A Naturally Minute Quantum Correction to the Cosmological Constant Descended from the Hierarchy Shu-Heng Shao1,3∗ and Pisin Chen1,2,3,4† 1. Department of Physics, National Taiwan University, Taipei 10617, Taiwan, R.O.C. 2. Graduate Institute of Astrophysics, National Taiwan University, Taipei 10617, Taiwan, R.O.C. 3. Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan, R.O.C. 4. Kavli Institute for Particle Astrophysics and Cosmology, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, U.S.A. (Dated: January 4, 2011) Wedemonstratethatanaturallysmallquantumcorrection, ortheCasimirenergy,tothecosmo- logical constant can arise from amassive bulkfermion field in the Randall-Sundrummodel. Under the assumption that the ground value of four-dimensional effective cosmological constant is zero, 1 which can only be attained by some other means, we show that its quantum correction originated 1 from theCasimir energy,that is thevacuumenergy associated with thenon-trivialtopology of the 0 extra dimension, can be as small as the observed dark energy scale without fine-tuningof thebulk 2 fermion mass. To ensure the stabilization of the system, we discuss two stabilization mechanisms n underthissetup. ItisfoundthattheGoldberger-Wisemechanismcanbesuccessfullyintroducedin a thepresence of a massive bulkfermion, without spoiling the smallness of thequantumcorrection. J 2 PACSnumbers: 04.50.-h,04.62.+v,11.10.Kk,11.25.-w,11.30.Pb ] h I. INTRODUCTION verse[3]impliesthatthecosmologicalconstant(CC)may t - be nonzero but minute, which corresponds to an energy p e The hierarchy between the Planck scale MPl ∼ 1019 density ρobs ≡ MC4C ∼ (10−3eV)4. Since the quantum h GeV and the electroweak scale M ∼ TeV has long vacuum energy naturally contributes to CC, which is at EW [ been a puzzle in high energy physics. In the past thePlanckscale,thelongstandingcosmologicalconstant 3 decade, there has been two famous solutions to the hier- problem (CCP) [4] can be casted as a naturalness prob- v archy problem, both involving the idea of extra dimen- lem: Why is ρobs smallerthan the Planckenergydensity 0 sions. The first one is the Arkani-Hamed-Dimopoulos- ρPl ∼(1019GeV)4 by ∼124 orders of magnitude? 2 Dvali (ADD) model [1], and the second is the Randall- 9 The CCP is actually more severe than the apparent Sundrum (RS) model [2]. In this Letter we shallonly fo- 1 minute value of CC. In the context of the quantum field cusonthelatter. InRSmodel,twoparallelflat3-branes, . theory,eachfieldcontributesavacuumenergyatthecut- 5 withequalbutopposite branetensions,areembeddedin 0 a five-dimensional Anti-de-Sitter (AdS) bulk. The RS offscale,say,thePlanckscale. Thisvacuumenergycould 0 bedroppedwithoutharmforthequantumfieldtheoryin metric reads 1 flat spacetime, since the interaction between particles is v: ds2 =e−2krc|φ|η dxµdxν +r2dφ2, (1) governedbythe gradientofthepotential,ratherthanits i µν c absolute value. On the contrary, in general relativity all X where k is the AdS curvature of the order M, the five- forms of energycontribute to the dynamics of the space- r dimensional Planck scale, and r is the radius of the time, hence the vacuum energy gravitates in the same a c S1/Z2 orbifold. xµ are the four-dimensional coordinates wayas the cosmologicalconstant. In addition, the spon- andφistheextradimensioncoordinaterangingfrom−π taneous symmetry breaking of the quantum fields also toπ. Thehiddenbraneislocatedatφ=0whilethe vis- gives rise to vacuum energy. Adding all these huge con- ible brane at φ = π, on which the standard model fields tributions from the quantum field theory, together with reside. Underthisconstruction,thehierarchyproblemis the bare cosmological constant, one is supposed to re- naturallysolvedviathewarpfactore−πkrc alongthefifth ducethesumtotheobservedcosmologicalconstantscale dimensionwithoutfine-tuningoftheparameters. Specif- ρobs ∼ (10−3eV)4. It is logically possible that the bare ically, the electroweak scale on the visible brane can be cosmologicalconstantmanagesitselfinanextremelydel- naturally bridged with the Planck scale on the hidden icate manner so that the resulting vacuum energy den- brane if kr ≃12. sity is 124 orders of magnitude smaller than their sepa- c There is another mysterious hierarchy problem in rate contributions, but such cancellation does not seem physics. The accelerating expansion of the present uni- appealing and natural. For further details, the CCP is summarized in [4] and references therein. General con- siderations of CCP in the brane-world scenario can be found in [5]. In particular, a “self-tuning” mechanism of ∗Electronicaddress: [email protected] the cosmologicalconstant is discussed in [6][7] under the †Electronicaddress: [email protected] brane-worldscenario [8]. 2 With regard to these two hierarchy problems, it is in- Apart from the order of the magnitude of the cosmo- teresting to notice that the hierarchy M /M ∼1016 logical constant, the sign is also crucial in order to give Pl EW betweenthePlanckscaleandtheTeVscaleisroughlythe rise to an accelerating universe instead of a decelerating square root of the hierarchy M /M ∼ 1031 between one. The sign of the Casimir energy has been discuss in Pl CC the Planck scale and the observedcosmologicalconstant detailsin[28](seealso[29][30]),anditispointedoutthat scale. The surprising numerical coincidence prompts us for a scalar field with periodic boundary condition, the to ask: Are these two hierarchy problems related? The sign is generically negative. Therefore in the present pa- idea that these two hierarchy problems are actually re- perweconsiderabulkfermionfieldwithmassminstead lated has been pursued by one of us (PC) and other au- of a scalar field. As we will see, the fermionic nature of thors[9–11],inwhichsomespecificmechanismswerede- thefielddrasticallychangethesignoftheCasimirenergy vised to induce a cosmologicalconstant scale of and therefore yield a desired positive value. There are several motivations for introducing a mas- M2 sive bulk fermion, summarized in [25]. In particular, the M = EW, (2) CC M bulk fermion arises naturally as the superpartner of the Pl radion field in a supersymmetric theory, especially the whichisroughlythecurrentobservedscale. Morespecif- string theory realization,of the brane-worldscenario. In ically,itwasproposedin[10,11]thatthe minute cosmo- the context of particle physics, Grossman and Neubert logical constant might actually be the result of a double used a massive bulk fermion to understand the neutrino suppression of the Planck scale by the same hierarchy mass hierarchy [31]. Using their result, Kitano demon- factor a=e−πkrc ≃10−16 in the RS model. That is, the stratedsome bounds onthe flavorchangingprocess [32]. expression Some other phenomenological studies have invoked bulk fermions as well [33–38]. M =a2M (3) CC Pl Weemphasizethatwewillmakenoattemptinexplain- ingthetraditionalCCP,thatis,thedelicatecancellation mightbeanaturalresultintheRSmodel. Followingthe between various vacuum energy contributions and the same spirit, in this Letter we provide an explicit mecha- bare cosmologicalconstant; rather, we focus on the pos- nism that actually proves this ansatz. sibility that an extremely small quantum correction can Our demonstration is associated with the Casimir en- be produced quite naturally from a massive bulk field, ergy of bulk fields in the RS geometry. We assume provided that the infinite part (or the very large part) that the ground value of four-dimensional effective CC, canbe properly renormalizedinto the counterterms. We namely,thesumofitsclassicalvalueaswellasthediver- alsodo notinclude the gravitonself-interaction,whichis gence (or the very large term) in the calculation of vac- in fact much larger than the observed CC. Nevertheless, uum energy, is identically zero provided by other mech- we consider it a step forward in demonstrating that a anisms. We then demonstrate that the inevitable contri- naturallyminutequantumcorrectiontoCCisattainable bution to CC resulted from the non-trivial topology of without fine-tuning, in a similar spirit as in [28]. RS geometry induced by a massive bulk fermion can be assmallastheobservedCCscale,M ,byvirtueofthe CC mass suppression. The Casimir energy of a bulk scalar II. CASIMIR ENERGY FOR A MASSIVE BULK field [12–20] or a bulk fermion field [21–26] in RS model FERMION FIELD have been investigated in great details. It is known that the presence of a massive bulk field would exponentially The one-loop effective potential for a massive bulk suppress the Casimir energy by a factor e−2mr in flat fermion is [25] spacetime [27], where r is the distance between two par- allel plates. We will show that this is also true for the V =VR+VRa4 eff h v non-flat RS geometry, and the above-mentioned ansatz k4a4 ∞ K (t)I (at) can be realized without fine-tuning. − dt t3 ln 1− µ µ , (4) It is insufficient, however, to simply demonstrate a 16π2 Z0 (cid:20) Iµ(t)Kµ(at)(cid:21) small quantum correction to the CC in our scheme. In where K (t) and I (t) are the modified Bessel functions order to give a satisfactory analysis of the CCP in the µ µ andVR are the shifts of the renormalizedbrane tension brane-worldscenario, it is necessary to consider in addi- h,v from their classical values, V0 = −V0 = 24M3k. The tionthe stabilizationissueinthe presenceofthe massive h v definition of µ is bulk fermion. The modulus stabilization in the brane- world scenario is known to be intimately related to the 1 m CCP [5]. In this paper we will discuss two stabilization µ= ± , (5) 2 k mechanisms under the present setup. In particular, we will show that the Goldberger-Wise mechanism can be where the ± stands for the type I and type II boundary successfully introducedin the presence ofa massivebulk condition, respectively [25]. (The minus sign is totally fermion, without spoiling the smallness of the quantum acceptable, since the Dirac spinor mass changes sign un- correction. der parity.) 3 The small a expansion of (4) reads: shifts VR are in general determined by the renormal- v,h ization conditions: k4a4 V =VR+VRa4+ f(µ)a2µ+O(a2µ+2), (6) eff h v 16π2 Veff(aobs)=Λobs, dV eff where (aobs)=0. (10) da 2 ∞ K (t) f(µ)≡ dt t2µ µ . (7) We then find, in this case, µΓ(µ)2 Z0 Iµ(t) f(2) Notice that a2µ in (6) is a significant suppression factor VhR ≃ 16π2a8k4+Λobs, (11) for generic values of µ=1/2±m/k. This is in complete f(2) analogy to the suppression factor e−2mr for the Casimir VvR ≃−8π2 a4k4. (12) energyofamassivefieldinflatspacetime[27],wherer is thedistancebetweentwoparallelplates. Thesuppression So the total values of the the brane tensions are would be effective, however, only if the function f(µ) is not exponentially large. This is indeed the case for V =V0+VR ≃(1019GeV)4+(10−4eV)4, (13) h h h m/k ∼ O(1), which is a natural choice for m since the V =V0+VR ≃−(1019GeV)4−(TeV)4. (14) 5DPlanckmassM(∼k)istheonlyfundamentalscalein v v v the RSmodel. Somevalues ofinterestforf(µ)arelisted It is interesting to note that the two terms in (11) are of below: the same order, in contrast to that for the massless bulk fermion. However, the fine-tuning problem reappears at f(0.6)∼2.1, f(1.3)∼8.7, f(2.0)∼29. (8) the stage of stabilization, as seen in (13) and (14). This motivates us to look for an alternative mechanism. For our purpose, we choose µ≃2, which corresponds to m≃±1.5k. Then we have B. Goldberger-Wise Mechanism f(2) V ≃VR+VRa4+ a8k4. (9) eff h v 16π2 In the Goldberger-Wise mechanism, a massive bulk The last term a8k4 in (9) corresponds to a mass scale scalarfield Φ with a brane self-interactioninduces a sta- a2k ∼ M , which is the desired order of magnitude bilizing potential [41]: CC forthe observedCCandisobtainedwithoutfine tuning. This is the main result of this paper. It should be men- VΦ(a)=4ka4 vv−vhaǫ 2, (15) tioned that as in the original RS model, the curvature k (cid:0) (cid:1) ischosentobesmallerthanM. Thusevenifthefermion where ǫ=m2/4k2 is treatedas a smallnumber andv Φ v,h mass m is larger than k, we may still choose it to be areofmassdimension3/2. Thispotentialisaddedto(9) smaller than M. We again emphasize that the quantum to ensure the stability. correction (the third term) in (9) is positive due to the There are two concerns about whether the introduc- fermionic nature of the field, in contrast to the negative tionofthebulkscalarfieldΦwouldoverwhelmthesmall Casimir energy for bulk scalar field found in [12]. fermionicCasimirenergy. First,themassmΦ ofthebulk scalar field is assumed to be small compared with k, but still roughly of the same order. Therefore its induced III. STABILIZATION Casimirenergymightbe largerthanthatbythe fermion field,whichissuppressedbyalargemassm≃1.5k. Sec- If we naively assume VR to be zero, i.e., the brane ond, since now the total effective potential is the sum of v,h (9) and (15), its minimum value might deviate signifi- tensions are not shifted by a finite amount due to quan- cantly from zero. It turns out that we are actually safe tum corrections,then the value of the effective potential from these problems. obtainedabovewouldindeedcorrespondtothe observed With regard to the first concern, let us consider the CC. However, such potential alone cannot stabilize the Casimirenergyofthebulkscalarfieldminimallycoupled orbifoldradius due to its monotonic dependence ona. A to the curvature (which is the case for [41]) [12], stabilization mechanism is therefore necessary. k4a4 VCas,Φ =−16π2g(ν)a2ν +O(a2ν+2), (16) A. Garriga-Pujol`as-Tanaka Mechanism where g(ν) is some unimportant numerical constant of Considerfirstthe effective potential(9) inducedsolely the orderunity. We note thatthe Casimirenergyis sup- by the massive bulk fermion [21]. The brane tension pressedby the factor a2ν, in a similar fashionas that for 4 the fermion. The crucial difference is that for the scalar whereC issomeunimportantconstantoftheorderunity. field case, It is clear that the second term in (20) is much smaller thanthe leadingterm,whichisindeedofthe scaleM4 . CC ν = 4+m2/k2. (17) Φ q So even for ǫ = m2Φ/4k2 ≪ 1 (but mΦ . k), ν is still slightly larger than 2, and it thus induces a Casimir en- IV. DISCUSSIONS ergy of the same order as that for the fermion. That is, the Casimir energyinduced by Φ does not overwhelm that by the bulk fermion. This is in contrast with the Insummary,wehavedemonstratedthatabulkfermion fermion case, where only a fermion with mass ∼ 1.5k with mass m≃±1.5k can induce a Casimir energy that would correspond to a Casimir energy about a8k4. The is of the observed cosmological constant scale. To en- origin of this difference lies in that the bulk scalar field sure the stability of the system, a self-interacting bulk Φ in [41] is minimally coupled to the curvature, rather scalarfield Φ with mass mΦ .k is introduced as in [41]. thanconformallycoupled. Hadweinvokedaconformally As shown above, the Casimir energy induced by Φ is of coupled bulk scalar field, ν would have been replaced by the same order as that by the fermion, and the resulting 1/4+m2/k2 [14]. Then the corresponding Casimir leading term of the minimum effective potential remains Φ penergy would be of the order a5k4 and would dominate to be of the desired order a8k4. over the fermionic contribution. It should be mentioned that by setting VR to zero, it v,h Thesecondconcernismorestraightforward. Thetotal actually implies that the infinite part of the brane ten- effective potential is sionshifts must be finely tuned to absorbthe divergence (orverylargeterm)inthe calculationofthe Casimiren- Veff(a)≃Bk4a8+4ka4 vv−vhaǫ 2, (18) ergies, in such a way that no residue is left behind. We (cid:0) (cid:1) made no attempt to explain this delicate cancellation in where B = [f(2)−g(2)]/(16π2) is a numerical constant the process of renormalization, but focused only on the of the order unity. Note that we have combined the possibilitythatanextremelyminute quantumcorrection Casimir energies for bulk scalar and fermion fields into can indeed be achieved without fine-tuning of the bulk the firstterm, which will be treatedas a perturbation to field mass. In this regard, our philosophy is similar to the second. In addition, we have set the brane tension that in [28]. shifts VR to zero. v,h It would be interesting to further pursue the possible To the zeroth order, amin = (vv/vh)1/ǫ ≡ a0 ∼ 10−16. relationsbetween the bulk fermion andscalarfields, and Assuming that the true minimum is amin = a0 + δ, also their phenomenological implications. In addition, putting this into the derivative of (18) and setting it to the dynamical origin for such massive bulk fermion field zero, we have is relevant and should be further investigated. dV eff 0= (amin =a0+δ) da =8Bk4a7+(56Bk4a6+8kv2a2ǫ+2ǫ2)δ+O(δ2). (19) 0 0 h 0 Acknowledgments The first term in the parenthesis can be dropped due to its a6 dependence. Further assuming that v areof the 0 v,h We thank D. Maity and A. Flachi for their useful sug- Planck scale, which is the only scale for the parameters in RS model, we find δ ∼a5−2ǫ. gestionsandcommentsonthesubject. Wearealsograte- 0 ful to K. Y. Su and C. I. 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