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A Quantum Gravitational Relaxation of the Cosmological Constant

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A Quantum Gravitational Relaxation of the Cosmological Constant SLAC-PUB-11132 hep-th/0503146 April 2005 A Quantum Gravitational Relaxation of The Cosmological Constant Stephon Alexander∗ Stanford Linear Accelerator Center and ITP, Dept of Physics , Stanford University, 2575 Sand Hill Rd., Menlo Park, CA 94025 USA (Dated: March 18, 2005) Similar to QCD, general relativity has a Θ sector due to large diffeomorphisms. We make explicit, for the first time, that the gravitational CP violating Θ parameter is non-perturbatively related to the cosmological constant. A gravitational pseudoscalar coupling to massive fermions gives rise to general relativity from a topological B ∧ F theory through a chiral symmetry breaking mechanism. We show that a gravitational Peccei-Quinn like mechanism can dynamically relax the cosmological constant. PACS numbers: 98.80.Cq, 98.70.Vc I. INTRODUCTION cosmological constant. Therefore, the question of fix- ing the Barbero-Immirizi parameter is related to a quan- It has always been a dream to solve the cosmologi- tum gravitational determination of the cosmological con- cal constant problem by relaxing it to the minimum of stant. Through this relationship we will propose a pos- a potential [1]. This hope has been especially unsuc- sible dynamical, background independent mechanism to cessful in conventional canonical quantum gravity. In relax the cosmological constant. QCD, the strong CP problem was solved by relaxing the Specifically, we will demonstrate that a Peccei-Quinn- Θ˜ parameter at the minimum of the potential associated like mechanism associated with a non-vanishing vev of with an axion field via the Peccei-Quinn mechanism. It fermion bilinears coupled to gravity will yield an effec- turns out that when quantum gravity is formulated in tive potential for a parameter which alters the ratio of the Ashtekar-Sen variables (LQG), the theory has a sem- Barbero-Immirizi parameter and the cosmological con- blance to Yang-Mills theory and the cosmological con- stant. We then use the value of the Barbero-Immirizi pa- stant problem becomes analogous to the strong CP prob- rameter determined from Black Hole quasinormal modes lem. It is the purpose of this paper to make this analogy to determine the conditions which relaxes the cosmologi- explicit and use a Peccei-Quinn like mechanism to pave a cal constant. This mechanism is deeply tied to the pres- possible route to solving the cosmological constant prob- ence of gravitational instantons in the state space of Loop lem. Quantum Gravity which arise naturally as exact solutions 1 Loop Quantum Gravity (LQG) has a one parameter of the (anti)self dual formulation of general relativity[7]. family of ambiguities which is labeled by γ, the Barbero- This paper is organized as follows. In section II we Immirizi parameter. This parameter plays a similar role discuss the theory and establish the relation between the to the QCD Θ˜ parameter which labels the unitarily in- cosmological constant and the CP violating term in gen- equivalent sectors of the quantum theory. These sectors eral relativity. In section III we show that there are phys- can be accessed by tunneling events due to instanton ical effects from the CP violating term in GR if massive field configurations. Θ˜ is also a measure of CP violation, fermions are included. In section IV we combine the re- which is constrained by the neutron electric dipole mo- sults of the previous sections to develop a quantum grav- ment to be Θ˜ ≤ 10−9. In LQG the Barbero-Immirizi pa- itational version of the Peccei Quinn mechanism to re- rameter is also a measure of CP violation, since it couples lax the cosmological constant. Finally in section V we to the first Pontrjagin class. Specifically, in the quantum conclude with some future directions for calculating the theory γ corresponds to unitarily inequivalent representa- chiral condensate expectation value in quantum gravity. tions of the algebra of geometric operators. For example, the simplest eigenvalues of the area operator Aˆs in the γ quantum sector is given by II. THE THEORY 2 A[j] = 8πγlP l jI (jI + 1) (1) General Relativity can be formulated in terms of a con- X p I strained topological field theory (TQFT). In this paper we shall study a BF-theory for an SO(5) gauge group. What fixes the value of the Barbero-Immirizi param- eter? In this letter we will show that this question is connected to another parameter in general relativity, the 1 While we do not perform any explicit instanton calculations in this paper, it was suggested by Soo [7] that, through a bundle reduction, instantons can reduce the value of cosmological con- ∗Electronic address: [email protected] stant. Work supported in part by the Department of Energy contract DE-AC02-76SF00515 2 BF theory is a topological field theory defined by the ei = lAi5, where l is a constant of dimension length, giv- IJ i wedge product between a curvature two form F (A) and ing rise to a four-dimensional metric gµν = eµeνj. This an SO(5) valued two form BIJ [3]. When the symme- gives the decomposition of the SO(5) curvature: try breaks to SO(4) the authors [5, 6] showed, that the 1 BF-theory becomes equivalent to general relativity with ij ij i j R (w)= F (A) − 2 e ∧ e (7) certain topological invariants. While these topological l terms do not affect the Einstein equations of motion clas- In terms of this decomposition we can rewrite (3) as: sically, we will argue that a quantum mechanical effect α ij kl can relax the cosmological constant in a manner similar SG = SP + R (w) ∧ R (w)ǫijkl to the Peccei-Quinn mechanism[4]. Z 4(α2 − β2) Let T IJ = −T JI be ten generators of the SO(5) Lie β C − Rij (w) ∧ R (w)+ (8) algebra, where I, J =1, ..., 5. The basic dynamical vari- 2(α2 − β2) ij β ables are the SO(5) connection AIJ and the SO(5) valued 2-form BIJ . A two-form carries a pair of anti-symmetric where Sp is the Palatini action of general relativity, as 2 internal indices IJ, with each index taking values from 0 will be discussed below and C is the Nieh-Yan class to 4 and the lower case indices i, j run from i, j =1...4. i ij C = dwe ∧ dwei − R ∧ ei ∧ ej . (9) The theory we will study is gravity coupled to massive fermions and a pseudoscalar field. Let us write the theory Notice that the parameter α in eq (8) corresponds to the down and motivate each part afterwards: magnitude of a fixed SO(5) vector vA. vA = (0, 0, 0, 0, α/2) (10) Stot = SG + SD + Sφ (2) We shall identify α with a pseudoscalar field φ. This field parameterizes the chiral rotations of the SO(5) group. IJ β IJ SG = − (B ∧ FIJ − B ∧ BIJ ZM 2 α 1 ij k l − B ∧ B ǫIJKL5). (3) SP = − (R (w) ∧ e ∧ e ǫijkl 4 IJ KL 2G Z 2 −Λei ∧ ej ∧ ek ∧ elǫ − Rij (w) ∧ e ∧ e ) (11) ijkl γ i j A A SD = e(ψ¯1Lτ EADψ1L + ψ¯2Lτ EADψ2L) (4) is the Palatini action of general relativity with a nonzero ZM cosmological constant and a nonzero Immirizi parameter γ, which is dimensionless. The initial parameters α, β and defined in eqs (10) and (2), respectively, are related to 1 φ the physical parameters as follows S = − ∂ φ∂µφ + ( )R ∧ Rij φ Z 2 µ M ij GΛ γGΛ M α = β = . (12) ¯1 µ 1 ¯2 µ 2 2 2 −if∂µφψLγ ψL − if∂µφψLγ ψL (5) 3(1 − γ ) 3(1 − γ ) Let us focus on the the other terms of the action. Cru- where E = Eµ∂ are the inverse vierbein vector fields, A µ cial for our mechanism is the second term in eq (8), the f is a dimensionless constant and F (A)IJ the curvature topological CP violating term. Using γ, we can rewrite 2-form this term as: IJ IJ I K F = dA + AK ∧ AJ (6) 3γ SCP = − R ∧ R. (13) GΛ ZM Let us motivate each part of the action and show how ˜ ˜ they conspire to relax the cosmological constant non- This term is analogous to the CP violating term,ΘF F , perturbatively. SG will be discussed in this section, SD in QCD. It has been proposed by Ashtekar, Balachan- in section III and Sφ in section IV. dran and Jo, that there will be an ambiguous Yang-Mills instanton Θ˜ angle term in the first order formulation be- The action SG in (3) is equivalent to general relativity with a non-zero cosmological constant via a Macdowell- cause of large SO(3, C) gauge transformations [2]. The Mansouri mechanism where topological F ∧F theory un- additional term in the action was suggested to be dergoes a symmetry breaking from SO(5) to SO(4). The case considered here arrives to general relativity from a Θ S = − R ∧ R (14) SO(5) B ∧ F theory[5, 6]. The authors of [6] showed CPA 384π2 Z that the B ∧ F theory spontaneously broken from SO(5) to SO(4) is equivalent to general relativity by introduc- ing the SO(4) connection wij = Aij and its curvature ij ij i kj 2 i i − ij R = dw + wk ∧ w . They also introduce a frame field In eq (9), dwe = de w ej . 3 Notice that the above action is identical to the second Comparing this with eq (14) we see that this phase shift term in 13 if the Θ parameter scales inversely propor- in the fermion fields is equivalent to shifting Θ by tional to the cosmological constant, i.e if one identifies Θ in (14) according to Θ → Θ+2 αf .
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