JHEP03(2021)212 Springer March 23, 2021 : January 13, 2021 February 15, 2021 : September 28, 2020 : : Revised Published Accepted Received a [email protected] , Published for SISSA by https://doi.org/10.1007/JHEP03(2021)212 and José W.F. Valle b [email protected] , . 3 Rahul Srivastava 2009.10116 a The Authors. c Beyond Standard Model, Neutrino Physics

GeV. We show, however, that the “dynamical” inverse seesaw with sponta- We investigate the stability of Higgs potential in inverse seesaw models. We , 10 [email protected] 10 ∼ E-mail: AHEP Group, Institut deCSIC/Universitat Física de Corpuscular, València, Parc CientíficC/ de Catedrático José Paterna, Beltrán, 2Department E-46980 of Paterna Physics, (Valencia), Indian Spain Bhopal Institute Bypass of Road, Science Bhauri, Education Bhopal, and India Research — Bhopal, b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: the Higgs quartic self-couplingscale goes negative wellneous lepton below number the violation can Standard leadup to Model to a the completely instability consistent Planck and scale. stable Higgs vacuum Abstract: derive the full two-loopcoupling, RGEs taking of thresholds into the account. relevant We parameters, find that such for relatively as large the Yukawa couplings quartic Higgs self- mechanism Sanjoy Mandal, Electroweak symmetry breaking in the inverse seesaw JHEP03(2021)212 Y 12 13 16 U(1) ⊗ 15 L SU(2) and Higgs quartic ⊗ t c 15 y 17 SU(3) 11 9 5 ] and the subsequent precise measurements 2 , 22 1 – 1 – 21 4 22 21 ) 21 23 Φ v ( 6 Φ O v = 6  σ 1 ] can be used to shed light on the electroweak symmetry breaking mecha- σ v 20 3 v couplings within the Standard Model. These, along with the B.1 Quartic scalarB.2 couplings Yukawa couplings A.1 Higgs quarticA.2 scalar self coupling Yukawa couplings 6.1 Sequential versus6.2 missing partner seesaw: Sequential versus electroweak vacuum missing stability partner seesaw: brief phenomenological discussion 5.1 Case I: 5.2 Case II: 3.1 Effective theory 3.2 Full theory SM nism. In particular, we canStandard now Model not only scalar determine potential thepossible at value new of the physics the electroweak all quartic scale, the couplingand way but of up Higgs the also to boson use Planck masses, scale. one itλ can Given to calculate the shed present the light measured corresponding top on Yukawa quark 1 Introduction The historical discovery of the Higgsof its boson properties [ [ B RGEs: inverse seesaw with majoron 8 Conclusions A RGEs: inverse seesaw 7 Impact of invisible Higgs decay on the vacuum stability 5 Vacuum stability in inverse seesaw with majoron 6 Comparing sequential and missing partner inverse seesaw 3 Higgs vacuum stability in inverse seesaw 4 The majoron completion of the inverse seesaw Contents 1 Introduction 2 The inverse seesaw mechanism JHEP03(2021)212 , , ]. 3 4 3 g , 2 g , GeV [ 1 4 g . 0 ]. Here we ± 16 – 18 . Here we adopt 9 10 SM λ . ) = 173 t SM ] or spontaneous vio- becomes negative at 1 m λ ( 17 ] for details. µ 0.126115 SM 5 15 λ 10 t y 0.93519 12 ] 10 ]. The Higgs vacuum stability problem ], to which we refer the reader for more details. 3 8 4 g SM GeV , the Higgs quartic coupling tends to run 1.16541 1 [ μ 9 – 2 – 10 gauge group. 2 3 g g Y 0.647737 6 U(1) and the quartic Higgs boson self-coupling t y t ⊗ 10 y are the central values. We use them as the input parameters for our L 1 SM 1 g λ as shown in table respectively, are the most important input parameters character- 1 1 3 0.462607 g GeV. This would imply that the Standard Model Higgs potential SU(2) 3 , g 2 10 2 ) ⊗ g t 10 10 c , g m 1 ( g

∼

µ 0.0 1.0 0.8 0.6 0.4 0.2 Couplings SU(3) ] and confine ourselves to the Standard-Model-based seesaw mechanism using 5 MS values of the input parameters at the top quark mass scale, . The renormalization group evolution of the Standard Model gauge couplings . The latter can be realized in “high-scale” schemes with explicit [ Moreover, despite its many successes, the Standard Model cannot be the final theory One sees that the Standard Model Higgs quartic coupling ]. Instead, these next-to-next-to-leading order analyses of the Standard Model Higgs MS scheme, taking the parameter values at low scale as input, see [ The numbers given in table 1 7 , follow ref. [ the simplest RGEs. The importance of errors has been studied in ref. [ potential suggest that the vacuum is actually metastable. of nature. One oferation, its needed main to shortcomings describe isin neutrino its neutrino oscillations inability mass [ to models account can for become neutrino worse mass than gen- in the Standard Model [ negative between the electroweak and Planck scales, as seenan in energy figure scale is unbounded from below.6 Hence, the Standard Model vacuum is not absolutely stable [ the gauge couplings izing the Standard Modelthese renormalization input group parameters, equations (RGEs). Given the values of Figure 1 the top quark Yukawa coupling Table 1 JHEP03(2021)212 ] ]. 2 21 we 20 7 , that 2 ], can substantially Φ Φ 19 , 18 This vacuum stability problem be- 2 , we describe neutrino mass generation tends to become negative sooner, much 2 ]. Indeed, we find that renormalization λ 11 L L ]. we show that the vacuum stability problem ]. Indeed, if the heavy mediator neutrino lies in – 3 – 5 22 3 3 singlet fermions with small Majorana mass terms, in addi- ]. These typicaly involve messenger masses much larger than Y , we compare the vacuum stability properties of the various 19 6 , U(1) ] included threshold effects, in the high-scale seesaw framework such effects 18 5 ⊗ how the majoron helps stabilize the Higgs vacuum, all the way up L , we then focus on the majoron completion of the inverse seesaw. We 5 4 Φ Φ SU(2) ⊗ c . Destabilizing effect of Weinberg’s effective operator on the Higgs quartic interaction. SU(3) The paper is organized as follows. In section Here we examine the consistency of the electroweak symmetry breaking vacuum within Any theory with massive neutrinos has an intrinsic effect, illustrated in figure In the presence ofNotice very that, specific symmetries while this ref. model-independent [ argument might be circumvented. 2 3 Figure 2 to Planck scale. Inmissing-partner-inverse-seesaw variants section with those of the sequential case. In section appear only at high energies, and do not affect low-scale physics. as the fermion sector of the theory. in the inverse-seesaw model.becomes worse within In the simplest section number. inverse-seesaw In extensions section with explicitlythen broken show lepton in section improve the electroweak vacuumof stability low-scale-majoron-seesaw properties. schemes plays Indeed, aThis the key sharpens extended role in scalar the improving sector group results their (RG) vacuum evolution presented stability. can in curein the ref. the vacuum stability [ presence problem of in threshold inverse seesaw effects. models These also can be associated both with the scalar as well the inverse seesaw mechanism.there will Apart in general from be the other,reverse model-dependent, destabilizing this and effect trend. possibly leading illustrated We contributions note in thatexistence that can of figure the a physical spontaneous Nambu-Goldstone violation boson, of dubbed lepton majoron number, [ implying the may potentially destabilize thecomes electroweak severe vacuum. in low-scale-seesaw schemes [ the TeV scale, its Yukawaseesaw. coupling will As run a for consequence,before much the the longer quartic Standard than coupling Model in instability the sets high-scale in. type-I This happens in theextra inverse seesaw mechanism. Leptontion number to is the conventional broken “right-handed” byor neutrinos. introducing spontaneous Again, lepton one number can violation have [ either explicit [ lation of lepton number [ the electroweak scale. Alternatively, neutrino massFor may result example, from “low-scale” the physics [ type-I seesaw mechanism can be mediated by “low-scale” messengers. JHEP03(2021)212 and (2.2) (2.3) (2.4) M can be made . As a result, S M µ transform under ) as, T ν S Y 2.2 1 ]. In contrast, being − ) 26 and H.c. (2.1) – T is invariant under lepton c + ν 24 , respectively. The M j ( M S S is the Standard Model Higgs ], to obtain the effective light +1 i µ S 1 has a block-diagonal form. Since 19 Φ ∼ ij S − , are expected to be naturally large. µ D S M 1 2    ν M S M D 0 Y + M µ 2 and j . Under this hierarchy assumption we 2 v M T S S D 1 0 c i µ = = m M ν ij , where U . T D T D .  ∼ − 0 0 U ν 8 – 4 – M m m c 1 D ν +    M − . , the symmetry breaking entries of ) m c j 0 T = T ν as U ˜ ν Φ  L M i → ( L M S S ij M are the lepton doublets, µ µ ν 1 U(1) Y 3 − , violates lepton number by two units. Light neutrino masses are ij 2 M X , ]. The relevant part of the Lagrangian is given by as S ] that follow from current LHC experiments. Finally, we conclude D µ = ν m 22 23 = 1 , m ≈ i −L is much smaller than the characteristic mediators scale 21 ; [ ν T S i  . Neutrino masses arise by block-diagonalizing eq. ( µ m S ν Y ν ` 2  v is the Dirac mass term. The two sets of fields and √ c i = = M ν i D L may also result from having a radiative origin associated to new physics such as m Furthermore, in contrast to conventional type-I seesaw, the scale of lepton number vio- S terms are both gauge invariant mass matrices, but only µ S lating parameter of supersymmetry, left-right symmetry orgauge and dark lepton-number matter invariant, the physicsThus elements [ of we obtainperform the the hierarchy standard seesawneutrino diagonalization mass procedure matrix [ through the unitary transformation matrix the lepton number is retorednaturally as small in the sense of t’Hooft. Apart from symmetry protection, the smallness with µ number symmetry, since generated through the tinybreaking, lepton the number effective violation. light neutrino Indeed, mass after matrix electroweak has symmetry the following form where doublet, the lepton number symmetry The issue of vacuumwe stability examine must it be in studiedinverse seesaw the on mechanism is a context realized by model-by-model of addingfermions two basis. inverse-seesaw sets of extensions electroweak In singlet of this “left-handed” the work Standard Model. The boson invisible decays [ and summarize our main results in section 2 The inverse seesaw mechanism briefly illustrate the interplay between vacuum stability and the restrictions on the Higgs JHEP03(2021)212 ]. 4 30 , upon 29 S and c come with the ν ]. It constitutes a S 27 ] two of the light , in addition to the ]. Finally, since the 17 and S 36 c – ]. In short, in contrast ν 32 and 41 – Note that, the small lepton c ν 4 allows the Yukawa couplings 37 lies in the TeV scale, without S µ M , as well as the scalars (in section S control the smallness of light neutrino S and µ c ν – 5 – we examine the quantitative differences between 6 . The smallness of we opt for the minimal (3,1,1) case, namely a single 3 6= 0 S µ , and there are three “heavy” quasi-Dirac leptons in addition to S , the global lepton number symmetry is restored, and as a result, in a given model. In the sequential inverse seesaw model the number 0 . These could be warm dark matter candidate if their mass lies in KeV S ’s are Standard Model gauge singlets, carrying no anomalies, there is S → µ S S and ], with stringent bounds, e.g. from the Delphi and L3 collaborations [ µ c 31 and ν – c ]. ν 28 42 matches that of ) responsible for the spontaneous breaking of lepton number. For the sake of simplicity, here we consider only the case where As In contrast to the high-scale type-I seesaw, in inverse-seesaw schemes one can have c 5 The concept of quasi-Dirac fermions was first suggested for the light neutrinos in [ to be sizeable, even when the messenger mass scale ν 4 ν the stability ofthresholds the associated electroweak with Higgs the extra vacuum.and fermions We take into account the effectcommon of feature of the all low-scale seesaw models. 3 Higgs vacuum stability inIn inverse the seesaw above preliminarythe considerations inverse we seesaw have model. briefly summarized We now the examine main the features effect of of the new fermions vacuum stability problem, in section pair of lepton mediators.neutrinos will In be such left massless. minimalthe In “missing-partner” section different seesaw multiplicity [ choicesbriefly concerning discuss the the issue phenomenological of viability of vacuum the stability. various options. Moroever, we light and heavy neutrinos,proportional the to spectrum willscale also [ contain intermediate states with mass same multiplicity. Moroever, since adding more fermion species will only worsen the Higgs no theoretical limit onnumber their of multiplicity. Manyof possibilities can arisethe depending three on light neutrinos. the For the case of different number of tentially large rates, unsuppressed bymediators the would small not neutrino take part masses intrix [ low-energy describing weak oscillations processes, the would light-neutrinoto be mixing the effectively ma- conventional non-unitary high-scale [ plethora seesaw, of the accessible inverse new seesaw physics mechanism processes, could that harbor could be a just rich around the corner. conflicting with the observed smallness of neutrino masses. a very rich phenomenologyexperiments. that For makes example, them the mediators potentiallyiments would testable be [ accessible in to current high-energyMoreover, or collider exper- they upcoming would induce lepton flavour and leptonic CP violating processes with po- number violating Majorana mass parametersmasses. in As all the three lightprotected” neutrinos by are the strictly tiny massless. valueY of Small neutrino masses are “symmetry- the heavy singlet neutrinos become quasi-Dirac-type fermions. JHEP03(2021)212 . ). Λ are and 2.1 (3.2) (3.3) ] S 5 — the SM λ 3) , and , which we 3 has negative S , c , respectively. ν κ (3 to the running S κ and c and will run. Neglecting ν c ] are given by [ κ ν and the 46 – ]. Hence, below the scale 2) 44 , 46 2 , , to distinguish it from the pure (3 κ 14 , κ given by their mass, see eq. ( κ . In this regime we have the full 5 λ λ Λ H.c. (3.1) ] associated, say, with dark matter. M + κ 26 ≈ 2 2 g 2 Λ 3 2 v effective coupling matrix. Unless they are Φ Φ + κ − 3 discussed above. slowly increases with the threshold scale κ κ ≡ LL × 2 t – 6 – λ κ ν y 3 2 κ m = = 6 . The contribution of the coupling κ Λ =5 β is the d ν 2 ) π T ν −L we will compare with the 16 can be. The Higgs quartic self-coupling in full UV-complete Y 1 6 ν , only the Standard Model couplings and , to distinguish it from the Standard Model coupling − Y ) Λ ], given by λ T 43 above the threshold scale is governed by the RGEs given in ap- M case provides the simplest reference scheme, that brings out all the will be almost the same as in the Standard Model. and thus negligible, as shown in [ ( ν S κ Y 2 below the scale 1) µ λ κ , 1 κ 2 1 − λ v , inverse seesaw. As mentioned, this of course is not — by itself — realistic, (3 M ν 1) Y , 1 , . Apart from the RG evolution, one must also take into account the threshold = ( (3 A κ is of order The running of For simplicity we will first study the case of just one species of κ λ , the evolution of Moreover, the relevant features. Insequential section inverse seesaw mechanim — with two and three species of pendix. theory will be denotedfrom by the effective theory quartic coupling call the as in this caseparameter only may one arise of from the a light different neutrinos mechanism obtains [ mass. However, the missing mass We now turn toUltra-Violet the (UV) region complete above theory. the Hencenew one threshold couplings must scale present take in intocoupling. account the the model, In RGEs particular, as of we they all willlarge the will see the affect that Yukawa coupling the the stability evolution of of the the electroweak vacuum Higgs limits quartic how quartic coupling of Λ 3.2 Full theory Due to theWe large denote top the Yukawa HiggsStandard coupling, quartic Model coupling case. in The this above case Weinberg as operator also gives a correction to the Higgs As a result, below the scale lepton and light quark Yukawa couplings, the one-loop RGEs [ where needed, in what follows wemass will dimension. suppress the The generationmatrix above indices. as Lagrangian Note leads that to a left-handed neutrino Majorana mass To begin with,integrated in out the we have effective aAs natural theory as threshold a where scale result, below the thisfive scale heavy Weinberg the operator theory singlet [ is the fermions Standard Model plus an effective dimension 3.1 Effective theory JHEP03(2021)212 λ λ 18 GeV 10 (3.5) (3.4) GeV, SM 3 5 λ 10 15 = 10 10 0.6 M ] in estimating 5 (Λ)= 12 λ ν ] GeV and 10 Λ = 3 GeV is the quartic coupling GeV. However, within [ μ 9 κ Y TeV, 100 λ 10 10 = M Λ = 10 almost coincides with that 10 Λ= κ ∼ λ 6 ν Y (solid-red) and Yukawa coupling 10 is dominated by the top quark . 2 4 λ | ) can dominate the evolution of ν Φ) becomes negative at around energy . It should also be noted that in the SM κ † Y 3 λ | . Here we follow ref. [ λ 2.1 . λ 4 2 10 (Φ 3 M π λ 5 0.8 0.6 0.4 0.2 0.0 0.2

32

- . We take into consideration this shift in Couplings 4 | Λ = − in eq. ( ν Φ) + † Y ν | at – 7 – 18 Y 2 (Λ) (Φ λ π 10 5 λ 2 Φ SM 32 λ µ → − − GeV for the threshold scale 15 at the threshold scale, taken to be 8 = = 10 6 (Λ) , as seen in figure . 10 λ V TH 0.6 M × λ = 0 12 λ (Λ)= ∆ ] 16 ν ν . 10 Λ = Y 3 GeV [ μ 9 Y TeV, 1 = 10 GeV (right panel). We see that M 5 Λ= GeV and 10 6 7 (red dashed), one sees how the Higgs vacuum stability problem becomes we have shown the RG evolution of the relevant coupling parameters assum- 10 10 3 on the stability of the Higgs vacuum. As already discussed, in the Standard SM × when solving the RGEs, ν ν λ . Evolution of the Higgs quartic self-coupling Y Y 3 27 M . κ 10 λ 3 In figure Having set up our basic scheme, let us start by looking at the impact of the Yukawa

0.0 0.2 0.8 0.6 0.4 0.2 , the SM quartic coupling, indicated by the dashed-red line.

(dotted-green) within the minimal (3,1,1) inverse seesaw scheme. - Λ = Couplings SM ν respectively. By comparingcoupling this with themore running acute of in the the inverse Standard seesawdestabilize model. the Model Higgs This Higgs vacuum, was as expected, quartic effective illustrated since theory in the regime new figure the fermions evolution tend of to the quartic coupling the inverse seesaw, theabove Yukawa the coupling threshold scale ing the Yukawa coupling (left panel) and scales coupling Model, the runningYukawa of coupling and the becomes Higgs negative around quartic energy coupling scale correction to the Higgs quarticthis coupling threshold correction as at tree-level Higgs potential is given by This will get correctionsas from from higher the extra loop fermions diagrams present of in Standard the Model inverse seesaw particles model. as It well introduces a threshold Y in the effective theoryλ with the Weinberg operator. For comparison, we alsocorrections, plot associated the with running of integrating the heavy fermions in the effective theory. The Figure 3 JHEP03(2021)212 . λ 58 . goes — the ν Y λ (Λ) = 1 ν Y one sees that at the threshold at the threshold 5 8 2 . 0 . | . ν | Y ν | Y | , the Higgs quartic coupling 5 L is relaxed to ν Y increases its value with increasing scale. ν c c Y ν ν – 8 – . From the left panel of figure λ value can lead to an unbounded potential already at and ν . values lead to large threshold corrections for Y L due to threshold corrections. RG running will further λ ν Φ Φ ν Y Φ Φ we show the result of demanding that neither Y We now turn to the issue of the general self-consistency of the 6 up to the Planck scale implies that remain within the perturbative region up to Planck scale. 3 π (Λ) = 0 4 goes negative up to 100 TeV. To quantify the implications of this λ √ λ < . The RG evolution of | π ν 4 Y | √ GeV. However, as one can see from figure GeV. Such large 3 one sees that a large 3 < GeV makes 5 | above the threshold scale, making the vacuum unstable. It is clear that threshold 3 ν λ . The destabilizing effect of right-handed neutrinos on the evolution of the Higgs quartic Y shows the evolution of | , one finds that the pertubativity limit on , due to the negligible effect of the Weinberg operator on its running. Finally, note 5 all the way up to the Planck scale does not ensure full consistency of the scalar Λ = 10 5 Λ = 10 As an example, in figure This highlights the importance of taking into account the threshold corrections for We start by examining the restrictions coming from perturbativity at tree-level, which ν SM requires one to take into account both RG evolution as well as the threshold corrections Λ = 10 Y λ becomes negative much before the Planck scale. Therefore, demanding pertubativity ν corrections are crucial whenY considering large Yukawa couplingsit and induces that on a the true quartic limit coupling on non-perturbative, nor From figure the threshold scale, even beforeat RG evolution. Takingdecrease the Yukawa coupling of potential. If one demandsfigure perturbativity only till, say, 100scale TeV, as shown in rightnegative panel jump of shown inRG the evolution. right panel — making it negative even before turning on its require Figure demanding that scale λ model, the tree-level couplings musta satisfy certain perturbative conditions, value, e.g. and allwe of take the them into potential should account have should thesection quantum be we corrections, analyze bounded these these conditions from modified also below. conditions get in corrected. more However, In detail. once this of that all couplings in figure Consistency restrictions. inverse seesaw mechanism. In order to ensure a perturbative and mathematically consistent Figure 4 coupling. JHEP03(2021)212 5 5 87 up . 10 10 only 0 π GeV 4 4 π . ] has a 4 √ ν √ ν 21 λ < Y Y < ν Y ν Λ = 10 also depends Y gets relaxed. ν 2 4 4 ν 1.02 Y λ Y 10 10 TeV). See text for (Λ)= (Λ)= ν ] ] negative even before ν λ GeV GeV to remain positive up to [ μ Y TeV, 1 [ μ Λ = 10 λ = Y TeV, 10 M = ν M Y reaches the perturbative limit, Λ= 3 Λ= 3 ν TeV ( Y 10 10 . The left panel requires GeV (left panel), and Λ = 1 ν 3 κ κ λ Y λ , before 0

1 2 3 3 2 1 0

1.0 0.8 0.6 0.4 0.2 0.0 - - - Couplings Couplings λ < Λ = 10 5 – 9 – 18 10 10 ν to remain perturbative and Y ν Y 15 10 4 0.84 0.87 λ is large enough that threshold effects make 10 12 λ ν ] (Λ)= ] (right panel). This illustrates that the limit on (Λ)= 10 ν Y ν GeV 02 GeV . by demanding 1 [ μ 9 [ μ Y TeV, 1 ν Y TeV, 1 = 10 = Y . M ν M ν Y Λ= Λ= Y 3 10 6 10 . Limiting κ . Perturbativity limits on the Yukawa coupling λ 3 κ 10 λ

3 2 1 0 1 2 3

0.0 0.8 0.6 0.4 0.2 1.0 - - - Couplings Couplings In the previous sectionin we order saw to that mediate the neutrinodestabilizing addition mass effect of generation on new via the fermions theare Higgs inverse to other seesaw vacuum. particles the mechanism in Standard This the [ Model theory problem providing can a be “positive” contribution potentially to cured the RGEs if governing there (right panel), respectively. With this(left combined panel) requirement and we obtainon the the limit choices of threshold scale, for higher threshold scales the limit4 on The majoron completion of the inverse seesaw Figure 6 100 TeV. Left (right)details. panel correspond to threshold scales demand, we have taken two threshold scales, to the Planck scale, soup that to only RG 100 TeV. evolution is Inrunning. relevant. this In The case both right panel casessee demands the text vacuum is for unstable, details. i.e. Figure 5 JHEP03(2021)212 , , ) Φ M , σ λ (4.5) (4.3) (4.4) (4.2) (Φ and V and dou- σ ], which is v σ provided . 19 ) , σ σ are all positive, † 18 [ σ Φ carrying two units J and λ σ σ Φ)( Φ † λ . √ (Φ H.c. (4.1) σ iJ gauge invariance, + Φ being the vacuum expecta- + 2 + λ j Y 0 2 σ 2 S σ T i σ ν + Φ v √ √ ]. Y λ 2 + 1 ) U(1) σS 22 = − σ σ ij ) S † ⊗ i v    and T Y σ σ S L ( h 0 . M µ σ → Φ + M ( λ λ j T σ D (1) S , 0 and v + c i σ m SU(2) M S ν 2 2 λ Φ Y ij ⊗ v ∼ O √ T D 1 Φ) 0 0 c − † ν m , σ M – 10 – = Y    M (Φ ! i + 0 ν ] here we focus on low-scale generation of neutrino Φ c j h Φ = Y λ h ν SU(3) 11 2 ν eV, are generated for reasonable choices of 2 Φ 0 + ˜ Φ + fields respectively. Again, within the standard seesaw v i √ Φ 1) M L σ . v σ † with ' ij

ν (0 σ ν Y σ 2 σ , we now add a complex scalar singlet 2 O 2 v 1 µ m S and S √ √ 3 gauge framework. To achieve this, in addition to the Stan- i,j , and sizeable Y X − S Φ Y → Y = Φ and † = 2 lepton number symmetry. Φ . The relevant Lagrangian is given by c Φ S are all positive. After the breaking of electroweak and lepton number σ ν U(1) −L 2 Φ µ σ µ , ⊗ 2 Φ U(1) − Φ λ 2 L v ν √ = − Y σ V λ = Φ SU(2) λ D 4 ⊗ m , the most general potential driving electroweak and lepton number symmetry break- c Φ This potential is bounded from below if Turning to the scalar sector, in the presence of the complex scalar singlet Light neutrino masses of Building up on the work of ref. [ and σ symmetries, we end up witha a pure physical gauge Goldstone singlet. boson, After the symmetry Majoron breaking one has, in the unitary gauge, As already noted, inalso addition has to a the global and has a minimum forλ non-zero vacuum expectation values of both blet ing is given by small Yukawa couplings where tion values (vevs) ofapproximation, the the effective neutrino mass is obtained as After the electroweak and lepton numberthe symmetry following breaking form the neutrino mass matrix has dard Model singlets of lepton number. Lepton numberexpectation symmetry value is of then spontaneously broken by the vacuum the dynamical version of the inverse seesaw mechanism [ mass through the inverseLepton seesaw number mechanism is withSU(3) promoted spontaneous lepton to number a violation. spontaneously broken symmetry within the minimal the evolution of the Higgs quartic coupling. A well-motivated way to do this is to assume JHEP03(2021)212 , Φ as λ α (4.6) (4.7) (4.8) (4.9) (4.11) (4.12) (4.13) (4.10) through T is given by ) ns h H M ( 2 2 ) ) σ σ GeV scalar discovered vv vv , σ σ 125 ! and the mixing angle Φ Φ 0 0 λ λ h σ 2 H m , , . + ( + ( ! , ) ) to solve for the parameters ! α α 2 2 α 2 h α α ) ) σ 2 2 2 H 2 2 σ σ v 2 σ m 4.7 v v v cos Φ sin σ σ cos sin , m cos v σ α 2 h λ λ )–( σ λ 2 H 2 H α Φ 2 m α − − m σ ( m λ 4.6 2 σ 2 Φ v 2 2 Φ Φ ) sin v v sin + σ + Φ v v 2 2 cos 2 H v respectively, are given by 2 Φ v − Φ Φ α diag α v Φ m λ λ

2 2 v is identified with the Φ ( ( = σ – 11 – λ − h ≡ Φ q q 2 h, H cos sin T R 2 h λ 2 h 2 h ! O − + 0 0 m

m m ( h 2 2 σ σ σ 2 ns v v =

) along with ( σ σ = = = M R . We start with the first possibility. As before, we fo- λ λ 2 R ns σ σ ) Φ O Φ λ 4.10 O Φ λ + + M v λ = 2 2 ( Φ Φ will mix, so the mass matrix for neutral scalar v v 0 quartic coupling receives a new 1-loop contribution through the ! of the scalars Φ Φ σ h λ λ . This “positive” contribution plays a crucial role in counteracting H Φ ≈ O ) and ( H as 7

= = σ R and 4.9 , m v 2 h ]. The rotation matrix satisfies 0 2 H O h h m m m 48 , ]. 2 47 , 1 and (ii) in terms of physical quantitites i.e. masses . σ Φ 4 v Φ . is the CP-even scalar mixing angle, and its range of allowed values is constrained , the RGE of the λ 6 σ  α σ Vacuum stability in this model can be studied in two different regimes namely We can use eqs. ( and v σ see figure (i) cus on the missingsection partner (3, 1, 1) inverse seesaw, other possibilities will be taken up in In this sectionnumber we symmetry will on explore the stabilityscalar the of consequences the of electroweak vacuum.diagram spontaneous shown Due in breaking to figure the of presencethe the of “negative” lepton the contribution coming from the extra fermions of the inverse seesaw model, 5 Vacuum stability in inverse seesaw with majoron λ The lighter of these twoat mass eigenstates the LHC [ where the masses Here by LHC data [ We can diagonalise the abovethe mass rotation matrix matrix to obtain the mass eigenstates The CP even fields JHEP03(2021)212 σ (5.1) (5.2) . This σ σ , have a 2 Φ λ 4 λ M . Below this = Λ H and . Both are very δλ M 0 Φ H λ = M is the same, since in , there is a tree-level ) Λ M Z in the effective theory is M , the first thing to note is ( 0 Φ Λ = M λ SM λ almost decouples, with its mass Λ = , . 2 and H form a quasi-Dirac pair with nearly σ σ ! ) 2 Φ λ 2 Z S 2 4 λ v Thus, below the threshold scale, we need σ M ( − − 5 κ and Φ Φ λ below the threshold scale is defined as c † λ ν 0 Φ ). They lead to a positive shift in value of the Φ = λ

– 12 – ]. As a result, at the scale κ 5.2 0 Φ λ 49 λ σ is essentially the same as that of quartic coupling due to its interaction with the singlet ≡ = quartic coupling, that can overcome the destabilizing effect Φ 0 Φ . Moreover, in this limit small neutrino masses require SM λ eff σ Φ λ v V Φ Φ Φ Φ σ λ of the effective theory with explicit lepton number breaking. 2 κ √ λ at tree-level [ will also be present in this effective theory. Even though massless or fairly ≈ ) . J σ Φ H ( the heavy CP-even Higgs boson 4 v M Φ . We assume, for simplicity of the analysis, that Re v 2  ≡ M . One can appreciate the jump in the value of the Higgs quartic coupling √ σ  H 8 v σ m . The evolution of the Higgs quartic coupling v , the RG evolution of 3 Λ is the effective quartic coupling for the case of explicit lepton number breaking, . One-loop correction to the , so the two heavy singlet fermions S κ Y λ given as Moving on to the full theory at the threshold scale Note that the majoron 5 H both cases one must reproduce the 125 GeV Higgsthe mass. impact of threshold corrections, eq. ( light, it will pratically decouple from the Higgs boson, and will not affect vacuum stability. see section shown in figure due to threshold corrections.the Since scale only the dimension-fiveclose Weinberg operator to the runs RG below runningMoreover, of at tree-level the numerical value of where the effective Higgs quartic coupling Here to integrate out threshold correction which induceswill a lead shift to in the the following Higgs effective Higgs quartic potential coupling, below the threshold scale small degenerate mass common value, so thatscale we we deal have with an just effectiveWeinberg operator one theory for with threshold neutrino the mass scale generation. Standard Model structure, suplemented by the of the fermions in figure 5.1 Case I: In the limit m Figure 7 that drives spontaneous leptona number “positive” violation term in in inverse the seesaw RGE models. of the This diagram leads to JHEP03(2021)212 ]. 18 41 and TeV 10 – 0.1 SM 0 Φ at the λ 28 λ ), which 1 (Λ)= . = 10 15 σ Φσ λ 10 H 4.13 = 0 (Λ)=λ M σ σ λ Φ = 12 Φ ] λ 10 , λ 0.45, M σ using the full RGEs GeV λ (Λ)= Φ ν [ μ 9 λ in the full theory above 10 Φσ Φ λ λ Y TeV, 100 6 = ν H , we give the two-loop RGEs of Y 10 m = B M , enhancing the chances of keeping Λ= 3 M ' Φ λ 10 , starting instead with eq. ( . We have taken H 45 0.2 0.1 0.0 0.1 0.6 0.5 0.4 0.3 at the threshold scale is coming from the Λ = .

- m Couplings λ is of the order of the electroweak scale. Hence = 0 both to the threshold corrections, as well as to and H ν . In appendix. 18 – 13 – Z Y σ is the effective Higgs quartic coupling below threshold, Λ that the dynamical variant of the inverse seesaw m 10 κ SM 0.1 M 8 λ λ ) at (Λ)= ). Notice that below threshold the running of ≡ 15 σ Φσ λ 0 Φ and evolve the quartic coupling 10 5.2 5.2 λ σ (Λ)=λ λ σ β λ . These could lead to a plethora of new phenomena [ 12 Φ , ) ] λ σ 10 Φ (1) Φ 0.45, v λ GeV ( β TeV for the left and right panels, respectively. For the sake of (Λ)= ν [ μ 9 O ∼ O 10 ν = Φσ Y λ one must perform the RG evolution of all parameters. Above the scale σ = 100 Y TeV, 10 v = (red-dashed). Here 6 ν we show the evolution of various couplings in the majoron inverse seesaw H M ]. Furthermore, to understand the evolution of Y H 10 m 8 5 = SM M M λ ). Λ = Λ= has been taken to be very small, it has no direct impact on vacuum stability. . The RG evolution of the quartic couplings and right-handed neutrino Yukawa couplings = 3 ' Φ 5.2 . The positive shift in the evolution of λ S 10 Y Λ M almost coincide with each other, due to the tiny effective Weinberg operator. Finally, In summary, it is clear from figure In figure 0.1 0.5 0.4 0.3 0.2 0.1 0.0 0.6

positive [

- Couplings one needs to include SM Φ dynamical variants can have a completely stable Higgs5.2 vacuum. Case II: In this case, the mass ofwe can the neglect heavy the scalar small range between to the positive contribution ofthe the RG evolution scalar of thenegative Higgs contribution quartic of coupling. the new These fermions effectsYukawa present couplings are in enough inverse to seesaw counteract model,Thus, the even for in sizeable contrast to the case of inverse seesaw with explicitly broken lepton number, the scale matching condition given in eq.λ ( since mechanism can be free from the Higgs vaccum instability problem. This is possible thanks model for given benchmark points. Weand have taken the threshold scale as comparison, the initial values ofThe other parameters Yukawa coupling have been has kept been the same fixed in at both panels. λ the scale Λ with the matching condition eq.the ( full theory. within the Majoron extensionevolution of of (3,1,1) inverse seesawsee scheme. eq. For ( comparison, we also show the Higgs quartic coupling above the threshold scale Figure 8 JHEP03(2021)212 18 18 SM SM λ λ 10 10 TeV 1 TeV 3 = = σ σ Φσ (0.8) for λ 15 15 σ v 0.1, v 0.2, 7 λ . 10 10 α= α= 0 is sufficiently starts running σ ≥ Φ ν 12 12 λ Y ] ] ν 10 10 0 GeV, 800 0 GeV, 400 Y = = H H . In these plots, we GeV GeV 9 [ μ [ μ 9 9 m 0.6, m 0.6, 10 10 between the two CP-even Φσ (Λ)= (Λ)= λ ν ν 1) . σ λ 6 6 (0 ν ν Y Y 10 10 Y TeV, 100 Y TeV, 100 ∼ O = = M M α 3 3 Φ Φ Λ= Λ= λ λ 10 10

0.8 0.6 0.4 0.2 0.0 0.8 0.6 0.4 0.2 0.0

). Thus in this case only the fermions are Couplings Couplings TeV in the left panel, and 100 TeV in the , while all the scalars remain in the resulting 5.2 18 18 M – 14 – SM SM λ λ 10 10 TeV 3 TeV 1 Λ = 10 = = Λ = σ σ Φσ λ 15 15 σ λ v 0.1, v 0.2, 10 10 α= α= , all scalars are part of the effective theory below threshold, taken 12 12 M ] ] 10 10 0 GeV, 800 0 GeV, 400 = = H H GeV GeV Λ = TeV (100 TeV), respectively, we get either unstable vacuum or non- [ μ [ μ 9 9 . We found that for large Yukawa couplings, ν m 0.5, m 0.5, 10 10 Y Φσ (Λ)= (Λ)= λ ν ν σ is shown in the red dashed curve. Here only the fermion singlets are integrated Λ = 10 λ 6 6 ν ν Y Y SM . This in turn implies a sizeable mixing 10 10 λ 1) Y TeV, 10 Y TeV, 10 . = = . Evolution of the quartic couplings and right-handed neutrino Yukawas within the M M (0 3 3 Φ Φ Λ= Λ= will also be large as there is not enough range, in terms of RGEs evolution, to O λ λ 10 10 ν The evolution of the couplings in this case is shown in figure

Y 0.0 0.6 0.4 0.2 0.0 0.8 0.6 0.4 0.2 0.8 Couplings Couplings only above threshold.of Notice that forsizeably relatively alter large mediator the threshold scale, scale the allowed value perturbative dynamics. large, Higgs bosons. have fixed the singletright panel. neutrino scale In contrast to the scalar couplings, the Yukawa coupling already includes the thresholdintegrated effect out of at eq. the ( theory threshold scale below threshold.better Thus chance the of scalar curingbefore, couplings the the evolve Higgs Higgs over vaccum vacuum a instability can instability larger be problem. range, avoided if and Needless the have mixed to quartic say that, as Figure 9 Majoron extension ofevolution the of missing partnerout at (3,1,1) the inverse threshold scale seesawas the scheme. weak scale. For comparison, the JHEP03(2021)212 , 4 α j . the 6= = 0 i H | m ν for Y | (dot-dash, should not ij TeV. Below ν σ Y and neglected v all the way up = 2 4 . σ = 10 n Φ case. For small H λ = 0 where we compare H m ii ν case. 10 m Y = and σ = 1 M and included the threshold λ n , S inverse seesaw scenarios with Φ GeV. In this case the scalars ], can be used to place a lower (solid, blue), ) Λ = λ 48 and , c = 1 , we can have a stable electroweak , n, n 47 ν = 500 2 , n (3 ) H ≥ σ within the Standard Model (dashed, red) ( m , while all off-diagonal ones, n 4 λ . Re 2 √ = 0 . Indeed, even in the higher (3,2,2) and (3,3,3) – 15 – ii H ν Y TeV and m . For (3,2,2) and (3,3,3) case, we have taken the 4 . , we can have positive = 1 and we illustrate the interplay between the vacuum stability 9 = 0 σ 7 α v | ν Y | matrix to be , e.g. coming from the LHC [ ν α Y . In section (dot, green). H inverse seesaw completions, with m = 3 we have taken the initial Yukawa coupling values in such a way as to ) , we display our vacuum stability results for the majoron inverse seesaw . In the left panel we have taken the case of n ij clearly illustrates that even for ν 10 11 Y , n, n 11 (3 the potential becomes unbounded from below at high energies. In other words, have worse Higgs vacuum stability properties than the H 1 Figure In figure In figure Moreover, as shown in figure m the right panel, weare have not fixed integrated outPlanck and scale. the quartic coupling runs smoothly from electroweakvacuum scale for till adequate choices of blue), (3,2,2) (dot-dash, magenta) and (3,3,3)As (dot, before, green) to majoron ensure inverse a seesaw consistentfor schemes. comparison, (3,1,1) we case, have while taken for the the Yukawaoff-diagonal (3,2,2) coupling and (3,3,3) cases, we have taken threshold we have integrated outeffects. the fields This leads to the jump in the quartic coupling seen in the figure. In contrast, for diagonal entries of the were neglected in then RGEs. > Clearly one sees how models. One can compare the Standard Model case (dashed, red) with the (3,1,1) (solid, with the magenta) and facilitate a proper comparisonfixed of the the Yukawa different coupling cases. To do this for (3,1,1) case, we have 6.1 Sequential versus missingAs partner already seesaw: mentioned, electroweak theaddition vacuum of problem stability extra of fermions. Higgsthe RG This vacuum evolution fact stability of is only the Higgs clearly gets quartic illustrated worse coupling in with figure the For simplicity we havebreaking so within far the simplest onlycompare (3,1,1) analyzed the missing the stability partner explicit inverse properties(3,3,3) seesaw inverse and of seesaw mechanism. dynamical this mechanisms. We lepton minimal now number construction with those of (3,2,2) and restrictions and the constraints oncurrent the LHC invisible experiments. width ofLandau There the poles Higgs we in boson also the that notebe running follow too that parameters, from small. the in lepton order number to breaking prevent scale the existence6 of Comparing sequential and missing partner inverse seesaw to the Planck scale, evenrequired for mixing sizeable angle Yukawa couplings. isor relatively We found large, that in forexperimental contrast small limits to on thelimit large on the mass JHEP03(2021)212 18 ) ) ) 10 ,3 3 3, 3, 2 2, 3, 1 1, 3, ( ( ( SM TeV 1 = 15 σ 10 ]. This lack 17 v 0.12, α= 12 ] 10 GeV 18 in the Standard Model in the Standard Model 0 GeV, 500 [ μ 9 λ λ 10 = H 10 ) ) ) 15 6 m TeV, 10 10 10 = ,3 3 3, 3, 2 2, 3, 1 1, 3, ( ( ( SM M Λ= 3 12 10 ] 10 0.05 0.00 0.05 0.10 0.20 0.15 0.10

GeV - - Φ λ [ μ 9 10 18 ) ) ) – 16 – 10 ,3 3 3, 3, 2 2, 3, 1 1, 3, ( ( ( SM 6 0.12 15 10 10 (Λ)= Φσ λ 3 12 0.1, ] 10 10 (Λ)= σ GeV λ [ μ 9 0.10 0.05 0.00 0.05 0.10 0.15

- - 10 λ 0TeV, 10 ]. This minimal scheme is simply the inverse seesaw embedding of the = H 8 6 m = 10 M Λ= . ν . Comparing the evolution of the quartic Higgs self-coupling . Comparing the evolution of the quartic Higgs self-coupling 3 Y 10 sion 0.05 0.10 0.05 0.00 0.15

- Φ λ inverse seesaw mechanism is phenomenologicallyto realistic. only The one reason massive neutrino isoscillation (lying that data say, (3,1,1) at [ leads the atmosphericminimum scale), “missing hence partner” inconsistent (3,1) with of see the saw mechanism solar of neutrino section mass III splitting in can ref. be [ avoided by the presence of a complementary vacuum can be kept stablecoupling all the way up to the Planck scale6.2 even for appreciable Yukawa Sequential versus missing partner seesaw: briefHere phenomenological we discus- note that neither the explicit nor the dynamical variant of the minimal (3,1,1) (red dashed) with the(blue), majoron (3,2,2) inverse is seesaw dot-dashed mechanism: (magenta) and the (3,3,3) minimal is (3,1,1) dotted is (green). denotedmajoron See in inverse seesaw, text solid the for positive details. contribution fromthe the negative new contribution scalar from is enough the to new overcome fermions of the inverse seesaw. In short, the Higgs Figure 11 Figure 10 (dashed, red) with various inverse-seesawdenoted extensions in solid with (blue), explicit (3,2,2) lepton dot-dashed number (magenta) violation: and (3,3,3) (3,1,1) dotted (green), see text for details. JHEP03(2021)212 , ]. is 8 ]. | 29 α 57 – ) that 55 sin | 4.13 ]. Finally, 36 ]. Moreover, – ]. A dedicated ), one needs a ) 32 59 † , 41 ν – Y 58 ν 37 Y † ν Y ν Y ( and vice-versa. Within dy- 6 ]. Tr . We see from eq. ( 2 H lead to sizeable invisible Higgs 53 H − ]. Finally, the sequential (3,3,3) m – ) 17 51 and and TeV ( ) one sees that in order to overcome h 4 , relatively large mixing angle t ) y 6 B.1 ∼ O − for larger TeV σ | ( v α . This in turn translates into a large mixing σ – 17 – Φ ∼ O in eq. ( sin λ | σ Φ v λ As a result, there is a lower bound on the neutrinoless 7 is the Majoron. The existence of such invisible decays ]. J 54 -even neutral Higgs bosons ] where ]. This would provide an elegant theory with a tree-level atmospheric 23 [ CP ], with stringent bounds, e.g. from the Delphi and L3 collaborations [ 50 31 JJ – 28 → h implies smaller mixing angle σ between the . From the RGE running of Φ 0 α λ collider [ Indeed, it has long been noted that models with spontaneous violation of global sym- There are other implications of low-scale seesaw schemes, such as our inverse-seesaw, Concerning neutrinoless double beta decay, here lies an important phenomenological Alternatively, one can generate non-zero tree-level masses for two neutrinos by going > − Modulo, of course, explaining the detailed pattern of mixingThis angles indicated feature by may the also oscillation be data [ implemented in some radiative schemes of scotogenic type, see e.g. [ 6 7 e ]. Likewise, they could produce interesting signatures at the LHC [ Φ + metries such as leptondecays, number i.e. at low scales Such a challenging task would require a family symmetry, whose detailed nature is not yet fully understood. relatively large mixed quarticangle coupling large namical low-scale seesaw schemes with expected. This is in potential conflict with the invisible Higgs decay constraints from LHC. 7 Impact of invisibleAs Higgs we decay saw on above, theλ vacuum vacuum stability stability is oftenthe threatened destabilizing by effect the coming violation from of fermions the ( condition since the heavy singlet neutrinos wouldfeatures not associated take to part unitarity in violation oscillations,study in these would the be could lepton required reveal new to mixing scrutinizemissing matrix whether partner [ these from signatures sequential could seesaw. be used to distinguish associated heavy neutrino mediators could bee accessible at high energy experiments30 such as these mediators would alsopotentially induce detectable lepton rates, flavour unsuppressed and by the leptonic small CP neutrino violation masses effects [ with missing partner seesaw there canamplitudes be leading to no the cancellation decay [ amongstdouble the beta individual decay light-neutrino rates that could be testable inthat the could upcoming be generation potentially of testable searches. in current or upcoming experiments. For example, the directly to the (3,2,2) “missingseesaw-analogue partner” of seesaw the scheme. (3,2) Again,inverse seesaw seesaw this mechanism mechanism would in will be ref. generate theof [ tree-level inverse- these masses would for be all three totally light consistent neutrinos. with neutrino Any oscillations. difference between the “missing partner” and the “sequential” seesaw mechanism. In the would need to invoke newdark physics. matter The sector latter [ could bescale, associated, and a say, to radiatively-induced the solarof presence neutrino of mass the scale, a bilinear very breaking much of analogous R-parity to in the case supersymmetry [ radiative mechanism. To implement such “radiative completion” of the minimal scheme one JHEP03(2021)212 , ]. π or for 4 62 and 0 [ | TeV, σ < α v ≤ ) GeV. µ sin ) 19% ( | µ σ Λ = 10 ( 130 ≤ σ or ) > λ is the running < λ σ , for fixed µ H Φ 0 0 | λ , m α π ≤ 4 ) Invisible for sin < where | µ 2 ( ) . → π Φ µ 0 4 ( h λ ( Φ < < | | = 1 TeV which lead to either ) α < λ σ µ ( v 0 sin ν | Y | for α ]. The tightest bound on invisible Higgs and π 61 4 and , leading to a stable potential (green), an unstable < H 60 | , α ) m – 18 – TeV one gets µ 47 ( σ = 1 Φ λ σ | v . , 0 0 = 1 TeV. Within the green region all couplings are perturbative ≤ > σ ) ) v µ µ ( ( σ σ and mixing angle and λ λ ) ) H 4 µ . µ ( m ( in this region we have a stable vacuum all the way up to the Planck shows the values of Φ Φ in this region the potential becomes unbounded from below at some = 0 λ λ 12 ν p p Y ) + 2 ) + 2 for the (3,1,1) missing partner seesaw with relatively large Yukawa coupling GeV. For example with µ µ . Values of ( ( . Figure 12 σ σ 4 . Φ Φ 130 λ mass scale. Red Region: high energy scale beforeif the any Planck (or scale. more) The of potential the is following unbounded conditions from is below realised: Green Region: scale, with all thethese couplings conditions are within implemented in the following perturbative ways: λ regime. In our numerical scan in such a way that one has consistency with the CMS constraint on invisible Higgs > So far in all of our discussions we have chosen the mixing angle • • = 0 H H ν stable/unstable potential or non-perturbative dynamics, as follows: m decay. However, thevacuum full stability parameter but disallowed space by of thein invisible the figure Higgs model decay constraints. containsY We regions illustrate this consistent with can be probed by theboson LHC decays experiments comes [ from theThis CMS upper experiment at limit the on LHC,m the BR invisible Higgs decay sets a tight constraint on Yukawa coupling and the vacuum is stable upfrom to below the Planck before scale. theLandau In Planck the poles) red scale. region at the The energydelimited potential becomes orange by scales unbounded region the below has black thedetails lines nonperturbative in Planck is couplings text. scale. ruled (including out The by region the LHC outside constraints the on horizontal invisible band Higgs decays. More Figure 12 potential (red) and non-perturbativemajoron dynamics inverse (orange). seesaw as Here the we reference, with take the the heavy (3,1,1) fermion threshold missing scale partner fixed as JHEP03(2021)212 , , ≤ π 4 and ) = 1 and σ n v | ≥ 12 ) . Hence ) A priori µ ν ( Y Φ schemes are Invisible † ν λ associated to | Y 2 ( σ → v ≥ h ( n by Tr 2 . Therefore, the LHC | σ ν with v Y | ) we have required vacuum we display the results for . Likewise, the same effect 13 13 12 , n, n . It is clear from figure (3 13 and figure . Note that the possibility of Landau poles 12 π 4 , the required mixing angle is large, in order to – 19 – ]. One may now wonder how this discussion will | ≥ H ) , the (3,1,1) and 50 m µ ) ( ν inverse seesaw schemes which do not require a “com- ν Y Y is more tightly restricted than in our reference † 2 | ν , 2 are larger than in figure Y ≥ , disappearing for high enough ( π σ 4 ≥ n v 13 Tr ; n ) this is the region disallowed due to the LHC restriction on | ≥ ≈ ) 2 µ | ( ν here one or more couplings become non-perturbative below the , n, n σ Y Φ | (3 λ | , π 4 seesaw with one sees that for small ) ]. | ≥ 12 62 ) [ µ , n, n ( that the allowed parameter space consistent with stability and LHC constraints σ (3 λ 19% Planck scale. This happens if| any one of the following conditionsis holds: also included inside the non-perturbative regions. Collider constraint: the Higgs invisible decay branching fraction which requires BR Orange Region: 13 Note that the restriction on the mixing angle gets stronger for lower values of The above discussion refers to our (3,1,1) majoron inverse seesaw reference case, tem- • • stability and perturbativityrequirement, all if the there way is other upstability new and to physics perturbativity at the only play.relaxing up Planck In to the scale. that a resulting case lower one restrictions. This energybeyond should the scale, All require will scope say vacuum of be of only these the an up issues present to over- require work. 100 TeV, a thus dedicated study, that lies weakens for higher valuesmeasurements of constitute a probeneutrino mass of generation. the Moreover, note leptonthe that lighter here number of we violation have the only scale the two considered CP possibility the case even that when scalars the isdiscussed. heavier CP identified Finally, as even in scalar the the is 125 GeV discussions the Higgs of 125 boson. GeV figure Higgs boson should also be note that, in termscorresponding of RGEs (see RGE appendix) evolution,as are there long the is as same not one byeffectively much replacing takes the difference same. between them. The figure in the case. However weparameter note regions that, where for electroweak moderateAlthough breaking in is values the consistent of above with we the discussed the Yukawa (3,1,1), LHC coupling, (3,2,2) measurements. and we (3,3,3) still cases have separately, one should pletion” so as tothe generate (3,2,2) the (left atmosphericeffect panel) scale. of and additional In (3,3,3) fermions figure unstable (right on red panel) the regions stability scenarios. of inis the figure As seen vacuum by expected, is comparing clearly the the visible. undesired left Indeed, and the right panels of figure vacuum consistency requirements of pertubativityout and stability. a Altogether, large these part can of rule the model parameter space. plate for the scoto-seesawchange mechanism in [ the higher From figure ensure a stable electroweakdecay vacuum. constraints. This As a is result, in one turn sees in that conflict these with constraints the are invisible complementary Higgs to the JHEP03(2021)212 10 , 9 , 8 , 3 matrix are fixed ν Y becomes negative λ ) has a destabilizing effect S , in which other mass scales, GeV. We have taken as our 3 , 10 and for the case of (3,2,2) (left) and 10 c = 2 ν 12 n ∼ . We found that for the inverse seesaw the quartic coupling λ with ν ) Y – 20 – , n, n (3 associated to neutrino mass generation. Its detailed σ v . We found that the LHC measurements constitute a probe of also be considered. We have also required vacuum stability and 13 and a priori 12 . See text. . Vacuum consistency constraints of figure 4 . . We showed how, in contrast to simplest inverse seesaw with explicit lepton number = 0 ii ν 11 Y All of these issues require a dedicated study, that lies beyond the scope of the present work. possibility should perturbativity all the way up topresence the of Planck scale. additional new This physics. isthe clearly The strong an latter CP over-requirement, could problem. in be the In associated suchonly say, case up to one to dark should a matter require lower intermediate or vacuum energy stability to scale, and perturbativity thus relaxing the restrictions we have obtained. a physical Nambu-Goldstone boson,is the given majoron. in The figures comparisonthe with lepton LHC number restrictions violationstudy, scale however, needs furtherof investigation. the two For CP example, even we scalars have to assumed be the the lighter 125 GeV Higgs boson. The alternative intriguing this reference case, in“higher” which inverse only seesaw one constructions oscillationsuch scale as is the generated atmospheric scale, atresults also tree-level, on arise with the from the stability theand of tree-level seesaw the mechanism. electroweakviolation, Our vacuum the main are stability summarized properties improve in when figures this violation is spontaneous, and there is on the running ofmechanism the with Higgs sizeable quartic Yukawamuch coupling coupling before thesimplest Standard benchmark Model neutrino instabilityit model scale has the the “incomplete” “best” (3,1,1) stability inverse properties seesaw within scheme, this as class of seesaw schemes. We compared We have examined theseesaw consistency mechanism. of electroweak We symmetryof have breaking the derived within relevant the the parametersinverse full inverse seesaw within two-loop with inverse renormalization explicit seesaw groupof violation schemes, of equations inverse examining seesaw. lepton number, both The as the addition well simplest as of the fermion majoron singlets extension ( Figure 13 (3,3,3) (right) inverse majoron seesawas mechanism. The diagonal entries of the 8 Conclusions JHEP03(2021)212 β 2 2 t t y ν 2 t λy (A.6) (A.2) (A.3) (A.4) (A.5) (A.1) 2 y Y 2 4 t † 2 2 t 2 g ν g y Y λy 2 2 2  45 ν 3  † g ν Y  + Y 8 − λ 45  λ ν 2 4 t 4 2 t † +225 ν  Y g y 540 + 2 t Y 2 3   λy 2 y t  ν 8 960 † − g 2 1 73 ν † 2 1 y ν Y g ν Tr g 2 1 Y − † Y 32 4 ν 2 − Y g ν 2 2 2 ν † g Y 17 ν Y λ g − 8 Y ν 3 4 17 2 2 Y  † 4 t +85 ν + Y g ν y 2 − + 2 2 Y t  1 Tr 2 Y 1 ν 2 λ y g †  g λ ν 4 2 † +675 Tr Y λ ν 8 5 . Y g 2 t  Y ν 20 90 9 4 y 117 +4 ν − Y +6 (2) c Tr 2 t Y +240 −  4 3 − + 9 4 β †  4 2 4 g t ν 540 2 t 2 λy λ g y  ) y Y − 4 1 Tr † − 2 2 2 ν ν 4 t g +120 2 2 108 2 1 g π 1 Y Y y g ν 230 2 270 g 1 +12 ν  2 1 − Y 4 g ] to perform the RG analysis. The 27 2 Y g − † − 2 3 (16  ν 200 † Tr λ 1887 2 2 ν g  63   63 10 3 Y − 2 g 2 10 2 † Y † † + ν ν ν ν λ 2  1 + g +223 ν + † Y Y Y g 6 − 2 ν Y  2 +24 t ν ν 48 ν λY (1) g † c † Y  ν ν y +9 Y 54 Y Y λ † ν β 4 1 − ν Y Y 2 2  2 3   are the one-loop and two-loop RG corrections. 2 Y g ν − ν Y 16 960  g g 305 – 21 –  π † ν 4 2 Y 1 1 Y 9 Tr 1 Tr ν Tr − g g 2 Y  (2) 2  c + Y 16 Tr − 171 100 ν g + +  β ν 4 2 2 2 15 19 21 Y Tr 2 2 Tr 2 λ 2 , Y g g − †  = 2 1 g Tr g ν 2 1  + ν 3 4 8 g 1 9 4 9 4 c g Y 15 (1) c +75 4 2 g Y λ 180 9 5 ν β Tr β +10 g  − − + Y λ † − is given by, 6 t 80 9 2 − 2 ν 1 1 2 4 2 ≡ 289 2 2  +2 23 y 2 3 312 100 g g  4 2 g Y c † g ν g †  g ν ν − 9 − − † Y 20 17 20 − ν dc 2 Y dµ 9 8 2 Y 2 15 2 2 2 ν 2 t Y ν  g +30 g µ λ − − Y y ν 4 + 1 Y 2 1 + 2 1280 2  +675 2 2 2 t 3  † g Y 2 Tr 2 † g g ν   ν g y ν λ 2 1 g  3 † t Y 8 3 9 Tr Y ν Y 2 g 1 20 y ν 2 † 1 ν  g Y ν Tr − Y g 400 ν 144 Y ν ν − + 1677 Y 2 t 9 † 3 8  Y Y ν +108 20 4 ν 5 Y 1 t − y +15 †  − g 2 y Y Y ν 3 2 + t + Tr 2 t + ν 6 2 1 1 λ Y 2 2 ν y t  Tr 2 4 g g 1 1 Y t ν 2 3 λy y Y g g 120 λ 600  y − 2 2 Y g † 3 3 1187 ν 2 1  4 g t g  g 279 + Y 5 Tr y t 27 1 ν 3 t 108 3 2 200 3411 2000 80  λ 6 y 180 y Y is the running scale and 1 +80 + − − − + + +20 − +800 3 2 2 80 3 µ = = + = + = = = ν ν t t (2) (1) (2) (1) (2) (1) λ λ y y Y Y β β β β β β A.2 Yukawa couplings The one-loop and two-loop RGsimplest corrections inverse seesaw for model the are most given relevant by Yukawa couplings in the where A.1 Higgs quarticThe scalar one-loop self and coupling two-loop RG corrections to the Higgs quartic self-coupling are given by In our work wefunction have of used a the given package parameter SARAH [ A RGEs: inverse seesaw JHEP03(2021)212 4 t y 6 Φ  2 λ t (B.2) (B.4) (B.5) (B.1) (B.6) (B.3) −  4 2 y  †  2 † g 2 ν † ν ν g  Y Y 8 2 Y 1 † ν  73 ν σ 2 ν t ν g ∗ Y  S y Y λ Y Y − †  † ν σ σ Y ν  ν  63 10 Φ † Y 2 Φ 2 Φ S Y  ν Y 2 t † Tr λ † ν λ ν λ Y + ν Y ν Tr y 2 2 Y 2  t Y ν Y Y Φ σ g 12 48 ν  y Y ν  2 1 λ 2 Φ  4 1 Tr +20 Y  g Y ∗ − − λ g † S Tr σ 2 t Tr ν   σ Y y +4 2 Φ Φ σ Tr Y ∗ 12 3 Φ S 20 2 S t 4 1 λ λ Φ ν Tr 117 λ 171 100 Y y g 2 − 2 Y 8 σ λ Y ∗ S g 2 Φ t + S 2 Φ  +60 9 2 11 − − y Y 9 λ Φ Y λ  2  2 Φ σ 100  15 − − Tr 4 ∗ λ S ∗ S S λ Φ 4 t σ 4 1 Y − + Y − σ Tr Y λ y g 2 Φ ∗ +12 2 t S 2 S  2 σ 3 Φ  S σ λ y † +10 † 2 Φ Y g Y λ λ ν Φ Y ν 2 2 6 t 2 Φ S λ  g Y  λ Y y 200 λ 80 60 Y ν 1887 ν  2 Tr  Tr Y Y σ +40 27 − − +3 + σ  2  t 2 Φ +24 3 σ 144 σ λ Tr σ +30 − 2 Φ y λ Φ λ σ 2 Φ 3 Φ Tr σ  Tr  λ − +40 λ 2 Φ λ † + ∗ λ σ 4 2 Φ t Φ ν S 2 2 Φ 2 1 λ 4  y Φ λ λ Y 240 g Y g λ − ∗ 2 2 2 1 Φ λ +256 ν S 9 S − 72 5 3 3 Φ − g g 2 2 λ Y Y  6 2 Y − λ g σ † 2 ∗ 3 2 − 3 S ∗ g + ν S 4 S 45 g λ Φ Φ Y σ Y 4 Y Y + +120 σ 15 – 22 – λ ∗ ν λ Φ S S 312 + S 16 2  1 σ 2 Φ 305 σ Y λ 2 † Y Y t + g Y λ λ ν − 4 +80 Φ 2  ∗ y S  S  9 5 + Y 2 g t 2 Φ λ † σ Y 20 ν 4 Y 2 ν y 2 Tr 2 λ Φ −  Tr g S Y Y g 2 Φ 2 +80 Φ − λ 2 σ σ 1 16 ν Y  145 g λ 2 σ σ λ 1 g Tr 2 Φ Φ Y  2 2 g Φ 3 Φ 8 λ λ  − Tr − g λ +72 λ 4 2 4 Tr 4 t σ − 80 + 17 8 289 24 g 2 Φ Tr σ y Φ 45  4 2 +108 σ 3 4 λ λ Φ ∗ − λ + g − − S Φ 2 σ Φ 2 +40 + 2 λ σ 2  2  9 8 Y − λ 2 Φ 2 g λ 2 t 3 † ∗ 2 Φ g ν S S  2 2  1 2 1 1 g y λ 4 1 + λ † +16  † Y − Y g Y g g ν 2 1 ν g Φ 2 2 2 2 † ν 4  ν t S  45 g 9 8 Y 3 4 Y g g λ † y Y Y ν 5 ν Y 2 ν 1 2 1 − 2 5 3 ν +  Tr + 108 Y  g Y 72 g Y 2 1 g 400 2 σ t Y ν σ  1677  g † 9 + +12 2 y Tr + λ Tr Φ Y ν 20 17 9 32 σ 2 σ σ Φ Φ Φ −  Tr λ Y Tr 2 Φ 4 6 + − 1 1 + − 2 λ λ λ λ λ ν 2 2 Φ +4 4 g g 1 2 4 λ g Tr t t σ σ σ σ σ Y λ  2 σ 2 2 g 1 1 y y σ Φ Φ Φ Φ Φ 2  σ λ 4 2 g g 2 1 2 Φ λ λ λ λ λ 48 Φ g g λ Tr 5 400 3 27 λ 10 − 12 1671 200 10 8 5 3411 2000 9 4 4 24 40 32 72 10 2  1 10 − − − − − − − − − − − = + =2 = = + = = + σ σ σ σ Φ Φ Φ Φ (2) (1) (2) (1) λ λ λ λ (1) (2) λ λ β β β β β β In the presence of thecouplings majoron in the the one- inverse and seesaw two-loop model RG corrections are for modified the to quartic scalar B.1 Quartic scalar couplings B RGEs: inverse seesaw with majoron JHEP03(2021)212  2 t y ν  (B.7) (B.8) (B.9) † Y ν (B.10) (B.11) (B.12)  † ν Y  ν 2 t 2 t Y † ν ]. y y ν Y Y 2 † 1 2 1 Y ν ν g  g  Y ∗ Y S ν Φ 8 † 17 ν 540 Y Y λ SPIRE Y S +85  − + ν IN Y 2 Φ ν 2 Φ Y ∗ 960 Tr S ][ λ Y λ  † Y ν − 90 (2012) 1 S 2 2 Y Tr +6 g Y − ν 9 4 σ 4 t  Y +240 † 2 Φ y − ν 716 σ Tr λ 4 t Y 2 Φ y 1 2 ν +675 λ 270 12 2 t Y 4 + 27 y − − 4 3   g − +20 † ν S 4 2  540 Y Y † +120 g ν arXiv:1207.7235 ν 108 − ν + Y [ 2 Y 1 Y ν −  g † 230  Phys. Lett. B ν 2 3 Y σ  , g Y −   λ Tr  2 2 ν 2 2  2 2 † g ν † Y g Tr g ν 2 +223 +8 1 2 Y 2 Φ Y g ν g   λ +9 ν (2012) 30 † ∗ Y ν 2 S 3 Y 8 54 +75  15 g Y Y  – 23 –  2 1 960 ν − S † + Tr g 716 ν 4 Y 1 Tr Y −  Y g  +  ν † + 19 15 ν ν 2 2 Y 2 2 21 Y Y Tr g Tr † + g ν ν   3 4 9 4 2 Y ν 9 4 Y  g ν 180 Y Tr  − S − ), which permits any use, distribution and reproduction in 2 Y 1 2 1 4 Y − ]. 2 1 2 23 2 S g Tr g ∗ 2 +2 3 g g S 2 Y 1 g −  ∗ 9 Y g S 20 † 2 2 Phys. Lett. B 17 20 ν S 3 8 Y +15 g , Observation of a new particle in the search for the Standard Model Y Y − 2 2 1 t S SPIRE − + ν 4 2 t +675 g 1280 y 2 Y 3 2 IN t Observation of a New Boson at a Mass of 125 GeV with the CMS Y y 2 ν 3  g − y 9 3  g 3 t Y 20 ][ 8 2 3 S CC-BY 4.0 † +4 y  ν g Y Tr − −  ν Y ∗ + ν 2 t ∗ 4 S 1  S ν Y This article is distributed under the terms of the Creative Commons 5 t y Y g σ Y Y +800 Y y † 3 + +20 ν Φ 2 S 2 1 2 t S 2 ν t  Y g λ Y y Y t y ν Y ∗ 2 2 120 600 S collaboration,  2 2 † y + 1187 Y g ν Y g  σ 2 279 Y +  Tr S collaboration, λ t ν 1 8 3  t S Y 45 80 y 180 y Y Y 1 3 2 80 3 2 +4 + + − +225 arXiv:1207.7214 =28 =2 = = + = = ATLAS Higgs boson with the[ ATLAS detector at the LHC CMS Experiment at the LHC S S ν ν t t (2) (1) (1) (2) (1) (2) Y Y y y Y Y [1] [2] β β β β β β Open Access. Attribution License ( any medium, provided the original author(s) and source areReferences credited. This work isPROMETEO/2018/165 (Generalitat supported Valenciana), Fundação by para a(FCT, the Ciência Portugal) e Spanish under a Tecnologia grant projectsolider MultiDark FPA2017-85216-P CERN/FIS-PAR/0004/2019, FPA2017-90566-REDC. and (AEI/FEDER, the UE), Spanish Red Con- Acknowledgments Likewise, in the presenceYukawas in of the the inverse seesaw majoron model the are modified one- to and two-loop RG corrections for the B.2 Yukawa couplings JHEP03(2021)212 22 ]. Adv. ]. 02 , ]. JHEP (2018) , (2017) (2017) ]. SPIRE SPIRE (2013) 089 98 96 IN JHEP IN 95 SPIRE ]. ]. , 12 ][ Naturalness, ][ IN SPIRE ][ ]. ]. Phys. Rev. D IN , SPIRE SPIRE ][ ]. IN JHEP IN , [ ][ SPIRE SPIRE ]. Phys. Rev. D IN IN Phys. Rev. D , Phys. Rev. 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