JHEP05(2010)010 , 6 − ing Springer and 10 May 4, 2010 9 : April 19, 2010 − : tic inflationary December 3, 2009 : Published lar potential, and no s. The inflaton decays Accepted 10.1007/JHEP05(2010)010 10 Received hysics experiments. The − of the light inflaton mass, nding on the inflaton mass, doi: y Universe and the inflaton –10 n the model. The inflaton-to- 9 a ences, − s can be produced per year with Published for SISSA by [email protected] , b - decays with branching ratios of orders 10 B 8 GeV, and discuss the ways to find the inflaton. The most promis . 1 0912.0390 . and D. Gorbunov χ Cosmology of Theories beyond the SM, Rare Decays We study the phenomenology of a realistic version of the chao a m [email protected] . Institute for Nuclear Research of60th the October Russian Anniversary Academy prospect 7a, of Moscow Sci 117312,E-mail: Russi Max-Planck-Institut Kernphysik, f¨ur P.O. Box 103980, 69029 Heidelberg, Germany b a Open Access Abstract: F. Bezrukov Light inflaton hunter’s guide 270 MeV model, which caninflaton be mixes fully with andadditional the directly scales Standard explored above Model in theHiggs Higgs electroweak coupling scale p is are via responsibleproduction present the for i in both sca particle reheating collisions. in the We find Earl the allowed range are two-body and Keywords: ArXiv ePrint: can be searched forfrom in several a billions beam-target to experiment,modern several where, tenths high-intensity depe beams. of millions inflaton respectively. The inflaton is unstable with the lifetime 10 JHEP05(2010)010 ). 1 3 6 10 12 14 16 16 2.10 ), ( 2.9 ersion of the simple ] for a review). This ompletely decoupled ng experiments 1 nflationary scale. This o observable signatures in le physics experiments can ns only light fields. In this . In this model, the inflaton ant at high energies in the ld coupled to the SM Higgs event any direct (laboratory) the case where the Higgs scalar see e.g. [ ng out the inflaton sector one itions generally permit only tiny strongly suppressed by both tiny expectation value is proportional s also below 1 TeV, see eqs. ( 10 GeV, where the lower limit actually comes – 1 – . χ m . 270 MeV 1 MSM extension ν The common assumption about the inflaton sector is that it is c The situation is quite different if the inflaton sector contai We will concentrate here on the model, which is a particular v In fact, there is another window for the inflaton mass, which i 1 is found to be light, coupling of the inflaton to any other fields including itself ( In this paper we presentdirectly an probe example of the how inflatonvery (low Early sector energy) Universe). partic (whose dynamics is import from the Standard Modelassumption (SM) appears at quite natural, energies since much the lower slow-roll than cond the i 1 Introduction 6 Limits from direct searches and7 predictions for forthcomi Conclusions A The 3 Inflaton decay palette 4 Inflaton from decays 5 Inflaton production in particle collisions Contents 1 Introduction 2 The model boson via a renormalisable operator.potential We confine is ourselves to scale-free at theto tree that level, of so the that inflaton its fieldχ vacuum and mixes with inflaton case even weak inflatonlaboratory coupling experiments. to the SM can lead t chaotic inflation with quartic potential and with inflaton fie implies within the perturbativeobtains approach at low that energies some by non-renormalisablecouplings integrati operators and high inflationaryinvestigation scale. of the These inflationary assumptions mechanism. pr JHEP05(2010)010 ] - A K 10 (see 8 GeV χ . + = 1 π flaton sector χ → m K ntensity and high- everal millions per ion, and in ref. [ utrinos, so that the he inflationary scales we study the inflaton riments. ]. 5 2K, NuMi, CNGS and uirement of a sufficient on mass from the results 15 ght SM Higgs boson was – o the inflaton, and give the ght , (very) weakly elopment for our case of a we describe the model and 11 ull realistic model of particle 2 [ the SM (for particular exam- quite heavy SM Higgs boson, is devoted to inflaton decays, ection ation of our model within the n asymmetry of the Universe scribed model (see the bounds hes of inflaton in decays of eriment with beam parameters e done since that time, and we t searches for and other ]. The interesting branchings . There we obtain lower limits 3 ly realistic extensions of the SM. o masses. This model is capable 5 4 for the decay – 2 9 − . There we also give predictions for 6 contains conclusions. In appendix 5 GeV). Models with larger masses are 7 ∼ and 10 χ χ m – 2 – + K + nothing and from the search for -like particles → π 190 GeV, and the inflaton mass is below B → < h K m for the decay -collisions and give production rates for operating high-i 500 MeV) to thousands (at 6 pp − ∼ )). Hence we conclude that in the model considered here the in ]). Phenomenology of these particles attracted some attent 9 χ ], the extension of the SM with three sterile right-handed ne – 2.8 m ). The production rate in the beam-target experiments is of s 6 16 4 - and searches for inflaton decays in beam-target expe 350–700 GeV. For the described model being consistent up to t B We present the estimates of meson decay branching ratios int The rest of the paper is organised as follows. In section In this respect it is worth mentioning, that hypothetical li ∼ h MSM [ section and in CHARM experiment, andreheating the after upper inflation. limit The most is promising related here to are the searc req estimate for the inflaton productionof rate in the beam-target exp T2K, NuMi, CNGS and NuTeV experiments [ hard to explore. However, thesem larger masses correspond to ν we discuss consequences of obtained results for a implement the Higgs mass should be from the searches for decays inflaton vacuum expectation value providesof sterile neutrin explaining oscillations,and, equipped dark with inflaton sector, andphysics. provides Thus baryo an the example inflation of can a be directly f tested also in ful production in the number of inflatonsNuTeV produced beams per in one a year beam-target of setup. operation of Section T outline the viable regionwhile in meson the decays parameter to space,on inflaton the section are inflaton considered mass from in existing section experimental results. In s very light bosons. These experimentsobtained are in relevant this for the paper).light de flavour At blind the same scalarstill time dates allowed experimentally. the back Quite theoretical to amostly dev few the used improvements in time, wer the when current very work the li results obtained in refs. coupled to the SM fields,ples are see present [ in various extensionsthe of list of relevant experiments is given in the section abou can be fully explored in the experiments. start from 10 year (at energy beams at JPARC, Fermilab andof CERN. Limits the on CHARM the inflat experiment are obtained in section (see eq. ( JHEP05(2010)010 ] 17 (2.6) (2.1) (2.3) (2.4) (2.5) (2.2) ]. The 16 ntial along , 2 , where  0 atter gets modified 2 β , X ≈ ] due to different values of λ α β introduced in [ χ 18 β 2 − m √ is the Higgs doublet. Note, level is the mass term with H onary and reheating stages. ) by the angle = † , H ctly the inflationary models in v below is simpler as compared to H minated stage, so the number of ) is α ] and in [ β , since the number of parameters 2  v, X α λ 16 2 λ 2.1 r . This particular choice is sufficient − − r . h X H 4 . χ . β v m 2 = 13 2 X 2 m i − √ = 4 √ β X X 10 χ h λ α − = × 2 ∼ = α 5 λ . – 3 – X 2 H 2 X = 246 GeV † r = 1 m λ v , m H X 0 = . 2 1 2 β h , m θ √ + , α 62. The normalisation of the matter power spectrum [ for the extra scalar the quartic as /m 2 βλ = XN X 2 χ 2 ≃ v µ 2 X h L √ r m e m m is a new neutral scalar field and + N X∂ = = − µ i X and v ∂ SM 2 1 H L θ ), h = = 2.1 with ) differs from the other values, presented in [ SM XN 2.6 X L L is the SM Lagrangian without the Higgs potential, while the l SM L The SM-like vacuum of the scalar potential in eq. ( The inflation in the model is supposed to be driven by a flat pote The estimate ( used there. 2 e the negative squared mass that it is supposed that the only scale violating term at tree in accordance with ( where and are rotated with respect to the gauge basis ( We consider the extension of the SM model with an inflaton field 2 The model to demonstrate the mainparticle statement: physics experiments. possibility to At the test samethose dire time, in the the algebra case ofis general smaller. inflaton-Higgs scalar potential generated during inflation fixes N to the leading order in Lagrangian of this extended model reads the direction The quartic couplingThus, dominates after inflation the the Universe potential expands asrequired during at e-foldings radiation inflati is do about The small excitations about this vacuum have the masses JHEP05(2010)010 , 3 0 g β − Pl 2 cles ] are 10 M (2.7) (2.9) (2.8) (2.10) . 17 & ]. This ∼ minimal β χ 21 ξ . ≤ , h 0 h β m 150 GeV yields 2 m 2 & 2, with > < r ] between predicted χ he framework of the R/ T χ 2 m 17 m e latter are discussed in ξX of scalar potential during eating temperature of the ameters determine the SM , is fixed by the amplitudes on [ elf-coupling is quite strong , on asymmetry within rele- y Universe. Strong mixing β n them are discussed below. n number is violated at mi- and for definiteness we choose s model happens at . We will suppose below, that GeV for es are rapid enough [ ), where both quartic coupling data. Though the limits [ ) is fixed at the electroweak vac- GeV for we account for the uncertainty in . 3 ), one, / 2.1 2 2.2 1 / 1  2.1 ,  13 0 − 13 β − · ), and lower bound 10 228 GeV β 10 ] with all current cosmological observations. × β 2.3 & × 5 20 . – 4 – h 5 1 , . m 1 = (1–2)  2 19 · ]:  β , 3 · > ]). In the model ( / 18 4 17 χ on the inflaton mass is  22  m ) at high energies with account of perturbative quantum as well. Thus, the Plank scale physics allows for considerin h h Pl 2.1 m m ]. The stronger is the mixing, the more efficient is the energy M 150 GeV 18 150 GeV .  lower bound  · χ · 114 GeV), the energy transfer from the inflaton to the SM parti 5 . 1 & . For our study of inflaton phenomenology at low energies non- 460 3 results in a larger value of the quartic coupling constant, . h − . ξ χ on the inflaton mass [ , m 10 β χ m m . ) implies lighter inflaton, see eq. ( ξ 1 for α . 150 GeV (for a review see [ 0 ≤ Thus, among four parameters in the Lagrangian ( The asymmetry of the Universe is unexplained within t & & r λ coupling to is irrelevant,the but in value of further estimates associated with its possible impact. upper bounds of primordial perturbations and another combination ( In this respect it is worth noting, first, that inflation in thi In the latter case the and and Higgs-to-inflaton mixing are( very weak, but Higgs boson s transfer and the higher(larger is the reheat temperature in the earl of particlecroscopic physics level (SM). in However, the primordialphenomena baryo plasma, is if often sphaleron exploitedvant process extensions by of mechanisms the generating SM.Universe bary This at places the a level lower somewhatT bound above the on electroweak the scale, reh uum by the value of theHiggs Fermi boson constant. mass Two and remaining the free inflaton par mass. Further constraints o due course. Second,tensor-to-scalar amplitudes for ratio the and fits quarticnot to dramatic inflation cosmological yet, there weak is non-minimal a coupling tensi toSwitching on Ricci scalar corrections only due to the gauge and Yukawa couplings. Thes is extremely inefficient [ makes the model fully consistent [ where gravity-induced corrections aresame expected (yet to unknown) be mechanism, large inflation, which operates guarantees at the flatness for 0 the same scalar potential ( JHEP05(2010)010 µ X the P (2.12) (2.11) (2.13) . With M ons may β d Yukawa ∼ ). Note, that X ground field onary potential . 2.13 turns to be large, roduction requires MeV is should be done, once α/λX 1 2 tions to the inflationary ) is quite promising, since p  s of the Higgs boson mass oupling constant r experimental constraints, tential. wever, in the larger part of 7 ounds are expected. Limits, t that quantum corrections, ic coupling driving inflation. g constant at the electroweak − 2.13 y inflaton, in the range given ring inflation are proportional α rgy collisions. In contrast, the 10 s, so ) only by ratios of the coupling ,  ) 2 1 ) and ( 2.12 X  2 gn for fermionic particles. ( ) is responsible for very tiny inflaton µ 2 cle, while the numeric coefficient changes with 2.8 13 − , m 2.12 7 10 − β log × ) 10 5 ) to obtain the Higgs field background, and 2 . – 5 – X . π 1 ( ) into the lower bound on the inflaton mass 4 2.5 α 64 m are, somewhat arbitrary, smaller than 10%, we get on the inflaton mass follows from the upper limit   2.3 β = , etc. This changes the lower bound ( λ h δV √ m g/ , 150 GeV Then, requiring that in the inflationary region λ  √ 3 lower bound / t 120 y . However, in some regions of parameter space these correcti )), all these bounds differ from ( > 6 χ 2.5 m ), follow from the requirement of smallness of the SM gauge an ) fixed at the electroweak scale the corrections to the inflati ), hopeless in foreseeable future. Indeed, inflaton direct p (see ( ) is the mass of the contributing particle in the inflaton back 2.1 2.12 ) can be converted using ( 2.10 X ), or even spoil the inflationary picture. ( 2.12 ), ( m 2.6 4 can be explicitly calculated and have the form α/λX / The weakness of Higgs-to-inflaton mixing ( The proper renormalisation group enhancement of the analys 2.9 Strictly speaking this is the contribution of a scalar parti 4 p 3 is the electroweak scale. (taking into account the flat direction ( corrections to the quartic coupling the number of degrees of freedom, and also changes the si from the contribution of the Higgs boson Limit ( which precludes large quantum corrections to inflaton quart hence so do the SM perturbative corrections to the inflaton po potential, which may change thescale value of ( the inflaton couplin Below this limit one should take into account quantum correc where In the former case the by ( to collect and studyopposite enormously case large of statistics light inflaton in in high the range ene between ( lead to much strongerthe effects Higgs — self-coupling for becomes example, small for at inflationary special scale range interaction with SM particles. This makes searches for heav similar to ( coupling corrections. As far asto all the SM particleconstants masses of du the form the exact value ofobtained this in bound section is not crucial due to the stronge any experimental evidence forthe the parameter light space inflaton no is significant changes found. to the Ho described b βX on the Higgs-inflaton mixing,originated appearing from from this the mixing, requiremen should not dominate over bare c the action ( JHEP05(2010)010 n ] in (3.2) (3.1) 11 (2.15) (2.14) , ]. Nev- f 10 GeV. As far as mions ]), or , 15 ]) 23  10 285 GeV [ − 14 lows from the Higgs π , ). Thus, its branching ∼ < + ss, its value determines tes of the light inflaton. 13 π 2.4 h )) with those of the light (  + Higgs boson mass may be equirement is the same, as m lision point. Then inflaton − cf. [ tistics is achievable. In this aton. 0 ) as it is, it should be valid 2.4 α/λX π ( π . n the SM Higgs boson mass is µ 0 p ¯ 2.1 ∂ 0 ff , π . π + χ ) to describe the inflation at high 1 2 0 π f χ  µ 2.1 · m m ∂ , π 190 GeV (see, e.g. [ . 2 π − β + . 2 π m 0 10 GeV ) to make our study applicable in a more + · h π p µ χ . m , π χ ∂ = 8 GeV 2.14 χ m 0 . − ). Considering for the Higgs boson mass the · 1 π µ m ¯ – 6 – ff µ β + ∂ . χ . 2 2.13 2 1 f χ , µ v p  m m − ), ( · e θ χ + + 1) 2.8 χ = m κ 30 MeV ]. In what follows we actually update the results of [ · ¯ ff (3 β 11 γγ, e χ 2 − L p 700 GeV as still possible, in what follows we study the inflato κ < = 2 h χππ < m L ) gives rise to the Yukawa-type inflaton coupling to the SM fer is the mass. Effective inflaton- interaction fol f 2.4 m In next sections we will estimate the decay and production ra Inflaton of the mass below 900 MeV decays mostly into can be produced inpaper beam-target we experiments, consider where low energy large phenomenology sta of this lightSince infl inflaton couplings do notonly depend viable on inflaton the mass SM range Higgs boson ( ma inflaton production does not require very high energy at a col where SM Higgs boson, studiedview in of [ further relevant developments and findings. general case. 3 Inflaton decay palette Light inflaton decays due to theratios mixing with coincide the (taking SM Higgs into boson account the small mixing angle ertheless, in extensions ofhigher, so this we model will discuss the the upper whole interval limit ( on the energies. Indeed, if(does one not considers become the strongly inflationary coupled) model up ( to the energy scale range 114 GeV Mixing ( the inflaton is veryfor weakly the coupled Standard with Model, the leading Higgs to field, the this bound r boson coupling to the trace of the energy-momentum tensor, ( low energy phenomenology for its mass in the interval Also, the fit to the electroweak data points at mass interval lower than 700 GeV, as we use the same scalar sector ( Actually, in the model under consideration the upper limit o JHEP05(2010)010 (3.8) (3.3) (3.7) (3.6) (3.4) (3.5) ]) 12 -boson and W responsible for , 2 π 2 χ m ecome more compli- m 4 − 1 ecay if the inflaton mass , . ion of each single fermion). e inflaton decays to e Lagrangian (cf. [ 1 1 s , 2 s come from fermion loops, so f , 2  n mixing results in the inflaton / F , . y > y < 2 π 2 χ 3  mion contributions almost cancel , 2 f 2 χ q m m ! µν  x π for for 2 2 f ) 8 F 2 χ 9 11 x , , y β m m , ) m µν colors 4 2 = 9 is the first coefficient in the QCD y + − f, 9 2  b − 2 9  χ F − π α P = y y 1 4  β χ · 

2 + h ]. Thus, in our numerical estimates we follow b − − 1 + (2 m 3 l χ 2 χ N √ 1 1  | 15 W – 7 – π 1 + (1 m y m 8 √ √ F F is a sum of loop contributions from  π | l βm = 2 4 y F α − = π F 2 = 4 = κ 1 1 + 1 βm ≈ − F 0 γγ − , π 1 = 0 = 2 + 3 = f log → y π ) is not correct far from the pion threshold, where strong χγγ χ ll ¯ f i eq √ W → Γ L F → χ 3.7 + F χ Γ π being the mass of the contributing particle; here also  = 2Γ 1 2 m ). − π = = Arctan 3.7 + x x π → , with χ 2 χ Γ -boson, which contributes to the triangle one-loop diagram /m W 2 with electric charge = 3 is the number of heavy flavours, m f being the pion mass and is the fine structure constant, h π ] to improve ( N α -boson contribution (the latter dominates over a contribut = 4 m 15 y The inflaton-to-SM fields couplings presented above yield th If inflaton is heavier than 900 MeV, its hadronic decay modes b W In fact, for the interesting range ofthe the inflaton mass the fer with the rates: with the inflaton decay to athat pair inflaton of decay . to Similar photons contribution can be described by the effectiv Note, the tree-level estimate ( final state interactions become importantref. [ [ cated. First, each flavour contributes to the inflaton d where and the inflaton decay to photons with the rate: beta function without heavycoupling . to Finally, Higgs-inflato the inflaton decays to with the rates: where and JHEP05(2010)010 ) ble 3.6 (3.9) -meson Strange ), which K 4 3.1 is the ), with sum only K e formula as ( 3.5 ctively proceeds via m hadronic scalar reso- . ]. We do not consider ve QCD scale the de- χ -meson mass. Lightest sons of heavier flavours 24 dronise later on. to coupling ( m D ve a quark threshold in- is well-justified, while in ton decays are described onding flavour. , see ( CD corrections (having in f .5 GeV can be obtained by sonances enter the game or f magnitude. Thus we con- ective Lagrangian describing F , where levant hadronic contributions ilar to what was expected for er limits of this “untractable” s” results. t here, and they coincide with rrections should be accounted , approach becomes unjustified s is applicable. Between these f s, where produced mesons are e-loop triangle diagram similar K is the ecay rates of leptonic, photonic P m ng off-shell. , for our estimates we adopt only d. D a µν 2 2 threshold [ = m G b ¯ > b F a µν χ m χ G ] to estimate the decay branching rate to β , where χ 2 15 D m √ m – 8 – 2 π s 8 > ). Farther from the thresholds final state interac- √ F α χ 4 3.2 discussed in section m ≈ β systems), then interference with such a resonance changes b ¯ χgg b L 5 GeV, both approximations are not quite correct and no relia . and ¯ c c 5–2 . 1 ≃ χ ] m 24 is the strong gauge coupling constant and s α Obviously, well above the heavy quark threshold and well abo Third, for the inflaton mass above 1 GeV the decay into mesons a Second, if the inflaton mass is close to the mass of some narrow Hence, for the inflaton decay rates to quarks we obtain the sam We do not discuss decays below thresholds with one hadron bei 4 mass. Charm quarks contribute, if exceeds the double mass of the lightest hadron of the corresp multiplied by the number ofmind colours, uncertainties 3, in and the by value a of factor due to Q those in the case of the SM Higgs boson.scription The of produced the inflatonthe ha opposite decays case in the terms effectiveranges, description where of in quarks terms and of meson gluons this decay is [ over quarks. Higher order QCD corrections are also importan here this case, though the calculations are straightforwar the inflaton--gluon coupling, which due toto the that quark contributing on to the inflaton decays to photons. The eff subsequently hadronise. To estimate thefor, decay which rate, are QCD-co the same as those for the SM Higgs boson of mass pions and . Thiswhile approach multimeson is final valid states untilin are other models negligible. flavour with re For inflaton pionsthe mass this approach above fails approximately not 1 GeV.not far from relativistic For me the but corresponding manyflaton threshold new decay can enter be the described game. as decay Far into abo a pair of quarks due quark starts to contribute for the inflaton mass tions become important and we follow ref. [ nance (e.g., oniums in where some interpolation between these “low-mass” and “high-mas description can be presented. Atdescribed the within same these time, two comparing approaches re validinterval, at we lower and observe upp deviationsclude, not that larger order-of-magnitude than estimates byand of an total lifetime order and hadronic o d modes for the inflaton mass interval 1.5–2 the SM Higgs boson if its mass would be close to the significantly inflaton hadronic decay rates in a way quite sim flavour states are mesonsby effective and interactions close similar to to the ( thresholds the infla JHEP05(2010)010 inflaton functions ]. − 25 10 10 − µ τ + + cc gg ss µ τ ) the lifetime can 2.7 as functions of the Lower panel: 1 , while for other values the on predominantly decays gure 0 ters reaches its maximum hold, its life-time is about β ecays into pions due to the ns decrease with increasing ominate the inflaton decay. s. For the inflaton with mass = ns and pions with comparable 9 r QCD-corrections [ mentum. Diphoton mode varies β , − χ π 4 β m 1 1 2 KK  s π ]). For decay rates into gluons we obtain α , GeV , GeV 4 χ χ − . Here we set  25 µ m m χ + 2 within the favourable interval ( | µ ππ – 9 – m β F | = gg . With → β χ Γ γγ - 0.1 0.1 e Inflaton branching ratios to various two-body final states as + e , for 1.5-2.5 GeV range see discussion in the main text; χ m 1

0.1

Upper panel:

0.01

Branching as a function of the inflaton mass

0.001 χ 1e-07 1e-08 1e-09 1e-10 1e-11 1e-05 1e-06 τ , s , . 0.0001 χ τ s and above this threshold it falls rapidly down to 10 The inflaton branching ratios and lifetime are presented in fi 5 − well above 1 GeV, heaviest kinematically allowed fermions d Figure 1 of about 2.5% justinto at , if threshold. its mass Thus, isbranching the below ratios. 200 sub-GeV MeV, inflat If otherwise to the10 muo inflaton mass is below the muon thres multiplied by the corresponding factor due to leading-orde inflaton mass. Note,inflaton that mass, inflaton while partial thecontribution widths of opposite kinetic into term behaviour fermio from is thewith trace observed of the for the energy-mo inflaton d mass and for the interesting range of parame leading order QCD corrections presented in [ of inflaton mass be two times smaller. lifetime life-time is inversely proportional to JHEP05(2010)010 s of is of (4.4) (4.5) (4.1) (4.3) (4.2) 2 θ , irtual quark ) , , is 2 2 V   2 i m ∗ id χ χ s. These are exactly is much smaller. For V that the models with m m 1 GeV, with branchings S c,t 2 boson was considered a | = K & X i 100 MeV 100 MeV red mixing angle χ K cays of mesons and obtain χ p 2   | M 2 · · 2 m √ π , ! 2 F   | 2 or π ) 0 0 16 is G β β β β χ 3 V M 2 i 2 ] one obtains for the amplitudes of + K 1 2   2 2 branching ratios of the kaon decays K − π · · θ θ m , 14 M 0 ∗ 2 χ id     β → ) + | | | | πM V m κ χ χ χ χ K K K K = + 16 c,t p p p p − | − | | | = , β K M M M M X i 2 2 2 2 1 (  (1    

A 2 | 120 MeV χ Im · · · · . The largest contribution comes from the third γ – 10 – κ ) + 2 3 9 8 3 ]. It follows from figure 2 . K + π γ 2 − − − − − v − χ 2 27 K 1 10 10 10 10 iM ( → m 1 , 1 2 γ × × × × + 2 26 √ ( 5 3 0 3 total π K . . . . F Γ 2 5 2 1 1 K 16 v G A M 3 ≈ ≈ ≈ = ≈ − θ θ   χ χ together with the relevant existing limits from the searche = ≃ ≃ ). For a model with 0 + 2    π π 0 + nothing [ χ χ 2.4 and much smaller 0 ( + → π → π χπ 7 7 π L − + − → → → K → K 10 S L –10 + K × 5 K K Br K − Br 1 ). At quark level this term is due to the inflaton emission by a v . . 3 is the inflaton 3-momentum. The branching ratio of 4 A A A 4.1 − χ ∼ p 1 γ First, light inflaton can be produced in two-body meson decay the order 10 the inflaton mass in the kinematically allowed range the squa where Decays into photons and electrons are strongly suppressed f the processes light inflatons, in the quark-W-boson loop. For the branching ratios we get the kaon decays with term in ( are presented in figure corresponding limits on the light inflaton mass. the processes widely discussed(sub)GeV in particle. the Making past, use of when the the results SM from Higgs [ below 10 4 Inflaton from hadron decays In this section we consider the inflaton production in rare de JHEP05(2010)010 ) ], . 31 2.6 χ (4.6) , m ),( 30 , . 2.4 2 ])  ! 28 2 χ 2 D , , decays of the χ m ] and ( m 14 ered. The most m

33 , ) which dominates). f 100 MeV )  32 4.1 and experimental bounds · µν  ers of magnitude smaller  0.35 n ( χ → are strongly suppressed as 0 . 0 β β e light inflaton. rk mass and the correspond- ) s region 150–250 MeV. Thus, π D x  these searches by one order of πχ 0.3 · → they are still quite small [ Br( L  log( → · | K 2 2 . From ref. [ χ η x θ 0.25 K p E949 E787 | 9 7 M πχ + + 12 2 · K K , expected expected ), Br  − 2 → L + 0.2 ) · χ ) ! K K 2 , GeV + 2 D K 2 χ χ 12 2 D x π − m m m /m − → 0.15 2 µ 4 D 10 – 11 –

+ m m f × )(1 2 K 2 F 8 − θ 0.1 . x G 9 1 2 (1 − + 2 µ ≈ ]. √ x decay rate is a bit more complicated because of the larger 10 2 m 8 26 0.05 2 θ × · − π 7 µνχ 6 . 96 5 − ] and [ → ≈ 10

27

·

) = ) = (1

D 1e-09 1e-10 1e-11 1e-06 1e-07 1e-08 πχ -meson with the branching ratio of the order (cf. [ Br(K-> ) | x η η ( χ f p eνχ M | ) from [ → ∼ χ ], but the rate is suppressed equally strong. On the contrary  D + 0 by negative results of these searches and models with 170 MeV π 30 χπ Br ( → . Expected branching ratios Br ( + → ]). K η 29 Two-body decays of charmed mesons similar to In case of a larger inflaton mass heavy mesons have to be consid excluded Br while the branching ratios(cf. [ of vector mesons are about two ord Figure 2 on Br ( beauty mesons are enhanced as compared to Three-body semileptonic decays have larger rates, however are magnitude would allow to explore thekaon light decays inflaton are in the the most mas promising processes to search forpromising th here is the 205 MeV are disfavoured. Further increase in sensitivity of compared to that decay becauseing of the CKM smallness matrix of elements the in up-qua the amplitude (cf. the third term i Similar formula for the muon mass [ JHEP05(2010)010 are (4.7) (5.1) s,c , section χ  ) t, as they s χX ation with the chance to probe → , pseudoscalar and aryon decays. We 2 B e as for the Higgs  eshold, the dominant 2 Br( θ . Note, that collected . c old escapes the detector 2 χ 1 uppressed by the mixing χ ! m sses similar to the case of clear mode being the muon 2 b 2 χ n particle collisions. If the m of inflaton production is nt for the inflaton production )) + 300 MeV m 40 mbarn is the total m he collision. Then the produc- χ le proton beams. ng fractions of relevant decays 0 − ≃ π   1 + nothing 0 ∗ β

β → total 4 K  L  pp, 2 K ], which is comparable to the expected t σ → W ! m 2 b 2 χ 34 M [ m m  7 25 Br( . − 2 − | 2 – 12 – ∗ ,B | 1 tb we present the estimate of this indirect inflaton V cb

3 ) + 0 V · ts ]. The inflaton production is calculated exactly as | χ V 6 and relative parts going into different flavours | + 38 − 3 π . pp 0 10 M → + nothing ≃ ≃ + ) s K K 300 MeV. → χX ∼ B χ 5 Br( 01–1, depending on the mass, see figure → . . m can be estimated as the product of the meson production cross (0 B s σ χ for several existing beams, and . For several available proton beams the result of the calcul ]. Here only strange and beauty mesons were taken into accoun with accuracy of about 10 Br ( 2 1 θ pp 10 − l ) giving the signatures M + 1 l ). Thus, an appropriate reanalysis of these data might give a stands for strange meson channel mostly saturated by a sum of ) = ∗ ( s 4.7 X K total σ In the models with the inflaton mass above the thr Inflaton can also be produced in other meson decays and heavy b → pp, σ we obtain for the light inflaton: source of the inflatonsthe is SM the Higgs directboson, boson. production i.e. in the The hard gluon-gluonthe main production fusion proce channel of [ for theangle the Higgs squared boson inflaton of is the the same sam mass, only it is s give the main contribution.production in In a figure beam-target experiment for several availab signal ( the inflaton of mass world statistics at b-factoriesB allowed to measure branchi where and the branching of the meson decay into the inflaton, Heavier inflaton can decay withinone the at detector, with the the level most of 0 where total hadron multiplicity tion cross section 5 Inflaton production inIn particle this collisions section wecollision discuss energy in is detailkinematically large allowed the enough, decays inflaton of the heavy production mesons, most produced i efficient in t mechanis do not discuss these channelsand considering less them as promising subdomina in direct searches for the light inflaton. vector kaons. The inflaton(see with figure the mass below the muon thresh cross section [ given in table JHEP05(2010)010 pp cross 8 7 ] − − -meson 10 12 37 pp η [ − − 10 10 b uon fusion · · 10 10 χ 3 2 3 − 4 3 5 ] − − − 10 37 [ · 10 10 10 c dominates below the · · · χ 45 collisions to the total 1 1 1 . 0 5 10 pp ers of several experiments. ] r small inflaton mass down 9 36 , 4 7 7 7 7 . Here we present the estimate / / / / . The lines are for several existing 4 8 35 1 1 1 1 [ 13 s − . χ =50 GeV) e, compared to the meson decay channel 7 10 0 =800 GeV) =400 GeV) =120 GeV) 3 collisions normalised to the total β lab × lab lab lab ] = 6 5 pp , GeV , GeV . 10 χ χ β [ m m 7 gg fusion, T2K 2 T2K (E 13 11 15 the account of the contribution from the = 1 5 pp NuMi (E β K CNGS (E Meson decay, T2K M – 13 – Meson decay, NuMi m 4 Meson decay, CNGS Meson decay, NuTeV 1 ] is given in figure ∼ 19 χ 39 3 m , 10 gg fusion:NuTeV (E 1 5 4.5 100 2 0 . POT 3 N

1e-14 1e-16 1e-18 1e-20 1e-22 1e-24 1e-26 1e-28 1e-20 1e-22 1e-24 1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 1e-18

pp,total

pp,total ). For the T2K case the estimate of the direct production by gl σ σ σ σ / / 40 mbarn, for inflationary 1 ∼ 50 800 400 120 , GeV E 40 mbarn. For all graphs ≃ ] pp total ] ] 5 σ ] 4 3 2 ) might somehow smooth the cross section change. 2 GeV, to illustrate the statement that indirect production . The inflaton production cross section in . The ratio of the inflaton direct production cross section in . Adopted values of relevant for inflaton production paramet pp total σ & 4.6 T2K [ NuMi [ χ CNGS [ NuTeV [ Experiment m Table 1 beam energies (see table cross section, section B-meson threshold, cf. figure Figure 4 Figure 3 of the direct contributionto to the inflaton production even fo of production. At the kaon threshold is also given, while for higher beam energies it is negligibl parton distribution functions from [ decay ( JHEP05(2010)010 . ] ) he 17 40 2.7 − (6.2) (6.3) (6.1) 10 , ≃ tes. The = 35 m is )  . 0 π 5 ng experi- σ cγ 0 detector N l detector ( l Γ e then get instead − N/ e bove we get an order M experiment (proper − roduction cross section. 1 ependence of the bound flaton. As far as in the etween the axion in [ m dump, and then decay in the detector is roughly  , ), and we adopt the simple higher values of masses for the CHARM results exclude cγ dec l earch of a penetrating particle  0 5.1 is the production cross section ]. In this article the search for all and a significant amount of sis can not be made by simple = 13 for the CNGS beam (see ] is given in figure plest possible way the resulting Γ β , detector sensitivity to different − X 40 40 e = γ pp σ )) β detector × M l µµ . Γ − we take ( → e χ we conclude, however, that the value of χ − σ 5 3, with 1 /  pp 270 MeV γ – 14 – dec ) + Br( l M & Γ ee χ = − e m ] is approximately reproduced for → X 40 σ χ total 0 280 MeV, while for the upper limit of the interval ( is the typical relativistic gamma factor of the axion, with pp, N & X /σ ≃ 0 χ π ) + Br( ]), Γ is the decay width (sum of eqs. (3) and (4) for and N m σ E/m γγ 40 ] can be reproduced by demanding that the number of decays in t = = 480 m is the decay length before the detector, → γ 40 χ yield: dec l 0 π ], and then change in the analysis the decay and production ra ]), (Br( 40 40 prod σ is the overall coefficient describing luminosity, 0 0 requires careful reanalysis of the CHARM data. N ). Then, the region forbidden by the experiment [ N ) 210 MeV is a conservative bound: within the simple approach a 0 1 in this region is sensitive to the proper analysis of the CHAR 10 GeV. Then, figure 4 from [ ≃ β Using the same logic we can obtain the bound for the inflaton. W One can see, that for the inflationary self coupling ments > 6.1 β ∼ N ∼ χ simulation of energy distribution ofdecay the produced modes, inflatons etc.). From the plot in figure where for the inflaton production cross section 6 Limits from direct searches and predictions for forthcomi of magnitude “safety margin”.β The definite conclusion about One should be somewhaton careful taking this limit. The exact d of ( table m we have estimate of the where electrons in [ bound from [ resulting picture in [ detector is larger thanestimated the as background. The number of decays and the light inflatonHere is we in will the makemost estimate interesting rough of region reanalysis of the for massesinflatons decay the the decay rate inflaton before case and lifetime they is of p rescaling reach sm of the the the detector, light result. this in So, reanaly first we will reproduce in the sim The light inflaton willfurther be into produced a in or large leptonof pair, amounts this depending in on type its any was mass. bea a performed S by generic the axion CHARM was experiment performed. [ However, the only difference b the detector length, the inflaton with masses E of the axion (eq. (5) in [ JHEP05(2010)010 ) , β χ m we esti- 5 the inflatons with 0 β = β hes. The region of ( ined in section 5 or the reference beams. All 0.4 that higher energy beams ng at designed luminosity 0.35 aon decays and one can win slightly 4 cause the dominant production tio by the number of on flationary and possible collimation or deflec- 0.3 es the lower limit on the inflaton mass 3 m. 0.25 , GeV χ , GeV χ m gg fusion, T2K 2 0.2 m 5 GeV it is about a thousand at best. Meson decay, T2K – 15 – Meson decay, NuMi ∼ Meson decay, CNGS Meson decay, NuTeV 0.15 χ 1 For small mass the number of the inflatons per year m 0.1 5 0.28 0 . This number should be taken with a grain of salt, as far as 1 0.05 1 100 0

10000 1e+14 1e+12 1e+10 1e+08 1e+06

β

produced

1e-12 1e-14 1e-16 N 1e-08 1e-10 β and table 6 ). . Number of the inflatons produced with one-year statistics f . Limits on the inflaton properties from the CHARM axion searc 6.3 For the allowed inflaton mass range and using the results obta This is not so for low mass inflatons, which can be produced in k 5 geometric factors are neglected.protect( The vertical line indicat because of usually higher luminosity of the lower energy bea Figure 6 Figure 5 mode is decays ofcan beauty exceed hadrons. several millions, while at for several experiments by multiplyingtarget, the see cross figure section ra we have totally neglected all possibletion geometrical of factors the producedare charged certainly kaons. preferable However one in can the conclude searches for the inflaton, be mate the number of inflatons produced during one year of runni surrounded by the curve is forbidden. One can see, that for in masses up to approximately 280 MeV are ruled out. JHEP05(2010)010 , ] s 16 n and ξ chanism of 8 GeV. Note, . 1 . nted in the frame- χ e possible, while the m if was supported in part ] may change the lower . een suggested in ref. [ the grant of the Russian 40 nflaton life-time is about amusing in fact, the first ional discussions through- model, which can be fully upper limit on the inflaton 00473a and 07-02-00820a), 957.2008.2, NS-1616.2008.2 th minimal (or very weakly e inflationary parameters — ts, V. Rubakov for valuable his model. This proves that ernment contract Π520), by nt [ m field theory can be directly s beyond the Standard Model s — dark matter and baryon lue of the self-coupling constant ght and can be hunted for in the ino produce the dark matter. In (see relations between the current WMAP data. In near ns of the SM. ξ MSM itself do not limit the interval for ν —with much better precision. This will allow MSM with Majorana masses, while decays of r ν – 16 – ]. In that joint model the inflaton vacuum expectation 42 , 41 MSM [ ]. ν 42 , and tensor-to-scalar ratio 41 s n s, so it decays close to the production point. The dominant me MSM extension we check, that the limits presented in ]). In case it is large, the laboratory detection will still b ν A ] the inflaton model considered in this paper has been impleme MSM [ 10 20 ν − – 16 18 –10 Neutrino oscillations is the only direct evidence of physic The paper analysed the quartic inflaton self-interaction wi Note also that the inflaton model discussed in this paper has b We are indebted to M. Shaposhnikov for valuable and inspirat The mass limits on the light inflaton are found to be 270 MeV 9 − and thus for stronger signal. in [ asymmetry of the Universe — which are not explained within SM of particle physics. Cosmology provides two other evidence In this paper weexplored presented in an particle example physics of experiments.beam-target a The experiments simple inflaton inflationary is and li in10 two-body meson decays. The i 7 Conclusions r work of value provides the three of analysis of the current work shouldβ be rescaled for larger va as an extension of the non-minimal) coupling to gravity, which is stillfuture allowed results by from the PLANCKspectral experiment will index determine th bound given here. A The (government contract 02.740.11.0244), bythe FAE program Scopes (gov grant ofScience Swiss Support National Foundation. Science Foundation and by inflation can be directly tested in fully realistic extensio out the work, P.comments. Pakhlov for D.G. discussion thanks onby EPFL recent the for Belle hospitality. Russian resul Foundation Theby for the work grants Basic of of D.G. Research the President (grants of 08-02- the Russian Federation MK-1 inflatons in the early Universeappendix to the lightest sterilethe neutr inflaton mass obtained inmass this constrains paper. the mass On the of contrary, dark the matter sterile neutrino in t stage of the Universe evolutiontested described with by means modern of experimental quantu techniques. inflaton production is the meson decays. Thus, and it is quite In ref. [ to fix the value of the non-minimal coupling constant that careful reanalysis of the results of the CHARM experime JHEP05(2010)010 I f (A.3) (A.1) (A.2) 2 antisym- . × tential implies , , o be viable dark ] coupled to the is 2 ]. GeV 3 uction due to the ǫ  16 43 ) is determined by .  ]  χ c . 3 coupling constants 16 m , follows from success- − ( 3 , χ + h 2 10 f I g and of sterile neutrino m e sterile neutrino is quite , where dressed within a neutrino ). Requiring again smaller f M X × ∗ constrain the inflaton mass. I ial plasma and decay to the 5 100 MeV alar field [ 2.6 . N os, ǫH 1 c  I . They provide contribution of MSM provides both early-time g ( · ¯ χ  cal time-scale implies extremely ν = N 3 2 ) three new singlet right-handed I m type see, e.g. ref. [ / ˜ 2  n, f H 1 ∼ 2.1  − 1 3 T , and small mass implies tiny coupling 13 ˜ − H M α − ) can be stable at cosmological time-scale I 1 10 10 keV F 10 N β ] and the function 2.1  α × × · ¯ L 44 5 . 5 . 1 13 αI – 17 – ], suggested as a minimal version of SM capable 1 ) in a primordial plasma at inflaton decays. It − F < 42 T  10 ( I , is a minimal, full and self-consistent extension of the β   f 4 for inflaton mass from 70 MeV to 500 MeV and for ∗ + . 6 × g 41 I 5 . N χ 1 µ · m ∂ 9 to 0 . ) µ χ γ 100 MeV MSM) [ I ν m  ¯ ( N S · i f to the energy density of the Universe today [ 6 3 with the Lagrangian . = is the Higgs doublet. , 1 1 270 2 N 3) are charged doublets and , H < = , 2 MSM i , ν N = 1 X L Ω are irrelevant for this study). The flatness of the inflaton po I h = 1 I , 1 GeV, and should not get thermalised in primordial plasma, t I f Iα 1 is a dilution factor accounting for a possible entropy prod α F N ( . = α 1 I S > L M M To begin with, let us discuss the allowed ranges of the Yukawa This extension implies addition to the model ( The lightest sterile neutrino in the model ( For a discussion of gauge hierarchy problem in models of this 6 metric matrix and (Yukawas smallness of the quantum corrections to the quartic couplin late decays of the heavier sterile [ where where and get for the sterile neutrino masses minimal Standard Model ( is a correct theory of gravity. These three problems can be ad and comprise the darklight, matter of the Universe. In this case, th than 10% contribution we obtain Standard Model of particle physics. neutrinos of explaining all ofSM the Higgs three. boson via Enlarged tree-level-scale-invariant by interactio the additional sc the number of degrees of freedom inflation and commonmasses. source This of one-energy electroweak scale model symmetry breakin matter. The latter issmall natural, (if since any) stability mixing at with cosmologi active neutrino via A lower limit on the masses of the two heavier sterile neutrin lightest sterile neutrino mostly at the temperature to inflaton. Light inflatons are in equilibrium in the primord changes monotonically from 0 fulness of the Nucleosynthesis (BBN), that does not the sterile neutrino JHEP05(2010)010 ) rsa, A.3 (A.5) (A.4) n elec- sses above . equation ( d ones. As far  be a Warm Dark χ m MSM. At the same . χ . Other mechanisms ν 2 m / nflaton mass obtained on the lightest sterile 3 keV 100 MeV 10 keV. 3)] 3 limit on the inflaton mass;  naturally / / a given & in their ranges to minimise 1 proportional to the squared χ 3 . inflaton mass range. Third, / 1 tes to sterile neutrinos. The n decays to SM fermions due an be somewhat shortened as ). Once the inflaton is found  if kinematically allowed. The 1 m the dark matter sector do not 2 ypical for 2 ) (invisible mode) has the width ( / M s to visible channels. 3  hat inflaton decays to the dark χ 3 , lower ∗ 3 9 2 , . 13 ! m 2 0 /g N ( upper limit − 2 f N 2 χ f 75 ). m 10 . m ) one concludes, that with the inflaton β 4  · × [10 ) fixes a A.1 − 3 A.4 5 . / 1 ≃ 1 1 A.5

)  ] of the dark matter phase space density in  b. χ 4 χ S 3 2 1 I 46 / m 1 – 18 – ( , πm  f βM   8 45 N 25 supersedes ( . = ) one obtains an Ω 0 1 χ I ). This decay mode can even be dominant, if the sterile f  N ) from direct searches, if m A.4 I 3 / 3.6 N 1 6.3 300 MeV →  ), the lightest sterile neutrino can χ  ) the lightest sterile neutrino is significantly lighter tha ) · ) imply that the dark matter is lighter than about 100 keV and ), one does not exceed the lower bound on the sterile neutrino Γ χ 2.13 m 13 2.15 ( A.4 S 2.15 f . ),( 4 ) on the dark matter neutrino mass will be settled. And vice ve 1 . ), ( 6 2.8 M  A.5 A.5 are the heaviest fermions between the kinematically allowe is disfavoured by the BBN, this can be relevant for inflaton ma 7 keV, from the study [ < . 3 1 π , 2 upper limit 1 m & N M 1 . 10 keV 3 M , 2 From the formulas above one concludes that the limits on the i There are several notes in order. First, from ( Note that inflaton can decay into sterile neutrinos M masses of the correspondinglightness fermions of similar the to darkmatter its matter neutrino decay sterile never ra neutrino suppresses guarantees, its t decay branching ratio in this paper are not affected due to additional constraints t tron (and other charged SMto fermions). mixing Since with the the light Higgs inflato boson, these partial decay rates are time, the limits ( this limit can supersede limit ( Matter candidate. Second, withr.h.s. model of parameters the tuned inequality with ( Hence, given the range ( heavier inflaton can be approximated as as neutrinos with account of thethe allowed Higgs upper boson limit mass ( once (see the section dark matter neutrinos are found, eq. ( mass, dwarf spheroidal galaxies. Hence,shrink in this the model allowed limits parametermaximising from region, r.h.s. and of in inequality particular, ( the mass in the range ( implies the This formula is very similar to ( approximately 300 MeV. As a consequence,compared inflaton to lifetime the c results presented in figure neutrino mass, can also contribute to the dark matter production. Hence, at The corresponding upper limit on decay mode into the not-dark-matter sterile neutrinos JHEP05(2010)010 nd nsity ommons ]. , , , , (2009) 330 ]. , . , . SPIRES mesons ]. searches 180 ][ η , SPIRES and MSM inflation (2009) 095002 ν ]. Low-energy theorems for k SPIRES ][ mmercial use, distribution, ]. r(s) and source are credited. ave background D 80 ]. SPIRES ]. ]. (1979) 711 [ akharov, Review of particle physics ][ hep-ph/0002049 nce/BeamPerfor.htm [ ]. SPIRES 30 DesignParameters.pdf Astrophys. J. Suppl. ][ 3 SPIRES , SPIRES SPIRES Phys. Rev. ][ A phenomenological profile of the Higgs ][ , V2 ][ ]. hep-ph/0012033 SPIRES [ Inflaton mass in the [ The decay of a light Higgs boson (2000) 176 Five-year Wilkinson Microwave Anisotropy Probe – 19 – MSM, inflation and dark matter hep-ph/9807278 ]. ]. [ SPIRES ν B 480 ]. Kaon physics with light sgoldstinos and (1985) 465 The Sov. J. Nucl. Phys. Light Higgs particle in decays of Density perturbations in generalized Einstein scenarios a astro-ph/0402185 , [ . SPIRES SPIRES hep-ph/0007325 (2001) 054008 [ [ [ arXiv:0809.1097 hep-ph/0604236 (1999) 1 [ [ Some low energy effects of a light stabilized radion in the Chiral perturbation theory: expansions in the mass of the ]. ]. SPIRES below the weak scale [ (1976) 292 [ B 250 . Particle physics models of inflation and the cosmological de 314 collaboration, C. Amsler et al., Phys. Lett. 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