Gravitational relaxation of electroweak hierarchy problem
KKEK-TH-1928
Gravitational relaxation of electroweak hierarchy problem
Hiroki Matsui ∗ and Yoshio Matsumoto † Department of Physics, Tohoku University, Sendai, 980-8578 Japan National Institute of Technology, Tsuyama College, Tsuyama, Okayama 708-0824, Japan
In the present paper, we discuss gravitational relaxation scenarios for the electroweak hierarchyproblem. We clearly show that modified gravity theory can naturally relax the electroweak hier-archy where conformal transformation provides a crucial information about what modified gravitytheories are favored for the electroweak hierarchy. The conformal transformation connects differenttheories and rescaling the metric changes the dimensional parameters like the Higgs mass or thecosmological constant in different frames drastically. When the electroweak scale is naturally real-ized by dynamical and running behavior of dilatonic scalar field or unknown scaling parameter, themodified gravity theories could relax the electroweak hierarchy problem. We discuss the theoreticalvalidity and the phenomenological constraints of the gravitational relaxation models.
I. INTRODUCTION
The electroweak hierarchy problem has been recog-nized as the most notorious difficulty for the high-energyphysics in the past decades and often rephrased as a prob-lem of naturalness [1–4]. The naturalness that the low-energy effective field theory should not be extremely sen-sitive to the high-energy theory is a theoretical and rea-sonable presumption. Actually, dimensional parameterslike scalar masses are as large as the ultraviolet (UV) cut-off scale without involving any special fine-tuning of theparameters or any symmetry. For the Standard Model(SM) case, the Higgs boson mass grows up to the UVcut-off scale M UV by the quadratically divergent quan-tum corrections δM H (cid:39) α (4 π ) M , (1)where δM H should not be much larger than the observedHiggs boson mass M obsH = 125 .
09 GeV [5–7]. The major-ity of theoretical efforts to solve the Higgs naturalness orelectroweak hierarchy problem assume a TeV-scale newphysics and many models has been proposed, e.g. super-symmetry, extra-dimensions and compositeness. How-ever, these prominent proposals has been suffered fromthe observed Higgs boson mass M obsH = 125 .
09 GeV andthe current experimental constraints on new physics. ∗ [email protected] † [email protected] On the other hand, the cosmological constant problem is moreserious from the viewpoint of the naturalness or hierarchy prob-lem. The quantum radiative corrections to the vacuum energydensity ρ vacuum which is dubbed zero-point vacuum energy en-larges up to the cut-off scale M UV as follows: δρ vacuum = 12 (cid:90) M UV d k (2 π ) (cid:112) k + m = M π + m M π + m π log (cid:32) m M (cid:33) + · · · , which is much larger than the dark energy 2 . × − eV in thecurrent Universe. Recently, theoretically different approaches to the elec-troweak hierarchy problem have been explored in Ref.[8–16] which is based on the cosmological relaxation mech-anism of Ref.[17] explaining dynamically the smallnessof the cosmological constant. This relaxation mechanismis based on the cosmological evolution of the Higgs fieldand the axion-like field in the inflationary Universe, andcan lead to the naturally small electroweak scale againstthe cut-off scale.In this paper we discuss other relaxation scenarios tothe electroweak hierarchy problem by involving the grav-itational modification which has been proposed and dis-cussed somewhat obscurely in Ref.[18–29]. The dilatonicscalar field non-minimally coupled to the gravity providesthe relaxation of the electroweak hierarchy or cosmolog-ical constant problem by its dynamical and running be-havior [18–23]. On the other hand, so-called vacuum en-ergy sequestering scenario [24–29] which minimally ex-tends the gravity action with the scaling parameter pro-poses the relaxation of the current vacuum energy andrecently has been discussed [30–33]. By using the con-formal transformation or rescaling the metric we clearlyshow that the extension of the gravity sector can relaxthe electroweak hierarchy. The conformal transformationconnects different gravity theories and rescaling the met-ric drastically changes dimensional parameters like theHiggs boson mass or cosmological constant in the differ-ent frame. We point out that the conformal transforma-tion provides a crucial information about what modifiedgravity theory can naturally relax the electroweak hier-archy problem. We consider the several gravitational re-laxation models, and discuss the theoretical validity andphenomenological constraints.The layout of this paper is the following: In Section IIwe introduce the basic formulation for the gravitationalrelaxation scenarios and discuss why the extended grav-ity theory relax the electroweak hierarchy problem usingthe conformal transformation. In Section III we considerthe running gravitational relaxation scenarios where thequantum equivalence between in the Einstein and Jordanframes is important. In Section IV we apply the vacuumenergy sequestering for the electroweak hierarchy prob- a r X i v : . [ h e p - ph ] A p r lem as a example of the gravitational modification. Fi-nally, in Section V we summarize the conclusion of ourwork. II. GRAVITATIONAL RELAXATION FOR THEELECTROWEAK HIERARCHY
In this section, we introduce the gravitational relax-ation scenario for the electroweak hierarchy problem. Weconsider modified gravity theory including the Higgs fieldΦ and the dilatonic scalar field χ non-minimally coupledto the gravity. The classic action is written by S ⊃ S gravity + S Higgs . (2)The action for the gravity sector including the dilatonicscalar field χ is given by S gravity = (cid:90) d x √− g (cid:18) F ( χ ) R − G ( χ )2 g µν ∇ µ χ ∇ ν χ − V ( χ ) (cid:19) , (3)The action for the Higgs sector is given by S Higgs = − (cid:90) d x √− g (cid:18) g µν ∇ µ Φ † ∇ ν Φ + V (cid:0) Φ † Φ (cid:1)(cid:19) , (4)where the (bare) Higgs potential can be written by V (cid:0) Φ † Φ (cid:1) = Λ b + M (cid:0) Φ † Φ (cid:1) + λ (cid:0) Φ † Φ (cid:1) , (5)Now, we assume that the Higgs mass M Φ and the cos-mological constant Λ b are the UV scale to be Λ / (cid:39) M Φ (cid:39) M UV . If there exists exact supersymmetry orconformal symmetry, such symmetries can force the cos-mological constant or the Higgs mass parameter to besmaller than the cut-off scale. However, such symmetriesare always broken in real world and the dimensional pa-rameters grow in proportion to the breaking scale. Thus,we eventually encounter the hierarchy problem via thesymmetry breaking scale, and that is the situation in thestandard SUSY models. Generally, it is difficult to pro-tect the dimensional parameters from both large classicaland quantum corrections of the high-energy physics andthat is the reason why the hierarchy problem is thoughtto be serious.Now, we rescale the metric via the conformal transfor-mations as follows [34]: g µν → g µν = Ω ( χ ) g µν , (6) g µν → g µν = Ω − ( χ ) g µν , (7) √− g → (cid:112) − g = Ω ( χ ) √− g. (8)The scalar curvature is transformed as follows: R = 1Ω ( χ ) (cid:20) R − (cid:3) Ω ( χ )Ω ( χ ) (cid:21) , (9) where (cid:3) denotes the covariant d’Alembertian operatorand satisfies, (cid:3) Ω = g µν ∇ µ ∇ ν Ω = 1 √− g ∂ µ (cid:2) √− gg µν ∂ ν Ω (cid:3) . (10)Thus, the action for the gravity sector can be transformedas follows: S gravity = (cid:90) d x (cid:112) − g (cid:18) M R − G ( χ )2 g µν ∇ µ χ ∇ ν χ − V ( χ ) (cid:19) , (11)where we write down the action in Einstein frame wherethe scalar curvature is not multiplied by the scalar field,and Ω ( χ ), F ( χ ), V ( χ ) and G ( χ ) are given byΩ ( χ ) = 2 F ( χ ) M , V ( χ ) = V ( χ )Ω ( χ ) , (12) G ( χ ) = G ( χ )Ω ( χ ) + 6 M Ω ( χ ) ∇ µ Ω ∇ ν Ω ∇ µ χ ∇ ν χ + · · · . (13)where M pl = 1 / (8 πG ) / = 2 . × GeV is the reducedPlanck mass and the related Newton’s constant has tightconstraints from the cosmological observations [35, 36].For the Higgs sector the action is given by S Higgs = − (cid:90) d x √− g (cid:18) g µν ∇ µ Φ † ∇ ν Φ+ Λ b + M (cid:0) Φ † Φ (cid:1) + λ (cid:0) Φ † Φ (cid:1) (cid:19) , (14)The Higgs field Φ are transformed asΦ → H = Ω − ( χ ) Φ , (15)The Higgs potential is transformed as follows: V (cid:0) H † H (cid:1) = Λ b Ω ( χ ) + M Ω ( χ ) (cid:0) H † H (cid:1) + λ (cid:0) H † H (cid:1) . (16)Note that the action in the SM is conformally invariantexcept for the Higgs potential. On the other hand, theaction of the gauge or fermion fields are only rescaled viathe conformal transformations and the couplings are notchanged. As the mathematical manipulation, there areseveral metric frames, e.g. Jordan frame (String frame)and Einstein frame. However, we comment that there isno consensus about physical equivalence of these framesover the years [37–44] and one should not determine aunique physical frame in modified gravity theory.The modified gravity theory like extra-dimensions orstring theory is often written by Jordan frame. Theconformal transformation from Jordan frame to Einsteinframe suppress the dimensional parameters of the ordi-nary SM via the scaling parameter. Our set-up is similarto the Randall-Sundrum model [45] where the large hi-erarchy is suppressed by the exponential warping factor e − kr c φ which depends on an addition extra-dimension .In this scenario, the four-dimensional components of thebulk metric g µν and the four-dimensional physical met-ric g µν have the relation g µν = e − kr c φ g µν where k isa Planck scale constant and φ is the extra-dimensionalcoordinate with the size r c . By using the rescalingphysical metric g µν instead of the bulk metric g µν , thisaction can be written as, S Higgs = − (cid:90) d x (cid:112) − ge − kr c φ (cid:18) e kr c φ g µν ∇ µ Φ ∇ ν Φ+ Λ b + M (cid:0) Φ † Φ (cid:1) + λ (cid:0) Φ † Φ (cid:1) (cid:19) = − (cid:90) d x (cid:112) − g (cid:18) g µν ∇ µ H ∇ ν H + Λ + M H (cid:0) H † H (cid:1) + λ (cid:0) H † H (cid:1) (cid:19) , (17)where H is the transformed Higgs field satisfying the re-lation H = e kr c φ Φ. The Higgs mass parameter M Φ inthe fundamental higher-dimensional theory can be sup-pressed when measured with the rescaling physical metric g µν and become the order of the electroweak scale with-out difficulty as the following, M H = e − kr c φ M Φ (18) A. Gravitational relaxation models
Let us discuss several gravitational relaxation models.If we take Ω ( χ ) = M /χ and F ( χ ) = M / χ , thetransformed Higgs potential is given by V (cid:0) H † H (cid:1) = Λ b M χ + M M χ (cid:0) H † H (cid:1) + λ (cid:0) H † H (cid:1) . (19)where the Higgs mass and cosmological constant are sup-pressed and screened off from the cut-off scale. Next, The five-dimensional metric in the Randall-Sundrum modeltakes the form ds = e − kr c φ η µν dx µ dx ν + r c dφ , where η µν is the 4D Minkowski metric. In the Randall-Sundrum model, the four-dimensional action forthe gravity sector is given by S gravity ⊃ (cid:90) d x (cid:90) dφ M r c e − kr c | φ | (cid:112) − g R, where M is the five-dimensional Planck scale and R is con-structed by the rescaling metric g µν . Ther 4D Planck scale M pl can be determined as follows: M = M r c (cid:90) dφ e − kr c | φ | = M k (cid:110) − e − kr c π (cid:111) . we consider Ω ( χ ) = χ /M , F ( χ ) = χ /
2, the trans-formed Higgs potential can be written as V (cid:0) H † H (cid:1) = M Λ χ + M M χ (cid:0) H † H (cid:1) + λ (cid:0) H † H (cid:1) . (20)which correspond to the screen scenario for the cosmolog-ical constant by so-called cosmon field [46–49]. When theclassic scalar field or vacuum expectation value (VEV)become larger and larger χ (cid:29)
1, the Higgs mass orthe cosmological constant are sufficiently suppressed andasymptotically vanish. Next we consider the specific dila-ton model [22] where the action can be written as follows: S gravity+Higgs = (cid:90) d x √− g (cid:18) M e χ/η R − g µν ∇ µ χ ∇ ν χ − λ χ (cid:0) χ − v χ (cid:1) − g µν ∇ µ Φ † ∇ ν Φ − Λ b − M (cid:0) Φ † Φ (cid:1) − λ (cid:0) Φ † Φ (cid:1) (cid:19) . (21)where we assume that M is the order of the cut-off scaleto be Λ / (cid:39) M Φ (cid:39) M UV . Let us transform this actioninto Einstein frame by rescaling the metric, g µν = M M e χ/η g µν (cid:39) M M e v χ /η g µν . (22)The Higgs mass parameter M Φ can be exponentially sup-pressed as the following M H (cid:39) M pl M Φ M UV e − v χ /η (23)where we can have two approaches and interpretations.If we regard g µν as the physical metric, the Planck-massscale emerges dynamically by the spontaneous symmetrybreaking of the dilaton symmetry as M pl (cid:39) M UV e v χ /η .We can solve the large hierarchy problem by assumingΛ / (cid:39) M Φ (cid:39) M UV (cid:39) M EW where M EW express theelectroweak scale. On the other hand, we can regard g µν as the physical metric and set M EW (cid:39) M pl e − v χ /η . Thisapproach resembles the Randall-Sundrum model and thesimplest possibility of gravitational relaxations althoughthe quantum gravity effects might appear above the elec-troweak scale [22]. In these models, the dimensional pa-rameters like the Higgs boson mass, or even the cosmo-logical constant are screened off from the cut-off scale.However, we can not solve both the electroweak hierarchyproblem and the cosmological constant problem at once.After all we encounter the fine-tuning problem betweenthe TeV scale and the dark energy although the physicalcosmological constant to be Λ / (cid:39) Ω − ( χ ) Λ / (cid:39) M EW might be more or less relaxed. Thus, we must require an-ther relaxation mechanism for the cosmological constant. III. RUNNING GRAVITATIONALRELAXATION FOR THE ELECTROWEAKHIERARCHY
In the previous section, we have discussed how mod-ified gravity can alleviate the large Higgs mass and thecosmological constant by using the conformal transfor-mation. In this section, we discuss the running gravita-tional relaxation scenario in which we treat the dilatonicscalar field χ as the quantum field and consider the renor-malisation group (RG) running behaviors .For simplicity, we consider the action for the Higgssector in Jordan frame as follows: S gravity+Higgs ⊃ (cid:90) d x √− g (cid:18) M χ R − g µν ∇ µ Φ † ∇ ν Φ − Λ b − M (cid:0) Φ † Φ (cid:1) − λ (cid:0) Φ † Φ (cid:1) (cid:19) , (24)where we assume that the dilatonic scalar χ satisfy G ( χ ) = 6 M /χ (cid:0) − M /χ (cid:1) + · · · , V ( χ ) = 0 . (25)This set-up is for simplifying our discussion and this ac-tion is consistent with induced gravity theories [50–63]Let us perform the conformal transformation Ω ( χ ) = M /χ and consider the action for the Higgs sector inEinstein frame S gravity+Higgs ⊃ (cid:90) d x (cid:112) − g (cid:18) M R − g µν ∇ µ χ ∇ ν χ − g µν ∇ µ H † ∇ ν H − Λ b M χ − M M χ (cid:0) H † H (cid:1) − λ (cid:0) H † H (cid:1) (cid:19) . (26)Here, we redefine the scalar field as χ → φ = χ √
12 andobtain the action for the Higgs sector as follows: S gravity+Higgs ⊃ (cid:90) d x (cid:112) − g (cid:18) M R − g µν ∇ µ φ ∇ ν φ − g µν ∇ µ H † ∇ ν H − Λ b M φ − M M φ (cid:0) H † H (cid:1) − λ (cid:0) H † H (cid:1) (cid:19) . (27)The transformed Higgs potential can be given by V (cid:0) H † H (cid:1) = λ Λ φ + λ M φ (cid:0) H † H (cid:1) + λ (cid:0) H † H (cid:1) , (28) Polyakov proposed that the cosmological constant could bescreened by the IR behavior of quantum gravity and the behav-ior can be translated by the RG running of the auxiliary gravita-tional field [18, 19]. The electroweak hierarchy is also discussedby Ref. [23]. where the Higgs mass parameter and the cosmologicalconstant become marginal operators with zero scaling di-mensions and they are fixed at the UV cut-off scale as λ Λ ( µ UV ) = Λ b M , λ M ( µ UV ) = M M . (29)The renormalisation group (RG) runnings of λ Λ , λ M and λ are determined by the one-loop β -functions [64, 65] β λ Λ = 1(4 π ) (cid:20) λ M + 20 λ (cid:21) , (30) β λ M = 1(4 π ) (cid:20) λ M (cid:16) y t − g (cid:48) − g (cid:17) +4 λ M (3 λ + 2 λ Λ ) + 4 λ M (cid:21) , (31) β λ = 1(4 π ) (cid:20) λ (cid:16) y t − g (cid:48) − g (cid:17) − y t + 34 g + 38 (cid:16) g (cid:48) + g (cid:17) + 24 λ + λ M (cid:21) . (32)where the one-loop running of the gauge coupling orthe Yukawa coupling are not changed by the conformaltransformation because the actions of the gauge sectorsor the fermion sectors are only rescaled via the confor-mal transformations. The transformed Higgs potentialhas classical conformal symmetry and naturally realizethe electroweak scale via the RG running behavior, i.e.the radiative symmetry breaking where these theories arefree from the electroweak hierarchy problem (see the ref-erence [66–69] as the more detailed discussion). Now,we found out that the classically conformal theory [70]which solves the electroweak hierarchy problem is closelyrelated with the modified gravity action of Eq. (27).Following Polyakov’s arguments [18] it is found thatthe RG running effects of the dilaton or graviton wouldsuppress the Higgs boson mass or cosmological constantin Eq. (27). Although the equivalence between the quan-tum theories in the Einstein and Jordan frames is still un-der debate, f ( R ) gravity theories in the Einstein and Jor-dan frame are equivalent on shell at the quantum level.The (off shell) quantum corrections are ambiguous, butthe equivalence of the effective potential or renormaliza-tion group equations have been shown (see, e.g. the ref-erence [71, 72]). Thus, the conformal transformation isalso effective for the RG running effects and the modifiedgravity would relax the electroweak hierarchy. IV. VACUUM ENERGY SEQUESTERING FORTHE ELECTROWEAK HIERARCHY
In this section, we clearly show that the vacuum energysequestering scenario can also be effective against theelectroweak hierarchy problem and discuss the relationbetween such a model and previous scenarios. The vac-uum energy sequestering scenario proposed in Ref.[24–29] is the simple model to solve the cosmological constantproblem, and can relax the large discrepancy between thevacuum energy density from quantum corrections and thecurrent observed value via the scaling parameter η . Thisscenario assumes a minimal modification of general rela-tivity to make all scales in the matter sector functionalsof the 4-volume element of the Universe. In the context ofthis scenario, the Universe should be finite in space-timeand a transient stage with the present epoch of accel-erated expansion before the big crunch [32], but it hasbeen shown that these models could be consistent withthe cosmological observation (there are similar models inthe context of the unimodular gravity and more detaileddiscussions are given by Ref.[73–81]). The mechanism ofthis scenario is almost the same as the previously dis-cussed one and the unknown scaling parameter η can beregarded as the VEV of the dilatonic scalar field. Thevacuum energy sequestering scenario is described by thefollowing action S gravity+Higgs = (cid:90) d x √− g (cid:34) M R − Λ b − η L (cid:0) η − g µν , Φ (cid:1)(cid:35) + σ (cid:18) Λ b η µ (cid:19) , where Λ b is the bare cosmological constant and η isthe scaling parameter relaxing the large hierarchy. Thefunction σ ( x ) is an adequate function to impose theglobal constraints and µ is a parameter with the mass di-mension. The Lagrangian density L (cid:0) η − g µν , Φ (cid:1) for thematter sector couples minimally to the rescaled metric g µν = η g µν and includes the Higgs potential as L (cid:0) η − g µν , Φ (cid:1) ⊃ V (cid:0) Φ † Φ (cid:1) = Λ b + M (cid:0) Φ † Φ (cid:1) + λ (cid:0) Φ † Φ (cid:1) . (33)The parameter η sets the hierarchy between the physicalscale (the electroweak scale or the dark energy scale) andthe UV cut-off scale M UV . Thus, the Higgs mass param-eter of the order of the UV cut-off scale are sufficientlysuppressed as M H M UV = η M Φ M UV , (34)where M H is the observed Higgs boson mass and η (cid:28) g µν as S gravity+Higgs = (cid:90) d x √− g (cid:32) M R − Λ b (cid:33) − (cid:90) d x (cid:112) − g L ( g µν , Φ) + σ (cid:18) Λ b η µ (cid:19) , (35)where (cid:112) − g L ( g µν , Φ) = √− gη L (cid:0) η − g µν , Φ (cid:1) . (36) The matter Lagrangian is written by the rescaling metric g µν as (cid:112) − g L ( g µν , Φ) = (cid:112) − g (cid:20) g µν ∇ µ Φ † ∇ ν Φ+ Λ + M (cid:0) Φ † Φ (cid:1) + λ (cid:0) Φ † Φ (cid:1) (cid:21) . Thus, by using the conformal transformation, the matterLagrangian can be written as (cid:112) − g L ( g µν , Φ) = √− gη L (cid:0) η − g µν , Φ (cid:1) = √− g (cid:20) g µν ∇ µ H † ∇ ν H + η Λ + η M (cid:0) H † H (cid:1) + λ (cid:0) H † H (cid:1) (cid:21) , where we assume H = η Φ. Therefore, if the Higgs sectoris sequestered from the gravitational sector via the scal-ing parameter η (cid:28)
1, the large Higgs mass parametercan be sufficiently suppressed. The unknown scaling pa-rameter η can be regarded as the dilatonic scalar field inthe scenarios previously discussed in Section II and Sec-tion III. Although the cosmological constraints on themodified gravity theory and the equivalence between theEinstein and Jordan frames should be carefully consid-ered, this mechanisms or scenarios would not significantlychange. In this paper we have focused on the possibilityrelaxing the electroweak hierarchy and left detail discus-sion of the limits for a forthcoming work. V. CONCLUSION AND DISCUSSION
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