Supersymmetric standard model inflation in the Planck era
aa r X i v : . [ h e p - ph ] S e p Supersymmetric standard model inflation in the Planck era
Masato Arai, Shinsuke Kawai,
2, 3 and Nobuchika Okada Institute of Experimental and Applied Physics, Czech Technical University in Prague,Horsk´a 3a/22, 128 00 Prague 2, Czech Republic Institute for the Early Universe (IEU), 11-1 Daehyun-dong, Seodaemun-gu, Seoul 120-750, Korea Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL35487, USA (Dated: July 30, 2018)We propose a cosmological inflationary scenario based on the supergravity-embedded StandardModel supplemented by the right-handed neutrinos. We show that with an appropriate K¨ahlerpotential the L - H u direction gives rise to successful inflation that is similar to the recently proposedgravitationally coupled Higgs inflation model but is free from the unitarity problem. The mass scale M R of the right-handed neutrinos is subject to the seesaw relation and the present 2- σ constraintfrom the WMAP7-BAO- H data sets its lower bound M R & M R . I. INTRODUCTION
Today observational cosmology is a precision science.Cosmological inflation, which is supported by all observa-tional data, is now an indispensable theoretical ingredientnot only in astrophysics but also in particle phenomenol-ogy. A remaining mystery of this otherwise extremelysuccessful paradigm is embedding into a particle theorymodel. By virtue of Occam’s razor, a plausible possi-bility may be that the fields responsible for cosmologi-cal inflation (inflatons) are those already included in theStandard Model (SM), or its (not too large) extension.The recently proposed SM Higgs inflation model [1] isan interesting idea to test this possibility. This modelis attractive due to its minimalistic nature and the re-markable agreement with the present day observationaldata. It also relates the dynamics of inflation with theelectroweak scale physics, making a prediction on the SMHiggs mass from the cosmological microwave background(CMB) data. A rather unfavourable feature of this typeof model is that it requires extremely large nonminimalcoupling to gravity, which could lead to violation of theunitarity bound [2]. The model also suffers from the hi-erarchy problem, which may be cured by supersymmetri-sation [3–5]. See [6] for related models.Certainly, there are more traditional ways of embed-ding inflation into supersymmetric SMs. It has beenknown for a while that the flat directions in the mini-mal supersymmetric Standard Model (MSSM), lifted bysoft supersymmetry breaking terms and other effects, canserve as inflatons (reviewed in [7]; more recent develop-ments include [8]). Another type of embedding is into asupersymmetric SM with right-handed neutrinos [9], inwhich one of the right-handed sneutrinos is identified asthe inflaton. These models are phenomenologically wellmotivated; the hierarchy problem is solved by supersym-metry, and the models with the right-handed neutrinosare furthermore consistent with the small but nonzeroneutrino masses indicated by neutrino oscillation. In this paper we present a new scenario of inflation,inspired by these developments. Our model has the fol-lowing features: (i) the scenario is based on the simplestsupersymmetric extension of the SM that includes theright-handed neutrinos, naturally explaining the smallneutrino masses through the seesaw mechanism [10]; (ii)the problem associated with the large nonminimal cou-pling that afflicts the SM Higgs inflation is alleviated; (iii)the CMB data gives predictions on the mass scale of theright-handed neutrinos through the seesaw relation; (iv)leptogenesis is naturally implemented; (v) the predictedcosmological parameters fit well in the present day obser-vational constraint, and (vi) the model can be tested bythe upcoming observational data from the Planck satel-lite. We discuss construction of the model and describethese features below.
II. THE SUPERSYMMETRIC SEESAW MODEL
Our model is based on the MSSM extended with theright-handed neutrinos, with the R -parity preserving su-perpotential W = W MSSM + 12 M R N cR N cR + y D N cR LH u , (1)where N R is the right-handed neutrino superfield (havingodd R -parity), M R the mass parameter for N R , and W MSSM = µH u H d + y u u c QH u + y d d c QH d + y e e c LH d , (2)is the MSSM part. Here, Q , u , d , L , e , H u , H d are theMSSM superfields, µ the MSSM µ -parameter, and y D , y u , y d , y e the Yukawa couplings (the family indices aresuppressed). As noted in [3], successful nonminimallycoupled Higgs inflation requires at least an extra fieldbesides those in the MSSM. Our crucial observation hereis that the model (1) is already such an extension, withthe L - H u direction playing the rˆole of inflaton. Duringinflation Q , u , d , e , H d do not play any part and we shalldisregard them. Parametrising the D-flat direction along L - H u as L = 1 √ (cid:18) ϕ (cid:19) , H u = 1 √ (cid:18) ϕ (cid:19) , (3)the superpotential becomes W = 12 M R N cR N cR + 12 y D N cR ϕ . (4)We assume supergravity embedding and chooseΦ = 1 − (cid:0) | N cR | + | ϕ | (cid:1) + 14 γ (cid:0) ϕ + c . c . (cid:1) + 13 ζ | N cR | , (5)with γ and ζ real parameters. The K¨ahler potential inthe superconformal framework is K = − R -parity violating term. For brevity’s sake,we shall set the reduced Planck scale M P = 2 . × GeV to be unity, take y D to be real and consider onlyone generation below.We introduce real scalar fields χ , N , α , α by ϕ = √ χe iα , N cR = N e iα . It can be checked that the scalarpotential is stable along the real axes of ϕ and N cR ; wethus assume α = α = 0 below. The scalar-gravity partof the Lagrangian in the Jordan frame reads (cf. [4]) L J = √− g J (cid:20)
12 Φ R J − g µν J ∂ µ χ∂ ν χ − κg µν J ∂ µ N ∂ ν N − V J (cid:21) , (6)whereΦ = M + ξχ , M ≡ − N + ζ N , ξ ≡ γ − . (7)The subscripts J indicate quantities in the Jordan frame,and κ = 1 − ζN is the nontrivial component of theK¨ahler metric. The F-term scalar potential is computedin the standard way [11]. In the Jordan frame it reads V J = 12 y D N χ + ( M R N + y D χ ) − ζN − N n M R N + γy D χ − ζN ( y D χ +4 M R N )2(1 − ζN ) o ζN − ζN + γχ ( γ − . (8)The scalar potential in the Einstein frame is V E = Φ − V J .In this model the Dirac Yukawa coupling y D and theright-handed neutrino mass M R are related by the see-saw relation [10] m ν = y D h H u i /M R , where m ν is themass scale of the light (left-handed) neutrinos. Using theneutrino oscillation data m ν ≈ ∆ m = 2 . × − eV [12] and the Higgs vev at low energy h H u i ≈
174 GeV,we find y D = (cid:18) M R . × GeV (cid:19) . (9)This puts an upper bound on M R since y D . O (1). - - N Χ FIG. 1: The scalar potential V E in the Einstein frame (left),and the inflaton trajectory in the contour plot of the samepotential (right). The red curve is the inflaton trajectory. Wehave chosen N e = 60, M R = 10 GeV and ζ = 100. For large y D (and thus large M R ) the inflationarymodel is very similar to the next-to-minimal supersym-metric SM [3, 4] or the supersymmetric grand unified the-ory model [5]. These two-field inflation models in generalhave nontrivial inflaton trajectories that can source theisocurvature mode. While such a scenario is certainly ofinterest, the analysis is rather involved; we thus allowthe quartic K¨ahler term in (5) to control the instabilityin the N -direction. For M R = 10 GeV we find ζ = 100keeps the deviation of N from N = 0 negligibly small( √ κ ∆ N/ ∆ χ . N e = 60e-folds). For M R ≤ GeV, ζ = 1 is enough. In Fig.1we show the potential and the inflaton trajectory of ourmodel, for M R = 10 GeV, N e = 60 and ζ = 100 (thenonminimal coupling is fixed by CMB as below). Oncethe trajectory is stabilised the cosmological parametersare insensitive to the value of ζ , and as the trajectory isnearly straight the model simplifies to single field infla-tion with the inflaton χ . The Lagrangian then becomes L J = √− g J (cid:20) M + ξχ R J − g µν J ∂ µ χ∂ ν χ − V J (cid:21) . (10) III. COSMOLOGICAL SCENARIO AND THEPREDICTION
Our model provides a cosmological scenario of slow-rollinflation: the slow roll parameters ǫ , η are small duringinflation, and inflation terminates when ǫ or η becomes O (1). The canonically normalised inflaton field ˆ χ in theEinstein frame is related to χ by d ˆ χ = p M + ξχ + 6 ξ χ M + ξχ dχ, (11)and the slow roll parameters in the Einstein frame are ǫ = 12 (cid:18) V E dV E d ˆ χ (cid:19) , η = 1 V E d V E d ˆ χ . (12)The inflaton value χ = χ ∗ at the end of the slow roll isrelated to the value χ = χ k at the horizon exit of the N e M R (GeV) ξ χ ∗ χ k n s r
257 0.0671 0.527 0.962 0.0042010 . . . × − . × −
306 0.0614 0.527 0.968 0.0029710 .
232 1.73 14.6 0.968 0.0050860 10 . . . × − . × − . × − ξ , the inflaton values at the end of theslow roll ( χ ∗ ) and at the horizon exit ( χ k ), the spectral index n s , and the tensor-to-scalar ratio r for e-folding N e = 50, 60and for various values of the right-handed neutrino mass M R .The coupling ξ is fixed by the amplitude of the curvatureperturbation. We used ζ = 100 for M R = 10 GeV and ζ = 1 . M R ≤ GeV. The last lines ( N e = 50, M R =644 GeV and N e = 60, M R = 378 GeV) correspond to theminimally coupled λφ model. comoving CMB scale k , through the e-folding number N e = R χ k χ ∗ dχV E ( d ˆ χ/dχ ) / ( dV E /d ˆ χ ). The potential V E atthe horizon exit is constrained by the power spectrum P R = V E / π ǫ of the curvature perturbation. We usedthe maximum likelihood value ∆ R ( k ) = 2 . × − fromthe 7-year WMAP data [13], which is related to the powerspectrum by ∆ R ( k ) = k π P R ( k ), with the normalisationfixed at k = 0 .
002 Mpc − . Apart from ζ which wasintroduced to keep the deviation of the trajectory from N = 0 small, the model contains only two parameters: ξ and y D . The former is fixed by the curvature pertur-bation P R , and the latter is related to the right-handedneutrino mass M R , through (9). Note that there existsa lower bound on y D , set by the minimal coupling limit ξ →
0. In this limit our model is essentially the chaoticinflation with quartic potential V E = y D χ , with y D fixed by P R . The corresponding value of M R at ξ = 0 is644 GeV for N e = 50 and 378 GeV for N e = 60.For a given value of M R the scalar spectral index n s ≡ d ln P R /d ln k = 1 − ǫ + 2 η and the tensor-to-scalar ratio r ≡ P gw / P R = 16 ǫ can be computed. Table I shows theseresults, evaluated for N e = 50, 60 and for several valuesof M R between the upper and lower bounds [22]. We seethat ξ . O (1) when M R . GeV. This shows thatin the wide parameter region our model is free from the ìì ìì ææççèèéé àà ŸŸ ææææææ æææææ n s H Primordial tilt L r H T e n s o r (cid:144) s ca l a rr a ti o L N e =
50 60NM - LH u è æ ΛΦ é ç m Φ Ÿ à M R =
378 GeV10 M R =
644 GeV HZAFD D n s FIG. 2: The scalar spectral index n s and the tensor-to-scalarratio r , with the 68% and 95% confidence level contours fromthe WMAP7+BAO+ H data [13]. The prediction of ourmodel (NM- LH u ) is indicated by • with corresponding M R values. The predictions of the Harrison-Zel’dovich (HZ), the λφ and m φ chaotic inflation models, as well as the A-termMSSM flat-direction (AFD) inflation models, are also shownfor comparison. ∆ n s is the expected Planck accuracy [21]. dangers [2] arising from the large nonminimal coupling.For small ξ , instead, a super-Planckian initial value ofthe inflaton field is inevitable. This feature is similar tothe model studied in [14].After the slow roll the inflaton oscillates around theminimum at N = χ = 0, and decays. The effect of non-minimal coupling on the reheating process can be impor-tant when ξ is large and the coupling between the inflatonand the matter field is small [15]. In our model, the in-flaton couples directly to the SM matter fields and thecoupling ξ does not have to be extremely large; we thusexpect the effect of ξ on the reheating to be limited. Theupper limit of the reheating temperature is estimatedas T rh ∼ GeV, assuming the Higgs component de-cay ϕ → b ¯ b (the slepton component decay may yieldslightly higher temperature [23] ). This is low enoughto avoid the gravitino problem. The generation of thebaryon asymmetry is due to the following mechanisms.If T rh & M R , the right-handed (s)neutrinos thermalise,leading to thermal leptogenesis [17] with the resonantenhancement effects [18]. If the reheating temperatureis lower T rh . M R , the mechanism of [9, 19] due to thedecay of oscillating sneutrinos can be operative; with N acquiring the vev at the end of the slow roll, as shownin Fig.1, the coherent oscillation in the direction of N produces lepton numbers. Interestingly, this mechanismdepends on the inflaton trajectory and thus on ζ . In ad-dition, the Affleck-Dine mechanism [20] can be operative.The prediction of n s and r in our model is shown inFig.2, along with the 68% and 95% confidence level con-tours from the WMAP7+BAO+ H data [13]. Also indi-cated are the predictions of two other inflationary mod-els arising from the same Lagrangian (1), namely the˜ N R chaotic inflation model [9], marked with (cid:4) , and theA-term inflation models [8] marked with (cid:7) (AFD). Theformer is essentially the standard m φ chaotic inflation.In the latter, the inflaton is u c d c d c , e c LL , or N cR LH u direction in the ( N R -extended) MSSM, and its typicalprediction is very small r and n s ≈ − /N e ; we used N e = 50 (thus n s = 0 .
92) as the e-folding cannot be large( N e .
50) in such low-scale inflation models. We see thatour model fits well with the present data unless M R istoo small. The 2- σ constraints roughly give M R & n s ≈ . N R -extended) MSSMwould clearly be discriminated. If our model turns outto be the likely scenario, the Planck data would also con-strain the mass scale of the right-handed neutrinos. IV. DISCUSSION
While the SM of particle theory is the greatest successin the twentieth century physics, it is not a completetheory. For one thing, the neutrino oscillation indicatesthat the right-handed neutrinos must be included. Also,in order to solve the hierarchy problem and to accountfor the dark matter in the universe, some extension, suchas supersymmetrisation, is necessary. In this paper we presented a new scenario of inflation, for which the right-handed neutrinos, supersymmetry, and the non-minimalcoupling are essential. Note that all of them naturallyarise in the supergravity embedding of the SM with theright-handed neutrinos. Not too large nonminimal cou-pling is also natural as we are dealing with quantum fieldtheory in curved spacetime.Our scenario is economical as it explains – apartfrom the standard issues that are solved by inflation –small nonvanishing neutrino masses and the origin of thebaryon asymmetry. The predicted values of n s and r are consistent with the present observation, and can betested by the Planck satellite data. What we find partic-ularly interesting is that it constrains the right-handedneutrino mass scale. The nature of the heavy neutrinosis mysterious; being gauge singlets, their detection in col-liders is virtually impossible, nevertheless they must bepresent for the seesaw mechanism and leptogenesis. Ifour scenario turns out to be correct, CMB would providea new window to the physics of right-handed neutrinos. Acknowledgements. — S.K. acknowledges helpful con-versation with Kari Enqvist. This work was supportedin part by the Research Program MSM6840770029,ATLAS-CERN International Cooperation (M.A.), theNational Research Foundation of Korea Grant No. 2012-007575 (S.K.) and by the DOE Grant No. DE-FG02-10ER41714 (N.O.). [1] F. L. Bezrukov and M. Shaposhnikov, Phys. Lett.
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