relativistic wave equations,” Physics Letters B,vol.690, no. 4, pp. 407–412, 2010. 𝑚 𝑁 1 𝜆 (𝑁−1)/2 [6] W. Chemissany, S. Das, A. F. Ali, and E. C. Vagenasc, “Effect 𝐾(𝑥󸀠󸀠,𝑡󸀠󸀠;𝑥󸀠,𝑡󸀠)=(√ ) ( ) 2𝑖𝜋ℏΔ𝑡 √𝑁 𝑖𝜋 of the generalized uncertainty principle on post-inflation pre- heating,” Journal of Cosmology and Astroparticle Physics,vol.12, 2 article 017, 2011. 𝑖𝜆𝑁 (𝑥 −𝑥0) ⋅ exp ( ) (20) [7] A. Kempf and G. Mangano, “Minimal length uncertainty 𝑁 relation and ultraviolet regularization,” Physical Review D,vol. 55, no. 12, pp. 7909–7920, 1997. −𝑖4𝛽𝑚 2 (0) [8] L. J. Garay, “Quantum gravity and minimum length,” Interna- ⋅ exp ( (𝐸𝑛 ) Δ𝑡) . ℏ tionalJournalofModernPhysicsA,vol.10,no.2,pp.145–166, 1995. Now, if we calculate the probability of detecting the particle Δ𝑥 𝑥󸀠󸀠 [9] A. N. Tawfik and A. M. Diab, “Review on generalized uncer- at a finite region , enclosing final point ,from(20),we tainty principle,” http://arxiv.org/abs/1509.02436. get [10] S. Gangopadhyay, A. Dutta, and M. Faizal, “Constraints on the 𝑚 generalized uncertainty principle from black-hole thermody- 𝐾(𝑥󸀠󸀠,𝑡󸀠󸀠;𝑥󸀠,𝑡󸀠)=(√ ) namics,” Europhysics Letters, vol. 112, no. 2, Article ID 20006, 2𝑖𝜋𝑁ℏΔ𝑡 2015. 2 [11] H. Hassanabadi, S. Zarrinkamar, and E. Maghsoodi, “Scattering 𝑖𝑚𝑁 (𝑥 −𝑥0) ⋅ exp ( ) (21) states of Woods–Saxon interaction in minimal length quantum 2ℏ𝑁Δ𝑡 mechanics,” Physics Letters B,vol.718,no.2,pp.678–682,2012. [12] S. Das and E. C. Vagenas, “Universality of quantum gravity 𝑖4𝛽𝑚 (0) 2 corrections,” Physical Review Letters,vol.101,no.22,ArticleID ⋅ exp ( (𝐸 ) Δ𝑡) . ℏ 𝑛 221301, 2008. 4 Advances in High Energy Physics

[13] K. Jahankohan, S. Zarrinkamar, and H. Hassanabadi, “Cornell potential in generalized uncertainty principle formalism: the case of Schrodinger¨ equation,” Quantum Studies: Mathematics and Foundations,vol.3,no.1,pp.109–114,2016. [14] K.Jahankohan,H.Hassanabadi,andS.Zarrinkamar,“Relativis- tic Ramsauer-Townsend effect in minimal length framework,” Modern Physics Letters A,vol.30,no.32,ArticleID1550173,8 pages, 2015. [15] S. Haouat, “Schrodinger¨ equation and resonant scattering in the presence of a minimal length,” Physics Letters B,vol.729,pp.33– 38, 2014. [16] D. Kothawala, S. Shankaranarayanan, and L. Sriramkumar, “Quantum gravitational corrections to the propagator in space- times with constant curvature,” http://arxiv.org/abs/1002.1132. [17] F. de Felice, M. T. Crosta, A. Vecchiato, M. G. Lattanzi, and B. Bucciarelli, “A general relativistic model of light propagation in the gravitational field of the solar system: the static case,” Astrophysical Journal,vol.607,no.1I,pp.580–595,2004. [18] A. Maas, “Some more details of minimal-Landau-gauge SU(2) Yang-Mills propagators,” Physical Review D,vol.91,ArticleID 034502, 2015. [19] H. Hassanabadi, S. Zarrinkamar, and E. Maghsoodi, “Scattering states of Woods-Saxon interaction in minimal length quantum mechanics,” Physics Letters B,vol.718,no.2,pp.678–682,2012. [20] S. Pramanik, M. Faizal, M. Moussa, and A. Farag Ali, “Path inte- gral quantization corresponding to the deformed Heisenberg algebra,” Annals of Physics,vol.362,pp.24–35,2015. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 4647069, 6 pages http://dx.doi.org/10.1155/2016/4647069

Research Article Fermions Tunneling from Higher-Dimensional Reissner-Nordström Black Hole: Semiclassical and Beyond Semiclassical Approximation

ShuZheng Yang,1 Dan Wen,1,2,3 and Kai Lin1,4

1 Department of Astronomy, China West Normal University, Nanchong, Sichuan 637002, China 2College of Physics and Information Science, Hunan Normal University, Changsha, Hunan 410081, China 3Faculdade de Engenharia de Guaratingueta,´ Universidade Estadual Paulista, 12516-410 Guaratingueta,´ SP, Brazil 4Instituto de F´ısica e Qu´ımica, Universidade Federal de Itajuba,´ 37500-903 Itajuba,´ MG, Brazil

Correspondence should be addressed to Kai Lin; [email protected]

Received 20 April 2016; Accepted 25 July 2016

Academic Editor: Mir Faizal

Copyright © 2016 ShuZheng Yang 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.

Based on semiclassical tunneling method, we focus on charged fermions tunneling from higher-dimensional Reissner-Nordstrom¨ black hole. We first simplify the Dirac equation by semiclassical approximation, and then a semiclassical Hamilton-Jacobi equation is obtained. Using the Hamilton-Jacobi equation, we study the Hawking temperature and fermions tunneling rate at the event horizon of the higher-dimensional Reissner-Nordstrom¨ black hole space-time. Finally, the correct entropy is calculation by the method beyond semiclassical approximation.

Hawking radiation is an important prediction in modern the Klein-Gordon equation of curved space-time, but also gravitation theory [1–5]. Recently, Kraus et al. proposed with the Dirac equation in curved space-time. Applying the quantum tunneling theory to explain and study Hawking Hamilton-Jacobi equation, we can then obtain semiclassical radiation [6–31], and then semiclassical Hamilton-Jacobi Hawking temperature and tunneling rate at the event horizon method is put forward to research the properties of scalar par- of higher-dimensional Reissner-Nordstrom¨ black hole. ticles’ tunnels [32–36]. In 2007,Kerner and Mann investigated In modern physics theory, the concept of an extra the 1/2 spin fermion tunneling from static black holes [37]. In dimension can help to solve some theoretical issues, so sev- their work, the spin up and spin down cases are researched, eral higher-dimensional metrics of curved space-time were respectively, and the radial equations are obtained, so that investigated. The metric of static charged (𝑛+2)-dimensional they can finally determine the Hawking temperature and Reissner-Nordstrom¨ black hole is given by [10, 51–55] tunneling rate at the event horizon. Subsequently, Kerr and 𝑑𝑠2 =−𝑓 𝑟 𝑑𝑡2 +𝑓−1 𝑟 𝑑𝑟2 +𝑟2𝑑Ω2, Kerr-Newman black holes cases, the charged dilatonic black ( ) ( ) 𝑛 (1) hole case, the de Sitter horizon case, the BTZ black hole 𝑑Ω2 𝑛 case, 5-dimensional space-time cases, and several nonsta- where 𝑛 is the metric of -dimensional sphere 𝑛 tionary black hole cases were all researched, respectively 2 𝑖𝑖 2 [38–48], and we used Hamilton-Jacobi method to study the 𝑑Ω𝑛 = ∑ℎ 𝑑𝜃𝑖 fermion tunneling from higher-dimensional uncharged black 𝑖=1 holes [49, 50]. However, up to now, no one has studied =𝑑𝜃2 + 2 𝜃 𝑑𝜃2 + 2 𝜃 2 𝜃 𝑑𝜃2 +⋅⋅⋅ higher-dimensional charged black holes cases, so we set 1 sin 1 2 sin 1sin 2 3 outtoresearchthatcase.Inourwork,wedevelopedthe 𝑛−1 2 2 Kerner and Mann method and proved that the semiclassical + ∏sin 𝜃𝑖𝑑𝜃𝑛, Hamilton-Jacobi equation can be obtained not only with 𝑖=1 2 Advances in High Energy Physics

𝜔 𝑀 𝜔 𝑄2 16𝜋 𝑓 (𝑟) =1− 𝑛 + 𝑛 ,𝜔= , Corresponding to the flat case, the gamma matrices can be 𝑛−1 2𝑛−2 𝑛 𝑟 2 (𝑛−1) 𝑉𝑛𝑟 𝑛𝑉𝑛 chosen as (2) 𝑖 𝛾𝑡 = 𝛾̃1 𝑚×𝑚 √𝑓 𝑚×𝑚 where 𝑀 and 𝑄 are mass and electric charge of black hole, and the electromagnetic potential is 𝑟 √ 2 (9) 𝛾𝑚×𝑚 = 𝑓𝛾̃𝑚×𝑚 𝑄 𝐴 = ( ,0,0,0,...) , 𝜂 −1√ 𝜂𝜂 𝜂 𝜇 𝑛−1 (3) 𝛾𝑚×𝑚 =𝑟 ℎ 𝛾̃𝑚×𝑚, 𝜂=3,4,5,...,𝑛+2. (𝑛−1) 𝑉𝑛𝑟 Now, let us simplify the Dirac equation via semiclassical where 𝑉𝑛 is volume of unit 𝑛-sphere (we can adopt the units 𝐺=𝑐=ℏ=1). The outer/inner horizon located at approximation, and rewrite the spinor function as 𝐴 (𝑡,𝑟,...,𝑥𝜂,...) 𝑚/2×1 (𝑖/ℏ)𝑆(𝑡,𝑟,...,𝑥𝜂,...) 𝜔 𝑛𝑄2 Ψ=( )𝑒 , (10) 𝑟𝑛−2 = 𝑛 [𝑀±√𝑀2 − ] . 𝐵 (𝑡,𝑟,...,𝑥𝜂,...) ± 2 8𝜋 (𝑛−1) (4) 𝑚/2×1 [ ] 𝜂 𝜂 where 𝐴𝑚/2×1(𝑡,𝑟,...,𝑥 ,...)and 𝐵𝑚/2×1(𝑡,𝑟,...,𝑥 ,...)are Obviously, at the horizons, the equation 𝑓(𝑟±)=0should column matrices with 𝑚/2 × 1 order and 𝑆 is classical action. be satisfied. However, the physical property near the inner Via semiclassical approximation method, Substituting (10) horizon cannot be researched, so we just study the fermion into (5) and dividing the exponential term and multiplying tunneling at the outer event horizon of this black hole. The by ℏ,wecanget charged Dirac equation in curved space-time is 𝐶𝐷 𝐴𝑚/2×1 𝜇 𝑚 ( )( )=0 (11) 𝛾 𝐷 Ψ+ Ψ=0, 𝜇=𝑡,𝑟,𝜃,...,𝜃 , 𝐸𝐹 𝐵 𝜇 ℏ 1 𝑛 (5) 𝑚/2×1 1 𝜕𝑆 where 𝐶=− ( +𝑞𝐴 )𝐼 √𝑓 𝜕𝑡 𝑡 𝑚/2×𝑚/2 (12) 𝑖𝑞𝐴𝜇 𝐷 =𝜕 +Γ + +𝑚𝐼 𝜇 𝜇 𝜇 ℏ 𝑚/2×𝑚/2 (6) 1 𝜕𝑆 Γ = [𝛾̃𝑎, 𝛾̃𝑏]𝑒 ]𝑒 , 𝐷=𝑖√𝑓 𝐼 𝜇 8 𝑎 𝑏];𝜇 𝜕𝑟 𝑚/2×𝑚/2 𝜕𝑆 (13) where 𝑚 and 𝑞 are mass and electric charge of the particles, − ∑𝑟−1√ℎ𝜂𝜂 𝛾̃𝜂−2 𝛼 𝜕𝑥𝜂 𝑚/2×𝑚/2 and 𝑒𝑏];𝜇 =𝜕𝜇𝑒𝑏] −Γ𝜇]𝑒𝑎𝑏 is the covariant derivative of tetrad 𝜂 𝑒 𝑏]. The gamma matrices in curved space-time need tobe 𝜕𝑆 satisfied 𝐸=𝑖√𝑓 𝐼 𝜕𝑟 𝑚/2×𝑚/2 𝜇 ] 𝜇] (14) {𝛾 ,𝛾 }=2𝑔 𝐼. (7) 𝜕𝑆 + ∑𝑟−1√ℎ𝜂𝜂 𝛾̃𝜂−2 𝜂 𝑚/2×𝑚/2 𝜂 𝜕𝑥 After the gamma matrices are defined, we choose the gamma matrices in (𝑛+2)-dimensional flat space-time as 1 𝜕𝑆 𝐹= ( +𝑞𝐴 )𝐼 √𝑓 𝜕𝑡 𝑡 𝑚/2×𝑚/2 𝐼 0 (15) 𝛾̃1 =(𝑚/2×𝑚/2 ) 𝑚×𝑚 +𝑚𝐼 . 0−𝐼𝑚/2×𝑚/2 𝑚/2×𝑚/2 0𝐼 2 𝑚/2×𝑚/2 Solving (11), we have 𝛾̃𝑚×𝑚 =( ) 𝐼𝑚/2×𝑚/2 0 −1 (8) (𝐸 − 𝐹𝐷 𝐶) 𝐴𝑚/2×1 =0 0𝑖𝛾̃𝜂−2 (16) 𝜂 𝑚/2×𝑚/2 −1 𝛾̃ =( ), (𝐹 − 𝐸𝐶 𝐷)𝑚/2×1 𝐵 =0. 𝑚×𝑚 ̃𝜂−2 −𝑖𝛾𝑚/2×𝑚/2 0 It is evident that the coefficient matrices of (16) must vanish, 𝜂=3,4,5,...,𝑛+2, when 𝐴𝑚/2×1 and 𝐵𝑚/2×1 have nontrivial solutions. Due to the fact that 𝐶𝐷 = 𝐷𝐶,wecanwritetheconditionthat ̃𝜂 where 𝐼𝑚/2×𝑚/2 and 𝛾𝑚/2×𝑚/2 are unit matrices and flat gamma determinant of coefficient vanish as (𝑛+2)/2 matrices with 𝑚/2 × 𝑚/2 order and 𝑚=2 is the order (𝐸𝐷 − 𝐹𝐶) =0. of the matrices in even (odd) dimensional space-time. det (17) Advances in High Energy Physics 3

𝜇 ] From the relation of flat gamma matrices {𝛾̃ , 𝛾̃ }=2𝛿𝜇],we However, above calculation is worked on semiclassical can obtain the semiclassical Hamilton-Jacobi equation in (𝑛+ approximation, because we ignored all higher order terms of 2)-dimensional Reissner-Nordstrom¨ space-time O(ℏ). Recently, Banerjee and Majhi proposed a new method beyond semiclassical approximation to research the quantum 1 𝜕𝑆 2 𝜕𝑆 2 𝜕𝑆 − ( +𝑞𝐴 ) +𝑓( ) +⋅⋅⋅+𝑔𝜂𝜂 ( ) tunneling, and their works show that the conclusion should 𝑓 𝜕𝑡 𝑡 𝜕𝑟 𝜕𝑥𝜂 be corrected [56–65], and this correct entropy may be applied (18) in quantum gravity theory. Now let us generalize this work 2 +⋅⋅⋅+𝑚 =0. in higher-dimensional Reissner-Nordstrom¨ black hole space- time. 𝑎 Using the Hamilton-Jacobi equation, in charged static space- Because the tetrad 𝑒𝜇 in the space-time are given by time, we can separate the variables for the action as 𝑛−1 𝜂 𝑎 1 𝑆=−𝜔𝑡+𝑅(𝑟) +𝑌(...,𝑥 ,...)+𝐾, 𝑒𝜇 = diag (√𝑓, ,𝑟,𝑟sin 𝜃1,...,𝑟∏ sin 𝜃𝑖) , (27) √𝑓 (19) 𝑖=1 (𝐾 is a constant) so that Γ𝜇 is and the Hamilton-Jacobi equation is broken up as 󸀠 𝑎 𝜇 1 𝑛 𝑓 𝛾̃ 𝑒𝑎 Γ𝜇 = 𝛾̃ √𝑓( + ) 1 𝑑𝑅 2 𝜆 2𝑟 4𝑓 − (𝜔 − 𝑞𝐴 )2 +𝑓( ) +𝑚2 = 𝑓 𝑡 𝑑𝑟 𝑟2 (20) (28) 1 𝑛−1 (𝑛−𝑘) 𝜃 + ∑𝛾̃𝑘+1 cot 𝑘 . 𝜕𝑌 2 2𝑟 ∏𝑘−1 𝜃 ∑ℎ𝜂𝜂 ( ) +𝜆=0, 𝑘=1 𝑖=1 sin 𝑖 𝜂 (21) 𝜂 𝜕𝑥 It means the Dirac equation becomes where (20) and (21) are radial and nonradial equations, 𝛾̃0 𝜕 𝑖𝑞𝐴 𝜕 𝑛 𝑓󸀠 𝜆 𝑖 ( + 𝑡 )Ψ+𝛾̃1√𝑓( + + )Ψ respectively, and is a constant. However, we only research on √𝑓 𝜕𝑡 ℏ 𝜕𝑟 2𝑟 4𝑓 the radial equation, because the tunnel at the event horizon is radial. From (20), we can get 𝑛−1 𝛾̃𝑘+1 𝜕 (𝑛−𝑘) 𝜃 + ∑ ( + cot 𝑘 )Ψ 𝑘−1 (29) 2 𝑟∏ 𝜃 𝜕𝜃𝑘 2 𝑑𝑅 (𝑟) √(𝜔 − 𝑞𝐴 ) 𝑟2 +𝑓(𝜆−𝑚2𝑟2) 𝑘=1 𝑖=1 sin 𝑖 =± 𝑡 . (22) 𝑑𝑟 𝑓𝑟 𝛾̃𝑛+1 𝜕Ψ 𝑚 + + Ψ=0, 𝑛−1 𝑟∏ 𝜃 𝜕𝜃𝑛 ℏ Near the event horizon, the radial action is given by 𝑖=1 sin 𝑖 and this equation can be simplified at event horizon 𝜋(𝜔−𝜔0) Im 𝑅± =± + Im 𝐶, (23) 󸀠 𝑓󸀠 (𝑟 ) 0 𝜕 𝑖𝑞𝐴0 1 𝜕 𝑓 + 𝑖𝛾̃ ( + )Ψ+𝛾̃ ( + )Ψ=0, (30) 𝜕𝑡 ℏ 𝜕𝑟∗ 4 where 𝑅+ is part of the outgoing solution, while 𝑅− is the part 𝑓→0 𝑑𝑟 = 𝑑𝑟/𝑓 of incoming solution, and because at event horizon, and ∗ is tortoise coordinate. On the other hand, the space-time background is 𝑄 Ψ 𝜔 =𝑞 . static, and canberewrittenas 0 𝑛−1 𝑉 𝑟𝑛−1 (24) ( ) 𝑛 + 𝐴 (𝑟) −(𝑖/ℏ)𝜔𝑡 Ψ=[ ]𝑒 , (31) So the tunneling rate is 𝐵 (𝑟) 𝐴(𝑟) 𝐵(𝑟) 𝑚/2 × 1 𝜔 Prob [out] exp (−2 Im 𝑆+) where and are matrices with and is Γ= = frequency or energy of Dirac particle. Finally, applying the Prob [in] exp (−2 Im 𝑆−) 0 1 definitions of 𝛾̃ and 𝛾̃ ,weget exp (−2 Im 𝑅+ + Im 𝐾) 󸀠 = (25) 𝜕 𝑓 (−2 𝑅 + 𝐾) 𝜔−𝜔0 ℏ𝑓 ( − ) exp Im − Im 𝜕𝑟 4𝑓 𝐴𝑞 ( )( )=0; 𝜕 𝑓󸀠 𝐵 (32) −4𝜋 (𝜔0 −𝜔 ) 𝑞 = ( ), ℏ𝑓 ( − )−(𝜔−𝜔0) exp 󸀠 𝜕𝑟 4𝑓 𝑓 (𝑟+)

here 𝐴𝑞 and 𝐵𝑞 are 𝑞th elements of matrices 𝐴(𝑟) and 𝐵(𝑟), where Im represents the imaginary part of the function, and respectively. Above equation becomes theHawkingtemperatureis 󸀠 󸀠 𝜕𝐵𝑞/𝜕𝑟 (𝜔 −0 𝜔 )𝐴𝑞 −ℏ(𝑓𝐵𝑞/4) 𝑓 (𝑟+) = , (33) 𝑇 = . (26) 𝜕𝐴 /𝜕𝑟 󸀠 𝐻 4𝜋 𝑞 −(𝜔−𝜔0)𝐵𝑞 −ℏ(𝑓𝐴𝑞/4) 4 Advances in High Energy Physics so Therefore, the determinants of matrices vanish:

󸀠 𝜕𝐴 𝜕𝐵 0 𝐾0 𝜔−𝜔 𝜕 2 2 𝑓 𝑞 𝑞 ℏ 𝑅 =±∫ 𝑑𝑟, 0 (𝐴 +𝐵 )−ℏ (𝐵 −𝐴 ) : 𝑞0± 𝑓 2 𝜕𝑟 𝑞 𝑞 4 𝑞 𝜕𝑟 𝑞 𝜕𝑟 (34) 𝐾 −𝑖𝑓󸀠 (𝑟 )/4 =0. ℏ1 𝑅 =±∫ 1 + 𝑑𝑟, : 𝑞1± 𝑓 (40) 𝑓󸀠(𝑟 ) =0̸ 𝑟 𝐾 At event horizon 0 and depends on the position 0, ℏ𝑘 𝑅 =±∫ 𝑘 𝑑𝑟, 𝑘≥2, so above equation implies : 𝑞𝑘± 𝑓 2𝜋𝐾 𝜕 2 2 𝑖 Im 𝑅𝑞𝑖 = Im 𝑅𝑞𝑖+ − Im 𝑅𝑞𝑖− = . (𝐴𝑞 +𝐵𝑞)=0, 󸀠 𝜕𝑟 𝑓 (𝑟+) (35) 𝜕𝐴 𝑞 𝜕𝐵𝑞 In order to calculate the tunneling rate and Hawking temper- 𝐵𝑞 −𝐴𝑞 =0, 𝑖 𝜕𝑟 𝜕𝑟 ature, we rewrite Im 𝑅𝑞𝑖 =(𝛽𝑖/𝐴ℎ) Im 𝑅𝑞0 (𝑖≥0, 𝐴ℎ is area of black hole, and 𝛽𝑖 are dimensionless constant parameters) and the solution is since the forms of 𝑅𝑞𝑗 arethesame,sothetotalradialaction is 2 2 𝐴𝑞 +𝐵𝑞 =0. (36) ∞ 𝑖 Im 𝑅𝑞 = Im 𝑅𝑞0 (𝑟) + ∑ℏ 𝑅𝑞𝑖 (𝑟) 𝑖=1 Above relation means 𝐴𝑞 and 𝐵𝑞 canberewrittenas (41) ∞ ℏ𝑖 (𝑖/ℏ)𝑅 (𝑟) =(1+∑𝛽 )𝑅 . 𝐴 =𝐶𝑒 𝑞 , 𝑖 𝑖 𝑞0 𝑞 𝑞 𝑖=1 𝐴ℎ (37) (𝑖/ℏ)𝑅𝑞(𝑟) 𝐵𝑞 =𝐹𝑞𝑒 , The tunneling rate of Dirac particle at event horizon is given by and 𝐶𝑞 =±𝑖𝐹𝑞 are constants. 2 ∞ ℏ𝑖 Next, let us use the method beyond semiclassical approx- Γ = [− (1 + ∑𝛽 )𝑅 ] 𝑅 (𝑟) 𝐾=𝜔−𝜔 ℎ exp ℏ 𝑖 𝑖 𝑞0 imation to expand 𝑞 and 0 as 𝑖=1 𝐴ℎ (42) ∞ 4𝜋𝐾 ∞ ℏ𝑖 𝑖 = [− 0 (1 + ∑𝛽 )] . 𝑅𝑞 =𝑅𝑞0 + ∑ℏ 𝑅𝑞𝑖 (𝑟) , exp 󸀠 𝑖 𝑖 ℏ𝑓 (𝑟+) 𝐴 𝑖=1 𝑖=1 ℎ (38) ∞ 𝑖 From the relation between tunneling rate and Hawking 𝐾=𝜔−𝜔0 =𝐾0 + ∑ℏ 𝐾𝑖, radiation, we get the temperature of black holes 𝑖=1 ∞ ℏ𝑖 𝑇 =(1+∑𝛽 )𝑇 . so we get ℎ 𝑖 𝑖 𝐻 (43) 𝑖=1 𝐴ℎ 𝐾 𝜕𝑅 −𝑖 0 𝑞0 Finally, the laws of black hole thermodynamics request 𝑓 𝜕𝑟 𝐶 0 𝑞 󵄨 ℏ : ( )( )=0, 𝑑𝑀 − 𝐴𝑑𝑄󵄨 𝜕𝑅𝑞0 𝐾 𝐹 󵄨 0 𝑞 𝑆ℎ = ∫ 𝑑𝑆ℎ = ∫ 󵄨 −𝑖 𝑇 󵄨 𝜕𝑟 𝑓 ℎ 𝑟=𝑟+

󸀠 𝐴 (44) 𝐾 𝜕𝑅𝑞1 𝑖𝑓 = ℎ +𝜋𝛽 (𝐴 )+⋅⋅⋅ −𝑖 1 + 1 ln ℎ 𝑓 𝜕𝑟 4𝑓 𝐶 4𝜋 ℏ1 ( )( 𝑞)=0, : 󸀠 𝜕𝑅 𝑖𝑓 𝐾 𝐹 =𝑆𝐻 +𝜋𝛽1 ln (𝑆𝐻)+⋅⋅⋅ 𝑞1 + −𝑖 1 𝑞 (39) 𝜕𝑟 4𝑓 𝑓 and 𝑆𝐻 =𝐴ℎ/4𝜋 is entropy of semiclassical approximation, 𝐾 𝜕𝑅 and this result shows the correction of entropy is logarithmic −𝑖 𝑘 𝑞𝑘 𝐶 correction. 𝑘 𝑓 𝜕𝑟 𝑞 ℏ : ( )( )=0, In this paper, we studied fermions tunneling from higher- 𝜕𝑅 𝐾 𝐹 𝑞𝑘 −𝑖 𝑘 𝑞 dimensional Reissner-Nordstrom¨ black holes and obtained 𝜕𝑟 𝑓 the Hamilton-Jacobi equation from charged Dirac equation. This work shows that semiclassical Hamilton-Jacobi equation 𝑘≥2. can describe the property of both 0 spin scalar particles Advances in High Energy Physics 5

and 1/2 spin fermions. In this work, we did not emphasize [12] J. Ren, J. Y. Zhang, and Z. Zhao, “Tunnelling effect and hawking dimensions of space-time larger than (3+1) dimensions, radiation from a Vaidya black hole,” Chinese Physics Letters,vol. so the method also can be used in the research of (3+1)- 23,no.8,pp.2019–2022,2006. dimensions and lower cases. [13] S. Iso, H. Umetsu, and F. Wilczek, “Anomalies, Hawking radia- As we all know, the information loss is an open problem tions, and regularity in rotating black holes,” Physical Review D, in black hole physics, and the information of particles maybe vol. 74, no. 4, Article ID 044017, 10 pages, 2006. vanishes at the singularity. In order to solve this difficulty, [14] S. Iso, T. Morita, and H. Umetsu, “Quantum anomalies at Horowitz and Maldacena proposed a boundary condition, horizon and Hawking radiations in Myers-Perry black holes,” which is called the black hole final state, at singularity of black JournalofHighEnergyPhysics,vol.2007,no.4,article068,14 hole to perfectly entangle the incoming Hawking radiation pages, 2007. particles and the collapsing matter [66, 67]. Due to the [15]S.Z.Yang,H.L.Li,Q.Q.Jiang,andM.Q.Liu,“Theresearch boundary condition, any particle which is falling into the on the quantum tunneling characteristics and the radiation black holes completely annihilates. It is a new and interesting spectrum of the stationary axisymmetric black hole,” Science in idea to investigate the black hole physics and quantum China Series G, vol. 50, no. 2, pp. 249–260, 2007. gravity, so we also will work on this area in the future. [16] E. C. Vagenas, “Are extremal 2D black holes really frozen?” Physics Letters B,vol.503,no.3-4,pp.399–403,2001. [17] E. C. Vagenas, “Two-dimensional dilatonic black holes and Competing Interests Hawking radiation,” Modern Physics Letters A,vol.17,no.10,pp. 609–618, 2002. The authors declare that they have no competing interests. [18] E. C. Vagenas, “Semiclassical corrections to the Bekenstein– Hawking entropy of the BTZ black hole via self-gravitation,” Acknowledgments Physics Letters B,vol.533,no.3-4,pp.302–306,2002. [19] A. J. M. Medved, “Radiation via tunnelling in the charged BTZ ThisworkissupportedinpartbyFAPESPno.2012/08934-0, black hole,” Classical and Quantum Gravity,vol.19,no.3,pp. CNPq, CAPES, and National Natural Science Foundation of 589–598, 2002. China (no. 11573022 and no. 11375279). [20] M. K. Parikh, “New coordinates for de Sitter space and de Sitter radiation,” Physics Letters B,vol.546,no.3-4,pp.189–195,2002. References [21] A. J. M. Medved, “Radiation via tunneling from a de Sitter cosmological horizon,” Physical Review D,vol.66,no.12,Article [1] S. W. Hawking, “Black hole explosions?” Nature,vol.248,no. ID 124009, 2002. 5443, pp. 30–31, 1974. [22] E. C. Vagenas, “Generalization of the KKW analysis for black [2]S.W.Hawking,“Particlecreationbyblackholes,”Communica- hole radiation,” Physics Letters B,vol.559,no.1-2,pp.65–73, tions in Mathematical Physics,vol.43,no.3,pp.199–220,1975. 2003. [3] S. P. Robinson and F. Wilczek, “Relationship between Hawking [23] M. K. Parikh, “Energy conservation and Hawking radiation,” radiation and gravitational anomalies,” Physical Review Letters, https://arxiv.org/abs/hep-th/0402166. vol. 95, no. 1, 011303, 4 pages, 2005. [24] J. Y. Zhang and Z. Zhao, “Massive particles’ Hawking radiation [4] T. Damoar and R. Ruffini, “Black-hole evaporation in the Klein- via tunneling,” Acta Physica Sinica,vol.55,p.3796,2006 Sauter-Heisenberg-Euler formalism,” Physical Review D,vol.14, (Chinese). no.2,p.332,1976. [25] J. Y. Zhang and Z. Zhao, “Charged particles’ tunnelling from the [5] S. Sannan, “Heuristic derivation of the probability distributions Kerr-Newman black hole,” Physics Letters B,vol.638,no.2-3,pp. of particles emitted by a black hole,” General Relativity and 110–113, 2006. Gravitation, vol. 20, no. 3, pp. 239–246, 1988. [26] E. T. Akhmedov, V. Akhmedova, D. Singleton, and T. Pilling, [6] P. Kraus and F. Wilczek, “Self-interaction correction to black “Thermal radiation of various gravitational backgrounds,” Inter- hole radiance,” Nuclear Physics B,vol.433,no.2,pp.403–420, national Journal of Modern Physics A,vol.22,no.8-9,pp.1705– 1995. 1715, 2007. [7] M. K. Parikh and F. Wilczek, “Hawking radiation as tunneling,” [27] E. T. Akhmedov, V. Akhmedova, and D. Singleton, “Hawking Physical Review Letters,vol.85,no.24,pp.5042–5045,2000. temperature in the tunneling picture,” Physics Letters B,vol.642, [8] S. Hemming and E. Keski-Vakkuri, “Hawking radiation from no. 1-2, pp. 124–128, 2006. AdS black holes,” Physical Review D,vol.64,no.4,ArticleID [28] B. D. Chowdhury, “Problems with tunneling of thin shells from 044006, 8 pages, 2001. black holes,” Pramana,vol.70,no.1,pp.3–26,2008. [9] Q.-Q. Jiang, S.-Q. Wu, and X. Cai, “Hawking radiation from [29] E. T. Akhmedov, T. Pilling, and D. Singleton, “Subtleties in the dilatonic black holes via anomalies,” Physical Review D,vol.75, quasi-classical calculation of hawking radiation,” International no.6,ArticleID064029,10pages,2007. Journal of Modern Physics D,vol.17,no.13-14,p.2453,2008. [10] S.-Q. Wu and Q.-Q. Jiang, “Hawking radiation of charged [30] V.Akhmedova, T. Pilling, A. de Gill, and D. Singleton, “Tempo- particles as tunneling from higher dimensional Reissner- ral contribution to gravitational WKB-like calculations,” Physics Nordstrom-de Sitter black holes,” https://arxiv.org/abs/hep-th/ Letters B,vol.666,no.3,pp.269–271,2008. 0603082. [31] V. Akhmedova, T. Pilling, A. de Gill, and D. Singleton, [11] R. Kerner and R. B. Mann, “Tunnelling, temperature, and Taub- “Comments on anomaly versus WKB/tunneling methods for NUT black holes,” Physical Review D,vol.73,no.10,ArticleID calculating Unruh radiation,” Physics Letters B,vol.673,no.3, 104010, 2006. pp. 227–231, 2009. 6 Advances in High Energy Physics

[32] K. Srinivasan and T. Padmanabhan, “Particle production and [52] S. J. Gao and J. P. Lemos, “Collapsing and static thin massive complex path analysis,” Physical Review D,vol.60,ArticleID charged dust shells in a Reissner-Nordstrom¨ black hole back- 024007, 1999. ground in higher dimensions,” International Journal of Modern [33] S. Shankaranarayanan, T. Padmanabhan, and K. Srinivasan, Physics A, vol. 23, no. 19, pp. 2943–2960, 2008. “Hawking radiation in different coordinate settings: complex [53] R. A. Konoplya and A. Zhidenko, “Stability of multidimensional paths approach,” Classical and Quantum Gravity,vol.19,no.10, black holes: complete numerical analysis,” Nuclear Physics B, pp. 2671–2687, 2002. vol.777,no.1-2,pp.182–202,2007. [34] M. Angheben, M. Nadalini, L. Vanzo, and S. Zerbini, “Hawking [54] R. A. Konoplya and A. Zhidenko, “Instability of higher- radiation as tunneling for extremal and rotating black holes,” dimensional charged black holes in the de Sitter world,” Physical Journal of High Energy Physics (JHEP),vol.505,p.14,2005. Review Letters,vol.103,no.16,ArticleID161101,4pages,2009. [35] S. Z. Yang and D. Y. Chen, “Hawking radiation as tunneling [55] R. A. Konoplya and A. Zhidenko, “Stability of higher dimen- from the Vaidya-Bonner black hole,” International Journal of sional Reissner-Nordstrom-anti-de¨ Sitter black holes,” Physical Theoretical Physics,vol.46,no.11,pp.2923–2927,2007. Review D,vol.78,no.10,ArticleID104017,9pages,2008. [36] D. Y. Chen and S. Z. Yang, “Hawking radiation of the Vaidya- [56] R. Banerjee and B. R. Majhi, “Quantum tunneling beyond Bonner-de Sitter black hole,” New Journal of Physics,vol.9,no. semiclassical approximation,” JournalofHighEnergyPhysics, 8, p. 252, 2007. vol.2008,no.6,article095,2008. [37] R. Kerner and R. B. Mann, “Fermions tunnelling from black [57] R. Banerijee and B. R. Majhi, “Quantum tunneling and trace holes,” Classical and Quantum Gravity,vol.25,no.9,095014, anomaly,” Physics Letters B,vol.674,no.3,pp.218–222,2009. 17 pages, 2008. [58] B. R. Majhi, “Fermion tunneling beyond semiclassical approx- [38] R. Kerner and R. B. Mann, “Charged fermions tunnelling from imation,” Physical Review D,vol.79,no.4,ArticleID044005, Kerr-Newman black holes,” Physics Letters B,vol.665,no.4,pp. 2009. 277–283, 2008. [59] R. Banerjee and B. R. Majhi, “Connecting anomaly and tunnel- [39] R. Li, J. R. Ren, and S. W. Wei, “Hawking radiation of Dirac ing methods for the Hawking effect through chirality,” Physical particles via tunneling from the Kerr black hole,” Classical and Review D,vol.79,no.6,ArticleID064024,5pages,2009. Quantum Gravity,vol.25,no.12,ArticleID125016,1250. [60] R. Banerijee, B. R. Majhi, and D. Roy, “Corrections to Unruh [40] R. Li and J.-R. Ren, “Dirac particles tunneling from BTZ black effect in tunneling formalism and mapping with Hawking hole,” Physics Letters B,vol.661,no.5,pp.370–372,2008. effect,” https://arxiv.org/abs/0901.0466. [41] D.-Y. Chen, Q.-Q. Jiang, and X.-T. Zu, “Fermions tunnelling [61] B. R. Majhi and S. Samanta, “Hawking radiation due to photon from the charged dilatonic black holes,” Classical and Quantum and gravitino tunneling,” Annals of Physics,vol.325,no.11,pp. Gravity,vol.25,no.20,ArticleID205022,2008. 2410–2424, 2010. [42] D.-Y. Chen, Q.-Q. Jiang, and X.-T. Zu, “Hawking radiation of [62] R. Banerijee and B. R. Majhi, “Hawking black body spectrum Dirac particles via tunnelling from rotating black holes in de from tunneling mechanism,” Physics Letters B,vol.675,no.2, Sitter spaces,” Physics Letters B,vol.665,no.2-3,pp.106–110, pp. 243–245, 2009. 2008. [63] R. Banerjee and B. R. Majhi, “Quantum tunneling and back [43] X.-X. Zeng and S.-Z. Yang, “Fermions tunneling from Reissner- reaction,” Physics Letters B,vol.662,no.1,pp.62–65,2008. Nordstrom¨ black hole,” General Relativity and Gravitation,vol. [64] R. Banerjee, B. R. Majhi, and S. Samanta, “Noncommutative 40, no. 10, pp. 2107–2114, 2008. black hole thermodynamics,” Physical Review D,vol.77,no.12, [44] K. Lin and S. Z. Yang, “Fermions tunneling of the Vaidya- ArticleID124035,8pages,2008. Bonner black hole,” Acta Physica Sinica,vol.58,no.2,pp.744– [65] S. K. Modak, “Corrected entropy of BTZ black hole in tunneling 748, 2009 (Chinese). approach,” Physics Letters B,vol.671,no.1,pp.167–173,2009. [45] R. Di Criscienzo and L. Vanzo, “Fermion tunneling from [66] G. T. Horowitz and J. Maldacena, “The black hole final state,” dynamical horizons,” Europhysics Letters,vol.82,no.6,Article JournalofHighEnergyPhysics,vol.2004,no.2,article008,15 ID 60001, 2008. pages, 2004. [46] L. H. Li, S. Z. Yang, T. J. Zhou, and R. Lin, “Fermion tunneling [67] G. T. Horowitz and E. Silverstein, “The inside story: quasilocal from a Vaidya black hole,” EPL (Europhysics Letters),vol.84,no. tachyons and black holes,” Physical Review D,vol.73,no.6, 2, p. 20003, 2008. Article ID 064016, 14 pages, 2006. [47] Q.-Q. Jiang, “Fermions tunnelling from GHS and non-extremal D1–D5 black holes,” Physics Letters B,vol.666,no.5,pp.517–521, 2008. [48] Q.-Q. Jiang, “Dirac particle tunneling from black rings,” Physical Review D,vol.78,no.4,ArticleID044009,8pages,2008. [49] K. Lin and S. Z. Yang, “Fermion tunneling from higher- dimensional black holes,” Physical Review D,vol.79,no.6, Article ID 064035, 2009. [50] K. Lin and S. Z. Yang, “Fermions tunneling of higher- dimensional Kerr-anti-de Sitter black hole with one rotational parameter,” Physics Letters B,vol.674,no.2,pp.127–130,2009. [51] F. R. Tangherlini, “Schwarzschild field inn dimensions and the dimensionality of space problem,” Il Nuovo Cimento,vol.27,no. 3, pp. 636–651, 1963. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 3650632, 10 pages http://dx.doi.org/10.1155/2016/3650632

Research Article Matter Localization on Brane-Worlds Generated by Deformed Defects

Alex E. Bernardini1 andRoldãodaRocha2 1 Departamento de F´ısica, Universidade Federal de Sao˜ Carlos, P.O. Box 676, 13565-905 Sao˜ Carlos, SP, Brazil 2Centro de Matematica,´ Computac¸ao˜ e Cognic¸ao,˜ Universidade Federal do ABC (UFABC), 09210-580 Santo Andre,´ SP, Brazil

Correspondence should be addressed to Roldao˜ da Rocha; [email protected]

Received 14 March 2016; Revised 21 April 2016; Accepted 16 June 2016

Academic Editor: Barun Majumder

Copyright © 2016 A. E. Bernardini and R. da Rocha. 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.

Localization and mass spectrum of bosonic and fermionic matter fields of some novel families of asymmetric thick brane configurations generated by deformed defects are investigated. The localization profiles of spin 0, spin 1/2, and spin 1 bulk fields are identified for novel matter field potentials supported by thick branes with internal structures. The condition for localization is constrained by the brane thickness of each model such that thickest branes strongly induce matter localization. The bulk mass terms for both fermion and boson fields are included in the global action as to produce some imprints on mass-independent potentials of the Kaluza-Klein modes associated with the corresponding Schrodinger¨ equations. In particular, for spin 1/2 fermions, a complete analytical profile of localization is obtained for the four classes of superpotentials here discussed. Regarding the localization of fermion fields, our overall conclusion indicates that thick branes produce a left-right asymmetric chiral localization of spin 1/2 particles.

1. Introduction have been further studied [11–15], and analogous scenarios in an expanding Universe have been approached [16, 17]. The The brane-world model is a prominent paradigm that has curvature nature of the brane-world, namely, to be a de Sitter, been addressed to solve several questions in physics. Within Minkowski, or anti-de Sitter one, is in general obtained a this framework, brane-worlds are required to render a con- posteriori,bysolvingthe5𝐷 Einstein field equations. In fact, 4𝐷 sistent physics of our Universe, at least up to certain the bulk and the brane cosmological constants depend upon sensible limits [1]. In the brane-world scenario all kinds of the brane and the bulk gravitational field content, governed matter fields should be localized on the brane. In the RS by curvature, and must obey the intrinsic fine-tuning, in the brane-world model [2], the brane is generated by a scalar field Randall-Sundrum-like models limit. coupled to gravity [3, 4], in a particular scenario which may The analytical study of stability can be uncontrollably be interpreted as the thin brane limit of thick brane scenarios. intricate, due to the involved structure of the scalar field Generically, a prominent test that thick brane-world models coupledtogravity.Tocircumventthecomplicatedandnot must pass, to be physically consistent, regards their stability, analytical approaches, linearized formulations have been with respect to tensor, vector, and scalar fluctuations of commonly worked out. In this context, supported by the the background fields that generate the field configurations, stability of deformed defect generated brane-world mod- namely, the thick brane itself. At least the zero modes of els, scalar, vector, and tensor perturbations are investigated Standard Model matter fields were shown to be localized on throughout this work. several brane-world models [5–10], suggesting that such kind Localization aspects of various matter fields with spin 0, of models is physically viable in high energy physics. Several spin 1/2, and spin 1 on analytical thick brane-world models are alternative scenarios, including Gauss-Bonnet terms, 𝑓(𝑅) indeed a main concern in deriving brane-world models, since gravity, tachyonic potentials, cyclic defects, and Bloch branes, they must describe our physical 4𝐷 world. The localization 2 Advances in High Energy Physics

𝐵𝑀 of the spin 1/2 fermions deserves a special attention, since where 𝑅 is the 5𝐷 scalar curvature, 𝑅𝑁𝑄 =𝑔 𝑅𝐵𝑁𝑄𝑀 is the 1/2 thereisnoscalarfieldtocouplewithinthismodel,in Ricci tensor, and 𝜅5 =(8𝜋𝐺5) denotes the 5𝐷 gravitational contrast to thick branes generated by deforming defect coupling constant, hereon set to be equal to unity, where 𝐺5 mechanisms [18]. Otherwise, Kalb-Ramond fields, although is the 5𝐷 Newton constant. The Einstein equations read already investigated [19], will not be the main aim here. The 1 spin 1/2 issue has been previously studied in some other 2 𝜁 𝑅𝑀𝑁 − 𝑅𝑔𝑀𝑁 =−Λ5𝑔𝑀𝑁 +𝜅5𝑇𝑀𝑁, (3) contexts [20], including further coupling of more scalar fields 2 in the action [21] and asymmetric brane-worlds generated by 𝜁 where 𝑇 denotes the energy-momentum tensor corre- a plenty of scalar field potentials [8, 22–25]. In particular, 𝑀𝑁 sponding to the matter Lagrangian, regarding the matter field asymmetric Bloch branes in the context of the hierarchy 𝜁.Aftersolvingthe5𝐷 Einstein field equations, the bulk problem have been addressed in [13]. cosmological constant turns out, in general, to be positive Our aim is to investigate the localization of bulk matter or negative, thus realising a de Sitter or anti-de Sitter brane- and gauge fields on the brane, in the context where the mass- world, respectively, generated by curvature. It realises and independent potentials of the corresponding Schrodinger-¨ emulates the interplay involving the 4𝐷 and 5𝐷 cosmological like equations, regarding the 1𝐷 quantum mechanical ana- constants. Some further possibilities are devised, for example, logue problem, can be suitably acquired from a warped met- in [14, 26]; however it is worth mentioning that an addi- ric. In particular, for a bulk mass proportional to the fermion tionalscalarfieldcanbestilladdedintheaction,whose mass term enclosed by the global action, the possibility of isotropisation will precisely define the nature of the brane- trapping spin 1/2 fermions on asymmetric branes is discussed world. This latter case is however beyond the scope of our and quantified. analysis.Obviously,whateverthepossibilitytobeconsidered, To accomplish this aim, this paper is organized as follows. the thin brane limit must obey the fine-tuning relation [27] In Section 2, a brief review of brane-world scenarios sup- 2 2 2 Λ 4 =(𝜅5/2)((1/6)𝜅5𝜎 +Λ5),amongtheeffective4𝐷 and ported by an effective action driven by a (dark sector) scalar 5𝐷 𝜎 field is presented. Warp factors and the corresponding inter- cosmological constants and the brane tension as well. nal brane structure are described for four different analytical Considering the real scalar field action, (2), one can models. In Section 3, the left-right-chiral asymmetric aspects compute the stress-energy tensor 1 of matter localization for spin 1/2 fermion fields on thick 𝑇𝜁 =𝜕 𝜁𝜕 𝜁+𝑔 𝑉 (𝜁) − 𝑔 𝑔𝐴𝐵𝜕 𝜁𝜕 𝜁, branes are investigated. Extensions to scalar boson and vector 𝑀𝑁 𝑀 𝑁 𝑀𝑁 2 𝑀𝑁 𝐴 𝐵 (4) bosonfieldsareobtainedinSections4and5,respectively. Final conclusions are drawn in Section 6. which, supposing that both the scalar field and the warp factor dynamics depend only upon the extra coordinate, 𝑦, leads to an explicit dependence of the energy density in terms 2. Brane-World Preliminaries and of the field, 𝜁, and of its first derivative, 𝑑𝜁/𝑑𝑦,as Some Analytical Models 2 𝜁 1 𝑑𝜁 2𝐴(𝑦) 𝑇 (𝑦) = [ ( ) +𝑉(𝜁)]𝑒 . (5) Let one start considering a 5𝐷 space-time warped into 4𝐷. 00 2 𝑑𝑦 The most general 5𝐷 metric compatible with a brane-world 𝜁 spatially flat cosmological background has the form given by With the same constraints on about the dependence on 𝑦, the equations of motion currently known from [3, 4], 2 𝑀 𝑁 2𝐴(𝑦) 𝛼 𝜇 ] 2 which arise from the above action, are 𝑑𝑠 =𝑔𝑀𝑁𝑑𝑥 𝑑𝑥 =𝑒 𝑔𝜇] (𝑥 )𝑑𝑥 𝑑𝑥 +𝑑𝑦 , (1) 𝑑2𝜁 𝑑𝐴 𝑑𝜁 𝑑 2𝐴(𝑦) +4 − 𝑉 (𝜁) =0, (6) where 𝑒 denotes the warp factor, and the signature 𝑑𝑦2 𝑑𝑦 𝑑𝑦 𝑑𝜁 (−++++) is employed, with 𝑀, 𝑁 = 0, 1, 2,. 3,5 𝑔𝜇] stands 𝜁 for the components of the 4𝐷 metric tensor (𝜇, ] =0,1,2,3). through a variational principle relative to the scalar field, , One can identify 𝑦≡𝑥4 as the infinite extra dimension and coordinate (which runs from −∞ to ∞) and notice that the 3 𝑑2𝐴 𝑑𝜁 2 𝑦 =−( ) , (7) normal to surfaces of constant is orthogonal to the brane, 2 𝑑𝑦2 𝑑𝑦 into the bulk (brane tension terms have been suppressed/ absorbed by the metric (c.f. Equations (24) and (25) from [10] through a variational principle relative to the metric, or for real scalar field Lagrangians in the context of thick brane equivalently to 𝐴, manipulated to result into solutions)). 𝑑𝐴 2 1 𝑑𝜁 2 The brane-world scenario examined here is set up by 3( ) = ( ) −𝑉(𝜁) , (8) an effective action, driven by a (dark sector) scalar field, 𝜁, 𝑑𝑦 2 𝑑𝑦 coupled to 5𝐷 gravity, given by after an integration over 𝑦. 1 For the scalar field potential written in terms of a 𝑆 =−∫ 𝑑𝑥5√ 𝑔 [ (𝜅−2𝑅−2Λ ) 𝑤 eff det 𝑀𝑁 4 5 5 superpotential, ,as (2) 2 1 1 𝑑𝑤 1 2 + 𝑔 𝜕𝑀𝜁𝜕𝑁𝜁−𝑉(𝜁)], 𝑉 (𝜁) = ( ) − 𝑤 , (9) 2 𝑀𝑁 8 𝑑𝜁 3 Advances in High Energy Physics 3 the above equations are mapped into first-order equations [3, Fromtheabovesuperpotentials,therespectivesolutions 4] as for 𝜁(𝑦) are set as

𝑑𝜁 1 𝑑𝑤 𝑦 = , (10) 𝜁I (𝑦) = √6 arctan [tanh ( )] , 𝑑𝑦 2 𝑑𝜁 2√2𝑎 𝑑𝐴 1 √3𝑦 =− 𝑤, (11) 𝜁II (𝑦) = 3 ( ), 𝑑𝑦 3 sech 2𝑎 (16) 𝑦 𝜁III (𝑦) = ( ), for which the solutions can be found straightforwardly arcsinh 𝑎 through immediate integrations [3] (see also [10] and refer- 𝑦 ences therein). The energy density follows from (9) as 𝜁IV (𝑦) = , √𝑎2 +𝑦2 2 𝜁 1 𝑑𝑤 1 2 2𝐴(𝑦) 𝑇00 (𝑦) = [ ( ) − 𝑤 ] 𝑒 . (12) 4 𝑑𝜁 3 where one has suppressed any additional (irrelevant) constant of integration for convenience, and one has just considered thepositivesolutions(in(16)therecouldbeexplicitconstant The analysis of localization aspects of brane-world scenar- of integration that amounts to letting 𝑦 󳨃→𝑦+𝐶, correspond- ios will be constrained by some known examples, I, II, III, and ing to the position of the brane in the extra dimension, for IV, for which the warp factor, 𝐴(𝑦), and the energy density, which one has set 𝐶=0). 𝑇00(𝑦), can be analytically computed. Model I is supported The obtained expressions for the warp factor as resulting by a sine-Gordon-like superpotential given by from (11) are, respectively, given by

2 2 𝑤I (𝜁) = (√ 𝜁) , 𝑦 √ sin 3 (13) 𝐴I (𝑦) = − ln [cosh ( )] , 2𝑎 √2𝑎

√3𝑦 2 √3𝑦 which reproduces the results from [4]. Model II corresponds II 4 𝐴 (𝑦) = tanh ( ) −2ln [cosh ( )] , to a deformed 𝜆𝜁 theory with the superpotential given by 2𝑎 2𝑎 (17) 1 𝑦2 𝑦 𝑦 3/2 𝐴III (𝑦) = [ (1 + )−2 ( )] , 3√3 𝜁2 3 ln 𝑎2 𝑎 arctan 𝑎 𝑤II (𝜁) = (1 − ) . (14) 𝑎 9 1 𝑦2 𝑦 𝑦 𝐴IV (𝑦) = − [ +3 ( )] , 12 𝑎2 +𝑦2 𝑎 arctan 𝑎 Models III and IV are deformed topological solutions from [28] supported by superpotentials like where integration constants are introduced as to set a normal- 𝐴(0) = 0 2 ization criterion for which . 𝑤III (𝜁) = [ (𝜁)], The solutions for 𝐴I and 𝐴II are depicted in Figure 1. The 𝑎 arctan sinh corresponding localized energy densities computed through 𝑤IV (𝜁) (12) are, respectively, given by (15)

1 2 𝜁 2 2 = [𝜁 (5 − 2𝜁 ) √1−𝜁2 +3arctan ( )] , 3 𝑦 𝑦 4𝑎 √1−𝜁2 𝑇I (𝑦) = sech ( ) [sech ( ) 00 4𝑎2 √2𝑎 √2𝑎 𝑦 2 where the parameter 𝑎 fixes the thickness of the brane −2 ( ) ], 2𝐴(𝑦) tanh described by the warp factor, 𝑒 . Besides exhibiting √2𝑎 analytically manipulable profiles, the above superpotentials 8 2 have already been discussed in the context of thick brane 9 √3𝑦 √3𝑦 𝑇II (𝑦) = sech ( ) tanh ( ) localization[4,5,10].ModelsIandIIare,respectively, 00 8𝑎2 2𝑎 2𝑎 4 motivated by sine-Gordon and 𝜆𝜁 theories, and models √ √ III and IV are obtained (also analytically) from deformed 3𝑦 2 3𝑦 4 ⋅[7cosh ( )−cosh ( )] versions of the 𝜆𝜁 model [12]. In particular, models III and 𝑎 𝑎 IVcanalsobemappedontotachyonicLagrangianversions 2 tanh(√3𝑦/2𝑎)2 of scalar field brane models [6, 10, 28]. ⋅𝑒 , 4 Advances in High Energy Physics

1.0 1.0 (y)] (y)] II 0.8 IV 0.8 [2A [2A 0.6 0.6 Exp Exp (y)], (y)],

I 0.4 0.4 III [2A 0.2 [2A 0.2 Exp Exp 0.0 0.0 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15 y y

(a) (b)

2𝐴(𝑦) 2𝐴(𝑦) Figure 1: (a) Warp factors, 𝑒 for models I (solid (black) lines) and II (dashed (red) lines). (b) Warp factors, 𝑒 for models III (solid (black) lines) and IV (dashed (red) lines). One has considered integer values of the brane width parameter 𝑎, running from 1 (thinnest line) to 4 (thickest line), corresponding to an increasing thickness.

−1/3 1 𝑦2 the following sections. Spin 0, spin 1/2, and spin 1 fields will 𝑇III (𝑦) = (1 + ) (1 00 𝑎2 𝑎2 evolvecoupledtogravityand,asusual,thebulkmatterfield contributiontothebulkenergywillbeneglected.Itmeans 4 𝑎2 𝑦 2 that the obtained solutions hold in the presence of the bulk − (1 + ) ( ) )𝑒(4𝑦/3𝑎) arctan(𝑦/𝑎), 3 𝑦2 arctan 𝑎 matter, without disturbing the bulk geometry.

𝑎4 3. Asymmetric Left-Right Matter Localization 𝑇IV (𝑦) = ( 00 (𝑎2 +𝑦2)3 for Spin 1/2 Fermion Fields To investigate the localization of bulk matter on the brane, 3 3 2 2 2 2 5𝑎 𝑦+3𝑎𝑦 +3(𝑎 +𝑦 ) arctan (𝑦/𝑎) one first considers that fermion localization on brane-worlds − ) 5𝐷 48𝑎2 (𝑎2 +𝑦2)4 is usually accomplished when the Dirac algebra is realised Γ𝑀 =𝑒𝑀Γ𝑀 𝑒𝑀 by the objects 𝑀 ,where 𝑀 denotes the funfbein¨ , 2 2 𝑀 𝑀 𝑁 𝑀𝑁 𝑀 ⋅𝑒−(𝑦/2𝑎)((𝑎𝑦/3(𝑎 +𝑦 ))+arctan(𝑦/𝑎)). Γ satisfy the Clifford relation {Γ ,Γ }=2𝑔 ,andΓ 5𝐷 (18) arethegammamatricesinthe flat space-time. Hereupon 𝑀, 𝑁,...= 0,1,2,3,5and 𝜇, ],... = 0,1,2,3denote the 5𝐷 and 4𝐷 local Lorentz indexes, respectively. The funfbein¨ is The brane scenarios for models from I to IV are depicted 𝑒𝑀 ={𝑒𝐴̃𝑒] ,𝑒𝐴} Γ𝑀 =𝑒−𝐴(𝛾𝜇,𝛾5) in Figure 1 for the warp factors and in Figure 2 for the energy provided by 𝑀 𝜇 ,where ,and 𝛾𝜇 = ̃𝑒𝜇𝛾] 𝛾5 4𝐷 densities, from which one can observe that models from I to ] and are, respectively, the gamma matrices IV give rise to thick branes, most of them with no internal and the 4𝐷 volume element, respectively. The Dirac action structures. In fact, only the potential that controls the scalar for a spin 1/2 fermion with a mass term can be expressed as field from model II allows the emergence of thick branes that [20, 25] host internal structures in the form of a layer of a novel phase 𝑆 enclosed by two separate interfaces, inside which the energy 1/2 density of the matter field gets more concentrated. It is related 5 𝑀 (19) to the extension/localization of the warp factor; namely, when = ∫ 𝑑 𝑥√−𝑔ΨΓ [ (𝜕𝑀 +𝜔𝑀)Ψ−𝑀𝐹(𝑧) ΨΨ] . the profiles depicted in Figure 1 approach a plateau form in the region inside the brane, the corresponding internal 𝑅𝑆 Here 𝜔𝑅 = (1/4)𝜔 Γ Γ is the spin connection, where structure is observed through its energy profile. 𝑅 𝑅 𝑆 The appearance of negative energy densities in the plots 𝑅𝑆 1 𝑇𝑅 𝑄𝑆 𝑇 1 𝑆[𝑅 𝑆] 𝜔 =− 𝑒 𝑒 𝜕 𝑒 𝑒 + 𝑒 𝜕 𝑒 , (20) for 𝑇00 may be related to a predominance of the scalar field 𝑅 2 [𝑇 𝑄]𝑇 𝑅 2 [𝑅 𝑆] potential over the kinetic-like term related to the coordinate 𝑦. Speculatively, it indicates that the vacuum minimal energy and 𝐹(𝑧) is some general scalar function, providing a mass canbeadjustedbytheinclusionofsomeadditionalterm, term with a kink-like profile, which from this point is written −𝐴(𝑦) eventually related to the cosmological constant. in terms of a conformal variable 𝑧 such that 𝑑𝑧 =𝑒 𝑑𝑦 The localization of bulk matter fields on thick branes regards a transformation to conformal coordinates. This kind generatedbyeachoneofthesemodelswillbeidentifiedin of mass term is introduced in the action, for it has played Advances in High Energy Physics 5

1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4 y) ( y) ( II I T T 0.2 0.2

0.0 0.0

−0.2 −0.2

−0.4 −0.4 −6 −4 −2 0 2 4 6 −6 −4 −2 0246 y y

(a) (b) 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4 y) ( (y)

0.2 IV III 0.2 T T 0.0 0.0

−0.2 −0.2

−0.4 −0.4 −6 −4 −2 0246 −6 −4 −2 0246 y y

(c) (d)

Figure 2: Energy density, 𝑇00(𝑦), for models from I (a) to IV (d), where integer values of the brane width parameter 𝑎, running from 1 (thinnest line) to 4 (thickest line), corresponding to an increasing thickness.

a critical role on the localization of fermionic fields on a the 𝐿𝑛(𝑧) and 𝑅𝑛(𝑧) functions should then satisfy the subse- Minkowski brane. The components of the spin connection quent coupled equations, 𝜔𝑀 with respect to (1) are 𝜔𝜇 = (1/2)(𝜕𝑧𝐴)𝛾𝜇𝛾5 + 𝜔̂𝜇,where 𝜇] [𝜕 −𝑒𝐴𝑀𝐹 (𝑧)]𝑅 (𝑧) =−𝑚𝐿 (𝑧) , 𝜔̂𝛼 = (1/4)𝜔𝛼 Γ𝜇Γ] is the spin connection derived from the 𝑧 𝑛 𝑛 𝑛 (23a) 𝜇 ] metric 𝑔𝜇] = ̃𝑒𝜇̃𝑒]𝜂𝜇]. Thus, the equation of motion corre- 𝐴 [𝜕𝑧 +𝑒 𝑀𝐹 (𝑧)]𝐿𝑛 (𝑧) =𝑚𝑛𝑅𝑛 (𝑧) . (23b) sponding to the action (19) reads The associated Schrodinger-like¨ equations can be thus 𝜇 5 𝐴 [𝛾 (𝜕𝜇 + 𝜔̂𝜇)+𝛾 (𝜕𝑧 +2𝜕𝑧𝐴) − 𝑒 𝑀𝐹 (𝑧)]Ψ=0. (21) acquired for the left- and right-chiral KK modes of fermions, respectively, as

The 5𝐷 Dirac equation can be hence studied by taking 2 2 (−𝜕𝑧 +𝑉𝐿 (𝑧))𝐿𝑛 =𝑚𝐿 𝐿𝑛, 4𝐷 𝑛 spinors with respect to effective fields. In this way the (24) chiral splitting yields (−𝜕2 +𝑉 (𝑧))𝑅 =𝑚2 𝑅 , 𝑧 𝑅 𝑛 𝑅𝑛 𝑛

−2𝐴(𝑧) 𝜇 𝜇 where the mass-independent potentials are given by Ψ=𝑒 (∑𝜓𝐿𝑛 (𝑥 )𝐿𝑛 (𝑧) +𝜓𝑅𝑛 (𝑥 )𝑅𝑛 (𝑧)), (22) 𝑛 2𝐴 2 2 𝐴 󸀠 𝑉𝐿 (𝑧) =𝑒 𝑀 𝐹 (𝑧) −𝑒 𝐴 𝑀𝐹 (𝑧) (25a) 𝐴 where 𝐿𝑛(𝑧) and 𝑅𝑛(𝑧) are the well-known KK modes, and −𝑒 𝑀𝜕 𝐹 (𝑧) , 𝜇 5 𝜇 𝜇 5 𝜇 𝑧 𝜓𝑅𝑛(𝑥 )=+𝛾𝜓𝑅𝑛(𝑥 ) [𝜓𝐿𝑛(𝑥 )=𝛾𝜓𝐿𝑛(𝑥 )]istheright- 4𝐷 2𝐴 2 2 𝐴 󸀠 chiral [left-chiral] component of a Dirac field, respec- 𝑉𝑅 (𝑧) =𝑒 𝑀 𝐹 (𝑧) +𝑒 𝐴 𝑀𝐹 (𝑧) tively. In addition, the sum over 𝑛 can be both continuous (25b) 𝜇 𝐴 and discrete. Assuming that 𝛾 (𝜕𝜇 + 𝜔̂𝜇)𝜓(𝑅,𝐿)𝑛 =𝑚𝑛𝜓(𝐿,𝑅)𝑛, +𝑒 𝑀𝜕𝑧𝐹 (𝑧) . 6 Advances in High Energy Physics

Note that the Schrodinger-like¨ equations (24) can be trans- asymptotic behaviors that tend to zero from up, as 𝑦→±∞, † 2 † 2 𝑦=0 formed into 𝑈 𝑈𝐿 𝑛 =𝑚𝑛𝐿𝑛 and 𝑈𝑈 𝑅𝑛 =𝑚𝑛𝑅𝑛, for all models from I to IV. In model I, at the potential 𝐴 𝑉 (𝑧) where 𝑈≡𝜕𝑧 +𝑒 𝑀𝐹(𝑧).Thisobservationisbasedupon 𝑅 attains its maximum positive value, a global maximum, supersymmetric quantum mechanics, implying that the mass for 𝑎=1.Thepotential𝑉𝑅(𝑧) changes to a volcano-type squared is nonnegative. profile along the interval of 1<𝑎<2,suchthatfor In order to lead these results to the standard 4𝐷 action for 𝑎 = 2, 3, 4, . . the point 𝑦=0regards a local minimum, a massless fermion and a series of massive chiral fermions, the which allows for producing unstable resonances, which can 4 𝜇 action 𝑆=∑𝑛 ∫𝑑 𝑥√−𝑔𝜓𝑛[𝛾 (𝜕𝜇 +𝜔̂𝜇)−𝑚𝑛]𝜓𝑛 is employed, be tunneled to the outside of the potential. Nevertheless, for orthonormalization conditions the potential 𝑉𝐿(𝑧) has the associated minima at 𝑦=0 +∞ +∞ for all positive integer values of 𝑎, 𝑎 = 2, 3, 4, .,andit . ∫ 𝐿𝑚𝐿𝑛𝑑𝑧𝑚𝑛 =𝛿 = ∫ 𝑅𝑚𝑅𝑛𝑑𝑧, creates the conditions for producing bound states. A very −∞ −∞ (26) similar behavior is exhibited by model III, in spite of showing +∞ different amplitudes. Model IV is quite similar to these ∫ 𝐿𝑚𝑅𝑛𝑑𝑧 = 0. −∞ models, with the only qualitative difference concerning the fact that, at 𝑦=0,thepotential𝑉𝑅(𝑧) attains its maximum In formulae (23a) and (23b), by setting 𝑚𝑛 =0,thusit positive value, a global maximum, for 𝑎=1and 𝑎=2. yields For model IV, the stability conditions created by the right- and left-chiral volcano-type potentials are more sensible to −𝑀 ∫ 𝑒𝐴𝐹𝑑𝑧 𝐿0 ∝𝑒 , theincreasingofthebranewidth(𝑎≳3), in comparison to (27) models I and II ones (𝑎≳2), inducing no mass gap to separate 𝑀∫𝑒𝐴𝐹𝑑𝑧 𝑅0 ∝𝑒 . the fermion zero mode from the excited KK massive modes. In these cases, there exist continuous spectra for the Kaluza- Hence, either the massless left- or right-chiral KK fermion Klein modes of fermions of both chiralities. These volcano- modes can be localized on the brane, being the other one type potentials imply the existence of resonant or metastable nonnormalizable. states of fermions which can tunnel from the brane to the 𝐹(𝑧) = 𝜁(𝑧) By taking , regarding (16), it yields bulk [9]. The left-chiral KK mode has a continuous gapless spectrum for models I, III, and IV, according to Figure 3. 2𝐴(𝑦) 2 2 𝑑𝐴 (𝑦) 𝑉𝐿 (𝑧 (𝑦)) =𝑒 (𝑀 𝜁 (𝑦) − 𝑀𝜁 (𝑦) Since the potential for left-chiral fermions presents a negative 𝑑𝑦 value at the brane location for these models, the zero modes (28a) 𝑅 (𝑦) 𝐿 (𝑦) 𝑑𝜁 (𝑦) of right- and left-chiral fermions, 0 and 0 ,arethe −𝑀 ), onlynecessaryingredienttobetestedtobelocalizedon 𝑑𝑦 the brane. For model II, both potentials for the left- and 𝑑𝐴 (𝑦) right-chiral fermions have positive values of the potential, 𝑉 (𝑧 (𝑦))2𝐴(𝑦) =𝑒 (𝑀2𝜁2 (𝑦) + 𝑀𝜁 (𝑦) 𝑦 𝑉 (𝑦) 𝑎≲1 𝑅 𝑑𝑦 irrespective of . However for both cases 𝑅,𝐿 ,when , an asymmetric behavior emerges and produces a totally odd (28b) 𝑑𝜁 (𝑦) symmetric well-barrier profile in the limit of 𝑎→0.Except +𝑀 ). 0<𝑎≲1 𝑑𝑦 for , the zero mode of left- and right-chiral fermions can not be trapped. All potentials for model II are asymmetric 𝑎=0 Equations (28a) and (28b) evince that when the mass term (except for , which is nonsense in the brane context), 𝑦=0 𝑦→±∞ in the action (19) regards 𝑀=0,thepotentialsforleft- have maxima at , and tend to zero at ,andthere and right-chiral KK modes 𝑉𝐿,𝑅(𝑧) vanish. Then both chiral is no bound state for right-chiral fermions. In particular, for 𝑉 (𝑦) 𝑎=1 𝑦 ∼ ±0.87 fermions cannot be localized on the thick brane. Moreover, if 𝑅,𝐿 when ,theminimaoccurat . 𝑉𝐿(𝑧) and 𝑉𝑅(𝑧) are demanded to be Z2-even with respect to the extra dimension 𝑧, then the mass term 𝑀𝐹(𝑧) must be an 4. Matter Localization for Spin 0 Scalar Fields odd function of 𝑧 [29]. In fact, some useful classes of brane- 1 world models have the extra dimension topology 𝑆 /Z2.If The localization of scalar fields on thick branes generated by the background scalar is an odd function of extra warped deformed defects can also be considered from this point. In dimension, the Yukawa coupling, between the fermion and particular, an interesting approach on domain walls can be the background scalar field, assures the localization mech- alsofoundin[30].Infact,amassivescalarfieldcoupledto anism for fermions [29]. For the majority of brane-world gravity can be described by the following action: models, the scalar field 𝜁 is, usually, a kink, being an odd 1 function of the extra dimension. Here we do not necessarily 𝑆 =− ∫ 𝑑5𝑥√−𝑔 (𝑔 𝜕𝑀Φ𝜕𝑁Φ+𝑚2Φ2), impose this condition, in order to not preclude asymmetric 0 2 𝑀𝑁 0 (29) solutions, with respect to the extra dimension. In what follows the profile of the above left-right poten- where 𝑚0 denotes the effective mass of a bulk scalar field, Φ, tials is depicted in Figure 3 for different values of the and from where one can check whether spin 0 matter fields localization parameter 𝑎. In fact, the potentials 𝑉𝐿,𝑅(𝑧) have can be trapped on the thick brane. By employing the metric Advances in High Energy Physics 7

1.0 10

8 0.5 6 (y) (y) I R II R V

0.0 V ,

, 4 (y) I (y) L II V L −0.5 V 2

0 −1.0 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15 y y

(a) (b) 1.0 1.0

0.5 0.5 (y) (y) III IV R R

0.0 V 0.0 , (y), V (y) III IV L L V −0.5 V −0.5

−1.0 −1.0 −15 −10 −5 0 5 10 15 −15 −10 −5 0 5 10 15 y y

(c) (d)

Figure 3: Associated Schrodinger-like¨ quantum mechanical potentials, 𝑉𝐿(𝑦) and 𝑉𝑅(𝑦), respectively, for left-chiral (solid (black) lines) and right-chiral (dashed (red) lines) KK modes of fermions, for models from I (a) to IV (d). Again, one has considered integer values of the brane width parameter 𝑎, running from 1 (thinnest line) to 4 (thickest line), corresponding to an increasing thickness. The fermion mass parameter has been assumed to be equal to unit, 𝑀=1.

(1), the associated equation of motion from the action in (29) The profile of the above scalar boson potential is depicted reads in Figure 4 (solid (black) lines) for different values of the 1 𝑎 𝑚 =1 𝜕 (√−𝑔𝑔𝜇]𝜕 Φ) + 𝑒−3𝐴𝜕 (𝑒3𝐴𝜕 Φ) − 𝑒2𝐴𝑚2Φ localization parameter .For 0 , only brane scenarios √−𝑔 𝜇 ] 𝑧 𝑧 0 with 𝑎≲2provide conditions to have a localized scalar field. (30) Even in this case, such localized states behave much more =0. as resonances than as bound states, given that they can be 𝜇 Hence, by the KK decomposition Φ(𝑥 ,𝑧) = tunneled out of the potential. Bound states appear only for 𝜇 −3𝐴/2 noninteger values of the brane width such that 𝑎<1,which ∑𝑛 𝜒𝑛(𝑥 )𝜉𝑛(𝑧)𝑒 ,where𝜉𝑛 is assumed to satisfy the 4𝐷 𝜇] 2 𝜇 will correspond to typical volcano-type potentials. Klein-Gordon equation [𝜕𝜇(√−𝑔𝑔 𝜕])/√−𝑔−𝑚𝑛]𝜉𝑛(𝑥 )= 0, 𝑚𝑛 being the 4𝐷 mass of the KK excitation of the scalar 𝜉 (𝑧) field. Then the scalar KK mode 𝑛 is ruled by the following 5. Matter Localization for Spin 1 Vector Fields equation: Onenowturnstospin1vectorfieldsandbeginswiththe5𝐷 [−𝜕2 +𝑉 (𝑧)]𝜉 (𝑧) =𝑚2𝜉 (𝑧) . 𝑧 0 𝑛 𝑛 𝑛 (31) action of a vector field: This equation is a Schrodinger¨ one, with effective potential given by 1 5 𝑀𝑁 𝑅𝑆 𝑆1 =− ∫ 𝑑 𝑥√−𝑔𝑔 𝑔 𝐹𝑀𝑅 𝐹𝑁𝑆, (33) 3 9 4 𝑉 (𝑧 (𝑦)) = 𝐴󸀠󸀠 (𝑧) + 𝐴󸀠2 (𝑧) +𝑒2𝐴(𝑧)𝑚2 0 2 4 0 𝐹 =𝜕[ 𝐴 ] 2 (32) where 𝑀𝑁 𝑀 𝑁 denotes the field strength tensor. A 3 𝑑2𝐴(𝑦) 15 𝑑𝐴 (𝑦) 5𝐷 =𝑒2𝐴(𝑧) ( + ( ) +𝑚2). spin 1 field can be now studied via the KK decomposition 2 𝑑𝑦2 4 𝑑𝑦 0 𝜌 (𝑛) 𝜌 𝐴𝑀(𝑥 ,𝑧)= ∑𝑛 𝑎𝑀 (𝑥 )𝜏𝑛(𝑧).Theactionofthe5𝐷 massless 8 Advances in High Energy Physics

1.5 1.5

1.0 1.0

0.5 0.5 (y) (y) II I 1 1 V V 0.0 ,

, 0.0 (y) (y) I 0 II 0

V −0.5 −0.5 V

−1.0 −1.0

−1.5 −1.5 −10 −5 0 5 10 −10 −5 0510 y y

(a) (b) 1.5 1.5

1.0 1.0

0.5 0.5 (y) (y) III IV 1 1 V V

, 0.0

, 0.0 (y) (y) III IV 0 −0.5 0 −0.5 V V

−1.0 −1.0

−1.5 −1.5 −10 −5 0510−10 −5 0510 y y

(c) (d)

Figure 4: Associated Schrodinger-like¨ quantum mechanical potentials, 𝑉0(𝑦) and 𝑉0(𝑦), respectively, for spin 0 (solid (black) lines) and spin 1 (dashed (red) lines) bosons for models from I (a) to IV (d). Again, one has considered integer values of the brane width parameter 𝑎 running from 1 (thinnest line) to 4 (thickest line), corresponding to an increasing thickness. The spin 0 boson mass parameter has been assumed to be equal to unit.

󸀠 vector field (33) is invariant under the following gauge where () =𝜕𝑧. Given a set of orthonormal functions 𝜏𝑛(𝑧), transformation: playing the role of spin 1 Kaluza-Klein modes, and the decom- 𝜌 (𝑛) 𝜇 −𝐴/2 position of the vector field 𝐴𝜇(𝑥 ,𝑧)=∑𝑛 𝑎𝜇 (𝑥 )𝜏𝑛(𝑧)𝑒 , 𝜌 ̃ 𝜌 𝐴𝑀 (𝑥 ,𝑧)󳨃󳨀→ 𝐴𝑀 (𝑥 ,𝑧) the action (36) reads (34) =𝐴 (𝑥𝜌,𝑧)+𝜕 𝐹(𝑥𝜌,𝑧), 1 4 1 (𝑛)𝜇] (𝑛) 2 (𝑛)𝜇 (𝑛) 𝑀 𝑀 𝑆1 =− ∑ ∫ 𝑑 𝑥√−𝑔 ( 𝑓 𝑓𝜇] +𝑚𝑛𝑎 𝑎𝜇 ), (37) 2 𝑛 2 𝐹(𝑥𝜌,𝑧) (𝑛) (𝑛) where denotes any arbitrary regular scalar function, where 𝑓𝜇] =𝜕[𝜇𝑎 stands for the 4𝐷 field strength tensor. 𝑀=𝜇,5 𝐴 (𝑥𝜌,𝑧) ]] for .Thefieldcomponent 5 equals zero [25], The KK modes 𝜏𝑛(𝑧) satisfy the Schrodinger¨ equation by this gauge. In fact, (34) yields 2 2 (−𝜕𝑧 +𝑉1 (𝑧))𝜏𝑛 (𝑧) =𝑚𝑛𝜏𝑛 (𝑧) , (38) 𝐴̃ (𝑥𝜌,𝑧)=∑𝑎(𝑛) (𝑥𝜌)𝜏 (𝑧) +𝜕 𝐹(𝑥𝜌,𝑧). 5 5 𝑛 𝑧 (35) where the mass-independent potential reads [31] 𝑛 1 󸀠2 1 󸀠󸀠 𝑉1 (𝑧) = 𝐴 (𝑧) + 𝐴 (𝑧) 𝜌 (𝑛) 𝜌 4 2 By choosing 𝐹(𝑥 ,𝑧) = 𝑛−∑ 𝑎5 (𝑥 )∫𝜏𝑛(𝑧)𝑑𝑧,[25]then 2 (39) 𝐴̃ =0 1 𝑑2𝐴(𝑦) 3 𝑑𝐴 (𝑦) 5 , and hence the action (33) leads to the space-time =𝑒2𝐴(𝑧) [ + ( ) ]. action 2 𝑑𝑦2 4 𝑑𝑦

1 The profile of the above vector boson potential is depicted 𝑆 =− ∫ 𝑑5𝑥√−𝑔 (𝐹𝜇]𝐹 +2𝑒−𝐴𝑔𝜇]𝐴󸀠 𝐴󸀠 ), 1 4 𝜇] 𝜇 ] (36) in Figure 4 (dashed (red) lines) for different values of Advances in High Energy Physics 9 the localization parameter 𝑎.Allthethickbranescenarios no. 300809/2013-1 and Grant no. 440446/2014-7). Roldao˜ da with 𝑎≳1provide localization conditions to have vector field Rocha is grateful to CNPq (Grants no. 303293/2015-2 and bound states. Increasing values of 𝑎 lead to more stable bound no. 473326/2013-2) and to FAPESP (Grant 2015/10270-0) for states. partial financial support. 6. Conclusions and Discussion References Thick branes driven by superpotentials supported by [1] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, deformed defects (c.f. (13)–(15)) for various bulk matter fields “New dimensions at a millimeter to a fermi and superstrings at of spin 0, spin 1/2, and spin 1 have been investigated. For aTeV,”Physics Letters B,vol.436,no.3-4,pp.257–263,1998. spin 1 gauge fields, the profile of the associated vector boson [2] L. Randall and R. Sundrum, “Large mass hierarchy from a small potential showed that, in the thick brane models for 𝑎≳1, extra dimension,” Physical Review Letters,vol.83,no.17,pp. localization conditions hold as to guarantee the existence of 3370–3373, 1999. vector field bound states. Quantitatively, increasing values of the brane thickness parameter, 𝑎,leadtomorestablebound [3]O.DeWolfe,D.Z.Freedman,S.S.Gubser,andA.Karch, “Modeling the fifth dimension with scalars and gravity,” Physical states. Review D,vol.62,no.4,ArticleID046008,2000. Concerning spin 0 (scalar) fields, the profile of the potential evinces that only thick brane scenarios with 𝑎≳ [4] M. Gremm, “Four-dimensional gravity on a thick domain wall,” Physics Letters B,vol.478,no.4,pp.434–438,2000. 4 provide localization conditions compatible to scalar field bound states. [5] D. Bazeia, C. Furtado, and A. R. Gomes, “Brane structure from The most intricate result is related to spin 1/2 fields. In a scalar field in warped spacetime,” Journal of Cosmology and Astroparticle Physics,vol.2004,no.2,article002,2004. fact, for fermionic fields, left-right potentials were deeply studied and for the four models considered here, the issue of [6] X.-H. Zhang, Y.-X. Liu, and Y.-S. Duan, “Localization of localization has been scrutinized. It is worth pointing out that fermionic fields on braneworlds with bulk tachyon matter,” Modern Physics Letters A, vol. 23, no. 25, pp. 2093–2101, 2008. models I, III, and IV admit volcano-type potentials, inducing no mass gap to separate the fermion zero mode from the [7]A.A.Andrianov,V.A.Andrianov,andO.O.Novikov,“Gravity excited KK massive modes. Hence continuous spectra for effects on thick brane formation from scalar field dynamics,” the Kaluza-Klein modes of fermions of both chiralities are European Physical Journal C, vol. 73, article 2675, 2013. allowed. A refined analysis of the values of the 𝑎 parameter [8] Y.-X. Liu, C.-E. Fu, L. Zhao, and Y.-S. Duan, “Localization and in these models, influencing the localization of fermionic mass spectra of fermions on symmetric and asymmetric thick fields, was provided. Model II, induced by the superpotential branes,” Physical Review D,vol.80,no.6,ArticleID065020, 2009. (14), reveals a peculiar behavior. In this model, right-chiral fermions have positive values of the potential irrespective of [9] Z.-H. Zhao, Y.-X. Liu, H.-T. Li, and Y.-Q. Wang, “Effects of the extra dimension when 𝑎=1.Hence,exceptforthisvalue, the variation of mass on fermion localization and resonances on thick branes,” Physical Review D,vol.82,no.8,ArticleID the zero mode of left- and right-chiral fermions can not be 084030, 10 pages, 2010. trapped. All potentials for model II are asymmetric and have maxima at 𝑦=0and minima at 𝑦→±∞,andthereisno [10] V. Dzhunushaliev, V. Folomeev, and M. Minamitsuji, “Thick brane solutions,” ReportsonProgressinPhysics,vol.73,no.6, bound state for right-chiral fermions, but, again, for 𝑉𝑅,𝐿(𝑦) Article ID 066901, 2010. when 𝑎=1. It is worth mentioning that, for the localization of a [11] D. Bazeia, L. Losano, and J. M. C. Malbouisson, “Deformed 𝑀𝐹(𝑧)ΨΨ defects,” Physical Review D,vol.66,no.10,ArticleID101701, fermion zero mode, the mass term was considered 2002. in the 5𝐷 action. An interesting approach concerning such mass term in (19) has been studied, corresponding to the [12] A. E. Bernardini and R. da Rocha, “Cyclically deformed defects and topological mass constraints,” Advances in High Energy so-called singular dark spinors [32, 33]. Such massive mass Physics, vol. 2013, Article ID 304980, 18 pages, 2013. dimension one quantum fields are prime candidates for the dark matter problem, also presenting possible signatures at [13]R.A.Correa,A.deSouzaDutra,andM.B.Hott,“Fermion localization on degenerate and critical branes,” Classical and LHC [34]. It generates a slightly different action responsible Quantum Gravity, vol. 28, no. 15, Article ID 155012, 2011. for spin 1/2 matter fields localization [35, 36]. Such approach can be also extended in the context of deformed defects here [14] G. German, A. Herrera-Aguilar, D. Malagon-Morejon, R. R. Mora-Luna, and R. da Rocha, “A de Sitter tachyon thick presented. braneworld and gravity localization,” Journal of Cosmology and Astroparticle Physics,vol.1302,p.035,2013. Competing Interests [15] G. German,´ A. Herrera–Aguilar, D. Malagon–Morej´ on,´ I. Quiros, and R. da Rocha, “Study of field fluctuations and The authors declare that they have no competing interests. their localization in a thick braneworld generated by gravity nonminimally coupled to a scalar field with the Gauss-Bonnet Acknowledgments term,” Physical Review D,vol.89,no.2,ArticleID026004,2014. [16] A. E. Bernardini, R. T. Cavalcanti, and R. da Rocha, “Spheri- The work of Alex E. Bernardini is supported by the Brazilian cally symmetric thick branes cosmological evolution,” General Agencies FAPESP (Grant 2015/05903-4) and CNPq (Grant Relativity and Gravitation,vol.47,article1840,2015. 10 Advances in High Energy Physics

[17] A. Ahmed and B. Grzadkowski, “Brane modeling in wraped models,” Physical Review D. Particles, Fields, Gravitation, and extra-dimension,” Journal of High Energy Physics,vol.2013, Cosmology,vol.91,no.8,ArticleID085008,6pages,2015. article 177, 2013. [36] Y. X. Liu, X. N. Zhou, K. Yang, and F. W. Chen, “Localization of [18] N. Barbosa-Cendejas, D. Malagon-Morej´ on,´ and R. R. Mora- 5D Elko spinors on Minkowski branes,” Physical Review D,vol. Luna, “Universal spin-1/2 fermion field localization on a 5D 86,ArticleID064012,2012. braneworld,” General Relativity and Gravitation,vol.47,article 77, 2015. [19] W. T. Cruz, R. V. Maluf, and C. A. S. Almeida, “Kalb-Ramond field localization on the Bloch brane,” The European Physical Journal C,vol.73,p.2523,2013. [20] I. Oda, “Locally localized gravity models in higher dimensions,” Physical Review D,vol.64,no.2,ArticleID026002,11pages, 2001. [21] C. A. Almeida, R. Casana, J. Ferreira, and A. R. Gomes, “Ferm- ion localization and resonances on two-field thick branes,” Physical Review D. Particles, Fields, Gravitation, and Cosmology, vol.79,no.12,125022,12pages,2009. [22] D. Bazeia, R. Menezes, and R. da Rocha, “Anote on asymmetric thick branes,” Advances in High Energy Physics,vol.2014,Article ID 276729, 8 pages, 2014. [23] D. Bazeia, L. Losano, R. Menezes, and R. da Rocha, “Study of models of the sine-Gordon type in flat and curved spacetime,” European Physical Journal C,vol.73,article2499,2013. [24]A.deSouzaDutra,G.P.deBrito,andJ.M.HoffdaSilva, “Asymmetrical Bloch branes and the hierarchy problem,” EPL (Europhysics Letters), vol. 108, no. 1, p. 11001, 2014. [25] H. Guo, A. Herrera-Aguilar, Y.-X. Liu, D. Malagon-Morej´ on,´ and R. R. Mora-Luna, “Localization of bulk matter fields, the hierarchy problem and corrections to Coulomb’s law on a pure de Sitter thick braneworld,” Physical Review D,vol.87,no.9, Article ID 095011, 21 pages, 2013. [26] A. Herrera-Aguilar, D. Malagon-Morej´ on,´ and R. R. Mora- Luna, “Localization of gravity on a de Sitter thick braneworld without scalar fields,” Journal of High Energy Physics,vol.2010, article 15, 2010. [27] R. Maartens and K. Koyama, “Brane-world gravity,” Living Reviews in Relativity,vol.13,article5,2010. [28] A. E. Bernardini and O. Bertolami, “Equivalence between Born- Infeld tachyon and effective real scalar field theories for brane structures in warped geometry,” Physics Letters B,vol.726,no. 1–3, pp. 512–517, 2013. [29] Y.-X. Liu, Z.-G. Xu, F.-W. Chen, and S.-W. Wei, “New localiza- tion mechanism of fermions on braneworlds,” Physical Review D,vol.89,no.8,ArticleID086001,5pages,2014. [30] D. Stojkovic, “Fermionic zero modes on domain walls,” Physical Review D,vol.63,no.2,ArticleID025010,2000. [31]Q.-Y.Xie,Z.-H.Zhao,Y.Zhong,J.Yang,andX.-N.Zhou, “Localization and mass spectra of various matter fields on scalar-tensor brane,” Journal of Cosmology and Astroparticle Physics,vol.2015,no.3,article014,2015. [32] D. V.Ahluwalia, C. Y. Lee, and D. Schritt, “Self-interacting Elko dark matter with an axis of locality,” Physical Review D,vol.83, no. 6, Article ID 065017, 2011. [33] R. da Rocha, A. E. Bernardini, and J. M. Hoff da Silva, “Exotic dark spinor fields,” Journal of High Energy Physics, vol. 2011, article 110, 2011. [34]M.Dias,F.deCampos,andJ.M.HoffdaSilva,“ExploringElko typical signature,” Physics Letters B,vol.706,no.4-5,pp.352– 359, 2012. [35] I. C. Jardim, G. Alencar, R. R. Landim, and R. N. Costa Filho, “Solutions to the problem of Elko spinor localization in brane Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 2341879, 9 pages http://dx.doi.org/10.1155/2016/2341879

Research Article The Effects of Minimal Length, Maximal Momentum, and Minimal Momentum in Entropic Force

Zhong-Wen Feng,1 Shu-Zheng Yang,2 Hui-Ling Li,1,3 and Xiao-Tao Zu1

1 School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China 2Department of Astronomy, China West Normal University, Nanchong 637009, China 3College of Physics Science and Technology, Shenyang Normal University, Shenyang 110034, China

Correspondence should be addressed to Zhong-Wen Feng; [email protected]

Received 24 April 2016; Revised 25 June 2016; Accepted 11 July 2016

Academic Editor: Giulia Gubitosi

Copyright © 2016 Zhong-Wen Feng 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.

The modified entropic force law is studied by using a new kind of generalized uncertainty principle which contains aminimal length, a minimal momentum, and a maximal momentum. Firstly, the quantum corrections to the thermodynamics of a black hole are investigated. Then, according to Verlinde’s theory, the generalized uncertainty principle (GUP) corrected entropic force is obtained. The result shows that the GUP corrected entropic force is related not only to the properties of the black holes but alsoto the Planck length and the dimensionless constants 𝛼0 and 𝛽0. Moreover, based on the GUP corrected entropic force, we also derive the modified Einstein’s field equation (EFE) and the modified Friedmann equation.

1. Introduction be derived from the first law of thermodynamics together with the relation between the entropy and the horizon area The existence of thermodynamics of black holes is a great of a black hole [12]. Subsequently, Padmanabhan pointed discovery for the foundations of physics [1–11]. This idea was out that, in the static spherically symmetric spacetimes, the proposed by Bekenstein who proved that the entropy of a 3 gravitational field equations on the horizon can be rewritten black hole is 𝑆=𝐴𝑘𝐵𝑐 /4ℏ𝐺,where𝐴 is the horizon area, as a form of the ordinary first law of thermodynamics [13]. 𝑘𝐵 is the Boltzmann constant, ℏ is the Planck constant, and Inspired by Padmanabhan’s idea, people found that Einstein’s 𝐺 is Newton’s gravitational constant, respectively [2]. Later, equation is a thermodynamic identity. Those works explain based on the entropy of black hole, Hawking showed that the why the field equations should encode information of horizon Schwarzschild (SC) black hole emits thermal radiation and thermodynamics [14–19]. the temperature of SC black hole is proportional to the surface In 2011, Verlinde proposed a remarkable new perspective gravity on the event horizon 𝜅;namely,𝑇=𝜅/2𝜋[5, 6]. Fur- on the relation between the gravity and the thermodynamics. thermore, in [9, 10], the authors proved that the entropy and Based on Sakharov’s idea [20] and the holographic principle, thetemperatureofblackholesatisfythelawsofthermody- Verlinde pointed out that the gravity is no longer a fun- namics. These discoveries indicate that the thermodynamics damental force; instead, it can be explained as an entropic of black hole have profound connection with the gravity. force which arises from the change of information when In order to investigate the deeper-seated relation between material bodies move away from the screens of holographic thermodynamics and the gravity, Jacobson assumed the systems [21]. In his paper, Verlinde showed various inter- spacetime as a kind of gas, its entropy is proportional to esting results. For example, the the second law of Newton the area, then using the fundamental Clausius relation and can be obtained by incorporating the entropic force with the equivalence principle, he demonstrated that the Einstein the Unruh temperature. Using the entropic force together field equation is nothing but a state equation of this kind of with the holographic principle and the equipartition law of gas. Following this agreement, the Einstein field equation can energy, one can yield Newton’s law of gravitation. Moreover, 2 Advances in High Energy Physics an astounding discovery should be mentioned that Einstein’s where Δ𝑥 and Δ𝑝 are the uncertainties for position and equation can be derived from the theory of entropic force. momentum, 𝛼0 and 𝛽0 are dimensionless constants of the This new proposal of the gravity has received wide attention order of unity that depends on the details of the quantum √ 3 −35 causingpeopletodomanyrelevantworkstodiscussthe gravity hypothesis, and ℓ𝑝 = ℏ𝐺/𝑐 ≈10 misthePlanck entropic force [22–25]. length, respectively [60]. From (1), one can easily obtain the On the other hand, a lot of works showed that the orig- minimal length Δ𝑥min =𝛼0ℓ𝑝,theminimalmomentum inal thermodynamics of black holes would not be held when Δ𝑝min =2𝛽0ℓ𝑝, and the maximal momentum Δ𝑝max =[1+ considering the quantum gravity effects [26–31]. Various √1−(1+𝛼2𝛽2ℓ4)]/𝛼 ℓ theories of quantum gravity suggest the existence of a 0 0 𝑝 0 𝑝. Under the standard limit, that is, minimal observable length, which can be identified with the Δ𝑥 ≫𝑝 ℓ , (1) becomes the Heisenberg uncertainty principle Planckscale.ThisviewissupportedbymanyGedanken (HUP). It is well known that GUP has a great effect on the experiments and is applied to different physical systems thermodynamics of black holes and the entropic force that [32–35]. The generalized uncertainty principle (GUP) is one would bring many new results. Therefore, in this paper, using of the most important theories, which is modified by the (1), we first investigate the modified thermodynamics ofa minimum measurable length. The applications of GUP has black hole. Then, following Verlinde’sviewpoint, the modified been widely studied [36–45]. In particular, the GUP effect number of bits 𝑁 is obtained. The modified number of bits on the micro black holes has been deeply discussed in [46– 𝑁 leads to the GUP corrected entropic force. Considering 48]. In [49], combining the GUP with the thermodynamics the GUP corrected entropic force, the gravitational force, of black holes, the authors investigated the modified Hawking the gravitational potential, Einstein’s field equation, and the temperatureandtheentropy;theirresultsshowedthatthe Friedmann equation are modified. GUP corrected thermodynamics of black holes are different This paper is organized as follows. The quantum correc- from those of the original cases. The GUP can stop the tions to the entropy and Hawking temperature are derived in Hawking radiation in the final stages of black holes’ evolution Section 2. According to Verlinde’s theory, the GUP corrected and lead to the remnants of black holes. Therefore, the GUP entropicforceiscalculatedinSection3.InSections4and5, is considered as a good tool to solve the information paradox using the GUP corrected entropic force, the modified Ein- problem of black holes. In [50–57], people have investi- stein’s field equation and the modified Friedmann equation is gated the GUP corrected entropic force by using the GUP investigated. The last section is devoted to the conclusions. corrected thermodynamics. The implications of modified entropic force have been investigated in many contexts such 2. The GUP Impact on the Thermodynamics of as cosmology [50–53], Hamiltonians of the quantum systems in quasispace [54], quantum walk [55], and quarkonium a Black Hole binding [56, 57]. In fact, the expression of GUP is not In order to calculate the GUP impact on the thermodynamics unique.However,inmostofthepapers,theexpressionof of a black hole, one needs to solve (1) as a quadratic equation GUP is limited to two forms. One form only has a minimal Δ𝑝 2 2 2 in . The result is given by length Δ𝑥Δ𝑝 ≥ ℏ[10 +𝛼 ℓ𝑝(Δ𝑝) ] [30]. The other form has a minimal length and a maximal momentum Δ𝑥Δ𝑝 ≥ ℏ[1 − 𝛼 ℓ Δ𝑝 +2 𝛼 ℓ2(Δ𝑝)2] Δ𝑥+ℏ𝛼 ℓ { 0 𝑝 0 𝑝 [58]. According to the previous Δ𝑝 ≥ 0 𝑝 1 2 2 { works, it is interesting to raise the question whether it is 2ℏ𝛼0ℓ𝑝 possible to derive a general form of GUP which contains { a minimal length, a minimal momentum, and a maximal (2) momentum. Actually, in [59], the authors pointed out that the 4ℏ2𝛼2ℓ2 [1 + 𝛽2 (Δ𝑥)2 ℓ2]} − √1− 0 𝑝 0 𝑝 , existence of a minimal length Δ𝑥min comes from the fact that 2 } (Δ𝑥+ℏ𝛼ℓ ) } a string cannot probe its distances smaller than the length, 0 𝑝 } and the maximal momentum Δ𝑝max originated in the doubly special relativity (DSR) that predicts there exists an upper where we choose the negative-signed solution since only it bound for the momentum of a particle. For the minimal can recover the HUP in the classical limit ℓ𝑝 →0.Employing Δ𝑝 momentum min, it is well known that the notion of a plane the Taylor expansion, (2) can be rewritten as wave does not appear in the a general curved spacetime; hence, it indicates that there exists a limit to the precision with which the corresponding momentum can be described. 1 { Δ𝑝 ≥ {1 According to this phenomenon, one can express a nonzero Δ𝑥 +0 𝛼 ℓ𝑝 minimal uncertainty in momentum measurement, that is, { (3) Δ𝑝min. Taking these facts into account, Nozari and Saghafi 𝛼2 } introduced the most general form of GUP, which admits [ 2 2 0 ] 2 4 + 𝛽0 (Δ𝑥) + ℓ𝑝 + O (ℓ𝑝)} , a minimal length, a minimal momentum, and a maximal 2 [ (Δ𝑥+𝛼0ℓ𝑝) ] } momentum. The GUP is given by ℏ=1 2 2 2 2 2 2 where we are setting .As[49,61]havepointedout,the Δ𝑥Δ𝑝 ≥ℏ [1−𝛼0ℓ𝑝Δ𝑝 +0 𝛼 ℓ𝑝 (Δ𝑝) +𝛽0ℓ𝑝 (Δ𝑥) ] , (1) uncertainty momentum Δ𝑝 can be defined as the energy 𝜔 of Advances in High Energy Physics 3 emitted photon from black hole. Therefore, based on (3), the In accordance with the idea of entropy-area law, we obtain a low bound for the energy is constant 𝑏/𝜆𝐵 =𝑘 /4. Putting this constant into the above- mentioned equation and expanding it and then integrating 1 { 𝛼2 the result, the GUP corrected entropy is obtained as follows: 𝜔≥ 1+[𝛽2 (Δ𝑥)2 + 0 ] ℓ2 { 0 2 𝑝 Δ𝑥+𝛼0ℓ𝑝 { [ (Δ𝑥+𝛼0ℓ𝑝) ] 𝐴𝑘 𝛼2𝜋 𝐴 𝐴𝛽2 𝑆= 𝐵 {1 − [ 0 ( )+ 0 ]ℓ2 (4) 4ℓ2 4𝐴 ln 4ℓ2 2𝜋 𝑝 } 𝑝 𝑝 4 (10) + O (ℓ𝑝)} . } 4 + O (ℓ𝑝)} . Next, assuming an emitted particle with energy 𝜔 and size 𝑅 , for any black hole absorbing or emitting this particle, the It is clear that the GUP corrected entropy is proportional to minimal change in the horizon area of a black hole can be the area of horizon 𝐴, Planck length ℓ𝑝,andthedimensionless expressed as constants 𝛼0 and 𝛽0.Whenignoring𝛼0 and 𝛽0,(10)reduces Δ𝐴 ≥8𝜋𝜔𝑅ℓ2. to the original entropy of the black hole. Moreover, it can be min 𝑝 (5) found that the first correction term of (10) is logarithmic in 𝐴 ℓ 𝛼 In the classical model, people can set 𝑅=0.However,accord- , 𝑝,and 0, which is coincident with previous findings [64– ing to the arguments of [2], the size of a quantum particle 66]. It should be noted that the second correction term is pro- 𝛽 𝐴2 𝛽 ≥ √2𝜋/𝐴 cannot be smaller than Δ𝑥. Therefore, it implies the existence portional to 0 and goes like ;if 0 ,thesecond of a finite boundary given by correction term is in principle larger than the leading term proportional to 𝐴. In order to avoid this paradoxical situation, 2 √ Δ𝐴min ≥ 8𝜋𝜔Δ𝑥ℓ𝑝. (6) it requires that 𝛽0 < 2𝜋/𝐴. Meanwhile, one can calculate the GUP corrected Hawking temperature based on (10): By substituting (4) into inequality (6), then, using the relation Δ𝑥 =𝛼ℓ 𝜅 𝑑𝐴 of minimal length min 0 𝑝,theboundaryturnsoutto 𝑇= be 8𝜋 𝑑𝑆 2 2 (11) Δ𝐴min 𝜅 𝐴𝛽 𝜋𝛼 = ℓ2 [1 + ( 0 + 0 )ℓ2 + O (ℓ4)] , 2𝜋𝑘 𝑝 𝜋 4𝐴 𝑝 𝑝 𝛼2 (7) 𝐵 ≥4𝜋ℓ2 {1 + [𝛽2 (Δ𝑥)2 + 0 ]ℓ2 + O (ℓ4)} . 𝑝 0 2 𝑝 𝑝 (2Δ𝑥) where 𝜅 is the surface gravity of black holes. For the SC black hole, one sets 𝜅 = 1/4𝑀,ifonesets𝛼0 =𝛽0 =0,the Now, consider the case that a photon is emitted by the SC GUP corrected temperature reduces to the original Hawking black hole [49, 61, 62]. Near the event horizon of SC black temperature. hole, the position uncertainty of a photon is the order of the radiusoftheblackhole;thatis,Δ𝑥 = 2𝑟𝑠,where𝑟𝑠 is the radius of SC black hole. According to the area of the SC black hole 3. The Modified Newton’s Law of 2 𝐴=4𝜋𝑟𝑠 ,therelationbetweenΔ𝑥 and 𝐴 can be expressed Gravitation due to the GUP 2 2 as (Δ𝑥) =4𝑟𝑠 =𝐴/𝜋. Substituting this relation into (7), the abovementioned equation can be rewritten as the following In this section, we will investigate the GUP impact on expression: Newton’s law of gravitation. For revealing the entropic force, Verlinde used the holographic principle and the first law of 𝐴 𝜋𝛼2 thermodynamics. When a test particle approaches a holo- Δ𝐴 ≃𝜆ℓ2 [1 + (𝛽2 + 0 )ℓ2 + O (ℓ4)] , min 𝑝 0 𝜋 4𝐴 𝑝 𝑝 (8) graphic screen, the entropic force of a gravitational system is expressed as with 𝜆 being an undetermined coefficient that is greater than 𝐹Δ𝑥 = 𝑇Δ𝑆, 4𝜋. In the previous works, people proved that the entropy of (12) black holes depended on the area of horizon. Moreover, the where 𝐹 is the entropic force, 𝑇 and Δ𝑆 are the temperature ideas of information theory also showed the minimal increase Δ𝑥 of entropy is conjectured related to the value of the area 𝐴. and the change of entropy on holographic screen, and is According to [2, 63], the fundamental unit of entropy as one the displacement of the particle from the holographic screen, respectively [21]. Equation (12) implies a nonzero force is bitofinformationcanbedenotedasΔ𝑆min =𝑏=ln 2,sothat one can easily obtain proportional to a nonzero acceleration. Using the argument of Bekenstein, that is, the change of entropy associated with 𝑑𝑆 Δ𝑆 Δ𝑆 = 2𝜋𝑘 = min the information on the boundary 𝐵,itisfoundthat 𝑑𝐴 Δ𝐴min the change in the entropy near the holographic screen is linear in Δ𝑥: 2 −1 (9) 𝑏 2 𝐴 𝜋𝛼0 2 4 = [1 + (𝛽 + )ℓ + O (ℓ )] . 2𝜋𝑘𝐵𝑚𝑐Δ𝑥 𝜆ℓ2 0 𝜋 4𝐴 𝑝 𝑝 Δ𝑆 = , (13) 𝑝 ℏ 4 Advances in High Energy Physics where Δ𝑥=ℏ/𝑚𝑐and 𝑚 and ℏ are the mass of elementary large scales; however, it becomes weak at small scales (recent component and the Planck constant, respectively. Equation experiments show that the Newtonian gravitational force is (13) is reminiscent of the osmosis across a semipermeable led down to 0.13 mm∼0.16 mm [68]). Meanwhile, it is hard membrane.Meanwhile,itshouldbenotedthatΔ𝑆 is pro- to combine the Newtonian gravitational force with quantum portional to the mass of the elementary component. In order mechanics. In (18), it is clear that the GUP corrected Newton’s to understand this idea, one can postulate that a particle law is dependent not only on Newton’s gravitational constant near the holographic screen is made up of two or more 𝐺,themassoftwobodies𝑀 and 𝑚, and the distances 𝑅 subparticles and each subparticle leads to the associated but also on the dimensionless constants 𝛼0 and 𝛽0 as well change in entropy after displacement. Because the mass of as the Planck length ℓ𝑝. Therefore, due to the effect of GUP, the elementary component and the entropy are additive, it the result shows that the Newtonian gravitational force is leads to the fact that Δ𝑆 is proportional to 𝑚.Basedonthe valid at scales which is smaller than the order of a millimeter. conclusions in [67], the horizon of black holes can be taken as When 𝛼0 =𝛽0 =0, (18) reduces to the original Newton’s law. a storage device for information. If one denotes the amount of Moreover,onecanobtaintheNewtonianpotentialfrom(18): information by 𝑁 bits, the information is proportional to the 3 𝐺𝑀𝑚 area 𝑁=𝐴𝑐/𝐺ℏ. With the help of the entropy-area law 𝑆= 𝑉 (𝑅) =− {1 3 𝑅 𝐴𝑘𝐵𝑐 /4ℏ𝐺, the number of bits obeys the following relation: 4𝑆 2 𝑁= . 𝛼0 1 𝜋𝑅 2 2 (14) +{ [ − ln ( )] − 2𝑅𝛽 ln 𝑅} ℓ (19) 𝑘𝐵 2 0 𝑝 32𝑅 32 ℓ𝑝 Obviously,thenumberofbitsisproportionaltotheentropy. 4 Substituting (10) into (14), the number of bits is modified as + O (ℓ𝑝)} . follows: 𝐴 𝛼2𝜋 𝐴 𝐴𝛽2 It is interesting to compare (19) with the predictions that came 𝑁= {1 − [ 0 ( )+ 0 ]ℓ2 + O (ℓ4)} 2 ln 2 𝑝 𝑝 from higher order corrections to the Newtonian potential ℓ𝑝 4𝐴 4ℓ𝑝 2𝜋 in the Randall-Sundrum II (RS II) [69]; the modification in 𝐴𝑐3 𝛼2𝜋 𝐴 𝐴𝛽2 Newton’s gravitational potential on brane is [70] = {1 − [ 0 ( )+ 0 ]ℓ2 𝐺ℏ 4𝐴 ln 4ℓ2 2𝜋 𝑝 (15) 4𝑙 𝑝 {𝐺𝑀𝑚 𝜇 { (1 + −⋅⋅⋅) for 𝑙𝜇 ≫𝑟 { 𝑟 3𝜋𝑟 𝑉 𝑅 ∼ + O (ℓ4)} , ( ) { (20) 𝑝 { 2𝑙 {𝐺𝑀𝑚 𝜇 (1 + −⋅⋅⋅) for 𝑙𝜇 ≪𝑟, { 𝑟 3𝜋𝑟2 2 3 where ℓ𝑝 =𝐺ℏ/𝑐. By setting the total energy of the black where 𝑟 and 𝑙𝜇 are the radius and the characteristic length hole (or holographic system) as 𝐸 and noting that the energy scaleofthetheory,respectively.Ourresultagreeswiththe is divided evenly over the bits 𝑁,itiseasytoobtainthenotion Newtonian potential in RS II when 𝑙𝜇 ≫𝑟.Hence,itsuggests that each bit carries an energy equal to 𝑘𝐵𝑇/2. According to that (𝛼0,𝛽0)∼𝑙𝜇 can help us to set a new upper bound on the the equipartition rule, the total energy can be expressed as dimensionless constants (𝛼0,𝛽0). Besides, both (20) and 𝑘 𝑁𝑇 the correction terms in (19) become susceptible at a short 𝐸= 𝐵 . (16) 2 distance; they indicate that GUP and brane world may predict the similar phenomena. 2 Using the relation 𝐸=𝑀𝑐 andthenputting(12)and(13) into the abovementioned equation, one gets 4. The Quantum Corrections to Einstein’s 4𝜋𝑐3𝑀𝑚 Field Equation 𝐹= . (17) ℏ𝑁 A lot of works predict that Einstein’s field equation can be Next, substituting 𝑁 from (15) into (17), the GUP corrected derived from entropic force. In this section, we will further Newton’s law of gravitation becomes investigate the laws of gravity and extend them to the rela- tivistic case, so that we can obtain the modified Einstein’s field 𝐺𝑀𝑚 𝛼2 𝜋𝑅 equation via the GUP corrected entropic force. According 𝐹= {1 + [ 0 ( )+2𝛽2𝑅] ℓ2 2 ln 2 0 𝑝 to the GUP corrected number of bits, the bit density on the 𝑅 16𝑅 ℓ𝑝 (18) screen can be expressed as 2 2 4 1 𝛼0𝜋 𝐴𝛽0 2 4 + O (ℓ𝑝)} . 𝑑𝑁 = [1−( + ) ℓ + O (ℓ )] 𝑑𝐴, 2 𝑝 𝑝 (21) ℓ𝑝 4𝐴 𝜋 2 In the abovementioned equation, one has 𝐴=4𝜋𝑅.Itiswell where 𝐴 represents the area of the holographic screen, and known that the Newtonian gravitational force dominates at we use the natural units 𝑐=𝑘𝐵 =1in the equation above. Advances in High Energy Physics 5

𝜙 Here, we assume that the energy associated with the mass 𝑀 With the help of 𝐹=−𝑚𝑒 ∇𝜙,onecanobtaintheGUPcor- is divided over 𝑁. Moreover, due to the equipartition law, it rected Einstein’s field equation by some manipulations: is easy to find that each bit carries 𝑇/2 mass. Therefore, the 1 total mass is 𝑅 =8𝜋𝐺(𝑇 − 𝑇𝑔 ) 𝑎𝑏 𝑎𝑏 2 𝑎𝑏 1 2 (29) 𝑀= ∫ 𝑇𝑑𝑁, (22) 𝛼2𝜋 𝐴𝛽2 ℓ 2 S ⋅[1−( 0 + 0 ) 𝑝 + O (ℓ4)] , 4𝐴 𝜋 2𝜋 𝑝 where S istheholographicscreen.Thelocaltemperaturecan be expressed as with the area of the holographic screen 𝐴.Obviously,thisfield equation is dependent not only on the geometry of the space ℏ𝑒𝜙𝑛𝑏∇ 𝜑 time and energy-momentum tensor but also on the GUP 𝑇= 𝑏 , (23) 2𝜋 terms. For large horizon area or 𝛼0 =𝛽0 =0,themodified Einstein’s field equation reduces to the original case. 𝜙 where 𝑒 istheredshiftfactorasthelocaltemperatureis measured by an observer from infinity [71]. Substituting (21) 5. The Quantum Corrections to and (23) into (22), one has the Friedmann Equation 1 𝑀= In[19,22,51,73–76],peopleanalyzedtheFriedmannequa- 4𝜋𝐺 tion by using the entropic force. Hence, we will study the (24) 𝛼2𝜋 𝐴𝛽2 effect of the GUP arising from (1) on the form of the Fried- ⋅ ∫ 𝑒𝜙∇𝜙 [1 −( 0 + 0 )ℓ2 + O (ℓ4)] 𝑑𝐴. mann equation. In the homogeneous and isotropic space- 4𝐴 𝜋 𝑝 𝑝 S time, the Friedmann-Robertson-Walker (FRW) universe is described by the line element: It is necessary to mention that the integral on the right side of 2 𝜇 ] 2 2 the equation above represents the modified Komar mass (the 𝑑𝑠 =ℎ𝜇]𝑑𝑥 𝑑𝑥 + ̃𝑟 𝑑Ω , (30) original Komar mass contained inside a volume in a static 𝑀 = (1/4𝜋𝐺) ∫ 𝑒𝜙∇𝜙 𝑑𝐴) 𝜇 2 2 2 2 curvedspacetimeisdefinedas 𝐾 S where ̃𝑟 = 𝑟𝑎(𝑡), 𝑥 =(𝑡,𝑟), 𝑑Ω =𝑑𝜃 + sin 𝜃𝑑𝜑 is the 2 [21, 71, 72]; hence, (24) is the modified Gauss law in general metric of two-dimensional unit sphere, ℎ𝜇] = diag[−1, 𝑎 /(1− 2 relativity. Using the Stokes theorem and the Killing equation 𝑘𝑟 )] is the two-dimensional metric with 𝜇=] =0,1,and𝑘 is ∇𝑎∇ 𝜉𝑏 =−𝑅𝑏𝜉𝑎 𝑎 𝑎 ,theKomarmassintermsoftheKilling the spatial curvature constant, respectively. Using the relation 𝜉𝑎 𝑅 𝜇] vector and Ricci tensor 𝑎𝑏 canberewrittenas[21,73] ℎ 𝜕𝜇̃𝑟𝜕]̃𝑟=0, the dynamical apparent horizon of the FRW 1 universe can be expressed as 𝑀 = ∫ 𝑅 𝑛𝑎𝜉𝑏𝑑𝑉. 𝐾 𝑎𝑏 (25) 1 4𝜋𝐺 Σ ̃𝑟=𝑎𝑟= , √𝐻2 +𝑘/𝑎2 (31) Therefore, (24) becomes where 𝐻=𝑎/𝑎̇ is the Hubble parameter. Now, supposing 1 1 𝑎 𝑏 𝜙 that the matter source in the FRW universe is a perfect fluid, 𝑀= ∫ 𝑅𝑎𝑏𝑛 𝜉 𝑑𝑉 + 𝑒 4𝜋𝐺 Σ 4𝜋𝐺 stress-energy tensor is (26) 𝛼2𝜋 𝐴𝛽2 0 0 2 4 𝑇𝜇] =(𝜌+𝑝)𝑢𝜇𝑢] +𝑝𝑔𝜇], (32) ∇𝜙 ∫ [− ( + )ℓ𝑝 + O (ℓ𝑝)] 𝑑𝐴, S 4𝐴 𝜋 where 𝑢𝜇 is the four velocities of the fluid. The conservation where Σ is the three-dimensional volume bounded by the law of energy-momentum leads to the following continuity 𝑎 holographic screen S and its normal is 𝑛 .In[72],theauthors equation: showed that 𝑀 can be expressed in terms of the stress-energy 𝜌+3𝐻̇ (𝜌+𝑝) =0. (33) tensor 𝑇𝑎𝑏: In order to investigate the GUP corrected Friedmann equa- 1 𝑎 𝑏 𝑀=2∫ 𝑑𝑉𝜇 (𝑇 ] − 𝑇𝑔𝜇])𝑛 𝜉 . (27) tion, one should consider a compact spatial region 𝑉= Σ 2 3 2 (4/3)𝜋̃𝑟 with a compact boundary Σ=4𝜋̃𝑟 .Bycombining As a result, substituting (27) into (26), one yields (18) with Newton’s second law, one has 𝜕2̃𝑟 𝑀𝑚 1 𝐹=𝑚( ) =𝑚𝑎𝑟̈ = −𝐺 𝜒 (ℓ ,𝛼 ,𝛽 ) , 𝑎 𝑏 2 2 𝑝 0 0 (34) ∫ [𝑅𝑎𝑏 −8𝜋𝐺(𝑇𝑎𝑏 − 𝑇𝑔𝑎𝑏)] 𝑛 𝜉 𝑑𝑉 𝜕𝑡 ̃𝑟 Σ 2 2 2 (28) 𝜒(ℓ ,𝛼 ,𝛽 )=1+[2𝛽2̃𝑟2 +𝛼2 (𝜋̃𝑟2/ℓ2)/16̃𝑟2]ℓ2 + 2 𝜙 𝛼0𝜋 𝐴𝛽0 4 where 𝑝 0 0 0 0 ln 𝑝 𝑝 =ℓ𝑝 ∫ 𝑒 ∇𝜙 [ + + O (ℓ𝑝)] 𝑑𝐴. 4 S 4𝐴 𝜋 O(ℓ𝑝) and 𝑚 represents the test particle near the holographic 6 Advances in High Energy Physics

4 screen; the higher order terms O(ℓ𝑝) can be ignored since ℓ𝑝 It should be noted that 𝑘 is the spatial curvature which takes isaverysmallvalue.Thetotalphysicalmass𝑀 inside the the values −1, 0, 1, and the values correspond to a close, volume V canbedefinedas flat,andopenFRWuniverse,respectively.Forcalculating the correction term of (40), we assume an equation of state 𝜇 ] 4 3 𝜔=𝑝/𝜌 𝜔 𝑀=∫ 𝑑𝑉𝜇 (𝑇 ]𝑢 𝑢 )= 𝜋̃𝑟 𝜌, (35) parameter is ,where is a constant independent of V 3 time (or redshift), so, integrating the continuity equation (33), it yields where 𝜌=𝑀/𝑉is the energy density of the matter in the spatial region 𝑉. Putting (34) into (33), one has the accelera- −3(1+𝜔) 𝜌=𝜌𝑎 , (42) tion equation: 0 𝜌 𝑎̈ 4 where 0 is an integration constant. Substituting (41) into (40) =− 𝜋𝐺𝜌𝜒(ℓ ,𝛼 ,𝛽 ). 𝑎 3 𝑝 0 0 (36) and integrating, the result is given by 𝑘 8𝜋𝐺 𝛼2 For deriving the Friedmann equation, it is necessary to use 𝐻2 + = 𝜌{1+(1+3𝜔) [ 0 the active gravitational mass (or Tolman-Komar mass) M 𝑎2 3 72̃𝑟2 (1+𝜔)2 instead of the total mass 𝑀 because the acceleration in a (43) 2𝛽2̃𝑟2 𝛼2 ̃𝑟√𝜋 dynamical background is produced by the active gravitational + 0 + 0 ( )] ℓ2}. 2 ln 𝑝 mass. According to [22], one can express the active gravita- 3𝜔 − 1 24̃𝑟 (1+𝜔) ℓ𝑝 tional mass in terms of energy-momentum tensor 𝑇𝜇]: With the help of (31), the equation above can be further 1 𝜇 ] 4 3 M =2∫ 𝑑𝑉 (𝑇𝜇] − 𝑇𝑔𝜇]) 𝑢 𝑢 = 𝜋̃𝑟 (𝜌+3𝑝) . (37) rewritten as V 2 3 𝑘 𝑀 (𝐻2 + ){1−(1+3𝜔) Replacing the total mass by the active gravitational mass 𝑎2 M,(35)canberewrittenas 2 𝑎̈ 4 2 𝛼0 2 𝑘 =− 𝜋𝐺 (𝜌 + 3𝑝)𝑝 𝜒(ℓ ,𝛼0,𝛽0). (38) ⋅ℓ [ (𝐻 + ) 𝑎 3 𝑝 72 (1+𝜔)2 𝑎2

The equation above is the GUP corrected acceleration equa- 2 −1 2𝛽 2 𝑘 tion for the dynamical evolution of the FRW universe. Using + 0 (𝐻 + ) (44) 3𝜔 − 1 𝑎2 continuity equation (32) and multiplying both sides of (37) 𝑎𝑎̇ 2 with andthenintegratingit,theresultis[51] 𝛼2 𝑘 𝑘 ℓ + 0 (𝐻2 + ) [(𝐻2 + ) 𝑝 ]]} 𝑑 8𝜋𝐺 𝑑 24 (1+𝜔) 𝑎2 ln 𝑎2 𝜋 (𝑎̇2)= [ (𝜌𝑎2)] 𝜒 (ℓ ,𝛼 ,𝛽 ). 𝑑𝑡 3 𝑑𝑡 𝑝 0 0 (39) 8𝜋𝐺 = 𝜌. Integrating both sides for each term of (38), one has 3

8𝜋𝐺 1 TheequationaboveistheGUPcorrectedFriedmannequa- 𝑎̇2 +𝑘= 𝜌𝑎2 {1 + tion of the FRW universe, which derived from the entropic 3 𝜌𝑎2 force. It should be noted that (43) is not only determined by the Hubble parameter 𝐻, the expansion scale factor 𝛼2𝜋 𝜋 (𝑟𝑎)2 ⋅ ∫ [2 (𝑟𝑎)2 𝛽2 + 0 ( )] of universe 𝑎,thespatialcurvature𝑘,andtheconstant𝜔 0 2 ln 2 (40) 16𝜋 (𝑟𝑎) ℓ𝑝 but also affected by the dimensionless constants 𝛼0 and 𝛽0 and the Planck length ℓ𝑝. For the present universe, (43) is ⋅ℓ2𝑑(𝜌𝑎2)} ; nothing but a usual Friedmann equation since the apparent 𝑝 horizon radius is very large. However, the correction terms make sense when the apparent horizon radius is at a short theaboveequationcanberewrittenas scale. Hence, one can use the GUP corrected Friedmann to investigate the early stage of the universe. Moreover, in 𝑘 8𝜋𝐺 1 [51, 52], the authors concluded that the impact of quantum 𝐻2 + = 𝜌{1+ 𝑎2 3 𝜌𝑎2 corrections at the early stage of the universe can affect the inflation, so that people may detect those consequences by 2 2 astronomical observation. 2 2 𝛼0𝜋 𝜋 (𝑟𝑎) ⋅ ∫ [2 (𝑟𝑎) 𝛽 + ln ( )] (41) 0 16𝜋 (𝑟𝑎)2 ℓ2 𝑝 6. Conclusions 2 2 In this paper, we studied the quantum corrections to the ⋅ℓ𝑝𝑑(𝜌𝑎)} . entropic force via a new kind of GUP that admits a minimal Advances in High Energy Physics 7 length, a minimal momentum, and a maximal momentum. [9] J. M. Bardeen, B. Carter, and S. W. Hawking, “The four laws Firstly, we derived the modified entropy-area law of a black of black hole mechanics,” Communications in Mathematical hole. Then, using the the modification of entropy, the GUP Physics,vol.31,pp.161–170,1973. correctednumberofbits𝑁 was obtained. Subsequently, [10]D.Y.Chen,H.T.Yang,andX.T.Zu,“Areaspectraofnear basedontheGUPcorrectednumberofbitsandVerlinde’s extremal black holes,” The European Physical Journal C,vol.69, conjecture about the entropic force, the GUP corrected no. 1, pp. 289–292, 2010. Newton’s law of gravitation and Einstein’s field equation [11] G.-M. Deng, “Hawking radiation of charged rotating AdS as well as the Friedmann equation have been investigated. black holes in conformal gravity for charged massive particles, The results showed that the GUP corrected Newton’s law complex scalar and Dirac particles,” General Relativity and of gravitation, the GUP corrected Einstein’s field equation, Gravitation,vol.46,no.6,ArticleID1757,2014. and the GUP corrected Friedmann equation are dependent [12] T. Jacobson, “Thermodynamics of spacetime: the Einstein equa- on the quantum correction terms 𝛼0 and 𝛽0 and the Planck tion of state,” Physical Review Letters,vol.75,no.7,pp.1260– 1263, 1995. length ℓ𝑝. These results agree with the original cases at a larger scale. However, when the length approaches the order [13] T. Padmanabhan, “Classical and quantum thermodynamics of of Planck scale, the corrected results depart from the original horizons in spherically symmetric spacetimes,” Classical and cases since the GUP effect becomes susceptible at a short Quantum Gravity,vol.19,no.21,pp.5387–5408,2002. scale. Besides, it is found that the GUP corrected Newton’s [14] A. Paranjape, S. Sarkar, and T. Padmanabhan, “Thermodynamic law of gravitation is working at the sub-𝜇mrange,anditcan route to field equations in Lanczos-Lovelock gravity,” Physical predict the similar phenomenon as the Randall-Sundrum II Review.D.ThirdSeries,vol.74,no.10,ArticleID104015,2006. [15] D. Kothawala, S. Sarkar, and T. Padmanabhan, “Einstein’s model; this can help us to set a new upper bound on (𝛼0,𝛽0). Moreover, we found that the GUP corrected Friedmann equations as a thermodynamic identity: the cases of stationary axisymmetric horizons and evolving spherically symmetric equation can help people to study the properties of the early horizons,” Physics Letters B,vol.652,no.5-6,pp.338–342,2007. universe, and the impact of GUP may be detected by the [16] R.-G. Cai and L.-M. Cao, “Thermodynamics of apparent hori- astronomical observation. zon in brane world scenario,” Nuclear Physics B,vol.785,no.1-2, pp.135–148,2007. Competing Interests [17] R.-G. Cai and N. Ohta, “Horizon thermodynamics and grav- itational field equations inorava-Lifshitzgravity,” H˘ Physical The authors declare that they have no competing interests. Review D,vol.81,no.8,ArticleID084061,8pages,2010. [18] T. Padmanabhan, “Thermodynamical aspects of gravity: new Acknowledgments insights,” ReportsonProgressinPhysics,vol.73,no.4,Article ID 046901, 2010. The authors would like to acknowledge useful discussions [19] S. H. Hendi and A. Sheykhi, “Entropic corrections to Einstein with Karissa Liu and Lorraine Boyd. This work is supported equations,” Physical Review D,vol.83,no.8,ArticleID084012, by the Natural Science Foundation of China (Grant no. 2011. 11573022). [20] A. D. Sakharov, “Vacuum quantum fluctuations in curved space and the theory of gravitation,” General Relativity and Gravitation,vol.32,no.2,pp.365–367,2000. References [21] E. Verlinde, “On the origin of gravity and the laws of Newton,” JournalofHighEnergyPhysics,vol.4,article29,27pages,2011. [1]F.R.Tangherlini,“Schwarzschildfieldin𝑛 dimensions and the dimensionality of space problem,” ILNuovo Cimento,vol.27,pp. [22] R.-G. Cai, L.-M. Cao, and N. Ohta, “Friedmann equations from 636–651, 1963. entropic force,” Physical Review D—Particles, Fields, Gravitation and Cosmology,vol.81,no.6,ArticleID061501,2010. [2] J. D. Bekenstein, “Black holes and entropy,” Physical Review. D. ParticlesandFields.ThirdSeries,vol.7,pp.2333–2346,1973. [23] I. Sakalli, “Dilatonic entropic force,” International Journal of Theoretical Physics,vol.50,no.8,pp.2426–2437,2011. [3] J. D. Bekenstein, “Generalized second law of thermodynamics [24] Y. S. Myung, “Does entropic force always imply the Newtonian in black-hole physics,” Physical Review D,vol.9,no.12,pp.3292– force law?” The European Physical Journal C,vol.71,no.2,article 3300, 1974. 1549, 2011. [4]S.W.Hawking,“Blackholeexplosions?”Nature,vol.248,no. [25] R. G. L. AragooandC.A.S.Silva,“EntropiccorrectedNewton’s˜ 5443, pp. 30–31, 1974. law of gravitation and the loop quantum black hole gravitational [5]S.W.Hawking,“Particlecreationbyblackholes,”Communica- atom,” General Relativity and Gravitation,vol.48,p.83,2016. tions in Mathematical Physics,vol.43,no.3,pp.199–220,1975. [26] Y. S. Myung, Y.-W. Kim, and Y.-J. Park, “Black hole thermody- [6] D. Y. Chen, H. T. Yang, and X. T. Zu, “Fermion tunneling from namics with generalized uncertainty principle,” Physics Letters anti-de Sitter spaces,” The European Physical Journal C,vol.56, B,vol.645,no.5-6,pp.393–397,2007. no. 1, pp. 119–124, 2008. [27]D.Y.Chen,H.W.Wu,andH.T.Yang,“Fermion’stunnelling [7]W.G.Unruh,“Notesonblack-holeevaporation,”Physical with effects of quantum gravity,” Advances in High Energy Review D,vol.14,no.4,pp.870–892,1976. Physics, vol. 2013, Article ID 432412, 6 pages, 2013. [8] M. Li and T. Yoneya, “Short-distance space-time structure and [28] D. Chen and Z. Li, “Remarks on remnants by fermions’ tun- black holes in string theory: a short review of the present status,” nelling from black strings,” Advances in High Energy Physics,vol. Chaos, Solitons & Fractals,vol.10,no.2-3,pp.423–443,1999. 2014, Article ID 620157, 9 pages, 2014. 8 Advances in High Energy Physics

[29]Z.W.Feng,L.Zhang,andX.T.Zu,“TheremnantsinReissner– [48] P. Jizba, H. Kleinert, and F. Scardigli, “Uncertainty relation on a Nordstrom–de¨ Sitter quintessence black hole,” Modern Physics world crystal and its applications to micro black holes,” Physical Letters A,vol.29,no.26,ArticleID1450123,2014. Review D,vol.81,no.8,ArticleID084030,13pages,2010. [30]Z.W.Feng,H.L.Li,X.T.Zu,andS.Z.Yang,“Quantumcor- [49] R. J. Adler, P. Chen, and D. I. Santiago, “The generalized uncer- rections to the thermodynamics of Schwarzschild-Tangherlini tainty principle and black hole remnants,” General Relativity and black hole and the generalized uncertainty principle,” The Gravitation,vol.33,no.12,pp.2101–2108,2001. European Physical Journal C,vol.76,article212,2016. [50] S. Ghosh, “Planck scale effect in the entropic force Law,” http:// [31] A. Ashour, M. Faizal, A. F.Ali, and F.Hammad, “Branes in Grav- arxiv.org/abs/1003.0285. ity’s rainbow,” The European Physical Journal C,vol.76,article [51] A. Awad and A. F. Ali, “Planck-scale corrections to Friedmann 264, 2016. equation,” Central European Journal of Physics,vol.12,no.4,pp. [32] F. Scardigli, “Generalized uncertainty principle in quantum 245–255, 2014. gravity from micro-black hole Gedanken experiment,” Physics [52] P. Jizba, H. Kleinert, and F. Scardigli, “Inflationary cosmology Letters B,vol.452,no.1-2,pp.39–44,1999. from quantum conformal gravity,” European Physical Journal C, [33] J. Magueijo and L. Smolin, “Generalized Lorentz invariance vol.75,no.6,article245,2015. with an invariant energy scale,” Physical Review D,vol.67,no. [53] A. Awad and A. F. Ali, “Minimal length, Friedmann equations 4, Article ID 044017, 12 pages, 2003. and maximum density,” Journal of High Energy Physics,vol. [34] J. Magueijo and L. Smolin, “Gravity’s rainbow,” Classical and 2014,article93,2014. Quantum Gravity,vol.21,no.7,pp.1725–1736,2004. [54] K. Nozari, P. Pedram, and M. Molkara, “Minimal length, [35] S. Das and E. C. Vagenas, “Universality of quantum gravity maximal momentum and the entropic force law,” International corrections,” Physical Review Letters,vol.101,no.22,ArticleID Journal of Theoretical Physics,vol.51,no.4,pp.1268–1275,2012. 221301, 2008. [55] M. M. Amaral, R. Aschheim, and K. Irwin, “Quantum walk on [36] A. Ashoorioon, A. Kempf, and R. B. Mann, “Minimum length spin network,” https://arxiv.org/abs/1602.07653. cutoff in inflation and uniqueness of the action,” Physical Review [56] D. E. Kharzeev, “Deconfinement as an entropic self-destruction: D—Particles, Fields, Gravitation and Cosmology,vol.71,Article a solution for the quarkonium suppression puzzle?” Physical ID 023503, 2005. Review D,vol.90,no.7,ArticleID074007,7pages,2014. [37] A. Ashoorioon and R. B. Mann, “On the tensor/scalar ratio in [57] H. Satz, “Quarkonium binding and entropic force,” The Euro- inflation with UV cutoff,” Nuclear Physics B,vol.716,no.1-2,pp. pean Physical Journal C,vol.75,article193,2015. 261–279, 2005. [58] P. Pedram, K. Nozari, and S. H. Taheri, “The effects of minimal [38] A. Ashoorioon, J. L. Hovdebo, and R. B. Mann, “Running length and maximal momentum on the transition rate of ultra of the spectral index and violation of the consistency rela- cold neutrons in gravitational field,” Journal of High Energy tion between tensor and scalar spectra from trans-Planckian Physics, vol. 2011, no. 3, article 093, 2011. physics,” Nuclear Physics B,vol.727,no.1-2,pp.63–76,2005. [59] H. Soltani, A. Damavandi Kamali, and K. Nozari, “Microblack [39] S. Das and R. B. Mann, “Planck scale effects on some low energy holes thermodynamics in the presence of quantum gravity quantum phenomena,” Physics Letters B,vol.704,no.5,pp.596– effects,” Advances in High Energy Physics,vol.2014,ArticleID 599, 2011. 247208, 12 pages, 2014. [40] M. Sprenger, P.Nicolini, and M. Bleicher, “Neutrino oscillations [60] K. Nozari and S. Saghafi, “Natural cutoffs and quantum tunnel- as a novel probe for a minimal length,” Classical and Quantum ing from black hole horizon,” JournalofHighEnergyPhysics,vol. Gravity,vol.28,no.23,ArticleID235019,2011. 2012, article 5, 2012. [41] A. F. Ali, S. Das, and E. C. Vagenas, “Proposal for testing [61] F. Scardigli, “Some heuristic semi-classical derivations of the quantum gravity in the lab,” Physical Review D—Particles, Fields, Planck length, the Hawking effect and the Unruh effect,” Il Gravitation and Cosmology,vol.84,ArticleID044013,2011. Nuovo Cimento B,vol.110,no.9,pp.1029–1034,1995. [42] T. Zhu, J.-R. Ren, and M.-F. Li, “Influence of generalized and [62] A. J. Medved and E. C. Vagenas, “When conceptual worlds col- extended uncertainty principle on thermodynamics of FRW lide: the generalized uncertainty principle and the Bekenstein- universe,” Physics Letters B,vol.674,no.3,pp.204–209,2009. Hawking entropy,” Physical Review D,vol.70,no.12,ArticleID [43] W.Chemissany,S.Das,A.F.Ali,andE.C.Vagenas,“Effectofthe 124021, 5 pages, 2004. generalized uncertainty principle on post-inflation preheating,” [63] C. Adami, “The physics of information,” Philosophy of Informa- JournalofCosmologyandAstroparticlePhysics,vol.2011,no.12, tion,vol.42,pp.609–683,2004. p. 17, 2011. [64] M.-I. Park, “The generalized uncertainty principle in (A)dS [44] M. Asghari, P. Pedram, and K. Nozari, “Harmonic oscilla- space and the modification of Hawking temperature from the tor with minimal length, minimal momentum, and maximal minimal length,” Physics Letters B,vol.659,no.3,pp.698–702, momentum uncertainties in SUSYQM framework,” Physics 2008. Letters B,vol.725,no.4-5,pp.451–455,2013. [65] R. Banerjee and B. R. Majhi, “Quantum tunneling and back [45] F. Scardigli and R. Casadio, “Gravitational tests of the gener- reaction,” Physics Letters B,vol.662,no.1,pp.62–65,2008. alized uncertainty principle,” The European Physical Journal C, [66] P. Bargueno˜ and E. C. Vagenas, “Semiclassical corrections to vol. 75, article 425, 2015. black hole entropy and the generalized uncertainty principle,” [46] F. Scardigli, “Generalized uncertainty principle in quantum Physics Letters B, vol. 742, pp. 15–18, 2015. gravity from micro-black hole gedanken experiment,” Physics [67] G. t’Hooft, “Dimensional reduction in quantum gravity,”https:// Letters, Section B: Nuclear, Elementary Particle and High-Energy arxiv.org/abs/gr-qc/9310026. Physics,vol.452,no.1-2,pp.39–44,1999. [68] S.-Q. Yang, B.-F. Zhan, Q.-L. Wang et al., “Test of the gravita- [47] F. Scardigli, “Glimpses on the micro black hole Planck phase,” tional inverse square law at millimeter ranges,” Physical Review https://arxiv.org/abs/0809.1832. Letters,vol.108,no.8,ArticleID081101,2012. Advances in High Energy Physics 9

[69] L. Randall and R. Sundrum, “An alternative to compactifica- tion,” Physical Review Letters,vol.83,no.23,pp.4690–4693, 1999. [70] P. Callin and F. Ravndal, “Higher order corrections to the New- tonian potential in the Randall-Sundrum model,” Physical Review D,vol.70,no.10,ArticleID104009,2004. [71] B. Majumder, “The effects of minimal length in entropic force approach,” Advances in High Energy Physics,vol.2013,Article ID 296836, 8 pages, 2013. [72] A. S. Sefiedgarr, “How can rainbow gravity affect on gravita- tional force?” http://arxiv.org/abs/1512.08372. [73] R. G. Cai and S. P.Kim, “First law of thermodynamics and Fried- mann equations of Friedmann Robertson Walker universe,” Journal of High Energy Physics,vol.2,article50,2005. [74] A. Sheykhi, “Entropic corrections to Friedmann equations,” Physical Review D, vol. 81, no. 10, Article ID 104011, 2010. [75] Y. F. Cai, J. Liu, and H. Li, “Entropic cosmology: a unified model of inflation and late-time acceleration,” Physics Letters B,vol. 697, pp. 213–219, 2010. [76] Y.-F. Cai and E. N. Saridakis, “Inflation in entropic cosmology: primordial perturbations and non-Gaussianities,” Physics Let- ters B,vol.697,no.4,pp.280–287,2011. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 5353267, 7 pages http://dx.doi.org/10.1155/2016/5353267

Research Article Spontaneous Symmetry Breaking in 5D Conformally Invariant Gravity

Taeyoon Moon1,2 and Phillial Oh1

1 Department of Physics and Institute of Basic Science, Sungkyunkwan University, Suwon 440-746, Republic of Korea 2Institute of Basic Science and Department of Computer Simulation, Inje University, Gimhae 621-749, Republic of Korea

Correspondence should be addressed to Phillial Oh; [email protected]

Received 4 May 2016; Revised 27 June 2016; Accepted 28 June 2016

Academic Editor: Barun Majumder

Copyright © 2016 T. Moon and P. Oh. 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.

We explore the possibility of the spontaneous symmetry breaking in 5D conformally invariant gravity, whose action consists of a scalar field nonminimally coupled to the curvature with its potential. Performing dimensional reduction via ADM decomposition, we find that the model allows an exact solution giving rise to the 4D Minkowski vacuum. Exploiting the conformal invariance with Gaussian warp factor, we show that it also admits a solution which implements the spontaneous breaking of conformal symmetry. We investigate its stability by performing the tensor perturbation and find the resulting system is described by the conformal quantum mechanics. Possible applications to the spontaneous symmetry breaking of time-translational symmetry along the dynamical fifth direction and the brane-world scenario are discussed.

1. Introduction If the scalar field spontaneously breaks the conformal invariance with a Planckian VEV, the theory reduces to the Conformal symmetry is an important idea which has 4D Einstein gravity with a cosmological constant [9, 10]. appeared in diverse area of physics, and its application to On the other hand, the spontaneous symmetry breaking of gravity has started with the idea that conformally invariant Lorentz symmetry [11] or gauge symmetry [12] in 5D brane- gravity in four dimensions (4D) [1–4] might result in a world scenario [13–16] was studied, but little is known in the unified description of gravity and electromagnetism. The context of 5D conformal gravity. In this paper, we explore Einstein-Hilbert action of general relativity is not confor- 5D conformal gravity with a conformal scalar and investigate mallyinvariant.Inrealizingtheconformalinvarianceof possible consequences in view of the spontaneous symmetry this, a conformal scalar field is necessary [5, 6] in order breaking. to compensate the conformal transformation of the metric, Let us consider 5D conformal scalar action of the form and a quartic potential for the scalar field can be allowed. Its higher dimensional extensions are straightforward. In 1 𝜔 𝑆=∫ 𝑑5𝑥√−𝑔 [ 𝜉𝜙2𝑅− 𝑔𝐴𝐵𝐷 𝜙𝐷 𝜙−𝑉(𝜙)] five dimensions (5D), conformal symmetry can be preserved 2 2 𝐴 𝐵 (1) with a fractional power potential [7, 8] for the scalar field. So far, it seems that little attention has been paid to the +𝑆𝑚, 5D conformal gravity with its fractional power potential. Such a potential renders a perturbative approach inaccessible, where 𝑅 is the five-dimensional curvature scalar, 𝑆𝑚 is the but nonperturbative treatment may reveal novel aspects. One action for some matter1,and𝐴, 𝐵 runover0,1,2,3,4.Here, can also construct a conformally invariant gravity with the 𝜉 is a dimensionless parameter describing the nonminimal Weyl tensor via 𝑅-squared gravity, but we focus on the couplingofthescalarfieldtothespacetimecurvature.Alsoa Einstein gravity with a conformal scalar. parameter 𝜔=±1with +(−) corresponds to canonical (ghost) 2 Advances in High Energy Physics

scalar. For 𝜔=±1,theconformalinvarianceofaction(1) along the extra dimension, and 𝑔𝜇] can be interpreted as the without matter term forces 𝜉 to be metric of the 4D spacetime. Using the metric ansatz (4), one obtains 󵄨 󵄨10/3 𝑉(𝜙)=𝑉0 󵄨𝜙󵄨 , ∗ ∗ 2 𝛼𝛽𝑔 3 (2) ∇ 𝑁 𝜖 𝑁 𝑔 𝛼𝛽 ∗∗ 𝜉=∓ , 𝑅(5) =𝑅(4) −2 + ( −𝑔𝛼𝛽 𝑔 16 𝑁 𝑁2 𝑁 𝛼𝛽 where 𝑉0 is a constant and the corresponding conformal (6) ∗ 2 transformation is given by ∗𝛼𝛽 ∗ 𝛼𝛽 𝑔 𝑔 (𝑔 𝑔 ) 3 𝛼𝛽 𝛼𝛽 2𝜎(𝑥) − − ), 𝑔𝐴𝐵 󳨀→ 𝑒 𝑔𝐴𝐵, 4 4 (3) 𝜙󳨀→𝑒−(3/2)𝜎(𝑥)𝜙 (𝑥) . ∗ where the asterisk denotes the differentiation with respect 𝜔=−1 2 𝜇 When ,theconformalscalarhasanegative to 𝑦 and ∇ =∇𝜇∇ isthefour-dimensionalLaplacian.Using kinetic energy term, but we regard it as a gauge artifact this, we find that action (1) becomes [17] which can be eliminated from the beginning through field redefinition. Even with no scalar field remaining after 𝜔=±1 [1 gauging away for both cases ( ), the physical mass scale 𝑆=∫ 𝑑5𝑥√−𝑔(4)𝑁 [ 𝜉𝜙2𝑅(4) can be set since the corresponding vacuum solution requires [2 introduction of a scale which characterizes the conformal [ symmetry breaking. In 4D conformal gravity, it is known ∗ that the conformal symmetry can be spontaneously broken 𝜉𝜙2 ∗𝛼𝛽 ∗ ∗ 2 8 𝜙 ∗ +𝜖 (𝑔 𝑔 +(𝑔𝛼𝛽𝑔 ) + 𝑔𝛼𝛽𝑔 ) at electroweak [10, 18] or Planck [9, 10] scale. In all cases, the 8𝑁2 𝛼𝛽 𝛼𝛽 𝜙 𝛼𝛽 action (1) becomes 5D Einstein action with a cosmological (7) 4/3 constant by redefinition of the metric, 𝑔̂𝐴𝐵 =𝜙 𝑔𝐴𝐵,butwe stick to the above conformal form (1) of the action to argue ∗ ∗ 2 2 𝜔 𝜇] 𝜔 ] with the spontaneous breakdown of the conformal symmetry. −𝜉∇ 𝜙 − 𝑔 ∇ 𝜙∇ 𝜙−𝜖 𝜙𝜙 −𝑉 (𝜙)] 2 𝜇 ] 2𝑁2 ] The paper is organized as follows. In Section 2, we per- form the dimensional reduction from five to four dimensions ] by using the ADM decomposition. In Section 3, we present +𝑆𝑚. exact solutions with four-dimensional Minkowski vacuum (𝑅(4) =0) and check if they can give a spontaneous One can check that the above action (7) is invariant with 𝜇 󸀠𝜇 breaking of the conformal symmetry. In Section 4, the respecttofour-dimensionaldiffeomorphism𝑥 →𝑥 ≡ 󸀠𝜇 󸀠 󸀠 gravitational perturbation and their stability for the solutions 𝑥 (𝑥) with 𝑁 (𝑥 , 𝑦) = 𝑁(𝑥, 𝑦). It is also invariant under 󸀠 󸀠 󸀠 󸀠 −1 are considered. In Section 5, we include the summary and 𝑦→𝑦 ≡𝑦(𝑦) and 𝑁→𝑁 ≡(𝑑𝑦/𝑑𝑦) 𝑁 apart from the discussions. matter action. Before going further, we would like to comment on the 2. Dimensional Reduction (5D to 4D) homogeneous solution to the equations of motion given in 5D conformal gravity (1) without matter term. To this end, In order to derive the 4D action from the 5D conformal grav- we first consider the Einstein equation for action (1), whose ity (1), we make use of the following ADM decomposition form is given by 2 where the metric in 5D can be written as 1 2 𝐴 𝐵 𝑅𝐴𝐵 − 𝑔𝐴𝐵𝑅=𝑇𝐴𝐵, (8) 𝑑𝑆 =𝑔𝐴𝐵𝑑𝑥 𝑑𝑥 2 𝜇 𝜇 ] ] 𝑇 =𝑔𝜇] (𝑥, 𝑦) (𝑑𝑥 +𝑁 𝑑𝑦) (𝑑𝑥 +𝑁 𝑑𝑦) (4) 𝐴𝐵

2 2 1 1 𝐶 +𝜖𝑁 (𝑥, 𝑦) 𝑑𝑦 . = (𝜕𝐴𝜙𝜕𝐵𝜙− 𝑔𝐴𝐵𝜕 𝜙𝜕𝐶𝜙−𝑉𝑔𝐴𝐵) 𝜉𝜙2 2 (9) Todescribethebackgroundsolution,wegotothe“comoving” 1 𝑁𝜇 =0 + (𝐷 𝐷 𝜙2 −𝑔 ◻(5)𝜙2) gauge and choose .Inthiscase,wecanrecoverour 𝜙2 𝐴 𝐵 𝐴𝐵 4D spacetime by going onto a hypersurface Σ𝑦 :𝑦=𝑦0 = constant, which is orthogonal to the 5D unit vector: and the scalar equation can be written as 𝛿𝐴 (5) 󸀠 𝑛̂𝐴 = 4 , 0=𝜔◻ 𝜙+𝜉𝜙𝑅−𝑉 (𝜙) , (10) 𝑁 (5) (5) 𝐴 𝐴 where 𝐷𝐴 is 5D covariant derivative, ◻ ≡𝐷𝐴𝐷 ,andthe 𝑛 𝑛 =𝜖, 󸀠 𝐴 prime denotes the differentiation with respect to 𝜙.Onecan Advances in High Energy Physics 3

∗ easily check that the homogeneous solution to (8) and (10) is 𝑁 𝜙2 ∗ ] given by −2 ( )𝑔𝛼𝛽𝑔 𝑔 ] , 𝜙2 𝑁 𝛼𝛽 𝜇] 𝑅𝐴𝐵 =Λ𝑔𝐴𝐵, ] 𝜙=𝜙, 𝜉 1 0 𝑇(3) =− [∇ 𝑁∇ 𝜙2 − 𝑔 ∇ 𝑁∇𝜌𝜙2], (11) 𝜇] 𝑁 (𝜇 ]) 2 𝜇] 𝜌 2𝑉 4/3 Λ= 0 (𝜙 ) . 𝜔 𝜔 1 3𝜉 0 𝑇(4) =[ ∇ 𝜙∇ 𝜙− 𝑔 ∇ 𝜙∇𝜌𝜙− 𝑉(𝜙)𝑔 ], 𝜇] 2 𝜇 ] 4 𝜇] 𝜌 2 𝜇] We note that this solution can be approached in diverse ways. 4/3 ∗ Firstly, field redefinition 𝑔̃ =(𝜙/𝜙0) 𝑔𝑀𝑁 necessitates ∗ ∗ ∗ 𝑀𝑁 [ 𝜙 𝜙𝜙 ] introduction of scale, which leads to five-dimensional Planck (5) 𝜖 [ 𝜙 𝜔 ] 2/3 𝑇𝜇] =− [𝜉 ( )𝑔𝜇] + 𝑔𝜇]] . mass 𝑀5 =𝜉𝜙0 . Secondly, solution (11) corresponds to a 𝑁 𝑁 4 𝑁 gaugefixedcase(𝜙=𝜙0) through the conformal transfor- [ ] mation (3). Lastly, it can be interpreted as a vacuum solution (14) obtained when considering an effective potential 𝑉eff = 2 10/3 Also the equation of motion for the scalar field can be written −𝜉𝑅𝜙 /2 + 𝑉0|𝜙| (we will study the effective potential 𝑉eff as for details at the end of the next section). In all cases, con- 𝜖𝜉𝜙 ∗𝛼𝛽 ∗ ∗ 2 formal symmetry is spontaneously broken with a symmetry (4) 2 𝑔 𝑔 𝛼𝛽𝑔 0=𝜉𝑁𝜙𝑅 −2𝜉𝜙∇ 𝑁+ ( 𝛼𝛽 −(𝑔 𝛼𝛽) breaking scale ∼𝜙0 =0̸ . In addition, they yield the physically 4𝑁 equivalent results: de Sitter 𝑉0/𝜉 > 0 or anti-de Sitter 𝑉0/𝜉 < ∗ ∗ 0. It is to be noticed that both cases are classically stable. ∗ ∗ 𝛼𝛽𝑔 𝑁 𝛼𝛽𝑔 𝜇 Also there is a huge degeneracy of vacuum solutions due to −4(𝑔 𝛼𝛽)+4 𝑔 𝛼𝛽)+𝑁(𝜔∇𝜇∇ 𝜙 (0) (0) 𝑁 conformal invariance such that if (𝑔𝐴𝐵(𝑥, 𝑦), 𝜙 (𝑥, 𝑦)) is a 2𝜎(𝑥,𝑦) (0) ̃ −(3/2)𝜎(𝑥,𝑦) (0) (15) solution, then (𝑔̃ =𝑒 𝑔 , 𝜙=𝑒 𝜙 )isalsoa ∗ ∗ 𝐴𝐵 𝐴𝐵 ∗ ∗ 𝜙 𝜎(𝑥, 𝑦) 𝜔 𝜇 󸀠 𝜖𝜔 𝛼𝛽𝑔 𝑁 solution for an arbitrary function . In the next section, + ∇𝜇𝑁∇ 𝜙−𝑉 (𝜙)) + (𝑔 𝛼𝛽 𝜙 −2 we will investigate the explicit solution form, starting from 𝑁 2𝑁 𝑁 the reduced action (7) without matter term. ∗∗ +2 𝜙 ). 3. Exact Solutions From action (7) without matter term, we find the equation of Now one can equate (12) and trace of (13) and then obtain motion for 𝑁 as 2 𝜉 2 2 𝜉 ] 2 3𝜉𝜙 2 𝜉𝜙2 𝑉(𝜙)=− ∇ 𝜙 −2 ∇𝜇𝑁∇ 𝜙 − ∇ 𝑁 0= 𝑅(4) 2 𝑁 2𝑁 2 ∗ ∗ ∗ ∗ ∗ 2 𝛼𝛽 2 𝛼𝛽 2 𝜖 [ 𝜙 𝜙 𝜙𝜙] 𝜖𝜉𝜙 ∗ ∗ 𝜖𝜉𝜙 ∗ ∗ 𝜇] ∗ 8 𝜙 𝛼𝛽 ∗ − [8𝜉 ( )+3𝜔 ] + [−3𝑔 𝑔 − (𝑔 𝑔 +(𝑔 𝑔 ) + 𝑔 𝑔 ) (12) [ ] 2 𝛼𝛽 8𝑁2 𝛼𝛽 𝜇] 𝜙 𝛼𝛽 2𝑁 𝑁 𝑁 8𝑁 (16) [ ] ∗ ∗ 2 2 𝜔 𝜇] 𝜖𝜔 ∗ ∗ −𝜉∇ 𝜙 − 𝑔 ∇ 𝜙∇ 𝜙+ 𝜙𝜙 −𝑉 (𝜙) ∗∗ 𝜙 ∗ ∗ 𝜇 ] 2 𝛼𝛽 𝑔 𝛼𝛽𝑔 𝑁 𝛼𝛽𝑔 ] 2 2𝑁 −6𝑔 𝛼𝛽 −4 𝑔 𝛼𝛽 +6 𝑔 𝛼𝛽 . 𝜇] 𝜙 𝑁 and the equation for four-dimensional metric 𝑔 is given by ] 1 1 Substituting the above equation back into (12), we arrive at 𝜉𝜙2 (𝑅(4) − 𝑔 𝑅(4))=𝑇(1) +𝑇(2) +𝑇(3) +𝑇(4) 2 𝜇] 2 𝜇] 𝜇] 𝜇] 𝜇] 𝜇] 2 2 (13) 𝜉𝜙 (4) 𝜉 2 2 𝜉 ] 2 3𝜉𝜙 2 𝜔 (5) 𝑅 = ∇ 𝜙 −2 ∇𝜇𝑁∇ 𝜙 − ∇ 𝑁+ +𝑇𝜇] , 2 2 𝑁 2𝑁 2 𝜉 ∗ 𝑇(1) = [∇ ∇ (𝑁𝜙2)−𝑔 ∇2 (𝑁𝜙2)] , ∗ 𝜇] 2𝑁 𝜇 ] 𝜇] 2𝜖𝜔 ∗ ∗ 4𝜖𝜉 𝜙 𝜙 ⋅𝑔𝜇]∇ 𝜙∇ 𝜙− 𝜙𝜙 − ( ) ∗ 𝜇 ] 𝑁2 𝑁 𝑁 𝜖𝜉𝜙2 [1 ∗𝛼𝛽 ∗ 𝑁 𝜙2 ∗ ∗∗ 𝑇(2) = [ 𝑔 𝑔 𝑔 +2 ( ) 𝑔 +2𝑔 (17) 𝜇] 8𝑁2 2 𝛼𝛽 𝜇] 𝜙2 𝑁 𝜇] 𝜇] 𝜖𝜉𝜙2 ∗𝛼𝛽 ∗ ∗ 2 ∗∗ [ + (−2𝑔 𝑔 +(𝑔𝛼𝛽𝑔 ) −6𝑔𝛼𝛽 𝑔 8𝑁2 𝛼𝛽 𝛼𝛽 𝛼𝛽 ∗ ∗ ∗𝛼𝛽 ∗ ∗ ∗ 𝛼𝛽𝑔 𝑔 𝑔 𝑔 𝛼𝛽𝑔 𝑔 +𝑔 𝛼𝛽 𝜇] + 𝑔𝛽𝜇 ]𝛼 +𝑔 𝛽] 𝜇𝛼 ∗ ∗ 4 𝜙 ∗ 6 𝑁 ∗ ∗ + 𝑔𝛼𝛽𝑔 + 𝑔𝛼𝛽𝑔 ). 1 ∗ 2 ∗ 𝛼𝛽 𝛼𝛽 − (𝑔𝛼𝛽𝑔 ) 𝑔 −2(𝑔𝛼𝛽𝑔 )𝑔 𝜙 𝑁 2 𝛼𝛽 𝜇] 𝛼𝛽 𝜇] 4 Advances in High Energy Physics

−2𝜇2(𝑦−𝑦 )2 𝜔= 0 Let us focus on the conformally invariant case with (ii) 𝑔𝜇] =𝑒 𝜂𝜇], 10/3 −16𝜉/3 and 𝑉(𝜙)0 =𝑉 |𝜙| . In order to find solutions for −𝜇2𝑦2 this case, we consider the following ansatz: 𝑁=𝑁0𝑒 ,

2 2 (3/2)𝜇2𝑦2 −2𝜇 (𝑦−𝑦0) 𝜙=𝜙𝑒 , 𝑔𝜇] (𝑥, 𝑦) =𝑒 𝑔̂𝜇] (𝑥) , 0 2 4 2 2 𝑦 𝜇 −𝑧1𝜇 𝑦 0 𝑁(𝑥,𝑦)=𝑁0𝑒 , (18) 𝑉 = −24𝜖𝜉 . 0 2 4/3 𝑁0 𝜙0 (3/2)𝑧 𝜇2𝑦2 𝜙(𝑥,𝑦)=𝜙 𝑒 2 , 0 (22) where 𝑁0 and 𝜙0 are constants. For this ansatz, (15)–(17) Nowweturntoanissuerelatedtothespontaneous become breaking of the conformal symmetry. We notice that the spontaneous symmetry breaking can be realized for a nega- 10 7/3 2(𝑧 −𝑧 )𝜇2𝑦2 𝜙 𝑅<0 𝑉 >0 𝑉 𝜙 =𝑒 1 2 0 tivevalueofthecurvaturescalarwith and 0 .Tosee 0 0 2 𝑉 3 𝑁0 this, we consider an effective potential eff for the canonical scalar field (𝜔=1)and𝑉0 >0in action (1) as 4 2 ⋅𝜖𝜉[80(1−𝑧2)(𝑧1 −1)𝜇 𝑦 (19) 1 2 󵄨 󵄨10/3 𝑉 =− 𝜉𝜙 𝑅+𝑉 󵄨𝜙󵄨 . (23) 4 2 eff 2 0 󵄨 󵄨 +80𝑧𝑦0𝑦(2−𝑧1 −𝑧2)𝜇 +40(1−𝑧2)𝜇 As was mentioned in (2), here 𝜉 is fixed as a negative value of −80𝑦2𝜇4], 0 𝜉=−3/16for the canonical scalar 𝜔=1, which preserves the 𝑉 = 2 conformal symmetry of action (1). Since solution (i) ( 0 10/3 2(𝑧 −𝑧 )𝜇2𝑦2 𝜙 0 𝜙 =0 𝑉 𝜙 =𝑒 1 2 0 ) with a stable equilibrium V does not provide a 0 0 2 𝑁0 symmetry-brokenphase,wefocusonthecase(ii)with𝜖=1 giving 𝑉0 >0(hereafter we fix 𝜖=1). ⋅𝜖𝜉[24(1−𝑧 )(𝑧 −1)𝜇4𝑦2 2 1 (20) It turns out that, for (ii) with the positive 5D curvature scalar 𝑅>0,wehaveonlyonevacuumsolutionof𝜙V =0, +24𝑦𝑦(2−𝑧 −𝑧 )𝜇4 +12(1−𝑧 )𝜇2 0 1 2 2 while for 𝑅<0, there exist two vacua of ±𝜙V with nonzero value: −24𝑦2𝜇4], 0 2 2 (3/2)𝜇 𝑦∗ 𝜙V =𝑒 𝜙0, (24) (4) 2𝑧 𝜇2𝑦2 2𝜖 4 2 𝑅 =𝑒 1 [24 (1 − 𝑧 )(𝑧 −𝑧 )𝜇 𝑦 𝑁2 2 1 2 𝑦 𝑦 =4𝑦/3 ± √3/𝜇2 + 16𝑦2/3 0 (21) where ∗ is given by ∗ 0 0 .In 4 2 this case, the conformal symmetry is spontaneously broken −4𝑦0𝑦(𝑧1 −𝑧2)𝜇 +12(1−𝑧2)𝜇 ]. with the symmetry breaking scale 𝜙V given by (24). This result is summarized in Figure 1 which shows that the 𝑅(4) 𝑦 The above equation (21) determines as a function of ; effective potential with 𝑅>0has only one minimum 𝜙V = 𝑦=𝑦 namely, each hypersurface has different values of 0 (a), while for 𝑅<0, it has two minima with ±𝜙V (b) (4) 𝑅 .But,wewillrestrictourattentiontofour-dimensional which corresponds to the case of spontaneous breaking of the Minkowski space. Then, we notice that (19)–(21) always allow conformal symmetry. (4) trivial vacuum solution 𝜙0 =0, 𝑅 = arbitrary, independent 𝑧 𝑧 of 1 and 2, in general. The search for nontrivial vacuum with 4. Tensor Perturbation 𝜙0 =0̸ is facilitated by the fact that the coefficients in (19) and (20) come out right so that the two equations are identical. In this section, we explore the stability of solutions (i), (ii) by (4) Finally,forthe4DMinkowskivacuum(𝑅 =0), we can performing a tensor fluctuation around the solutions. Note obtain two solutions: (i) the 𝑟.ℎ.𝑠 of (19)–(21) vanishes when that the impact of conformal invariance shows up in the (4) 𝑦0 =0and 𝑧2 =1,whichyields𝑉0 =0, 𝑅 =0and in this perturbation theory. One can always go to the unitary gauge 𝜑=0 𝜙=𝜙 +𝜑 case, 𝑧1 is arbitrary; (ii) for 𝑉0 =0̸ and 𝑦0 =0̸ ,theyallowthe and choose in the scalar perturbation with 0 . 𝜔/𝜉 =−16/3̸ solution of 𝑧1 =𝑧2 =1.Wesummarizethe4DMinkowski On the other hand, when ,itisnolongerpossible solutions as follows: to gauge away the fluctuation, and 𝜑 is a dynamical field coupled to other fluctuations. −2𝜇2𝑦2 (i) 𝑔𝜇] =𝑒 𝜂𝜇], To investigate a tensor fluctuation around solutions (i), (ii), we need to consider the tensor perturbation of the metric 2 2 −𝑧1𝜇 𝑦 𝑁=𝑁0𝑒 , as 2 2 𝜇 ] 2 2 (3/2)𝜇2𝑦2 𝑑𝑠 =𝑎 (𝜂𝜇] +ℎ𝜇])𝑑𝑥 𝑑𝑥 +𝑁 𝑑𝑦 (25) 𝜙=𝜙0𝑒 , 2 𝜇 ] 2 =𝑎 [(𝜂𝜇] +ℎ𝜇])𝑑𝑥 𝑑𝑥 +𝑑𝑧 ], (26) 𝑉0 =0 Advances in High Energy Physics 5

V eff Veff

R>0 R<0

−𝜙 𝜙 𝜙

𝜙

𝜙 =0 (a) (b)

2 10/3 Figure 1: The effective potential 𝑉eff =−𝜉𝑅𝜙/2 + 𝑉0|𝜙| with 𝜉<0and 𝑉0 >0, where (a) [unbroken phase] corresponds to the effective potential with a positive value of the curvature scalar 𝑅, while (b) [broken phase] is a case of 𝑅<0. where a new variable 𝑧 given in a conformally flat metric (26) which guarantees that the graviton mode along the fifth satisfies 𝑁𝑑𝑦 = 𝑎𝑑𝑧. It is found that the equation of motion dimension with 𝑔=15/4is stable. for the tensor modes ℎ𝜇] is given by Before closing the section, we remark that the graviton mode along the fifth dimension preserves the residual con- 2 (4) 𝜕𝑧ℎ𝜇] +𝐴(𝑧) 𝜕𝑧ℎ𝜇] −◻ ℎ𝜇] =0, (27) formal 𝑆𝑂(2, 1) symmetry. It is well known that the quantum mechanical system of Hamiltonian (33) is conformally invari- ℎ where 𝜇] satisfy the transverse-traceless gauge conditions ant [21], being referred to as conformal quantum mechanics 𝜕𝜇ℎ =0 ℎ=𝜂𝜇]ℎ =0 𝐴(𝑧) ( 𝜇] , 𝜇] )and is given by (CQM). To see this, we first construct the CQM action for 𝐻 3 𝜕𝑎 2 𝜕𝜙 (33) from the Lagrangian formalism: 𝐴 (𝑧) = + . 𝑎 𝜕𝑧 𝜙 𝜕𝑧 (28) 1 𝑔 𝑆 = ∫ 𝑑𝑡 (𝑧̇2 − ), CQM 2 𝑧2 (35) Consideration of a separation of variables ℎ𝜇](𝑥, 𝑧) = 𝑓(𝑧)𝐻 (𝑥) 𝜇] splits (27) into two parts: which is invariant under the nonrelativistic conformal trans- 2 2 formations: [−𝜕𝑧 +𝑉QM]𝑓=𝑚 𝑓, (29) 󸀠 𝛼𝑡 +𝛽 (4) 2 𝑡 = , ◻ 𝐻𝜇] =𝑚𝐻𝜇]. (30) 𝛾𝑡 +𝛿 𝑚 𝑧 Here is the mass of four-dimensional Kaluza-Klein modes 𝑧󸀠 = (36) and the corresponding quantum mechanical potential 𝑉QM 𝛾𝑡 +𝛿 reads with 𝛼𝛿 − 𝛽𝛾 =1. 1 𝜕𝐴 𝐴2 𝑉QM = + . (31) 2 𝜕𝑧 4 In this case, it is known that three generators, that is, 𝐻 (time translation), 𝐷 (dilatation), and 𝐾 (special conformal) One can check easily that 𝑉QM vanishes for solution (i), which yieldsjustplanewavesolutionwithconstantzeromode.On generators, can act with the transformation rules: the other hand, for solution (ii), it gives an inverse square 𝐻;󸀠 𝑡 =𝑡+̃𝑡, potential3 as 15 󸀠 ̃2 𝑉 = , 𝐷; 𝑡 = 𝑑 𝑡, QM 4𝑧2 (32) (37) 𝑡 𝐾;󸀠 𝑡 = , where the corresponding Hamiltonian can be written as 𝑘𝑡̃ +1 1 2 −2 15 𝐻= (𝑝 +𝑔𝑧 ) 𝑔= . (33) ̃ ̃ 2 𝑧 with 4 where ̃𝑡, 𝑑, 𝑘 are some constants and the generators 𝐷 and 𝐾 at 𝑡=0in addition to 𝐻 (33) are given by Itisknownin[19,20]that,fortheSchrodinger¨ equation (29) 2 with the inverse square potential (𝑉QM =𝑔/𝑧), the stability 1 𝐷=− (𝑧𝑝 +𝑝 𝑧) , of the mode 𝑓 is determined by the condition4 4 𝑧 𝑧 (38) 1 1 𝑔>− , 𝐾= 𝑧2. 4 (34) 2 6 Advances in High Energy Physics

These generators obey 𝑆𝑂(2, 1) commutations rules given by in the Lagrangian but generated by the particular form of the vacuum. On the other hand, it should be pointed out [𝐻, 𝐷] =𝑖𝐻, that since the well-defined ground state described by the 𝐻 [𝐾,] 𝐷 =𝑖𝐾, Hamiltonian which generates the time translation is not (39) present, it may lead to a spontaneous symmetry breaking [𝐻, 𝐾] =2𝑖𝐷, of time-translational invariance along the dynamical fifth direction [22–25]. whose Casimir invariant C is given by C =(𝐻𝐾+𝐾𝐻)/2− We conclude with the following remark. We see that both 2 𝐷 =3/4. solutions (i) and (ii) can be characterized by Gaussian warp factor [26–28], where the maximum value is located at 𝑦= 0 𝑦=𝑦 𝜙 5. Summary and Discussion and 0, respectively. But, the vacuum mode V of (24) of the scalar field and the massless mode of gravity for In this paper, we considered the conformally invariant gravity the broken phase (𝑅<0, 𝑉0 >0)arenotlocalizedon in 5D, which consists of a scalar field nonminimally coupled the Gaussian brane, because a value of 𝑦∗ in (24) cannot be to the curvature with its potential. We found two solutions equivalent to 𝑦0 and zero mode (40) is written as 𝑓(𝑦)𝑚2=0 = 2 2 −5𝜇 𝑦0𝑦 3𝜇 𝑦0𝑦 (i) and (ii) giving 4D Minkowski vacuum. By analyzing the ̃𝑐1𝑒 + ̃𝑐2𝑒 in 𝑦-coordinate. Thus, it seems that it dynamics of the metric perturbations around the solutions, is hard to describe the brane-world scenario with the current we showed that two solutions are stable, since the former approach. One possible alternative to apply our result to the yields a plane wave solution with the constant zero mode, scenariowouldbetotreattheconformalgravitydiscussed whereas the latter gives an inverse square potential. In par- in this study (or its variation) as the conformal matter sector ticular, it was shown for solution (ii) that one has unbroken and introduce 5D Einstein-Hilbert action separately. Then, 𝑅>0𝑉 >0 𝑅<0𝑉 >0 phase when , 0 , while for , 0 the there could be a possibility of addressing some of the related spontaneous breaking of the conformal symmetry can be issues, especially the brane stabilization as a consequence 𝜙 realized with the scale V given by (24). ofthespontaneoussymmetrybreaking.Thedetailswillbe We point out that solution (ii) may lead to a different reported elsewhere. mechanism which allows the possibility of a spontaneous breaking of translational invariance along the extra dimen- Competing Interests sion. To this end, we consider a solution explicitly to (29) for 2 𝑚 =0as The authors declare that they have no competing interests.

5/2 −3/2 𝑓 (𝑧)𝑚2=0 =𝑐1𝑧 +𝑐2𝑧 (40) Acknowledgments with arbitrary constants 𝑐1 and 𝑐2.Thefirsttermofthe𝑟.ℎ.𝑠 This work was supported by Basic Science Research Pro- in (40) is not normalizable since the function 𝑓(𝑧) diverges at gram through the National Research Foundation of Korea infinity, while the second term cannot lead to a normalizable (NRF) funded by the Ministry of Education (Grant no. solution due to its divergence at the origin. Thus, there is no 2015R1D1A1A01056572). normalizable zero mode solution. To resolve this problem, we R 𝐾 define a new evolution operator [21] given in terms of Endnotes (38) and 𝐻 (33): 1 1 1. We assume that the matter is confined on a hypersurface R ≡ ( 𝐾+𝑎𝐻), 𝑦=𝑦 𝑦 2 𝑎 (41) at 𝑚,where is the fifth coordinate. It is known that this matter in the brane has a geometrical origin which yields the eigenvalues of R as follows: in space-time-matter (STM) theory or induced-matter theory (IMT) [29–31]. Also, the IMT has been extended 3 to the modified Brans-Dicke theory (MBDT) of type (1), 𝑟 =𝑟 +𝑛, 𝑟 = . (42) 𝑛 0 0 2 where the induced matter exhibits interesting cosmolog- ical consequences [32, 33]. However, in this paper we are 𝑎 Here, is some constant with the length dimension. It interested in the phenomena of spontaneous symmetry R turns out that the new evolution operator (41) provides a breaking and, thus, we will be neglecting the matter 𝑓 normalizable ground state 0: sector. 𝜇 5/2 −𝑧2/2 2. 𝑥 are the coordinates in 4D and 𝑦 is the noncompact 𝑓0 (𝑧) =𝑐0𝑧 𝑒 ,𝑧≥0, (43) coordinate along the extra dimension (see [32, 33] and (−, +, where a constant 𝑐0 is given by 𝑐0 =1,beingobtainedfrom references therein). We use spacetime signature ∞ 2 +, +, ±),while𝜖=±1denotes spacelike or timelike extra the normalization condition ∫ |𝑓0(𝑧)| 𝑑𝑧 =1.Importantly, 0 dimension. even if we have the normalizable ground state (43) by introducing the new evolution operator R (41), it implements 3. The inverse square potential (32) corresponds to a the spontaneous breaking of the conformal symmetry in the repulsion from the origin. For an attractive potential −2 sense that the fundamental length scale 𝑎 is not included (𝛼<−1/4)casewith𝑉(𝑥) = 𝛼𝑥 ,itwasshownin Advances in High Energy Physics 7

[34] that the quantum mechanical system has infinite [17] I.Bars,P.Steinhardt,andN.Turok,“Localconformalsymmetry continuous bound states from negative infinity to zero. in physics and cosmology,” Physical Review D,vol.89,no.4, Article ID 043515, 2014. 4. This condition yields exactly the BF bound [35] given in the 𝑑-dimensional AdS spacetime for the one-dimen- [18] H. Cheng, “Possible existence of Weyl’s vector meson,” Physical Review Letters, vol. 61, no. 19, p. 2182, 1988. sional Schodinger¨ equation: [19] F. Calogero, “Solution of a three−body problem in one dimen- 𝑚2 +(𝑑2 −1)/4 sion,” Journal of Mathematical Physics,vol.10,no.12,p.2191, −𝜕2𝜓+ =𝐸𝜓󳨀→𝑚2 ≥𝑚2 1969. 𝑥 𝑥2 BF (∗) [20] S. Moroz, “Below the Breitenlohner-Freedman bound in the 𝑑2 nonrelativistic AdS/CFT correspondence,” Physical Review D, =− , 4 vol. 81, Article ID 066002, 2010. [21] V. de Alfaro, S. Fubini, and G. Furlan, “Conformal invariance 2 2 when replacing 𝑔 with 𝑚 +(𝑑 − 1)/4. in quantum mechanics,” Il Nuovo Cimento A,vol.34,no.4,pp. 569–612, 1976. [22] S. Fubini, “A new approach to conformal invariant field theo- References ries,” Il Nuovo Cimento A,vol.34,no.4,pp.521–554,1976. [1] H. Weyl, “Gravitation and electricity,” Sitzungsberichte der [23] E. D’Hoker and R. Jackiw, “Space-translation breaking and com- Preussischen Akademie der Wissenschaften zu Berlin (Physika- pactification in the Liouville theory,” Physical Review Letters, lisch-Mathematische),vol.1918,p.465,1918. vol.50,no.22,pp.1719–1722,1983. [2]H.Weyl,“EineneueErweiterungderRelativitatstheorie,”¨ [24] C. Bernard, B. Lautrup, and E. Rabinovici, “Monte Carlo study Annalen der Physik,vol.364,no.10,pp.101–133,1919. of the breakdown of translational invariance in a Liouville model,” Physics Letters B,vol.134,no.5,pp.335–340,1984. [3] V. H. Weyl, “Eine neue erweiterung der relatiritastheorie,”¨ Surveys in High Energy Physics,vol.5,no.3,pp.237–240,1986. [25] E. Rabinovici, “Spontaneous breaking of space-time symme- tries,” Lecture Notes in Physics, vol. 73, p. 573, 2008. [4]H.Weyl,“Eineneueerweiterungderrelativitatstheorie,”¨ Annalen der Physik,vol.364,no.10,pp.101–133,1919. [26]E.E.Flanagan,S.H.H.Tye,andI.Wasserman,“Braneworld models with bulk scalar fields,” Physics Letters. B,vol.522,no. [5] P. A. M. Dirac, “A new basis for cosmology,” Proceedings of the 1-2, pp. 155–165, 2001. Royal Society of London A,vol.165,pp.199–208,1938. [27] J. Alexandre and D. Yawitch, “Branon stabilization from [6] P. A. Dirac, “Long range forces and broken symmetries,” fermion-induced radiative corrections,” Physics Letters B,vol. ProceedingsoftheRoyalSociety.London.SeriesA.Mathematical, 676, no. 4-5, pp. 184–187, 2009. Physical and Engineering Sciences, vol. 333, pp. 403–418, 1973. [28] I. Quiros and T. Matos, “Gaussian warp factor: towards a prob- [7] R. Jackiw and S.-Y.Pi, “Tutorial on scale and conformal symme- abilistic interpretation of braneworlds,” https://arxiv.org/abs/ tries in diverse dimensions,” JournalofPhysics.A.Mathematical 1210.7553. and Theoretical, vol. 44, no. 22, Article ID 223001, 2011. [8] T. Moon, J. Lee, and P. Oh, “Conformal invariance in Einstein- [29] P. S. Wesson, Space-Time-Matter: Modern Kaluza-Klein Theory, Cartan-Weyl space,” Modern Physics Letters A,vol.25,no.37,pp. World Scientific, Hackensack, NJ, USA, 2007. 3129–3143, 2010. [30] J. Ponce De Leon, “Equivalence between space-time-matter and [9] T. Padmanabhan, “Conformal invariance, gravity and massive brane-world theories,” Modern Physics Letters A,vol.16,no.35, gauge theories,” Classical and Quantum Gravity,vol.2,no.5, pp. 2291–2303, 2001. pp. L105–L108, 1985. [31]N.Doroud,S.M.M.Rasouli,andS.Jalalzadeh,“Aclassofcos- [10] D. A. Demir, “Nonlinearly realized local scale invariance: mological solutions in induced matter theory with conformally gravity and matter,” Physics Letters B,vol.584,no.1-2,pp.133– flat bulk space,” General Relativity and Gravitation,vol.41,no. 140, 2004. 11, pp. 2637–2656, 2009. [11] O. Bertolami and C. Carvalho, “Lorentz symmetry derived from [32] S. M. M. Rasouli, M. Farhoudi, and P. V. Moniz, “Modified Lorentz violation in the bulk,” Physical Review D,vol.74,no.8, Brans-Dicke theory in arbitrary dimensions,” Classical and Article ID 084020, 2006. Quantum Gravity, vol. 31, no. 11, Article ID 115002, 26 pages, 2014. [12] O. Bertolami and C. Carvalho, “Spontaneous symmetry break- ing in the bulk and the localization of fields on the brane,” [33] S. M. M. Rasouli and P.V.Moniz, “Exact cosmological solutions Physical Review D, vol. 76, no. 10, Article ID 104048, 13 pages, in modified Brans-Dicke theory,” Classical and Quantum Grav- 2007. ity,vol.33,no.3,ArticleID035006,2016. [13] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, “The hierarchy [34] K. M. Case, “Singular potentials,” Physical Review Letters,vol. problem and new dimensions at a millimeter,” Physics Letters B, 80, pp. 797–806, 1950. vol. 429, no. 3-4, pp. 263–272, 1998. [35] P.Breitenlohner and D. Z. Freedman, “Positive energy in anti-de [14] L. Randall and R. Sundrum, “Large mass hierarchy from a small Sitter backgrounds and gauged extended supergravity,” Physics extra dimension,” Physical Review Letters,vol.83,no.17,pp. Letters B,vol.115,no.3,pp.197–201,1982. 3370–3373, 1999. [15] L. Randall and R. Sundrum, “An alternative to compactifica- tion,” Physical Review Letters,vol.83,no.23,pp.4690–4693, 1999. [16] P. D. Mannheim, Branelocalized Gravity, World Scientific, Hackensack, NJ, USA, 2005. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 9897051, 7 pages http://dx.doi.org/10.1155/2016/9897051

Research Article On the UV Dimensions of Loop Quantum Gravity

Michele Ronco1,2 1 Dipartimento di Fisica, Sapienza-Universita` di Roma, Piazzale A. Moro 2, 00185 Roma, Italy 2INFN,Sez.Roma1,PiazzaleA.Moro2,00185Roma,Italy

Correspondence should be addressed to Michele Ronco; [email protected]

Received 1 May 2016; Accepted 22 June 2016

Academic Editor: Ahmed Farag Ali

Copyright © 2016 Michele Ronco. 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.

Planck-scale dynamical dimensional reduction is attracting more and more interest in the quantum-gravity literature since it seems to be a model independent effect. However, different studies base their results on different concepts of space-time dimensionality. Most of them rely on the spectral dimension; others refer to the Hausdorff dimension; and, very recently, the thermal dimension has also been introduced. We here show that all these distinct definitions of dimension give the same outcome in the case of the effective regime of Loop Quantum Gravity (LQG). This is achieved by deriving a modified dispersion relation from the hypersurface-deformation algebra with quantum corrections. Moreover, we also observe that the number of UV dimensions can be used to constrain the ambiguities in the choice of these LQG-based modifications of the Dirac space-time algebra. In this regard, introducing the polymerization of connections, that is, 𝐾→sin(𝛿𝐾)/𝛿, we find that the leading quantum correction gives 𝑑UV = 2.5. This result may indicate that the running to the expected value of two dimensions is ongoing, but it has not been completed yet. Finding 𝑑UV at ultrashort distances would require going beyond the effective approach we here present.

1. Introduction description from the fundamental discrete blocks, which characterize the Planckian realm. It is a common expectation There is an increasing interest in the quantum-gravity litera- that this procedure should leave some traces in a semiclassical ture about the effect of dynamical dimensional reduction of regime where the emerging picture would be given in terms space-time. It consists in scale dependence of the dimension of quantum space-time. This reduction has the advantage of 𝑑 that runs from the standard IR value of four space- allowingustorecoveratleastsomeofourmorefamiliarphys- 𝑑≃2 time dimensions to the lower value at Planckian ical observables or, when it would not be possible, analogous energies. Remarkably, despite the fact that quantum-gravity ones with potential departures from their classical counter- approaches start from different conceptual premises and parts. The dimension belongs to this latter set of semiclassical adopt different formalism types, this dimensional running observables because the usual Hausdorff dimension is ill- has been found in the majority of them, such as Causal Dyn- defined for quantum space-time [12]. In the CDT approach amical Triangulation (CDT) [1], Horava-Lifshitz gravity [2], [1, 11], it was recognized for the first time that a proper Causal Sets [3], Asymptotic Safety [4], Space-Time Noncom- “quantum analogue” could be the spectral dimension 𝑑𝑆, mutativity [5], and LQG [6–8], which is here of interest. which is the scaling of the heat-kernel trace, and it reproduces However, in quantum gravity, even the concept of space- the standard Hausdorff dimension when the classical smooth time dimension is a troublesome issue and it requires some space-time is recovered. What is more, it was found that in carefulness. In fact, nonperturbative, background indepen- the UV 𝑑𝑆 ≃2(see however [13] for recent CDT simulations dent approaches (e.g., LQG [9, 10] and CDT [11]) generally favouring a smaller value of the dimension), which is now a rely on nongeometric quantities and they have discreteness recurring number in the literature [14–17]. In the Asymptotic as their core feature. For this reason, in order to extract phe- Safety program, such a value is also intimately connected to nomenological predictions, a coarse-graining process would the hope of having a fixed point in the UV. In fact, it has be necessary aimed at deriving a more manageable effective been proven that renormalizability is accomplished only if 2 Advances in High Energy Physics the dimension runs to two [18]. Furthermore, this prediction Another significant observation we make is that, follow- finds support in a recently developed approach [19] that has ing our analysis “in reverse,” we can get information about the advantage of relying on a minimal set of assumptions. the LQG quantum-geometric deformations, which affect the Provided that quantum gravity will host an effective limit Dirac algebra, from the value of the UV dimension. The characterized by the presence of a minimum allowed length importance of this recognition resides in the fact that these (identified with the Planck length), then it is shown [19] that modifications are subjected to many sources of possible the Euclidean volume becomes two-dimensional near the ambiguities [34–36] coming, for example, from the regular- Planck scale (the author is grateful to Thanu Padmanabhan ization techniques used to formally quantize the Hamiltonian for pointing this out). constraint. These ambiguities are far from being resolved However, in a recent paper [20], the physical significance and it is still not clear whether they may affect potentially of 𝑑𝑆 hasbeenquestioned.Suchaconcernisbasedontwo physical outcomes [37]. Thus, they are usually addressed observations: the computation of 𝑑𝑆 requires preliminary only on the basis of mere theoretical arguments or being Euclideanization of the space-time and also it turns out to guided by a principle of technical simplification. The main be invariant under diffeomorphisms on momentum space. sources of ambiguities are the spin representations of the Both of these features are regarded as evidence of the fact quantumstatesofgeometryaswellasthechoiceofthe that 𝑑𝑆 is an unphysical quantity [20]. Given that, it has space lattice and, in effective models we will here consider, been proposed to describe the phenomenon of dimensional they correspond, respectively, to holonomy and inverse-triad reduction in terms of the thermal (or thermodynamical) corrections. We here partially fix them with what is believed dimension 𝑑𝑇, which can be defined as the exponent of the to be a phenomenological prediction, that is, the number of Stefan-Boltzmann law. Then, the UV flowing of 𝑑𝑇 is realized UV dimensions. Notably, we notice that the leading-order through a modified dispersion relation (MDR) that affects the correction provided by the holonomy corrections of homo- partition function used to compute the energy density (see geneous connections [38, 39], which are often implemented (𝛿𝐾)/𝛿 𝐾 [20] or Section 3 for further details). Thus, the value of 𝑑𝑇 near by simply taking the expression sin instead of ,for the Planck scale depends crucially on the specific form of the example, in Loop Quantum Cosmology (LQC) [40, 41] (see MDR. Furthermore, it has been recently noticed (see [21, 22]) also [42] for a recent review on symmetry reduced models of that, from the MDR, it is also possible to infer the Hausdorff LQG), is compatible with 𝑑UV = 2.5.Thus,aswewouldhave dimension 𝑑𝐻 of energy-momentum space. If the duality expected, the number of dimensions is correctly flowing to between space-time and momentum space is preserved in lowervaluesevenifithasnotalreadyreachedthevalueof2, quantum gravity, this framework should provide another a value which is favoured in the quantum-gravity literature alternative characterization of the UV running. In this way, for the aforementioned reasons. On the basis of the steps we are in presence of proliferation of distinct descriptions of we sketch out in the analysis, we are here reporting that it the UV dimensionality of quantum space-time. These pic- should be possible to exclude all the deformation functions tures make use of very different definitions of the dimension 𝑓(𝐾) that are not consistent with 𝑑UV =2. Remarkably, those and, in principle, there is no reason why they should give the quantum corrections, which are related to LQC as well as to same outcome. On the other hand, they all coincide in the IR- the semiclassical limit of the theory, seem to point toward low-energy regime where they reduce to 4 and, thus, we could the right UV flowing. As already stated, the prediction of expect that this should happen also in the UV. Planckian dimensional reduction has been also confirmed by In this paper, we show that this advisable convergence previous LQG analyses [6, 7] but more refined computations canbeachievedinthesemiclassicallimitofLQGunder of[8]haverevealedthatthe“magic”numberof2canbe rather general assumptions. The insight we gain is based on the recently proposed quantum modifications of the reproduced only focusing on specific superposition of kine- hypersurface-deformation algebra (or the algebra of smeared matical spin-networks states (see [8] for the details). Relying constraints) [23–27], which reduces to a correspondingly on the recently developed effective methods for LQG, we here deformed Poincare´ algebra in the asymptotic region, as provide further support to the idea that the effective space- shownin[28].ThesePlanckiandeformationsoftherelativis- time of LQG may be two-dimensional at ultra-Planckian tic symmetries are a key feature of the Deformed Special Rel- scales. ativity scenario [29, 30], as already pointed out in [28], and, what is more, it has been recently shown in [31] that they are consistent with 𝜅-Minkowski noncommutativity of the space- 2. Modified Dispersion Relation time coordinates [32, 33]. We here exploit them to compute the MDR, thereby linking the LQG-based quantum correc- We start considering the classical hypersurface-deformation tions to the deformation of the dispersion relation. The gen- algebra (HDA), which was first introduced by Dirac [43]. eral form of the MDR we derive allows us to find that the spec- It is the set of Poisson brackets closed by the smeared tral, the thermal, and the Hausdorff (notice that we are always constraints of the Arnowitt-Deser-Misner formulation of referring to the Hausdorff dimension of momentum space General Relativity (GR) [44]. The Dirac algebra of constraints since, as we mentioned, that of quantum space-time cannot is the way in which general covariance is implemented once be defined) dimensions follow the same UV flowing; that is, the space-time manifold has been split into the time direction 𝑑𝑆 =𝑑𝑇 =𝑑𝐻. and the three spatial surfaces, that is, M = R ×Σ. Advances in High Energy Physics 3

It is given by one has to consider just the local point-wise holonomies √ ℎ𝑖(𝐴) = cos(𝛿𝐴/2)I + sin(𝛿𝐴/2)𝜎𝑖 (where 𝛿∝𝑙𝑃 = 𝐺≈ 𝑘 𝑗 𝑘 −35 {𝐷 [𝑀 ],𝐷[𝑁 ]} = 𝐷 [L𝑀⃗ 𝑁 ], 10 misconnectedtothesquarerootoftheminimum

𝑘 eigenvalue of the area operator [41]). These are the kinds of {𝐷 [𝑁 ],𝐻[𝑀]}=𝐻[L𝑁⃗ 𝑀] , (1) quantum effects considered in effective (semiclassical) LQG theories as well as both in spherically reduced models and 𝑗𝑘 {𝐻 [𝑁] ,𝐻[𝑀]} =𝐷[ℎ (𝑁𝜕𝑗𝑀−𝑀𝜕𝑗𝑁)] , in cosmological contexts [42]. In particular, for spherically symmetric LQG (see, e.g., [28, 50, 51]), the deformation where 𝐻[𝑁] is the Hamiltonian (or scalar) constraint, while 𝑘 function depends on the homogeneous angular connection 𝐷[𝑁 ] is the momentum (or vector) constraint. The function 𝐾𝜙 and it is directly related to the second derivative of 𝑁 is called the lapse and it is needed to implement time 𝑓(𝐾𝜙) 𝛽= 𝑖 the square of the holonomy correction ;thatis, diffeomorphisms, while 𝑁 is the shift vector necessary to (1/2)(𝑑2𝑓2(𝐾 )/𝑑𝐾2) 𝑖𝑗 𝜙 𝜙 . Then, the important contribution of move along a given hypersurface and, finally, ℎ is the inverse 𝑘 [28] has been to establish a link between these LQG-inspired metric induced on Σ.Thus,𝐻[𝑁] and 𝐷[𝑁 ] have to be quantum corrections and DSR-like deformations of the rela- understood as the generators of gauge transformations which, tivistic symmetries, thanks to the recognition that the angular inthecaseofGR,arespace-timediffeomorphisms.Forthe connection 𝐾𝜙 ∝𝑃𝑟 is proportional to the Brown-York purposes of our analysis, it is relevant to point out the well radial momentum [52] that generates spatial translations at established fact (see [45]) that when the spatial metric is flat infinity (see [28, 31] for the details). In fact, it has been shown ℎ =𝛿 𝑁=Δ𝑡+V 𝑥𝑎 V 𝑖𝑗 𝑖𝑗 ,ifwetake 𝑎 (where 𝑎 is the infinitesimal that, taking the Minkowski limit of (2) as we sketched above 𝑘 𝑘 𝑘 𝑙 𝑘 boost parameter) and 𝑁 =Δ𝑥 +𝑅𝑙 𝑥 (where 𝑅𝑙 is the for the classical case, the LQG-deformed HDA produces a matrix that generates infinitesimal rotations), we can infer the corresponding Planckian deformation of the Poincarealge-´ Poincare´ algebra from the Dirac algebra [45]. This classical bra: relationisexpectedtoholdalsoatthequantumlevel. [𝐵 ,𝑃]=𝑖𝑃𝛽(𝑙 𝑃 ), OneoftheopenissuesintheLQGresearchisthesearch 𝑟 0 𝑟 𝑃 𝑟 (3) forfullyquantizedversionsoftheconstraints𝐻[𝑁] and where the other commutation relations remain unmodified. 𝐷[𝑁𝑘] on a Hilbert space. While it is known how to treat Here, 𝐵𝑟 is the generator of radial boosts and 𝑃0 is the energy. spatial diffeomorphisms and also how to solve the momen- The explicit form of 𝛽 is unknown and as we already stressed tum constraint [46, 47] thereby obtaining the kinematical it is affected by ambiguities. In light of this, for our analysis, Hilbert space of the theory, the finding of a Hamiltonian we assume a rather general form: operator is far from being completed. However, over the last 𝛾 𝛾 fifteen years, several techniques have been developed, using 𝛽 (𝜆𝑃𝑟) =1+𝛼𝑙𝑃𝑃𝑟 (4) both effective methods and discrete operator computations. In this way, some candidates for an effective scalar constraint which is motivated by the above considerations and, obvi- 𝑄 𝛽(𝑙 𝑃 )=1 𝐻 [𝑁] have been identified [48–50]. For the analysis we are ously, satisfies the necessary requirement: lim𝑙𝑃→0 𝑃 𝑟 ; here reporting, the interesting fact is that the semiclassical that is, we want to recover the standard Poincare´ algebra in corrections introduced in the Hamiltonian leave trace in the the continuum limit. We leave the constants of order one 𝛼 𝛾 algebra of constraints. Remarkably, even if these calculations and that parametrize the aforementioned ambiguities use different formalism types and they are based on different unspecified. These parameters should encode at least the assumptions, the general form of the modified HDA turns out leading-order quantum correction to the Poincarealgebra.´ tobethesameinallthesestudies;thatis,onlythePoisson Using (3) and (4) and taking into account the fact that [𝐵 ,𝑃]=𝑖𝑃 [𝑃 ,𝑃]=0 bracket between two scalar constraints is affected by quantum 𝑟 𝑟 0 and 𝑟 0 , a straightforward computation effects [39, 51]: givesusthefollowingMDR:

𝑄 𝑄 𝑖𝑗 2 2 2𝛼 𝛾 𝛾+2 {𝐻 [𝑀] ,𝐻 [𝑁]}=𝐷[𝛽ℎ (𝑀𝜕𝑗𝑁−𝑁𝜕𝑗𝑀)] , (2) 𝐸 =𝑝 + 𝑙 𝑝 . 𝛾+2 𝑃 (5) where the specific form of the deformation function 𝛽 as This completes the analysis started in [28] and carried on in well as its dependence on the phase space variables, which 𝑎 𝑖 [31], which aimed at building a bridge between the formal are (ℎ𝑖𝑗 ,𝜋𝑖𝑗 ) if we use the metric formulation or (𝐴 ,𝐸 ) if 𝑖 𝑎 structures of loop quantization to the more manageable DSR we use Ashtekar’s one, varies with the quantum corrections 𝐻𝑄[𝑁] scenario with the objective of enhancing the possibilities of considered to define . linking mathematical constructions to observable quantities. One of the causes of these quantum modifications of the The remarkable fact of having derived (5) from the LQG- scalar constraint is the fact that LQG cannot be quantized 𝐴𝑎 deformed algebra of constraints (2) is that it will give us the directly in terms of the Ashtekar variables 𝑖 ,whichhaveto opportunity to constrain experimentally the formal ambi- be replaced with their parallel propagators (or holonomies) 𝑖 𝑎 guities of the LQG approach exploiting the ever-increasing −∫ 𝑡 𝐴𝑖 𝜏𝑎 [9,10,38,39]ℎ𝛼(𝐴) = P𝑒 𝛼 (where P is the path- phenomenological implications of MDRs (see [53] and the 𝑖 ordering operator, 𝑡 the tangent vector to the curve 𝛼,and references therein). We are often in the situation in which 𝜏𝑎 = −(𝑖/2)𝜎𝑎 the generators of 𝑆𝑈(2)). If 𝑎 is the spatial index quantum-gravity phenomenology misses clear derivation of a direction along which space-time is homogeneous, then from full-fledged developed approaches to quantum gravity 4 Advances in High Energy Physics or, on the contrary, the high complexity of these formalism which can be obtained as usual deriving the logarithm of the types does not allow inferring testable effects. Following thermodynamical partition function [20] with respect to the the work initiated in [31], we are here giving a further temperature 𝑇. Evidently, in our case, we have that 𝛾𝑡 =0and contribution to fill this gap. We also find it interesting to 𝛾𝑥 =𝛾/2and, thus, we find 𝑑𝑇 =1+6/(2+𝛾);thatis,the notice that our MDR confirms a property of two previously thermal dimension agrees with the spectral dimension 𝑑𝑇 ≡ proposed MDRs (see [54, 55]); that is, LQG corrections affect 𝑑𝑆. only the momentum sector of the dispersion relation leaving Finally, we can calculate also the Hausdorff dimension of the energy dependence untouched. Therefore, this property, momentum space that if the duality with space-time is not which has rigorous justification in the spherically symmetric broken by quantum effects should agree with both 𝑑𝑆 and 𝑑𝑇. framework [28, 50, 51] we are here adopting, seems to be a As pointed out in [21], a way to compute 𝑑𝐻 is to find a set recurring feature of LQG. Moreover, all the precedent anal- of momenta that “linearize” the MDR. Given (5), a possible yses were confined to the kinematical Hilbert space of LQG, choice is given by whilewehavehereobtained(5)fromtheflat-space-timelimit of the full HDA including also the semiclassical Hamiltonian 2𝛼 𝑘=√𝑝2 + 𝑙𝛾 𝑝𝛾+2. constraint (2). Thus, even if we are working off shell (i.e., we 𝛾+2 𝑃 (10) do not solve the constraint equations), the MDR (5) should contain at least part of the dynamical content of LQG. In In terms of these new variables (𝐸, 𝑘),theUVmeasureon the next section, we will see that the form of (5) is crucial to momentum space becomes prove that the running of dimensions does not depend on the 2 (4−𝛾)/(𝛾+2) chosen definition of the dimension. 𝑝 𝑑𝑝𝑑𝐸 󳨀→𝑘 𝑑𝑘𝑑𝐸. (11) 𝑑 3. Dimensions and Quantum Corrections From (11), we can read off 𝐻: 4−𝛾 6 Our next task is to use the MDR (5) we derived in Section 2 𝑑𝐻 =2+ =1+ (12) in order to show that, regardless of the value of the unknown 𝛾+2 2+𝛾 parameters 𝛼 and 𝛾, the different characterizations of the UV 𝑑 𝑑 flowing introduced in the literature predict the same number which, evidently, coincides with both 𝑆 and 𝑇.Thus,no of dimensions if we consider the effective regime of LQG matter which definition of dimensionality is used, in the in the sense introduced in [28, 50, 51] and sketched in the semiclassical limit (or in symmetry reduced models) of LQG, previous section. To see this, we start by the computation of the UV running is free of ambiguities since we have found 𝑑 ≡𝑑 ≡𝑑 the spectral dimension, which is defined as follows: that 𝑆 𝑇 𝐻. Thelastconsiderationwewanttomakeconcernswhatthe 𝑑 𝑃 (𝑠) log number of UV dimensions can teach us about LQG. In the 𝑑𝑆 =−2lim , (6) 𝑠→0 𝑑 log (𝑠) analysis we here reported, the value of 𝛾 should be provided 𝐻𝑄[𝑁] where 𝑃(𝑠) is the average return probability of a diffusion pro- by the LQG corrections used to build , which, though, cess in Euclidean space-time with fictitious time 𝑠.Following are far from being unique. On the other hand, we mentioned that the number of dimensions runs to two in the UV, a [20, 56–60], we compute 𝑑𝑆 fromtheEuclideanversionofour MDR(5)whichisad’Alembertianoperatoronmomentum prediction that seems to be model independent. Moreover, space: support in LQG has been also found by the studies of [6, 8] under certain assumptions. It is evident from (8) and (12) that 𝐸 2 2 2𝛼 𝛾 𝛾+2 in order to reproduce such a shared expectation we should Δ =𝐸 +𝑝 + 𝑙𝑃𝑝 . (7) 𝛾+2 take 𝛾=4. A recurring form of holonomy corrections both in spherically symmetric LQG [24, 28, 39, 61] and in LQC Then, a lengthy but straightforward computation (see [58– [40, 41] is represented by the choice 𝑓(𝐾) = sin(𝛿𝐾)/𝛿 that 60]) leads to the following result: implies 𝛽=cos(2𝛿𝐾). In light of the above arguments, if we 6 𝑑 =1+ . keep restricted to mesoscopic scales where the MDR is well 𝑆 2+𝛾 (8) approximated by the first-order correction, it is easy to realize that this implies 𝛾=2(see [31] for the explicit computation). 𝑑 𝛼 Notice that the value of 𝑆 does not depend on but only In this way, we would have 𝑑=2.5at scales near but below 𝛾 on , that is, only on the order of Planckian correction to the thePlanckscale.Wefindthefactthatweobtainavaluewhich dispersion relation (see (5)). We will use this fact later on. is greater than 2 rather encouraging, since such an outcome Nowwewanttoshowthatalsothethermaldimension may signal that the descent from the classical value of 4 to the 𝑑 𝑇 isgivenby(8).Tothisend,werecallthedefinitionof UVvalueisinprogressbutithasnotbeencompletedyet.In 𝑑𝑇 introduced in [20]. If you have deformed Lorentzian 𝐿 2 2 2𝛾 2(1+𝛾 ) 2𝛾 2(1+𝛾 ) fact, at ultra-Planckian energies, the Taylor expansion of the Δ =𝐸 −𝑝 +𝑙 𝑡 𝐸 𝑡 −𝑙 𝑥 𝑝 𝑥 𝑓(𝐾) 𝛿 d’Alembertian 𝛾𝑡𝛾𝑥 𝑡 𝑥 ,then correction function in series of powers of is no more 𝑑𝑇 is the exponent of the temperature 𝑇 in the modified reliable and, as a consequence, it cannot fully capture the Stefan-Boltzmann law: flowing of 𝑑. Therefore, we are led to conclude that the much-

1+3((1+𝛾𝑡)/(1+𝛾𝑥)) used polymerization of homogeneous connections, which is 𝜌𝛾 𝛾 ∝𝑇 (9) 𝑡 𝑥 a direct consequence of evaluating the holonomies in the Advances in High Energy Physics 5 fundamental 𝑗=1/2representation of 𝑆𝑈(2), realizes at least constraints, is able to generate the running of the dimension. partially the expected running of dimensions. Polymerizing In this way, we have provided further evidence that the configuration variables is used not only in LQC and in sym- phenomenon of UV dimensional reduction can be realized metry reduced contexts but also in the definition of the semi- alsointheLQGapproach,therebyconfirmingtheresultsof classicalregimeofLQG,whichisbasedontheintroductionof previous studies. Remarkably, the value of 𝑑UV is sensible to spin-network states peaked around a single representation, in the specific choice of quantum corrections which are consid- the majority of cases 𝑗=1/2.Wehavehereshownthatthese ered in the model. Therefore, we pointed out that there is an quantum modifications can be related to the phenomenon of observablewhosevaluecanbeusedtoselectaparticularform dimensional reduction. This link we have established gives for the quantum correction functions, thereby reducing the us the possibility of constraining part of the quantization LQG quantization ambiguities. In particular, we showed that ambiguities in LQG. In fact, the value of 𝑑UV fixes a specific the evaluation of the Hamiltonian constraint over semiclas- choice of the parameter 𝛾 in (4), thereby restricting the form sical states peaked at 𝑗=1/2(that can be also implemented of the allowed deformation functions 𝐾→𝑓(𝐾).Inlight with the substitution 𝐾→sin(𝛿𝐾)/𝛿 at an effective level) of our analysis, one should select only those modifications corresponds to 𝑑UV = 2.5. Since we inferred such a value from which are compatible with the prediction of two-dimensional the parametrization of the first LQG correction (4) (which space-time at ultrashort distances, that is, those giving 2≲ corresponds to a second-order correction in the Planck 2 𝑑<4(or equivalently 0<𝛾≲4) to a first approximation. length ∼𝑙𝑃), then it is reasonable to regard our result as a first approximation that cannot capture the ending outcome of 4. Outlook the dimensional running. From this perspective, we observed that obtaining a dimension greater than 2 at energies near In the recent quantum-gravity literature, basically three but below the Planck scale might be significant, because it different definitions of dimension have been proposed with can be read as a hint that the dimension is flowing to the the aim of generalizing this notion for quantum space-time. It “magic” number of 2 that we can expect to be reached in the is well known that they all provide a possible characterization deep UV. Developing the off-shell constraint algebra in full of the UV running but, in the majority of cases, they also generality without any symmetrical reduction or semiclassi- give different outcomes for the value of the dimension. This cal approximation would be of fundamental importance to clashes with the growing consensus on the fact that the extend our observations to the full LQG framework. phenomenon of dynamical dimensional reduction is a model independent feature of quantum gravity that gives the unique Competing Interests predictions 𝑑UV ≃2.Thus,if𝑑𝑆,𝑑𝑇,and𝑑𝐻 are really proper definitionsoftheUVdimension,wewouldliketohavethat The author declares that he has no competing interests. 𝑑𝑆 ≡𝑑𝑇 ≡𝑑𝐻. In this paper, we showed that this striking convergence canbeaccomplishedinthecaseoftheLQGapproach. Acknowledgments Toachievesucharesult,wereliedonthedeformationsof The author is grateful to Giovanni Amelino-Camelia and the constraint algebra recently proposed in the framework Giorgio Immirzi for useful and critical discussions during the of both effective spherically symmetric LQG and LQC. first stages of the elaboration of this work. He also wants to Remarkably, in the former case, it has been proven that these thank Gianluca Calcagni for carefully reading the paper and quantum corrections leave traces in the Minkowski limit giving many helpful comments that improved this work. in terms of a DSR-like Poincare´ algebra where the relevant deformations are functions of the spatial momentum. We here exploited these LQG-motivated deformations of the References relativistic symmetries to infer the generic form of the MDR. [1] J. Ambjørn, J. Jurkiewicz, and R. Loll, “Reconstructing the Interestingly,ourMDRisqualitativelyofthesametypeof universe,” Physical Review D,vol.72,no.6,ArticleID064014, the two previously proposed MDRs for LQG [54, 55]; namely, 2005. the modifications affect only the momentum sector. From the [2] P. Hoˇrava, “Spectral dimension of the universe in quantum MDR, we derived the spectral, the thermal, and the Hausdorff gravity at a Lifshitz point,” Physical Review Letters,vol.102,no. dimensions proving that they all agree. Thus, in the top-down 16, Article ID 161301, 4 pages, 2009. approach of LQG, the desirable convergence of different [3] S. Carlip, “Dimensional reduction in causal set gravity,” Classical characterization of the concept of dimensionality is accom- and Quantum Gravity,vol.32,article23,ArticleID232001,2015. plished. On the other hand, the analysis we here reported may [4] M. Niedermaier and M. Reuter, “The asymptotic safety scenario provide a guiding principle for the construction of bottom-up in quantum gravity,” Living Reviews in Relativity,vol.9,p.5, approaches. 2006. Moreover, our analysis led us to give a contribution [5] D. Benedetti, “Fractal properties of quantum spacetime,” Phys- toward enforcing the fecund bond between theoretical for- ical Review Letters, vol. 102, no. 11, Article ID 111303, 4 pages, malism types and phenomenological predictions. In fact, we 2009. found that the simple polymerization of connections, which [6] L. Modesto, “Fractal spacetime from the area spectrum,” Clas- is much used both in midi-superspace models and in LQC sical and Quantum Gravity,vol.26,no.24,ArticleID242002,9 where semiclassical states are exploited to compute effective pages, 2009. 6 Advances in High Energy Physics

[7] G. Calcagni, D. Oriti, and J. Thurigen,¨ “Spectral dimension of [26] C. Tomlin and M. Varadarajan, “Towards an anomaly-free quantum geometries,” Classical and Quantum Gravity,vol.31, quantum dynamics for a weak coupling limit of Euclidean no. 13, Article ID 135014, 2014. gravity: diffeomorphism covariance,” Physical Review D,vol.87, [8] G. Calcagni, D. Oriti, and J. Thrigen, “Dimensional flow in Article ID 044040, 2013. discrete quantum geometries,” Physical Review D,vol.91,no.8, [27]A.Barrau,M.Bojowald,G.Calcagni,J.Grain,andM.Kagan, Article ID 084047, 11 pages, 2015. “Anomaly-free cosmological perturbations in effective canon- [9] A. Ashtekar and J. Lewandowski, “Background independent ical quantum gravity,” JournalofCosmologyandAstroparticle quantum gravity: a status report,” Classical and Quantum Physics,vol.2015,no.5,article051,2015. Gravity,vol.21,no.15,pp.R53–R152,2004. [28]M.BojowaldandG.M.Paily,“Deformedgeneralrelativity,” [10] C. Rovelli, “Loop quantum gravity,” Living Reviews in Relativity, Physical Review D,vol.87,ArticleID044044,2013. vol. 11, article 5, 2008. [29] G. Amelino-Camelia, “Testable scenario for relativity with [11] J. Ambjørn, A. Gorlich,¨ J. Jurkiewicz, and R. Loll, “Nonpertur- minimum length,” Physics Letters B,vol.510,no.1–4,pp.255– bative quantum gravity,” Physics Reports,vol.519,no.4-5,pp. 263, 2001. 127–210, 2012. [30] G. Amelino-Camelia, “Relativity in spacetimes with short- [12] S. Carlip, “Spontaneous dimensional reduction in short−dis- distance structure governed by an observer-independent tance quantum gravity?” in AIP Conference Proceedings,vol. (Planckian) length scale,” International Journal of Modern 1196, p. 72, Wroclaw, Poland, 2009. Physics. D. Gravitation, Astrophysics, Cosmology,vol.11,no.1,pp. [13] J. Ambjorn, D. Coumbe, J. Gizbert-Studnicki, and J. Jurkiewicz, 35–59, 2002. “Recent results in CDT quantum gravity,” http://arxiv.org/abs/ [31] G. Amelino-Camelia, M. M. Da Silva, M. Ronco, L. Cesarini, 1509.08788. and O. M. Lecian, “Spacetime-noncommutativity regime of [14] O. Lauscher and M. Reuter, “Fractal spacetime structure in Loop Quantum Gravity,” https://arxiv.org/abs/1605.00497. asymptotically safe gravity,” JournalofHighEnergyPhysics,vol. [32] S. Majid and H. Ruegg, “Bicrossproduct structure of 𝜅-Poincare 2005, no. 10, article 050, 2005. group and non-commutative geometry,” Physics Letters B,vol. [15] G. Calcagni, A. Eichhorn, and F. Saueressig, “Probing the quan- 334, no. 3-4, pp. 348–354, 1994. tum nature of spacetime by diffusion,” Physical Review D,vol. [33] J. Lukierski, H. Ruegg, and W. J. Zakrzewski, “Classical and 87,no.12,ArticleID124028,14pages,2013. quantum mechanics of free 𝜅-relativistic systems,” Annals of [16] D. Benedetti and J. Henson, “Spectral geometry as a probe of Physics,vol.243,no.1,pp.90–116,1995. quantum spacetime,” Physical Review D,vol.80,no.12,Article [34] D.-W. Chiou and L.-F. Li, “Loop quantum cosmology with ID13,124036pages,2009. higher order holonomy corrections,” Physical Review D. Parti- [17] S. Carlip, “Spontaneous dimensional reduction in quantum cles, Fields, Gravitation, and Cosmology,vol.80,no.4,043512, gravity,” Third award, 2016 Gravity Research Foundation Essay 19 pages, 2009. contest, https://arxiv.org/abs/1605.05694. [35] A. Laddha and M. Varadarajan, “Hamiltonian constraint in [18] R. Percacci and D. Perini, “On the ultraviolet behaviour of polymer parametrized field theory,” Physical Review D,vol.83, Newton’s constant,” Classical and Quantum Gravity,vol.21,no. Article ID 025019, 2011. 22, pp. 5035–5041, 2004. [36] A. Corichi and P. Singh, “Is loop quantization in cosmology [19] T. Padmanabhan, S. Chakraborty, and D. Kothawala, “Space- unique?” Physical Review D: Particles, Fields, Gravitation, and time with zero point length is two-dimensional at the Planck Cosmology,vol.78,no.2,ArticleID024034,13pages,2008. scale,” General Relativity and Gravitation,vol.48,article55, [37] H. Nicolai, K. Peeters, and M. Zamaklar, “Loop quantum 2016. gravity: an outside view,” Classical and Quantum Gravity,vol. [20] G. Amelino-Camelia, F. Brighenti, G. Gubitosi, and G. Santos, 22, no. 19, pp. R193–R247, 2005. “Thermal dimension of quantum spacetime,” https://arxiv.org/ [38] E. Wilson-Ewing, “Holonomy corrections in the effective equa- abs/1602.08020. tions for scalar mode perturbations in loop quantum cosmol- [21] G. Amelino-Camelia, M. Arzano, G. Gubitosi, and J. Magueijo, ogy,” Classical and Quantum Gravity,vol.29,no.8,085005,19 “Dimensional reduction in the sky,” Physical Review D— pages, 2012. Particles, Fields, Gravitation and Cosmology,vol.87,no.12, [39] M. Bojowald and G. M. Paily, “Deformed general relativity and Article ID 123532, 2013. effective actions from loop quantum gravity,” Physical Review D, [22] G. Amelino-Camelia, M. Arzano, G. Gubitosi, and J. Magueijo, vol. 86, Article ID 104018, 2012. “Planck-scale dimensional reduction without a preferred frame,” Physics Letters. B,vol.736,pp.317–320,2014. [40] A. Ashtekar and P. Singh, “Loop quantum cosmology: a status report,” Classical and Quantum Gravity,vol.28,no.21,213001, [23] M. Bojowald, G. M. Hossain, M. Kagan, and S. Shankara- 122 pages, 2011. narayanan, “Anomaly freedom in perturbative loop quantum gravity,” Physical Review D. Particles, Fields, Gravitation, and [41] M. Bojowald, “Loop quantum cosmology,” Living Reviews in Cosmology,vol.78,no.6,063547,31pages,2008. Relativity,vol.8,p.11,2005. [24] T. Cailleteau, J. Mielczarek, A. Barrau, and J. Grain, “Anomaly- [42] A. Ashtekar, “Symmetry reduced loop quantum gravity: a bird’s free scalar perturbations with holonomy corrections in loop eye view,” https://arxiv.org/abs/1605.02648. quantum cosmology,” Classical and Quantum Gravity,vol.29, [43] P. A. M. Dirac, “The theory of gravitation in hamiltonian form,” no.9,ArticleID095010,17pages,2012. Proceedings of the Royal Society of London A,vol.246,no.1246, [25] A. Perez and D. Pranzetti, “On the regularization of the 1958. constraint algebra of quantum gravity in 2+1dimensions with [44] R. L. Arnowitt, S. Deser, and C. W. Misner, “Republication a nonvanishing cosmological constant,” Classical and Quantum of: the dynamics of general relativity,” General Relativity and Gravity,vol.27,no.14,ArticleID145009,2010. Gravitation,vol.40,no.9,pp.1997–2027,2008. Advances in High Energy Physics 7

[45] T. Regge and C. Teitelboim, “Role of surface integrals in the Hamiltonian formulation of general relativity,” Annals of Phy- sics, vol. 88, pp. 286–318, 1974. [46] A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao, and T. Thiemann, “Quantization of diffeomorphism invariant theories of connections with local degrees of freedom,” Journal of Mathematical Physics,vol.36,no.11,pp.6456–6493,1995. [47] J. Lewandowski, A. Okolow, H. Sahlmann, and T. Thiemann, “Uniqueness of diffeomorphism invariant states on holonomy- flux algebras,” Communications in Mathematical Physics,vol. 267, no. 3, pp. 703–733, 2006. [48] M. Bojowald and R. Swiderski, “Spherically symmetric quan- tum geometry: hamiltonian constraint,” Classical and Quantum Gravity,vol.23,no.6,pp.2129–2154,2006. [49] K. Banerjee and G. Date, “Discreteness corrections to the effective Hamiltonian of isotropic loop quantum cosmology,” Classical and Quantum Gravity,vol.22,no.11,pp.2017–2033, 2005. [50] M. Bojowald, M. Kagan, H. H. Hernandez,´ and A. Skirzewski, “Effective constraints of loop quantum gravity,” Physical Review D: Particles, Fields, Gravitation, and Cosmology,vol.75,no.6, Article ID 064022, 25 pages, 2007. [51] S. Brahma, “Spherically symmetric canonical quantum gravity,” Physical Review D. Particles, Fields, Gravitation, and Cosmology, vol.91,no.12,124003,12pages,2015. [52] J. D. Brown and J. W.York Jr., “Quasilocal energy and conserved charges derived from the gravitational action,” Physical Review D,vol.47,no.4,pp.1407–1419,1993. [53] G. Amelino-Camelia, “Quantum-spacetime phenomenology,” Living Reviews in Relativity,vol.16,p.5,2013. [54] R. Gambini and J. Pullin, “Nonstandard optics from quantum space-time,” Physical Review D,vol.59,no.12,ArticleID124021, 4 pages, 1999. [55]J.Alfaro,H.A.Morales-Tecotl,´ and L. F. Urrutia, “Loop quantum gravity and light propagation,” Physical Review D,vol. 65,no.10,ArticleID103509,2002. [56] L. Modesto, “Spectral dimension of a quantum universe,” Physical Review D,vol.81,ArticleID104040,2010. [57] G. Calcagni, L. Modesto, and G. Nardelli, “Quantum spectral dimension in quantum field theory,” International Journal of Modern Physics D,vol.25,no.5,ArticleID1650058,26pages, 2016. [58]T.P.Sotiriou,M.Visser,andS.Weinfurtner,“Fromdispersion relations to spectral dimension—and back again,” Physical Review D, vol. 84, Article ID 104018, 2011. [59] M. Arzano, G. Gubitosi, J. Magueijo, and G. Amelino-Camelia, “Anti-de Sitter momentum space,” Physical Review D,vol.92, no.2,ArticleID024028,12pages,2015. [60] G. Calcagni, “Diffusion in multiscale spacetimes,” Physical Review E,vol.87,no.1,ArticleID012123,20pages,2013. [61] M. Campiglia, R. Gambini, J. Olmedo, and J. Pullin, “Quantum self-gravitating collapsing matter in a quantum geometry,” http://arxiv.org/abs/1601.05688. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 1632469, 8 pages http://dx.doi.org/10.1155/2016/1632469

Research Article Charged Massive Particle’s Tunneling from Charged Nonrotating Microblack Hole

M. J. Soleimani,1 N. Abbasvandi,2 Shahidan Radiman,2 and W. A. T. Wan Abdullah1

1 Department of Physics, University of Malaya, 50603 Kuala Lumpur, Malaysia 2SchoolofAppliedPhysics,FST,UniversityKebangsaanMalaysia,43600Bangi,Malaysia

Correspondence should be addressed to M. J. Soleimani; [email protected]

Received 30 March 2016; Revised 23 June 2016; Accepted 26 June 2016

Academic Editor: Barun Majumder

Copyright © 2016 M. J. Soleimani 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.

In the tunneling framework of Hawking radiation, charged massive particle’s tunneling in charged nonrotating TeV-scale black hole is investigated. To this end, we consider natural cutoffs as a minimal length, a minimal momentum, and a maximal momentum through a generalized uncertainty principle. We focus on the role played by these natural cutoffs on the luminosity of charged nonrotating microblack hole by taking into account the full implications of energy and charge conservation as well as the backscattered radiation.

1. Introduction pioneering work [25] constructed a procedure to describe the Hawking radiation emitted from a Schwarzschild black hole One of the most exciting consequences of models of low scale as a tunneling through its quantum horizon. The emission gravity [1–4] is the possibility of production of small black rate (tunneling probability) which arising from the reduction holes [5–7] at particle colliders such as the Large Hadron of the black hole mass is related to the change of black hole Collider (LHC) as well as in Ultrahigh Energy Cosmic Ray entropies before and after the emission. In this paper, charged Air Showers (UECRAS) [8–10]. Incorporation of gravity in particle’s tunneling from charged nonrotating microblack quantum field theory supports the idea that the standard hole is investigated. Weconsider a more general framework of Heisenberg uncertainty principle should be reformulated GUP that admits a minimal length, minimal momentum, and by the so-called generalized uncertainty principle near the maximal momentum to study the effects of natural cutoffs Planck scale [11–13]. In particular, the existence of a minimum on the tunneling mechanism and luminosity of charged observable length is indicated by string theory [14], TeV- nonrotating TeV-scale black holes with extra dimensions in scale black hole physics [15], and loop quantum gravity [16]. Arkani-Hamed, Dimopoulos, and Dvali (ADD) model [1] Moreover, some black hole Gedanken experiments support in the context of this GUP. The calculation shows that the the idea of existence of a minimal measurable length in emission rate satisfies the first law of black hole thermody- a fascinating manner [17, 18] On the other hand, Doubly namics. The paper is organized as follows: in Section 2, we Special Relativity theories [19–24] suggest that a test particle’s introduce a generalized uncertainty principle with minimal momentum cannot be arbitrarily imprecise and there is length, minimal momentum, and maximal momentum. In an upper bound for momentum fluctuation. It means that Section 3, we obtain an expression for emission rate of there is also a maximal particle momentum. It has been charged particle from charged nonrotating microblack hole shown that incorporation of quantum gravity effects in black based on the ADD model and the mentioned GUP. We hole physics and thermodynamics through a generalized consider the backscattering of the emitted radiation taking uncertainty principle (GUP) with the mentioned natural into account energy and charge conservation to evaluate the cutoffs modifies the result dramatically, specially, the final luminosity of TeV-scale black hole in presence of natural stage of black hole evaporation. Parikh and Wilczek on their cutoffs.Thelastpartisthediscussionandcalculation. 2 Advances in High Energy Physics

2. Generalized Uncertainty Principle 3. Tunneling Mechanism The existence of a minimal measurable length of the order The idea of large extra dimensions might allow studying −35 of the Planck length, 𝑙𝑝 ∼10 m, was indicated by most interactions at Trans-Planckian energies in particle colliders of quantum gravity approaches [26, 27] which modifies the and the ADD model used d new large space-like dimen- Heisenberg uncertainty principle (HUP) to the so-called sions. So, in order to investigate the Hawking radiation via generalized (gravitational) uncertainty principle (GUP). The tunneling from charged nonrotating TeV-scale black holes of minimal position uncertainty, Δ𝑥0, could be not made arbi- higher dimensional, a natural candidate is that of Reissner- trarily small toward zero [12] in the GUP framework due to Nordstrom d-dimensional modified solution in presence its essential restriction on the measurement precision of the of generalized uncertainty principle. In this case, the line particle’s position. On the other hand, Doubly Special Rela- element of d-dimensional Reissner-Nordstrom solution of tivity (DSR) theories [21–24] have considered that existence Einstein field equation is given by [35] of a minimal measurable length would restrict a test particle’s 𝑑𝑠2 =𝑓(𝑟) 𝑐2𝑑𝑡2 −𝑓−1 (𝑟) 𝑑𝑟2 −𝑟2𝑑Ω2 momentum to take arbitrary values and therefore there is an 𝑑−2 (6) upper bound for momentum fluctuation [28, 29]. So there =𝑔 𝑑𝑥𝜇𝑑𝑥], is a maximal particle’s momentum due to the fundamental 𝜇] structureofspace-timeatthePlanckscale[30,31].Basedon 𝑑−2 where Ω𝑑−2 is the metric of the unit 𝑆 as Ω𝑑−2 = the above arguments, the GUP that predicts both a minimal (𝑑−1)/2 length and a maximal momentum can be written as follows 2𝜋 /Γ((𝑑 − 1)/2) and [19, 20]: 2 𝜔𝑑−2𝑀 𝜔𝑑−2𝑄 𝑓=𝑓(𝑀, 𝑄, 𝑟) =1− + , (7) ℏ 2 2 𝑟𝑑−3 2 (𝑑−3) Ω 𝑟2(𝑑−3) Δ𝑥Δ𝑝 ≥ [1 − 𝛼 ⟨Δ𝑝⟩ +2𝛼 ⟨Δ𝑝⟩ ]. (1) 𝑑−2 2 where 𝜔𝑑−2 =16𝜋/(𝑑−2)Ω𝑑−2.Here𝑀 and 𝑄 are the mass The relation (1) can lead us to the following commutator and electric charge of the black hole, respectively; units 𝐺𝑑 = relation: 𝑐=ℏ=1are adopted throughout this paper. The black hole 2 2 has an outer/inner horizon located at [𝑥, 𝑝] = 𝑖ℏ (1 − 𝛼𝑝 +2𝛼 𝑝 ), (2) 𝜔 (𝑑−2) 𝑄2 𝛼 𝑟𝑑−3 = 𝑑−2 [𝑀±√𝑀2 − ] . where is GUP dimensionless positive constant of both ± 2 8𝜋 (𝑑−3) (8) minimal length and maximal momentum that depends on [ ] the details of the quantum gravity hypothesis. It has been developed that particle’s momentum cannot be zero if the Therefore, the event horizon shrinks, and the inner one curvature of space-time becomes important and its effects appears; when the black hole becomes charged the inner radius is related to the amount of charge and the outer one are taken into account [32, 33]. In fact, there appears to be a 𝑟 limit to the precision of which the corresponding momentum + corresponds to the radius of Schwarzschild black hole. In can be expressed as a nonzero minimal uncertainty in this case, (8) can be rewritten as follows: momentum measurement. Based on this more general frame- 1/(𝑑−3) 2 work as a consequence of small correction to the canonical 𝜔𝑑−2 (𝑑−2) 𝑄 𝑟 =( [𝑀+√𝑀2 − ]) . (9) commutation relation, this GUP can be represented as [34] + 2 8𝜋 (𝑑−3) [ ] Δ𝑥Δ𝑝 ≥ ℏ (1 −𝛼𝑙 Δ𝑝 +2 𝛼 𝑙 2 (Δ𝑝)2 +𝛽2𝑙 2 (Δ𝑥)2) 𝑝 𝑝 𝑝 (3) In order to apply the semiclassical tunneling analysis, one can find a proper coordinate system for the black hole metric which in extra dimensions can be written as follows: wherealltheconstantlinesareflatandthetunnelingpath is free of singularities. In this manner, Painlevecoordinates´ Δ𝑥𝑖Δ𝑝𝑖 aresuitablechoices.Inthesecoordinates,thed-dimensional 2 2 2 2 2 2 (4) ≥ℏ(1−𝛼𝑙𝑝 (Δ𝑝𝑖)+𝛼 𝑙𝑝 (Δ𝑝𝑖) +𝛽 𝑙𝑝 (Δ𝑥𝑖) ). Reissner-Nordstrom metric is given by 2 2 √ 2 2 2 Here, 𝛼 and 𝛽 are dimensionless positive coefficients which 𝑑𝑠 =−𝑓𝑑𝑡 ±2 1−𝑓𝑑𝑡𝑑𝑟+𝑑𝑟 +𝑟 𝑑Ω𝑑−2 (10) are independent of Δ𝑥 and Δ𝑝. In general they may depend on expectation value of 𝑥 and 𝑝. According to the generalized which is stationary, nonstatic, and nonsingular at the horizon Heisenberg algebra, we suppose that operators of position and plus (minus) sign corresponds to the space-time line and momentum obey the following commutation relation: element of the outgoing (incoming) particles across the event horizon, respectively. The trajectory of charged massive 2 2 2 2 [𝑥, 𝑝] = 𝑖ℏ (1 − 𝛼𝑝+𝛼 𝑝 +𝛽 𝑥 ). (5) particles as a sort of de Broglie s-wave can be approximately determined as [36, 37] In what follows, we use this more general framework of GUP 𝑑𝑟 𝑔 𝑓 tofindthetunnelingrateofemittedparticlesthroughcharged 𝑟=̇ =− 𝑡𝑡 =± , 𝑑𝑡 2𝑔 2√1−𝑓 (11) nonrotating TeV-scale black holes. 𝑡𝑟 Advances in High Energy Physics 3

𝜇] where the plus (minus) sign denotes the radial geodesics of where 𝐿𝑒 = −(1/4)𝐹𝜇]𝐹 is the Lagrangian function of the outgoing (incoming) charged particles tunneling across the electromagnetic field corresponding to the generalized the event horizon, respectively. We incorporate quantum coordinates 𝐴𝜇 =(𝐴𝑡,0,0,0)[38]. gravity effects in the presence of the minimal length, minimal We assume the tunneling mechanism as a semiclassical momentum, and maximal momentum via the GUP which method producing Hawking radiation. In this case, using motivates modification of the standard dispersion relation in WKB approximation, the emission rate of tunneling massive the presence of extra dimensions based on ADD model. If the charged particle can be obtained from the imaginary part of GUP is a fundamental outcome of quantum gravity proposal, theparticleactionatthestationaryphaseforthetunneling it should appear that the de Broglie relation is as follows: trajectory; namely, [39, 40] 𝜆 ± Γ∼exp (−2 Im 𝐼) . (17)

2 2 2 2 2 𝑝 4𝛽 𝑙 (1 − 𝛼𝑙𝑝𝑝𝑖 +𝛼 𝑙 𝑝 ) (12) Assuming the generalized coordinate 𝐴𝑡 is an ignorable one, = 𝑖 (1 ± √1− 𝑝 𝑝 𝑖 ). 2𝛽2𝑙2 𝑝2 to eliminate this degree of freedom completely, we can obtain 𝑝 𝑖 the action of the matter-gravity system as

𝑡 Onecanfindeasilythatpositivesigndoesnotrecover 𝑓 𝐼=∫ (𝐿 − 𝑃 𝐴̇ )𝑑𝑡 ordinary relation in the limits 𝛼→0and 𝛽→0.Sowe 𝐴𝑡 𝑡 𝑡𝑖 consider the minus sign as (18) 𝑟 (𝑝 ,𝑝 ) 𝑓 𝑟 𝐴𝑡 𝑑𝑟 2 4 2 3 2 2 󸀠 ̇ 󸀠 3𝛽 𝑙 2𝛽 𝑙 𝛽 𝑙 = ∫ [∫ (𝑟𝑑𝑝̇ 𝑟 − 𝐴𝑡𝑑𝑝 𝐴 )] , 𝑝 2 2 𝑝 𝑝 𝑡 𝑟̇ 𝜆 =( +𝑝𝑙 )𝛼 −( +𝑙 )𝛼+ 𝑟𝑖 (0,0) − 𝑖 𝑝 2 𝑝 3 𝑝𝑖 𝑝𝑖 𝑝𝑖 (13) where 𝑟𝑖 and 𝑟𝑓 are the location of the event horizon 1 + corresponding to 𝑡𝑖 and 𝑡𝑓,respectively,beforeandafterthe 𝑝𝑖 𝜀 𝑞 𝑝 particle of energy and charge tunnels out, in which 𝐴𝑡 and 𝑝𝑟 are the canonical momentum conjugate to the coordinates or equivalently 𝐴𝑡 and 𝑟,respectively. 𝜀 = (3𝐸𝛽2𝑙4 +𝐸3𝑙2 )𝛼2 − (2𝛽2𝑙3 +𝐸2𝑙 )𝛼+𝐸 In order to consider the effect of quantum gravity, the 𝑝 𝑝 𝑝 𝑝 commutation relation between the radial coordinate compo- 𝛽2𝑙2 (14) nents and conjugate momentums should be modified based + 𝑝 . on (1) and (2) of the expressed GUP as follows [41]: 𝐸 [𝑟, 𝑝 ]=𝑖(1−𝛼𝑙 𝑝 +𝛼2𝑙2 𝑝2). Here, for investigating Hawking radiation of charged massive 𝑟 𝑝 𝑟 𝑝 𝑟 (19) particles from the event horizon of charged nonrotating microblack hole, we use this more general uncertainty prin- So as it is clear from the more general GUP and based on (5) ciple and take into consideration the response of background the commutation relation should be modified as geometry to radiated quantum of energy E with GUP cor- 2 2 2 2 2 2 rection, that is, 𝜀.Theemittedparticlewhichcanbetreated [𝑟,𝑟 𝑝 ]=𝑖(1−𝛼𝑙𝑝𝑝𝑟 +𝛼 𝑙𝑝𝑝𝑟 +𝛽 𝑙𝑝𝑟 ). (20) as a shell of energy 𝜀 and charge q moves on the geodesics of a space-time with central mass 𝑀−𝜀substituted for M In the classical limit it is replaced by Poisson bracket as and charge parameter 𝑄−𝑞replaced with 𝑄.Wesetthe follows: total Arnowitt-Deser-Misner (ADM) mass, M,andtheADM 2 2 2 2 2 2 charge of the space-time to be fixed but allow the hole mass {𝑟,𝑟 𝑝 } = (1−𝛼𝑙𝑝𝑝𝑟 +𝛼 𝑙𝑝𝑝𝑟 +𝛽 𝑙𝑝𝑟 ) . (21) andchargetofluctuateandreplaceM by 𝑀−𝜀and 𝑄 by 𝑄−𝑞 in both the metric and the geodesic equation. So the outgoing Now, we apply the deformed Hamiltonian equation: radial geodesics of the charged massive particle tunneling 󵄨 out from the event horizon and the nonzero component of 𝑑𝐻󵄨 𝑟=̇ {𝑟, 𝐻} ={𝑟,𝑝𝑟} 󵄨 , electromagnetic potential are 𝑑𝑟 󵄨𝑟 𝑓(𝑀−𝜀,𝑄−𝑞,𝑟) 𝑑𝐻|(𝑟,𝐴 ,𝑝 ) =𝑑(𝑀−𝜀) , 𝑟=̇ , 𝑡 𝑡 (22) 2√1−𝑓(𝑀−𝜀,𝑄−𝑞,𝑟) 𝑑𝐻 𝑑𝐸󸀠 (15) 𝐴̇ = = 𝑄 ,𝑑𝐻| =𝐴𝑑(𝑄−𝑞). 𝑡 (𝐴𝑡,𝑟,𝑝𝑟) 𝑡 𝑄−𝑞 𝑑𝑝𝐴 𝑑𝑝𝐴 𝐴 = . 𝑡 𝑡 𝑡 (𝑑−3) Ω 𝑟𝑑−3 𝑑−2 󸀠 2 In (18) as the Hamiltonian is 𝐻=𝑀−𝜀,onecanset𝑝 ≃ 󸀠2 󸀠 So the Lagrangian for the matter-gravity system is 𝜀 and 𝑝≃𝜀 and eliminate the momentum in favor of the 𝐿=𝐿 +𝐿 , energy in integral (18) and switching the order of integration 𝑚 𝑒 (16) yields the imaginary part of the action as follows: 4 Advances in High Energy Physics

𝑟 (𝑀−𝜀,𝑄−𝑞) 󸀠 𝑓 𝑄−𝑞 𝑑𝑟 𝐼= ∫ ∫ [(1 − 𝛼𝑙 𝜀󸀠 +𝛼2𝑙2 𝜀󸀠2 +𝛽𝑙2 𝑟2)𝑑(𝑀−𝜀󸀠)− 𝑑(𝑄−𝑞󸀠)] Im Im 𝑝 𝑝 𝑝 𝑑−3 𝑟̇ 𝑟𝑖 (𝑀,𝑄) (𝑑−3) Ω𝑑−2𝑟 (23) 𝑟 (𝑀−𝜀,𝑄−𝑞) 󸀠 󸀠 󸀠 𝑓 2√1−𝑓(𝑀−𝜀,𝑄−𝑞 ,𝑟) 𝑄−𝑞 = ∫ ∫ [(1 − 𝛼𝑙 𝜀󸀠 +𝛼2𝑙2 𝜀󸀠2 +𝛽𝑙2 𝑟2)𝑑(𝑀−𝜀󸀠)− 𝑑(𝑄−𝑞󸀠)] 𝑑𝑟. Im 󸀠 󸀠 𝑝 𝑝 𝑝 𝑑−3 𝑟𝑖 (𝑀,𝑄) 𝑓(𝑀−𝜀 ,𝑄−𝑞 ,𝑟) (𝑑−3) Ω𝑑−2𝑟

r integral can be evaluated by deforming the contour of the (24) is complicated, one can find such terms as an example single pole at the outer horizon. During r integral first, we for 𝑑=5as follows: find 2 2 4 Im 𝐼𝑑=5 ≈ ⋅ ⋅ ⋅ + 4096𝛼𝑙𝑝𝜋𝑀 − 8192𝛼𝑙𝑝𝜋𝑀 (3𝐸𝛽 𝑙𝑝 𝜀 󸀠 +𝐸3𝑙2 )𝛼2 + 8192𝛼𝑙 𝜋𝑀 (2𝛽2𝑙3 +𝐸2𝑙 )𝛼 Im 𝐼=Im ∫ 2 (−𝜋𝑖) 𝑟+ (𝑀 − 𝜀 ,𝑄−𝑞) 𝑝 𝑝 𝑝 𝑝 0 𝛽2𝑙2 ⋅(1−𝛼𝑙 𝜀󸀠 +𝛼2𝑙2 𝜀󸀠2 +𝛽𝑙2 )𝑑(−𝜀󸀠) − 8192𝛼𝑙 𝜋𝑀𝐸 − 8192𝛼𝑙 𝜋𝑀 𝑝 𝑝 𝑝 𝑝 𝑝 𝑝 𝐸 𝑄−𝑞 (24) 󸀠 2 4 3 2 2 (25) − ∫ 2 (−𝜋𝑖) 𝑟+ (𝑀−𝜀,𝑄−𝑞) + 4096𝛼𝑙 𝜋 ((3𝐸𝛽 𝑙 +𝐸 𝑙 )𝛼 0 𝑝 𝑝 𝑝

󸀠 (𝑄 − 𝑞 ) 2 ⋅ 𝑑(𝑄−𝑞󸀠). 𝛽2𝑙2 (𝑑−3) 𝜋𝑟𝑑−3 − (2𝛽2𝑙3 +𝐸2𝑙 )𝛼+𝐸+ 𝑝 ) − 192𝛼𝑙 𝑄2 𝑝 𝑝 𝐸 𝑝

This allows us to consider the leading order correction to be +⋅⋅⋅ . just proportional to second order of 𝛼𝑙𝑝 andalsosecondorder of 𝛽𝑙𝑝 for simplicity without loss of generality. In this regard, Substituting (24) into (17), the tunneling probability of we can finish the integration by applying Taylor series and charged particles from charged nonrotating TeV-scale black obtain the imaginary part of the action. Although integral holes is obtained as

𝜀 󸀠 󸀠 2 2 󸀠2 2 󸀠 Γ=exp (−2 Im 𝐼) ≃ exp [Im ∫ 2 (−𝜋𝑖) 𝑟+ (𝑀 − 𝜀 ,𝑄−𝑞)(1−𝛼𝑙𝑝𝜀 +𝛼 𝑙𝑝𝜀 +𝛽𝑙𝑝)𝑑(−𝜀 ) 0 (26) 𝑄−𝑞 (𝑄 −󸀠 𝑞 ) − ∫ 2 (−𝜋𝑖) 𝑟 (𝑀 − 𝜀, 𝑄 −𝑞󸀠) 𝑑(𝑄−𝑞󸀠)] = (Δ𝑠) , + 𝑑−3 exp 0 (𝑑−3) 𝜋𝑟

where Δ𝑠 is the difference in black hole entropies before of microblack holes radiation from ordinary thermal radia- and after emission [42–47]. It was shown that the emission tion. rates on the high energy scales correspond to differences between the counting of states in the microcanonical and in 4. Backscattering and Luminosity the canonical ensembles [48, 49]. By performing integration on (24), one can find that the first order of E in the exponential It has been shown [51] that black holes radiate a thermal gives a thermal, Boltzmannian spectrum. The existence of spectrum of particles. So microblack holes emit black body extra terms in relation (25) shows that the radiation is not radiation at the Hawking temperature. Following a heuristic Δ𝐸 ≈ completely thermal. In fact, these extra terms enhance the argument [52], the energy of the Hawking particles is 𝑐Δ𝑝 nonthermal character of the radiation. Also, it is easy to find and it is deduced for the Hawking temperature of black that the tunneling rate should be greater than the ordinary holebasedonLEDscenario, oneinanystageoftunnelingprocess.Thistunnelingrate (𝑑−3) Δ𝑝 𝑇 ≃ , (27) compared to the tunneling rate which is calculated in [50] 𝐻 4𝜋 obviously shows that, by considering all natural cutoffs in generalized uncertainty principle relation, many additional where (𝑑 − 3)/4𝜋 is a calibration factor in d-dimensional terms appeared. The additional terms show strong deviation space-time. By saturating inequality (4), one can find Advances in High Energy Physics 5

2 𝛾𝜀 𝛾𝜀 momentum uncertainty in terms of position uncertainty as where 𝑁 =𝑒 /(𝑒 −1)is a normalization constant and 𝑘 follows: is the surface gravity. This quantum state is transformed with respect to an observer outside the horizon. In order to obtain 𝛼𝑙𝑝 +Δ𝑥𝑖 𝜀 Δ𝑃 =( ) the average particle number in the energy state with respect 𝑖 2 2 4𝛼 𝑙𝑝 to an observer, one can trace out the inside degrees of freedom to yield the reduced density matrix of the form 2 2 2 2 2 (28) 4𝛼 𝑙𝑝 (1 + 4𝛽 𝑙𝑝 (Δ𝑥𝑖) ) ∞ ⋅(1±√1− ). 2𝜋𝜀 −𝛾𝑛𝜀 󵄨 (𝑅) 2 𝜌 =(1− (− )) ∑𝑒 󵄨𝑛 ⟩ (𝛼𝑙 +Δ𝑥) reduced exp ℏ𝑘 󵄨 out 𝑝 𝑖 𝑛=0 (32) 󵄨 ⊗⟨𝑛(𝑅)󵄨 . SothemodifiedblackholeHawkingtemperatureinthe out󵄨 presence of natural cutoffs becomes In this regard, the number distribution with respect to 𝜀 is given by (𝑑−3)(2𝑟 +𝛼𝑙 ) + 𝑝 1 𝑇𝐻 = (1 16𝜋𝛼2𝑙2 ⟨𝑛𝜀⟩=trace (𝑛𝜌reduced)= , (33) 𝑝 𝑒𝛾𝜀 −1 (29) 𝛾=1/𝑇 2 2 2 2 2 where 𝐻. Whenever a particle is radiated from the 4𝛼 𝑙 (1 + 16𝛽 𝑙 𝑟+ ) − √1− 𝑝 𝑝 ). microblack hole event horizon, its wave function satisfies a 2 (2𝑟 +𝛼𝑙 ) wave equation with an effective potential that depends on + 𝑝 outer event horizon. As the potential represents a barrier to the outgoing radiation, one part of the radiation is Based on (29), GUP give rise to the existence of a minimal backscattered. mass of a charged nonrotating microblack hole given by In this way, it can be shown that the distribution ⟨𝑛𝜀⟩ for the Hawking radiation will be modulated by grey body factor 𝑀GUP min [54] which for a charged nonrotating TeV-scale black hole is 2𝑑−6 given as (1 + 2 1−12𝛽2𝛼2𝐿4 )𝐿 𝛼 (𝑑−2) Ω √ 𝑝𝑙 𝑝𝑙 = (( ) 2 2 16𝜋 2 2 4 Λ=4𝜀𝑟+, (34) 32𝛽 𝛼 𝐿𝑝𝑙 −2 which Λ is the standard approximated grey body factor. In this way, one can take energy and charge conservation into 8𝜋𝑄2 + ) (30) account[54]andgetthestraightforwardresultbysubstituting (𝑑−2)(𝑑−3) (14) into (34). So we obtain

2 2 2 Λ 𝐸𝐶 =4[𝐸(1−𝛼𝑙𝑝𝐸+𝛼 𝑙𝑝𝐸 ) 𝑑−3 −1 2 2 4 ⋅ (1 + 2√1−12𝛽 𝛼 𝐿𝑝𝑙)𝐿𝑝𝑙𝛼 (( ) ) . 𝛽2𝑙2 2 32𝛽2𝛼2𝐿4 −2 𝑝 2 2 2 2 (35) 𝑝𝑙 ⋅(1+ (1 − 𝛼𝑙 𝐸+𝛼 𝑙 𝐸 ))] 𝑟 (𝑀 − 𝜀, 𝑄 𝐸2 𝑝 𝑝 +

Therefore, there are some black hole remnants without −𝑞). radiation based on (30). A radiated particle state correspond- ing to an arbitrary finite number of virtual pairs inside the On the other hand, if we consider the full consequences blackholeeventhorizonisasfollows[53]: of energy and charge conservation, for total flux, including 󵄨 −𝜋𝑛𝜀/ℏ𝑘 󵄨 (𝐿) 󵄨 (𝑅) backscattering [55], the luminosity modulated according to 󵄨𝜓⟩ = 𝑁 ∑ 𝑒 󵄨𝑛 ⟩⊗󵄨𝑛 ⟩, 󵄨 󵄨 out 󵄨 out (31) the grey body factor has to be written as

𝑀−𝑀GUP 𝑑 1 min 𝐿 (𝑀) = ∫ ⟨𝑛𝜀⟩Λ𝐸𝐶𝜀𝑑𝜀 2𝜋 0

2 𝑀−𝑀GUP 2 2 2 2 2 2 2 2 2 2 1 min 4[𝐸(1−𝛼𝑙𝑝𝐸+𝛼 𝑙 𝐸 )(1+(𝛽 𝑙 /𝐸 )(1−𝛼𝑙𝑝𝐸+𝛼 𝑙 𝐸 ))] 𝑟 (𝑀 − 𝜀, 𝑄 − 𝑞) 𝑑𝜀 (36) = ∫ 𝑝 𝑝 𝑝 + . −1 2𝜋 0 2 2 2 √ 2 2 2 2 2 exp ((16𝜋𝛼 𝑙𝑝𝜀/ (𝑑−3) (2𝑟+ +𝛼𝑙𝑝)) [1 − 1−4𝛼 𝑙𝑝 (1 + 𝑙𝑝𝛽 𝑟+)/(2𝑟+ +𝛼𝑙𝑝) ] )−1 6 Advances in High Energy Physics

0.07 0.15

0.06

0.05 0.10

0.04 H H T T 0.03 0.05 0.02

0.01

0 2468101214 051015 M M d=5 d=9 Q=0 Q=1 d=6 d=10 Q=±1/3 Q=±4/3 d=7 d=11 Q=±2/3 d=8 Figure 1: Hawking temperature for different amount of charges in Figure 2: Hawking temperature with respect to the mass in terms presence of GUP. of GUP.

It is important to remark that the total luminosities for microblack hole as it evaporates respecting energy and charge microblackholewouldbetentimesbiggerifweneglect conservation,wehavealsomodifiedthegreybodyfactor, backscattering effect. We are taking into account in the which allows considering the effect of the backscattered integration limits that the maximum energy of a radiated emittedradiation.WehavecalculatedHawkingtemperature GUP particle could be 𝑀−𝑀min . Equation (36) gives larger basedontheGUPwhichadmittedaminimallength,amax- luminosities for smaller masses. The results show that, in imal momentum, and a minimal momentum. The adopted large extra dimension scenario, Hawking temperature of GUP predict a minimal mass remnant with respect to the charged black hole increases and leads to faster decay and less charge of black hole. So we have been able to derive an classical behaviors for black holes (Figure 2). On the other expression for the luminosity that takes into account natural hand,ithasbeenshown[56,57]thattheallowedparticles cutoffs in presence of large extra dimension based on ADD forming the black hole at the LHC are quarks, antiquarks, scenario for different amount of charge of black hole (Figures and gluons which formed nine possible electric charge states: 2 and 1). The investigation implies that, considering natural ±4/3, ±1, ±2/3, ±1/3, 0. In this case, as far as the electric cutoffs in the presence of LED, information conservation of charge of the black hole increases, the minimum mass and charged nonrotating microblack hole is still possible. its order of magnitude increase and the temperature peak is displaced to the lower temperature (see Figure 1). As (36) Competing Interests isrelatedtotheblackholetemperature,basedontheabove arguments, the luminosity of charged nonrotating TeV-scale The authors declare that they have no competing interests. black hole has different amount with respect to the charge of black hole and also extra dimensions. Acknowledgments 5. Conclusion and Discussion M. J. Soleimani would like to give special thanks to E. C. Vagenasforhisusefuldiscussion.Thepaperissupportedby In this paper, we have investigated Hawking radiation of University of Malaya (Grant no. Ru-023-2014) and University the charged massive particles as a semiclassical tunneling Kebangsaan Malaysia (Grant no. FRGS/2/2013/ST02/UKM/ process from the charged nonrotating microblack hole. In 02/2 coordinated by Dr. Geri Kibe Gopir). this respect, we considered possible effect of natural cutoffs as a minimal length, a maximal momentum, and a minimal References momentum on the tunneling rate. We have shown that, in the presence of generalized uncertainty principle, the tunneling [1] N. Arkani-Hame, S. Dimoulos, and G. Dvali, “The hierarchy rate of charged massive particle is deviated from thermal problem and new dimensions at a millimeter,” Physics Letters emission. In order to study the evolution of the TeV-scale B,vol.429,no.3-4,pp.263–272,1998. Advances in High Energy Physics 7

[2] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, [23] G. Amelino-Camelia, “Doubly-special relativity: first results “New dimensions at a millimeter to a fermi and superstrings at and key open problems,” International Journal of Modern aTeV,”Physics Letters B, vol. 436, no. 3-4, pp. 257–263, 1998. Physics D,vol.11,no.10,pp.1643–1669,2002. [3] L. Randall and R. Sundrum, “Large mass hierarchy from a small [24] J. Kowalski-Glikman, “Introduction to doubly special relativity,” extra dimension,” Physical Review Letters,vol.83,no.17,pp. in PlanckScaleEffectsinAstrophysicsandCosmology,vol.669, 3370–3373, 1999. pp. 131–159, Springer, Berlin, Germany, 2005. [4] L. Randall and R. Sundrum, “An alternative to compactifica- [25] M. K. Parikh and F. Wilczek, “Hawking radiation as tunneling,” tion,” Physical Review Letters,vol.83,no.23,pp.4690–4693, Physical Review Letters,vol.85,no.24,pp.5042–5045,2000. 1999. [26] M. Maggiore, “A generalized uncertainty principle in quantum [5] P.C.Argyres,S.Dimopoulos,andJ.March-Russell,“Blackholes gravity,” Physics Letters B,vol.304,no.1-2,pp.65–69,1993. and sub-millimeter dimensions,” Physics Letters B,vol.441,no. [27] M. Maggiore, “The algebraic structure of the generalized uncer- 1–4, pp. 96–104, 1998. tainty principle,” Physics Letters B,vol.319,no.1–3,pp.83–86, [6]R.Emparan,G.T.Horowitz,andR.C.Myers,“Blackholes 1993. radiate mainly on the brane,” Physical Review Letters,vol.85, [28] J. Magueijo and L. Smolin, “Generalized Lorentz invariance no. 3, pp. 499–502, 2000. with an invariant energy scale,” Physical Review D,vol.67,no. [7] S. B. Giddings and S. Thomas, “Black hole bound on the number 4, Article ID 044017, 12 pages, 2003. of species and quantum gravity at CERN LHC,” Physical Review [29] J. L. Corte’s and J. Gamba, “Quantum uncertainty in doubly D,vol.77,no.4,ArticleID045027,2008. special relativity,” Physical Review D,vol.71,no.6,ArticleID [8] J. L. Feng and A. D. Shapere, “Black hole production by cosmic 065015, 2005. rays,” Physical Review Letters, vol. 88, no. 2, Article ID 021303, 4 [30] J. Magueijo and L. Smolin, “Lorentz invariance with an invariant pages, 2001. energy scale,” Physical Review Letters, vol. 88, no. 19, Article ID [9] A. Ringwald and H. Tu, “Collider versus cosmic ray sensitivity 190403,4pages,2002. to black hole production ,” Physics Letters B,vol.525,no.1-2,pp. [31] J. Magueijo and L. Smolin, “String theories with deformed 135–142, 2002. energy-momentum relations, and a possible nontachyonic [10] M. Cavagli’a, “Black hole and brane production in tev gravity: bosonic string,” Physical Review D,vol.71,no.2,ArticleID a review,” International Journal of Modern Physics A,vol.18,no. 026010,6pages,2005. 11, p. 1843, 2003. [32] H. Hinrichsen and A. Kempf, “Maximal localization in the pres- [11] G. Veneziano, “A stringy nature needs just two constants,” ence of minimal uncertainties in positions and in momenta,” Europhysics Letters (EPL),vol.2,no.3,pp.199–204,1986. Journal of Mathematical Physics,vol.37,no.5,pp.2121–2137, 1996. [12] A. Kempf, G. Mangano, and R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” [33] M. Zarei and B. Mirza, “Minimal uncertainty in momentum: the Physical Review D,vol.52,no.2,pp.1108–1118,1995. effects of IR gravity on quantum mechanics,” Physical Review D, vol. 79, Article ID 125007, 2009. [13] A. Kempf and G. Mango, “Minimal length uncertainty relation [34] P. Pedram, K. Nozari, and S. H. Taheri, “The effects of minimal and ultraviolet regularization,” Physical Review D,vol.55,no.12, length and maximal momentum on the transition rate of ultra pp. 7909–7920, 1997. cold neutrons in gravitational field,” Journal of High Energy [14] D. Amati, M. Ciafaloni, and G. Veneziano, “Can spacetime be Physics, vol. 2011, article 93, 2011. probed below the string size?” Physics Letters B,vol.216,no.1-2, [35] R. C. Myers and M. J. Perry, “Black holes in higher dimensional pp.41–47,1989. space-times,” Annals of Physics,vol.172,no.2,pp.304–347,1986. [15] K. A. Meissner, “Black-hole entropy in loop quantum gravity,” [36] Q.-Q. Jiang and S.-Q. Wu, “Hawking radiation of charged Classical and Quantum Gravity,vol.21,no.22,pp.5245–5251, particles as tunneling from Reissner-Nordstrom-de¨ Sitter black 2004. holes with a global monopole,” Physics Letters B,vol.635,no. [16] L. J. Gary, “Quantum gravity and minimum length,” Interna- 2-3, pp. 151–155, 2006. tional Journal of Modern Physics A,vol.10,no.2,p.145,1995. [37] J. Y.Zhang and Z. Zhao, “Hawking radiation of charged particles [17] F. Scardigli, “Generalized uncertainty principle in quantum via tunneling from the Reissner-Nordstrom¨ black hole,” Journal gravity from micro-black hole gedanken experiment,” Physics of High Energy Physics,vol.2005,no.10,article055,2005. Letters B,vol.452,no.1-2,pp.39–44,1999. [38] J. Makel¨ a¨ and P. Repo, “Quantum-mechanical model of the [18] R. J. Adler, “Six easy roads to the Planck scale,” American Journal Reissner-Nordstrom¨ black hole,” Physical Review D,vol.57,no. of Physics,vol.78,no.9,pp.925–932,2010. 8, pp. 4899–4916, 1998. [19] A. F. Ali, S. Das, and E. C. Vagenas, “Proposal for testing [39] K. Srinivasan and T. Padmanban, “Particle production and quantum gravity in the lab,” Physical Review D,vol.48,no.4, complex path analysis,” Physical Review D,vol.60,no.2,Article Article ID 044013, 2011. ID 024007, 1999. [20] S. Das and E. C. Vagenas, “Universality of quantum gravity [40] J. L. Feng, A. Rajaraman, and F. Takayama, “Probing grav- corrections,” Physical Review Letters,vol.101,no.22,ArticleID itational interactions of elementary particles,” International 221301, 2008. JournalofModernPhysicsD, vol. 13, no. 10, pp. 2355–2359, 2004. [21] G. Amelino-Camelia, “Relativity in spacetimes with short- [41] K. Nozari and S. Saghafi, “Parikh-Wilczek tunneling from distance structure governed by an observer-independent noncommutative higher dimensional black holes,” Journal of (Planckian) length scale,” International Journal of Modern High Energy Physics,vol.2012,no.11,article005,2012. Physics D,vol.11,no.1,pp.35–59,2002. [42] P. Kraus and F. Wilczek, “Self-interaction correction to black [22] G. Amelino-Camelia, “Relativity: special treatment,” Nature, hole radiance,” Nuclear Physics B,vol.433,no.2,pp.403–420, vol. 418, no. 6893, pp. 34–35, 2002. 1995. 8 Advances in High Energy Physics

[43] E. C. Vagenas, “Are extremal 2D black holes really frozen?” Physics Letters B,vol.503,no.3-4,pp.399–403,2001. [44] E. C. Vagenas, “Two-dimensional dilatonic black holes and Hawking radiation,” Modern Physics Letters A,vol.17,no.10,pp. 609–618, 2002. [45] E. C. Vagenas, “Semiclassical corrections to the Bekenstein- Hawking entropy of the BTZ black hole via self-gravitation,” Physics Letters B,vol.533,no.3-4,pp.302–306,2002. [46] A. J. M. Medved, “Radiation via tunnelling in the charged BTZ black hole,” Classical and Quantum Gravity,vol.19,no.3,article 589, 2002. [47] E. C. Vagenas, “Generalization of the KKW analysis for black hole radiation,” Physics Letters B,vol.559,no.1-2,pp.65–73, 2003. [48] M. Parikh, “A secret tunnel through the horizon,” International Journal of Modern Physics D,vol.13,no.10,pp.2351–2354,2004. [49] E. T. Akhmedov, V. Akhmedova, and D. Singleton, “Hawking temperature in the tunneling picture,” Physics Letters B,vol.642, no. 1-2, pp. 124–128, 2006. [50] S.-Q. Wu and Q.-Q. Jiang, “Hawking radiation of charged parti- cles as tunneling from higher dimensional reissner-nordstrom- de sitter black holes,” https://arxiv.org/abs/hep-th/0603082. [51] S. W. Hawking, “Particle creation by black holes,” Communica- tions in Mathematical Physics,vol.43,no.3,pp.199–220,1975. [52] R. J. Adler, P. Chen, and D. I. Santiago, “The generalized uncer- tainty principle and black hole remnants,” General Relativity and Gravitation, vol. 33, no. 12, pp. 2101–2108, 2001. [53] S. K. Modak, “Backreaction due to quantum tunneling and modification to the black hole evaporation process,” Physical Review D,vol.90,no.4,ArticleID044015,2014. [54]R.Torres,F.Fayos,andO.Lorente-Esp´ın, “Evaporation of (quantum) black holes and energy conservation,” Physics Letters B,vol.720,no.1–3,pp.198–204,2013. [55] A. Fabbri and J. Navario-Salas, Modeling Black Hole Evapora- tion, Imperial College Press, London, UK, 2005. [56] D. M. Gingrich, “Quantum black holes with charge, color and spin at the LHC,” Journal of Physics G: Nuclear and Particle Physics, vol. 37, no. 10, Article ID 105008, 2010. [57] G. Landsberg, “Black holes at future colliders and beyond,” Journal of Physics G,vol.32,no.9,p.337,2006. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 7058764, 7 pages http://dx.doi.org/10.1155/2016/7058764

Research Article Lorentz Invariance Violation and Modified Hawking Fermions Tunneling Radiation

Shu-Zheng Yang,1 Kai Lin,2,3 Jin Li,4 and Qing-Quan Jiang1

1 Department of Astronomy, China West Normal University, Nanchong, Sichuan 637002, China 2Instituto de F´ısica e Qu´ımica, Universidade Federal de Itajuba,´ 37500-903 Itajuba,´ MG, Brazil 3Instituto de F´ısica, Universidade de Sao˜ Paulo, CP 66318, 05315-970 Sao˜ Paulo, SP, Brazil 4Department of Physics, Chongqing University, Chongqing 400030, China

Correspondence should be addressed to Kai Lin; [email protected]

Received 20 April 2016; Accepted 8 June 2016

Academic Editor: Giulia Gubitosi

Copyright © 2016 Shu-Zheng Yang 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.

Recently the modified Dirac equation with Lorentz invariance violation has been proposed, which would be helpful to resolve some issues in quantum gravity theory and high energy physics. In this paper, the modified Dirac equation has been generalized in curved spacetime, and then fermion tunneling of black holes is researched under this correctional Dirac field theory. We also use semiclassical approximation method to get correctional Hamilton-Jacobi equation, so that the correctional Hawking temperature and correctional black hole’s entropy are derived.

1. Introduction the Hawking temperature can be determined. People have applied this method to research Hawking tunneling radiation In 1974, Hawking proved black holes could radiate Hawking of several static, stationary, and dynamical black holes. radiation once considering the quantum effect near the However, since the Hamilton-Jacobi equation is derived horizons of black holes [1, 2]. This theory indicates that black from Klein-Gordon equation, original Hamilton-Jacobi holewouldbeviewedasthermodynamicsystem,sothat method just can be valid for scalar particles in principle. blackholephysicscanbeconnectedcloselywithgravity, Therefore, Kerner and Mann studied fermions tunneling quantum theory, and thermodynamics physics. According of black hole by a new method [22–32], which assumes the to the view point of quantum tunneling theory, the virtual wave function of Dirac equation Ψ as spin-up and spin-down particlesinsideofblackholecouldcrossthehorizondueto and then calculates the fermions tunneling, respectively. the quantum tunneling effect and become real particles and Nevertheless, this method is still impossible to apply in then could be observed by observers as Hawking radiation. arbitrary dimensional spacetime. Our work in 2009 showed Wilczek et al. proposed a semiclassical method to study that the Hamilton-Jacobi equation can also be derived the quantum tunneling from the horizon of black hole [3– from Dirac equation via semiclassical approximation, so we 21]. Along with this method, the Hamilton-Jacobi method proved that the Hamilton-Jacobi equation can be used to was applied to calculate the Hawking tunneling radiation. study the fermions tunneling directly [33–35]. According to the Hamilton-Jacobi method, the wave func- On the other hand, as the basis of general relativity and Φ= tion of Klein-Gordon equation can be rewritten as quantum field theory, Lorentz invariance is proposed to be 𝐶 (𝑖𝑆/ℏ) 𝑆 exp (where is semiclassical action) and Hamilton- spontaneously violated at higher energy scales. A possible Jacobi equation is obtained via semiclassical approximation. deformed dispersion relation is given by [36–44] Using the Hamilton-Jacobi equation, the tunneling rate could be calculated by the relationship Γ∼exp (−2 Im 𝑆) (where Γ 2 𝛼 2 𝑝2 = 𝑝⃗ +𝑚2 −(𝐿𝑝 ) 𝑝⃗ , is the tunneling rate at the horizon of black hole), and then 0 0 (1) 2 Advances in High Energy Physics

where 𝑝0 and 𝑝⃗ are the energy and momentum of particle small, so that we can neglect the terms with ℏ after dividing and 𝐿 is “minimal length” with the order of the Plank length. by the exponential terms and multiplying by ℏ. Therefore, (4) The work of spacetime foam Liouville string models has is rewritten as 𝛼=1 introduced this relation with , and people also proposed [𝑖𝛾𝜇 (𝜕 𝑆+𝑒𝐴 )+𝑚−𝜎𝛾𝑡 (𝜔 − 𝑒𝐴 )𝛾𝑗 (𝜕 𝑆+𝑒𝐴 )] quantum equation of spineless particles by using this relation. 𝜇 𝜇 𝑡 𝑗 𝑗 (6) Recently, Kruglov considers the deformed dispersion relation ⋅𝜁(𝑡,𝑥𝑗)=0. with 𝛼=2and proposes modified Dirac equation [45]:

𝜇 𝑡 𝑗 Considering the relationship [𝛾 𝜕𝜇 +𝑚−𝑖𝐿(𝛾 𝜕𝑡)(𝛾 𝜕𝑗)] 𝜓 = 0, (2) 𝛾𝜇 (𝜕 𝑆+𝑒𝐴 )=−𝛾𝑡 (𝜔 − 𝑒𝐴 )+𝛾𝑗 (𝜕 𝑆+𝑒𝐴 ), 𝑎 𝜇 𝜇 𝑡 𝑗 𝑗 (7) where 𝛾 is ordinary gamma matrix and 𝑗 is space coordinate, while 𝜇 is spacetime coordinate. The effect of the correctional we can get term would be observed in higher energy experiment. [𝑖Γ𝜇 (𝜕 𝑆+𝑒𝐴 )+𝑀]𝜁(𝑡,𝑥𝑗)=0, In this paper, we try to generalize the modified Dirac 𝜇 𝜇 (8) equation in curved spacetime and then study the correction of Hawking tunneling radiation. In Section 2, the modified where 𝜇 𝑡 𝜇 Dirac equation in curved spacetime is constructed and then Γ = [1 + 𝑖𝜎 (𝜔 −𝑒𝐴𝑡)𝛾]𝛾 , the modified Hamilton-Jacobi equation is derived via semi- (9) 𝑡𝑡 2 classical approximation. We apply the modified Hamilton- 𝑀=𝑚−𝜎𝑔 (𝜔 − 𝑒𝐴𝑡) . Jacobi equation to the fermions tunneling radiation of 2+ ] 1-dimensional black string and higher dimensional BTZ- Now, multiplying both sides of (9) by the matrix −𝑖Γ (𝜕]𝑆+ like black strings in Sections 3 and 4, respectively, and 𝑒𝐴]),wecanobtain Section 5 includes some conclusion and the discussion about ] 𝜇 the correction of black hole’s entropy. Γ (𝜕]𝑆+𝑒𝐴])Γ (𝜕𝜇𝑆+𝑒𝐴𝜇)𝜁 (10) −𝑖𝑀Γ] (𝜕 𝑆+𝑒𝐴 )𝜁=0. 2. Modified Dirac Equation and ] ] Hamilton-Jacobi Equation in The second term of the above equation could be simplified Curved Spacetime againby(8),sotheaboveequationcanberewrittenas Γ]Γ𝜇 (𝜕 𝑆+𝑒𝐴 )(𝜕 𝑆+𝑒𝐴 )𝜁+𝑀2𝜁=0, As we all know, the gamma matrix and partial derivative ] ] 𝜇 𝜇 (11) 𝑎 shouldbecomegammamatrixincurvedspacetime𝛾 and where we can prove the relation covariant D𝑎 derivative, respectively, namely, Γ]Γ𝜇 =𝛾]𝛾𝜇 +2𝑖𝜎(𝜔−𝑒𝐴 )𝑔𝑡]𝛾𝜇 + O (𝜎2). 𝛾𝑎 󳨀→ 𝛾 𝑎, 𝑡 (12) 2 𝑖 (3) We always ignore O(𝜎 ) terms because 𝜎 is very small. Now, 𝜕 󳨀→ D =𝜕 +Ω + 𝑒𝐴 , 𝑎 𝑎 𝑎 𝑎 ℏ 𝑎 let us exchange the position of 𝜇 and ] in (11) and consider 𝑎 𝑏 𝑎𝑏 the relation of gamma matrices {𝛾 ,𝛾 }=2𝑔 𝐼;thenwecan 𝑎 𝑎 𝑏 𝑎 𝑏 𝑏 𝑎 where 𝛾 satisfy the relationship {𝛾 ,𝛾 }=𝛾𝛾 +𝛾𝛾 = obtain 𝑎𝑏 2𝑔 𝐼, 𝑒𝐴𝑎 is charged term of Dirac equation, and Ω𝜇 = 𝛼 𝛽 𝛼 𝛽 𝛾 𝛾 +𝛾 𝛾 2 (1/8)(𝛾𝑎𝛾𝑏 −𝛾𝑏𝛾𝑎)𝑒](𝜕 𝑒 −Γ𝑐 𝑒 ) { (𝜕 𝑆+𝑒𝐴 )(𝜕 𝑆+𝑒𝐴 )+𝑚 𝑎 𝜇 𝑏] 𝜇] 𝑏𝑐 is spin connection. 2 𝛼 𝛼 𝛽 𝛽 According to this transformation, we can construct the 𝑡𝑡 2 modifiedDiracequationincurvedspacetimeas −2𝜎𝑚𝑔 (𝜔 − 𝑒𝐴𝑡)

𝜇 𝑚 𝑡 𝑡 𝑗 𝑗 [𝛾 D𝜇 + −𝜎ℏ(𝛾D )(𝛾 D )] Ψ = 0, (4) 𝑡𝜌 𝜇 ℏ + 2𝑖𝜎 (𝜔 −𝑒𝐴𝑡)𝑔 (𝜕𝜌𝑆+𝑒𝐴𝜌)𝛾 (𝜕𝜇𝑆+𝑒𝐴𝜇)} where we choose 𝑐=1but ℏ =1̸ ,while𝑐=ℏ=1in (1) and (13) ⋅𝜁+O (𝜎2)={𝑔𝛼𝛽 (𝜕 𝑆+𝑒𝐴 )(𝜕 𝑆+𝑒𝐴 ) (2). It is assumed that 𝜎≪1, so that the correctional term 𝛼 𝛼 𝛽 𝛽 𝑡 𝑡 𝑗 𝑗 𝜎ℏ(𝛾 D )(𝛾 D ) is very small. +𝑚2 −2𝜎𝑚𝑔𝑡𝑡 (𝜔 − 𝑒𝐴 )2 Now let us use the modified Dirac equation to derive the 𝑡 modified Hamilton-Jacobi equation. Firstly, we rewrite the + 2𝑖𝜎 (𝜔 −𝑒𝐴 )𝑔𝑡𝜌 (𝜕 𝑆+𝑒𝐴 )𝛾𝜇 (𝜕 𝑆+𝑒𝐴 )} 𝜁 wave function of Dirac equation as [33–35] 𝑡 𝜌 𝜌 𝜇 𝜇 𝑖 + O (𝜎2)=0. Ψ=𝜁(𝑡,𝑥𝑗) [ 𝑆(𝑡,𝑥𝑗)] , exp ℏ (5) Namely, 𝜁(𝑡, 𝑥𝑗) Ψ 𝑚×1 𝜕 𝑆=−𝜔 where and are matrices and 𝑡 .In [𝑖𝜎𝛾𝜇 (𝜕 𝑆+𝑒𝐴 )+M]𝜁(𝑡,𝑥𝑗)=0, semiclassical approximation, we can consider that ℏ is very 𝜇 𝜇 (14) Advances in High Energy Physics 3

where where 𝐵, 𝑀3,and𝜆 are, respectively, the dilaton field, 3- M dimensional Planck mass, and the parameter with mass dimension. The static black string solution is given by 𝛼𝛽 2 𝑡𝑡 2 𝑔 (𝜕𝛼𝑆+𝑒𝐴𝛼)(𝜕𝛽𝑆+𝑒𝐴𝛽)+𝑚 −2𝜎𝑚𝑔 (𝜔 − 𝑒𝐴𝑡) (15) = . 𝑟 𝑟 −1 2𝑔𝑡𝜌 (𝜕 𝑆+𝑒𝐴 )(𝜔−𝑒𝐴 ) 𝑑𝑠2 =− ( )𝑑𝑡2 + ( ) 𝑑𝑟2 +𝑑𝑦2, 𝜌 𝜌 𝑡 ln 𝑟 ln 𝑟 𝐻 𝐻 (21) Using the idea of (10)-(11) again, we can multiply both sides ] 𝐵 = 𝜆𝑟. of (15) by the matrix −𝑖𝛾 (𝜕]𝑆+𝑒𝐴]),sothattheequation becomes It is evident that the horizon of this black hole is 𝑟𝐻,but 𝜎𝛾] (𝜕 𝑆+𝑒𝐴 )𝛾𝜇 (𝜕 𝑆+𝑒𝐴 )𝜁 ] ] 𝜇 𝜇 the black string is unstable as 𝑟𝐻 ≲ L (where L is scale of (16) compactification), so it is assumed that 𝑟𝐻 ≫ L. −𝑖M𝛾] (𝜕 𝑆+𝑒𝐴 )𝜁=0. ] ] Now we research the fermions tunneling of this black The second term of the above equation could be simplified hole, so the modified Hamilton-Jacobi equation in this again by (15). Then, exchange 𝜇 and ] and use the relationship spacetime is given by {𝛾𝑎,𝛾𝑏}=2𝑔𝑎𝑏𝐼 ,sotheaboveequationcanberewrittenas 𝑟 −1 𝑟 𝑑𝑅 2 − (1−2𝜎𝑚) ( ) 𝜔2 + ( )( ) 𝛾]𝛾𝜇 +𝛾𝜇𝛾] ln 𝑟 ln 𝑟 𝑑𝑟 [ 𝜎2 (𝜕 𝑆+𝑒𝐴 )(𝜕 𝑆+𝑒𝐴 )+M2] 𝐻 𝐻 2 ] ] 𝜇 𝜇 (22) 𝑑𝑌 2 +( ) +𝑚2 =0, ⋅𝜁(𝑡,𝑥𝑗) 𝑑𝑦 (17) 2 𝜇] 2 𝑗 𝑆 = −𝜔𝑡 + 𝑅(𝑟) +𝑌(𝑦) =[𝜎𝑔 (𝜕]𝑆+𝑒𝐴])(𝜕𝜇𝑆+𝑒𝐴𝜇)+M ]𝜁(𝑡,𝑥 ) wherewehaveset , yielding the radial Hamilton-Jacobi equation as =0. 𝑟 −1 𝑟 𝑑𝑅 2 − (1−2𝜎𝑚) ( ) 𝜔2 + ( )( ) +𝜆 The condition that (17) has nontrivial solution required the ln 𝑟 ln 𝑟 𝑑𝑟 0 determinant of coefficient in (17) should vanish, so we can 𝐻 𝐻 (23) directly get the equation +𝑚2 =0, 2 𝜇] 2 𝜎 𝑔 (𝜕]𝑆+𝑒𝐴])(𝜕𝜇𝑆+𝑒𝐴𝜇)+M =0. (18) where the constant 𝜆0 is from separation of variables, and (23) Consider the square root for left side of (18) and ignore all finally can be written as O(𝜎2) terms, so we can directly get the modified Hamilton- 𝑟 −1 Jacobi equation: 𝑅 (𝑟) =±∫ ( ) ± ln 𝑟 𝜇] 2 𝐻 𝑔 (𝜕]𝑆+𝑒𝐴])(𝜕𝜇𝑆+𝑒𝐴𝜇)+𝑚 (24) (19) 𝑟 𝑡𝑡 2 ⋅ √ 1−2𝜎𝑚 𝜔2 − ( )(𝜆 +𝑚2). −2𝜎𝑚𝑔 (𝜔 − 𝑒𝐴 ) =0. ( ) ln 0 𝑡 𝑟𝐻 Therefore, we find that the modified Dirac equation At the horizon 𝑟𝐻 oftheblackstring,theaboveequationis from Lorentz invariance violation could lead to the modified integrated via residue theorem, and we can get Hamilton-Jacobi equation, and the correction of Hamilton- Jacobi equation depends on the energy and mass of radiation 𝑅± (𝑟) =±𝑖𝜋(1−𝜎𝑚) 𝑟𝐻𝜔 (25) fermions. Using the modified Hamilton-Jacobi equation, we then investigate the fermions Hawking tunneling radiation of and the fermions tunneling rate 2+1-dimensional black string and 𝑛+1-dimensional BTZ- 2 2 like string in the following two sections. Γ= (− 𝑆) = [− ( 𝑅 − 𝑅 )] exp ℏ Im exp ℏ Im + Im − (26) −(4𝜋/ℏ)(1−𝜎𝑚)𝑟 𝜔 −𝜔/𝑇 3. Fermions Tunneling of =𝑒 𝐻 =𝑒 𝐻 . 2+1-Dimensional Black String The relationship between tunneling rate and Hawking tem- The research of gravity in 2+1dimension can help people perature required further understand the properties of gravity, and it is also important to construct the quantum gravity. Recently, Murata 1+𝜎𝑚 𝑇 =ℏ = (1+𝜎𝑚) 𝑇 , et al. have researched the 2+1-dimensional gravity with 𝐻 0 (27) 4𝜋𝑟𝐻 dilatonfield,whoseactionisgivenby[46] 𝑇 𝑇 𝜆2 where 𝐻 and 0 are the modified and nonmodified Hawking 𝐼=𝑀 ∫ 𝑑3𝑥√−𝑔 (𝐵𝑅 + ), temperature of the 2+1-dimensional black string, respec- 3 𝐵 (20) tively, and 𝜎 term is the correction. 4 Advances in High Energy Physics

4. Fermions Tunneling of Higher Dimensional where 𝜔0 =𝑒𝐴𝑡(𝑟𝐻). This means that the fermions tunneling BTZ-Like Black Strings rate is 2 2 As we all know that the linear Maxwell action fails to satisfy Γ= (− 𝑆) = [− ( 𝑅 − 𝑅 )] exp ℏ Im exp ℏ Im + Im − the conformal symmetry in higher dimensional spacetime (35) ¨ ´ −(4𝜋/ℏ)(1−𝜎𝑚)((𝜔−𝜔 )/𝑓󸀠(𝑟 )) −((𝜔−𝜔 )/𝑇 ) [47, 48], so Hassaıne and Martınez proposed gravity theory =𝑒 0 𝐻 =𝑒 0 𝐻 with nonlinear Maxwell field in arbitrary dimensional space- time: and the Hawking temperature is 1 2 𝑠 𝐼=− ∫ 𝑑𝑛+1𝑥√−𝑔𝑅+ [ −𝛽(𝛼𝐹 𝐹𝜇]) ] , 2 𝜇] (28) 1+𝜎𝑚 󸀠 16𝜋 𝑀 𝑙 𝑇 =ℏ 𝑓 (𝑟 ) 𝐻 4𝜋 𝐻 −2 (36) where Λ≡−𝑙 is cosmological constant. Hendi researched 1+𝜎𝑚 𝑛𝑟 =ℏ ( 𝐻 −2𝑛/2𝑄𝑛𝑟1−𝑛)=(1+𝜎𝑚) 𝑇 , 𝑛+1-dimensional static black strings solution with 𝛽=1, 4𝜋 𝑙2 𝐻 0 𝛼=−1,and𝑠=𝑛/2, and it is charged BTZ-like solutions [49], whose metric is where 𝑇𝐻 and 𝑇0 are the modified and nonmodified Hawking 𝑛+1 2 temperature of the -dimensional BTZ-like black string, 𝑑𝑟 2 𝜎 𝑑𝑠2 =−𝑓(𝑟) 𝑑𝑡2 + +𝑟2∑ (𝑑𝑥𝑘) , respectively, and term is the correction. 𝑓 (𝑟) (29) 𝑘 5. Conclusions where In this paper, we consider the deformed dispersion relation 𝑟2 𝑓 (𝑟) = −𝑟2−𝑛 (𝑀 +𝑛/2 2 𝑄𝑛−1𝐴 ), withLorentzinvarianceviolationandgeneralizethemodified 2 𝑡 (30) 𝑙 Dirac equation in curved spacetime. The fermions tunneling radiation of black strings is researched, and we find that the and the electromagnetic potential is modified Dirac equation could lead to Hawking temperature’s 𝑟 correction, which depends on the correction parameter 𝜎 and 𝐴=𝐴𝑑𝑡 =𝑄 ( )𝑑𝑡. 𝑡 ln 𝑙 (31) particle mass 𝑚 in the modified Dirac equation. Next we will discuss the correction of black hole entropy in this theory. As 𝑛=2, this solution is no other than the static charged BTZ The first law of black hole thermodynamics requires solution. We will study the Hawking radiation and black hole 𝑑𝑀 = 𝑇𝑑𝑆 + Ξ𝑑𝐽 +𝑈𝑑𝑄, temperature at the event horizon 𝑟𝐻 of this black string. In (37) 𝑆 = −𝜔𝑡 + 𝑅(𝑟)𝑘 +𝑌(𝑥 ) 𝑥𝑘 (19), we can set ,where are the Ξ 𝑈 space coordinates excluding the radial coordinate, so that the where and are electromagnetic potential and rotating modified Hamilton-Jacobi equation is given by potential, so the nonmodified entropy of black hole is [1, 2, 50, 51] 2 −1 2 𝑑𝑅 𝑑𝑀 − Ξ𝑑𝐽 −𝑈𝑑𝑄 − (1−2𝜎𝑚) 𝑓 (𝑟) (𝜔 − 𝑒𝐴𝑡) +𝑓(𝑟) ( ) 𝑑𝑟 𝑑𝑆0 = . (38) 𝑇0 (32) 1 𝑑𝑌 2 + ∑ ( ) +𝑚2 =0, From the above results, we know that the relationship 𝑟2 𝑑𝑥𝑘 𝑘 between modified and nonmodified Hawking temperature is 𝑇𝐻 =(1+𝜎𝑚)𝑇0, since the nonmodified black hole entropy and the radial equation with constant 𝜆0 is is given by 𝑑𝑅 2 𝜆 𝑑𝑀 − Ξ𝑑𝐽 −𝑈𝑑𝑄 − (1−2𝜎𝑚) 𝑓−1 (𝑟) (𝜔 − 𝑒𝐴 )2 +𝑓(𝑟) ( ) + 0 𝑆 = ∫ 𝑑𝑆 = ∫ 𝑡 𝑑𝑟 𝑟2 𝐻 𝐻 (1+𝜎𝑚) 𝑇 (33) 0 (39) 2 +𝑚 =0. 2 =𝑆0 −𝑚∫ 𝜎𝑑𝑆0 + O (𝜎 ), 𝑟 𝑓(𝑟 )=0 Therefore, at the horizon 𝐻 of the black string, 𝐻 2 and we finally get where we can ignore O(𝜎 ) because 𝜎≪1.Equation(39) shows that the correction of black hole entropy depends on 𝜎 𝑅 (𝑟) =±∫ 𝑓 (𝑟)−1 , which is independent from time and space coordinates. ± However, it is possible that 𝜎 depends on other parameters in curved spacetime, and it is very interesting that 𝜎 depends on √ 2 2 𝑆 𝜎=𝜎/𝑆 +⋅⋅⋅ ⋅ (1−2𝜎𝑚) (𝜔 − 𝑒𝐴𝑡) −𝑓(𝑟) (𝜆0 +𝑚 ) (34) 0.Inparticular,as 0 0 ,wecangetthelogarithmic correction of black hole entropy 𝜔−𝜔 =±𝑖𝜋(1−𝜎𝑚) 0 , 󸀠 𝑓 (𝑟𝐻) 𝑆𝐻 =𝑆0 −𝑚𝜎0 ln 𝑆0 +⋅⋅⋅. (40) Advances in High Energy Physics 5

In quantum gravity theory, the logarithmic correction has [5] S. Sannan, “Heuristic derivation of the probability distributions been researched in detail [52–69], and according to [70], it is of particles emitted by a black hole,” General Relativity and required that the coefficient of logarithmic correction should Gravitation, vol. 20, no. 3, pp. 239–246, 1988. be −(𝑛 + 1)/2(𝑛 −1) in 𝑛+1-dimensional spacetime, so it [6] P. Kraus and F. Wilczek, “Self-interaction correction to black indicates that 𝜎0 could be (𝑛 + 1)/2𝑚(𝑛. −1) hole radiance,” Nuclear Physics B,vol.433,no.2,pp.403–420, On the other hand, from the deformed dispersion relation 1995. (1) with 𝛼=2, it is implied that the Klein-Gordon equation [7] M. K. Parikh and F. Wilczek, “Hawking radiation as tunneling,” could be given by Physical Review Letters, vol. 85, article 5042, 2000. [8] S. Hemming and E. Keski-Vakkuri, “Hawking radiation from 2 2 2 2 2 2 2 (−𝜕𝑡 +𝜕𝑗 +𝑚 −𝜎 ℏ 𝜕𝑡 𝜕𝑗 )Φ=0, (41) AdS black holes,” Physical Review D,vol.64,no.4,ArticleID 044006, 8 pages, 2001. so the generalized uncharged Klein-Gordon equation in [9] Q.-Q. Jiang, S.-Q. Wu, and X. Cai, “Hawking radiation from static curved spacetime is dilatonic black holes via anomalies,” Physical Review D,vol.75, Article ID 064029, 2007. 𝑡𝑡 2 𝑗𝑗 2 2 2 2 𝑡𝑡 2 𝑗𝑗 2 [𝑔 ∇𝑡 +𝑔 ∇𝑗 +𝑚 +𝜎 ℏ (𝑔 ∇𝑡 )(𝑔 ∇𝑗 )] Φ = 0, (42) [10] S. Iso, H. Umetsu, and F. Wilczek, “Anomalies, Hawking radia- tions, and regularity in rotating black holes,” Physical Review D, and using the semiclassical approximation with Φ= vol. 74, no. 4, Article ID 044017, 10 pages, 2006. 𝐶 exp (𝑖𝑆/ℏ),themodifiedHamilton-Jacobiequationinscalar [11] A. J. Medved, “Radiation via tunneling from a de Sitter cosmo- field is given by logical horizon,” Physical Review D,vol.66,ArticleID124009, 2002. 2 𝑡𝑡 2 𝜇] 2 2 𝑡𝑡 2 4 (1+𝜎 𝑔 𝜔 ) 𝑔 𝜕𝜇𝑆𝜕]𝑆+𝑚 −𝜎 (𝑔 ) 𝜔 =0. (43) [12] M. K. Parikh, “Energy conservation and hawking radiation,” in Proceedingsofthe10thMarcelGrossmannMeeting,pp.1585– Namely, 1590, February 2006.

𝜇] 2 2 𝑡𝑡 2 2 𝑡𝑡 2 4 [13] M. K. Parikh, J. Y. Zhang, and Z. Zhao, “Hawking radiation of 𝑔 𝜕𝜇𝑆𝜕]𝑆+𝑚 −𝜎 𝑔 𝜔 (𝑚 +𝑔 𝜔 )+O (𝜎 ) charged particles via tunneling from the Reissner-Nordstrom¨ (44) black hole,” JournalofHighEnergyPhysics,vol.2005,no.10, =0. article 055, 2005. [14] J. Zhang and Z. Zhao, “Charged particles’ tunnelling from the Contrasting (19) and (44) as uncharged case, we find that Kerr-Newman black hole,” Physics Letters B,vol.638,no.2-3, the correctional terms of Dirac field and scalar field are very pp. 110–113, 2006. different. The fact implies that the corrections of Hawking [15] E. T. Akhmedov, V. Akhmedova, and D. Singleton, “Hawking temperature and black hole entropy from Hawking tunneling temperature in the tunneling picture,” Physics Letters B,vol.642, radiation with different spin particles could be different, and no. 1-2, pp. 124–128, 2006. this conclusion could be helpful to suggest a new idea to [16] V.Akhmedova, T. Pilling, A. de Gill, and D. Singleton, “Tempo- research the black hole information paradox. Work in these ral contribution to gravitational WKB-like calculations,” Physics fields is currently in progress. Letters B,vol.666,no.3,pp.269–271,2008. [17] K. Srinivasan and T. Padmanabhan, “Particle production and complex path analysis,” Physical Review D,vol.60,no.2,Article Competing Interests ID024007,20pages,1999. The authors declare that they have no competing interests. [18] S. Shankaranarayanan, T. Padmanabhan, and K. Srinivasan, “Hawking radiation in different coordinate settings: complex paths approach,” Classical and Quantum Gravity,vol.19,no.10, Acknowledgments pp. 2671–2687, 2002. [19] M. Angheben, M. Nadalini, L. Vanzo, and S. Zerbini, “Hawking This work was supported by FAPESP no. 2012/08934-0, radiation as tunneling for extremal and rotating black holes,” CNPq,CAPES,andNationalNaturalScienceFoundation Journal of High Energy Physics,vol.2005,no.5,article014,18 of China no. 11573022, no. 11205254, no. 11178018, and no. pages, 2005. 11375279. [20] S. Z. Yang and D. Y. Chen, “Hawking radiation as tunneling from the vaidya-bonner black hole,” International Journal of References Theoretical Physics, vol. 46, no. 11, pp. 2923–2927, 2007. [21] Y. Shu-Zheng and C. De-You, “Tunnelling effect of the non- [1] S. W. Hawking, “Black hole explosions?,” Nature,vol.248,no. stationary Kerr black hole,” Chinese Physics B,vol.17,no.3,pp. 5443, pp. 30–31, 1974. 817–821, 2008. [2]S.W.Hawking,“Particlecreationbyblackholes,”Communica- [22] R. Kerner and R. B. Mann, “Fermions tunnelling from black tions in Mathematical Physics,vol.43,no.3,pp.199–220,1975. holes,” Classical and Quantum Gravity,vol.25,no.9,ArticleID [3] S. P. Robinson and F. Wilczek, “Relationship between Hawking 095014, 2008. radiation and gravitational anomalies,” Physical Review Letters, [23] R. Kerner and R. B. Mann, “Charged fermions tunnelling from vol. 95, no. 1, Article ID 011303, 4 pages, 2005. kerr-newman black holes,” Physics Letters B,vol.665,no.4,pp. [4] T. Damoar and R. Ruffini, “Black-hole evaporation in the Klein- 277–283, 2008. Sauter-Heisenberg-Euler formalism,” Physical Review D,vol.14, [24] R. Li, J.-R. Ren, and S.-W. Wei, “Hawking radiation of Dirac no.2,p.332,1976. particles via tunneling from the Kerr black hole,” Classical and 6 Advances in High Energy Physics

Quantum Gravity,vol.25,no.12,ArticleID125016,5pages, [44]T.Jacobson,S.Liberati,andD.Mattingly,“Astrongastrophys- 2008. ical constraint on the violation of special relativity by quantum [25] R. Li and J.-R. Ren, “Dirac particles tunneling from BTZ black gravity,” Nature,vol.424,pp.1019–1021,2003. hole,” Physics Letters B,vol.661,no.5,pp.370–372,2008. [45] S. I. Kruglov, “Modified Dirac equation with Lorentz invariance [26] D.-Y. Chen, Q.-Q. Jiang, and X.-T. Zu, “Fermions tunnelling violation and its solutions for particles in an external magnetic from the charged dilatonic black holes,” Classical and Quantum field,” Physics Letters B,vol.718,no.1,pp.228–231,2012. Gravity,vol.25,no.20,ArticleID205022,2008. [46]K.Murata,J.Soda,andS.Kanno,“Evaporating(2+1)- [27] D.-Y. Chen, Q.-Q. Jiang, and X.-T. Zu, “Hawking radiation of dimensional black strings,” http://arxiv.org/abs/gr-qc/0701137. Dirac particles via tunnelling from rotating black holes in de [47] M. Hassa¨ıne and C. Mart´ınez, “Higher-dimensional black holes Sitter spaces,” Physics Letters B,vol.665,no.2-3,pp.106–110, with a conformally invariant Maxwell source,” Physical Review 2008. D,vol.75,ArticleID027502,2007. [28] R. Di Criscienzo and L. Vanzo, “Fermion tunneling from [48] M. Hassa¨ıne and C. Mart´ınez, “Higher-dimensional charged dynamical horizons,” Europhysics Letters,vol.82,no.6,Article black hole solutions with a nonlinear electrodynamics source,” ID 60001, 2008. Classical and Quantum Gravity,vol.25,no.19,2008. [29] H. Li, S. Yang, T. Zhou, and R. Lin, “Fermion tunneling from a [49] S. H. Hendi, “Charged BTZ-like black holes in higher dimen- Vaidya black hole,” Europhysics Letters,vol.84,no.2,ArticleID sions,” The European Physical Journal C,vol.71,article1551,2007. 20003, 2008. [50] J. D. Bekenstein, “Black holes and the second law,” Lettere al [30] Q.-Q. Jiang, “Fermions tunnelling from GHS and non-extremal Nuovo Cimento,vol.4,no.15,pp.737–740,1972. D1–D5 black holes,” Physics Letters B,vol.666,no.5,pp.517–521, [51] J. D. Bekenstein, “Black holes and entropy,” Physical Review D. 2008. ParticlesandFields.ThirdSeries, vol. 7, pp. 2333–2346, 1973. [31] Q.-Q. Jiang, “Dirac particle tunneling from black rings,” Physical [52] R. Banerjee and B. R. Majhi, “Quantum tunneling beyond Review D,vol.78,ArticleID044009,2008. semiclassical approximation,” JournalofHighEnergyPhysics, [32] K. Lin and S. Z. Yang, “Quantum tunneling from apparent vol.2008,no.6,article095,20pages,2008. horizon of rainbow-FRW universe,” International Journal of [53] R. Banerjee and B. R. Majhi, “Quantum tunneling and back Theoretical Physics,vol.48,no.7,pp.2061–2067,2009. reaction,” Physics Letters B,vol.662,no.1,pp.62–65,2008. [33] Y. Zhou, “D4 brane probes in gauge/gravity duality,” Physical [54] R. Banerjee, B. R. Majhi, and S. Samanta, “Noncommutative Review D, vol. 79, Article ID 066005, 2009. black hole thermodynamics,” Physical Review D,vol.77,Article [34] K. Lin and S. Yang, “Fermions tunneling of higher-dimensional ID 124035, 2008. Kerr-anti-de Sitter black hole with one rotational parameter,” [55] R. Banerjee and R. B. Majhi, “Connecting anomaly and tunnel- Physics Letters B,vol.674,no.2,pp.127–130,2009. ing methods for the Hawking effect through chirality,” Physical [35] K. Lin and S. Z. Yang, “A simpler method for researching Review D,vol.79,ArticleID064024,2009. fermions tunneling from black holes,” Chinese Physics B,vol.20, [56] S. K. Modak, “Corrected entropy of BTZ black hole in tunneling no. 11, Article ID 110403, 2011. approach,” Physics Letters B,vol.671,no.1,pp.167–173,2009. [36] G. Amelino-Camelia, “Relativity in spacetimes with short- [57] R. Banerjee and B. R. Majhi, “Quantum tunneling and trace distance structure governed by an observer-independent anomaly,” Physics Letters B,vol.674,no.3,pp.218–222,2009. (Planckian) length scale,” International Journal of Modern [58] R.-G. Cai, L.-M. Cao, and Y.-P. Hu, “Corrected entropy-area Physics D: Gravitation, Astrophysics, Cosmology,vol.11,article relation and modified Friedmann equations,” Journal of High 35, 2002. Energy Physics,vol.2008,no.8,article090,12pages,2008. [37] G. Amelino-Camelia, “Phenomenology of Planck-scale [59] J. Y. Zhang, “Black hole quantum tunnelling and black hole Lorentz-symmetry test theories,” New Journal of Physics,vol.6, entropy correction,” Physics Letters B,vol.668,no.5,pp.353– p. 188, 2004. 356, 2008. [38] J. Magueijo and L. Smolin, “Lorentz invariance with an invariant [60] R. Banerjee and S. K. Modak, “Exact differential and corrected energy scale,” Physical Review Letters, vol. 88, no. 19, Article ID area law for stationary black holes in tunneling method,” Journal 0112090, 2002. of High Energy Physics,vol.2009,no.5,article063,32pages, [39] J. Magueijo and L. Smolin, “Generalized Lorentz invariance 2009. with an invariant energy scale,” Physical Review D,vol.67,no. [61] K. Lin and S. Yang, “Quantum tunnelling in charged black holes 4, Article ID 044017, 2003. beyond the semi-classical approximation,” Europhysics Letters, [40] J. Ellis, N. E. Mavromatos, and D. V.Nanopoulos, “String theory vol. 86, no. 2, Article ID 20006, 2009. modifies quantum mechanics,” Physics Letters B,vol.293,no.1- [62] K. Lin and S. Z. Yang, “Fermion tunnels of higher-dimensional 2, pp. 37–48, 1992. anti-de Sitter Schwarzschild black hole and its corrected [41] J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, “A micro- entropy,” Physics Letters B,vol.680,no.5,pp.506–509,2009. scopic Liouville arrow of time,” Chaos, Solitons & Fractals,vol. [63] S. W. Hawking, “Zeta function regularization of path integrals 10, no. 2-3, pp. 345–363, 1999. in curved spacetime,” Communications in Mathematical Physics, [42] J. R. Ellis, N. E. Mavromatos, and A. S. Sakharov, “Synchrotron vol.55,no.2,pp.133–148,1977. radiation from the Crab Nebula discriminates between models [64] R. K. Kaul and P. Majumdar, “Logarithmic correction to the of space-time foam,” Astroparticle Physics,vol.20,no.6,pp. Bekenstein-Hawking entropy,” Physical Review Letters,vol.84, 669–682, 2004. article 5255, 2000. [43] S. I. Kruglov, “Modified wave equation for spinless particles [65] D. V. Fursaev, “Temperature and entropy of a quantum black and its solutions in an external magnetic field,” Modern Physics hole and conformal anomaly,” Physical Review D,vol.51,Article Letters A,vol.28,ArticleID1350014,2013. ID R5352, 1995. Advances in High Energy Physics 7

[66] S. Mukherjee and S. S. Pal, “Logarithmic corrections to black hole entropy and ADS/CFT correspondence,” Journal of High Energy Physics,vol.2002,no.5,article026,2002. [67] V. P. Frolov, W. Israel, and S. N. Solodukhin, “One-loop quantum corrections to the thermodynamics of charged black holes,” Physical Review D,vol.54,no.4,pp.2732–2745,1996. [68] M. R. Setare, “Logarithmic correction to the Cardy-Verlinde formula in topological Reissner-Nordstrom¨ de Sitter space,” Physics Letters B,vol.573,pp.173–180,2003. [69] M. R. Setare, “Logarithmic correction to the Brane equationin topological Reissner-Nordstrom¨ de Sitter space,” The European Physical Journal C—Particles and Fields,vol.38,no.3,pp.389– 394, 2004. [70] S. Das, P. Majumdar, and R. K. Bhaduri, “General logarithmic corrections to black-hole entropy,” Classical and Quantum Gravity,vol.19,no.9,pp.2355–2367,2002. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 7248520, 10 pages http://dx.doi.org/10.1155/2016/7248520

Research Article Thermodynamical Study of FRW Universe in Quasi-Topological Theory

H. Moradpour and R. Dehghani

Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran

Correspondence should be addressed to H. Moradpour; [email protected]

Received 8 March 2016; Accepted 19 April 2016

Academic Editor: Ahmed Farag Ali

Copyright © 2016 H. Moradpour and R. Dehghani. 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.

By applying the unified first law of thermodynamics on the apparent horizon of FRW universe, we get the entropy relation forthe apparent horizon in quasi-topological gravity theory. Throughout the paper, the results of considering the Hayward-Kodama and Cai-Kim temperatures are also addressed. Our study shows that whenever there is no energy exchange between the various parts of cosmos, we can get an expression for the apparent horizon entropy in quasi-topological gravity, which is in agreement with other attempts that followed different approaches. The effects of a mutual interaction between the various parts of cosmos on the apparent horizon entropy as well as the validity of second law of thermodynamics in quasi-topological gravity are perused.

1. Introduction to this expectation and show that such geometrical fluids and their interaction with other parts of cosmos may also affect Since observational data indicates an accelerating universe the horizon entropy of FRW universe [33–36]. [1–4], one should either consider a nontrivial fluid in the String theory together with AdS/CFT correspondence Einstein relativity [5–7] or modifying the Einstein theory conjecture gives us the motivation to study space-times [7–10]. Moreover, there are some observational evidences with dimensions more than 4 [37–39]. Moreover, brane which permit a mutual interaction between the dark sides of scenarios also encourage us to study (4 + 1)-dimensional cosmos [11–25], including the dark energy and dark matter space-times [40, 41]. The backbone of Einstein-Hilbert action [26]. The thermodynamic consequences of such interactions is producing the second-order field equations of motion. in the Einstein framework are addressed in [27]. Recently, it is In looking for a suitable Lagrangian for higher dimensions, argued that since the origin of dark energy is unknown, some which also keeps the field equations of motion for metric dark energy candidates may affect the Bekenstein entropy of of second-order, one reaches to Lovelock Lagrangian instead apparent horizon in a flat FRW universe [28–31]. In addition, of the usual (𝑛 + 1) version of Einstein-Hilbert action by following [32], one can see that all of the dark energy [42]. It is shown that the corresponding gravitational field candidates and their interaction with other parts of cosmos equations of static spherically symmetric space-times in the may affect the apparent horizon of a flat FRW universe in Einstein and Lovelock theories are parallel to the availability the Einstein framework. There are additional terms to the of the first law of thermodynamics on the horizons of Einstein tensor in modified theories of gravity. Since these considered metric [43]. For a FRW universe, where the field additional terms can be interpreted as a geometrical fluid, equations govern the universe expansion history, one can a mutual interaction between the dark sides of cosmos may also reach the corresponding Friedmann equations in the be an interaction between these geometrical and material Einstein and Lovelock theories by applying the first law of terms [10]. Therefore, one may expect that such modifications thermodynamics on the apparent horizon of FRW universe to Einstein theory and their interaction with other parts of [44]. Indeed, such generalization of Einstein-Hilbert action cosmos may also affect the horizon entropy. Indeed, there are leads to additional terms besides the Einstein tensor in space- some new attempts in which authors give a positive answer times with dimensions more than 4. Authors in [33] interpret 2 Advances in High Energy Physics suchadditionaltermsasthegeometricalfluidsandshowthat in which the geometrical fluid interacts with other parts of these terms and their interactions with other parts of cosmos cosmos. The results of attributing the Cai-Kim temperature affect the horizon entropy. to the apparent horizon in an interacting universe are also By considering cubic and higher curvature interactions, pointed out in Section 4. Section 5 includes some notes about some authors try to find new Lagrangian which keeps the the availability of second law of thermodynamics in interact- field equation of motion for the metric of second-order [45– ing cosmos described by quasi-topological gravity theory. The 47]. Their theory is now called quasi-topological gravity final section is devoted to summary and concluding remarks. which affects the gravitational field equations in space-time with dimensions more than 4. Thermodynamics of static 2. A Brief Overview of black holes are studied in this theory which leads to an expression for the horizon entropy [45–50]. Moreover, it is Quasi-Topological Gravity shownthatifoneassumesthattheblackholeentropyrelation, A natural generalization of the Einstein-Hilbert action to derived in [46–50], is available for the apparent horizon of higher dimensional space-time, and higher-order gravity FRW universe, then, by applying the unified first law of with second-order equation of motion, is the Lovelock action thermodynamics on the apparent horizon of FRW universe, one can get the corresponding Friedmann equations in quasi- 1 𝑚 𝐼 = ∫ 𝑑𝑛+1𝑥√−𝑔∑ ( 𝑐 L + L ), topological gravity [51]. Thereinafter, authors point out the 𝐺 𝑖 𝑖 𝑀 (1) 16𝜋𝐺𝑛+1 𝑖=1 second law of thermodynamics and its generalized form [51]. In fact, this scheme proposes that, in quasi-topological where 𝑐𝑖’s are Lovelock coefficients, L𝑖’s are dimensionally gravity, one may generalize the black hole entropy to the extended Euler densities, and L𝑀 is the matter Lagrangian. cosmological setup by inserting the apparent horizon radii In the above action because of the topological origin of instead of the event horizon radii. Is it the only entropy the Lovelock terms, the term proportional to 𝑐𝑖 contributes relation for the apparent horizon? Does a mutual interaction to the equations of motion in dimensions with 𝑛≥2𝑚, between the geometrical fluid, arising from the terms besides where 𝑚 is the order of Lovelock theory. Generally, although theEinsteintensorinthefieldequations,andotherpartsof the equations of motion of 𝑘th-order Lovelock gravity are cosmos affect the horizon entropy? second-order differential equations, the 𝑘th-order Lovelock Our aim in this paper is to study the thermodynamics of termhasnocontributiontothefieldequationsin2𝑘 and apparent horizon of FRW universe in the quasi-topological lower dimensions. For example, the cubic term L3 does gravity theory to provide proper answers for the above- not have any dynamical effect in five dimensions. Recently, mentioned questions. In this investigation, we point to the a modification of Lovelock gravity called quasi-topological results of considering the Hayward-Kodama and Cai-Kim gravity has been introduced, which has contribution to the temperatures for the apparent horizon. In order to achieve field equations in five dimensions from the 𝑚th-order (𝑚= this goal, by looking at the higher-order curvature terms, 3) term in Riemann tensor. Several aspects of 𝑚th-order arising in quasi-topological gravity, as a geometrical fluid quasi-topological terms which have at most second-order and applying the unified first law of thermodynamics on derivatives of the metric in the field equations for spherically the apparent horizon, we get an expression for the apparent symmetric space-times in five and higher dimensions except horizon entropy in this theory, while the geometrical fluid 2𝑝 dimensions have been investigated. does not interact with other parts of cosmos. Moreover, we The action of 4th-order quasi-topological gravity in (𝑛+1) focusonaFRWuniverseinwhichthegeometricalfluid dimensions can be written as follows: interacts with other parts of cosmos. This study signals us 1 to the effects of such mutual interaction on the apparent 𝐼 = ∫ 𝑑𝑛+1𝑥√−𝑔 [𝜇 L +𝜇 L +𝜇 X 𝐺 16𝜋𝐺 1 1 2 2 3 3 horizon entropy of a FRW universe with arbitrary curvature 𝑛+1 (2) for quasi-topological gravity theory. As the noninteracting +𝜇 X + L ], case, the second law of thermodynamics is also studied in an 4 4 matter interacting universe. which not only works in five dimensions but also yields The paper is organized as follows. In the next section, second-order equations of motion for spherically symmetric we give an introductory note about the quasi-topological space-times. gravity, the corresponding Friedmann equations, and some In action (2), L1 =𝑅is just the Einstein-Hilbert properties of FRW universe, including its apparent horizon 𝑎𝑏𝑐𝑑 𝑎𝑏 2 Lagrangian, L2 =𝑅𝑎𝑏𝑐𝑑𝑅 −4𝑅𝑎𝑏𝑅 +𝑅 is the second- radii, surface gravity, and thus the corresponding Hayward- order Lovelock (Gauss-Bonnet) Lagrangian, and Lmatter is Kodama temperature. Section 3 is devoted to a brief discus- the Lagrangian of the matter field. X3 is the curvature-cubed sion about the Unified First Law (UFL) of thermodynamics. Lagrangian given by [46] In Section 4, bearing the Hayward-Kodama temperature together with the unified first law of thermodynamics in 𝑐𝑑 𝑒𝑓 𝑎𝑏 X3 =𝑅𝑎𝑏𝑅𝑐𝑑𝑅𝑒𝑓 mind, we study the effect of a geometrical (curvature) fluid 1 3 (3𝑛 − 5) on the apparent horizon entropy in a FRW universe with + ( 𝑅 𝑅𝑎𝑏𝑐𝑑𝑅 arbitrary curvature parameter for quasi-topological theory. (2𝑛 − 1)(𝑛−3) 8 𝑎𝑏𝑐𝑑 The results of considering the Cai-Kim temperature are also 𝑎𝑏𝑐 𝑑𝑒 𝑎𝑐 𝑏𝑑 addressed. Thereinafter, we generalize our study to a universe −3(𝑛−1) 𝑅𝑎𝑏𝑐𝑑𝑅𝑒 𝑅 +3(𝑛+1) 𝑅𝑎𝑏𝑐𝑑𝑅 𝑅 Advances in High Energy Physics 3

3 (3𝑛 − 1) 2 3 2 +6(𝑛−1) 𝑅𝑏𝑅𝑐 𝑅𝑎 − 𝑅𝑏𝑅𝑎𝑅 𝑐7 =64(𝑛−2)(𝑛−1) (4𝑛 −18𝑛 + 27𝑛 − 9) , 𝑎 𝑏 𝑐 2 𝑎 𝑏 4 3 2 3 (𝑛+1) 𝑐8 =−96(𝑛−1)(𝑛−2) (2𝑛 −7𝑛 +4𝑛 +6𝑛−3), + 𝑅3), 8 3 4 3 2 𝑐9 =16(𝑛−1) (2𝑛 −26𝑛 +93𝑛 − 117𝑛 + 36) , (3) 5 4 3 2 𝑐10 =𝑛 −31𝑛 + 168𝑛 − 360𝑛 + 330𝑛 − 90, and X4 is the fourth-order term of quasi-topological gravity 𝑐 = 2 (6𝑛6 −67𝑛5 + 311𝑛4 − 742𝑛3 + 936𝑛2 − 576𝑛 [47]: 11 + 126) , 𝑐𝑑𝑒𝑓 ℎ𝑔 𝑎𝑏 𝑎𝑏𝑐𝑑 𝑒𝑓 X4 =𝑐1𝑅𝑎𝑏𝑐𝑑𝑅 𝑅𝑒𝑓 𝑅ℎ𝑔 +𝑐2𝑅𝑎𝑏𝑐𝑑𝑅 𝑅𝑒𝑓 𝑅 5 4 3 2 𝑐12 = 8 (7𝑛 −47𝑛 + 121𝑛 − 141𝑛 +63𝑛−9), 2 +𝑐𝑅𝑅 𝑅𝑎𝑐𝑅𝑏 +𝑐 (𝑅 𝑅𝑎𝑏𝑐𝑑) 3 𝑎𝑏 𝑐 4 𝑎𝑏𝑐𝑑 2 𝑐13 =16𝑛(𝑛−1)(𝑛−2)(𝑛−3) (3𝑛 −8𝑛+3), +𝑐𝑅 𝑅𝑎𝑐𝑅 𝑅𝑑𝑏 +𝑐𝑅𝑅 𝑅𝑎𝑐𝑅𝑑𝑏 5 𝑎𝑏 𝑐𝑑 6 𝑎𝑏𝑐𝑑 7 6 5 4 3 𝑐14 =8(𝑛−1) (𝑛 −4𝑛 − 15𝑛 + 122𝑛 − 287𝑛 𝑎𝑐 𝑏𝑒 𝑑 𝑎𝑐𝑒𝑓 𝑏 𝑑 +𝑐7𝑅𝑎𝑏𝑐𝑑𝑅 𝑅 𝑅 +𝑐8𝑅𝑎𝑏𝑐𝑑𝑅 𝑅 𝑅 (4) 𝑒 𝑒 𝑓 + 297𝑛2 − 126𝑛 + 18) . +𝑐𝑅 𝑅𝑎𝑐𝑅 𝑅𝑏𝑒𝑑𝑓 +𝑐 𝑅4 +𝑐 𝑅2𝑅 𝑅𝑎𝑏𝑐𝑑 9 𝑎𝑏𝑐𝑑 𝑒𝑓 10 11 𝑎𝑏𝑐𝑑 (5) 2 𝑎𝑏 𝑎𝑏𝑒𝑓 𝑐 𝑑𝑔 +𝑐12𝑅 𝑅𝑎𝑏𝑅 +𝑐13𝑅𝑎𝑏𝑐𝑑𝑅 𝑅𝑒𝑓𝑔 𝑅 In the context of universal thermodynamics, our universe should be a nonstationary gravitational system while, from 𝑎𝑒𝑐𝑓 𝑔𝑏ℎ𝑑 +𝑐14𝑅𝑎𝑏𝑐𝑑𝑅 𝑅𝑔𝑒ℎ𝑓𝑅 . the cosmological point of view, it should be homogeneous and isotropic. Therefore, the natural choice is the FRW 𝑐 universe, a dynamical spherically symmetric space-time, The coefficients 𝑖 intheabovetermaregivenby having only inner trapping horizon (the apparent horizon), which is described by the line element 7 6 5 4 3 𝑐1 =−(𝑛−1) (𝑛 −3𝑛 −29𝑛 + 170𝑛 − 349𝑛 2 𝑎 𝑏 2 2 𝑑𝑠 =ℎ𝑎𝑏𝑑𝑥 𝑑𝑥 + ̃𝑟 𝑑Ω , (6) 2 + 348𝑛 − 180𝑛 + 36) , 0 1 where 𝑥 =𝑡, 𝑥 =𝑟, ̃𝑟=𝑎(𝑡)𝑟, 𝑎(𝑡) is the scale factor 6 5 4 3 2 oftheuniversewiththecurvatureparameter𝑘 with values 𝑐2 =−4(𝑛−3) (2𝑛 − 20𝑛 + 65𝑛 −81𝑛 +13𝑛 −1, 0, 1 corresponding to the open, flat, and closed universes 2 2 2 +45𝑛−18), respectively, ℎ𝑎𝑏 = diag(−1, 𝑎(𝑡) /(1 − 𝑘𝑟 )),and𝑑Ω is the metric of (𝑛 − 1)-dimensional unit sphere. 2 2 𝑐3 =−64(𝑛−1) (3𝑛 −8𝑛+3)(𝑛 −3𝑛+3), Varying the action (2) with respect to metric leads to [51]

𝑐 =−(𝑛8 −6𝑛7 +12𝑛6 − 22𝑛5 + 114𝑛4 − 345𝑛3 𝑚 𝑘 𝑖 16𝜋𝐺 4 ∑𝜇̂ 𝑙2𝑖−2 (𝐻2 + ) = 𝑛+1 𝜌. 𝑖 2 (7) 𝑖=1 𝑎 𝑛 (𝑛−1) + 468𝑛2 − 270𝑛 + 54) ,

4 3 2 This is the Friedmann equation of arbitrary-order quasi- 𝑐5 =16(𝑛−1) (10𝑛 − 51𝑛 +93𝑛 −72𝑛+18), topological cosmology where 𝐻 is the Hubble parameter. 𝐺 =1 𝜇̂ 2 2 2 From now on, we set 𝑛+1 for simplicity. Moreover, 𝑖’s 𝑐6 =−32(𝑛−1) (𝑛−3) (3𝑛 −8𝑛+3), are dimensionless parameters as follows:

𝜇̂1 =1, (𝑛−2)(𝑛−3) 𝜇̂ = 𝜇 , 2 𝑙2 2

(𝑛−2)(𝑛−5) (3𝑛2 −9𝑛+4) (8) 𝜇̂ = 𝜇 , 3 8 (2𝑛 − 1) 𝑙4 3

𝑛 (𝑛−1)(𝑛−2)2 (𝑛−3)(𝑛−7) (𝑛5 −15𝑛4 +72𝑛3 − 156𝑛2 + 150𝑛 − 42) 𝜇̂ = 𝜇 . 4 𝑙6 4 4 Advances in High Energy Physics

The dynamical apparent horizon is determined by the relation The term 𝑊𝑑𝑉 in the first law comes from the fact that we 𝑎𝑏 ℎ 𝜕𝑎̃𝑟𝜕𝑏̃𝑟=0. It is a matter of calculation to show that the have a volume change for the total system enveloped by the radius of the apparent horizon for the FRW universe is [52] apparent horizon. For a pure de Sitter space, 𝜌=𝑝,and the work term reduces to the standard 𝑝𝑑𝑉;thusweobtain 1 𝑑𝐸 = 𝑇𝑑𝑆− ̃𝑟 = . exactly the standard first law of thermodynamics, 𝐴 √𝐻2 +𝑘/𝑎2 (9) 𝑝𝑑𝑉.Itisamatterofcalculationtoshowthat(14)canbe written as In cosmological context, various definitions of temperature are used to get the corresponding Friedmann equations on 𝜌+𝑝 𝜌+𝑝 𝐴Ψ = −𝐴𝐻̃𝑟𝐴 ( )𝑑𝑡+𝐴𝑎( )𝑑𝑟𝐴, (16) the apparent horizon. First, we use the original definition 2 2 of temperature (Hayward-Kodama temperature) together 𝑇𝑑𝑆 =𝑑𝑄𝑚 with the Clausius relation ( 𝐴 )aswellasthe and thus unified form of the first law of thermodynamics to extract an expression for the entropy of apparent horizon in the quasi- 3𝑉 (𝜌 + 𝑝)𝐻 topological cosmology. The Hayward-Kodama temperature 𝐴Ψ = − 𝑑𝑡 𝑇 =𝜅/2𝜋 2 associated with the apparent horizon is defined as ℎ , (17) where 𝜅 is the surface gravity which can be evaluated by using 𝐴(𝜌+𝑝) √ √ 𝑎𝑏 + (𝑑̃𝑟 − ̃𝑟 𝐻𝑑𝑡) , 𝜅=(1/2 −ℎ)𝜕𝑎( −ℎℎ 𝜕𝑏̃𝑟) [53–58]. Therefore, the surface 2 𝐴 𝐴 gravity at the apparent horizon of the FRW universe has the following form: on the apparent horizon of FRW universe. In obtaining the ̇ last equation, we used 𝐴̃𝑟𝐴 =3𝑉relation. 1 ̃𝑟𝐴 𝜅=− (1 − ), (10) ̃𝑟𝐴 2𝐻̃𝑟𝐴 4. Thermodynamics of Apparent Horizon in which leads to Quasi-Topological Gravity Theory ̇ 𝜅 1 ̃𝑟𝐴 Our universe is undergoing an accelerating expansion which 𝑇ℎ = =− (1 − ), (11) 2𝜋 2𝜋̃𝑟𝐴 2𝐻̃𝑟𝐴 represents a new imbalance in the governing Friedmann equations. Physicists have addressed such imbalances either for the Hayward-Kodama temperature of apparent horizon. by introducing new sources or by changing the governing Then, using the Cai-Kim temperature and the Clausius equations. The standard cosmology model addresses this relation, we get the entropy of apparent horizon. In this imbalance by introducing a new source (dark energy) in the approach, the horizon temperature is [44, 59] Friedmann equations. On the contrary, a group of physicists have explored the second route, that is, a modified gravity 1 approach, that, at large scales, Einstein theory of general 𝑇= . (12) 2𝜋̃𝑟𝐴 relativity breaks down and a more general action describes the gravitational field. In this study, we follow the second In obtaining the above relation, we consider an infinitesimal approach. time interval, so the horizon radius will have a small change The Friedmann equation of FRW universe in quasi- and we can use the 𝑑̃𝑟𝐴 =0approximation [44, 59]. topological gravity is represented by (7). In modified gravity theories, the modified Friedmann equations can be written as 3. Unified First Law of Thermodynamics and 𝑘 16𝜋 Horizon Entropy 𝐻2 + = 𝜌 , 𝑎2 𝑛 (𝑛−1) 𝑡 (18) The unified first law of thermodynamics can be expressed as 𝑘 8𝜋 [52, 60, 61] 𝐻−̇ =− (𝜌 +𝑝). 𝑎2 (𝑛−1) 𝑡 𝑡 (19) 𝑑𝐸 = 𝐴Ψ + 𝑊𝑑𝑉, (13) 𝜌 𝐸 In the above equations 𝑡 is the total energy density which where is the total baryonic energy content of the universe includes two noninteracting fluid systems: one is the usual inside an 𝑛-sphere of volume 𝑉, while 𝐴 is the area of the 𝜌 𝑝 Ψ fluid of energy density and thermodynamic pressure , horizon. The energy flux is termed as the energy supply while the second one is termed as effective energy density vector and 𝑊 is the work function which are, respectively, 𝜌𝑒 due to curvature contributions and its corresponding defined as pressure 𝑝𝑒.So,wehave 𝑏 𝐴Ψ = 𝐴𝑎 (𝑇 𝜕𝑏̃𝑟𝐴 +𝑊𝜕𝑎̃𝑟𝐴), (14) 𝜌𝑡 =𝜌+𝜌𝑒, 1 𝑎𝑏 (20) 𝑊=− 𝑇 ℎ𝑎𝑏. (15) 2 𝜌𝑡 +𝑝𝑡 =(𝜌+𝑝)+(𝜌𝑒 +𝑝𝑒), Advances in High Energy Physics 5 where universe. The energy conservation law leads to the continuity equation in the form 𝑛 (𝑛−1) 𝑚 𝜇̂ 𝑙2𝑖−2 𝜌 =− ∑ 𝑖 , 𝜌̇ +𝑛𝐻(𝜌 +𝑝)=0. 𝑒 16𝜋 2𝑖 (21) 𝑡 𝑡 𝑡 (24) 𝑖=2 ̃𝑟𝐴 For a noninteracting system it breaks down to 𝜀 (𝑛−1) 𝑚 𝑖𝜇̂ 𝑙2𝑖−2 𝜌 +𝑝 =− ∑ 𝑖 , 𝑒 𝑒 4𝜋 2𝑖 (22) 𝜌+𝑛𝐻(𝜌+𝑝)=0,̇ 𝑖=2 ̃𝑟𝐴 (25) 𝜌̇ +𝑛𝐻(𝜌 +𝑝)=0, ̇ ̇ 𝑒 𝑒 𝑒 where we have defined 𝜀=̃𝑟𝐴/2𝐻̃𝑟𝐴,inwhich̃𝑟𝐴 =𝑑̃𝑟𝐴/𝑑𝑡, and, in terms of the horizon radius ̃𝑟𝐴,wehave[54] which implies that (21) and (22) are available. Differentiating (18), we reach 𝑘 2𝜀 𝐻−̇ =− . 2 16𝜋 16𝜋 2 2 (23) 𝑎 ̃ − 𝑑̃𝑟𝐴 − 𝑑𝜌𝑒 = 𝑑𝜌. 𝑟𝐴 3 𝑛 (𝑛−1) 𝑛 (𝑛−1) (26) ̃𝑟𝐴 Finally, we should note that, by combining (21) and (22), we Bearing (25) in mind, we have get 𝜌̇𝑒 +𝑛𝐻(𝜌𝑒 +𝑝𝑒)=0.Itmeansthat(21)and(22)are ̇ ̇ 𝑑̃𝑟𝐴 8𝜋 8𝜋 only valid if the 𝜌𝑒 +𝑛𝐻(𝜌𝑒 +𝑝𝑒)=0and 𝜌 + 𝑛𝐻(𝜌 + + 𝑑𝜌 = 𝐻 (𝜌+𝑝) 𝑑𝑡. 3 𝑛 (𝑛−1) 𝑒 𝑛−1 (27) 𝑝) = 0 conditions are simultaneously available. These results ̃𝑟𝐴 demonstrate that there is no energy exchange between the −𝑇 geometrical and material fluids. Multiplying both sides of the above equation by ( ), one obtains 𝑑̃𝑟 8𝜋 4.1. Noninteracting Case. The terms due to the curvature (−𝑇) ( 𝐴 + 𝑑𝜌 ) formally play the role of a further source term in the field 3 𝑛 (𝑛−1) 𝑒 ̃𝑟𝐴 equations whose effect is the same as that of an effective fluid (28) of purely geometrical origin. Indeed, since such terms modify 8𝜋 1 ̃𝑟̇ = 𝐻(𝜌+𝑝)𝑑𝑡[ (1 − 𝐴 )] . the Einstein theory, they lead to changing the Bekenstein rela- 𝑛−1 2𝜋̃𝑟𝐴 2𝐻̃𝑟𝐴 tion for the entropy of black hole horizon [46–49], which is in agreement with the result obtained by applying the first law Assume that the total energy content of the universe inside 𝑛 of thermodynamics on the black hole horizon [50]. In order an 𝑛-sphere of radius ̃𝑟𝐴 is 𝐸=𝜌𝑉,where𝑉=Ω𝑛̃𝑟𝐴 is to study the thermodynamics of apparent horizon of FRW thevolumeenvelopedbyan𝑛-dimensional sphere. Taking universe in quasi-topological gravity, authors in [51] assumed differential form of the total energy, after using the continuity that the black hole entropy expression, derived in [46–50], equation (25), we obtain is also valid for the apparent horizon of FRW universe. In 𝑛−1 𝑛 addition, by applying the UFL of thermodynamics on the 𝑑𝐸 = 𝑛Ω𝑛𝜌̃𝑟𝐴 𝑑̃𝑟𝐴 −𝑛𝐻Ω𝑛̃𝑟𝐴 (𝜌 + 𝑝) 𝑑𝑡, (29) apparent horizon, they could get the Friedmann equation in the quasi-topological gravity [51]. In fact, their recipe is which leads to a way of proposing an expression for the apparent horizon 𝑑𝐸 𝜌𝑑̃𝑟 (𝜌 + 𝑝) 𝑑𝑡= − + 𝐴 . entropy instead of a way for deriving the corresponding 𝑛 (30) 𝑛𝐻Ω𝑛̃𝑟 𝐻̃𝑟𝐴 entropy. Therefore, the inverse of their recipe may indeed be 𝐴 considered theoretically as a more acceptable way of getting Substituting (30) into (28), simple calculations lead to the apparent horizon entropy. Finally, based on approach [51], 𝑑̃𝑟 8𝜋 it seems that their proposal for the apparent horizon entropy 𝑇( 𝐴 + 𝑑𝜌 ) is independent of any interaction between the cosmos sectors. 3 𝑛 (𝑛−1) 𝑒 ̃𝑟𝐴 It is also worthy to note that the mutual interaction between 𝑒 𝑚 (31) the geometrical (𝑇 ) and nongeometrical (𝑇 ) sectors of 4𝑑𝐸 2(𝜌−𝑝)𝑑̃𝑟 𝜇] 𝜇] = − 𝐴 , cosmos in various modified theories of gravity affects the 𝑛+1 2 𝑛 (𝑛−1) Ω𝑛̃𝑟𝐴 (𝑛−1) ̃𝑟𝐴 apparent horizon entropy [33–36]. In fact, depending on the scales, it is such a curvature fluid which may play the role and consequently of dark energy. In Einstein general relativity framework, a 𝑑̃𝑟 8𝜋 dark energy candidate and its interaction with other parts 𝑇( 𝐴 + 𝑑𝜌 ) 3 𝑛 (𝑛−1) 𝑒 of cosmos affect the Bekenstein entropy [28–32]. Therefore, ̃𝑟𝐴 the results obtained in [33–36] are in line with those of [28– (32) 4 32]. In order to get the effects of mutual interaction between = 𝑑𝐸 − 𝑊𝑑𝑉 , 𝑛+1 [ ] the geometrical and nongeometrical parts of cosmos, we 𝑛 (𝑛−1) Ω𝑛̃𝑟𝐴 need to establish a recipe to find out the horizon entropy 𝑛−1 by starting from the Friedmann equations and applying the wherewehaveused𝑑𝑉 =𝑛 𝑛Ω ̃𝑟𝐴 𝑑̃𝑟𝐴.Inthisequationthe UFL of thermodynamics on the apparent horizon of FRW work density 𝑊 = (𝜌 − 𝑝)/2 is regarded as the work done 6 Advances in High Energy Physics

when the apparent horizon radius changes from ̃𝑟𝐴 to ̃𝑟𝐴 + Consequently the modified entropy on the event horizon has 𝑑̃𝑟𝐴.TheClausiusrelationis[32] the explicit form 𝑚 𝑇𝑑𝑆𝐴 =𝛿𝑄 =𝐴Ψ, (33) 𝐴 𝑚 (𝑛−1) 𝜇̂ 𝑙2𝑖−2 𝑚 𝑆 = ∑𝑖 𝑖 +𝑆 , where 𝛿𝑄 is the energy flux crossing the horizon during the 𝐴 2𝑖−2 0 (40) 4 𝑖=1 (𝑛−2𝑖+1) ̃𝑟 universe expansion. Considering (13), (32), and (33), we have 𝐴 𝑛 (𝑛−1) Ω ̃𝑟𝑛−2𝑑̃𝑟 whichiscompatiblewiththeresultsobtainedbyconsidering 𝑑𝑆 = 𝑛 𝐴 𝐴 +2𝜋Ω ̃𝑟𝑛+1𝑑𝜌 . (34) 𝐴 4 𝑛 𝐴 𝑒 the Hayward-Kodama temperature (36). As before, since entropy is not an absolute quantity, without loss of generality, Using(25)leadsto we can set 𝑆0 to zero. Setting 𝑛=3and 𝑖=1in the above 𝑛−2 result leads to the Bekenstein-Hawking entropy formula. 𝑛 (𝑛−1) Ω𝑛̃𝑟𝐴 𝑑̃𝑟𝐴 𝑑𝑆𝐴 = 4 (35) 4.2. Interacting Universe. In order to take the interaction −2𝑛𝐻𝜋Ω ̃𝑟𝑛+1 (𝜌 +𝑝)𝑑𝑡. 𝑛 𝐴 𝑒 𝑒 into account, for the quasi-topological gravity, consider the Now, by either inserting (21) into (34) or (22) into (35) and energy-momentum conservation law in the forms integrating the result, we get 𝜌+𝑛𝐻(𝜌+𝑝)=𝑄,̇ 𝑚 2𝑖−2 (41) 𝐴 (𝑛−1) 𝜇̂𝑖𝑙 𝑆𝐴 = ∑𝑖 +𝑆0, (36) ̇ 4 (𝑛−2𝑖+1) 2𝑖−2 𝜌𝑒 +𝑛𝐻(𝜌𝑒 +𝑝𝑒) =−𝑄, (42) 𝑖=1 ̃𝑟𝐴 for the horizon entropy. This result is in full agreement with where 𝑄 is the mutual interaction between different parts previous proposal in which authors assumed that the event of the cosmos. The latter means that (21) and (22) are not horizon entropy is extendable to the apparent horizon of simultaneously valid for an interacting universe. Using (26) FRW universe [51]. 𝑆0 is an integration constant and we can and (41), we get consider it zero without loss of generality. It is also useful to mention here that we considered the Clausius relation in 𝑑̃𝑟 8𝜋 8𝜋𝐻 𝑇𝑑𝑆 =𝛿𝑄𝑚 𝐴 + (𝑑𝜌 +𝑄𝑑𝑡) = (𝜌+𝑝) 𝑑𝑡. the form 𝐴 to obtain this result, while authors 3 𝑛 (𝑛−1) 𝑒 𝑛−1 (43) 𝑚 ̃𝑟𝐴 in [51] took into account the 𝑇𝑑𝑆𝐴 =−𝛿𝑄 form of the Clausius relation. This discrepancy is due to our different (−𝑇) way of defining the horizon temperature. We have used Multiplying both sides of the above equation by ,one the Hayward-Kodama definition of temperature (11), while obtains authors in [51] used its absolute value to avoid attributing 𝑑̃𝑟 8𝜋 negative temperatures on the apparent horizon and thus the 𝑇( 𝐴 + (𝑑𝜌 +𝑄𝑑𝑡)) 3 𝑛 (𝑛−1) 𝑒 Hawking radiation. ̃𝑟𝐴 (44) 2𝐻 (𝜌 +𝑝) 2(𝜌+𝑝) The Cai-Kim Approach.Here,wefocusontheCai-Kim =− 𝑑𝑡+ 𝑎𝑑𝑟 , (𝑛−1) ̃𝑟 2 𝐴 approach to get the effects of geometrical fluid on the horizon 𝐴 (𝑛−1) ̃𝑟𝐴 entropy. When one applies the first law on the apparent horizon to calculate the surface gravity and thereby the ̇ wherewehaveused̃𝑟𝐴𝑑𝑡= 𝑎𝑑𝑟 𝐴 +𝑟𝐴𝐻𝑑𝑡. Multiplying the temperature and considers an infinitesimal amount of energy 𝑛(𝑛 − 1)Ω ̃𝑟𝑛+1/4 crossing the apparent horizon, the apparent horizon radius result by the factor 𝑛 𝐴 , we arrive at ̃𝑟𝐴 should be regarded to have a fixed value (𝑑𝑉=0)[59]. ̃ 𝑑̃𝑟 8𝜋 Bearing (17) together with 𝑑𝑟𝐴 =0approximation in mind, 𝑇( 𝐴 + (𝑑𝜌 +𝑄𝑑𝑡)) 3 𝑒 by using (25) and the Clausius relation in the form 𝑇𝑑𝑆𝐴 = ̃𝑟 𝑛 (𝑛−1) 𝑚 𝑚 𝐴 −𝛿𝑄 =−𝐴Ψ,where𝛿𝑄 is the energy flux crossing the 𝑑𝑡 𝑛 (𝑛−1) Ω ̃𝑟𝑛+1 horizon during the infinitesimal time interval ,weget ⋅( 𝑛 𝐴 ) (45) 4 𝑇𝑑𝑆𝐴 =−𝑉𝑑𝜌. (37) 𝜌+𝑝 𝜌+𝑝 Substituting 𝑑𝜌 from (26) into the above equation, we are led 𝑛 =𝑛Ω𝑛̃𝑟𝐴 [− 𝐻𝑑𝑡+ 𝑎𝑑𝑟𝐴]. to 2 2̃𝑟𝐴 𝑉 𝑛 (𝑛−1) 𝑑̃𝑟 𝑑𝑆 =− 𝑑𝜌 = 2𝜋Ω ̃𝑟𝑛+1 [ 𝐴 +𝑑𝜌] , 𝐴 𝑇 𝑛 𝐴 3 𝑒 (38) Simple calculations lead to 8𝜋̃𝑟𝐴 𝑛 𝑛 (𝑛−1) Ω ̃𝑟𝑛−2𝑑̃𝑟 wherewehaveused𝑉=Ω𝑛̃𝑟𝐴 and the Cai-Kim temperature 𝑛 𝐴 𝐴 𝑛+1 𝑇( +2𝜋Ω𝑛̃𝑟𝐴 (𝑑𝜌𝑒 +𝑄𝑑𝑡)) (𝑇 = 1/2𝜋̃𝑟𝐴)[44,59].Finally,weobtain 4 (46) 𝑛 (𝑛−1) Ω ̃𝑟𝑛−2𝑑̃𝑟 𝜌+𝑝 𝜌+𝑝 𝑑𝑆 = 𝑛 𝐴 𝐴 +2𝜋Ω ̃𝑟𝑛+1𝑑𝜌 . (39) =−𝐴𝐻 ̃𝑟 𝑑𝑡+ 𝐴𝑎 𝑑𝑟 . 𝐴 4 𝑛 𝐴 𝑒 2 𝐴 2 𝐴 Advances in High Energy Physics 7

The right-hand side of (46) is nothing but the energy flux Using(26)and(42),wehave 𝑚 (𝛿𝑄 )crossingtheapparenthorizon(16).BearingtheClaus- 𝑛 (𝑛−1) 𝑑̃𝑟 ius relation (33) in mind, we get 𝑑𝜌 − 𝑄𝑑𝑡= − 𝐴 +𝑛𝐻(𝜌 +𝑝) 𝑑𝑡. 3 𝑒 𝑒 (54) 𝑛−2 8𝜋̃𝑟𝐴 𝑛 (𝑛−1) Ω𝑛̃𝑟𝐴 𝑑̃𝑟𝐴 𝑑𝑆𝐴 = 4 (47) Substituting it into (53) leads to 𝑛+1 +2𝜋Ω𝑛̃𝑟𝐴 (𝑑𝜌𝑒 +𝑄𝑑𝑡). 𝑛−2 𝑛 (𝑛−1) Ω𝑛̃𝑟𝐴 𝑑̃𝑟𝐴 𝑑𝑆𝐴 = Now, use (42) to obtain 4 (55) 𝑛 (𝑛−1) Ω ̃𝑟𝑛−2𝑑̃𝑟 𝑛+1 𝑛 𝐴 𝐴 −2𝜋𝑛𝐻Ω𝑛̃𝑟𝐴 (𝜌𝑒 +𝑝𝑒)𝑑𝑡, 𝑑𝑆𝐴 = 4 (48) 𝑛+1 which is in full agreement with (48). −2𝜋𝑛𝐻Ω𝑛̃𝑟𝐴 (𝜌𝑒 +𝑝𝑒)𝑑𝑡, which leads to 5. Second Law of Thermodynamics 𝐴 𝑆 = −2𝜋𝑛Ω ∫ 𝐻̃𝑟𝑛+1 (𝜌 +𝑝)𝑑𝑡+𝑆 , 𝐴 4 𝑛 𝐴 𝑒 𝑒 0 (49) The time evolution of the entropy in a noninteracting universe governed by quasi-topological gravity is already which is in agreement with the results obtained by authors considered by authors in [51]. Now, we are interested in [33] for the Lovelock theory but in a different way. In fact, examining the validity of the second law of thermodynamics since in the interacting universes 𝜌𝑒 and 𝜌𝑒 +𝑝𝑒 cannot in an interacting universe. This law states that the horizon simultaneously meet (21) and (22), respectively, we can not entropy should meet the 𝑑𝑆𝐴/𝑑𝑡≥ 0 condition [62]. For this use them to integrate with the RHS of (47) and (48) to get an propose, from (48), we have expression for the horizon entropy. Now, consider a special 𝑛−2 ̇ situation in which 𝜌𝑒 satisfies (21); from (47) we get 𝑑𝑆𝐴 𝑛 (𝑛−1) Ω𝑛̃𝑟𝐴 ̃𝑟𝐴 𝑛+1 = −2𝜋𝑛𝐻Ω𝑛̃𝑟𝐴 (𝜌𝑒 +𝑝𝑒). (56) 𝑛−2 𝑑𝑡 4 𝑛 (𝑛−1) Ω𝑛̃𝑟𝐴 𝑑̃𝑟𝐴 𝑛+1 𝑑𝑆𝐴 = +2𝜋Ω𝑛̃𝑟𝐴 𝑑𝜌𝑒 4 (50) However, using (42) and (43), one gets 𝑛+1 +2𝜋Ω𝑛̃𝑟𝐴 𝑄𝑑𝑡. ̃𝑟̇ 8𝜋𝐻 8𝜋𝐻 𝐴 − (𝜌 +𝑝)= (𝜌 + 𝑝) , 3 𝑛−1 𝑒 𝑒 𝑛−1 (57) In the above equation, the first two terms of the right-hand ̃𝑟𝐴 side are exactly the same as (39). In other words, which leads to the Raychaudhuri equation (𝑆 ) =(𝑆 ) +2𝜋Ω ∫ ̃𝑟𝑛+1𝑄𝑑𝑡. 𝐴 interaction 𝐴 non-interaction 𝑛 𝐴 (51) ̇ 8𝜋𝐻 3 ̃𝑟𝐴 = (𝜌+𝑝+𝜌𝑒 +𝑝𝑒) ̃𝑟𝐴. (58) It is crystal clear that the second term of RHS of this 𝑛−1 equation counts the effects of mutual interaction between Substituting it into (56), we arrive at the geometrical and nongeometrical fluids on the horizon entropy. Loosely speaking, the above expression for horizon 𝑑𝑆 𝑛+1 𝐴 =2𝜋𝑛𝐻Ω̃𝑟 (𝜌+𝑝) . (59) entropy is no longer the usual Bekenstein entropy formula; 𝑑𝑡 𝑛 𝐴 rather there are two correction terms which are due to higher- order curvature terms, arising from the quasi-topological The above result indicates that the second law of thermody- nature of gravity theory, along with the mutual interaction namics (𝑑𝑆𝐴/𝑑𝑡≥ 0 ) is valid for the apparent horizon under between the geometrical and nongeometrical parts. Note the condition (𝜌 + 𝑝) >0 and if we define the state parameter that, in this situation, due to (42), the 𝜌𝑒 +𝑝𝑒 term does not 𝜔=𝑝/𝜌, this law is valid whenever the state parameter obeys meet (22). As another example, consider a situation in which 𝜔≥−1. 𝜌𝑒 +𝑝𝑒 satisfies (22), which also means that, due to (42), 𝜌𝑒 does not obey (21). For this case, by combining (42) and (47) 6. Summary and Conclusions and integrating the result, we obtain After giving a brief review of the quasi-topological gravity (𝑆𝐴)interaction =(𝑆𝐴)non interaction , (52) - theory, by considering a FRW universe, we pointed out to the which means that the mutual interaction 𝑄 does not affect the corresponding Friedmann equation in this modified gravity horizon entropy in this special case. theory. In Section 2, we mentioned the apparent horizon ofFRWuniverseasthepropercausalboundary,itssurface The Cai-Kim Approach.Therestofthissectionisdevotedto gravity, and the Hayward-Kodama temperature of this hyper- achieving (48) by using the Cai-Kim approach [44, 59]. For surface. Motivated by recent works on the thermodynamics an interacting universe the UFL is in the form [31] of horizon of FRW universe in some modified gravity theories 𝑉 [33–36], we considered the terms other than Einstein tensor 𝑑𝑆 =− (𝑑𝜌 − 𝑄𝑑𝑡) . (53) 𝐴 𝑇 asafluidwithenergydensity𝜌𝑒 and pressure 𝑝𝑒. In fact, since 8 Advances in High Energy Physics these terms may play the role of dark energy and because a Competing Interests dark energy candidate may also affect the horizon entropy [28–32], such justification at least is not forbidden. Then, The authors declare that they have no competing interests. we showed that if we use either the Hayward-Kodama or the Cai-Kim temperature, the same result for the horizon Acknowledgments entropy is obtainable ((36) and (40)) which is in agreement with previous work [50], in which authors, by using different This work has been supported financially by Research Insti- definitions for the horizon temperature and generalizing the tute for Astronomy & Astrophysics of Maragha (RIAAM). black hole entropy to the cosmological horizon setup, could propose the same expression as ours for the apparent horizon References entropy in quasi-topological gravity. Moreover, it is useful to [1] A. G. Riess, A. V. Filippenko, P. Challis et al., “Observational note that (36) is valid only if there is no energy-momentum 𝑇𝑒 evidence from supernovae for an accelerating universe and a exchange between the geometrical ( 𝜇]) and nongeometrical cosmological constant,” The Astronomical Journal,vol.116,pp. 𝑚 (𝑇𝜇])fluids. 1009–1038, 1998. Finally, we generalized our investigation to an interacting [2] S. Perlmutter, G. Aldering, G. Goldhaber et al., “Measurements Ω Λ FRW universe with arbitrary curvature parameter (𝑘)in of and from 42 high-redshift supernovae,” The Astrophysical Journal,vol.517,no.2,article565,1999. quasi-topological gravity. We got a relation for the horizon [3]R.A.Knop,G.Aldering,R.Amanullahetal.,“Newconstraints entropy (see (49)) which showed the effects of an energy- on Ω𝑀, ΩΛ,and𝑤 from an independent set of 11 high-redshift momentum exchange between the geometrical and nongeo- supernovae observed with the Hubble Space Telescope,” The metrical fluids on the apparent horizon entropy. Thereinafter, Astrophysical Journal,vol.598,no.1,article102,2003. 𝜌 westudiedthespecialcaseinwhich 𝑒 meets (21) to perceive [4] P.deBernardis,P.A.R.Ade,J.J.Bocketal.,“AflatUniversefrom clearly the effects of such mutual interaction on the horizon high-resolution maps of the cosmic microwave background entropy. We got (51) for the horizon entropy, which showed radiation,” Nature,vol.404,pp.955–959,2000. two terms besides the Bekenstein entropy due to higher-order [5] S. Nojiri and S. D. Odintsov, “The new form of the equation of curvature terms, arising from the quasi-topological nature state for dark energy fluid and accelerating universe,” Physics of model, and mutual interaction between the geometrical Letters B,vol.639,no.3-4,pp.144–150,2006. and nongeometrical fluids. Our investigation also showed [6] M. Li, X. D. Li, S. Wang, and Y. Wang, “Dark energy,” The Universe,vol.1,no.4,pp.24–45,2013. that whenever 𝜌𝑒 +𝑝𝑒 satisfies (22), the mutual interaction 𝑄 does not affect the horizon entropy and therefore the result [7]K.Bamba,S.Capozziello,S.Nojiri,andS.D.Odintsov,“Dark ofnoninteractingcaseisobtainable.Wehavealsopointed energy cosmology: the equivalent description via different theoretical models and cosmography tests,” Astrophysics and out the validity of second law of thermodynamics in an Space Science,vol.342,no.1,pp.155–228,2012. interacting case and found out that, independent of curvature [8] S. Nojiri and S. D. Odintsov, “Unified cosmic history in modi- 𝑘 𝜌+𝑝 ≥0 parameter ( ), whenever ,thesecondlawis fied gravity: from F(R) theory to Lorentz non-invariant models,” obtainable. Physics Reports, vol. 505, no. 2–4, pp. 59–144, 2011. Our study shows that, in quasi-topological gravity theory, [9] S. Capozziello and V.Faraoni, Beyond Einstein Gravity,Springer, a geometrical fluid with energy density 𝜌𝑒 and pressure New York, NY, USA, 2011. 𝑝𝑒, independent of its nature, affects the apparent horizon [10] F. S. N. Lobo, “Beyond Einstein’s general relativity,” Journal of entropy as Physics: Conference Series, vol. 600, no. 1, Article ID 012006, 2015. 𝐴 𝑛+1 [11] W. Zimdahl, D. Pavon,´ and L. P. Chimento, “Interacting quin- 𝑆𝐴 = −2𝜋𝑛Ω𝑛 ∫ 𝐻̃𝑟 (𝜌𝑒 +𝑝𝑒) 𝑑𝑡, (60) 4 𝐴 tessence,” Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics,vol.521,no.3-4,pp.133–138,2001. which is in full agreement with other studies [28–36]. Since [12] L. P. Chimento, A. S. Jakubi, D. Pavon,´ and W. Zimdahl, “Inter- such correction in quasi-topological gravity has geometrical acting quintessence solution to the coincidence problem,” Physi- nature, it is indeed inevitable. Moreover, the effects of mutual cal Review D,vol.67,ArticleID083513,2003. interaction between the dark energy candidate and other [13] S. del Campo, R. Herrera, and D. Pavon,´ “Late universe expan- parts of cosmos are stored in the second term of RHS of sion dominated by domain walls and dissipative dark matter,” this equation. Indeed, based on our approach, one may Physical Review D,vol.70,no.4,ArticleID043540,2004. find the apparent horizon entropy of FRW universe with [14] D. Pavon´ and W. Zimdahl, “Holographic dark energy and cos- arbitrary curvature parameter (𝑘) in modified theories of mic coincidence,” Physics Letters, Section B: Nuclear, Elementary gravity by interpreting the terms other than Einstein tensor Particle and High-Energy Physics,vol.628,no.3-4,pp.206–210, 2005. in the corresponding Friedmann equation as a geometrical [15] G. Olivares, F. Atrio-Barandela, and D. Pavon,´ “Observa- fluid and applying the UFL of thermodynamics on the tional constraints on interacting quintessence models,” Physical apparenthorizon.Attheend,weshouldstressthat,inthe Review D,vol.71,no.6,ArticleID063523,pp.1–7,2005. absence of a mutual interaction, the results of interacting [16] G. Olivares, F. Atrio-Barandela, and D. Pavon,´ “Matter density universes in quasi-topological theory converge to those of the perturbations in interacting quintessence models,” Physical noninteracting case. Review D,vol.74,ArticleID043521,2006. Advances in High Energy Physics 9

[17] B. Wang, Y. Gong, and E. Abdalla, “Transition of the dark [36] S. Saha, S. Mitra, and S. Chakraborty, “A study of universal energy equation of state in an interacting holographic dark thermodynamics in massive gravity: modified entropy on the energy model,” Physics Letters B,vol.624,no.3-4,pp.141–146, horizons,” General Relativity and Gravitation,vol.47,no.4,Art. 2005. 38,17pages,2015. [18] B. Wang, Ch.-Y. Lin, and E. Abdalla, “Constraints on the inter- [37] J. Polchinski, String Theory I & II, Cambridge University Press, acting holographic dark energy model,” Physics Letters B,vol. Cambridge, UK, 1998. 637, no. 6, pp. 357–361, 2006. [38] J. M. Maldacena, “The large N limit of superconformal field [19] B. Wang, J. Zang, C.-Y.Lin, E. Abdalla, and S. Micheletti, “Inter- theories and supergravity,” Advances in Theoretical and Math- acting dark energy and dark matter: observational constraints ematical Physics,vol.2,no.2,pp.231–252,1998. from cosmological parameters,” Nuclear Physics B,vol.778,no. [39] J. Maldacena, “The large-N limit of superconformal field 1-2,pp.69–84,2007. theories and supergravity,” International Journal of Theoretical [20] B. Wang, C.-Y. Lin, D. Pavon,´ and E. Abdalla, “Thermodynam- Physics, vol. 38, no. 4, p. 1113, 1999. ical description of the interaction between holographic dark [40] L. Randall and R. Sundrum, “Large mass hierarchy from a small energy and dark matter,” Physics Letters B,vol.662,no.1,pp. extra dimension,” Physical Review Letters,vol.83,no.17,pp. 1–6, 2008. 3370–3373, 1999. [21] S. Das, P.S. Corasaniti, and J. Khoury, “Superacceleration as the [41] L. Randall and R. Sundrum, “An alternative to compactifica- signature of a dark sector interaction,” Physical Review D,vol. tion,” Physical Review Letters,vol.83,no.23,pp.4690–4693, 73, no. 8, Article ID 083509, 2006. 1999. [22] L. Amendola, “Coupled quintessence,” Physical Review D,vol. [42] D. Lovelock, “The einstein tensor and its generalizations,” 62, no. 4, Article ID 043511, 2000. Journal of the Mathematical Physics,vol.12,no.3,p.498,1971. [23] L. Amendola and D. Tocchini-Valentini, “Stationary dark [43] A. Paranjape, S. Sarkar, and T. Padmanabhan, “Thermodynamic energy: the present universe as a global attractor,” Physical route to field equations in Lanczos-LOVelock gravity,” Physical Review D,vol.64,no.4,ArticleID043509,2001. Review D,vol.74,no.10,ArticleID104015,9pages,2006. [24] L. Amendola and C. Quercellini, “Tracking and coupled dark [44] R. G. Cai and S. P.Kim, “First law of thermodynamics and Fried- energy as seen by the Wilkinson Microwave Anisotropy Probe,” mann equations of Friedmann-Robertson-Walker universe,” Physical Review D,vol.68,no.2,ArticleID023514,2003. Journal of High Energy Physics, vol. 2005, no. 2, article 050, 2005. [25] L. Amendola, S. Tsujikawa, and M. Sami, “Phantom damping [45] J. Oliva and S. Ray, “A new cubic theory of gravity in five of matter perturbations,” Physics Letters, Section B: Nuclear, dimensions: black hole, Birkhoff’s theorem and C-function- Elementary Particle and High-Energy Physics, vol. 632, no. 2-3, function,” Classical and Quantum Gravity,vol.27,no.22,Article pp.155–158,2006. ID225002,16pages,2010. [26] M. Roos, Introduction to Cosmology, John Wiley & Sons, Chich- ester, UK, 2003. [46] R. C. Myers and B. Robinson, “Black holes in quasi-topological gravity,” JournalofHighEnergyPhysics,vol.2010,article67,2010. [27] M. Jamil, E. N. Saridakis, and M. R. Setare, “Thermodynamics of dark energy interacting with dark matter and radiation,” [47] M.H.Dehghani,A.Bazrafshan,R.B.Mann,M.R.Mehdizadeh, Physical Review D,vol.81,no.2,ArticleID023007,6pages,2010. M. Ghanaatian, and M. H. Vahidinia, “Black holes in (quartic) quasitopological gravity,” Physical Review D,vol.85,no.10, [28] C.-J. Feng, X.-Z. Li, and X.-Y. Shen, “Thermodynamic of the Article ID 104009, 12 pages, 2012. ghost dark energy universe,” Modern Physics Letters A,vol.27, no.31,ArticleID1250182,11pages,2012. [48] M. H. Dehghani and M. H. Vahidinia, “Surface terms of qua- sitopological gravity and thermodynamics of charged rotating [29] A. Sheykhi, “Entropy and thermodynamics of ghost dark black branes,” Physical Review D,vol.84,no.8,ArticleID energy,” Canadian Journal of Physics,vol.92,no.6,pp.529–532, 084044, 2011. 2014. [30] H. Moradpour, M. T. Mohammadi Sabet, and A. Ghasemi, [49]W.G.Brenna,M.H.Dehghani,andR.B.Mann,“Quasitopo- “Thermodynamic analysis of universes with the initial and final logical Lifshitz black holes,” Physical Review D,vol.84,no.2, de Sitter eras,” Modern Physics Letters A,vol.30,no.31,Article Article ID 024012, 12 pages, 2011. ID 1550158, 2015. [50] A. Sheykhi, M. H. Dehghani, and R. Dehghani, “Horizon ther- [31] H. Moradpour and M. T. Mohammadi Sabet, “Thermodynamic modynamics and gravitational field equations in quasi-topo- descriptions of polytropic gas and its viscous type as dark energy logical gravity,” General Relativity and Gravitation,vol.46,no. candidates,” Canadian Journal of Physics,vol.94,no.3,pp.334– 4, pp. 1–14, 2014. 341, 2016. [51] M. H. Dehghani, A. Sheykhi, and R. Dehghani, “Thermody- [32] H. Ebadi and H. Moradpour, “Thermodynamics of universe namics of quasi-topological cosmology,” Physics Letters B,vol. with a varying dark energy component,” International Journal 724, no. 1–3, pp. 11–16, 2013. of Modern Physics D, vol. 24, no. 14, Article ID 1550098, 2015. [52] S. A. Hayward, “Unified first law of black-hole dynamics and [33] S. Mitra, S. Saha, and S. Chakraborty, “Universal thermodynam- relativistic thermodynamics,” Classical and Quantum Gravity, ics in different gravity theories: conditions for generalized sec- vol. 15, no. 10, pp. 3147–3162, 1998. ond law of thermodynamics and thermodynamical equilibrium [53] D. Bak and S.-J. Rey, “Cosmic holography,” Classical and Quan- on the horizons,” Annals of Physics,vol.355,pp.1–20,2015. tum Gravity,vol.17,no.15,pp.L83–L89,2000. [34] S. Mitra, S. Saha, and S. Chakraborty, “Astudy of universal ther- [54] R.-G. Cai and L.-M. Cao, “Unified first law and the thermody- modynamics in Lanczos-LOVelock gravity,” General Relativity namics of the apparent horizon in the FRW universe,” Physical and Gravitation,vol.47,article69,2015. Review D,vol.75,no.6,ArticleID064008,11pages,2007. [35] S. Mitra, S. Saha, and S. Chakraborty, “A study of universal [55] R.-G. Cai and L.-M. Cao, “Thermodynamics of apparent hori- thermodynamics in brane world scenario,” Advances in High zon in brane world scenario,” Nuclear Physics. B,vol.785,no. Energy Physics,vol.2015,ArticleID430764,9pages,2015. 1-2, pp. 135–148, 2007. 10 Advances in High Energy Physics

[56] D. W. Tian and I. Booth, “Apparent horizon and gravitational thermodynamics of the universe: solutions to the temperature and entropy confusions and extensions to modified gravity,” Physical Review D,vol.92,ArticleID024001,2015. [57] P.Binetruy´ and A. Helou, “The apparent Universe,” Classical and Quantum Gravity, vol. 32, no. 20, Article ID 205006, 12 pages, 2015. [58] A. Helou, “Dynamics of the cosmological apparent hori- zon: surface gravity & temperature,” http://arxiv.org/abs/1502 .04235#. [59] R. G. Cai, L. M. Cao, and Y. P. Hu, “Hawking radiation of an apparent horizon in a FRW universe,” Classical and Quantum Gravity,vol.26,no.15,ArticleID155018,2009. [60] S. A. Hayward, “Gravitational energy in spherical symmetry,” Physical Review D,vol.53,no.4,pp.1938–1949,1996. [61] S. A. Hayward, “Energy conservation for dynamical black holes,” Physical Review Letters, vol. 93, no. 25, Article ID 251101, 4pages,2004. [62] S. W. Hawking, “Gravitational radiation from colliding black holes,” Physical Review Letters,vol.26,no.21,pp.1344–1346, 1971. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 5101389, 8 pages http://dx.doi.org/10.1155/2016/5101389

Research Article The Minimal Length and the Shannon Entropic Uncertainty Relation

Pouria Pedram

Department of Physics, Islamic Azad University, Science and Research Branch, Tehran 1477893855, Iran

Correspondence should be addressed to Pouria Pedram; [email protected]

Received 5 January 2016; Accepted 13 March 2016

Academic Editor: Barun Majumder

Copyright © 2016 Pouria Pedram. 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.

In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation 2 relation [𝑋, 𝑃] = 𝑖ℏ(1 +𝛽𝑃 ),where𝛽 is the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, Bialynicki-Birula, and Mycielski (BBM) depends on both the algebra and the used representation, we show that using the formally self-adjoint representation, that is, 𝑋=𝑥and 𝑃=tan(√𝛽𝑝)/√𝛽,where[𝑥, 𝑝] ,theBBM= 𝑖ℏ inequality is still valid in the form 𝑆𝑥 +𝑆𝑝 ≥1+ln 𝜋 as well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.

1. Introduction relativistic quantum mechanics [20–22], and cosmological problems [23–26]. On the other hand, in the classical domain, The existence of a minimal observable length proportional to deformed classical systems in phase space [27, 28], Keplerian √ 3 −35 the Planck length ℓP = 𝐺ℏ/𝑐 ≈10 mismotivatedby orbits [29], composite systems [30], and the thermostatistics various proposals of quantum gravity such as string theory, [31, 32] have been investigated in the presence of the minimal loop quantum gravity, noncommutative geometry, and black- length. hole [1–3]. Indeed, several schemes have been established In the last decade, many applications of information to investigate the effects of the minimal length range from theoretic measures such as entropic uncertainty relations, as astronomical observations [4, 5] to table-top experiments [6]. alternatives to Heisenberg uncertainty relation, appeared in In particular, a measurement method is proposed recently various quantum mechanical systems [33–48]. The concept to detect this fundamental length scale which is based on of statistical complexity was introduced by Shannon in 1948 the possible deviations from ordinary quantum commutation [49], where the term uncertainty can be considered as a relation at the Planck scale within the current technology [6]. measure of the missing information. In particular, it is shown Deformed commutation relations have attracted much that the information entropies such as Shannon entropy may attention in recent years and several problems range from be used to replace the well-known quantum mechanical classical to quantum mechanical systems have been studied uncertainty relation. The first entropic uncertainty relation exactly or approximately in the context of the general- for position and momentum observables was proposed ized (gravitational) uncertainty principle (GUP). Among by Hirschmann [50] and is later improved by Beckner, these investigations in quantum domain, we can mention Bialynicki-Birula, and Mycielski (BBM) [51–54]. the harmonic oscillator [7–9], Coulomb potential [10–12], In the context of the generalized uncertainty principle, singular inverse square potential [13], coherent states [14], there exists two questions. (i) Are the wave functions in Dirac oscillator [15], Lamb’s shift, Landau levels, tunneling position space and momentum space related by the Fourier current in scanning tunneling microscope [16], ultracold transform in arbitrary GUP algebra in the form [𝑋, 𝑃] = neutrons in gravitational field [17, 18], Casimir effect [19], 𝑖ℏ𝑓(𝑋, 𝑃)? (ii) If in a particular algebra these wave functions 2 Advances in High Energy Physics

2 arenotrelatedbytheFouriertransform,istheBBMinequal- which results in Δ𝑋Δ𝑃 ≥ (ℏ/2)[1 + 𝛽(Δ𝑃) ] (generalized ity still valid? Note that the validity of the BBM entropic uncertainty principle) and for 𝛽→0the well-known uncertainty relation depends on both the algebra and the commutation relation in ordinary quantum mechanics is used representation. For instance, consider the modified recovered. Notice that, Δ𝑋 cannot take arbitrarily small 2 2 commutation relation in the form [𝑋, 𝑃] = 𝑖ℏ(1 +𝛽𝑃 +𝛼𝑋 ) values and the absolutely smallest uncertainty in position is which implies a minimal length and a minimal momentum given by (Δ𝑋)min =ℏ√𝛽. proportional to ℏ√𝛽 and ℏ√𝛼,respectively.ThisformofGUP Now, consider the formally self-adjoint representation [9] has no formally self-adjoint representation in the form 𝑋=𝑥 and 𝑃=𝑓(𝑝). So the momentum space and coordinate space 𝑋=𝑥, wave functions are not related by the Fourier transform in tan (√𝛽𝑝) (3) any representation and the BBM uncertainty relation does not 𝑃= , hold in this framework. √𝛽 The well-known Robertson uncertainty principle for two noncommuting observables is given by where [𝑥, 𝑝] ,= 𝑖ℏ −𝜋/2√𝛽<𝑝<𝜋/2√𝛽, and it exactly 1 󵄨 󵄨 satisfies (2). In this representation, the ordinary nature of the Δ𝐴Δ𝐵 ≥ 󵄨⟨𝜓 |[𝐴,]| 𝐵 𝜓⟩󵄨 , 2 󵄨 󵄨 (1) position operator is preserved and the inner product of states takes the following form: where [𝐴, 𝐵] denotes the commutator of 𝐴 and 𝐵 and Δ𝐴 and Δ𝐵 +𝜋/2√𝛽 are their dispersions. However, this uncertainty relation ∗ suffers from two serious shortcomings [55, 56]. (i) For ⟨𝜓 | 𝜙⟩ = ∫ 𝜓 (𝑝) 𝜙 (𝑝) 𝑑𝑝. (4) −𝜋/2√𝛽 two noncommuting observables of a finite 𝑁-dimensional † Hilbert space, since the right-hand side of (1) depends on Note that although the position operator obeys 𝑋 = the wave function 𝜓, it is not a fixed lower bound. Indeed, † 𝑋 = 𝑖ℏ𝜕/𝜕𝑝 in momentum space, we have D(𝑋) ⊂ D(𝑋 ). if 𝜓 is the eigenstate of the observable 𝐴 or 𝐵,theright- So 𝑋 is merely symmetric and it is not a true self-adjoint hand side of (1) vanishes and there is no restriction on Δ𝐴 or operator. However, based on the von Neumann’s theorem, Δ𝐵 by this uncertainty relation. (ii) The dispersions cannot † for the momentum operator we obtain 𝑃=𝑃 and D(𝑃) = be considered as suitable measures for the uncertainty of † D(𝑃 )={𝜙∈Dmax(R)} [9]. Thus, 𝑃 is indeed a self-adjoint two complementary observables with continuous probability operator. Moreover, the scalar product and the completeness densities. This problem is more notable when their corre- relation read sponding probability densities contain several sharp peaks. 󸀠 󸀠 Among various proposals for the uncertainty relations that ⟨𝑝 |𝑝⟩=𝛿(𝑝−𝑝), (5) arenotsufferedfromtheseshortcomings,wecanmentionthe information-theoretical entropy instead of the dispersions +𝜋/2√𝛽 󵄨 󵄨 which is a proper measure of the uncertainty. ∫ 󵄨𝑝⟩⟨𝑝󵄨 𝑑𝑝 = 1. (6) −𝜋/2√𝛽 In this paper, we study the effects of the minimal length on the entropic uncertainty relation. In this scenario, the In momentum space, the eigenfunctions of the position position and momentum operators obey the modified com- 2 operator are given by the solutions of the eigenvalue equation mutation relation [𝑋, 𝑃] = 𝑖ℏ(1 +𝛽𝑃 ),where𝛽 is the deformation parameter. Using formally self-adjoint represen- 𝑋𝑢𝑥 (𝑝) =𝑥𝑢𝑥 (𝑝) . (7) tation of the algebra, we show that the coordinate space and momentum space wave functions are related by the Fourier Here, 𝑢𝑥(𝑝) = ⟨𝑝 |𝑥⟩ which can be expressed as transform and consequently the BBM inequality is preserved. 1 𝑖𝑝 However, as we will show, in the quasiposition representation 𝑢 (𝑝) = (− 𝑥) . 𝑥 √ exp ℏ (8) themomentumspaceandquasipositionspacewavefunctions 2𝜋 are not related by the Fourier transformation and the BBM Now, using (6) and (8), coordinate space wave function can inequality does not hold. As an application, we obtain the be written as generalized Schrodinger¨ equation for the harmonic oscillator +𝜋/2√𝛽 and exactly solve the corresponding differential equation in 1 𝑖𝑝𝑥/ℏ 𝜓 (𝑥) = ∫ 𝑒 𝜙(𝑝)𝑑𝑝. (9) momentum space. Then, we find information entropies for √2𝜋 −𝜋/2√𝛽 the two lowest energy eigenstates and explicitly show the validityoftheBMMinequalityinthepresenceoftheminimal Moreover, 𝜙(𝑝) is given by the inverse Fourier transform of length. the coordinate space wave function; namely,

+∞ 1 −𝑖𝑝𝑥/ℏ 2. The Generalized Uncertainty Principle 𝜙(𝑝)= ∫ 𝑒 𝜓 (𝑥) 𝑑𝑥. (10) √2𝜋 −∞ In one-dimension, the deformed commutation relation reads [7] To this end, by taking 𝜙(𝑝) =0 for |𝑝| > 𝜋/2√𝛽 we can [𝑋, 𝑃] =𝑖ℏ(1+𝛽𝑃2), formally extend the domain of the momentum integral (9) (2) to −∞ < 𝑝 < ∞ without changing the coordinate space Advances in High Energy Physics 3 wave function 𝜓(𝑥). Therefore, 𝜓(𝑥) is the Fourier transform representation, the relation between the momentum space of 𝜙(𝑝). Now, using the Babenko-Beckner inequality [51, 52] wave functions and quasiposition wave functions is given by and following Bialynicki-Birula and Mycielski [53] we obtain 𝜓 (𝜉) (see [54] for details)

+∞ 𝑆 +𝑆 ≥1+ 𝜋, 2√𝛽 1 −1 (19) 𝑥 𝑝 ln (11) = √ ∫ 𝑒𝑖𝜉 tan (√𝛽𝑝)/ℏ√𝛽𝜓(𝑝)𝑑𝑝. 3/2 𝜋 −∞ (1 + 𝛽𝑝2) where +∞ Thus, the quasiposition wave functions are not Fourier 󵄨 󵄨2 󵄨 󵄨2 𝑆𝑥 =−∫ 󵄨𝜓 (𝑥)󵄨 ln 󵄨𝜓 (𝑥)󵄨 𝑑𝑥, transform of the momentum space wave functions and the −∞ quasiposition entropy +∞ (12) 2 2 −1 󵄨 󵄨 󵄨 󵄨 2 𝑆𝑝 =−∫ 󵄨𝜙(𝑝)󵄨 ln 󵄨𝜙(𝑝)󵄨 𝑑𝑝, 𝑆 =−(8𝜋ℏ√𝛽) −∞ 𝜉

+∞ 𝜙(𝑝) =0 |𝑝| > 𝜋/2√𝛽 󸀠 −1 subject to for . Note that, in this ⋅ ∭ 𝑒𝑖(𝜉−𝜉 )(tan (√𝛽𝑝)/ℏ√𝛽)𝜓∗ (𝜉) 𝜓(𝜉󸀠) (20) representation, the expression for the entropic uncertainty −∞ relation is similar to the ordinary quantum mechanics. 󵄨 󵄨2 󵄨 󸀠 󵄨 󸀠 However, as we will see in the next section, the presence of ⋅ ln 󵄨𝜓(𝜉 )󵄨 𝑑𝑝 𝑑𝜉 𝑑𝜉 the minimal length modifies the Hamiltonian, its solutions, does not represent the continuous Shannon entropy and and the values of 𝑆𝑥 and 𝑆𝑝.Butsince𝜓(𝑥) and 𝜙(𝑝) are still related by the Fourier transform, the lower bound for the contains plenty of overcounting (because of the nonorthog- |𝜓𝑚𝑙⟩ entropic uncertainty relation will not be modified. onality of 𝜉 )thatshouldbeavoided.Theseresultsshow that the information entropies 𝑆𝑝 (16) and 𝑆𝜉 (20) are not proper measures of uncertainty. However, the information 3. Quasiposition Representation entropies (12) based on formally self-adjoint representation Another possible representation that exactly satisfies (2) is [7] do not suffer from these shortcomings and they can be considered as proper measures of uncertainty in the presence 2 of the minimal length. 𝑋𝜙 (𝑝) = 𝑖ℏ (1 +𝛽𝑝 )𝜕𝑝𝜙(𝑝), (13) 𝑃𝜙 (𝑝) = 𝑝𝜙 (𝑝). 4. Quantum Oscillator The corresponding scalar product and completeness relations The Hamiltonian of the harmonic oscillator is given by 2 2 2 read 𝐻=𝑃/2𝑚 + (1/2)𝑚𝜔 𝑋 . So the generalized Schrodinger¨ equation in momentum space using the representation (3) ⟨𝑝󸀠 |𝑝⟩=(1+𝛽𝑝2)𝛿(𝑝−𝑝󸀠), (14) reads 2 +∞ 1 󵄨 󵄨 1 𝑑2𝜙(𝑝) tan (√𝛽𝑝) ∫ 󵄨𝑝⟩ ⟨𝑝󵄨 𝑑𝑝 = 1. 2 2 2 󵄨 󵄨 (15) − 𝑚ℏ 𝜔 + 𝜙(𝑝)=𝐸𝜙(𝑝). (21) −∞ 1+𝛽𝑝 2 𝑑𝑝2 2𝑚𝛽

Now, since the measure in the integral (15) is not flat, the Using the new variable 𝜉=√𝛽𝑝,theaboveequationcanbe momentum space entropy written as 2 +∞ 𝑑 𝜙 (𝜉) 𝑉 1 󵄨 󵄨2 󵄨 󵄨2 + (𝜖− ) 𝜙 (𝜉) =0, 𝑆 =−∫ 󵄨𝜙(𝑝)󵄨 󵄨𝜙(𝑝)󵄨 𝑑𝑝, 2 2 (22) 𝑝 2 󵄨 󵄨 ln 󵄨 󵄨 (16) 𝑑𝜉 cos 𝜉 −∞ 1+𝛽𝑝 −2 where 𝑉=(𝑚𝛽ℏ𝜔) and 𝜖 = 𝑉(1 + 2𝑚𝛽𝐸).Now,taking has no proper form of the continuous Shannon entropy 𝜆 relation in this representation. 𝜙𝑛 (𝜉) =𝑃𝑛 (𝑠) cos 𝜉 (23) The quasiposition wave function is defined as [7] results in 𝜙 (𝜉) ≡⟨𝜓𝑚𝑙 |𝜙⟩, 𝑑2𝑃 (𝑠) 𝑑𝑃 (𝑠) 𝜉 (17) (1 − 𝑠2) 𝑛 −𝑠(1+2𝜆) 𝑛 𝑑𝑠2 𝑑𝑠 (24) where 2 +(𝜖−𝜆 )𝑃𝑛 (𝑠) =0, 2√𝛽 −1/2 −1 𝜓𝑚𝑙 (𝑝) = √ (1 + 𝛽𝑝2) 𝑒−𝑖(𝜉 tan (√𝛽𝑝)/ℏ√𝛽) (18) where 𝑠=sin 𝜉 and 𝜉 𝜋 𝑉=𝜆(𝜆−1) , denotes the maximal localization states. These states satisfy 𝑚𝑙 𝑚𝑙 1 4 (25) ⟨𝜓𝜉 |𝑋|𝜓𝜉 ⟩=𝜉and (Δ𝑋)|𝜓𝑚𝑙⟩ =ℏ√𝛽 and are not mutually 𝜉 𝜆= [1 + √1+ ]. 𝑚𝑙 𝑚𝑙 󸀠 2 𝑚2𝛽2ℏ2𝜔2 orthogonal; that is, ⟨𝜓𝜉󸀠 |𝜓𝜉 ⟩ =𝛿(𝜉−𝜉̸ ).Inthis 4 Advances in High Energy Physics

Itisknownthatthesolutionsoftheaboveequationfor𝜖= where 𝐻𝑛 denotes Hermite polynomials. So in this limit the 2 𝜆 (𝑛 + 𝜆) are given by the Gegenbauer polynomials 𝐶𝑛 (𝑠). solutions read Therefore, the exact solutions read −𝑝2/2𝜔 𝑒 𝐻𝑛 (𝑝/√𝜔) 𝜆 𝜆 lim 𝜙𝑛 (𝑝) = , 𝜙 (𝑝) = 𝑁 𝐶 ( (√𝛽𝑝)) (√𝛽𝑝) , 𝛽→0 (34) 𝑛 𝑛 𝑛 sin cos (26) √2𝑛𝑛!√𝜋𝜔

1 𝜂2 𝜂 1 which are normalized eigenstates of the ordinary harmonic 𝐸 =ℏ𝜔(𝑛+ )(√1+ + )+ ℏ𝜔𝜂𝑛2, 𝑛 2 4 2 2 oscillator as we have expected. (27) 𝑛=0,1,2,..., 5. Information Entropy where 𝜂=𝑚𝛽ℏ𝜔, 𝑁𝑛 is the normalization coefficient, and the The information entropies for the position and momentum Gegenbauer polynomials are defined as [57] spaces can be now calculated for the harmonic oscillator in the GUP framework using (12). In ordinary quantum [𝑛/2] Γ (𝑛−𝑘+𝜆) 𝐶𝜆 (𝑠) = ∑ (−1)𝑘 (2𝑠)𝑛−2𝑘 . mechanics and in the position space, the information entropy 𝑛 Γ (𝜆) 𝑘! (𝑛−2𝑘)! (28) 𝑘=0 can be obtained analytically for some quantum mechanical systems. However, since the momentum wave functions 𝛽→0 Note that for we obtain the ordinary energy spectrum are derived from the Fourier transform, the corresponding 𝐸 = ℏ𝜔(𝑛 + 1/2) of the harmonic oscillator; that is, 𝑛 .The momentum information entropies are rather difficult to Gegenbauer polynomials also satisfy the following useful obtain. For our case, as we will show, we find 𝑆𝑝 analytically formula [57]: for the two lowest energy states and obtain 𝑆𝑥 numerically. 1 1−2] ]−1/2 𝜋2 Γ (𝑛+2]) Thus, because of the difficulty in calculating 𝑆𝑥 and 𝑆𝑝,we ∫ (1 − 𝑥2) [𝐶] (𝑥)]2 𝑑𝑥 = , 𝑛 2 −1 𝑛! (𝑛+]) [Γ (])] only consider the two lowest energy eigenstates. (29) First consider the Fourier transform of the momentum 1 space ground state (26) which gives the following state in the Re ] >− . 2 position space (ℏ=1): 𝑠= 𝜉 𝑁 Since sin and 𝑛 is given by the normalization condition 𝜆 [Γ (𝜆)]2 sin [(𝜋/2) (𝑥/√𝛽 − 𝜆)] +𝜋/2√𝛽 2 √ ∫ |𝜙(𝜉)| 𝑑𝑝 =1 𝜓0 (𝑥) = −𝜋/2√𝛽 ,wefind 𝜋2√𝛽Γ (2𝜆) 𝑥/√𝛽−𝜆

√𝛽𝑛! (𝑛+𝜆) [Γ (𝜆)]2 1 𝑥 𝑁 = √ . (30) ⋅ 𝐹 [ ( −𝜆), (35) 𝑛 𝜋21−2𝜆Γ (𝑛+2𝜆) 2 1 2 √𝛽

The solutions can be also written in terms of relativistic 1 𝑥 Hermite polynomials using the relation [58] −𝜆, ( − (𝜆−2)),1]. 2 √𝛽 𝜆 √ 𝑛! 2 𝑛/2 𝜆 𝑢 𝐻𝑛 ( 𝜆𝑢) = (1 + 𝑢 ) 𝐶𝑛 ( ), (31) 𝑛=1 𝜆𝑛/2 √1+𝑢2 Also, for the first excited state ( )wehave

𝜆 𝜓1 (𝑥) where 𝐻𝑛 (𝑧) denotes relativistic Hermite polynomials. 𝜆 𝑛/2 𝐶 ( (√𝛽𝑝)) = (𝜆 / 2 Thus, we obtain 𝑛 sin (𝜆+1) [Γ (𝜆)] sin [(𝜋/2) (𝑥/√𝛽−(𝜆+1))] 𝑛 √ 𝜆 √ √ =𝑖𝜆√ { 𝑛!)cos ( 𝛽𝑝)𝐻𝑛 ( 𝜆 tan( 𝛽𝑝)) which results in 𝜋2√𝛽Γ (2𝜆 + 1) 𝑥/√𝛽−(𝜆+1) 𝜙 (𝑝) 𝑛 1 𝑥 ⋅ 2𝐹 [ ( − (𝜆+1)), 𝑁 𝜆𝑛/2 (32) 1 2 √𝛽 𝑛 𝜆+𝑛 √ 𝜆 √ √ = cos ( 𝛽𝑝)𝑛 𝐻 ( 𝜆 tan ( 𝛽𝑝)) . 𝑛! 1 𝑥 −𝜆, ( − (𝜆−1)),1] For the small values of the deformation parameter we have 2 √𝛽 (36) (𝑚=ℏ=1) 1 sin [(𝜋/2) (𝑥/√𝛽−(𝜆−1))] lim 𝜆= , 𝛽󳨀→0⇐⇒𝜆󳨀→∞, − 𝛽→0 𝜔𝛽 𝑥/√𝛽−(𝜆−1)

2 1 𝑥 𝜆+𝑛 𝑝 ⋅ 𝐹 [ ( − (𝜆−1)), lim cos (√𝛽𝑝) = exp (− ), 2 1 2 √𝛽 𝜆→∞ 2𝜔 (33) 𝑝 1 𝑥 lim √𝜆 tan (√𝛽𝑝) = , −𝜆, ( − (𝜆−3)),1]}. 𝜆→∞ √𝜔 2 √𝛽 𝜆 Figure 1 shows the resulting ground state and first excited lim 𝐻𝑛 (𝑧) =𝐻𝑛 (𝑧) , Ref. [58] , 𝜆→∞ state in position space for 𝛽 = {0,0.5,1}. Notice that, Advances in High Energy Physics 5

0.8 0.6

0.6 0.4

0.2 0.4 (x) 1 (x) 0.0 0 𝜓 −i𝜓 0.2 −0.2 −0.4 0.0 −0.6 −6 −4 −2 0462 −10 −50 5 10 x x

𝛽=0 𝛽=0 𝛽 = 0.5 𝛽 = 0.5 𝛽=1 𝛽=1

Figure 1: Plots of the position space wave functions for 𝑚=ℏ=𝜔=1and 𝑛=0,1.

n=0 n=1 0.00 0.00

−0.05 −0.05

−0.10 −0.10

−0.15 −0.15 (p) (p) s s

𝜌 −0.20 𝜌 −0.20

−0.25 −0.25

−0.30 −0.30

−0.35 −0.35 −1.5 −1.0 −0.5 0 0.5 1.0 1.5 −1.5 −1.0 −0.5 00.5 1.0 1.5 p p

𝛽=0 𝛽=0 𝛽 = 0.5 𝛽 = 0.5 𝛽=1 𝛽=1

Figure 2: Plots of the momentum space entropy densities for 𝑚=ℏ=𝜔=1and 𝑛=0,1.

for 𝛽→0, the solutions tend to the simple har- For the first excited state, 𝑆𝑝 is given by monicoscillatorwavefunctions;thatis,lim𝛽→0𝜓𝑛(𝑥) = 1 2 𝑆 = (1+𝜆) 𝐻 −𝜆𝐻 + √𝜋−2 √ 𝑛 −𝜔𝑥 /2 𝑝 𝜆+1 𝜆−1/2 ln √𝜔/2 𝑛!√𝜋𝑒 𝐻𝑛(√𝜔𝑥). For the ground state, the analytical expression for 𝑆𝑝 reads √𝛽Γ (𝜆+2) (39) − ( ), ln 2Γ (𝜆+1/2) √𝛽𝜆Γ (𝜆) 𝑆0 =𝜆𝐻 −𝜆𝐻 + √𝜋− ( ), 𝛽→0 𝑝 𝜆 𝜆−1/2 ln ln Γ (𝜆+1/2) (37) and for it reads

1 1 1 1 lim 𝑆𝑝 = lim 𝑆𝑥 = ln 𝜋+ln 2+𝛾− , 𝑛 𝛽→0 𝛽→0 2 2 (40) where 𝐻𝜆 denotes the harmonic number 𝐻𝑛 = ∑𝑘=1 1/𝑘.Itis easy to check that, for the small values of 𝛽, the momentum where 𝛾 ≈ 0.5772 is the Euler constant. information entropy tends to the ordinary harmonic oscilla- The information entropy densities are defined as 𝜌𝑠(𝑥) = tor information entropy; namely, 2 2 2 2 |𝜓(𝑥)| ln |𝜓(𝑥)| and 𝜌𝑠(𝑝) = |𝜙(𝑝)| ln |𝜙(𝑝)| .Thebehavior of 𝜌𝑠(𝑥) and 𝜌𝑠(𝑝) is illustrated in Figures 2 and 3 for 𝑛= 0 0 1 0, 1 lim 𝑆𝑝 = lim 𝑆𝑥 = (1+ln 𝜋) . and several values of the deformation parameter. Now, 𝛽→0 𝛽→0 2 (38) using the numerical values for 𝑆𝑥,weobtaintheleft-hand 6 Advances in High Energy Physics

n=0 n=1 0.00 0.00

−0.05 −0.05

−0.10 −0.10 −0.15 −0.15 (x) (x) s s 𝜌 −0.20 𝜌 −0.20

−0.25 −0.25

−0.30 −0.30

−0.35 −0.35 −4 −2 024 −4 −2 024 x x

𝛽=0 𝛽=0 𝛽 = 0.5 𝛽 = 0.5 𝛽=1 𝛽=1

Figure 3: Plots of the position space entropy densities for 𝑚=ℏ=𝜔=1and 𝑛=0,1.

Table 1: The numerical results that establish the BBM entropic momentum. Since this algebra has no formally self-adjoint uncertainty relation for 𝑚=ℏ=𝜔=1and various values of 𝛽. representation in the form 𝑋=𝑥and 𝑃=𝑓(𝑝),the momentum space and coordinate space wave functions are 𝑛𝛽𝑥 𝑆 𝑆𝑝 𝑆𝑥 +𝑆𝑝 1+ln 𝜋 not related by the Fourier transform and the BBM uncertainty 0.1 1.12153 1.02361 2.14515 2.14473 relationisnotvalidforthisformofGUP.Forthecaseof 0 0.5 1.30251 0.85220 2.15471 2.14473 the harmonic oscillator, we exactly solved the generalized 1.0 1.49095 0.68153 2.17248 2.14473 Schrodinger¨ equation in momentum space and found the 0.1 1.40656 1.24992 2.65648 2.14473 solutions in terms of the Gegenbauer polynomials. Also, for 1 0.5 1.62672 0.97566 2.60238 2.14473 the two lowest energy eigenstates, we obtained the solutions 1.0 1.84423 0.74892 2.59315 2.14473 in position space in terms of the hypergeometric functions. Then, the analytical expressions for the information entropies arefoundinthemomentumspacewithproperlimiting values for 𝛽→0. Using the numerical values for the side of (11). In Table 1, we have reported the position and position information entropy, we explicitly showed that the momentum space information entropies for 𝛽 = {0.1, 0.5, 1} BBM inequality holds for various values of the deformation and showed that they obey the BBM inequality. These results parameter. To check the validity of the BBM inequality for indicate that the position space information entropy increases other potentials, we need to solve the generalized Schrodinger¨ with the GUP parameter 𝛽 andviceversaforthemomentum equation which contains higher order differential terms. space information entropy but their sum stays above the value However, for the small anharmonic potential terms, the 1+ln 𝜋. perturbationtheorycanbeusedtofindtheapproximate solutions. Also, For other types of GUPs, if a formally self- 6. Conclusions adjoint representation is viable, the BBM inequality is still valid. In this paper, we studied the Shannon entropic uncertainty relation in the presence of a minimal measurable length Competing Interests proportional to the Planck length. We showed that using the formally self-adjoint representation, since the coordinate The author declares that he has no competing interests. space and momentum space wave functions are related by the Fourier transformation, the measure in the information entropic integral is flat and the lower bound that is predicted References by the BBM inequality is guaranteed in the GUP framework. [1] S. Hossenfelder, “Minimal length scale scenarios for quantum It is worth mentioning that the BBM inequality does not gravity,” Living Reviews in Relativity,vol.16,article2,2013. hold for all wave functions. In fact, its validity depends on [2] P. Pedram, “A higher order GUP with minimal length uncer- both the deformed algebra and its representation. As we tainty and maximal momentum,” Physics Letters B,vol.714,no. have indicated, this inequality does not hold in quasiposition 2–5, pp. 317–323, 2012. [𝑋, 𝑃] = 𝑖ℏ(12 +𝛽𝑃 ) representation of our algebra .As [3] P. Pedram, “A higher order GUP with minimal length uncer- another example, we mentioned the algebra [𝑋, 𝑃] = 𝑖ℏ(1 + tainty and maximal momentum II: applications,” Physics Letters 2 2 𝛼𝑋 +𝛽𝑃 ) that implies both a minimal length and a minimal B,vol.718,no.2,pp.638–645,2012. Advances in High Energy Physics 7

[4] G.Amelino-Camelia,J.Ellis,N.E.Mavromatos,D.V.Nanopou- [23] P. Pedram, “Generalized uncertainty principle and the confor- los, and S. Sarkar, “Tests of quantum gravity from observations mally coupled scalar field quantum cosmology,” Physical Review of big gamma-ray bursts,” Nature,vol.393,pp.763–765,1998. D,vol.91,no.6,ArticleID063517,10pages,2015. [5] U. Jacob and T. Piran, “Neutrinos from gamma-ray bursts as [24] H. R. Sepangi, B. Shakerin, and B. Vakili, “Deformed phase a tool to explore quantum-gravity-induced Lorentz violation,” space in a two-dimensional minisuperspace model,” Classical Nature Physics,vol.3,pp.87–90,2007. and Quantum Gravity,vol.26,no.6,ArticleID065003,21pages, [6] I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and C.ˇ 2009. Brukner, “Probing planck-scale physics with quantum optics,” [25] B. Vakili, “Dilaton cosmology, noncommutativity, and gener- Nature Physics,vol.8,no.5,pp.393–397,2012. alized uncertainty principle,” Physical Review D,vol.77,no.4, [7] A. Kempf, G. Mangano, and R. B. Mann, “Hilbert space Article ID 044023, 10 pages, 2008. representation of the minimal length uncertainty relation,” [26]S.Jalalzadeh,S.M.M.Rasouli,andP.V.Moniz,“Quantum Physical Review D,vol.52,no.2,pp.1108–1118,1995. cosmology, minimal length, and holography,” Physical Review [8] L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, “Exact D,vol.90,no.2,ArticleID023541,2014. solution of the harmonic oscillator in arbitrary dimensions with [27] S. Benczik, L. N. Chang, D. Minic, N. Okamura, S. Rayyan, minimal length uncertainty relations,” Physical Review D,vol. and T. Takeuchi, “Short distance versus long distance physics: 65,no.12,ArticleID125027,8pages,2002. the classical limit of the minimal length uncertainty relation,” [9] P. Pedram, “New approach to nonperturbative quantum Physical Review D,vol.66,no.2,ArticleID026003,2002. mechanics with minimal length uncertainty,” Physical Review [28] A. M. Frydryszak and V.M. Tkachuk, “Aspects of pre-quantum D,vol.85,no.2,ArticleID024016,12pages,2012. description of deformed theories,” Czechoslovak Journal of [10] T. V. Fityo, I. O. Vakarchuk, and V. M. Tkachuk, “One- Physics,vol.53,no.11,pp.1035–1040,2003. dimensional Coulomb-like problem in deformed space with [29] Z. K. Silagadze, “Quantum gravity, minimum length and Kep- minimal length,” Journal of Physics A: Mathematical and Gen- lerian orbits,” Physics Letters A,vol.373,no.31,pp.2643–2645, eral,vol.39,no.9,pp.2143–2149,2006. 2009. [11] P.Pedram, “Anote on the one-dimensional hydrogen atom with [30] C. Quesne and V.M. Tkachuk, “Composite system in deformed minimal length uncertainty,” Journal of Physics A: Mathematical space with minimal length,” Physical Review A,vol.81,no.1, and Theoretical, vol. 45, no. 50, Article ID 505304, 11 pages, 2012. Article ID 012106, 2010. [12] P. Pedram, “One-dimensional hydrogen atom with minimal [31] B. Vakili and M. A. Gorji, “Thermostatistics with minimal length uncertainty and maximal momentum,” Europhysics Let- length uncertainty relation,” Journal of Statistical Mechanics: ters, vol. 101, no. 3, Article ID 30005, 2013. Theory and Experiment,vol.2012,no.10,ArticleIDP10013,2012. [13] D. Bouaziz and M. Bawin, “Regularization of the singular [32] M. Abbasiyan-Motlaq and P.Pedram, “Generalized uncertainty inverse square potential in quantum mechanics with a minimal principle and thermostatistics: a semiclassical approach,” Inter- length,” Physical Review A, vol. 76, Article ID 032112, 2007. national Journal of Theoretical Physics,vol.55,no.4,pp.1953– [14] P. Pedram, “Coherent states in gravitational quantum mechan- 1961, 2016. ics,” International Journal of Modern Physics D,vol.22,no.2, [33] S. Wehner and A. Winter, “Entropic uncertainty relations—a Article ID 1350004, 14 pages, 2013. survey,” New Journal of Physics,vol.12,ArticleID025009,2010. [15] C. Quesne and V. M. Tkachuk, “Dirac oscillator with nonzero [34] V.Majernik and T. Opatrny, “Entropic uncertainty relations for minimal uncertainty in position,” JournalofPhysicsA.Mathe- a quantum oscillator,” Journal of Physics A,vol.29,no.9,pp. matical and General,vol.38,no.8,pp.1747–1765,2005. 2187–2197, 1996. [16] S. Das and E. C. Vagenas, “Universality of quantum gravity [35] S. Dong, G.-H. Sun, S.-H. Dong, and J. P. Draayer, “Quantum corrections,” Physical Review Letters,vol.101,ArticleID221301, information entropies for a squared tangent potential well,” 2008. Physics Letters A,vol.378,no.3,pp.124–130,2014. [17] P. Pedram, K. Nozari, and S. H. Taheri, “The effects of minimal [36]R.Atre,A.Kumar,C.N.Kumar,andP.Panigrahi,“Quantum- length and maximal momentum on the transition rate of ultra information entropies of the eigenstates and the coherent state cold neutrons in gravitational field,” Journal of High Energy of the Poschl-Teller¨ potential,” Physical Review A,vol.69,no.5, Physics, vol. 2011, no. 3, article 093, 2011. Article ID 052107, 6 pages, 2004. [18] P.Pedram, “Exact ultra cold neutrons’ energy spectrum in grav- [37] E. Romera and F. de los Santos, “Identifying wave-packet itational quantum mechanics,” The European Physical Journal C, fractional revivals by means of information entropy,” Physical vol.73,no.10,article2609,2013. Review Letters,vol.99,ArticleID263601,2007. [19] A. M. Frassino and O. Panella, “Casimir effect in minimal length [38] J. S. Dehesa, A. Mart´ınez-Finkelshtein, and V. N. Sorokin, theories based on a generalized uncertainty principle,” Physical “Information-theoretic measures for Morse and Poschl-Teller¨ Review D,vol.85,no.4,ArticleID045030,2012. potentials,” Molecular Physics,vol.104,no.4,pp.613–622,2006. [20] P.Pedram, “Dirac particle in gravitational quantum mechanics,” [39] E. Aydiner, C. Orta, and R. Sever, “Quantum information Physics Letters B,vol.702,no.4,pp.295–298,2011. entropies of the eigenstates of the morse potential,” International [21] P. Pedram, “Nonperturbative effects of the minimal length Journal of Modern Physics B,vol.22,no.3,pp.231–237,2008. uncertainty on the relativistic quantum mechanics,” Physics [40] A. Kumar, “Information entropy of isospectral Poschl-Teller¨ Letters B,vol.710,no.3,pp.478–485,2012. potential,” Indian Journal of Pure and Applied Physics,vol.43, [22] P. Pedram, M. Amirfakhrian, and H. Shababi, “On the (2+1)- no.12,pp.958–963,2005. dimensional Dirac equation in a constant magnetic field with [41] J. Katriel and K. D. Sen, “Relativistic effects on information a minimal length uncertainty,” International Journal of Modern measures for hydrogen-like atoms,” Journal of Computational Physics D,vol.24,no.2,ArticleID1550016,8pages,2015. and Applied Mathematics,vol.233,no.6,pp.1399–1415,2010. 8 Advances in High Energy Physics

[42] S. H. Patil and K. D. Sen, “Net information measures for modified Yukawa and Hulthen´ potentials,” International Journal of Quantum Chemistry,vol.107,no.9,pp.1864–1874,2007. [43] K. D. Sen, “Characteristic features of Shannon information entropy of confined atoms,” TheJournalofChemicalPhysics,vol. 123, no. 7, Article ID 074110, 2005. [44] D. Dutta and P. Roy, “Information entropy of conditionally exactly solvable potentials,” Journal of Mathematical Physics,vol. 52, no. 3, Article ID 032104, 2011. [45] H. Grinberg, “Quadrature squeezing and information entropy squeezing in nonlinear two-level spin models,” International Journal of Modern Physics B,vol.24,no.9,pp.1079–1092,2010. [46] M. Abdel-Aty, I. A. Al-Khayat, and S. S. Hassan, “Shannon information and entropy squeezing of a single-mode cavity QED of a raman interaction,” International Journal of Quantum Information,vol.4,no.5,pp.807–814,2006. [47] H. G. Laguna and R. P. Sagar, “Shannon entropy of the wigner function and position-momentum correlation in model systems,” International Journal of Quantum Information,vol.8, no. 7, pp. 1089–1100, 2010. [48] H. H. Corzo, H. G. Laguna, and R. P. Sagar, “Localization- delocalization phenomena in a cyclic box,” Journal of Mathe- matical Chemistry,vol.50,no.1,pp.233–248,2012. [49] C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal,vol.27,no.4,pp.623–656, 1948, (Reprinted in: C. E. Shannon, Claude Elwood Shannon: Collected Papers, IEEE Press, New York, NY, USA, 1993). [50] I. I. Hirschmann, “A note on entropy,” American Journal of Mathematics,vol.79,no.1,pp.152–156,1957. [51] K. I. Babenko, “An inequality in the theory of Fourier integrals,” Izvestiya Rossiiskoi Akademii Nauk, Seriya Matematicheskaya, vol. 25, no. 4, pp. 531–542, 1961. [52] W. Beckner, “Inequalities in Fourier analysis,” Annals of Mathe- matics,vol.102,no.1,pp.159–182,1975. [53] I. Bialynicki-Birula and J. Mycielski, “Uncertainty relations for information entropy in wave mechanics,” Communications in Mathematical Physics,vol.44,no.2,pp.129–132,1975. [54] I. Bialynicki-Birula and L. Rudnicki, “Entropic uncertainty relations in quantum physics,” in Statistical Complexity: Appli- cations in Electronic Structure, pp. 1–34, Springer, Berlin, Ger- many, 2011. [55] J. Sanchez-Ruiz,´ “Maassen-Uffink entropic uncertainty relation for angular momentum observables,” Physics Letters A,vol.181, no. 3, pp. 193–198, 1993. [56] J. B. M. Uffink and J. Hilgevoord, “Uncertainty principle and uncertainty relations,” Foundations of Physics,vol.15,no.9,pp. 925–944, 1985. [57] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, NY, USA, 5th edition, 1994. [58] B. Nagel, “The relativistic Hermite polynomial is a Gegenbauer polynomial,” Journal of Mathematical Physics,vol.35,no.4,pp. 1549–1554, 1994. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 7404218, 5 pages http://dx.doi.org/10.1155/2016/7404218

Research Article An Attempt for an Emergent Scenario with Modified Chaplygin Gas

Sourav Dutta, Sudeshna Mukerji, and Subenoy Chakraborty

Department of Mathematics, Jadavpur University, Kolkata, West Bengal 700032, India

Correspondence should be addressed to Sourav Dutta; [email protected]

Received 17 December 2015; Accepted 21 February 2016

Academic Editor: Ahmed Farag Ali

Copyright © 2016 Sourav Dutta 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.

The present work is an attempt for emergent universe scenario with modified Chaplygin gas. The universe is chosen as spatially flat FRW space-time with modified Chaplygin gas as the only cosmic substratum. It is found that emergent scenario is possible for some specific (unrealistic) choice of the parameters in the equation of state for modified Chaplygin gas.

1. Introduction matter with equation of state 𝑝=𝑤𝜌,(−1/3≤𝑤≤1), whose behavior similar to that of an emergent universe The origin of the universe is a controversial issue in cos- was highlighted. Then, in starobinsky model, Mukherjee et mology. It may start from the big bang singularity or there al. [4] derived solutions for flat FRW space-time having are proposals for nonsingular model of the universe. The emergent character in infinite past. Subsequently, Mukherjee inability of Einstein’s general theory of relativity at zero and associates [5] presented a general framework for an volume leads to the well known big bang singularity in emergent universe model with an ad hoc equation of state standard cosmology. To overrule this initial discomfortable connecting the pressure and density, having exotic nature situation various cosmological scenarios have been proposed in some cases. These models are interesting as they can be and are classified as bouncing universes or the emergent cited as specific examples of nonsingular (i.e., geometrically universes. Here, we will choose the second option which complete) inflationary universes. Also, it is worth mentioning results from searching for singularity-free inflationary sce- here that entropy considerations favour the Einstein static nario in the background of classical general relativity. In a model as the initial state for our universe [6, 7]. Thereafter, a word, emergent universe is a model universe, ever existing series of works [8–16] have been done to formulate emergent (𝑡 → −∞) with almost static behavior in the infinite past universe in different gravity models and also for various (gradually evolves into inflationary stage) and having no types of matter. Very recently, emergent scenario has been time-like singularity. Also, the modern and extended version formulated with some interesting physical aspects. The idea of of the original Lemaitre-Eddington universe can be identified quantum tunneling [17] has been used for the decay of a scalar astheemergentuniversescenario. field having initial static state as false vacuum to a state of true Long back in 1967, Harrison [1] showed a model of the vacuum. Secondly, a model of an emergent universe has been closed universe containing radiation, which approaches the formulated in the background of nonequilibrium thermody- state of an Einstein static model asymptotically (i.e., 𝑡→ namical prescription with dissipation due to particle creation −∞). This kind of model was again reinvestigated after a mechanism [18]. Very recently, Paul and Majumdar [19] have long gap by Ellis and collaborators [2, 3]. Although they formulated emergent universe with interacting fields. Finally, were not able to obtain exact solutions, they presented closed Pavonetal.[20,21]havestudiedtheemergentscenario universes with a minimally coupled scalar field 𝜙 having from thermodynamical view point. They have examined the typical self-interacting potential and possibly some ordinary validity of the generalized second law of thermodynamics 2 Advances in High Energy Physics during the transition from a generic initial Einstein static (i) When the Scale Factor “𝑎”IsVerySmall.For small “𝑎”, 𝜌 phase to the inflationary phase and also during the transition can be approximated from (4) and 𝑝 can be approximated from inflationary era to the standard radiation dominated from (1) as follows: era. 𝜌 1/(𝑛+1) 𝜌≅( 0 ) 𝑎−3(𝐴+1) 𝐴+1 2. Chaplygin Gas and Possible Solution 𝐵 + 𝑎3(1+𝐴)𝑛 Mixed exotic fluid known as modified Chaplygin gas [21] has 1/(𝑛+1) (𝑛/(𝑛+1)) (𝑛+1)(𝐴+1) 𝜌0 the equation of state [22, 23] ≡𝜌 +𝜌 , 𝐵 1𝑖 2𝑖 𝑝=𝐴𝜌− , 0<𝑛≤1. (6) 𝜌𝑛 (1) 𝐴𝜌1/(𝑛+1) 𝑝≅ 0 (𝐴+1)1/(𝑛+1) 𝑎3(𝐴+1) Thisequationofstateshowsbarotropicperfectfluid 𝑝=𝐴𝜌 𝑎(𝑡) 𝐵 [1+𝑛(𝐴+1)] , at very early phase (when the scale factor is − 𝑎3𝑛(1+𝐴) vanishingly small), while it approaches Λ𝐶𝐷𝑀 model when (𝑛+1)(𝐴+1)1/(𝑛+1) 𝜌𝑛/(𝑛+1) the scale factor is infinitely large. It shows a mixture at all 0 stages. Note that at some intermediate stage the pressure ≡𝑝1𝑖 +𝑝2𝑖. vanishes and the matter content is equivalent to pure dust. Further, this typical model is equivalent to a self-interacting 𝑎 scalar field from field theoretic point of view. It should be (ii) When the Scale Factor “ ”HasInfinitelyLargeValue. 𝑎 𝜌 𝑝 noted that the Chaplygin gas was introduced in the context of Similarly, for large “ ”, and are approximated from (4) aerodynamics. In the present paper, we will examine whether and (1), respectively, as follows: emergent scenario is possible for FRW model of the universe 𝐵 1/(𝑛+1) 𝜌 𝐵 1/(𝑛+1) with matter content as modified Chaplygin gas (MCG). 𝜌≅( ) + 0 ( ) 𝑎−3𝜇 For homogeneous and isotropic flat FRW model of the 𝐴+1 (𝑛+1) 𝐵 𝐴+1 universe, the Einstein field equations are (choosing 8𝜋𝐺 =1) ≡𝜌1𝑓 +𝜌2𝑓, (7) 3𝐻2 =𝜌, 1 𝑛+(𝑛+1) 𝐴 𝜌 𝑝≅− + 0 𝑎−3𝜇 (2) 1/(𝑛+1) 1/(𝑛+1) (𝑛+1) 𝐵 2𝐻=−(𝜌+𝑝),̇ (𝐴+1) (𝐴+1)

≡𝑝1𝑓 +𝑝2𝑓. with energy conservation relation: Thus, in the asymptotic limits, the components of energy 𝜌+3𝐻̇ (𝜌+𝑝) =0. (3) density and pressure can be expressed as sum of two nonin- teracting barotropic fluids having equation of states: Using (1) in (3), one can integrate 𝜌 as 𝑤1𝑖 =𝐴, 𝐵 𝑐 1/(𝑛+1) 𝜌=[ + ] , (4) 𝑤 =−[1+𝑛(𝐴+1)] , 1+𝐴 𝑎3𝜇 2𝑖 𝑤 =−𝐵−(1/(𝑛+1)), (8) with 𝑐>0, a constant of integration. 1𝑓 Now, using this 𝜌 in the first Friedmann equation in (2), 𝑛+(𝑛+1) 𝐴 𝑤 = . one can integrate to obtain cosmic time as a function of the 2𝑓 𝐵(1/(𝑛+1)) scale factor as Thus, MCG can be considered in the asymptotic limits √3 (1+𝐴) 𝑐𝛼 (𝑡 − 𝑡 ) as two barotropic fluids of constant equation of state of 2 0 which one is exotic in nature. However, one can consider that (5) 𝐵 the two fluids in question (in the asymptotic limit) may be =𝑎3(1+𝐴)/2 𝐹 [𝛼,𝛼,1+𝛼,− 𝑎3(1+𝐴)/2𝛼], interacting with separate equation of state as 2 1 𝐶 (1+𝐴) 𝜌1𝑖̇ +3(𝜌1𝑖 +𝑝1𝑖)𝐻=𝑄𝑖, where 𝛼 = 1/2(1 + 𝑛) and 𝐹1 is the usual hypergeometric 2 𝜌 ̇ +3(𝜌 +𝑝 )𝐻=−𝑄, function. 2𝑖 2𝑖 2𝑖 𝑖 (9) 𝜌1𝑓̇ +3(𝜌1𝑓 +𝑝1𝑓)𝐻=𝑄𝑓, 3. Asymptotic Analysis and Equivalent Two Fluid Systems 𝜌2𝑓̇ +3(𝜌2𝑓 +𝑝2𝑓)𝐻=−𝑄𝑓,

We will now analyze the two asymptotic cases. where 𝑄𝑖 and 𝑄𝑓 represent the interaction term. Advances in High Energy Physics 3

𝑄𝑖 >0indicates a flow of energy from fluid 2 (having 5 energy density 𝜌2𝑖) to fluid 1 (having energy density 𝜌1𝑖) and similarly for 𝑄𝑓 also. Further, one can rewrite the conservation equations (9) as 4

eff 𝜌1𝑖̇ +3𝐻(1+𝑤1𝑖 )𝜌1𝑖 =0, 3 eff t 𝜌2𝑖̇ +3𝐻(1+𝑤2𝑖 )𝜌2𝑖 =0, (10) 𝜌 ̇ +3𝐻(1+𝑤eff )𝜌 =0, 1𝑓 1𝑓 1𝑓 2

eff 𝜌2𝑓̇ +3𝐻(1+𝑤2𝑓)𝜌2𝑓 =0, 1 with

eff 𝑄𝑖 𝑤 =𝑤 − , −1 012345 1𝑖 1𝑖 3𝐻𝜌 1𝑖 a 𝑄 𝑤eff =𝑤 + 𝑖 , n=1 2𝑖 2𝑖 n=1/2 3𝐻𝜌2𝑖 n = 0.1 𝑄 (11) eff 𝑓 𝑎(𝑡) 𝑡 𝑤1𝑓 =𝑤1𝑓 − , Figure 1: The Figure represents the scale factor against for 3𝐻𝜌1𝑓 𝐴 =−1̸ . 𝑄 𝑤eff =𝑤 + 𝑓 . 2𝑓 2𝑓 8 3𝐻𝜌2𝑓

The above conservation equations show that the fluids 7 may be considered as noninteracting at the cost of variable equation of state. 6

5

4. Emergent Scenario and a(t) Thermodynamical Analysis 4 Oneshouldnotethatinintegrating(3)tohave(4)weassume that 𝐴 =−1̸ .Now,wewilldiscussthesituationwhen𝐴=−1. 3 The expression for energy density now becomes 2 𝑎 1/(𝑛+1) 𝜌=[3 (𝑛+1) 𝐵 ln ( )] , (12) 𝑎0 −1 −0.5 0 0.5 which from the first Friedmann equation gives t n=1/2 (1/(1−𝛼)) 𝑎=𝑎0 exp [𝑏0 (𝑡 −0 𝑡 ) ], n=1/3 n=1/10 (1/(1−𝛼)) (13) √3 𝑏 =( 𝐵 (2𝑛 + 1)) . Figure 2: The Figure shows the graphical representation of the scale 0 2 factor 𝑎(𝑡) against 𝑡 for 𝐴=−1.

From the solutions (5) and (13), we see (Figures 1 and 2) that 𝑎→0as 𝑡→−∞, so it is not possible to have emergent Assuming the validity of the first law of thermodynamics scenario with the usual modified Chaplygin gas. However, if at the horizon (having area radius 𝑅ℎ), we have the Clausius we choose −1<𝑛<−1/2,then𝛼>1and we have from relation: solution (13) 𝑎→𝑎0 as 𝑡→−∞(see Figure 3). Hence, it is possible to have emergent scenario with this revised form of −𝑑𝐸 =𝑇𝑑𝑆 , MCG. ℎ ℎ ℎ (14) We will now discuss the thermodynamics of the emergent scenario with this revised form of MCG as the cosmic where 𝑇ℎ is the temperature of the horizon and 𝑠ℎ is the substratum. entropy of the horizon. In the above, 𝐸ℎ is the amount of 4 Advances in High Energy Physics

Case 1 (apparent horizon). The area radius for apparent horizon is given by 1.5 1 𝑅 = , 𝐴 𝐻 (20) 1.4 so that 𝐻̇ 4𝜋𝐺 (𝜌 +𝑝) 𝑅̇ =− = . 𝐴 2 2 (21) 1.3 𝐻 𝐻 a(t) Hence, 2 2 1.2 𝑑𝑆 (4𝜋) 𝐺(𝜌+𝑝) 𝑇 = >0. 4 (22) 𝑑𝑡 𝑇𝐴𝐻 1.1 Thus, generalised second law of thermodynamics (GSLT) is always true at the apparent horizon.

Case 2 (event horizon). The area radius for event horizon is −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 given by t ∞ 𝑑𝑡 𝑅 =𝑎∫ . n = −0.6 𝐸 𝑎 (23) n=−2/3 𝑡

Figure 3: The Figure shows the graphical representation of the scale The above improper integral converges for accelerating factor 𝑎(𝑡) against 𝑡 for 𝐴=−1. phase of the FRW model. Hence, in the present scenario, it is very much relevant. From the above definition ̇ 𝑅𝐸 =𝐻𝑅𝐸 −1, (24) energy crossing the horizon during time 𝑑𝑡 and is given by [24–26] so from (19) 𝑑𝑆 (4𝜋) 𝑅2 (𝜌 + 𝑝) 3 𝑇 = 𝐸 (𝐻𝑅 −1) −𝑑𝐸ℎ =4𝜋𝑅ℎ𝐻(𝜌+𝑝)𝑑𝑡. (15) 𝐸 𝑑𝑡 𝑇𝐸 So,using(15)in(14),wehavetherateofchangeofthe (4𝜋) 𝑅2 𝐻 𝐵 = 𝐸 [(𝐴+1) 𝜌− ](𝑅 −𝑅 ) horizon entropy as 𝑛 𝐸 𝐴 (25) 𝑇𝐸 𝜌 3 2 𝑑𝑆ℎ 4𝜋𝑅ℎ𝐻(𝜌+𝑝) (4𝜋) 𝑅 𝐻 𝑐 (1+𝑎) = . (16) = 𝐸 (𝑅 −𝑅 ). 𝑑𝑡 𝑇 3𝜇 𝑛 𝐸 𝐴 ℎ 𝑇𝐸 𝑎 ⋅𝜌

To obtain the entropy of the inside fluid, we start with the Hence, the validity of GSLT is possible if 𝑅𝐸 >𝑅𝐴 (as Gibbs equation [26, 27] 𝛼>1). In the above, temperature is chosen as the hawking temperatureonthehorizonas[28,29] 𝑇 𝑑𝑆 =𝑑𝐸 +𝑝𝑑𝑉, ℎ 𝑓 𝑓 (17) 1 𝑇𝐴 = , 2𝜋𝑅𝐴 where 𝑆𝑓 is the entropy of the fluid bounded by the horizon (26) and 𝐸𝐹 is the energy of the matter distribution. Here, for 𝑅 𝑇 = 𝐸 . thermodynamical equilibrium, the temperature of the fluid 𝐸 2 2𝜋𝑅𝐴 istakenasthatofthehorizon,thatis,𝑇ℎ. 3 3 Now, using 𝑉=4𝜋𝑅ℎ/3, 𝐸𝑓 =(4𝜋𝑅ℎ/3)𝜌,andthe Friedmann equations, the entropy variation of the fluid is Additional Points given by Inthepresentwork,wehaveexaminedthecosmologyofthe 𝑑𝑆 4𝜋𝑅2 emergent scenario for modified Chaplygin gas as the cosmic 𝑓 ℎ ̇ 𝐴 =−1̸ = (𝜌+𝑝)(𝑅ℎ −𝐻𝑅ℎ) . (18) fluid. It is found that for both the solutions (with 𝑑𝑡 𝑇ℎ and 𝐴=−1) the model does not exhibit emergent scenario at early epochs. So, one can conclude that it is not possible to Thus, combining (16) and (18), the variation of the total have emergent scenario with MCG. However, if 𝑛 is chosen to (𝑆 ) entropy 𝑇 is given by be negative, that is, −1<𝑛<−1/2,then𝑎→𝑎0 as 𝑡→−∞; thatis,initialbigbangsingularityisavoided. 4𝜋𝑅2 𝑑𝑆𝑇 𝑑 ℎ ̇ Finally, thermodynamical analysis of the emergent sce- = (𝑆ℎ +𝑆𝑓) = (𝜌+𝑝) 𝑅ℎ. (19) 𝑑𝑡 𝑑𝑡 𝑇ℎ nario has been presented. Advances in High Energy Physics 5

Competing Interests [16] S. Mukerji, N. Mazumder, R. Biswas, and S. Chakraborty, “Emergent scenario and different anisotropic models,” Interna- The authors declare that they have no competing interests. tional Journal of Theoretical Physics,vol.50,no.9,pp.2708–2719, 2011. Acknowledgments [17] P. Labrana,˜ “Emergent universe by tunneling,” Physical Review D,vol.86,no.8,ArticleID083524,2012. Subenoy Chakraborty thanks Inter-University Center for [18] S. Chakraborty, “Is emergent universe a consequence of particle Astronomy and Astrophysics (IUCAA), Pune, India, for their creation process?” Physics Letters B,vol.732,pp.81–84,2014. warm hospitality as a part of the work was done during a visit. [19] B. C. Paul and A. Majumdar, “Emergent universe with interact- Also Subenoy Chakraborty thanks UGC-DRS programme at ing fluids and the generalized second law of thermodynamics,” the Department of Mathematics, Jadavpur University. Sourav Classical and Quantum Gravity, vol. 32, no. 11, Article ID 115001, Dutta thanks the Department of Science and Technology 2015. (DST), Government of India, for awarding Inspire research [20] S. D. Campo, R. Herrera, and D. Pavon, “The generalized second fellowship. law in the emergent universe,” Physics Letters B,vol.707,no.1, pp. 8–10, 2012. [21] D. Pavon, S. del Campo, and R. Herrera, “The generalized sec- References ond law and the emergent universe,” http://arxiv.org/abs/1212 [1] E. R. Harrison, “Classification of uniform cosmological mod- .6863. els,” Monthly Notices of the Royal Astronomical Society,vol.137, [22] U. Debnath, A. Banerjee, and S. Chakraborty, “Role of modified no. 1, pp. 69–79, 1967. Chaplygin gas in accelerated universe,” Classical and Quantum [2] G. F.Ellis and R. Maartens, “The emergent universe: inflationary Gravity,vol.21,no.23,pp.5609–5617,2004. cosmology with no singularity,” Classical and Quantum Gravity, [23] H. Benaoum, “Accelerated universe from modified chaplygin vol.21,no.1,pp.223–232,2004. gas and tachyonic fluid,” http://arxiv.org/abs/hep-th/0205140. [3]G.F.R.Ellis,J.Murugan,C.G.Tsagasetal.,“Theemergent [24] R.-G. Cai and S. P. Kim, “First law of thermodynamics and universe: an explicit construction,” Classical and Quantum Friedmann equations of Friedmann-Robertson-Walker uni- Gravity,vol.21,no.1,p.233,2004. verse,” Journal of High Energy Physics,vol.2005,no.2,article [4] S.Mukherjee,B.C.Paul,S.D.Maharaj,andA.Beesham,“Emer- 050, 2005. gent universe in starobinsky model,” http://arxiv.org/abs/gr-qc/ [25] R. S. Bousso, “Cosmology and the S matrix,” Physical Review D, 0505103. vol. 71, no. 6, Article ID 064024, 2005. [5] S. Mukherjee, B. C. Paul, N. K. Dadhich, S. D. Maharaj, and A. [26]N.MajumderandS.Chakraborty,“Doesthevalidityofthefirst Beesham, “Emergent universe with exotic matter,” Classical and law of thermodynamics imply that the generalized second law Quantum Gravity,vol.23,no.23,pp.6927–6934,2006. of thermodynamics of the universe is bounded by the event [6] G. W. Gibbons, “The entropy and stability of the universe,” horizon?” ClassicalandQuantumGravity,vol.26,no.19,Article Nuclear Physics B,vol.292,no.4,pp.784–792,1987. ID 195016, 2009. [7] G. W. Gibbons, “The entropy and stability of the universe,” [27] G. Izquierdo and D. Pavon,´ “Dark energy and the generalized Nuclear Physics B,vol.292,pp.784–792,1987. second law,” Physics Letters B,vol.633,no.4-5,pp.420–426, [8]D.J.Mulryne,R.Tavakol,J.E.Lidsey,andG.F.R.Ellis,“An 2006. emergent universe from a loop,” Physical Review D,vol.71,no. [28] S. Chakraborty, “Is thermodynamics of the universe bounded 12, Article ID 123512, 2005. by event horizon a Bekenstein system?” Physics Letters B,vol. [9] A. Banerjee, T. Bandyopadhyay, and S. Chakraborty, “Emergent 718, no. 2, pp. 276–278, 2012. Universe in the brane-world scenario,” General Relativity and [29] S. Saha and S. Chakraborty, “A redefinition of Hawking tem- Gravitation,vol.40,no.8,pp.1603–1607,2008. perature on the event horizon: thermodynamical equilibrium,” [10] A. Banerjee, T. Bandyopadhyay, and S. Chakraborty, “Emergent Physics Letters B,vol.717,no.4-5,pp.319–322,2012. universe in brane world scenario with Schwarzschild-de Sitter bulk,” General Relativity and Gravitation,vol.40,no.8,pp. 1603–1607, 2008. [11] N. J. Nunes, “Inflation: a graceful entrance from loop quantum cosmology,” Physical Review D,vol.72,no.10,ArticleID103510, 2005. [12] J. E. Lidsey and D. J. Mulryne, “Graceful entrance to braneworld inflation,” Physical Review D,vol.73,no.8,ArticleID083508, 2006. [13] U. Debnath, “Emergent universe and the phantom tachyon model,” Classical and Quantum Gravity,vol.25,no.20,Article ID 205019, 8 pages, 2008. [14] B. C. Paul and S. Ghose, “Emergent universe scenario in the Einstein-Gauss-Bonnet gravity with dilaton,” General Relativity and Gravitation, vol. 42, no. 4, pp. 795–812, 2010. [15] U. Debnath and S. Chakraborty, “Emergent universe with exotic matter in brane world scenario,” International Journal of Theoretical Physics, vol. 50, no. 9, pp. 2892–2898, 2011.