Starobinsky-Like Inflation and Neutrino Masses in a No-Scale

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This content was downloaded from IP address 131.169.4.70 on 14/12/2017 at 22:40 JCAP11(2016)018 hysics P le d ic t ar h doi:10.1088/1475-7516/2016/11/018 strop A Natsumi Nagata, c and Keith A. Olive e . Content from this work may be used 3 osmology and and osmology C Marcos A.G. Garcia, 1609.05849 a,b inﬂation, neutrino masses from cosmology, supersymmetry and cosmology, baryon

Using a no-scale supergravity framework, we construct an SO(10) model that rnal of rnal Article funded byunder SCOAP the terms of the Creative Commons Attribution 3.0 License . ou An IOP and SISSA journal An IOP and Academy of Athens, Division28 of Panepistimiou Natural Avenue, Sciences, 10679 Athens,William Greece I. Fine Theoretical Physicsof Institute, Minnesota, School of Physics116 and Church Astronomy, University Street SE, Minneapolis,E-mail: MN, john.ellis@cern.ch 55455,[email protected], U.S.A. , [email protected] , [email protected] [email protected] Astroparticle Physics Group, HoustonMitchell Advanced Campus, Research Woodlands, Center 77381 (HARC), Texas, U.S.A. Theoretical Particle Physics andDepartment Cosmology of Group, Physics, King’s CollegeTheoretical London, Physics WC2R Department, 2LS CERN, London,CH-1211 U.K. Geneva 23, Switzerland Physics and Astronomy Department,6100 Rice Main University, Street, Houston, TXDepartment of 77005, Physics, U.S.A. University ofBunkyo-ku, Tokyo Tokyo, 113-0033, Japan George P. and Cynthia W.Texas A&M Mitchell University, Institute College for Station, Fundamental 77843 Physics Texas, and U.S.A. Astronomy, b c e g d a h f Any further distribution ofand this the work must title maintain of attribution the to work, the author(s) journal citation and DOI. Received September 26, 2016 Accepted October 26, 2016 Published November 8, 2016 Abstract. makes predictions forStarobinsky cosmic model of microwave inflation,consistent background and with incorporates observables oscillation a experiments similar and double-seesawto late-time model to the cosmology. for behaviour We those neutrino pay of particular masses of the attention scalar the fieldsKeywords: during inflation and theasymmetry subsequent reheating. ArXiv ePrint: Dimitri V. Nanopoulos

John Ellis, J Starobinsky-like inflation and neutrino masses in a no-scale SO(10) model JCAP11(2016)018 (1) in natural units [20, O in the perturbation spectrum, r – 1 – A central challenge in inflationary model-building 1 problem. One exception to this ‘holy’ rule is provided by and tensor-to-scalar ratio s η n modification of minimal Einstein gravity. Another model that is consistent 2 R + 2.2.1 Model4 2.2.2 Vacuum2.2.36 conditions Doublet-triplet2.2.48 splitting Proton9 decay R Among the leading frameworks for physics beyond the Standard Model at the TeV scale This model is disfavoured by current measurements of the top and Higgs masses, which indicate that the 4.1 Yukawa unification4.2 and its violation Neutrino masses 18 20 2.1 No-scale2.22 framework SO(10) GUT construction4 1 effective Higgs potential becomes negativeby at large new field physics. values [11], unless the Standard Model is supplemented 1 Introduction Recent measurements of theconstraints cosmic on microwave background the (CMB) scalar [1–3] tilt provide important 3 Realization of inflation 4 Yukawa couplings and neutrino masses 5 Reheating and leptogenesis 6 Summary 18 10 23 24 Contents 11 Introduction 2 An SO(10) inflationary model set-up in no-scale2 supergravity which in turn provide important6]. restrictions Among on the possible modelson models that fit of an the cosmological data inflation very [4– well is the Starobinsky model [7–9] that is based 21], an obstacle knownno-scale supergravity as [22–24], the which offers an effective potential that is positive semi-definite and above is supersymmetry.astrophysical It dark has matter, many offers advantages newcould for mechanisms also particle for stabilize physics, generating the could the small15]. provide potential baryon the parameters asymmetry, In and required cosmological applications, in it genericsupergravity is models framework essential of to [16–19]. inflation combine [12– supersymmetrycosmology, However, with since generic gravity in their supergravity the effective models potentials are contain not ‘holes’ suitable of for depth with the CMBStandard data Model is Higgs Higgs fieldis inflation therefore to [10], the gravity. which constructionalso of assumes plausible a a candidates model non-minimal for thatdark new coupling incorporates matter, physics not and of beyond, only the the the baryon such asymmetry Standard as of Model neutrino the but masses Universe. and oscillations, JCAP11(2016)018 (2.1) Starobinsky model of 2 R + ..., R + ... + 2 | 3 φ | − ∗ – 2 – T + T 3 ln problem can be avoided] [26 and it is natural, therefore, to η 3 − K , should not lead to overproduction of gravitinos. s n of the gauge group and the inflaton is identified with a singlet of SO(10). 16 ). However, achieving this result makes non-trivial demands on the structure of the ) that are similar to those of the Starobinsky model [41]. In particular, no-scale models The structure of this paper is as follows. In section2 we set up our inflationary model, We find parameters for the no-scale SO(10) GUT model that are compatible with all One issue is the behaviour of the GUT non-singlet scalar fields during inflation, which In this paper we take an alternative approach to the construction of a no-scale GUT Consequently [41], there has been continuing interest in constructing no-scale super- , r , r s s n n inflation [41]. In thebe minimal written two-field case as 74] [ useful for inflation, the Kählerpotential can including the no-scale andthis SO(10) model aspects is ofits described its predictions in framework. resemble section3, those Theof paying of realization neutrino particular the of masses attention Starobinsky inflationReheating in model. to in and this leptogenesis the Section4 after model, requirementsexplorespaid inflation that to as is the the discussed generation they in gravitino are abundance. section5, with generated Finally, particular our via attention conclusions a are summarized double-seesaw in2 mechanism. section6. An SO(10) inflationary2.1 model set-up in No-scale no-scale framework supergravity No-scale supergravity provides afor remarkably simple the field-theoretic CMB realization of observables predictions that are similar to those of the these cosmological and neutrino constraints,model providing an of existence particle proof physics for andno-scale a cosmology more supergravity than complete models has of been inflation. provided in previous Starobinsky-like we require tothe be form such of that themixes neutrino with the the mass model doublet matrix.double-seesaw (left-handed) predictions structure, and In are which singlet our must Starobinsky-like. (right-handed) model,light satisfy neutrino the (mainly certain fields, Another superpartner leading conditions left-handed) to issue if oflate-time neutrinos a it the is cosmology. that is inflaton are to Finally, field wethe give compatible masses also baryon with for asymmetry consider oscillation the following thePlanck experiments inflation, issue constraints and which, on of in reheating addition and to the being generation compatible of with the SO(10) model, which we study in this paper. model of inflation, namely weis consider embedded a in supersymmetric a SO(10) GUTWe in show which the that sneutrino ( this model also makes Starobinsky-like predictions for the CMB variables have been constructed in whichsneutrino the [43, inflaton68, could69], beflaton and identified is also identified with no-scale with a GUT a singletHiggs models supersymmetric (right-handed) inflation Higgs have boson, been [72, avoiding constructed73]. the in problems which of conventional the in- gravity models of inflation–71], [42 ( which lead naturally to predictions for the CMB variables at the tree level, andtheory has [25]. the added In motivationconsider this that inflationary case, it models the appears in in this compactifications context of [27–40]. string JCAP11(2016)018 and (2.2) (2.6) (2.7) (2.3) (2.4) (2.5) c = i T . , Re h . φ T c µ µ . 1. Alternatively, if ∂ ∂ ≡ ) + h N ∗ φ 1 − W πG 3 3 / − ) = 8 ∗ ∗ ∗ φ 2 T φ , − P W 2 − , ∗ + 2 ) 0 M φ represent possible additional matter φ T ( 3) 2) 3 ( 3 / / T / 2 1 φ 2 | W ... ) 3 λ φ √ − ∗ is converted to a canonical field. In fact, − 1 3 T − e T ˆ − | V T µ ( 2 ∗ + − φ , ∂ T 2 ˆ ∗ 2 µ | – 3 – Mφ (1 φ T + 2 3 µ = √ W ∂ T . M | ( ( ) 4 3 = ∗ 2 W ), the effective potential becomes T = = 3) in this context can be achieved through quartic terms in in natural units with W / + V V 2 T, φ 5 ∂W/∂T T | T ( − φ ( 3, as may be seen after a field redefinition to a canonically- ≡ 3 1 or W 10 c/ 3 − | T φ appeared as a singlet. In this model, reheating takes place when = 0), and the superpotential is given by [76] + . In order to obtain the correct amplitude for density fluctuations, ∗ p ≈ 0 W 2 φ R T φ

3 is ﬁxed with a vacuum expectation value (vev) 2 ν µ/ + √ T and ∂φ = ˆ : ∂W T µ/

( φ µ = = are complex scalar fields and the ≡ and M φ ˆ KE V ∂W/∂φ , and a specific no-scale supersymmetric GUT [80–86] model based on SU(5) was T L R is fixed (with ≡ ν 3 where and φ φ µ/ T = 0, as was shown in ref. [41], the Starobinsky inflationary potential W i In order to achieve reheating the inflaton must be coupled to matter. In no-scale models, If the modulus = T λ Im the field the inflaton decays into themay left-handed occur sneutrino simultaneously and with Higgs leptogenesis (or [34, neutrino87, and88 ]. Higgsino), and For a general superpotential normalized inflaton field proposed, in which the supergravity couplings of thetrivial inflaton are coupling strongly through suppressed thematter [78], gauge and sector kinetic require through either functioncould a the [69, be non- 78, superpotential. associated79], withsuperfield or the It scalar a was component direct of proposed coupling the to in right-handed the (SU(2)-singlet) ref. neutrino [43] that the inflaton we must take the Kählerpotential [42, 44, 57, 68 –70, 77]. if the Starobinsky potential (2.5)there is is a found large when The class stabilization of of superpotentials either that all lead to the same inflationary potential [42]. where with where would be obtained with the following Wess-Zumino choice of superpotential [75]: fields; untwisted if in thecase log, twisted for if the outside.scalar moment, Restricting fields our the attention Kählerpotential to the) (2.1 the yields two-field the following kinetic terms for the h JCAP11(2016)018 54 (2.8) representations , namely, where int 16 . M doublet components EM = L and 16 representation that breaks as usual. The MSSM Higgs U(1) GUT d ⊗ M H 210 C representation at the GUT scale symmetry via vevs of the minimal and u SU(3) representations that break the theory EM H 210 d for the Higgs field would typically be of fields as we will see below. As a result, 16 ,H µ u U(1) 16 H −−−−→ ⊗ and by vevs of a pair of C Y SM – 4 – 16 , and G representation whose SU(2) needed for inflation. Secondly, Starobinsky-type inwe obtain after the SO(10) symmetry breaking de- representation of SO(10), and there are no gauge- 16 U(1) 5 16 , , 10 − via a vev of a int ⊗ 16 16 L 10 −−−→ G 10 int ¯ Φ are the ∼ int G is the G , and to SU(3) SU(2) was considered in ref. [71], where different possible directions H . We also consider the case where int ⊗ 210 −−→ C M 210 210 couplings in SO(10). In principle, one could imagine using either a 3 SO(10) = SU(3) cannot be associated with the right-handed neutrino. This is because, in 16 is included in the were considered as inflaton candidates. There are however, two major hurdles φ R or SM ν 2 G 210 16 representation which do allow both quadratic and cubic couplings in the superpoten- . The intermediate-scale gauge group is subsequently broken to the Standard Model We are therefore led to consider a construction with an SO(10)-singlet inflaton field. 210 GUT The intermediate gauge symmetry pends on the vevthe of the SO(10) gaugeWe symmetry use is the following broken notationsSO(10) into for at the the the SO(10) Standard GUT fields: Model scale, Σ Φ is gauge and the symmetry directly. to the MSSM, respectively, the symmetry-breaking chain we consider is given by at the intermediate scale invariant order the GUT scale rather than SO(10), the While there iscouple no problem it writing toand a matter respect superpotential for the as other reheatingsee in phenomenological that in eq. constraints. the such (2.6) In fermionicdouble-seesaw for a the partner structure, a way model of and singlet, as discussed the theacceptable one below, to inflaton reheating twin must we temperature mixes preserve requirements will place with of itsmatrix constraints the whose Planck-compatible inflationary on consistency neutrino inflation the evolution with sector, and parameters experimental leading an of data to the we2.2.1 discuss. neutrino a mass Model We consider this scenariointermediate-scale within gauge an group SO(10) model of grand unification that breaks to an 2.2 SO(10) GUTWe construction consider here possibilitiestion for [89–94]. no-scale inflation Weinflaton, immediately in then observe the that, context of if SO(10) we grand consider unifica- the superpotential2.6) ( foror the tial. Indeed, itHiggs might fields seem used natural to topossibility utilize break utilizing one SO(10) the of downwithin these to the fields, some which intermediate arein gauge often this group. approach. present An as The interesting first is thatflation the drives mass the scale field towardcorrespond zero to vacuum SO(10) expectation symmetry valueof restoration. (vev), which early Then in one problems this isbreaking case associated left SO(10) would with with after the inflation, degenerate problemof whereas (reminiscent vacua vacuum normally is in it determined is supersymmetricmodel during assumed with inflation. GUTs that the Finally, [95–97]) the inflaton weature appropriate associated of note is choice with that very a reheating high, GUT-scale is leading Higgs so to field efficient the that in overproduction the a of reheating gravitinos temper- [98–103]. M (SM) group supersymmetric Standard Model (MSSM) Higgsfields fields are given by mixtures of the JCAP11(2016)018 . - i R S . S (2.9) κ , , and rather u + doublet , ¯ : it may 2 i -parity of Q 16 L Σ , 2 R 3 W e S 2 Σ γ 4! symmetry is a K/ Λ 4! e L h 3) are the MSSM + denote the baryon , Σ + 2 − = 2 term induces an , where ¯ s ψ , ¯ Σ d 2 B ¯ ¯ / Φ d ψ 3) denote the SO(10) Σ 3 0 , u 4! Φ b 4! = 1 2 m m ¯ , i Φ, and Σ can acquire non- , and H ¯ Φ couplings are forbidden ( 0 L fields–94], [92 as we see in i α , , and ¯ = 1 ψ ψψ , much lighter than the GUT ¯ ¯ d B ΦΦ + d , 16 i -parity. On the other hand, the d ( ¯ H ΦΦΣ + cS R H LQ i η 4! , S and ¯ e Φ and ) + and 1 and ψ u LL ψψ u Φ 0 ¯ H ΦΦ + doublet lepton field. This is because Φ is , and there is one singlet per generation, H α Φ [104], where L 16 m fields do not generate Majorana mass terms s ¯ Φ + + )+2 16 ¯ Φ ψ and – 5 – L α ¯ − Φ ¯ Φ terms induce mixing among the SU(2) ) B 16 and ¯ Φ 3( bS ¯ H 1) is the SU(2) ΦΦ + ¯ 16 + α − α ( L ( M H ≡ + ( ¯ + Φ, and by appropriately choosing these couplings we can make the generation index, and to break the intermediate scale–116]. [105 A supersymmetric -dependent Wess-Zumino superpotential terms that reproduce R 2 S i ΦΦ and ¯ -parity of each SO(10) representation is uniquely determined. , where R 126 L SH αH u yHψψ SH H , Φ, and + λ 3 H singlet charged lepton, doublet quark, singlet up-type quark, and singlet + S 2 ¯ Φ to yield the MSSM Higgs fields 3 λ L H are replaced by a pair of and SM fields: the magnitude of these couplings determines the neutrino masses − H -parity and thus its vev spontaneously breaks 2 S multiplets with R m S superfields, where one of these fields will be identified as inflaton. The 126 2 + m 16 -parity-violating operators in the MSSM, i.e., 1 R = The field content is similar to the SO(10) GUT in ref. [92–94], which uses a We consider the following generic form for the superpotential of the theory: and To obtain the Starobinsky inflationary potential, we drop a possible term linear in the singlet field 2 W are the SU(2) is tuned to yield a weak-scale gravitino mass through the relation ¯ than the more common version of this “minimal”126 theory was discussed in ref. [117]. In this version of SO(10), the d subgroup of SO(10), the down-type quark fields, respectively, areκ not generated at renormalizable level. The constant singlet each field is defined asnumber, usual: lepton number, and spin of the field, respectively. Since the by gauge symmetry, the vevs of the section4. In principle, onlyfor one leptogenesis such and singlet the ismasses non-zero needed for neutrino for all mass inflation, three differences, neutrinos. whereas and two three are for needed non-zero neutrino where for simplicity wegeneration have indices. omitted the We tensor assume structure of that each there term is and no suppressed the mixing among the singlet superfields other be generated by the presence of a separate supersymmetry-breaking sector such as a Polonyi one of which is identified as thefor inflaton. right-handed Since neutrinos the vialight renormalizable neutrino couplings. masses are However, induced in via our the model, mixing non-zero of two linear combinations of these fields, denoted by odd under mix with Φ and components inside and intermediate scales [117],vevs thereby of realizing these the fieldsscale desirable then as doublet-triplet break in splitting. the the SM MSSM. The gauge In group addition, at after the Φ electroweak acquires symmetry a breaking vev, the matter The first two termsthe are the predictions ofdetermines Starobinsky the inflation SM Yukawa couplings. inthe The no-scale inflaton fourth and supergravity tenth [41, termsand include the].42 couplings decay rate between of The thevanishing inflaton. third vevs The through term SM the singlet components couplingsfields of included develop Φ, in vevs, the the fifth through eighth terms. After these parity-violating term JCAP11(2016)018 , 0 R R b int ¯ e ν φ , G c (2.11) (2.10) , and 0 α quantum . We will R -parity as , φ R R , M , i , , ) i , just as in the yHψψ ¯ Φ, and 0 for the 3 ) SU(2) , 2 SH EM 1 , Σ λ ⊗ , † 1 , we obtain different , , L Σ 15 λ 4 R 1 U(1) ( ¯ 4! φ ψ Σ( , Φ, and ⊗ h h S SU(2) symmetry. Thus we are, in ¯ = = Φ + † ⊗ 2 , and R ¯ Φ Z C e ν R φ , Φ + † ω = 0; otherwise, a non-zero vev of , a ¯ , ω Φ develop vevs, as we mentioned above. R + Φ , i , e ν via the Yukawa coupling p ) ¯ ψ Φ, and Σ are stabilized at the origin with i † ) 1 vanish after inflation; one of them which is L , 2 ψ u i , 1 , S 1 , Φ, + ¯ H Φ develop vevs of the order of the electroweak -parity violation in this theory. , – 6 – H 15 4 R symmetry (besides the SO(10) gauge symmetry) H ), and we denote these vevs by † , 2 ¯ Σ( Φ( S S H h h Z + = = , Φ, and S R ∗ ¯ H φ S is a new quantum number: 1 for χ 3 1 (as well as − ψ ∗ , , a , SO(10) is first broken into an intermediate gauge symmetry i i T at the GUT scale, and it is then subsequently broken by ) ) = 0 minimum is in fact stable with a positive mass-squared if either -parity violating term 2 1 , where + ω , , R R χ e ν ¯ 1 1 Φ, and p, a, ω T + , , s 4 1 )+2 , and Φ( Σ( 3 ln L R a h h ¯ − φ − , , B = = p R = 3( p , as shown in eq. (2.8). We will discuss possible values of these vevs as well as the R φ φ is non-zero. Depending on the values of 1) -parity is spontaneously broken when Φ and K SM 0 . In ref. [117], there is an extra − R In the supersymmetric limit, all of the other components have vanishing vevs. After the The superpotential in eq. (2.9) contains several terms that are additional to those in b G γ = ( . In addition, we will consider the cases where or supersymmetry-breaking effects are transmitted totions the of visible the sector, doublet certain linear components combina- by vevs of into corresponding intermediate gauge symmetries in what follows. scale to breakMSSM. the SM The rest gauge of symmetry the spontaneously components to in SU(3) see below that the b symmetry-breaking patterns. Ifgauge all group of is these brokenhand, values directly if are into of the the SM same gauge order, group then at the the SO(10) GUT scale. On the other GUT- or intermediate-scale masses. where we express thenumbers. component fields We in assume terms thatregarded of all as the of SU(4) inflaton thesection, is vevs while automatically of the driven into otherm zero two can after also inflationgives be rise as to stabilized we a at see large the in origin the by next the quadratic coupling the minimal model in ref. [117], notably those with the couplings which mimics the two-field prototype (2.2). For the most part,their we possible will effects assume onthanks these our to terms results. the to non-renormalization be Thisin property assumption absent the of is or following the discussion stable small, on superpotential against but the radiative terms. will effect corrections comment2.2.2 We of on also comment Vacuum conditions The SO(10) andcomponents intermediate of gauge the above symmetries fieldscontained are without in breaking Σ, spontaneously the Φ, broken SM gauge by group. SM Such singlet components are sector [118]. The SO(10) no-scale Kählerpotential is then taken to be [74] R other fields. However,the for singlet Starobinsky-type inflaton, inflation, thus weprinciple, we must allowed do have (even not a obliged)case, to introduce term write such that down an is the extra cubic additional in couplings in eq. (2.9). In this that forbids these couplings, which is obtained by modifying the definition of and JCAP11(2016)018 ⊗ 3, / R 2 (2.21) (2.22) (2.14) (2.15) (2.16) (2.20) (2.17) (2.18) (2.19) (2.12) κ . ' Σ 3. In this SU(2) , with the ) + is realized 2 ⊗ 16 /ηm aω ' − L Φ -terms of these , Σ m F p, a, ω ) + 6 -term part of the 2 2 F x SU(2) /ηm , pω Φ ) ) (2.13) ⊗ φ 2 2 ω -terms. This solution is m x x C + 3 )(1 + D = 0. This solution can be x 3 − x − + 6 , a 3 x R , , , , , , , − a U(1). In this case, the vevs (1 3, we need Λ ¯ Φ 2 Σ φ − / 1 ⊗ 1 m − + 3 = = 0 = 0 = 0 = 0 (1 = SU(3) ηm Λ 1 x ' p ) = 0 2 R R R R )] = 0 )] = 0 ( ) + 2Λ( ) 2 ¯ ¯ ¯ Σ int x φ 2 φ φ φ 2 Σ ω ω R 1) ω ω Λ Λ ¯ R R R [115, 116] rather than a ω G φ m m = η ) . = 0. We also require that the non-zero − 2 + 6 + 6 − 2 + 6 R + 6 ηφ ηφ ηφ = SU(5) x | ω R + 6 ν i a a 2 a 0 e ν = = 126 + + 2 b a = a int ) + W 2 R = ω | 2 + 3 + 3 + a + 3 ¯ ) ω 3 φ G 3 = ( + 3 ω a S p p p R R + 3 ( ( = 2 ( K/ − 0 2 η η 2 – 7 – p b p ηφ + 2 ( x p e + 2 + 6Λ + + γ 2 )( + p p = a + ( Φ Φ 0, eq. (2.21) is satisfied for Λ -independent superpotential, the + Σ 3Λ and + 14 γS M V R m m T / R m ' Λ, and we obtain , a ν 2 − Σ ¯ − − 2 φ + / x [ [ R ) + 2Λ( x R -parity [119–123]. For Σ 2 + 2Λ( ¯ m Σ φ 2 R R R 15 ω ) ) x a ¯ ¯ D m φ φ cφ φ . From these equations we find that m 5 x Σ 0 Σ − bS − b , respectively. As discussed in refs. [115, 116], these equations − ' − m − − 3 do not break supersymmetry. Therefore, the m R + ( 2 + x 2 = e ν in order to ensure the vanishing of the ω (1 8 (1 R R a ¯ x φ φ φ M [116]. For ( , and denotes R is a solution of the cubic equation x , φ Σ ¯ i ' − φ Λ − Σ | ≡ , and M x ) D m ± in eq. (2.21) and a sign change in the right-hand side of eq. (2.22). The Λ a 3 R S , m 0 and , and ¯ − cS φ S , γ ω | ' R 3 R λ , = ' + = φ ¯ c φ = p p ω , Φ φ − | , ω , where 2 R R m ' ( , φ φ S D | a 0, 1/3, or , ω 2 − m ⊗ , ω ' p L ' , = a − x p In no-scale supergravity with a , B W p solutions in eq. (2.21) satisfythe eqs. (2.14)–(2.18). parameters The conditions (2.19) and (2.20) then restrict As we discussed above, we study vacua where possess a variety ofalong with solutions the that SO(10)-preservingparametrized lead vacuum in to the different, form degenerate symmetry-breaking vacua, where the parameter and where identical to that foundchange in in SO(10) sign models for using a when fields should vanish, leading to the following set of algebraic equations: vevs of for case, U(1) which leads to scalar potential has the simple form [74, 77] To study the scalarfields, potential, with we the write rest the of the superpotential fields2.9) ( set in to terms zero: of the SM singlet JCAP11(2016)018 . ). 2, P Λ, 2). / / and int / : 1 M Σ D α (2.26) (2.27) (2.24) (2.25) (2.23) − M case is ( im 2 (1 i O / to ensure 1 and κ has a posi- = ' ± − ' ± U 2 and , R ω / ω x = L e , ν in units of φ + 2 , Λ, Y | L p / ¯ R φ /η Σ , e ν = +1 | Σ )] a ) contribution to su- : m 4∆ a m . R 2 2 Y ) to be of the order of 3 − | 5115, in which case we e ν 0 Φ − . b p GUT 5 − ¯ 2 | Φ) with m L ' − GUT p L )] M , ( √ ¯ ( with φ ω H a η M ' − ( O + 46 2 + H 2 Λ, 0 O ); i.e., + 3 Σ | µ 3985 . This minimum with vanishing / . Φ b ) p | int Σ 1 0 a ( of 0 m µ 3 η m /ηm M κ ' GUT -term in the MSSM to be of the order ( 3 Φ → − + [ µ φ M = 0 at the vacua. To check that this is O + 2 0. 2 p = ' m R

¯ ( αφ . Among the possibilities which lead to ¯ R H φ p η ' U e ν L − L m x ∂ω − ¯ + [ φ ∂W with κ ) B

L Φ q ω ) cause rapid proton decay. The ¯ H + /η ∓ ¯ – 8 – α int Φ 2 ¯

+ 3 U(1) αφ ) H m M − ) when the conditions (2.17) and) (2.18 are applied. 3471 and Λ ω p αφ m doublet component in Φ ( − ( . ⊗ ( ω m R 0 ∂a − η L . We note, however, that one particular solution is ob- ∂W O R φ 2

+ 3 doublet components of L + 3 ) 0630) p , whose sizes are . φ + ( p L ' − H L 0 , eq. (2.24) leads to ( ω αφ L SUSY 2 η U(1) ,

, where we obtain η φ m x , φ − − i M L ⊗ 6 αH = 0, and thus is indeed stabilized at the origin in the vacua + 2 ¯ ∂p H L ∂W ± 0 T 2120 − H .

b ) = 2 3 D 0 a L , and m = ( , + 3 ¯ = a ) is the SU(2) H 2 µ L

, − ' − L b 1 2 SU(2) ¯ p φ W p R H Σ 0138 ( ( ⊗ ¯ . = are the SU(2) φ η H 0 ∂W L ∂ 2 C , L

develop a vev, φ m − + 1 /ηm ( ¯ H µ Φ + R Φ 3 cannot break SU(5), and thus the intermediate gauge symmetry is broken by ¯ φ 2 ' m m

) W R and ¯ , we consider here only the case with φ = SU(3) and ∂S ∂W L

R H int p, a, ω and φ In the above analysis we have assumed that Generically, the non-zero vevs of these fields lead to an p, a, ω G R φ couplings in eq. (2.9). To see this, let us first write down the relevant superpotential terms: ¯ indeed the case, we consider the scalar potential terms that contain superpotential is found for [71] supersymmetry-breaking scale, a weak-scale gravitino mass, i.e., if we impose The mass matrix in eq. (2.25) may be diagonalized using two unitary matrices with tained if we further require that the GUT sector does not require a fine-tuning of and have ( where After We note that φ of This intermediate gauge symmetry looksSU(5) phenomenologically gauge implausible, bosons however, whose since massesrealized the for are Λ where we have used the conditions)–(2.20). (2.14 We see immediately that persymmetry breaking, which may be fine-tuned with respectively, and the difference among α tive mass term unless considered above. However, we do not consider this particular2.2.3 solution here. Doublet-triplet splitting After the above fieldselectroweak symmetry acquire breaking vevs, correctly, we we needof obtain the the the soft mass MSSM scale, ascalled which doublet-triplet an is splitting, assumed effective and to theory. in be this much To lower model realize than we the can GUT realize scale. this This by fine-tuning is the the so- JCAP11(2016)018 , ) ω 3 1 GUT µ (2.29) (2.28) (2.30) M ' − ( p O -term for the µ -term of ) µ unless , we have SUSY φ GUT , M , ( Φ /M O m , L 2 L int ) to achieve the fine-tuning. U(1) [124, 125]. In this case ¯ φ H H M ⊗ m GUT † . ' D 2 M . Therefore, even though we have . ( H L = ) ¯ αφ O ¯ φ ω α ηm via the conditions (2.17) and (2.18)), are much smaller than the GUT scale 0 d d − η Φ + 3 H H ) values of and ) p m ω ( L have η vevs, the vector-like mass term for the color- φ and GUT 0 develop vevs to break electroweak symmetry, d + 3 M H Φ d + 2 – 9 – p can also be ( , ( H m 210 H O and/or H H GeV with great accuracy, which implies the absence and m L m L H ), this does not result in an 16 0 ¯ ηm u φ and H ' m H 10 2 u † µ SUSY H × , and U (1). ω M ∆ = 2 ( O , = p O ), then the color-triplet Higgs masses may also be much lighter ¯ 0 ), Φ. Due to the u u are of H H 1 GUT D µ ). This can be realized by cancelling the first and second terms in GUT , on the other hand, we need M ( M 8. In this case, are to be regarded as the MSSM Higgs fields with a ( -term of order of the soft mass scale, we need to fine-tune ∆ to be much and O / 2 GUT GUT ¯ µ Φ is different from that for remain at the origin. O d U 7) M 0 M d H ( = √ i H O couplings in eq. (2.9) also induce mixing between the color-triplet components φ ± α and , and in many supersymmetric GUTs this turns out to be the dominant con- int and ¯ ν (3 ), while the heavier states u 0 + u M and ¯ H ' H K with those in Φ and The exchange of the color-triplet Higgs multiplets leads to proton decay, e.g., via The eigenstates of the matrix in eq. (2.25) are related to the doublet fields via Finally we note that the fine tuning discussed above could potentially be avoided if GUT ' α x M H φ → ( -terms for the color-triplet components. On the other hand, if (some of) these values are and where smaller than Thus, to obtain a eq. (2.28). If If i.e., (notice that there is a relation between O of an intermediatemass scale matrix below should the bemixing GUT of angles the scale. in same order In in any order for case, the all cancellation to of occur, the and components thus the in the tribution [127–129].lifetime If tends supersymmetry to breaking beto is too suppress TeV-scale, short this the [130, contribution. resultant131], proton A and decay simple thus some way to additional mechanism evade is the required proton decay bound is to take p which may be phenomenologicallyconcentrate dangerous on as models we inwe discuss which will below. the see is intermediate ForThis beneficial scale this case for is reason, is proton also close we decay favoredMSSM to will in and meet the terms the each of GUT other evolution gauge scale, around the coupling 2 Higgs which unification, fields as as the during gauge inflation. couplings in the µ much smaller than than the GUT scale. instead of SO(10), the GUT gauge group were flipped SU(5) After supersymmetry iswhile broken, the doublet-triplet separation isnote below, solved several by of amodel, the wanted missing though features we partner discussed do mechanism below not [126]. could work be out2.2.4 carried As such over a we to model will a in Proton flipped any decay The detail here. fine-tuned ∆ to obtain color-triplet multiplets. In particular, for in triplets in Φ and JCAP11(2016)018 = 0, 0 (2.31) A to itself , namely S S , 2, such a large / = 8 TeV, ) years 2 β / 1 2 plays the role of the m S (tan [148], which shows that is favored to realize the R 3 b 2 A | ' ω The GUT-scale gauge boson /m 4 β t + denotes collectively the GUT- 3 m p | X = 10 TeV, sin 2 ' 0 M / X GeV yrs [140] is evaded if these vevs are + 2 β m 2 M | 16 34 as well as the unified gauge coupling; ω 10 10 φ + × a 4 | 7 4 . 1 , and p ω 25 > – 10 – / GUT , , all break the scale symmetry associated with the 0 1 α a γ GUT π g , + p e ) components (in terms of the SM quantum numbers) × 5 6 , and is a renormalization factor. → , 34 = 0 and comment later on the effects if they are non-zero, 2 p R , 10 b, c γ A 3 ( × = 5 c ⊕ yrs, [140 141] is satisfied for ) ' ) is the unified gauge coupling, 6 5 33 ) π 0 − , 10 (4 π 2 / + , the color-triplet Higgs masses tend to be as light as the intermediate × , should be non-zero as it also enters into the neutrino mass matrix, as we e decay amplitude is proportional to 1 3 6 b . is the dominant decay channel. The lifetime of the decay channel is approx- ¯ ν 2 GUT 6 → GUT g + 0 p > π ( M K = ) + τ ¯ e ν + → → GUT φ in the multi-TeV region [86, ];132–139 for instance, in the CMSSM, the current limit K p GeV. α p = 5 [86]. In the present scenario, however, the proton decay bound may become enhances the proton decay rate by orders of magnitude. Secondly, as we see above, if ' Of course, the exchange of the GUT-scale gauge bosons also induces proton decay, 16 → β β The one-loop renormalization factors of the Kählertype proton-decay operators for each intermediate p 3 10 int SUSY ( p gauge symmetry inref. supersymmetric [143]. theories are Belowref. given, [144 the in145]. supersymmetry-breaking ref.level scale, Below [142]. in the renormalization ref. electroweak factors [146]. For scale, a are The we relevant two-loop-level given use hadron computation, at the matrix QCD see one-loop elements are renormalization level evaluated factors in in computed ref. at [147]. two-loop where imated by tan M scale. Since the protonHiggs decay mass, rate this is again inverselycuss proportional reduces these to issues the further the proton in squarebecause lifetime this of of paper, by the a simply orders color-triplet very assuming ofsuppress high that magnitude. supersymmetry-breaking the the proton scale color-triplet We decay and/or doautomatically limit Higgs some is not solved additional exchange evaded dis- in mechanism contribution. to awhich the flipped As intermediate SU(5) scale noted model,proton is earlier, decay. close but these to here or issues we at are will the GUT concentrate scale on to models minimize in the latter effect on of the SO(10) gauge boson has a mass potential. Therefore in orderbe to small. realize suitable For now, inflation,while we we noting take must that require thesediscuss couplings in to the following section. M τ where scale gauge boson masses,masses and can be expressed in terms of the current experimental bound & 3 Realization of inflation As was explained ininflaton. the previous The section, shapeand to in of the our its Higgs modeltwo effective sector. terms the potential Strictly in singlet speaking, is eq.those the dependent (2.13) proportional Starobinsky on whereas to potential couplings the is the realized other couplings via terms of the in first the superpotential involving SO(10) relation for theto Yukawa couplings. the Since the wino (higgsino) exchange contribution for instance, the ( tan more severe. First, in SO(10) GUT models, a large tan JCAP11(2016)018 , s (3.7) (3.6) (3.1) (3.2) (3.3) (3.4) (3.5) 3 the √ m/ . s . = 3 i / 24. . 2 2 λ | 5 sufficiently close √ 2 2 x γω 2 − − | = 0: the potential s > e 12 ) 2 | b bφ , 2 | 3 if the corresponding | m 2 + that for φ √ + | . In this case, the scalar 2 ) | s } 2 2 | 0 → + 2 γa φ 2 1 | ) 6 , φ 2 | | 0 S and the superpotential param- − ω /V + + 2 | x , ω s 6) to zero during and after inflation, 2 0 2 . | | V -folds of expansion, where √ ( ) , the values that give rise naturally + 6 φ ω , a , e i | 1 2 γp 0 2 s/ , η | 2 p ± 6) | Φ a 50 = ) factor; for example, | + 6 √ 2 + b 2 ' , m | ( 2 s/ tanh( | ∗ + 3 a Σ }'{ | = 50(60) is found for an initial value of O ˆ V, V 2 3 and N | cφ m 2 will drive | / ∗ ; p : | 1 + 3 , the instantaneous minima of the singlets during b x m ( , b ( + ∆ – 11 – 2 1 3 + 2 3 tanh( | f 2 2 1 p 2 | 2 6) √ | − = 0 √ p, a, ω, φ b s √ { + 3 0 ≡ ∗ bφ all vanish in the limit N > N x 6) / | , defined along the real direction as s ' 2 2 s/ h s | Z S √ ( 2 | S √ 2 δφ | − s/ − ( S ( | e . We have checked numerically that, for 3 1 2 -folds = is the canonically normalized inflaton. For the Starobinsky e + − int ∗ − s 2 p, a, ω, φ 3 tanh 1 | N 1 M tanh 2 × 2 − λS ' m 1 3 4 − V ), where GUT = ' s 45). Thus, to realize Starobinsky-like inflation, we must ensure that any mS ( . M | V V = 24(5 ˆ . V is a (somewhat complicated and long) function of = 5 = 0, we find in terms of the canonically-normalized field f ∗ Sufficient inflation would require at least During inflation, the GUT-breaking Higgs fields are displaced from their vacuum val- For a non-vanishing but small value of Let us for now assume that the Higgs fields are displaced a negligible amount from γ > s = i scalar potential takes the form significant deviation from the Starobinsky potential occurs at values of s derivatives with respect to ues (2.21). These displacements would be exponentially small for for a potential potential, a total number of values of the Higgsthe singlet canonically components normalized are inflaton given by (2.21). For a finite, but large, value of the instantaneous deviation from the vacuum vev isinflation proportional are to perturbed relative to) (2.21 by an potential during inflation takes the simple form where eters. This functionto is the divergent hierarchy for their vacuum values during inflation, to these singular points,preventing any the finite spontaneous value breaking of of the intermediate gauge group. where This shows that, in orderneed to to recover constrain the predictions independentlyc of the no-scale values Starobinsky-like of inflation, we the squared moduli inside the brackets. For JCAP11(2016)018 ) 8 . + 4 as 7 . 2 3 and and | − 7 − . (3.8) (3.9) a s 3 | are of (3.11) (3.12) (3.10) . The 10 K n − 7 . (10 ω 5 + 3 10 using the − 1 2 . | 3 . bφ < 8 p 10 . ≈ − ) 7 s − 10 ≡ | 3 K and/or / b < = 0. In order to 2 K = 0. is p, a p b K < bφ b ( 2 = 0. As we have fixed for , . bφ sinh , K ∗ for 2 = 50 = 60 | here is not to be confused with K φ ∗ ∗ KN | K. . ∆ /V ∆ 12 + 2 ss ,N ,N 32 81 . The solid curves show the positions and the quantity ∆ 2 64 27 ' V b | b + ω + ≡ | 2 is increased from 0 to 10 ∗ ∗ η 00078 00043 N . . N + 6 0 0 2 bφ 2 2 | ( η , r a as | = 50 and 60. ≤ m bφ m bφ ∗ – 12 – 4 + 2 2 is shown in figure3. As one can see, so long as ) N + 3 8 3 2 6 | bφ 3 bφ 32 p 6 − + | , we can approximate numerically the limits on ∆ + 1 ∗ -folds of inflation. 10 s 2 e 2 K ∗ N 3 ' × / 12 N 2 s 1, the inflationary parameters can be approximated analyti- and the bounds from Planck are always satisfied. The scalar n 1. This is the cause of the offset when √ = 0 corresponds to the exact Starobinsky result. The dashed ' ' − − = 50 and 60, 7 − + (2 . and ∆ e given by (the quantity 3 2 s ∗ r -folds of inflation. Since the vev of Φ is no larger than the GUT bφ b − r K = n e m N K ∆ 10 x 1 2 ∆ , the potential is indistinguishable from the Starobinsky potential out 5 ) consistent with Planck, we must require that the product < ∼ . + 7 . In each figure, we plot the slope of the perturbation spectrum, bφ, , r 2 − | 5 needed for 60 K s -folds of inflation. If the largest vevs associated with . V e n 50 (60) 5 , the most severe constraint we have on the coupling 10 bφ , we show here only the analytic result. In this case, in order to obtain values 3 | . ' ∼ 2 3 4 ∼ K in ∆ − ∗ s ) plane for 2 m N | 10 GeV, ∆ 5 , r = . φ s 50 (60) 2 | 16 < ∼ n ) consistent with Planck, we must require that the quantity ∆ V − ' 8. Figure4 also shows the potential for other choices of ∆ φ ∆ , r ) for + 2 10 ∗ We see in figure2 the corresponding effect of varying ∆ We see in1 figure the effect of a non-zero value of For generic values of s = 0, and recall that 8 ≈ 2 n N The dotted lines simply interpolate between | − by < ∼ s 4 K ω | would be obtainedobtain for values of ( scale, to the value analytical approximation for the∆ potential given by eqs. (3.7)lines and are (3.8). derived Here we from have a taken full numerical evolution.(10 For these solutions, ∆ the value of ∆ of ( bφ in the ( scalar potential for several choices of for order 10 potential for this caseto is shown by the dashed curve in figure4, and is Starobinsky-like out bφ where We show in1 figures 2 and 6 the effects of the coupling the tensor-to-scalar ratio, the superpotential coupling): The orange (purple) shaded regionsIn correspond the to the limit 68 where cally (95) % by CL limits from Planck [1,2]. JCAP11(2016)018 0 , ' 6 . See − x (3.13) (3.14) (3.15) P denotes M b ¯ 5 a ; 2) = 10 − 1.5 Figure . The dotted K , bφ GUT , b = 10 b M 8 . . ···} m , bφ = R = 0 ω and = 5 during inflation, for the − a , η . ω 60, with the 68 and 95% CL 2 , . = , 0 a = 10 p = 0 − , = 0. The dashed curves show the β p , and we indicate the corresponding = = 50 b T, S, p, a, ω, φ ¯ P K , ∗ φ ¯ sin Ψ 1, for the same range of φ Λ = M N ∂V ν ∂ ≡ { − , f b ¯ s . a a , = ¯ for d 4 c K x 1 and − K bφ − + b 10 c ∂ ¯ = d ˙ Ψ × a b x 2 ˙ K . Ψ – 13 – eV assuming a bc = 4 = 8 − a bc + Γ Σ Γ a , quantified in units of ˙ . As was discussed previously, for case 1) the differences Ψ bφ , m H 2 GUT − + 3 10 a ) curves as functions of < M ¨ Ψ × , r s 3 int . n M = 3 Φ , i.e., , m 5 − . Parametric ( So far we have relied on the assumption that the Higgs fields track the instantaneous We consider two types of solutions: 1) = 10 φ < p, a, ω m minimum during inflation.classical equations We of have motion, verified given this by behavior by integrating numerically the section4 for more details of the relation between the light neutrino masses and curves illustrate particular values of left-handed neutrino masses in units of 10 Planck constraints shown inusing the the background. analytical The approximationpower solid spectrum (3.7) parameters curves and calculated illustrate (3.8) numerically the with assuming parametric ∆ dependence Figure 1 Here the indices run overthe all inverse field Kählermetric, components, and with the Ψ connection coefficients are given by displays the numerical solutions for the SM singlets between the instantaneous valuesare of negligibly the small, Higgs and fields inflation can during be inflation realized and for a their wide vacuum range vevs of values of and following set of parameters: JCAP11(2016)018 ' m , assuming 2 | s φ | + 2 0 = 0 here. 10 2 m | 4 K - ω | 10 + 6 2 | m 3 a | - 8 + 3 2 | , in units of the inflaton mass p bφ ≡ | 6 K m 10 . As one can see the evolution of all fields 2 6 - − 10 – 14 – × 4 8 . 3 GeV. The resulting inflationary parameters are illus- ≤ 16 b m 10 1 ≤ - 10 2 = 0 is the Starobinsky potential. We assume ∆ = ϕ bφ b ) equal to 10 2 16 m ( / V 1.0 0.8 0.6 0.4 0.2 1.2 16 and = 0. These parameter values are chosen to obtain vevs for the singlet components γ . As in1, figure but for different values of ∆ . The inflationary potential for different values of . The curve labeled = 210 P c M 5 = 0. − Figure 3 10 with trated in figure1 in the range 0 of the Figure 2 bφ JCAP11(2016)018 . 2 / 3 , for ) 2 GeV. . We | . The Φ 6 6 φ | 12 − − xm 10 ( 10 + 2 2 × /η & = 10 | 2 4 ω b b b | 2 / ' , thus eventually 1 Σ -folds of inflation. would in principle + 6 ω p and e m 2 x | 2 3 = a − − | 60 a ' − for any . This is illustrated in 10 ∼ 1 = ω x + 3 − ∝ 1. We note that the initial = 10 2 p | does not differ significantly 10 p − Φ | δφ m × = GeV. ≡ | bφ = 0, , | x 4 16 φ and K − few GeV and Φ 10 14 . = 10 × 10 0 ω /xηm × = = 4 2 φ/φ 3 a b 0004. In this case, as discussed at the begin- may be problematic for proton decay as we Σ that would lead to Planck-compatible infla- . 2 evolves over the last = | ' x − m p s = 0 φ bφ 10 | – 15 – = 2 for case 1 with x . ∝ φ ) = 0 GeV, 0 = Λ = 0, 16 that leads successfully to the SM vacuum. In this case, η p, a, ω i 10 ( φ/φ Σ δ ' − a i h Φ = 0 by h S U(1)-preserving minimum [116]. Figure6 illustrates a particular real- ⊗ . The inflationary potential for different values of ∆ rolling of the inflaton is seeded by the kinetic energy of the oscillations of the Higgs As an example of case 2), we consider The initial conditions for the Higgs fields chosen for the numerical solution shown in is too small, this deviation can no longer be considered a perturbation, and it can be x = 0. The black dashed line corresponds to At the end of inflation, all fields begin oscillations about the low energy vacuum. Figure 4 ning of this section, we findative to that the its instantaneous position minimum at during inflation is displaced rel- track very smoothly their local minimum as b If shown numerically that the Higgsrolling fields into are a driven towards SU(5) ization of the hierarchy the parameters used correspond to vevs in turn correspond to In this particular case, thethat Higgs the excursions inflationary during potential does inflationmust are not be have not the negligible, followed simple which to formtion. implies constrain (3.4), and Nevertheless, the a as value numerical of from6 figure approach thedemonstrates, analytical the approximation based bounddrive on on the eq. intermediate (3.8). scale A vevHiggs smaller lower, fields value but of fail it to can be lead shown to numerically a that SM in this minimum case if the we choose a smaller figures5 and6 coincide with thethat position of inflation the and instantaneous minimum, thethe but successive we initial have evolution checked conditions towards the are GUT-breaking perturbed vacuum by are up stable to if ∆ also note in passingdiscussed that above. solutions with small figure7 for an initial deviation ∆ uphill JCAP11(2016)018 during φ, p, a, ω are all equal to } 0 10 10 10 10 , φ 10 10 10 10 0 × × × × 1.5 , ω o 1.5 1.5 1.5 1.5 , a 0 p { = 1. P M and the SM singlets 10 10 10 10 s 10 10 10 10 × × × × 1.0 1.0 1.0 1.0 1.0 ) ) ) ) ) 7 7 7 7 7 10 10 10 10 10 (× (× (× (× (× t t t t t 1. The Higgs vevs – 16 – − 9 9 9 9 10 10 10 10 = × × × × 0.5 x 5.0 5.0 5.0 5.0 0 0 0 0 0.0

1 0 5 4 3 2 0 0 0 0 1 2 3 1 2 3 1 2 3 2 4 6 8

s ------

ϕ ϕ p p (× - (× ) (× ) (× - ) ) 0 0 0

10 10 a a 10 10 ω ω - -

- - - - 10 10 10 10 . Evolution of the canonically-normalized inﬂaton GeV. For simplicity we display values in Planck units with 16 10 inﬂation, for the parameters (3.15) and Figure 5 JCAP11(2016)018 ' p during ' − ω and φ, p, a, ω 4 − 10 × 1 . 2.0 2.0 2.0 1 2.0 2.0 ' φ , 3 − 10 and the SM singlets 1.5 1.5 1.5 1.5 × 1.5 s 4 ) ) ) ' − 7 7 7 ) ) 7 7 a 10 10 10 10 10 (× (× (× (× (× t t t 1.0 1.0 1.0 t t 1.0 1.0 – 17 – 0.5 0.5 0.5 0.5 0.5 = 1. P 0.0 0.0 0.0 M 0.0 0.0

1 0 5 4 3 2 0.5 0.0 0.0 1.0 0.5 0.0 1.0 0.5 0 1.0 4 2 0.5 1.0 0.5 1.0 0.5 1.0 2 4

s ------

ϕ ϕ p p (× - (× ) (× ) (× - ) ) 0 0 0

10 10 a a 10 10 ω ω - -

- - - - 6 6 6 5 0004. The Higgs vevs are given by . = 0 x in units with . Evolution of the canonically-normalized inflaton 6 − 10 × 6 . inflation, for Figure 6 1 JCAP11(2016)018 + 2 | 6= 0, (4.1) larger b (3.17) (3.16) . The cφ | c H allowed by are reduced = 0. In this c c b will be always . 3 . φ ¯ , the corrections Φ, and − c and 10 ··· b 1 and × ¯ + e , 3 L . Similarly to the d in our model are given 2 c ˙ & x φ H d e c , one would be tempted to f H c , which are not independent Higgs singlets will be driven + 2 γ − 2 , implying that ¯ d all ω i and Q and ± -folds compared to an unperturbed d u , c + 6 ˙ e ) H H 2 d x , and a ( f 2 ) g 3 and / − 2 + 2˙ are rapidly damped, and the subsequent , with a sufficiently large 1 cS / c R 2 , ( 1 x ˙ φ p ∼ Lν Σ Φ = 0 u ) with a constraint on the combination 2 GeV, this occurs for s x | m m H 1 case, the naive expectation would result in the – 18 – 3 ν 16 / − f bφ | 2 ; for example, c η = + c doublet components in the fields Φ, p = 10 ¯ u ' x L Q ω when both are non-vanishing must be checked numerically u δφ sinh( = c H u a 3 f 1 √ = and 2 = , only a factor of four below the maximum value of ) is divergent for p b 4 x . However, it can be verified numerically that values of − ( = 2 | g ' − 10 φ c Yukawa γω ˙ implies an increased total number of × Ψ , whereas a numerical calculation shows that 95% Planck compatibility b close to these points. For any s 7 W 7 increases, the oscillations of 12 ˙ | − Ψ x s . 1 and S bc + 10 c Γ − 2 1, the Higgs singlet components are displaced from their vevs during inflation | × = 5 γa x ) is another (somewhat complicated and long) function of 6 s | . x ( c + g So far we have neglected the effects of the couplings As the analytic approximation (3.5) is valid for very small The specific limits on 2 | γp = 0 case, the function 2 fields through the connection-dependent terms in (3.13), namely: case, for where c driven to zero for will be large dueto to zero the during induced inflation, mass-squared leavingcase the with universe in an SO(10)-symmetric state. In the particular by linear combinations ofYukawa coupling the terms SU(2) in the low-energy effective theory are then written as by corrections that depend linearly on As the value of initial condition. Similarly,of for the case fields 2 with welarge, respect the are to subsequent not evolution their required may minima. to well take fine-tune However, the if the theoryas these initial they to perturbations positions satisfy an are the SO(10)-symmetric relation initially vacuum. (2.19). too Let us for simplicity assume that relate directly the Planck constraint on due to the simultaneous effect of both couplings. 4 Yukawa couplings and neutrino4.1 masses Yukawa unification andAs its discussed violation in section 2.2.3, the MSSM Higgs fields evolution resembles that shown in5. figure transient growth Note in the difference in timescale in this figure. The on a case-by-case basis. Nevertheless, it is clear that the allowed values of | bound is retained for symmetry breaking. than the value thatPlanck one data would can naivelyrespect have still expected to lead to their to beexample, vevs Planck-compatible in the compensate the results; maximum the previously-discussed compatible the expected with deviations deformation of of the the fields inflaton with potential. For JCAP11(2016)018 during φ, p, a, ω 1.5 1.5 1.5 1.5 1.5 and the SM singlets s 1.0 ) 1.0 1.0 1.0 1.0 ) 3 ) ) ) 3 3 3 3 10 10 10 10 10 (× (× 1. Here we consider a perturbed initial condition (× (× (× t t t t t − = – 19 – x 0.5 0.5 0.5 0.5 0.5 are as in figure6. } 0 , φ 0 , ω 0.0 o 0.0 0.0 0.0 0.0 8 6 12 10 1 0 0 3 2 1 0 3 2 1 0.0 3 2 1.0 0.5 1 2 3 1 2 3 1 2 3 0.5 1.0

, a

s ------

ϕ ϕ p p p (× - (× ) (× ) (× - ) ) 0 0 0

10 10 a a 10 10 ω ω - -

- - - - 4 4 4 3 { 2. Here . . Evolution of the canonically-normalized inflaton = 0 0 φ/φ ∆ inflation, for the set of parameters defined by Figure 7 JCAP11(2016)018 ¯ is Φ γ (4.2) (4.3) (4.4) (4.5) at the ν and f c unification = e e - (10)% level in f d O = 0, then Φ and from mixing with d and f → S µ 0 = - s u , α 0 , f , 11 11 . D U M, b ) ) 11 f f D y 3∆ 3∆ = − − . e y y in this case. This feature distinguishes induces mixing between right-handed f ) ω in order to prevent b ψψ . 11 = = ( = ( Σ d + D e ν SH a H 6= ( λ f P f while the deviations in P ∆ 11 ∆ c M 5 U c M – 20 – , f should be small in order to avoid over-production , f , f = 11 , and = 11 11 SH U γ U D f eff λ y , ) ) c ∆ , since W f f = α ν (1), the GUT relations for the first- and second-generation 6= ¯ f + ∆ + ∆ O α , which are the fermionic component of the singlet superfields = y y i = ˜ u S f f = ( = ( ∆ d c u f f ) for 3 − (100)%. -parity violating operators, and thus is phenomenologically desirable as we . As we have seen above, for the inflaton field, the smallness of R O (10 H O = f These GUT relations are modified if there exist higher-dimensional operators suppressed quark mass before its discovery: see the third paper in [89–91]. The corresponding relation in non-supersymmetric SU(5) GUT actually led to a successful prediction of ¯ b Φ, Σ, and 5 . We also suppress the couplings i GUT scale. by the Planck scaleis–151]. [149 expected Among to such give operators, the the leading following contribution: dimension-five operator our model from other SO(10) GUT models, where one usually has the These equations show that weYukawa couplings expect at the the unification GUT ofYukawa scale, couplings down-type as are quark in also and SU(5) unified. charged-lepton be GUTs, These different and two from the classes each up-type of other quark the if and Yukawa neutrino couplings may, however, of gravitinos as we will see in the next section. In this case, mixing occurs only among the After Σ develops a vev, thisconditions operator in leads to this the case Yukawa couplings are in eq. given (4.1). by The matching with Since ∆ Φ, required by successful inflation, while neutrinos and the singlinos discuss below. A non-zero value of the coupling where the Yukawa couplings are relatedthe to the following corresponding GUT-scale GUT matching Yukawa couplings conditions: through are as large as 4.2 Neutrino masses We now investigate thehave mass no matrix mixing for neutrinos. withsuppresses the neutrinos; If we we take consider this limit for simplicity. NoteS that this limit Yukawa couplings may beerator. modified For significantly the insignificant. third-generation the Intriguingly, Yukawa presence couplings, this oftrum; on is experimentally, the the consistent bottom dimension-five and with other op- tau the hand, Yukawa unification observed its is quark effects realized and are at lepton the less mass spec- most of the parameter space in the MSSM, JCAP11(2016)018 , 5 ). − m , 10 (4.8) (4.9) (4.6) (4.7) 12 eV (4.10) (4.12) (4.11) .     ' 0 and M M M L R ν ˜ f ν ν b < S are shown ν     P m      M 2 (recall we have , . A similar form ) 5 P i m b − bφ d (    1 eV is obtained if 0 . 10 H c L R ˜ S h 0 ν + ν / i '    m . u 2   H m L ˜ S, bφ β m 0 β ≡ h − . sin , β bφ v β sin L L bφ ν ν v ν f . ν sin β β f bφ m 0 0 0 v 2 − m ν 2 m , sin − sin ) 2 f β v − v c − R m ν bφ ν 2 ν ( ) sin β m β bφ bφ bφ v ( bφ f m f ν sin 2 − sin f ) 0 0 + + v for the second and third generations are almost 2 v – 21 – β ν bφ ν ˜ c R S f ( m ν β m 3 MeV [153] and the GUT relation (4.2) implies sin − . m bφ m f v , the normal hierarchical neutrino mass spectrum is sin ' m   bφ bφ 5 0 0 0 v ν = 2 23 eV, and an even stronger bound of and ν + − . − f ν ˜ − f u 0 S b m c R L m ˜ , a left-handed neutrino mass S ν ν c R < ν P m = 10 ν ' ' ' L      M m 5 ν m . 8 M M M R L forest power spectrum obtained by BOSS in combination with CMB P M − ˜ ν ν S , making the neutrino mass shown in figure1 correspond to the − ˜ S 9 α 10 and − Currently, the Planck 2015 data [1,2] imposes a constraint on the sum = M R ∼ 10 6 ν /v . u 5 mass − M L bφ m L ν 10 = bφ × as we have seen in the previous section. In this case, the couplings satisfy − 174 GeV is the Standard Model Higgs vev and tan 9 β b = ' . sin v β ν Values of the mass eigenvalues calculated numerically for For the first-generation neutrinos, the requirement of successful inflation restricts the mass f We note that the up-quark mass L 6 sin ν in figure8.f If arbitrary, we may explainchoosing the these current parameters neutrino (as oscillation well data as [155, flavor-changing couplings156] corresponding by to appropriately data [154]. Since the parameters of the neutrino masses: comes from the Lyman and the corresponding mass eigenstates are approximately For the secondsuppressed and all third generation generations,still indices) on given can by the be other arbitrary, hand, but the the masses coupling for light neutrinos are which is consistent with the above limit. coupling right- and left-handed neutrinosthe and neutrino-singlino the fermion singlinos. mass matrix Disregarding takes Planck-suppressed the factors, form [92–94] for the mass matrix is found in flipped SU(5) [126, 152]. the hierarchy For obtained if where and thus the diagonal mass matrix has a double-seesaw form given approximately by JCAP11(2016)018 ) a 3 − (4.15) (4.13) (4.14) p ( 0 b , and below P + 13 M 10 8 M . leads to a quasi- 7 − = , bφ L ∗ ¯ φ = 10 L , and the light neutrino ν r KN bφ K. M 12 ∆ ∆ 10 and 81 32 . s 27 64 0 limit would correspond to a , n + )) + → 2 GeV ∗ a ∗ 0 3 N b 13 N 2 − − 2 ] . For these parameters, the allowed range p 10 L 11 R . A smaller value of ( β 0 bφ β m m × b 10 10 GeV − 2 [ GeV in figure8). On the other hand, the sin + sin m m 9 ' 10 v ϕ v b ν M L ν 10 × ( f , one can compute, e.g., the contribution of the f m 7 – 22 – L H ν ∆ ' m 10 ν bφ > ν ' bφ m 10 are different from zero (and related by the minimiza- m m m 0 , as a function of b ) 5 L 8 3 − 3 φ 32 ( lightest 10 + and ψ + ∗ × 2 2 ∗ N together with M 9 S 12 N L = 2 m 10 ¯ φ β , ' ' − L sin s r , φ ν n L f will in general result in significantly mixed mass eigenstates. As a crude ¯ H , φ 0 5 5 and L b 15 15 - - 10 H P

10 10

, , M GeV . The neutrino mass spectrum in the double-seesaw scenario arising from (4.6), assuming S R L ] [ m state. This contribution has the form p 5 − R L and correspondingly the neutrino masses, is located in the upper left region, bounded to the ν ν When the couplings gauge eigenstate to the lightest state, which in the bφ M = 10 L ˜ φ or approximation, if one assumesponents that only of the ‘left-handed’ fields mix, i.e., thepure fermionic com- tion condition (2.20)), theceases fermion to mass matrix be for block-diagonal, the and SM potentially singlets large and uncharged terms doublets such as Figure 8 m for left by the vertical dashed line showing the 95% CL Planck upper limit by the horizontal dashed line corresponding to lightest active neutrino (similarly for neutrino mass spectrum is inverted if degenerate mass spectrum. Theneutrino oscillation latter data two types [155, in156] of and future mass the spectrum experiments. CMB arebetween observations the constrained It [1,2, CMB by has154], observations, both as and notmasses, quantified will in escaped that be the becomes our values tested apparent of attention if that we there write (3.10) is and a3.11) ( strong as correlation JCAP11(2016)018 , the (5.3) (5.2) (5.6) (5.1) (5.4) (5.5) c . For a -term via yields the at the soft µ GUT b µ and neutrino- to be small in µ/M H 4 for the MSSM. / & ,M ¯ Φ- 0 , 0 b b 2 / via the mixing2.29). ( 1 ) to keep = 915 L , ∗ u P 2 g | GUT M would mix the Higgs sector SH 5 m − γ 21 SHL µ/M U 10 C | b . and π 4 0 m 4 L, / , , c b 1 u φ − M M R ν H m ˜ ) = ν f for S M 4 0 L b − / u S m bφ induces an inflaton-Higgs-neutrino coupling m ∗ bφ ˜ g GeV]. [158 However, these couplings vanish H ν 21 ∝ + 6 − f 915 U term leads to SHL → M M R – 23 – C | ψ b ˜ S ν S − ¯ Φ = ' ' = SHL bS R S C ˜ ν ) + Γ( int SHL ˜ L L × | C u H GeV → 15 S , comparable to the electron Yukawa coupling. 10 Γ( 5 )]. This implies that sizable mixing can occur for − ), the fine-tuning condition for the doublet-triplet splitting is modified, ' ω has been defined in), (2.28 and is related to the weak-scale R . However, we have verified that, in the Planck-allowed range for M +3 ˜ T S p ( 2 GUT η M +2 . denotes the effective number of degrees of freedom, and H (and thus ∗ 0 m ,M [ g 0 b b As discussed previously, non-vanishing values of In the present context, the inflaton is once again a singlet, and the coupling µ ' with where we have disregarded weak-scaleinteraction terms, cf., eqs. (4.9)–(4.11). As a result, we obtain an where and the inflatino mass spectrum, and in particular the left-handed neutrino5 state, are negligibly affected. Reheating and leptogenesis In the absence of amodels direct coupling almost between the always inflaton proceedsto and matter, a through reheating minimal the in supergravity reheat minimal temperature gravitational of couplings order [157], 10 leading This results in the inflaton decay rate in no-scale supergravity [78]matter and or reheating a must proceed couplingidentification of either to the though inflaton gauge a with fieldsing, the direct as right-handed through coupling sneutrino reheating the has to takes appeared gauge placeor to naturally neutrino/Higgsino kinetic be through pairs very term. the promis- [43, duction, decays68]. For of it this the In was inflaton reason, fact,sneutrino) to necessary the to of sneutrino/Higgs to avoid order set excessive 10 reheating a and limit gravitino on pro- Yukawa coupling of the inflaton (right-handed where ∆ ∆ larger and it turns outmass that scale. ∆ In should anyorder case, also to as grow ensure we (∆ a seethan good in the dark the age matter subsequent of section, candidatelarge the we (the Universe), need lightest and to neutralino thus with we a do lifetime not longer consider further in this paper the case of which leads to a reheat temperature direct coupling of the inflaton tosinglino Standard mixings. Model matter For fields the throughFor the former, the the latter, the neutrino Yukawa coupling through the scalar mixing JCAP11(2016)018 12, . (5.7) 0 (5.9) (5.8) (5.10) (5.11) < 2 h χ , i 1 M M -violating decay, so the L i 2 1 ) in our setup, this bound 3 ) are similar to those of the − , , r † SHL s , (10 C n O to be zero since they make it decay = χ SHL GeV R , 0 b C T . m R 10 1TeV : 5 m T − 10 m/φ and 10 SHL 13 Im ∼ − 3 C , 12 . M 3 S R | − , 10 n T =2 0 X in eq. (5.3) is clearly an i ), and – 24 – α 10 × 5 S 11 − SHL 4 ∼ × . C 4 4 | L . s (10 n 2 . O † SHL 2 ' 1 C / = s 3 2 / is a measure of the C and CP violation in the decay, which is n ν s 3 f SHL n is weak enough to ensure that the lifetime of the lightest su- C . For a more detailed discussion of reheating, see [166]. b 5 are the masses of the singlets, where the lightest is assumed to be (1), − -parity is violated in this model though, as described in [43], the 3 π 3 O 8 R 10 ,M = . 2 | ' − b M 21 . 5 U − In order to obtain the correct baryon asymmetry, we should place additional | ∼ | 10 m 6= 0, the inflaton can decay into a pair of higgsinos as well. Similarly to the 7 . is the number density of inflatons at the time of their decay. This lepton (or = SHL | SH 1 C is the entropy density and we have assumed that the gravitino is much heavier than S | λ SH n s M ) asymmetry then generates a baryon asymmetry [168, 169] through sphaleron inter- λ As noted earlier, If The abundance of gravitinos is determined by the reheat temperature [98–101, 159–164]: | L See ref. [152] for a related discussion in the context of flipped SU(5). 7 − at tree level. The form of the coupling of B actions [170–173]. The factor determined by loops inis which given one by or [174–176] both of the remaining singlet states is exchanged, and reheating process may well lead to a lepton asymmetry given by [157, ] 167 where where the gluino. In orderwe to must satisfy ensure the upper that limit [165] on the abundance of neutralinos: Ω Since we expect implies above case, to evadethat over-production of gravitinos, we need toviolation suppress via this coupling the such persymmetric coupling particle is muchsupersymmetric longer particle, than we the need age to of the take Universe. To stabilize the lightest where the inflaton. constraints on thefurther couplings here. and masses of the heavier6 singlets, which we Summary do notIt discuss has beencouplings shown of previously aflation that singlet whose no-scale inflaton predictions supergravity field for with provides the an inflationary bilinear observables economical and ( way trilinear to self- realize a model of in- which leads to an upper limit on the coupling JCAP11(2016)018 U(1) model [126] ⊗ of Higgses are needed ¯ 5 , 5 , Yukawa couplings, and similar 10 responsible for breaking SO(10), , τ 10 210 representations of SO(10), but there parity is not conserved, so one might and b 16 R – 25 – considered here). 16 simultaneously with the inflaton. One of the important phenomeno- , and 16 16 , couplings in SO(10). We therefore consider an SO(10) GUT model with a 3 multiplet. This model has the Kählerpotential shown in (2.10) and the 210 16 16 or 2 16 We have shown that inflation can be realized in such a framework, studying numerically Two specifically supersymmetric issues are gravitino production during reheating at the The no-scale SO(10) GUT scenario for inflation described here has many attractive We have discussed the fermion masses in this model, point out that it predicts the In this paper we have addressed these issues in a supersymmetric SO(10) GUT model. In the behaviours of the scalarevolution fields of during the the the inflationary three Standard epoch. Model In singlets particular, in the we tracked the end of inflation andimposes the a nature reasonable constraint ofcomparable on dark to the matter. that inflaton of Avoiding Yukawa coupling, the the which electron. overproduction should of In be gravitinos this at model most features, since itwith many combines phenomenological desiderata. Starobinsky-like We predictionsinflationary therefore consider model-building, for it while a admitting the significant that stepThus inflationary it forward there in has perturbations is some issues, still notably significant proton scope stability. for furtherAcknowledgments improvement. The work ofST/L000326/1. J.E. The work13ER42020 was of and supported D.V.N. in was partof in supported by M.A.G.G., the in N.N., part Alexander and partUniversity by K.A.O. S. by of was Onassis Minnesota. the the supported Public DOE in Benefit U.K. part grant Foundation. by DE-FG02- STFC DOE The grant work via DE-SC0011842 at the the research grant are no superpotential shown in (2.9). Asof discussed symmetry in breaking, section2, we paying consider careful various possible attention patterns to the vacuum conditions in each case. the single in the Higgs where the Higgs structureinstead of is the greatly simplified (only a (phenomenologically successful) unificationunification of between the the Yukawaneutrino couplings masses in have a the double-seesaw structureand up-type involving the the quark left- singlino and and partner right-handed of neutrinomasses neutrinos the impose sectors. inflaton on field. The this We structure, have and explored the shown constraints that that it neutrino can lead to successful leptogenesis. fear for the stabilitysupersymmetric of particle is supersymmetric typically darka much longer matter. plausible than candidate the However, for age the dark of matter. lifetime the of Universe, so the this lightest is still singlet inflaton field, inby which there a is Higgs an intermediate stage of symmetry breaking provided logical issues in constructing suchdiscussed, a the GUT proton model stability is constraintscale doublet-triplet requires mass and/or either splitting. some a As additional verytribution. we high have mechanism supersymmetry-braking These to issues suppress may the be color-triplet more Higgs easily exchange resolved con- in a flipped SU(5) general, sneutrino inflation is anGUT, attractive because scenario, but sneutrinos this are cannot be embedded realized in in matter an SO(10) Starobinsky model. 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