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SLAC-PUB-11132 hep-th/0503146 April 2005 A Quantum Gravitational Relaxation of The

Stephon Alexander∗ Stanford Linear Accelerator Center and ITP, Dept of , Stanford University, 2575 Sand Hill Rd., Menlo Park, CA 94025 USA (Dated: March 18, 2005) Similar to QCD, has a Θ sector due to large diffeomorphisms. We make explicit, for the first , that the gravitational CP violating Θ parameter is non-perturbatively related to the cosmological constant. A gravitational pseudoscalar coupling to massive gives rise to general relativity from a topological B ∧ F theory through a chiral 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 . 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 field via the Peccei-Quinn mechanism. It 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 (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 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 . (22) Xf Θ 3γ = . (15) 384π2 GΛ This phase rotation of the fermion fields also changes the mass terms in the Lagrangian Now that we have discussed the connection between the gravitational Θ parameter and the cosmological con- stant we are ready to discuss the physical relevance of this 1 1 L = − M ψ¯ ψ − M∗ ψ¯ ψ (23) connection. As stated before, the Θ sector in quantum m 2 f f,L f,L 2 f f,R f,R gravity corresponds to a one parameter family of vacuum Xf Xf states. However, at this level Θ is fixed. To allow Θ to The phase redefinition changes the mass as follows be altered and dynamical, we introduce a set of chiral fermions. Mf → exp(2iαf )Mf (24)

III. CHIRAL ROTATIONS AND It is important for our purposes to note that changing GRAVITATIONAL AXION the path integration variables will not have any physical effect, so observable quantities will only depend on the combination of Θ and the mass parameters M Consider now the covariant coupling of gravity to chiral f fermions. Recall from equation (2), exp(−iΘ)Πf Mf (25)

i A SD = − eΨ¯γ EADΨ+ h.c. (16) restoring the dependence on γ and Λ this becomes. 2 ZM γ1154π2 where the covariant derivative with respect to the spin exp(−i )Πf Mf (26) connection, w, is 2GΛ From this expression and eq (15) we see that the grav- µ 1 BC DΨ= dx (∂µ + wµBC S )Ψ (17) itational CP violating parameter, Θ, is proportional to 2 the ratio of the and the cosmological AB 1 A B constant. In the case of QCD, if the masses were with the generators S = 4 [γ ,γ ]. We can work in a chiral basis by expressing the Dirac to vanish, the Theta angle would have no effect and there bispinor Ψ in terms of two-component left and right would be no CP non-conservation Yang-Mills. This ob- handed Weyl spinors ψL,R. The chiral Dirac action which servation will also hold in the gravitational case, hence couples to the Palatini action now becomes we must have massive fermions coupling to gravity to implement our mechanism. A A It is well known that if we define quark fields in QCD SD = −i e(ψ¯1Lτ EADψ1L + ψ¯2Lτ EADψ2L) (18) Z so that all Mf are real, then a non-zero theta angle would produce a P and T non-conserving neutron electric dipole where 2 moment proportional to |Θ| and mπ of the order 2 ∗ ψR = −iτ ψ2L. (19) 2 3 −24 dn ≃ |Θ|emπ/mN ≃ 10 ecm. (27) To understand the physical effects of the Θ term, con- In order to explain in a natural way why Θ is so small, sider the effect on the redefinition of all the fermion fields Peccei and Quinn proposed that a pseudo-scalar field coupled to gravity. could dynamically relax to a minimum of the effective ψ → exp(iγ α )ψ , (20) potential at which CP and P could be conserved. Later f 5 f f Weinberg and Wilczek noted that it would require the ex- istence of a light spinless particle, the axion. The basic where f is a flavor index and αf are a set of real phases. The effect on the measure of for path integrals over idea behind the axion theory is that there is a U(1) sym- fermion fields is [8] metry which is spontaneously broken at energies much higher than those associated with QCD and is also broken by the anomaly involving the gluon fields. In our context −i we want to propose a similar mechanism whose effective [dψ][dψ¯] → exp( Rij∧R α )[dψ][dψ¯] 384π2 Z ij f theory relaxes the gravitational Θ angle and therefore the Xf cosmological constant by the presence of gravitational in- (21) stantons. 4

IV. QUANTUM GRAVITATIONAL PQ where MECHANISM MKγ φ′ =<φ> + (36) We shall now apply the Peccei-Quinn mechanism to 2GΛ general relativity. Let us first briefly outline the mecha- nism. We begin by introducing a pseudoscalar field in the Because the pseudoscalar Lagrangian has a shift symme- try φ′ → φ′ +α we can redefine φ′ at the minimum of the action Sφ (5). This field will play the role of the axion in QCD and will couple with two massive left handed Weyl potential to yield the condition on the cosmological con- fermions. This pseudoscalar field arises from breaking stant in terms of the pseudoscalar field and the symmetry an SO(5) topological BF theory to SO(4) as discussed in breaking scale. Therefore, we can dynamically relax the Section II. According to arguments in the previous sec- cosmological constant. At its minimum <φ> take the tion, a shift in the mass term introduces a CP violating value. gravitational Θ parameter which is inversely proportional KγM to the cosmological constant. By generating a vev for <φ>∼ . (37) the fermions, a potential is acquired for the pseudoscalar GΛ which relaxes the cosmological constant. The above condition is not sufficient to cancel the Recall from Section III, gravity couples to left handed bare cosmological constant, since according to eq(35) fermions via. the height of the potential has to balance the differ- i ence between the bare and effective cosmological con- S = − e(ψτ¯ AE Dψ + h.c.) (28) D 2 Z A stant. Hence, it allows the bare value of the cosmological constant to be shifted according to where Λ =Λ+ m∆cos(φ′). (38) τ a eff D = d − iAa (29) 2 When the fermionic chiral condensate acquires a vev, writing this terms of Weyl spinors we obtain a potential for the pseudoscalar is generated. Subse- quently, the pseudoscalar minimizes its potential and gets a vev yielding ∆ mcos (φ′) = −∆m for the potential. S = −i e(ψ† τ AE Dψ + ψ† τ AE Dψ ) (30) D Z 1L A 1L 2L A L2 Therefore when the potential of the pseudoscalar is min- imized, eq(38) becomes where we have rewritten right handed spinors in terms of left handed ones.

2 ∗ Λeff − Λ = ∆m (39) ψR = −iτ ψ2L (31)

These chiral fermions couple covariantly to the pseu- We view this as a consistency relation for quantum doscalar field through the following aciton. gravity to relax the cosmological constant. Hence, the theory should provide the value of the bare cosmologi- 1 φ Kγ cal constant and the dynamics giving rise to the value L = − ∂ φ∂µφ + [ + ]Rij ∧ R φ 2 µ M 2GΛ ij of the gap ∆ which cancels the bare cosmological con- µ µ stant. With these two relations (37) and (39) we have a −i∂ φψ¯1 γ ψ1 − i∂ φψ¯2 γ ψ2 + L . (32) µ L L µ L L m picture of how the cosmological constant can be dynam- where K = 1154π2 as in (26). ically suppressed in quantum gravity: As stated in the last section we can eliminate the Θ In order to solve the cosmological constant problem, term in eq(8) by redefining the mass term in the fermion the dynamics of the ground state of quantum gravity Lagrangian density to must self tune the vev of the gravitational fermionic con- densate ∆ to balance the magnitude of the bare cosmo- f MKγ f MKγ −i M ( 2GΛ +φ)γ5) i M (( 2GΛ +φ)γ5) logical constant. This dynamical tuning must manifest Lm = mψ¯1Le ψ1L+mψ¯2Le ψ2L (33) itself in the amplitude of the potential energy which is where m is the fermion mass. Now if the fermions con- deep enough to cancel the bare cosmological constant dense and get a vev at some scale M when the pseudoscalar falls to its minimum. In other words, this vev signify the formation of the energy gap < ψ¯1 ψ1 >=< ψ¯2 ψ2 >= ∆ (34) of the fermion condensate whose mass ’eats’ up the bare L L L L cosmological constant according to eq (39). It is interest- where ∆ ∼ M 3. Then equation (33) becomes ing that the physics which sets the scale of the symmetry breaking (ie the condensation of the fermion bilinear) and 1 fφ′ the dynamics of the pseudoscalar field mutually work to- L = − ∂ φ∂µφ + m∆cos( ) (35) m 2 µ M gether to relax the cosmological constant. 5

V. DISCUSSION We would like to end with a physical speculation mo- tivated by [20] which we believe might give insight into We have presented a non-perturbative dynamical why quantum gravity can relax the cosmological con- mechanism which relaxes the cosmological constant. By stant. The authors of [20] argued that at UV scales realizing the connection between the cosmological con- Minkowski space-time is unstable with respect to pertur- stant, the gravitational Θ parameter and the Immirzi bative exchange. This statement is similar to parameter, we were able to implement some wisdom the instability of the free fermi surface in superconduc- from QCD in solving the strong CP problem, namely tivity due to weak phonon exchange; and it is now under- the Peccei-Quinn mechanism. The solution of the cos- stood that non-perturbative physics takes over and gener- mological constant can now be reformulated in-terms of ates Cooper pairs from these exchanges between fermions a hierarchy between the mass of the gravitational axion while the system is driven to a true ground state that is and value of the chiral condensate energy gap ∆. purely non-perturbative. In our case we can think of the In light of this mechanism we can answer the question: vev of the fermion bilinears which couple to the gravita- Why does not the cosmological constant gravitate? Sim- tional field as Cooper pairs which signify the breakdown ply put, there is a direct non-perturbative channel for of a perturbative definition of the cosmological constant. a huge vacuum energy to transmute itself into a conden- It is intriguing to know to what extent this analogy can sate. In doing so an energy gap opens up which offsets the be applied to Spin Networks. value of the cosmological constant to a vanishing value and with respect to this new ground state the cosmolog- ical constant does not gravitate. For this mechanism to work, one needs to calculate the VI. ACKNOWLEDGEMENTS chiral condensate gap ∆ in quantum gravity. We propose that this calculation can be carried out using the Kodama State Ψ(A)K which corresponds to quantum gravity with It is a pleasure to thank Abhay Ashtekar, BJ Bjorken, a positive cosmological constant as well as de Sitter space, , Mohammad Sheikh-Jabbari and Marvin semiclassically[17, 18]. We conjecture that a modified Weinstein for discussions. I would especially like to thank Kodama state which includes the gravitational axion is Michael Peskin and Lee Smolin for looking over a few the basis to evaluate the chiral condensate drafts of this paper and for challenging questions. Even though the author has never met him in person, he would < ψγ¯ 5ψ>=< Ψ(A, φ)K |ψγ¯ 5ψ|Ψ(A, φ)K > (40) like to thank Chopin Soo for writing the influential series of papers on CTP properties of chiral gravity which im- Recently, a modified normalizable Kodama State that pacted this work greatly. Finally, I dedicate this work describes Inflation was found, which included a scalar to my late grandfathers Stephen Belfon and Mustafa field [19]. We leave it up to a future investigations to see Mohammed-Ali. I am grateful to the US DOE for fi- if this state is a candidate to evaluate the gap and hence nancially supporting this work under grant DE-AC03- see if the consistency relation (39) is satisfied. 76SF00515.

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