arXiv:1108.1237v3 [hep-th] 7 Jan 2013 epwrcutn eomlzbe h rtclexponent critical (HL) the Horava-Lifshitz renormalizable, the power-counting the so be as [13], to ( field In referred scalar gravity. Lifsitz’s often to is similar theory quite is This xiisasrn nstoybtensaeadtime, and distances space short between anisotropy at strong scaling a whose exhibits be system can gravitational but considered energies, a Horava low energies, at high at one absent emergent fundamentally an as [12]. only universe pear flat gravi- a spin-0 to the leads eliminates also only but not tons projectability 11] the (with [10, and symmetry [9]; condition) U(1) origins a geometric of purely inclusion constraint the the its [8]; Hamiltonian have constant” can local integration sector an the dark as of matter “dark lack to the leads infla- 7]; without [6, problem perturbations horizon tion scale-invariant the to solves universe leads scaling bouncing and anisotropic a spa- the to order 5]; rise high give [4, particular, can In terms a derivative [3]. attracted tial attention has re- of and deal it features standard great cosmology, remarkable the various to of in Applied language sults the [2]. in formulas metric field path-integral spacetime elementary the the frame- with Ho- theory, the as field Recently, in gravity quantum decades quantum of past of work [1]. theory the paths a in proposed different rava Physics several in following forces by driving main the nvriy ao X77871,USA 76798-7316, TX Waco, University, ∗ ()smer n lmnto fsi- rvtn nHorava in gravitons spin-0 of elimination and symmetry U(1) nLaefo CPCSE,PyisDprmn,Baylor Department, Physics GCAP-CASPER, from Leave On ihteprpcieta oet ymtymyap- may symmetry Lorentz that perspective the With of one been has fields gravitational of Quantization olg fMathematics of College nttt o dacdPhysics Advanced for Institute d rbyipoe.Tesrn opigpolmi h atrs matter the in problem scale coupling predi energy the strong whereby an The limit, dow IR improved. 70 healthy erably than a more possesses (from and break reduced complete softly significantly condition be balance can constants detailed gravitat the the in with speeds addition, different and coupling, strong ghost, rsrigdffoopim oicuealclU1 symmetry U(1) local extendin a by include eliminated to be diffeomorphisms preserving can condition projectability the eiaietrs n Λ and terms, derivative ASnmes 46.m 04.50.Kd 04.60.-m; numbers: PACS )dmnin,i re o h hoyto theory the for order in 1)-dimensions, + nti ae,w hwta h pn0gaiosapaigin appearing gravitons spin-0 the that show we paper, this In .INTRODUCTION I. x → b − 1 x & t , M hsc,CogigUiest fPssadTelecommunicat and Posts of University Chongqing Physics, ∗ othat so , → b ω − a h,QagW,adAzogWang Anzhong and Wu, Qiang Zhu, Tao h ol-esrn opigeeg scale. energy coupling strong would-be the z & t. ahmtc,Zein nvriyo ehooy aghu3 Hangzhou Technology, of University Zhejiang Mathematics, h rjcaiiycondition projectability the M ∗ < Dtd coe 7 2018) 27, October (Dated: Λ ω where , uWnShu Fu-Wen (1) M ∗ ob eoe yDiff( by denoted be to as function lapse con- the coupling introduced, and were independent conditions the two of stants, number the [16]. the only 60 reduce than particular, To more In are terms derivative the theory. spatial limit sixth-order the potentially of cou- could powers independently which prediction of 15], proliferation [2, constants a pling to rise gives hand, z o sboe rmtegnrldffoopim onto down ones, diffeomorphisms foliation-preserving general the the from broken is now eursta h rvttoa oeta hudb ob- be should superpotential a potential from gravitational tained the that requires nitga ftegaiainlCenSmn emover term space, Chern-Simons 3-dimensional gravitational a the of integral an epr ftesuperpotential coun- the the dimension(s). of thermodynamics, lower terpart non-equilibrium more or the one boundary in the- has Yet, conformal which field the space, quantum on this a of defined to cor- gravity de- equivalent AdS/CFT gravity without is the and ory space theory of one string on spirit For a fined same where the 17]. [18], in [2, respondence is features it remarkable balance example, detailed several The has in- five condition [2]. only constants contains coupling now dependent action general the conditions, rp,while tropy, rbes[] nldn h ntblt,got[,2] and 21], several [2, with ghost instability, plagued the is including conditions) [3], two problems these (with ory by recently the proposed on as lights further [20]. forces, Verlinde shed gravitational might of This nature [19]. force tropic eoe h upeso nryo ihorder high of energy suppression the denotes utbe must bnoigtegnrldffoopim,o h other the on diffeomorphisms, general the Abandoning ept falteaoermral etrs h the- the features, remarkable above the all of Despite ealdbalance detailed oa etraeatmtclyrsle.In resolved. automatically are sector ional n,tenme fidpnetcoupling independent of number the ing, h ag ymtiso h foliation- the of symmetries gauge the g sarsl,tepolm fstability, of problems the result, a As . to oeso h hoyaeconsid- are theory the of powers ction o1) hl h hoyi tl UV still is theory the while 15), to n z co a ecrdb introducing by cured be can ector ≥ δW δt oaaLfht rvt without gravity Horava-Lifshitz N d = g 2 4.Teascae ag symmetry gauge associated The 14]. [2, /δg eafnto of function a be − ∗ f ij ( 2.Tefre eursta the that requires former The [2]. M, t Lfht rvt without gravity -Lifshitz ) ersnstecrepnigen- corresponding the represents W os hnqn 005 China 400065, Chongqing ions, δx , g F ∼ ). i W R W = Σ g g where , ω − t 3 ζ ly h oeo en- of role the plays ny hl h latter the while only, Γ.Wt hs two these With (Γ). 03,China 10032, i ( t, x ) W , g projectibility sgvnby given is (2) 2

ˆijlk 2 k strong coupling [16, 22]. However, all of them are closely +2Kij a(l k)α + 2(1 λ)K α + a kα ,(6) related to the existence of the spin-0 gravitons [3]. Be- G ∇ − ∇ ∇
o cause of their presence, another important question also where ˆijkl = ijkl , ijkl(λ) (gikgjk + gilgjk)/2 raises: Their speeds are generically different from those G G λ=1 G ≡ − λgij glk, G R Rg /2 and f = (f + f )/2. of gravitational waves, as they are not related by any ij ij ij (ij) ij ji R (R) is the≡ Ricci − tensor (scalar) of g . To have symmetry. This poses a great challenge for any attempt ij ij the U(1) symmetry, we introduce the gauge field A [10], to restore Lorentz symmetry at low energies where it which transforms as has been well tested experimentally. In particular, one i needs a mechanism to ensure that in those energy scales δαA =α ˙ N iα. (7) all species, including gravitons, have the same effective − ∇ speed and light cones. With these in mind, Horava and Then, adding 1 Melby-Thompson (HMT) [10] extended the symmetry (2) to include a local U(1), S = ζ2 dtd3x√gA(R 2Λ ), (8) A − − g U(1) ⋉ Diff(M, ). (3) Z F to Sˆg, one finds that its variation (with Λg = 0) with With this enlarged symmetry, the spin-0 gravitons are respect to α exactly cancels the first term in Eq.(6). To eliminated [10, 23, 24], and the theory has the same long- repair the rest, we introduce the Newtonian prepotential distance limit as that of general relativity. This was ini- ϕ [10], which transforms as tially done in the special case λ = 1, and soon general- ized to the case with any λ [11], where λ is a coupling δαϕ = α. (9) constant defined below. Although the spin-0 gravitons − are also eliminated in the general case, the strong cou- Under Eq.(4), the variation of the term, pling problem raises again [12]. However, it can be solved 2 3 ij by the Blas-Pujolas-Sibiryakov (BPS) mechanism [15], Sϕ = ζ dtd x√gN ϕ 2Kij + i j ϕ + ai j ϕ G ∇ ∇ ∇ in which a new energy scale M is introduced, so that Z ∗ 2 n i 2 2 i
M < Λω, where M denotes the suppression energy of + (1 λ) ϕ + ai ϕ +2 ϕ + ai ϕ K ∗ ∗ − ∇ ∇ ∇ ∇ high order derivative terms of the theory, and Λω the h i 1 ˆijlk

would-be strong coupling energy scale [25]. + 4 ( i j ϕ) a(k l)ϕ +5 a(i j)ϕ a(k l)ϕ Note that in [10, 11] the projectability condition was 3G ∇ ∇ ∇ ∇ ∇ h assumed. In this paper, we shall study the case without +2 ϕ a ϕ +6K a ϕ ,
(10) ∇(i j)(k∇l) ij (l∇k) it, and our goals are twofold: (i) Extend the symmetry io (3) to the case without the projectability condition, so will exactly cancel
the rest in Eq.(6) as well as the one that the spin-0 gravitons are eliminated. (ii) Reduce sig- from the term Λ A in Eq.(8), where G +Λ gij g Gij ≡ ij g nificantly the number of the coupling constants by the and aij iaj . Hence, the total action detailed balance condition, whereby the prediction pow- ≡ ∇ ˆ ers of the theory can be improved considerably. To have Sg = Sg + SA + Sϕ, (11) a healthy IR, we allow it to be broken softly by adding is invariant under (3). all the low dimensional relevant terms [17]. We also note Equation (3) does not uniquely fix . To our sec- that in [26] a U(1) extension of F(R) HL gravity was V ond goal, we introduce the “generalized”L detailed balance recently studied. condition, ˆ = E ijklE g AiAj , (12) II. THE MODEL LV ij G kl − ij where Under the U(1) transformations, the metric coefficients ij 1 δWg i 1 δWa transform as [10], E = , A = , √g δgij √g δai δ N =0, δ N = N α, δ g =0, (4) α α i ∇i α ij 1 2 Wg = 2 Tr Γ dΓ+ Γ Γ Γ , where α is the U(1) generator, Ni the shift vector, and w Σ ∧ 3 ∧ ∧ Z   i the covariant derivative of gij . Since [15] 1 ∇ 3 i n Wa = d x√ga bn∆ ai, (13) ˆ 2 3 Sg = ζ dtdx N√g ( K V (gij ,ak)), (5) Z n=0 L − L X Z where a (ln N), , K Kij λK2 and K = i ≡ i LK ≡ ij − ij ( g˙ij + iNj + j Ni) /(2N), the potential V is invari- − ∇ ∇ L 1 Note the difference between the notations used here and the ones ant under (4), while the kinetic part transforms as HMT used in [10, 11]. In particular, we have Kij = −Kij , Λg = R ΩHMT , ϕ = −νHMT , G = ΘHMT , where quantities with the δS = ζ2 dtd3xN√g α˙ N i α +2αG Kij ij ij K − ∇i N ij superindices “HMT” were used in [10, 11]. Z n
3 with ∆ gij and b being arbitrary constants. λ 1 ∂2B = 1 3λ ψ,˙ (19) ≡ ∇i∇j n − − Now, some comments are in order. First, the term ðφ =2℘ψ, (20) 1/2 i

ai∆ a in principle can be included into Wa, which will 2 2 give rise to fifth order derivative terms. In this paper we where ℘ 1 ζ− β7∂ . Eq.(17) shows that ψ is not shall discard them by parity. Second, it is well-known propagating,≡ and− with proper boundary conditions, one that with the detailed balance condition a scalar field is can set ψ = 0. Then, Eqs.(18)-(20) show that B, A, not UV complete [4], and the Newtonian limit does not and φ are also not propagating and can be set to zero. exist [27]. Following [17], we break it softly by adding Hence, we obtain ψ = B = A = φ = 0, that is, the all the low dimensional relevant terms to ˆV , so that the scalar perturbations vanish identically in the Minkowski potential finally takes the form, L background, similar to that in general relativity. Thus, with the enlarged symmetry (3), the spin-0 gravitons are 2 i 2 2 ij V = γ0ζ β0aia γ1R + ζ− γ2R + γ3Rij R indeed eliminated even without the projectability condi- L − − tion. 2  i 2  i 2  i j  +ζ− β1 aia + β2 a i + β3 aia a j ijh
i
ij
i +β4a aij + β5 aia R + β6aiaj R + β7Ra i IV. STRONG COUPLING AND THE BLAS-PUJOLAS-SIBIRYAKOV MECHANISM 4 ij
i 2 i +ζ− γ5Cij C + β8 ∆a , (14)

h2 4 4
i 4 2 Since the spin-0 gravitons are eliminated, the ghost, where β0 b0, γ5 ζ /w , β8 ζ b1, and Cij de- ≡ ≡ ≡ − ikl instability, strong coupling and different speed problems ij e j notes the Cotton tensor, defined as C k R in the gravitational sector do not exist any longer. But, ≡ √g ∇ l − j the self-interaction of matter fields and the interaction 1 Rδ . All the coefficients, β and γ , are dimensionless 4 l n n between a matter field and a gravitational field can still and arbitrary, except β0 0,γ5 0 and β8 0, as indi- ≥ ≥ ≤ lead to strong coupling, as shown recently in [25] for the cated by their definitions. γ0 is related to the cosmolog- theory with the projectability condition. In the following 2 ical constant by Λ = ζ γ0/2, while the IR limit requires we shall show that this is also the case here. Since the γ = 1, ζ2 =1/(16πG), where G is the Newtonian con- 1 − proof is quite similar to that given in [25], in the following stant. From the above, we can see that with the “gen- we just summarize our main results, and for detail, we eralized” detailed balance condition softly breaking, the refer readers to [25]. A scalar field χ with detailed bal- number of independent coupling constants is significantly ance condition softly breaking is described by Eqs.(3.11) reduced from more 70 to 15: G, Λ, λ, βn, γs, (n = and (3.12) of [17]. In the Minkowski background, we have 0, ..., 8; s =2, 3, 5). In the following, we shall show that χ¯ =0= V (0) = V ′(0). Considering the linear pertur- such a setup is UV complete and IR healthy. bations χ = δχ, we find that the quadratic part of the scalar field action reads, III. ELIMINATION OF THE SPIN-0 GRAVITONS (2) 3 1 2 1 2 1 2 S = dtd x fχ˙ V ′′χ (1+2V ) (∂χ) χ 2 − 2 − 2 1 Z " To show this, it is sufficient to consider linear pertur- 2 2 2 2 4 2 2 4 bations in the Minkowski background, given by +c1A∂ χ V2(∂ χ ) V4′χ∂ χ + σ3 ∂ χ∂ χ , (21) − − # N = 1+ φ, N = ∂ B, g = (1 2ψ)δ +2E , i i ij − ij ,ij A = δA, ϕ = δϕ. (15) where f is a constant, usually chosen to be one [17]. Since (2) Sχ does not depend on ψ, B and φ explicitly, the varia- Choosing the gauge, E =0= δϕ, we find that (2) (2) (2) tions of the total action S = Sχ +Sg with respect to (2) 2 3 2 2 them will give the same Eqs.(18) - (20), while Eq.(17) is S = ζ dtd x (1 3λ)(3ψ˙ +2ψ∂˙ B) 2 g − replaced by ψ = c1χ/(4ζ ). Integrating out φ, ψ, B, A, Z n2 2 2 2 2 we obtain +(1 λ)(∂ B) φð +4β ζ− ∂ ψ ∂ φ − − 7 2 2 2 2(ψ 2φ +2A +α1ψ∂ )∂ ψ ,
(16) S(2) = M 2 dtd3x χ˙ 2 α ∂χ m2 χ2 − − − 0 − χ o Z " 2 ð 2 2
where α1 ζ− (8γ2 +3γ3) and β0 +ζ− (β2 +β4)∂ 2 4 4 ≡ (2) ≡ − 2 4 4 6 2 ℘ χ ζ− β8∂ . Variations of Sg with respect to A, ψ, B, and M − χ∂ χ + M − χ∂ χ + γχ∂ , (22) − A B ð φ yield, respectively,   # ∂2ψ =0, (17) 2 2 2 2 2 where M 2πGc1/ cψ + f/2, γ 4πGc1/M , α0 2 2 2 ≡ 2 | |2 2 ≡2 2 ≡ 1 2 2∂ (A + ψ + α1∂ ψ) 2℘∂ φ 1+2V1 4πGc1 /(2M ), MA M / 2πGα1c1 + V2 + ψ¨ + ∂ B˙ + = , (18) 4 − 2 2 2 ≡ 2 2 3 3(1 3λ) 3(1 3λ) V4′ , MB M /σ3, mχ V ′′/(2M ), and cψ = (1 − − ≡
≡ −
4

λ)/(3λ 1). Clearly, the scalar field is ghost-free for MA & MB, the sixth-order derivative terms become dom- f > 0,− and stable in the UV and IR. In fact, it can be inant for processes with E & M , and the quadratic made stable in all energy scales by properly choosing the action (22) now is invariant only under∗ the anisotropic coupling coefficients Vn. rescaling, To consider the strong coupling problem, one needs to 3 i 1 i calculate the cubic part of the total action, which takes t b− t, x b− x , χ χ. (26) the form, → → → Then, one finds that under the new rescaling all the terms (3) 3 1 2 1 ,i of λs in Eq.(23) become scaling-invariant, while the rest S = dtd x λ1 2 χ¨ χ∂ χ + λ2 2 χ¨ χ,iχ δ ( ∂ ∂ scale as b− with δ > 0. The former are strictly renor- Z     malizable, while the latter are superrenormalizable [28]. 2℘ 2℘ χ˙ +λ χ˙ 2 1 χ + λ χ∂˙ i 1 χ∂ Therefore Eq.(25) makes the strong coupling problem dis- 3 ð − 4 ð − i ∂2       appeared. ∂i∂j ∂ 2℘ V. CONCLUSIONS +λ χ˙ i χ˙ ∂ +3 χ 5 ∂2 ∂2 j ð       In this paper, we have shown that the spin-0 gravitons ∂i +λ χχ˙ ,i χ˙ + ... , (23) in the HL theory even without the projectability condi- 6 ∂2   ) tion can be eliminated by the enlarged symmetry (3). An immediate result is that all the problems related to them where “...” represents the terms that are independent of disappear, including the ghost, instability, strong cou- λ, so they are irrelevant to the strong coupling prob- pling and different speeds in the gravitational sector. In lem. λs are functions of λ, f, ζ and ci. In particular, addition, the requirements [15], λ 1 β , 0 <β < 2, λ = c3/(64ζ4 c 4), which will yield the strong coupling | − |≃ 0 0 5 1 | ψ| now become unnecessary, which might help to relax the energy Λω given below. The exact expressions of other observational constraints. Moreover, it is exactly because coefficients are not relevant to Λω, so will not be given of this elimination that softly breaking detailed balance here. For a process with energy E MA,MB, the first condition can be imposed, whereby the number of inde- two terms in Eq.(22) are dominant,≪ and S(2) is invariant 1 i 1 i pendent coupling constants is significantly reduced from under the relativistic rescaling t b− t, x b− x ,χ → (3) → → more than 70 to 15. This considerably improves the pre- bχ. Then, all the terms of λs in S scale as b. Asa diction powers of the theory. Note that, without the result, when the energy of a process is greater than a elimination of the spin-0 gravitons, the strong coupling certain value, say, Λω, the amplitudes of these terms are problem will appear in the gravitational sector and can- greater than one, and the theory becomes nonrenormal- not be solved by the BPS mechanism, because this condi- izable [28]. In the present case, it can be shown that tion prevents the existence of sixth-order derivative terms (3) 3/2 in S . M g Λ pl M c 5/2. (24) These results put the HL theory with/without the pro- ω ≃ c pl| ψ|  1  jectability condition in the same footing, and provide However, if a very promising direction to build a viable theory of quantum gravity, put forwards recently by HMT [10]. Certainly, many challenging questions [3] need to be an- M < Λω, (25) ∗ swered before such a goal is finally reached. where M = Min.(MA,MB), one can see that before Acknowledgements: This work was supported ∗ the strong coupling energy Λω is reached, the high or- in part by DOE Grant, DE-FG02-10ER41692 (AW); der derivative terms in Eq.(22) become large, and their NNSFC No. 11005165 (FWS); and NNSFC No. effects must be taken into account. In particular, for 11047008 (QW, TZ).

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