Radiative Inflation and Dark Energy
Pasquale Di Bari, Stephen F. King, Christoph Luhn, Alexander Merle, Angnis Schmidt-May
aa r X i v : . [ h e p - ph ] O c t Radiative Inflation and Dark Energy
Pasquale Di Bari a ∗ , Stephen F. King a † , Christoph Luhn a ‡ ,Alexander Merle bc § , and Angnis Schmidt-May b ¶ a School of Physics and Astronomy, University of Southampton,Southampton, SO17 1BJ, United Kingdom b Max-Planck-Institut f¨ur Kernphysik,Postfach 10 39 80, 69029 Heidelberg, Germany c Department of Theoretical Physics, School of Engineering Sciences,Royal Institute of Technology (KTH) – AlbaNova University Center,Roslagstullsbacken 21, 106 91 Stockholm, Sweden
October 12, 2018
Abstract
We propose a model based on radiative symmetry breaking that combines inflationwith Dark Energy and is consistent with the WMAP 7-year regions. The radiativeinflationary potential leads to the prediction of a spectral index 0 . . n S . . . . r . . ∗ email: [email protected] † email: [email protected] ‡ email: [email protected] § email: [email protected] ¶ email: [email protected] Introduction
Although modern cosmology seems to require both inflation and Dark Energy there arerelatively few models which attempt to unify these two ideas [1, 2]. One of the mostinteresting attempts to achieve such a unification has relatively recently been discussed [3],based on the earlier ‘schizon model’ [4, 5, 6]. This model was, however, essentially basedon ϕ chaotic inflation, which was significantly threatened by the Wilkinson MicrowaveAnisotropy Probe (WMAP) 5-year data [7] (if not ruled out). However, the model hassome nice features as, e.g., naturally generating a pseudo Nambu-Goldstone boson (PNGB),which receives a potential via gravitational effects [9] and can then be used as quintessencefield. Other attempts can provide a better match to the data by invoking hybrid inflation,with [10] or without [11] using a PNGB as quintessence field (see Refs. [12, 13] for anextensive discussion of that subject).In this letter we propose a simple new model which can overcome the difficulties of ϕ chaotic inflation but which can also lead to a PNGB quintessence field. The new model isbased on the idea of a massive complex scalar field whose mass squared is driven negativeclose to the Planck scale by radiative effects, leading to a model of Radiative Inflation andDark Energy (RIDE). The complex scalar field Φ has a potential which is invariant undera global U (1)-symmetry which, in turn, is broken by radiative effects leading to an almostmassless PNGB. Close to the Planck scale the inflaton field ˜ η = √ | Φ | rolls slowly downa simple potential that resembles ϕ chaotic inflation for high field values. After inflation,however, it settles at its minimum, thereby breaking the global U (1) and generating theNambu-Goldstone boson which would be massless in the absence of gravitational effects.Including gravitational effects generates a potential for the PNGB, so that it can thenplay the role of the quintessence field. The RIDE model leads to interesting predictionsfor inflation which are fully consistent with WMAP 7-year data [14] but which allow themodel to be ruled out or confirmed by the Planck experiment.Radiative corrections have been studied before in both, the context of inflation (see, e.g.,Refs. [15, 16, 17, 18, 19, 20, 21, 22, 23, 24]) and in the context of quintessence (see, e.g.,Refs. [25, 26, 27, 28, 29, 30, 31]). In contrast to previous studies, however, the radiativecorrections here are responsible for symmetry breaking via a scalar mass squared beingdriven negative at a high scale close to the Planck scale, allowing us to relate inflation toquintessence.The remainder of this letter is organised as follows. After introducing the model in Sec. 2,we perform analyses of inflation and quintessence in Secs. 3 and 4, respectively. Finally, wesketch a possible scenario in which our model could be realised in Sec. 5, before concludingin Sec. 6. More details on the effects of radiative corrections on the potential can be foundin the Appendix. Note, however, that the situation of the model from Ref. [3] looks much better in the case of smallfield inflation [8]. The Model
The model is based on a complex scalar field Φ = √ ˜ ηe iφ/f (with f = h ˜ η i ), whose potentialin the absence of radiative corrections has a simple quadratic form, V ≈ M Φ † Φ. Thebasic idea is that radiative corrections then drive the mass squared negative at some scaleΛ not too far below the Planck scale. This radiative symmetry breaking mechanism isperhaps most familiar in the minimal supersymmetric standard model (MSSM) where topand stop loops drive the Higgs mass squared negative at the TeV scale [32], but has beenrecently used elsewhere in different contexts where a mass squared is driven negative at amuch higher scale [33, 34]. Such a radiative potential may be parametrised as in [33, 34], V ≈ M Φ † Φ ln (cid:18) Φ † ΦΛ (cid:19) = M η ln (cid:18) ˜ η (cid:19) . (1)This leads to a vacuum expectation value (VEV) of f = q e Λ for ˜ η . In such a potential,inflation can completely take place in a region where ˜ η ≫ Λ, in which the ln-term in Eq. (1)is well behaved and the inflaton field ˜ η only feels a potential that is very similar to the oneused for quadratic inflation. Later on, the field will settle at its VEV. As the potential is symmetric under a global U (1),either imposed or accidental, the VEV will break this global symmetry, thereby generatinga massless Nambu-Goldstone boson φ = f arg(Φ). This field has no mass term and infact no potential at all. The original U (1) symmetry of Φ translates to a shift symmetry φf → φf + α , with α being a continuous real parameter. However, the continuous shiftsymmetry can be broken by gravitational effects [3, 9], dubbed gravitational instantons ,which, similar to the case of the axion [38, 39], can generate a mass term. Although theybreak the shift symmetry, these gravitational effects leave invariant a discrete subgroupof transformations, namely those for which α = 2 πn , with n ∈ N denoting the windingnumber of equivalent vacua which one can freely choose also in the presence of gravitationalcorrections. Hence, any potential V q ( φ ) that is generated by such effects must still beinvariant under φf → φf + 2 πn . In order to have a mass term for φ in its Taylor expansion,the potential must be an even 2 πn -periodic function. The most general such function is asum of cosines whose arguments are integer multiples of φf . It is possible to argue, see [40],that the dominant contribution is obtained from the lowest harmonic ∝ cos (cid:16) φf (cid:17) , so that Note that, when trying to relate such a potential to a concrete particle physics model, it has to beverified that a possible (Φ † Φ) -term is absent or at least suppressed. Indeed, such a framework can berealised in certain scenarios, see Sec. 5 for an example. Note that we concentrate on the corrections due to the renormalization group evolution, just as donein Refs. [17, 18, 23], which is the dominant contribution of the Coleman-Weinberg correction [35] in thecase of broken supersymmetry [36]. See also Ref. [37] for experimental constraints on such corrections. V q H Φ L Quintessence V q H Φ L = m H + cos Φ f LX Φ \ =Π f Φ init Figure 1: The shape of the inflationary potential describing the inflaton ˜ η (left panel) andthe potential describing the quintessence field φ (right panel). Both fields originate fromthe complex scalar field Φ = √ ˜ ηe iφ/f described by the potential in Eq. (1).the resulting potential for the quintessence field φ reads V q ( φ ) = m (cid:20) (cid:18) φf (cid:19)(cid:21) . (2)Both potentials, V (˜ η ) and V q ( φ ), as well as the field dynamics are schematically depictedin Fig. 1. Note that the dynamics of both sectors can be easily disentangled, as the kineticterm simplifies to ( ∂ µ Φ) ∗ ( ∂ µ Φ) = 12 ( ∂ µ ˜ η )( ∂ µ ˜ η ) + ˜ η f ( ∂ µ φ )( ∂ µ φ ) , (3)with the φ -part being negligible during inflation and ˜ η already sitting at its (constant)VEV f during quintessence. Due to this separation of the dynamics of the two fields, weshould be safe from potentially dangerous corrections due to (iso-) curvature fluctuationsthat can appear in multi-field inflation models [41], since we are practically dealing with asingle-field potential. Note that, in certain settings, it might be necessary to protect this potential against too large radiativecorrections, see Refs. [4, 5]. Inflation
The inflaton potential of Eq. (1) depends on two parameters, M and Λ. In this sectionwe show that they can be chosen such as to be consistent with the WMAP 7-year data.Assuming Λ close to the Planck scale, we defineΛ = k M P , (4)and fix k to take a particular value, like e.g. 0 .
01. With this specific choice, all dimensionfulquantities, including the parameter M , can be expressed in terms of M P only.To determine M , and subsequently the scalar spectral index n S as well as the tensor toscalar ratio r , it is convenient to start with the slow-roll parameters, ǫ = M P π (cid:18) V ′ V (cid:19) = M P π ˜ η (cid:18) L (cid:19) and η = M P π " V ′′ V − (cid:18) V ′ V (cid:19) = M P π ˜ η (cid:18) L − L (cid:19) , (5)where L = ln (cid:16) ˜ η (cid:17) . The field value ˜ η e at the end of inflation is calculated numericallyby setting ǫ = 1. Note that, in the interesting part of the parameter space, ˜ η e is alwaysvery well above f , though not necessarily by orders of magnitude. The next quantity wedetermine is the field value ˜ η N , N e -folds before the end of inflation. Since there is againno simple approximation, we determine ˜ η N by numerically solving N ≃ π M P Z ˜ η N ˜ η e V (˜ η ) V ′ (˜ η ) d ˜ η = 2 π " ˜ η N − ˜ η e M P − e (cid:18) Λ M P (cid:19) [Ei(1 + L N ) − Ei(1 + L e )] , (6)where Ei( z ) is the exponential integral Ei( z ) = − R ∞− z e − t t dt , L i = ln (cid:16) ˜ η i (cid:17) , and N lieswithin the interval N ∈ [46 , η N , the parameter M in Eq. (1) is constrained by the size of the scalar perturbations in the Cosmic MicrowaveBackground (CMB) [14], P / R = H (˜ η N ) M P p πǫ (˜ η N ) ≃ . · − , with H ≃ πV M P , (7)leading to M ≃ [10 − M P , − M P ]. We have checked that this result is nearly independentof Λ, which only enters logarithmically. Hence our predictions for inflation are very stablewith respect to adjustments of Λ which, as we will show later, are necessary to satisfy theconstraints coming from the quintessence side.The above discussion shows how to determine the parameter M for certain values of k = Λ M P and N . With this the potential of Eq. (1) is completely fixed and we can calculatethe scalar spectral index, n S = 1 − ǫ (˜ η N ) + 2 η (˜ η N ), as well as the tensor to scalar ratio, r = 16 ǫ (˜ η N ), for different values of k = Λ M P and N . The corresponding predictions are inthe ranges 0 . . n S . .
967 and 0 . . r . . , (8)4 æ æ æà à à à - n S r WMAP7 yearsV = M ΗŽ ln ΗŽ L % %L M P = = L M P = = æ æ æ æà à à à Figure 2: The predictions of the RIDE model for the spectral index n S and tensor to scalarratio r as compared to the WMAP 7-year data [14], with the inset showing a blow-up ofthe interesting region. The red squares (black circles) are for k = 1 ( k = 0 .
01) for valuesof N = 46 − N = 50 − . M P (or slightly larger) are perfectly fine for our model,as we will see in the next section. The quintessence part of the potential, Eq. (2), arises from (non-perturbative) gravitationaleffects, as indicated in Sec. 2. Such corrections induced by gravity, though hard to avoid,are expected to be exponentially suppressed [9]. Although this might make them soundnegligible, in the absence of other corrections, such gravitational corrections will determinethe potential for the pseudo Nambu-Goldstone boson, leading to a suitable quintessenceinterpretation. The Dark Energy scale m will be determined by m = e − S M P f , where f = h ˜ η i and S ∼ π M P M is a potentially large instanton action [9, 10, 43, 44], with M string being the scale of string theory. Assuming m ∼ − eV the ratio M P M string is required tobe around 10, which is not unreasonable. The message is that, although not giving a5rediction, such considerations give at least a motivation for why m should be small inthe first place.Assuming the smallness of m ∼ V q ( φ ) ∼ ρ φ, to be given, this quantity must be of theorder of the current critical density of the Universe, ρ c, = H M P π , so that the currentDark Energy fraction Ω Λ , = ρ Λ , ρ c, equals 0 . +0 . − . [14]. A further constraint arises fromthe requirement that the quintessence field must not have settled at its VEV today, whichtranslates into a bound on its mass, M φ = m f . H [3, 6]. Both these conditions leadto a bound on f (and thus also on Λ = p e f ) which should be f & . M P [3] or, if onewants to avoid too much tuning, even f & . M P [13]. Similarly as for inflation, one mightquestion values of f too close to the Planck scale, a problem that can be cured by, e.g.,invoking extra spatial dimensions [45]. We have analysed the potential in Eq. (2) with f = M P / √ π , using an extended version of the SuperCosmology package [46], where wehave also included the cosmological evolution of radiation. This means that we numericallysolve the acceleration equation (which is, for a flat Universe, equivalent to the Friedmannequation), ¨ aa = ˙ H + H = − π M P ( ρ tot + 3 p tot ) , (9)where ρ tot = ρ rad + ρ mat + ρ φ and p tot = p rad + p mat + p φ are the total energy density and thetotal pressure. Conveniently, we can immediately insert the known evolutions of radiationand matter, ρ rad = ρ rad , init a , p rad = 13 ρ rad , ρ mat = ρ mat , init a , p mat = 0 , (10)where a is the scale factor. The energy density and the pressure of the quintessence field,however, are only known as functions of φ , ρ φ = ˙ φ + V q and p φ = ˙ φ − V q . Note thatit is often convenient to use the so-called equation of state (EoS) parameter w , which isalways defined as the ratio between pressure and energy density. For example, w rad = and w mat = 0. To find the evolution of the quintessence field φ , we have to solve thecorresponding equation of motion,¨ φ + 3 H ˙ φ + V ′ q ( φ ) = 0 , (11)supplemented by the definitions of the field momentum P and the Hubble parameter H , P = a ˙ φ and H = ˙ aa . (12)This gives a total of 4 ordinary first order differential equations to determine the 4 functions φ , P , a , and H . The key ingredient to this system of equations is the quintessence potential,Eq. (2), which determines the qualitative evolution of the Universe. Note that, however,due to Γ = V q V ′′ q ( V ′ q ) ≪ >
1) [47], which means that specific initial conditions have to beimposed at the beginning of Big Bang cosmology, i.e., after reheating. Starting withinitial values of Ω rad , init = 0 .
99, Ω mat , init = 0 .
01, and Ω φ, init = 10 − (where the smallness6 - - H t Φ @ f D LSSRECBBN Φ init = × Π f, Φ init = = M P (cid:144) Π observations today - - - - - - H t W r a d , W m a t , W Φ , w Φ , w t o t W rad W mat W Φ W Φ w Φ w tot w tot Figure 3: Evolutions of the field φ (left panel) and of the energy density parameters formatter, radiation, and the quintessence field, as well as of the quintessence and total EoSparameters (right panel), including the constraints on Ω φ from big bang nucleosynthesis(BBN), recombination (REC), and structure formation (LSS).of the latter is related to the tiny value of m ), we have solved the evolution equationsnumerically, where we have determined the current time by matching the Dark Energydensity parameter to its current value Ω φ, = Ω Λ , . In order to also hit the other rangesfrom the WMAP 7-year data at 1 σ or 68% C.L., i.e. Ω mat , = 0 . ± . w Λ , = − . ± . ∼ /H , with H = 70 . +1 . − . / sMpc [14],we had to choose m ∼ ρ c, / < φ init < . · πf .The initial speed of the field does not influence the cosmological evolution remarkably [13],and we therefore set ˙ φ init = 0. The constraint for the total EoS parameter of the Universe, w tot , = w Λ , · Ω Λ , = − . ± . Λ , and w Λ , , under the assumptions of a flat Universe and negligible radiation. The result of theanalysis is displayed in Fig. 3: On the left, the evolution of the quintessence field φ isplotted as a function of time, whereas on the right the evolution of the whole Universe isdisplayed. The field trivially falls into its minimum and starts a damped oscillation aroundit. The evolution of all important energy densities and EoS parameters also behaves asexpected. For early times, radiation remains dominant, which can be clearly seen fromthe total equation of state parameter w tot that is close to . Later on, matter starts todominate and w tot is pulled closer and closer to zero. The current time is marked by thegrey line, which also indicates the current obervational bounds on the quantities underconsideration. Dark Energy remains subdominant until shortly before today, but will later(when w tot finally fuses with w φ ) become the only component that matters. This alsoexplains the oscillatory behavior of the EoS parameter: The field will have zero velocity,˙ φ = 0, at the turning points, which leads to w φ = ˙ φ + V q ˙ φ − V q = −
1, whereas at the minimum7 - - H t Ρ Φ @ M P × H Π D RIDE Ρ L = const.10 - - - - - - H t L og @ a H t L D Figure 4: The evolution of the Dark Energy density ρ φ ( t ) and the scale factor a ( t ) inthe RIDE model compared to a cosmological constant. We have explicitly verified that adifferent normalisation of the scale factor a ( t ) does not change our results.of the potential we would have V q = 0 and w φ = +1 accordingly.On the right panel, we have also indicated the bounds for early Dark Energy coming frombig bang nucleosynthesis (Ω Λ . .
14, [2]), from recombination (Ω Λ . .
1, [48]), and fromstructure formation (Ω Λ . .
2, [49]), at times [1] t ∼ t ∼ · y, and t ∼ y,respectively, which all essentially indicate that Dark Energy should have become importantonly now. Note that the constraint arising from the formation of nuclei should actually beimposed at H t ∼ − [1], which is not displayed but indicated by the arrow in the rightpanel of Fig. 3. In the numerical analysis, we have normalised the evolution equationsin such a way that the present Hubble constant H equals one, and time is measured ininverse Hubble units. The age of the Universe is found to agree with 1 /H within 5%. Thedeviation from a cosmological constant becomes obvious in Fig. 4, where we have plottedthe Dark Energy density on the left, which would simply be a horizontal line in the caseof constant vacuum energy. The right panel shows the deviation of the scale factor from ascenario with a cosmological constant. The deviation at later times again comes from thebehavior of the quintessence field φ : It rolls down the potential towards its minimum andthen performs a damped oscillation around that point (cf. right panel of Fig. 1 and leftpanel of Fig. 3).We would like to conclude this section, just noticing that the energy density at recombi-nation is predicted to be a standard mixture of matter and radiation. The remaining question is whether there are realistic scenarios that include our model.The example that we present here is a sequestered scenario which is consistent with theMSSM and has already been discussed in the literature in a similar fashion. It can be usedas (toy) realization of RIDE. 8onsider a superpotential, W = W obs + W seq , (13)where W obs represents the observable sector, e.g. the MSSM spectrum, while W seq repre-sents a sequestered sector. Sequestered is more hidden than usual hidden sectors, sincethere will be no Planck scale suppressed operators that couple it to the observable sector.In practice this is achieved as described in Ref. [50], as we discuss now. The idea is thatthere are two 3-branes, an observable one and a sequestered one, separated by an extradimension. We live on the observable brane, along with the MSSM particles, while ourRIDE model lives on the sequestered brane. Supersymmetry (SUSY) is badly broken inthe sequestered sector, in our RIDE model, but the SUSY breaking is not easily transmit-ted to the observable sector, since the two branes are separated by the extra dimensioncoordinate, where the separation is sufficiently large and only gravity is in the bulk. Thereare then no operators of order 1 /M P connecting the observable sector to the hidden sector,which is called “sequestered”. So SUSY can be badly broken in the RIDE model withoutspoiling the observable MSSM.However, as discussed in [50] there will always be the gravity anomaly contribution tosoft masses in the observable sector which gives soft masses m ∼ m / π due to a loopsuppression, where m / is the gravitino mass. So we need to ensure m / .
100 TeV.Let us see how this could work in an example. The following example will also addressthe questions of the absence of the quartic scalar coupling and the origin of the radiativesymmetry breaking of the scalar field.We shall take W obs = W MSSM for definiteness (although any SUSY model in the observablesector would suffice equally well) and the sequestered superpotential as follows: W seq = M ΦΦ + λ Φ ψχ, (14)where Φ , Φ , ψ, χ are independent superfield degrees of freedom, and we drop hats on su-perfields, which should not be confused with their scalar components. The global SUSY F -terms include F Φ = M Φ , (15)so that the potential includes a term V = | F Φ | = M Φ † Φ , (16)of the kind that we began with in Sec. 2. Note that there is no quartic term in the potential,since in SUSY theories quartic terms arise from D -terms, and here the fields are supposedto carry no gauge charges. Note that the second term in Eq. (14) could, in principle, leadto dissipative effects [51] in case it caused a decay of the quintessence field. However, wedisregard this possibility here because of three reasons: First, the fields ψ and χ carryno Standard Model charges and are barely coupled to any active fields (even the couplingto neutrinos could be easily switched off by a suitable symmetry). Second, these fieldscan be assumed to obtain very heavy masses such that they effectively decouple in thequintessence phase. And third because any treatment of a quintessence field decay is, in9eneral, very model-dependent and beyond the scope of a toy model as the one presentedin this paper.We now want the mass squared to be driven negative radiatively. This is achieved by thesecond term on the right-hand side of Eq. (14) proportional to the Yukawa couplings λ .Loops of ψ and χ will tend to drive M negative in pretty much the same way as the Higgsmass squared is driven negative by top quark loops in the MSSM. The main difference isthat here we need M ∼ GeV, and we require its square to be driven negative close tothe Planck scale. In Appendix A, we show that large soft masses of ψ, χ (for example softmasses of order 10 GeV in the considered example) are required to drive M negativeclose to the Planck scale and result in a running scalar mass term of the kind that weparametrised in Eq. (1). Such large soft masses are consistent with SUSY being broken inthe hidden sector at very large scales as we now discuss. Assuming the radiative symmetry breaking mechanism just described, the VEV h Φ i ∼ Λ ∼ − M P results in a very large value for h F Φ i ∼ M Λ ∼ GeV . (17)This large F-term VEV is also consistent with other F-term VEVs which are required togenerate the large soft masses for the ψ, χ fields responsible for driving h Φ i in the firstplace. Without the sequestering such large soft masses in the observable sector (far inexcess of the TeV scale) would render the MSSM so badly broken as to be not relevantfor the LHC, Dark Matter, the hierarchy problem, gauge unification, and so on. However,assuming sequestering, the observable sector soft masses may be at the TeV scale, and theonly requirement is that the gravitino mass does not exceed about 100 TeV, as discussedabove. The gravitino mass arising from the sequestered sector is given by m / = e K/ M P h W seq i M P ∼ h M ΦΦ i M P ∼ M Λ h Φ i M P <
100 TeV . (18)Here K denotes the K¨ahler potential which, in the canonical form, is just Φ † Φ. Since h Φ i < M P , it is approximately correct to disregard the exponential. Inserting the value ofEq. (17) into Eq. (18), we find the constraint h Φ i M P < − . (19) Note that the presence of extra dimensions could significantly lower the estimated SUSY breaking scaleas follows: Suppose that the sequestered brane has a number of extra dimensions which are parallel to it,as opposed to the extra dimension orthogonal to it which serves to separate it from the observable brane.The fields ψ and χ feel these extra dimensions, while the field Φ does not, and their Kaluza-Klein (KK)excitations give rise to a large multiplicity of states which all enter the loop corrections of M , helping todrive it negative at a scale Λ close to the Planck scale. The separation of the KK states depends on thesize R of the parallel extra dimension. Large R corresponds to many different KK states within a givenenergy interval. Moreover, the number of KK states increases multiplicatively by adding more parallelextra dimensions. Such large multiplicity factors would serve to lower the above estimate of the SUSYbreaking scale. We have proposed a model based on radiative symmetry breaking that combines inflationwith Dark Energy and is consistent with the WMAP 7-year regions. The RIDE model leadsto the prediction of a spectral index 0 . . n S . .
967 and a tensor to scalar ratio 0 . . r . . w φ = − w φ = − .
98 in our numerical example), withthe expansion of the Universe differing from the case of a cosmological constant in futureepochs. Finally, we have presented an example scenario in which a RIDE toy model couldarise. A next step of investigation could be to search for more realistic examples of RIDE,and to put them to a thorough test.
Acknowledgments
We would like to thank M. Garny and L. Sorbo for providing useful information, aswell as M. Lindner for useful discussions. This work has been supported by the DFG-Sonderforschungsbereich Transregio 27 ‘Neutrinos and beyond – Weakly interacting par-ticles in Physics, Astrophysics and Cosmology’. The work of AM is supported by theRoyal Institute of Technology (KTH), under the project no. SII-56510, , and by the G¨oranGustafsson foundation. SFK and CL acknowledge support from the STFC Rolling GrantST/G000557/1. SFK is grateful to the Royal Society for a Leverhulme Trust Senior Re-search Fellowship and a Travel Grant. PDB acknowledges financial support from the NExTInstitute and SEPnet.
A On the Renormalization Group Evolution
From the superpotential in Eq. (14), it is easy to derive the SUSY-preserving Lagrangianin terms of component fields, supplemented by soft breaking terms [52]. Using this, onecan calculate the divergent part of the correction to the self-energy of ˜ η , just as for theMSSM Higgs case, which results inΠ div˜ η ˜ η = 4 λ π (cid:0) m b − m f (cid:1) ǫ , (A.1)where m f ( m b ) are essentially the masses of the fermionic (bosonic) components of thesuperfield Φ, and where we have neglected the tri-linear coupling arising from soft break-ing. Indeed this correction vanishes in the supersymmetric limit, m f = m b . Using the11caling of the 4-scalar coupling λ in dimensional regularization, one can easily derive thecorresponding renormalization group (RG) equation that describes the dependence of themass square m η on the energy scale µ [23]: d ( m η ) d ln ˜ η = µ d ( m η ) dµ = 8 λ π (cid:0) m b − m f (cid:1) . (A.2)Indeed, the right-hand side of this equation has just the form expected for the general β -function of the scalar field under consideration (cf. Eq. (3) in Ref. [23], where in our case C = 0, due to the absence of gauge interactions, and D = 8). Approximating the left-handside by a difference quotient, one obtains m η ( µ = f ) = m + 8 λ π ∆ m ln (cid:18) fM P (cid:19) , (A.3)where m = M , ∆ m = m b − m f , and f = q e Λ is the VEV of ˜ η . Taking Λ ∼ . M P (cf. Sec. 3), M P = 1 . · GeV, and hence M ∼ GeV [cf. Eq. (17)], the requirementof the right-hand side of Eq. (A.3) being negative results in the constraint λ q ∆ m > ∼ · GeV , (A.4)and hence, e.g., p ∆ m ∼ GeV for λ ∼ − , which is indeed far above the TeVscale, as anticipated. References [1] W. H. Kinney (2009), .[2] E. J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod. Phys.
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