Consistency of the Dynamical High-Scale Type-I Seesaw Mechanism

PHYSICAL REVIEW D 101, 115030 (2020)

Consistency of the dynamical high-scale type-I seesaw mechanism

† ‡ Sanjoy Mandal ,1,* Rahul Srivastava,2, and José W. F. Valle1, 1AHEP Group, Institut de Física Corpuscular—CSIC/Universitat de València, Parc Científic de Paterna. C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia), Spain 2Department of Physics, Indian Institute of Science Education and Research—Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462066, India

(Received 6 March 2020; revised manuscript received 1 June 2020; accepted 5 June 2020; published 22 June 2020) ð3Þ ⊗ ð2Þ ⊗ We analyze the consistency of electroweak breaking within the simplest high-scale SU c SU L ð1Þ U Y type-I seesaw mechanism. We derive the full two-loop renormalization group equations of the relevant parameters, including the quartic Higgs self-coupling of the Standard Model. For the simplest case of bare “right-handed” neutrino mass terms we find that, with large Yukawa couplings, the Higgs quartic self-coupling becomes negative much below the seesaw scale, so that the model may be inconsistent even as an effective theory. We show, however, that the “dynamical” type-I high-scale seesaw with spontaneous lepton number violation has better stability properties.

DOI: 10.1103/PhysRevD.101.115030

ð3Þ ⊗ ð2Þ ⊗ ð1Þ I. INTRODUCTION of the SU c SU L U Y Standard Model with an ungauged lepton number [12].For“sizeable” Yukawa The discovery of a scalar particle with 125 GeV mass coupling, Yν ∼ Oð1Þ, in order to reproduce the required plays a central role within particle physics [1,2].In neutrino masses, heavy neutrinos must lie at mass scale particular, the precise Higgs boson mass measurement M ∼ Oð1014 Þ determines the value of the quartic coupling in the scalar N GeV . This characterizes the case of genuine “high-scale” type-I seesaw constructions. We stress that potential at the electroweak scale and allows one to study ð3Þ ⊗ ð2Þ ⊗ ð1Þ SU c SU L U Y seesaw extensions can be for- its behavior all the way up to high energies. Given the “ ” measured values of Standard Model parameters such as the mulated with any number of right-handed neutrinos, since top quark and Higgs boson masses, we know that the Higgs they carry no anomaly. Here for definiteness, we start from quartic coupling remains perturbative after renormalization the minimalistic (3,1) model containing only one right- handed neutrino, in addition to the 3 known left-handed group equations (RGEs) are used to evolve it to high “ ” energies. However, the stability of the fundamental vacuum neutrinos [12]. We start from this missing partner seesaw, may fail at mass scales below the fundamental Planck aware that, by itself, it does not provide a fully realistic scale [3]. picture, since only one neutrino mass scale arises at the tree Another most important milestone in particle physics has level [12]. However, there are interesting and realistic variants where the “missing partner” seesaw scale can be been the discovery of neutrino oscillations [4,5]. This “ ” “ ” implies the existence of neutrino masses [6] and hence identified with the atmospheric scale , while the solar new physics that can produce them [7]. Electroweak mass scale is generated by radiative corrections that could arise, for example, from a “dark matter” sector [13]. vacuum stability can be substantially affected in the “ ” presence of a dynamical seesaw mechanism [8,9].1 We therefore take such missing partner seesaw as our Here we examine more closely the issue of the consis- reference scheme. tency of the Higgs vacuum within type-I seesaw extensions We show that, although it has better stability properties than the fully sequential (3,3) seesaw mechanism, for sizeable magnitudes of the Yukawa couplings the Higgs *[email protected] † potential of the minimal (3,1) seesaw becomes unstable [email protected] ‡ even below the seesaw scale. The situation can only get [email protected] 1Here we will focus on stability within high-scale seesaw. For worse by having more right-handed neutrinos, in the (3,2) discussions of low-scale seesaw see [10,11]. seesaw or in the “sequential” (3,3) seesaw. In this sense, too, it makes sense to take such a “missing partner” (3,1) Published by the American Physical Society under the terms of seesaw as the reference scheme. An important implication the Creative Commons Attribution 4.0 International license. of this missing partner type-I seesaw is the existence of a Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, lower bound on the neutrinoless double beta decay rate and DOI. Funded by SCOAP3. even for normal ordered neutrino masses [14].

2470-0010=2020=101(11)=115030(15) 115030-1 Published by the American Physical Society MANDAL, SRIVASTAVA, and VALLE PHYS. REV. D 101, 115030 (2020)

We then show explicitly that vacuum stability can be TABLE I. MS values of the main input parameters at the top ¼ 173 0 4 improved naturally if one implements spontaneous viola- quark mass scale, mt . GeV. tion of lepton number. This is characterized by the g1 g2 g3 y λ existence of a physical Nambu-Goldstone boson, dubbed t SM μð Þ majoron [15,16]. We show how the extra scalars required to mt 0.462607 0.647737 1.16541 0.93519 0.126115 implement spontaneous lepton number violation play a key role to improve stability properties. Indeed, their couplings can easily restore stability of the electroweak symmetry the high energy behavior of the Standard Model. As an breaking even if the lepton number violation scale is high, example, in this work we address the stability of Higgs as required to fit neutrino masses in this case. We also vacuum at energies far above the electroweak scale. analyze the scale at which instability sets in as a function of The detailed analysis of the Higgs vacuum within the – the magnitude of the Yukawa coupling relevant for gen- Standard Model has been carried out in [22 27].For erating neutrino mass in (3,1), as well as the conventional completeness here we revisit this analysis. This serves us (3,3) seesaw case. to calibrate our Renormalization Group analysis against This work is organized as follows: In Sec. II we revisit known results. Although in dedicated Standard Model the vacuum stability problem in the Standard Model studies there are some partial 3-loop results [27],to showing that the Higgs quartic coupling becomes negative compare the seesaw and Standard Model results it will when RGEs-evolved to high scales. In Sec. III, we describe suffice for us to stay at the two-loop level. In our analysis the neutrino mass generation in type-I seesaw and type-I we adopt the MS scheme, taking the parameter values at seesaw with majoron extensions. We then show in Sec. IV low scale as the input values [3]. In particular, the Higgs that the vacuum stability problem becomes worse in high- pole mass is taken as the current best fit value of ¼ 125 18 0 16 scale type-I seesaw Standard Model extensions. We then mH . . GeV, the top quark pole mass is ¼ 173 0 4 focus on the majoron extension of the canonical type-I taken as mt . GeV, and the strong coupling α ð Þ¼0 1184 0 0007 seesaw. We show how the majoron helps stabilize the Higgs constant s MZ . . . Using these exper- vacuum, which can be made stable all the way up to Planck imental values, we adopt the “On-Shell” renormalization scale. In addition the majoron could provide a viable dark scheme in order to express the renormalized parameters matter candidate [17–21], thereby solving another basic directly in terms of the physical observables and then relate problem in particle physics. Finally, we conclude and the on-shell parameters to the MS parameters in a way summarize our main results in Sec. VIII. similar to [27]. In Table I we list the MS input values of the relevant parameters at the top mass mt scale. II. HIGGS VACUUM IN THE STANDARD MODEL Taking the initial MS values of Table I as input values, Let us briefly revisit the status of the electroweak (EW) we then RGEs-evolve the Standard Model parameters to vacuum within the Standard Model. For a long time the higher scales as shown in Fig. 1. Higgs boson was the “last” missing piece of the theory. The Taking into account the updated input parameter values, our two-loop results are in good agreement with earlier discovery of a scalar particle with mass mH ≈ 125 GeV at the Large Hadron Collider (LHC) is very suggestive that it ones. Tiny differences arise mainly due to the increased could be the long-awaited Standard Model Higgs boson. current precision of the experimental numbers. We stress While further work is still needed to unambiguously establish this, current data indicates that its couplings and decay properties are close to the Standard Model Higgs expectations. If, indeed, this is the case, the next question is, given that so far we have not seen any evidence for new particles at the LHC, whether the Standard Model can be the final theory. The answer is obviously no, since the Standard Model predicts neutrinos to be massless and there is no viable Standard Model candidate for cosmological dark matter. For the moment we put these two issues aside, and ask ourselves whether there are other compelling hints that the Standard Model cannot be the final theory up to Planck scale. Indeed, there are several other theoretical and aesthetical arguments against this being the case. For example, achieving the unification of forces and the FIG. 1. The RGEs evolution of the Standard Model gauge improving the hierarchy/fine-tuning/naturalness problem. couplings g1, g2, g3, the top quark Yukawa coupling yt, and the “ ” λ However, the Higgs discovery has facilitated us to study quartic Higgs boson self-coupling SM.

115030-2 CONSISTENCY OF THE DYNAMICAL HIGH-SCALE TYPE-I … PHYS. REV. D 101, 115030 (2020) that an in-depth reanalysis of the Standard Model Higgs is as a result of postulating new mediator particles. A very not the goal of our paper, but rather the comparison of simple “UV-completion” is the type-I seesaw mechanism. Standard Model and seesaw scenarios. Hence, we refrain from performing a sensitivity analysis of Higgs vacuum B. Type-I seesaw mechanism stability and its dependence on the input parameter errors. The most general “type-I seesaw” mechanism is the one Indeed, in the seesaw scenarios of interest to us, such tiny ð3Þ ⊗ ð2Þ ⊗ ð1Þ formulated in terms of just the SU c SU L U Y effects are negligible when compared to the effects of the structure characterizing the Standard Model, without extra new Yukawa couplings. gauge symmetry [12]. One postulates the existence of Notice from Fig. 1 that the Standard Model Higgs quartic gauge singlet “right-handed neutrinos”, ν , i ¼ 1; 2; …n, coupling λ becomes negative at μ ≃ 1010 GeV. This Ri SM whose mass term is obviously gauge invariant. Neutrino would imply that the Higgs potential is unbounded from masses arise from the exchange of “right-handed neutrino” below and the Higgs vacuum would be unstable. A mediators whose multiplicity is arbitrary since, as gauge dedicated analysis shows that, in fact, the Standard singlets, they carry no anomaly. The relevant part of the Model Higgs vacuum is not unstable, but rather metastable2 Lagrangian is written as with a very long lifetime [27]. X 1 X ai ¯ a ˜ ij c −L ¼ Yν l HνR þ M ν νR þ H:c: ð2Þ L i 2 R Ri j III. NEUTRINO MASS GENERATION a;i i;j As already mentioned, the Standard Model cannot be the la ¼ðνa a ÞT ¼ 1 where L L;lL with a , 2, 3 denotes the three final theory up to the Planck scale, as it has massless families of left-handed lepton doublets, while i; j ¼ neutrinos and no viable candidate for dark matter. Hence 1; 2; …n labels the right-handed singlet neutrinos, and, the vacuum stability issue must be reconsidered. We now as before, H is the Standard Model Higgs doublet. After do this adopting simple seesaw extensions of the Standard electroweak symmetry breaking the full neutrino mass Model. We show that, above the seesaw scale, the Higgs matrix is expressed as vacuum stability can be completely dominated by the new couplings. Hence it suffices for our purposes to discuss 0 mD Mν ¼ ð Þ electroweak vacuum stability at the two-loop level. T ; 3 mD MR

A. Dimension-five operator ¼ pYνffiffi “ ” where mD 2 v is the Dirac mass matrix . Within the Standard Model neutrinos are massless. ð3Þ ⊗ ð2Þ ⊗ ð1Þ Being SU c SU L U Y invariant, the right- However, as first noted by Weinberg [28], nonzero masses “ ” ij handed neutrino Majorana mass matrix MR entries can will arise from an effective nonrenormalizable dimension- j ijj ≫ be much larger than the EW scale, MR v, implying five operator characterizing lepton number nonconserva- aj mD tion. The effective Lagrangian reads j ij j ≪ 1. Hence the mass matrix in Eq. (3) can be block- MR diagonalized perturbatively in an exponential series, 1 3 d¼5 ¯ T c Eq. (3.1) in [16]. The two diagonalized blocks correspond −Lν ¼ ðlLHÞ:κ:ðH l ÞþH:c:; ð1Þ 2 L to “light” and “heavy” neutrino mass matrices, denoted as ab ab mν and M , respectively, which can be written symboli- κ 3 3 N where is the × symmetric coupling matrix with cally as: negative mass dimension, and, for brevity, we have sup- h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i pressed the generation indices. When the electroweak 1 ab ab ¼ ab − 2 þ 4 2 ; symmetry breaking occurs, the Higgs gets vacuum expect- mν 2 MR MR mD h i¼pvffiffi ¼ 246 h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i ation value (vev) H 2 with v GeV, H being the 1 ab ab ¼ ab þ 2 þ 4 2 ð Þ Standard Model Higgs doublet. This leads to a Majorana MN 2 MR MR mD : 4 mass matrix for the left-handed neutrinos given as To leading order the mass matrix elements for light ab ab 2 neutrinos mν and heavy neutrinos M are given as ≡ κ v N mν 2 ; ab ≃− aið −1Þijð T Þjb þ ð Þ mν mD MR mD higher order terms; 5 which leads to light neutrino masses and lepton number ab ≃ ab þ aið −1Þijð T Þjb þ ð Þ violation by two units. There are many ways to generate κ MN MR mD MR mD higher order terms; 6

2Standard Model vacuum stability is sensitive to input param- 3We are now using the series expansion in Ref. [16] for the eter values, in particular the top-quark mass. case of explicit lepton number violation.

115030-3 MANDAL, SRIVASTAVA, and VALLE PHYS. REV. D 101, 115030 (2020) where the negative sign in (5) can be absorbed4 through field redefinition. The full expression for the diagonalizing matrix is found in [16], as Eq. (3.5). The light neutrino mass matrix in (5) is further diagonalized by a unitary matrix Uν in the light neutrino sector νa; a ¼ 1, 2, 3. This famous type-I seesaw formula links the smallness of the light neutrino masses to the heaviness of the right-handed neutrinos νR.

C. The missing partner type-I seesaw mechanism FIG. 2. Amplitude for neutrinoless double beta decay in As already mentioned, since νR’s are Standard Model gauge singlets, their number n need not match the number missing partner seesaw. The horizontal bands represent the limits of left-handed ones. Depending on the value of “n” many from current experiments, while the horizontal dashed lines show the maximum reach of future experiments (see text). possibilities can be envisaged. Here we consider the case n ≤ 3 of “high-scale” constructions.5 The observation of neutrino oscillations [4,5] proves atmospheric scale is seesaw-induced, while the solar scale that two of the three “active” neutrinos are massive [6]. has scotogenic origin [13]. However, there is so far no indication for a finite mass for An implication of the missing partner seesaw schemes is the lightest neutrino. Indeed, the Katrin experiment has a prediction for the parameter mββ describing the amplitude derived an upper limit of 1.1 eV (at 90% C.L.) on the for neutrinoless double beta decay versus the (relative) absolute mass scale of neutrinos [34] from the Tritium massive neutrino Majorana phase, shown in Fig. 2. endpoint spectrum. This bound applies irrespective of The lower band corresponds to normal mass ordering whether neutrinos are Dirac or Majorana particles. On and the upper one to inverted. Their narrow widths reflect Pthe other hand, cosmological observations indicate that the small allowed spread in neutrino oscillation parameters ma ≤ 0.12 eV [35,36]. Hence, for “sizeable” Yukawa [6]. Notice that, in contrast to the general “complete” (3,3) coupling values, Yν ∼ Oð1Þ, this bound is satisfied for seesaw, in the missing partner there can be no cancellation, 14 heavy neutrino masses MN ∼ Oð10 GeVÞ. so that nonzero neutrinoless double beta decay is predicted, In order to account for the current oscillation evidence even if neutrinos are normal-ordered. The horizontal bands for neutrino mass it suffices to have a “missing partner” in Fig. 2 show the reach of present experiments: CUORE seesaw mechanism with n ¼ 2, since in this case both solar (green, limits: 0.11–0.52 eV) [37], EXO-200 (grey, limits: and atmospheric scales can be produced by the seesaw. 0.147–0.398 eV) [38], Gerda-II (yellow, limits: 0.120– Following the general formulation in [12] we call such a 0.260 eV) [39] and KamLAND-Zen (cyan, limits: 0.061– scheme in which one left-neutrino has no right partner, 0.165 eV) [40]. The horizontal lines indicate the maximum (3,2) seesaw. In this case each right-handed neutrino estimated experimental sensitivities6 of upcoming experi- mediates the generation of the corresponding scale, solar, ments: SNOþ Phase-II (0.019 eV) [41], LEGEND − 1000 or atmospheric. (0.015) [42] and nEXO − 10 yr (0.0057) [43]. One sees Notice, however, that the minimal type-I seesaw mecha- from Fig. 2 that, although the upcoming experiments are nism is the one in which only one right-handed neutrino is only sensitive to inverted ordering, the detectability chan- added to the Standard Model. This (3,1) scheme is the ces improve in the “missing partner” as compared to the minimal “missing partner” seesaw, in which two left- expectations of a generic “complete” seesaw mechanism. neutrinos lack a right-partner and remain massless. It This brings hope that upcoming experiments may be able to can be viable as part of a bigger scheme in which, for measure, for the first time, the relevant Majorana phase. example, the solar scale arises radiatively, hence accounting To sum up, in what follows we take the missing partner for the small solar/atmospheric scale ratio. Such scheme is seesaw as our reference benchmark, because of its minimal- easily obtained by “cloning” the seesaw with some other ity and notational simplicity, and also because of the fact sector associated, for example, with dark matter. An that having extra fermions can only worsen stability of the interesting realization is the scotoseesaw, in which the Higgs potential. In addition, in the “complete” seesaw picture one looses the neutrinoless double beta decay 4Note that, even though this negative sign in (5) is physical, prediction in Fig. 2. related to the CP properties of the light neutrinos, for our purposes However, in Sec. VII we explicitly compare our results they will not be relevant. with those obtained for the sequential (3,3) seesaw. 5 Here we discard cases with n>3 since having extra fermions Moreover, all the relevant renormalization group equations can only worsen stability. An interesting example would be the (3,6) seesaw scheme, which includes the template for the sequential “low-scale” seesaw, including both the inverse seesaw 6Here we made the most optimistic assumptions concerning [29,30] and the linear seesaw mechanisms [31–33]. nuclear matrix element uncertainties.

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FIG. 3. Effect of Weinberg’s effective operator on the Higgs quartic interaction in the effective theory.

FIG. 4. One-loop correction to the Higgs quartic interaction in given in the Appendix assume the conventional (3,3) the full seesaw theory. seesaw picture.

B. Higgs vacuum stability in type-I seesaw IV. HIGGS VACUUM STABILITY AND NEUTRINO MASS In what follows, we will be mostly concerned with the effects of sizable Yukawa couplings in the context of We saw how the Standard Model vacuum is not high-scale missing partner seesaw and their impact on the absolutely stable. Instead, with the present measured values stability of the Higgs vacuum. Building upon the dis- of Higgs and top masses, it is metastable; as the quartic cussion of the previous section, we now turn to the region λ ΛSM ∼ 1010 coupling SM becomes negative around I GeV, above the scale μ ¼ M . In this case one has the full the Standard Model instability scale. Before discussing R theory in which the running of Yukawa coupling Yν will Higgs vacuum stability within type-I seesaw embeddings of have an impact on the running of the Higgs quartic the Standard Model we first consider the effective coupling which we now call λ. This is done so as to Weinberg operator. distinguish it from both the Standard Model case as well as from the regime where the renormalization group A. Dimension-five operator running is performed only with the effective Weinberg We now discuss the running of the quartic scalar operator. Within the type-I seesaw picture, below and μ ¼ coupling characterizing the EW symmetry breaking sector above the seesaw scale MR, there will be contribu- λ of the Standard Model in the context of a dimension-five tions on Higgs quartic coupling from the Figs. 3 and 4, λ operator picture that results effectively at low-energies from respectively. Hence, in order to describe the running of we need to take into account the matching condition at the a UV-complete type-I seesaw. Below the scale μ ¼ MR we scale μ ¼ MR. integrate out the heavy neutrino νR, so that the theory is the Standard Model plus an effective dimension-five For the reasons mentioned in Sec. III B here we focus on d¼5 −1 T the simplest missing partner type-I seesaw mechanism Weinberg operator Lν (with κ ¼ YνM Yν ). Below the R containing a single right-handed neutrino. It provides a scale μ ¼ MR, only the Standard Model couplings and κ will run. Neglecting the contribution from lepton and light clear picture of the impact of seesaw extensions on the Higgs vacuum stability in the simplest possible setting. quark Yukawa couplings, the one-loop RGEs are given by μ ¼ [44–46] (see also Ref. [10]), As we discussed earlier, below the MR scale, the theory is an effective Standard Model supplemented by

2 2 2 the dimension-five Weinberg operator. However, above the 16π βκ ¼ 6y κ − 3g2κ þ λκκ: ð7Þ t μ ¼ MR scale the theory is UV-complete, so that all the new couplings in the model like the neutrino Yukawa coupling As the Standard Model case, here yt also denotes the top Yν will take part in the system of renormalization group ð2Þ Yukawa coupling and g2 is the SU L gauge coupling. We equations and will affect the running of the Standard Model denote the Higgs quartic coupling in this case as λκ to couplings, especially that of λ. distinguish it from the pure Standard Model case. Hence, As a result, the stability of the electroweak vacuum will due to the large top Yukawa coupling yt, the coupling κ set a potential limit on how large Yν can be. As the new slowly increases with the energy scale μ. As seen in Fig. 3 Yukawa coupling Yν runs only above the threshold scale [10], the same operator which generates the neutrino MR, this can be technically implemented by replacing mass below the scale μ ¼ MR, also provides a correction Yν → Yνθðμ − MRÞ in the right hand side of the RGEs of to the Higgs quartic self-coupling λκ below that scale. The the full theory, given in Appendix B. Here θðxÞ¼1;x>0 contribution from the coupling κ on the running of and θðxÞ¼0;x<0 are the step functions. 2 2 the Higgs quartic coupling λκ is of order v κ and thus Integrating out the heavy neutrinos also introduces negligible [47]. As a result, in the effective theory, the threshold corrections to the Standard Model Higgs quartic λ μ ¼ running of λκ below the scale μ ¼ MR will be almost the coupling [48] at the scale MR. The tree level Higgs same as in the Standard Model. potential in the Standard Model is given by

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FIG. 5. The continuous (red) curve gives the evolution of the FIG. 6. Same as Fig. 5 but now with Yν ¼ 1 and MR ¼ Higgs quartic self-coupling within the minimal (3,1) Type I 3 × 1014. In contrast to Fig. 5, λ shows a marked deviation from λ ¼ 1 λ seesaw scheme. The gauge and Yukawa couplings g1, g2, g3, yt, SM as a result of the larger Yukawa value, Yν . As before, SM and Yν are also indicated by the dashed lines. The light neutrino and λκ nearly coincide as the Weinberg operator has negligible mass is fixed at mν ¼ 0.1 eV, corresponding to a heavy neutrino impact on RGEs of the quartic Higgs coupling. 13 mass MR of 7.5 × 10 GeV. For comparison we show the λ evolution of the Standard Model coupling SM, seen as the red dashed line. Finally, λκ denotes the Higgs quartic coupling in the 5 4 λðM Þ → λðM Þ − jYνj : ð Þ effective theory including neutrino mass through the Weinberg R R 32π2 11 operator, while λ is the corresponding quartic in the minimal missing-partner Type I seesaw theory. Having set up our basic scheme, let us start by looking at the impact of the right-handed neutrinos on the stability of the Higgs vacuum. As we discussed at length in Sec. II, the † † † 2 VðH HÞ¼−μHH H þ λðH HÞ : ð8Þ Standard Model RGEs running of the Higgs quartic scalar λ coupling SM are dominated by the top Yukawa, which is the largest coupling present in the theory. As we saw, in this This will get corrections from higher loop diagrams λ case the Standard Model SM coupling becomes negative of Standard Model particles and extra fermion from the around the scale μ ∼ 1010 GeV. However, within the see- ð Þ type-I seesaw. The one-loop effective potential V1 h has saw completion, above the scale μ ¼ M the neutrino the form R Yukawa couplings Yν of (2) can completely dominate the RGEs behavior of λ as shown in Figs. 5 and 6. Figures 5 SM ν and 6 illustrate the effect of the new neutrino Yukawa V1ðhÞ¼V1 ðhÞþV1ðhÞ; ð9Þ coupling Yν on various other couplings. For illustration we have taken two representative values of Yukawa couplings SM where V1 ðhÞ is the usual one loop Standard Model Yν ¼ 0.5 and 1. One sees how the problem of Higgs potential. The one loop potential from the neutrino sector vacuum stability becomes more acute in a type-I seesaw is given by [48,49] completion of the Standard Model. This was expected, since the addition of new fermions tends to destabilize the X 2 2 Higgs vacuum. Notice that, in the regime below the onset of 1 mν M ν 4 i 4 N the seesaw mechanism, μ ≤ M , the running of the Higgs V1ðhÞ¼− mν log þ θðμ − M ÞM log : R 32π2 i μ2 R N μ2 λ λ i quartic coupling κ nearly coincides with SM. This follows due to the negligible effect of the Weinberg operator on the ð10Þ running of the Higgs quartic coupling. The small negative shift in the λ running at the scale μ ¼ MR results from the The matching of the complete and effective theory at matching condition, which becomes clearly visible for threshold requires one to introduce a threshold contribution larger Yukawa couplings, Yν ∼ 1. Notice that in Figs. 5 1 M2 Δ ¼ − ð 4 N Þ and 6 we have chosen a larger seesaw scale, with below MR, THV 32π2 MN log 2 , whose expansion MR “ ” 2 correspondingly larger Dirac neutrino Yukawa coupling gives the threshold corrections to the μ and λ parameters values, in order to make the running coupling effects visible Δ μ2 ¼ 1 j j2 2 Δ λ ¼ − 5 j j4 as TH 16π2 Yν MR, TH 32π2 Yν . Hence, we in the plots. 10 −3 need to consider this shift in λ at μ ¼ MR when solving the For MN ≤ 10 GeV, the Yukawa coupling is Yν ≤ 10 , RGEs as hence too small to alter the running of λ significantly. As a

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ð3Þ ⊗ ð2Þ ⊗ ð1Þ result, the vacuum instability problem will persist. SU c SU L U Y gauge invariance, in the However, if regarded as an effective one, the theory remains unbroken phase, the theory is also invariant under lepton mathematically self-consistent. For larger MN, for example number. The above potential can develop a minimum for MN close to the unification scale, the type-I seesaw relation nonzero vacuum expectation values of both H and σ if λH, λ 4λ λ − λ2 (5) implies that Yν should also be sizeable. Such a large σ and H σ Hσ are all positive. The vevs break both the Yukawa coupling will have a destabilizing effect, worsen- electroweak and lepton number symmetries, three of the 7 ing the metastability of the Standard Model vacuum. In degrees of freedom are eaten by the massive Standard fact, now the vacuum can be completely unstable, making Model gauge bosons, while the imaginary part of the σ the model inconsistent. corresponds to the physical majoron J ¼ Imσ. The real In conclusion, in seesaw scenarios the stability properties parts of H and σ will mix with each other to give two CP- of the electroweak vacuum will at best be those of the even mass eigenstates h1 and h2. The lighter of these is Standard Model Higgs vacuum. In order to enhance Higgs identified with 125 GeV Higgs boson [1,2]. vacuum stability it is desirable to further extend or embed the type-I seesaw [50]. A natural way to do this is to assume spontaneous breaking of lepton number, as we do next. VI. VACUUM STABILITY IN TYPE-I SEESAW WITH MAJORON V. THE MAJORON COMPLETION Here we take again the simplest majoron extension of the type-I seesaw mechanism based on the (3,1) missing partner We now consider the type-I seesaw extensions of the 8 scheme considered above. We adopt the high-scale seesaw Standard Model, in which lepton number is promoted to a limit vσ ≫ v . In this limit, mass of the heavier CP-even spontaneously broken symmetry within the SUð3Þ ⊗ H c scalar boson and right neutrino are approximately given as SUð2Þ ⊗ Uð1Þ gauge framework [15,16]. In addition pffiffiffiffiffiffiffiffiffiffiffiffi L Y ≡ ≈ 2λ ≈ pYRffiffi mh2 M σvσ and MN vσ. The light and heavy to the right-handed neutrinos νR we add a complex scalar 2 singlet σ carrying two units of lepton number. The relevant Higgs sectors will be almost decoupled, though we can still λ α Lagrangian is given by allow appreciable Hσ with very small mixing angle , see X 1 X Appendix A. For simplicity, we consider nearly degenerate ai ¯ a ˜ ij c M and M , such that we have only one threshold scale L ¼ − Yν l HνR − Y σν νR þ H:c: ð12Þ N L i 2 R Ri j a;i i;j μ ¼ MN or M, below which the theory is an effective one. Above that scale we have the full theory with all the new ν ν The resulting neutrino mass matrices in L and R basis is couplings running. given by When going from energies above M to energies below M 0 1 we need to integrate out the massive scalar degree of 0 YpνvffiffiH B 2 C freedom at tree level as described in Ref. [53]. This leads to M ¼ @ A ð Þ ν T : 13 Ypν vffiffiH YpRvffiffiσ a tree level threshold effect that arises from the matching 2 2 conditions at the energy scale μ ¼ M. We will now briefly describe this procedure. We can write the scalar potential The effective light neutrino mass obtained by perturbative for the case of type-I seesaw with majoron extensions as diagonalization of the above mass matrix is of the form 2 2 2 2 2 v vσ −1 T vH ¼ λ † − H þ λ σ†σ − mν ≃ YνY Yν pffiffiffi : ð14Þ V0 H H H σ R 2 2 2 vσ 2 2 σ þ λ † − vH σ†σ − vσ ð Þ In the presence of the complex scalar singlet , the most Hσ H H 2 2 : 16 general Higgs potential that can drive electroweak and lepton number symmetry breaking is given by [51] In our case, vσ ≫ vH, therefore M is much larger than ðσ Þ¼−μ2 † − μ2σ†σ þ λ ð † Þ2 μ ¼ V ;H HH H σ H H H the Higgs mass mh1 . As a result, below the scale M,we † 2 † † can integrate out the field σ using the following equation of þ λσðσ σÞ þ λ σðH HÞðσ σÞ: ð15Þ H motion (apart from derivative terms)

This potential is bounded from below if λσ, λH and pffiffiffiffiffiffiffi 8 λHσ þ 2 λσλ are all positive. In addition to the standard Note that vacuum stability in a seesaw majoron model was discussed in [52]. However, the majoron in that paper was completely detached from the neutrino sector, lacking any solid 7 14 For example, if MN ∼ 10 GeV (which implies Yν ∼ 1 for motivation. Moreover, the low scale choice for vσ was artificial, mν ∼ 0.1 eV), the vacuum lifetime is less than the age of the requiring tiny Dirac Yukawa couplings. In our opinion, it is best universe τU, hence Standard Model metastability is worsened by to present the discussion within a genuine low-scale neutrino the effect of this large Yukawa coupling. mass generation mechanism, as in Ref. [10].

115030-7 MANDAL, SRIVASTAVA, and VALLE PHYS. REV. D 101, 115030 (2020) 2 2 ¼ λ0 † − vH ð Þ Veff H H H 2 ; 18 λ0 where H is identified as

λ2 λ0 ≡ λ ¼ λ − Hσ ð Þ H κ H : 19 4λσ

Notice that, since only the dimension-five Weinberg oper- FIG. 7. One-loop correction to the Higgs quartic interaction in μ ¼ type-I seesaw with majoron. ator is running below the scale M, one has that the λ0 λ running of H is essentially the same as that of κ in the effective type-I seesaw. Moreover, at tree-level the numeri- 2 2 0 † vσ † vH cal value of λ ðM Þ and λ ðM Þ is the same, since in both 2λσ σ σ − þ λ σ H H − ¼ 0 H Z SM Z 2 H 2 cases one must reproduce the 125 GeV Higgs mass. 2 2 Equation (19) suggests that the matching condition at the vσ λ σ v σ†σ ¼ − H † − H ð Þ μ ¼ 2 2λ H H 2 : 17 scale M induces a shift in the Higgs quartic coupling, σ λ2 δλ ¼ Hσ . This corresponds to a larger Higgs quartic coupling 4λσ above the scale M and improves the chances of keeping λH positive all the way. In order to obtain the effective potential below the scale In Appendix C, we give the two-loop RGEs of the full μ ¼ M, we use Eq. (17) in Eq. (16), leading to the effective theory. Note that below the scale M the RGEs are the ones Higgs potential expressed as β β λ λ0 with λHσ and λσ removed and H replaced by H. Above M

FIG. 8. Evolution of the scalar quartic couplings λH, λσ, and λHσ in type-I seesaw with majoron. We include in purple dashed color the Dirac Yukawa coupling Yν, see Eqs. (12) and (15). For comparison with Standard Model we have also shown the RGE for λSM, red λ0 dashed curve. Here H is the effective Higgs quartic coupling below the mass threshold of the heavy particles and is essentially the same λ λ0 λ as κ in the effective type-I seesaw, see Eq. (19). Since, in this regime the RGE of H differs from that of SM only due to the tiny λ0 λ contribution of the effective Weinberg operator, H and SM almost coincide with each other. See text for more detailed discussion of various key features of the plots.

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β β λ one needs to include λHσ , λσ and find H using the full RGEs with the boundary condition as in Eq. (19) at μ ¼ M. As far as the new Yukawa couplings are concerned, they can be obtained by substituting Yν ¼ θðμ − MNÞYν and YR ¼ θðμ − MÞYR on the right side of the RGEs of the full theory. Figure 7 shows that positive contribution to the RGEs of the Higgs quartic coupling (left panel) is accompanied by the destabilizing effect of RH neutrinos through the 1-loop diagram (right panel). Our results for the (3,1) type-I seesaw mechanism with majoron are shown in Fig. 8, where we have taken 9 M ≈ MN ≈ 10 GeV, such that the threshold effects start contributing positively to λH before the Standard Model Λ ≈ 1010 λ ¼ 0 1 instability scale SM GeV. We have taken σ . at the scale M. The renormalization group evolution in FIG. 9. Zoomed view of the evolution of the quartic Higgs self- coupling λ in the Standard Model (red-dashed) and in the (3,1), Fig. 8 is shown for four Yukawa coupling values Yν ¼ 0.1, (3,2), and (3,3) seesaw extensions (blue dot-dashed, magenta- 0.3, 0.4, and 0.5. It shows that, indeed, the stability dashed, and green solid, respectively). In the (3,1) case we have properties can substantially improve due to the presence aj taken Yν ¼ 0.5, while for ð3;n¼ 2; 3Þ we took Yν ¼ 0.5; a ¼ of the new scalar. In fact, for appreciable Yukawa cou- aj j ¼ 1; …;n and Yν ¼ 0 for a ≠ j. plings, one can have positive λH all the way up to Planck scale. For the Yukawa couplings Yν ¼ 0.1, 0.3, 0.4, and 0.5, the required values of minimum λ σ are 0.08, 0.1, 0.12, aj H value of Yν ¼ 0.5; a ¼ j ¼ 1, 2, 3 and taken the off- and 0.15, respectively. diagonal terms to be zero for the (3,3) case. Note that such a choice is unrealistic vis-a-vis neutrino oscillation data. VII. COMPARING STANDARD AND MISSING However, taking the Yukawa texture consistent with PARTNER TYPE-I SEESAW neutrino oscillation data will not change our conclusions. Therefore, for the sake of simplicity, we have taken this So far we have taken the missing partner seesaw simple choice. mechanism based on the (3,1) construction as our bench- In contrast, going to the (3,3) majoron type-I seesaw with mark. This choice was made for the reasons given at the end three right handed neutrinos, we find that the Higgs vacuum of Sec. III C. Such scheme can be made phenomenologi- can be still kept stable up to Planck scale for appreciable cally viable in the presence of radiative corrections asso- 9 Yukawa couplings. Of course, the presence of additional ciated, for example, to a dark matter completion. ai fermions means that the maximum values of Yν , for which Here we compare the stability properties of this simplest Higgs vacuum stability can be achieved up to Planck scale, benchmark with those of a missing partner seesaw based on (3,2) construction and with those of the standard (3,3) is somewhat reduced. In Fig. 10 we compare the Higgs type-I seesaw mechanism. For completeness we also compare with the Standard Model stability results. As already mentioned, the problem of Higgs vacuum stability in type-I seesaw extensions only gets worse with the addition of extra right-handed neutrinos. This fact is clearly illustrated in Fig. 9, where we compare the evolution of the Higgs quartic self-coupling λ within the Standard Model within the (3,n) seesaw completions, with n ¼ 1, n ¼ 2, and n ¼ 3. Note that, for the general ð3;nÞ 2 † seesaw scheme jYνj should be replaced as TrðYνYνÞ and Eq. (11) should be replaced as

2 5n † 2 λðM Þ → λðM Þ − ðYνYνÞ ; ð Þ R R 32π2 Tr 20 FIG. 10. Zoomed view of the evolution of the quartic Higgs where n is the number of right handed neutrinos. self-coupling λ in the Standard Model (red-dashed) and (3,1), For simplicity, in Fig. 9, we have fixed the benchmark (3,2), and (3,3) majoron seesaw (blue dot-dashed, magenta- dashed, and green solid, respectively). In the (3,1) case we have aj 9 ¼ 0 3 ð3 ¼ 2 3Þ ¼ 0 3 ¼ An explicit scotogenic model of this type has been proposed taken Yν . , while for ;n ; we took Yν . ; a aj in [13]. j ¼ 1; …;n and Yν ¼ 0 for a ≠ j.

115030-9 MANDAL, SRIVASTAVA, and VALLE PHYS. REV. D 101, 115030 (2020) vacuum stability of the (3,3) majoron seesaw case with its one can show that it can provide a viable warm dark matter (3,1) analogue as well as with Standard Model. candidate. It decays to neutrinos, with a tiny strength For Fig. 10 we have taken Yν ¼ 0.3 for the (3,1) case, proportional to their mass [16]. Hence, it is naturally ai while for the (3,3) case we have taken Yν ¼ 0.3; long-lived on a cosmological scale, as required, with a ¼ i ¼ 1, 2, 3, with all the off-diagonal entries taken to lifetime τJ larger than the age of the Universe 17 zero. The remaining parameters are kept the same as t0 ¼ 13.8 Gyr ≃ 4 × 10 s. Such majoron dark matter described previously for the (3,1) majoron seesaw case. scenario has been shown to be consistent with cosmic In short, phenomenologically realistic type-I seesaw microwave background data for adequate choices of the majoron models can have a stable vacuum all the way relevant parameters [18,20,55], the majoron decay lifetime up to Planck scale. constraints ranging from τJ > 50–160 Gyr. Using N-body simulations one can also show that majoron dark matter VIII. SUMMARY AND OUTLOOK provides a viable alternative to the ΛCDM scenario, with predictions that can differ substantially on small scales We have examined the consistency of electroweak [21]. Finally we also mention that, in addition to dark symmetry breaking within the context of the simplest matter, the majoron picture may also provide new insights high-scale type-I seesaw mechanism. We have derived to other cosmological challenges of the Standard Model, the full two-loop RGEs for the relevant parameters, such such as inflation [56] and leptogenesis [57]. as the quartic Higgs self-coupling λ of the Standard Model within the schemes of interest. These are compared, for calibration, with the Standard Model results. The addition ACKNOWLEDGMENTS of fermionic fields like “right-handed” neutrinos, has a This work is supported by the Spanish Grants destabilizing effect on the Higgs boson vacuum. For the No. FPA2017-85216-P (AEI/FEDER, UE), No. PROMETEO/ simplest type-I seesaw with bare mass term for the right- 2018/165 (Generalitat Valenciana), and the Spanish Red handed neutrinos, we found that for sizeable Yukawa Consolider MultiDark FPA2017-90566-REDC. We thank couplings the Higgs quartic self-coupling λ becomes Martin Hirsch and Werner Porod for useful discussions. negative much before reaching the seesaw scale. For such “large” Yukawas the type-I seesaw may be inconsistent even as an effective theory. We have taken as our simplest APPENDIX A: HIGGS SECTOR IN benchmark neutrino model the “incomplete” (3,1) seesaw MAJORON MODEL scheme with a single right-handed neutrino, as it has the The scalar potential for the majoron type-I seesaw is “best” stability properties within the class of high-scale given by, type-I seesaw schemes. We compared this case, in which ¼ −μ2 † − μ2σ†σ þ λ ð † Þ2 þ λ ðσ†σÞ2 only one oscillation scale is generated at tree level, with the V HH H σ H H H σ “higher” (3,2) type-I seesaw, in which the other mass scale † † þ λ σðH HÞðσ σÞ: ðA1Þ also arises from the tree level seesaw mechanism. In both H “missing partner” type-I seesaw schemes, (3,1) and (3,2), The Standard Model gauge singlet scalar σ carries two units the neutrinoless double beta decay prediction given in of lepton number and its vev hσi¼pvσffiffi breaks the lepton Fig. 2 holds. We also studied the stability of the electro- 2 ð1Þ Z weak vacuum for the canonical sequential (3,3) type-I number symmetry U L to a 2 subgroup. After sym- seesaw, in which all three neutrinos get tree-level mass. We metry breaking one has, in the unitary gauge showed how the stability properties improve in the case of 0 1 0 vσ þ σ spontaneous lepton number violation due to the presence of H → pffiffiffi ; σ → pffiffiffi : ðA2Þ 2 þ 0 2 a Nambu-Goldstone boson, the majoron. vH h To sum up, our results show how, in contrast to the type-I The scalars h0 and σ0 will mix with each other, their mass seesaw with explicit breaking of a lepton number, the eigenvalues are given by, majoron version can have stable electroweak vacuum all the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ λ 2 þ λ 2 − ðλ 2 − λ 2Þ2 þðλ Þ2 way up to Planck scale for reasonable Yukawa coupling mh1 HvH σvσ HvH σvσ HσvHvσ ; choices. Thus, the majoron completion of type-I seesaw ðA3Þ schemes can be considered as fully consistent theories. Before concluding we should note the cosmological qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ λ 2 þ λ 2 þ ðλ 2 − λ 2Þ2 þðλ Þ2 advantages of the majoron completion. The first is that it mh2 HvH σvσ HvH σvσ HσvHvσ : can also provide a dark matter candidate, namely the ð Þ majoron [17], providing an alternative to the ΛCDM A4 0 0 paradigm. The majoron is assumed to get mass from The mass eigenstates h1, h2 are related to the fields h ; σ gravitational effects that explicitly violate the global lepton by the mixing matrix parameterized by the angle α and number [54]. Assuming that its mass lies in the keV range given by

115030-10 CONSISTENCY OF THE DYNAMICAL HIGH-SCALE TYPE-I … PHYS. REV. D 101, 115030 (2020) 0 dc 1 ð1Þ 1 ð2Þ h1 cos α − sin α h ≡ β ¼ β þ β ¼ ð Þ c 2 c 2 2 c ; 0 ; A5 dt 16π ð16π Þ h2 sin α cos α σ βð1Þ βð2Þ where the mixing angle α is given by, where c are the one-loop RGEs corrections and c are the two-loop RGEs corrections. λ σv vσ sin 2α ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH H ; ðλ 2 − λ 2Þ2 þðλ Þ2 HvH σvσ HσvHvσ 1. Higgs quartic scalar self coupling 2 2 λ v − λσvσ cos 2α ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH H : ðA6Þ For the most general (3,n) seesaw the one-loop and two- ðλ 2 − λ 2Þ2 þðλ Þ2 HvH σvσ HσvHvσ loop RGEs corrections to the Higgs quartic self-coupling are given by: One can see from (A6) that in the limit vσ ≫ vH the α → 0 mixing angle , irrespective of the value of the quartic 27 9 9 9 couplings. βð1Þ ¼þ 4 þ 2 2 þ 4 − 2λ − 9 2λ þ 24λ2 λ 200 g1 20 g1g2 8 g2 5 g1 g2 2 † 4 † † APPENDIX B: RGEs: TYPE I SEESAW þ 12λyt þ 4λTrðYνYνÞ − 6yt − 2TrðYνYνYνYνÞ; We have used the package SARAH [58] to do the RGEs ðB1Þ analysis in our work. The β function of a given parameter c is given by

3411 1677 289 305 1887 117 73 108 βð2Þ ¼ − 6 − 4 2 − 2 4 þ 6 þ 4λ þ 2 2λ − 4λ þ 2λ2 λ 2000 g1 400 g1g2 80 g1g2 16 g2 200 g1 20 g1g2 8 g2 5 g1 171 63 9 17 45 þ 108 2λ2 − 312λ3 − 4 2 þ 2 2 2 − 4 2 þ 2λ 2 þ 2λ 2 þ 80 2λ 2 − 144λ2 2 g2 100 g1yt 10 g1g2yt 4 g2yt 2 g1 yt 2 g2 yt g3 yt yt 9 3 3 3 15 − 4 ð †Þ − 2 2 ð †Þ − 4 ð †Þþ 2λ ð †Þþ 2λ ð †Þ − 48λ2 ð †Þ 100 g1Tr YνYν 10 g1g2Tr YνYν 4 g2Tr YνYν 2 g1 Tr YνYν 2 g2 Tr YνYν Tr YνYν 8 − 2 4 − 32 2 4 − 3λ 4 − λ ð † †Þþ30 6 þ 10 ð † † †Þ ð Þ 5 g1yt g3yt yt Tr YνYνYνYν yt Tr YνYνYνYνYνYν : B2

2. Yukawa Couplings

The one-loop and two-loop RGEs corrections to Yν in the (3,n) seesaw are given by ð1Þ 3 † 2 9 2 9 2 † β ¼ YνYνYν þ Yν 3y − g − g þ TrðYνYνÞ ; ðB3Þ Yν 2 t 20 1 4 2

ð2Þ 1 2 † 2 † † † † † 2 β ¼ ð279g YνYνYν þ 675g YνYνYν − 960λYνYνYν þ 120YνYνYνYνYν − 540YνYνYνy Yν 80 1 2 t † † 4 2 2 4 2 2 2 2 2 − 180YνYνYνTrðYνYνÞþ2Yνð21g1 − 54g1g2 − 230g2 þ 240λ þ 85g1yt þ 225g2yt 2 2 2 † 2 † 4 † † þ 800g3yt þ 15g1TrðYνYνÞþ75g2TrðYνYνÞ − 270yt − 90TrðYνYνYνYνÞÞÞ: ðB4Þ

The RGEs corrections to the top-Yukawa coupling yt are given by ð1Þ 3 3 2 2 17 2 9 2 † β ¼ y þ y 3y − 8g − g − g þ TrðYνYνÞ ; ðB5Þ yt 2 t t t 3 20 1 4 2

ð2Þ 1 5 3 2 † 2 2 2 β ¼þ ð120y þ y ð1280g − 180TrðYνYνÞþ223g − 540y þ 675g − 960λÞÞ yt 80 t t 3 1 t 2 1187 9 23 19 17 45 þ 4 − 2 2 − 4 þ 2 2 þ 9 2 2 − 108 4 þ 6λ2 þ 2 2 þ 2 2 yt 600 g1 20 g1g2 4 g2 15 g1g3 g2g3 g3 8 g1yt 8 g2yt 3 15 27 9 þ 20 2 2 þ 2 ð †Þþ 2 ð †Þ − 4 − ð † †Þ ð Þ g3yt 8 g1Tr YνYν 8 g2Tr YνYν 4 yt 4 Tr YνYνYνYν : B6

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APPENDIX C: RGEs: TYPE I SEESAW WITH MAJORON

1. Quartic scalar couplings

The scalar sector of the majoron model is given in Eq. (15). It contains three scalar quartic couplings λH; λHσ; λσ whose one-loop and two-loop RGEs are given by

ð1Þ 27 4 9 2 2 9 4 2 9 2 2 2 2 † 4 † † βλ ¼þ g1 þ g1g2 þ g2 þ λ σ − g1λH − 9g2λH þ 24λ þ 12λHyt þ 4λHTrðYνYνÞ − 6yt − 2TrðYνYνYνYνÞ; ðC1Þ H 200 20 8 H 5 H

ð2Þ 3411 6 1677 4 2 289 2 4 305 6 3 1887 4 117 2 2 73 4 βλ ¼ − g1 − g1g2 − g1g2 þ g2 − 4λ σ þ g1λH þ g1g2λH − g2λH H 2000 400 80 16 H 200 20 8 108 171 63 − 10λ2 λ þ 2λ2 þ 108 2λ2 − 312λ3 − λ2 ð Þ − 4 2 þ 2 2 2 Hσ H 5 g1 H g2 H H HσTr YRYR 100 g1yt 10 g1g2yt 9 17 45 9 3 − 4 2 þ 2λ 2 þ 2λ 2 þ 80 2λ 2 − 144λ2 2 − 4 ð †Þ − 2 2 ð †Þ 4 g2yt 2 g1 Hyt 2 g2 Hyt g3 Hyt Hyt 100 g1Tr YνYν 10 g1g2Tr YνYν 3 3 15 − 4 ð †Þþ 2λ ð †Þþ 2λ ð †Þ − 48λ2 ð †Þ − 3λ ð † Þ 4 g2Tr YνYν 2 g1 HTr YνYν 2 g2 HTr YνYν HTr YνYν HTr YRYνYνYR 8 − 2 4 − 32 2 4 − 3λ 4 − λ ð † †Þþ2 ð † † Þþ2 ð † T Þ 5 g1yt g3yt Hyt HTr YνYνYνYν Tr YRYνYνYνYνYR Tr YRYνYνYRYν Yν 6 † † † þ 30yt þ 10TrðYνYνYνYνYνYνÞ; ðC2Þ

ð1Þ 9 2 9 2 2 † βλ ¼ − g1λHσ − g2λHσ þ 4λ σ þ 8λHσλσ þ 12λHσλH þ λHσTrðYRY Þþ2λHσTrðYνYνÞ Hσ 10 2 H R þ 6λ 2 − 4 ð † Þ ð Þ Hσyt Tr YRYνYνYR ; C3

ð2Þ 1671 4 9 2 2 145 4 3 2 2 2 2 3 2 2 βλ ¼þ g1λHσ þ g1g2λHσ − g2λHσ þ g1λ σ þ 3g2λ σ − 11λ σ − 48λ σλσ − 40λHσλσ Hσ 400 8 16 5 H H H H 72 þ 2λ λ þ 72 2λ λ − 72λ2 λ − 60λ λ2 − 2λ2 ð Þ − 8λ λ ð Þ 5 g1 Hσ H g2 Hσ Hσ H Hσ H HσTr YRYR Hσ σTr YRYR 17 45 3 þ 2λ 2 þ 2λ 2 þ 40 2λ 2 − 12λ2 2 − 72λ λ 2 þ 2λ ð †Þ 4 g1 Hσyt 4 g2 Hσyt g3 Hσyt Hσyt Hσ Hyt 4 g1 HσTr YνYν 15 7 þ 2λ ð †Þ − 4λ2 ð †Þ − 24λ λ ð †Þþ λ ð † Þ 4 g2 HσTr YνYν HσTr YνYν Hσ HTr YνYν 2 HσTr YRYνYνYR 3 27 9 − λ ð Þ − λ 4 − λ ð † †Þþ14 ð † † Þ 2 HσTr YRYRYRYR 2 Hσyt 2 HσTr YνYνYνYν Tr YRYνYνYνYνYR 41 39 þ ð † Þþ8 ð † T Þþ ð † Þ ð Þ 8 Tr YRYνYνYRYRYR Tr YRYνYνYRYν Yν 8 Tr YRYRYRYνYνYR ; C4

βð1Þ ¼ 20λ2 þ 2λ2 þ 2λ ð Þ − ð Þ ð Þ λσ σ Hσ σTr YRYR Tr YRYRYRYR ; C5

ð2Þ 12 2 2 2 2 3 2 3 2 2 2 β ¼þ g1λ σ þ 12g2λ σ − 8λ σ − 20λ σλσ − 240λσ − 20λσTrðY Y Þ − 12λ σy λσ 5 H H H H R R H t 13 − 4λ2 ð †Þ − 6λ ð † Þþλ ð Þþ ð † Þ HσTr YνYν σTr YRYνYνYR σTr YRYRYRYR 4 Tr YRYνYνYRYRYR 3 þ ð † Þþ4 ð Þ ð Þ 4 Tr YRYRYRYνYνYR Tr YRYRYRYRYRYR : C6

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2. Yukawa Couplings

The one-loop and two-loop RGEs of the Yukawa couplings Yν, yt and YR are given by ð1Þ 1 † 2 9 2 9 2 † β ¼þ ð3YνYνYν þ YνY Y ÞþYν 3y − g − g þ TrðYνYνÞ ; ðC7Þ Yν 2 R R t 20 1 4 2

ð2Þ 1 2 † 2 † † † † † β ¼ ð279g YνYνYν þ 675g YνYνYν − 960λ YνYνYν − 160λ σYνY Y þ 120YνYνYνYνYν − 10YνY Y YνYν Yν 80 1 2 H H R R R R − 10 þ 140 T − 30 ð Þ − 540 † 2 − 180 † ð †Þ YνYRYRYRYR YνYRYν YνYR YνYRYRTr YRYR YνYνYνyt YνYνYνTr YνYν þ 2 ð21 4 − 54 2 2 − 230 4 þ 20λ2 þ 240λ2 þ 85 2 2 þ 225 2 2 þ 800 2 2 þ 15 2 ð †Þ Yν g1 g1g2 g2 Hσ H g1yt g2yt g3yt g1Tr YνYν þ 75 2 ð †Þ − 30 ð † Þ − 270 4 − 90 ð † †ÞÞÞ ð Þ g2Tr YνYν Tr YRYνYνYR yt Tr YνYνYνYν ; C8 ð1Þ 3 3 2 2 17 2 9 2 † β ¼ y þ y 3y − 8g − g − g þ TrðYνYνÞ ; ðC9Þ yt 2 t t t 3 20 1 4 2

ð2Þ 1 5 3 2 † 2 2 2 β ¼þ ð120y þ y ð1280g − 180TrðYνYνÞþ223g − 540y þ 675g − 960λ ÞÞ yt 80 t t 3 1 t 2 H 1187 9 23 19 1 17 þ 4 − 2 2 − 4 þ 2 2 þ 9 2 2 − 108 4 þ λ2 þ 6λ2 þ 2 2 yt 600 g1 20 g1g2 4 g2 15 g1g3 g2g3 g3 2 Hσ H 8 g1yt 45 3 15 3 27 9 þ 2 2 þ 20 2 2 þ 2 ð †Þþ 2 ð †Þ − ð † Þ − 4 − ð † †Þ ð Þ 8 g2yt g3yt 8 g1Tr YνYν 8 g2Tr YνYν 4 Tr YRYνYνYR 4 yt 4 Tr YνYνYνYν ; C10

1 ð1Þ † T β ¼ YRTrðYRY ÞþYRYνYν þ YRY YR þ Yν YνYR; ðC11Þ YR 2 R R 1 ð2Þ 2 T 2 T T β ¼ ð−320λσYRY YR þ 51g1Yν YνYR þ 255g2Yν YνYR − 160λHσYν YνYR YR 40 R − 10 † † − 10 † þ 70 − 10 T YRYνYνYνYν YRYνYνYRYR YRYRYRYRYR YRYRYν YνYR þ 160 T † − 10 T T − 30 ð Þ − 180 T 2 Yν YνYRYνYν Yν YνYν YνYR YRYRYRTr YRYR Yν YνYRyt † 2 2 2 † T † þ YRYνYνð−160λHσ − 180yt þ 255g2 þ 51g1 − 60TrðYνYνÞÞ − 60Yν YνYRTrðYνYνÞ þ 10 ð16λ2 − 3 ð Þþ4λ2 − 6 ð † ÞÞÞ ð Þ YR σ Tr YRYRYRYR Hσ Tr YRYνYνYR : C12

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