Is the Higgs Boson Composed of Neutrinos?

Home , Infrared fixed point, Landau pole, Topcolor

Jens Krog1, ∗ and Christopher T. Hill2, † 1CP3-Origins, University of Southern Denmark Campusvej 55, 5230 Odense M, Denmark

2Fermi National Accelerator Laboratory P.O. Box 500, Batavia, Illinois 60510, USA

(Dated: January 16, 2020) We show that conventional Higgs compositeness conditions can be achieved by the running of large Higgs-Yukawa couplings involving right-handed neutrinos that become active at ∼ 1013 − 1014 GeV. Together with a somewhat enhanced quartic coupling, arising by a Higgs portal interaction to a dark matter sector, we can obtain a Higgs boson composed of neutrinos. This is a ”next-to-minimal” dynamical electroweak symmetry breaking scheme.

PACS numbers: 14.80.Bn,14.80.-j,14.80.Da

I. INTRODUCTION treatment indicates that a tt composite Higgs boson requires (i) a Landau pole at scale Λ in the running top HY Many years ago it was proposed that the top quark coupling constant, yt(µ), (ii) the Higgs-quartic coupling λH must also have a Landau pole, and (iii) compositeness Higgs-Yukawa (HY) coupling, yt, might be large and 4 conditions must be met, such as λH (µ)/gt (µ) → 0 and governed by a quasi-infrared-fixed point behavior of the 2 renormalization group [1, 2]. This implied, using the min- λH (µ)/gt (µ) →(constant) as µ → Λ, [5]. This predicts a imal ingredients of the Standard Model, a top quark mass Higgs boson mass of order ∼ 250 GeV with a heavy top quark of order ∼ 220 GeV, predictions that come within of order 220−240 GeV for the case of a Landau pole in yt at a scale, Λ, of order the GUT to Planck scale. In light a factor of 2 of reality. of the observed 173 GeV top quark mass, the fixed point While the tt minimal composite Higgs model is ruled prediction is seen to be within 25% of experiment. This out, it remains of interest to ask, “can we rescue an NJL– suggests that small corrections from new physics might RG composite Higgs boson scenario with new physics?” bring the prediction into a more precise concordance with and if so, “what are the minimal requirements of new experiment. physics needed to maintain a composite Higgs boson sce- One of the main interpretations of the quasi-infrared nario?” In the present paper we address this issue and fixed point was the compositeness of the Higgs boson. revisit a composite Higgs boson model based upon an In its simplest form, the Higgs boson was considered to attractive idea of S. P Martin, [9] (this has also been be a bound state containing a top and anti-top quark considered in a SUSY context by Leontaris, Lola and [3–6]. This was amenable to a treatment in a large-N Ross [10]). Martin pointed out that the top quark HY c i Nambu–Jona-Lasinio model [7], defined by a 4-fermion is sensitive to right-handed neutrinos, νR, that become interaction at a scale Λ, with a a large coupling constant, active in loops above the large Majorana mass scale, M. and a strong attractive 0+ channel. The theory requires The right-handed neutrinos are assumed to have HY cou- i drastic fine-tuning of quadratic loop contributions, which plings, yν ≥ O(1), and also have a Majorana mass of or- 13 is equivalent to a fine-tuning of the scale-invariant NJL der M ∼ 10 GeV, thus leading to the neutrino seesaw coupling constant (and may require a novel insight into model at low energies [11]. Turning on the neutrino loops will generally pull a large yt(mt) to a Landau pole at a arXiv:1506.02843v2 [hep-ph] 15 Jan 2020 scale symmetry, such as [8]). By tuning the NJL cou- 15 19 pling close to criticality, the Higgs boson mass becomes scale of order Λ ∼ 10 − 10 GeV, and the large top small, creating an infrared hierarchy between the com- quark mass becomes intertwined with neutrino physics positeness scale, Λ, and the electroweak scale embodied above M. The strong dynamics that forms the bound- in m . Tuning the coupling slightly supercritical yields state Higgs boson for us is the dominant large coupling, h i a vacuum instability and the Higgs boson acquires its yν . VEV. Martin’s model preserved some of the features of the Once the infrared hierarchy has been tuned, the re- tt composite Higgs model, but extends the Higgs com- maining structure of the theory is controlled by renor- positeness structure to become an entanglement of neu- malization group (RG) running of couplings [5]. The RG trinos and top quark. Martin’s model considered one large neutrino HY interaction, while we will presently consider Nf = 3 right-handed neutrinos with degener- ate HY couplings to the corresponding left-handed dou- ∗Electronic address: [email protected] blets, yν . Hence, our present model becomes a large-Nf †Electronic address: [email protected] fermion bubble NJL model as we approach the compos- 2

i iteness scale Λ. The yν become active above the scale M pling constant in our scheme is g and h < g/Nf . We and are chosen to be large enough to have Landau poles will have additional smaller couplings involving the other at the scale Λ. quarks associated with light fermion mass generation The top quark HY coupling is pulled up by the large and flavor physics, as well as charge conjugated terms i i jk C yν to a Landau pole, but we find that the ratio of like (LLiνR)g (νRjLLk) . These generate the charged yt(µ)/yν (µ) → (constant) as µ → Λ. This implies that lepton and quark masses and mixing angles, which we the top quark couples to the dynamics that forms the presently ignore. Higgs boson at the scale Λ, which we treat as a Nambu– We follow [5] and factorize the NJL interactions to Jona-Lasinio model, but this is merely a comparatively write: weak extension of the dynamics to give mass to the top 0 i 0 a 2 † (and presumeably all other quarks and leptons). The LΛ = gLiLHνR + g T aLHtR − Λ H H (2) Higgs doublet (Higgs scalar) in our scheme is primarily Here we define g0 = h2/g. Here we have introduced an composed of P L νi ,(P ν νi), where L = (ν , ` ) i iL R i i iL i i L auxillary field H that regenerates eq.(1) by H equation of is the left-handed lepton doublet, and summing over motion. This is the Lagrangian at the scale Λ, where the N = 3 generations. The top quark HY coupling is then f auxilliary field H will become the dynamical Higgs boson only a spectator to this physics. boundstate at lower energies. We have ignored terms of A second and important demand of an NJL-composite order g02 which are generated when H is integrated out Higgs model is the behavior of the running of the Higgs to recover eq.(1). quartic coupling, λ . It is show in ref.[5] that λ will H H We now use the RG to run the Lagrangian down to the have a Landau pole at the scale Λ, but λ (µ)/y4(µ) → 0 H t Majorana mass scale, M, of the right-handed neutrinos, and λ (µ)/y2(µ) → (constant) as µ → Λ. Engineer- H t using only fermion loops. The result is formally: ing this is more challenging issue than that of the Lan- dau poles in the Yukawa couplings, as we are confronted 0 2 2 † λeH † 2 by the small value of λH in the standard model, and L = ZH |DH| − Mf H H + (H H) M 2 the apparent RG behavior λH → 0 for scale of order 12 i 0 a C j ∼ 10 GeV. For general gauge-Yukawa theories contain- +[gLiLHνR + g T aLHtR + νRiMijνR + h.c.] (3) ing a scalar bilinear, the divergent behavior is readily obtainable [12], but for the single doublet of the standard where the Majorana mass matrix, Mij, is now incorpo- model, this is not easily constructed. rated by hand. The Higgs boson has acquired a loga- There is, however, a simple remedy available to us here: rithmic kinetic term and a quartic interacton due to the the Higgs portal interaction. The main point is that if fermion loops, and the Higgs mass has run quadratically: there exists new physics coupled to the Higgs via a portal Z = (4π)−2(g2N + g02N ) ln(Λ2/M 2) interaction, e.g., a sterile dark matter boson, then the H f c 2 2 −2 2 02 2 2 Higgs Yukawa coupling that we observe, λ is actually only Mf = Λ − (4π) (2g Nf + 2g Nc)(Λ − M ) effective, and is replaced typically by a larger value near −2 4 04 2 2 λeH = (4π) (2g Nf + 2g Nc) ln(Λ /M ) (4) the TeV scale due to the mixing via the portal interaction [13]. We presently exploit this mechanism. The quantities appearing in eq.(3) are, of course, un- We note that many authors have considered various renormalized. The renormalized couplings at the present neutrino-composite Higgs boson schemes, many in the level of approximation are: context of a fourth generation and some with overlap to 0 our present case [14]. We turn presently to a Nambu– g g λeH y = √ y = √ λ = (5) Jona-Lasinio schematic model of our mechanism. ν t H 2 ZH ZH ZH

2 2 and we see that in the large (Nf ,Nc) limit the ratio yν /yt II. NJL-MODEL is a constant. For simplicity, we take the Majorana mass matrix to be The eﬀective UV model we have in mind is a variation diagonal, M = diag(M1,M2,M3). In the large M/vweak on the Nambu–Jona-Lasinio model [7] and top conden- limit, where vweak ∼ 175 GeV, the masses of the three sation models [3–5]. We adapt this to a neutrino conden- light neutrino states are given by the seesaw mechanism: sate with the four-fermion interaction Lagrangian: y2v2 mi = ν weak , (6) g2 h2 ν M L0 = (L νi )(ν Lj )+ (L νi )(t T a)+h.c. (1) i Λ2 Li R Rj L Λ2 Li R Ra L Assuming that yν is ∼ O(1), and ∼ eV masses for the i i i 13 where LL = (ν , ` )L (νRi) are left-handed lepton dou- light neutrinos, we expect Mi ∼ 10 GeV. Thus, in blets (right-handed neutrino singlets), and TL (tR) is the RG evolution of the system, loops containing right- the top quark doublet (singlet); (i, j, ..) are generation handed neutrinos occur only above the scales Mi. As an i indices running to Nf = 3 and (a, b, ..) are color in- approximation, take the threshold of the νR in loops to dices running to Nc = 3. The dominant large cou- be at a common Majorana mass scale M. 3

2 Note that the renormalized λH = λeH /ZH has the lim- the form: 4 2 its λH /y → 0 and λH /y → (constant) as µ → Λ. The ν ν β 9 9 17 extent to which the top quark participates in the binding 2 yt 2 2 2 2 2 (4π) = yt − 8g3 + 3 θM (µ − M)yν − g2 − g1 , of the Higgs boson relative to the neutrinos is determined yt 2 4 12 by g02N /g2N which we assume is of order 1/N , and c f f 2 βyν 9 2 2 9 2 3 2 thus the dominant coupling at the UV scale is g2. While (4π) = θM (µ − M) yν + 3yt − g2 − g1 , yν 2 4 4 we could keep the order g02 terms in the factorization 2 βg1 41 2 2 βg2 19 2 of eq.(3), this would make a weakly boundstate doublet, (4π) = g1 , (4π) = − g2 H0 composed mainly of tt, but since g02 is subcritical g1 6 g2 6 this state would remain a heavy dormant doublet with 2 βg3 2 2 2 (4π) = −7g3 (9) m ∼ Λ . g3 Below the Majorana mass scale M the neutrinos de- couple and the only significant running in the fermion where g1, g2, and g3 are the gauge couplings of the U(1)Y , loop approximation is the top quark. The electroweak SU(2)L, and SU(3)c symmetries respectively, yt is the scale is tuned by the choice of critical couplings. The top HY coupling, and yν the HY coupling of the lepton doublets to right-handed neutrinos. We have introduced quadratic running to a zero mass Higgs boson, Mf2 = 0 defines the critical coupling: a step-function, θM = θ(µ − M), where θ(x) = 1; x ≥ 0 and θ(x) = 0; x < 0. The step-function models the threshold of the turn on of the right-handed neutrinos at M 2 g2N + g02N = 8π2 1 + (7) the scale of the Majorana mass matrix. f c Λ2 In Fig. 1 we demonstrate the running of the HY coupling for the top quark and the neutrino as described. The criticality, we assume, is due principally to the 2 We use the initial conditions for the gauge couplings large value of g and is only slightly modified by the g (m ) = 0.36, g (m ) = 0.65, g (m ) = 1.16, for the top quark. We then choose g2 slightly supercritical to 1 Z 2 Z 3 Z HY couplings: yt(mt) = 0.99, and yν (M) = 1, and produce the phenomenological tachyonic Higgs potential, for the masses m = 91.2 GeV, m = 173.2 GeV, and 2 2 Z t Mf = −MH ZH (Λ/MH ). M = 1013 GeV. The NJL model is schematic, and must itself be an The evolution in Fig. 1 clearly indicates the existence approximation to some new dynamics in the UV. This of a Landau pole for the HY couplings at a scale Λ ∼ 1020 structure suggests a new gauge interaction which leads to GeV, in accord with what one would expect if the Higgs eq.(1) upon Fierz rearrangement, in analogy to topcolor is a fermion pair condensate. models [15], as: The Landau pole of the neutrino HY coupling is seen to pull the top HY towards a Landau pole at Λ. The g2 g2 λA λA L νi ν Lj = − L γµL ν γ ν + ... , neutrino HY coupling is always significantly larger than Λ2 Li R Rj L Λ2 L 2 L R 2 µ R the top coupling for the scales where the perturbative (8) result is valid. For the displayed example we find the A where the Gell-Mann matrices λ now act on the flavor ratio y /y ≥ 3 for the region very close to the Landau 0 ν t indices. The g term then requires some extension of the pole. theory. A model such as this assigns an SU(3) gauge group to lepton family number, and therefore gauge 13 charges to the νRi, i.e., the νRi are no longer sterile. M = 10 GeV This would imply that the Majorana mass matrix must be generated by a VEV associated, e.g., with additional 5.0 SU(3) scalar fields. With νRi in the triplet represen- ation, this requires {3} and/or {6} scalar condensates, and would dictate the neutrino mass and mixing angle 2.0 structure. Construction of this kind of model will be yn done elsewhere. 1.0 yt 0.5 L III. YUKAWA SECTOR 100 106 1010 1014 1018 m GeV The above discussion is the Wilsonian renormalization group approach. To improve the calculation, we turn FIG. 1: The RG evolutions ofê the top (solid) and neutrino to the full RG equations which are used below the scale (dashed) HY couplings with contributions from right-handed Λ, together with the matching conditions dictated by the neutrinos for renormalization scales above the neutrino Ma- fermion bubble approximation [5]. The full RG equations jorana mass; µ > M = 1013 GeV. (for Nf = 3 these are a slight modification of ref.[9]) take 4

To verify the consistency of this behavior of the top exp quark, consider the region below, but near, Λ. Here the mΝ 1019 0.1 eV RG equations for top and neutrino HY couplings can be 0.3 eV 1.0 eV approximated in the large (Nf ,Nc) limit by: 1018

L GeV d ln y d ln y 17 (4π)2 t ≈ (4π)2 ν ≈ N y2 + N y2, (10) 10 d ln µ d ln µ f ν c t 1016 hence: 1015 d ln(y /y ) 13 14 14 16 (4π)2 t ν = 0 (11) 5´10 1´10 5´10 1´1015 5´10151´10 d ln µ M GeV

This implies that yν (µ)/yt(µ) → (constant), as we ap- FIG. 2: Numerical results displaying the relation between 0 proach the scale Λ. The ratio yt/yν ∼ g /g, so the role the Majorana mass and the scale associated with the Landau of the top quark role is only that of a spectator. pole for the neutrino HY coupling for different values of the In this simplified setup, inserting an experimental neu- neutrino mass. trino mass in (6) yields yν (M) as a function of M. For a chosen M, this value may be used as an initial con- dition in the RG equation for y , and the scale Λ may ν IV. SCALAR SECTOR be read off from the solution to the RG equations. A simple analytic estimate is given by setting to zero all couplings except yν , in which case one finds for the one In the minimal version of a single composite Higgs bo- loop solution son, the physical Higgs mass prediction is larger than the observed ∼ 125 GeV. The Higgs mass is controlled by the 2 (4πvweak) electroweak VEV, vweak, and the quartic coupling. The Λ = M exp exp . (12) 9 mν M Higgs compositeness conditions predict a Landau pole for the quartic scalar coupling at the compositeness scale Λ exp Here v is again the Higgs VEV and mν is the exper- [5]. However, the quartic coupling constant in the stan- imentally measured neutrino mass. The estimate (12) is dard model is to be too low to match these conditions, in good agreement with the full numerical solution due and indeed, appears to decrease with scale potentially, to the fact that the neutrino coupling itself is what drives becoming negative at ∼ 1012 GeV [16]. the divergence at Λ. The relation (12) also gives a lower To achieve compositeness of the Higgs boson, we em- bound on the possible compositeness scale Λmin for any ploy a simple modification by which the observed Higgs neutrino mass given by quartic coupling, λ, becomes only a low energy effective coupling, while the true quartic coupling, λH , is larger mexp −1 Λ ' 1.5 × ν × 1015 GeV. (13) and can have the requisite Landau pole. The actual quar- min eV tic coupling needs only be about 2× the observed λ to achieve this, but requires additional physics at the ∼ 1 We perform the numerical analysis as before using the TeV scale. RG equations above, and obtain the scale associated with We extend the scalar sector to include a complex sin- the Landau pole for different values of M given a specific glet [13], S and the new Higgs potential becomes: mass of the light neutrino states in the eV range. In Fig. 2 we show numerical results concerning the relation λ 2 λ 2 V = H H†H − v2 + S S†S − u2 between the Majorana mass and the Λ scale for different 2 2 values of the neutrino mass. The perturbative nature of + λ H†H − v2 S†S − u2 , (14) our analysis does not allow us to extrapolate to infinite HS coupling values, so we instead take the naive estimate where we have assigned the vacuum expectation values of the Λ scale to be defined by yν (Λ) = 30. We stress that this analysis is meant to provide a demonstration of hH†Hi = v2, hS†Si = u2. (15) principles rather than high precision results. Two distinct behaviors are exhibited in Fig. 2: For The VEVs (15) are the global minima of the potential smaller values of the Majorana mass, the scale Λ is very 2 when λH , λS > 0 and λH λS > λHS. sensitive to the choice of neutrino and Majorana mass. Expanding about the minimum of eq.(15), one finds This is due to the fact that yν (mt) is quite small for the mass matrix for the massive scalars to be these values, and more RG time is needed to run to the 2 2 Landau pole. For larger values of the masses, yν (mt) also ∂ V λH v λHSvu = 2 2 , grows large in accordance with (6) and the Landau pole ∂φiφj λHSvu λSu is shifted closer to the scale where the neutrino coupling becomes active in the RG equations. where φi refers to the direction of the vev in H and S. 5

13 The eigenvalues are M = 510 GeV , mΝ = 1 eV 100 2 2 2 m± = λH v + λSu ± κ, 10 p 2 2 2 2 2 2 where κ = (λH v − λSu ) + 4λHSv u . In the limit 2 2 where λH v λSu , the lightest state mostly resides 1 within H, and the mass can be approximated by ΛH

2 4 2 2 λHS 2 λH v 0.1 ΛHS mH = m− = 2 λH − v + O 2 . (16) λS λSu Λ 0.01 S The effective quartic coupling, measured from the Higgs 100 5 8 11 14 17 mass, is now: 10 10 10 10 10 Μ GeV λ2 λ = λ − HS , (17) H λ FIG. 3: The RG evolution for the quartic couplings for a S specific choice of initial conditions at the UV scale. The IR phenomenology features a large quartic for the Higgs, while which is intrinsically smaller than the coupling λH . Thus, the effective coupling leads to a light Higgs mass mH ∼ 130 the composite picture with a suitable Landau pole in λH is now possible. GeV.

A. Singlet scalar extension 18 yields a divergence of yν around Λ = 10 GeV, under the assumption that M = 5 × 1013 GeV. At the scale We now analyze the RG evolution of the full theory where yν (µ) = 10, we then define the initial conditions with an eye to the landau pole in Λ . Assuming S is H for the quartic couplings λH (µ) = 98, in accordance with an electroweak SU(2) singlet, and U(1) sterile, the RG Y (22), and the somewhat arbitrary choices λHS(µ) = 23, equations for the scalar sector are given by λS(µ) = 1.7. The assumed value for u = 1 TeV. The 2 2 2 4 4 IR phenomenology features a large value for the Higgs βλH = (12yt + 12 θM yν − 3g1 − 9g2)λH − 12(yt + θM yν ) quartic coupling λH ∼ 0.7, while the effective coupling is 3 4 3 2 2 9 2 2 2 considerably smaller λ ∼ 0.28 corresponding to a Higgs + g1 + g1g2 + g2 + 12λH + 2 θuλHS, (18) 4 2 4 mass mH ∼ 130 GeV. 3 9 β = 6y2 + 12 θ y − g2 − g2 + 6λ λ (19) The RG system involves some degree of tuning to en- λHS t M ν 2 1 2 2 H HS sure the proper behavior of the two new quartic couplings. Specifically, we must tune λ to be small, to +4 θu(λS + λHS)λHS , S 2 2 ensure a large correction in λ as seen in (16) while λHS βλS = 4λHS + 10 θuλS, (20) is also tuned, such that λH > λHS > λS is satisfied for where we have included the Heaviside function all RG scales, in order to ensure a valid value of λ at small scales. θu = θ(µ − u), to adjust for the fact that loops involving the S state are not taking into account for scales This model should merely serve as a proof of concept, below the vev hSi = u which generates the mass for the displaying the possibility that the UV behavior of the S state. Higgs quartic coupling can include a Landau pole. In To accommodate the composite scenario, as first de- this setup we have looked at the simplest possible scalar scribed in [5], both the quartic coupling and the HY cou- extension of the standard model with the standard Higgs pling for the condensating fermions must diverge at a mechanism in play for both scalars. The issues of tuned scale Λ. Furthermore, the nature with which the scalar scalar couplings may then be alleviated if a different becomes propagating at lower energy scales, sets the re- mechanism for symmetry breaking or a more complex quirement scalar sector is considered. For a large class of more general gauge-Yukawa theories, a composite limit due to four 4 fermion interactions at high energies is easily obtainable, lim λH /yν = 0, (21) µ→Λ as shown in [12], while we will focus on the simplest alternative solutions below. and we expect the common divergence to yield

2 lim λH /yν = O(1). (22) µ→Λ V. ALTERNATIVE SCALAR EXTENSIONS In Fig. 3 we demonstrate the evolution of the quartic coupling for a specific choice of initial conditions. We As we have introduced a tuning between the dimen- exp choose a mass for the active neutrino mν = 1 eV, which sionless coupling constants of the scalar sector in addi- 6 tion to the usual tuning for the Higgs mass parameter, consider a dark1 scalar doublet S, gauged under a new 2 it would be beneficial to find a mechanism to stabilize SU(2)X group . Since we want all mass scales to be the IR phenomenology towards changes in the initial UV generated dynamically, the potential is given as conditions. We expect that this might be found by con- λH † 2 † † λS † 2 necting the symmetry breaking mechanisms for the scalar V = (H H) + λHS(H H)(S S) + (S S) , (23) sector. 2 2

In the previous example, the role of the “portal” cou- where we will investigate the cases where λHS < 0. Just pling λHS was to supply a correction to the quartic Higgs as before, the requirement for stability of the potential is coupling in the eﬀective coupling by connecting the two 2 scalar sectors, while the symmetry breaking mechanism λH > 0, λS > 0, λH λS > λHS. (24) is that of the standard Higgs boson for both the H and Spontaneous symmetry breaking then occurs dynami- S scalars. cally in this setup via the Coleman-Weinberg mechanism when the RG evolution brings the system of coupling constants into violation of the stability conditions (24). The driving force behind the symmetry breaking in this A. Negative portal coupling setup is the new gauge coupling gx, related to the SU(2)X gauge symmetry. As this coupling becomes large at some We can expand the role of the portal coupling by scale due to asymptotic freedom, the quartic coupling λS letting the portal interaction communicate symmetry will be driven negative in the IR, due to the form of its breaking in the dark S sector to the standard model. beta function which is positive for any nonzero value of Setting λHS < 0 and assuming hSi= 6 0 can trigger spon- the couplings: taneous symmetry breaking in the standard model, even 9 if the mass term for the Higgs m2 ≥ 0, since the portal 2 2 4 2 H βλS = 4λHS + 12λS + gx − 9gxλS. (25) interaction will add a negative squared mass contribution 4 ∗ for H. If the portal coupling is very small, then there can Denoting by s the scale at which λS = 0, and per- be a large hierarchy between the vevs of S and H, and forming the approximation close to this scale that λS ' the validity of eq.(16) is guaranteed. s βλS ln s∗ , the estimated value for the vev of S coming The change from a positive portal coupling to a neg- from the associated Coleman-Weinberg symmetry break- ative one can thus change the nature of the symmetry ing mechanism is given by breaking for the Higgs particle. It allows for other val- ∗ −1/4 ues of the Higgs mass parameter, and speciﬁcally one hSi = u = s e . (26) can choose m2 = 0 and still obtain a second order phase H In return, the negative portal coupling λHS induces a vev transition due to the portal interaction. The actual anal- for H: ysis of this alternative model is however almost identical r to the original, since the stability constraint and mass −λHS 2 hHi = v = u . (27) prediction only involves λHS. The measured Higgs mass λH is still obtainable together with a Landau pole for λH , albeit tuning between the scalar couplings is needed. At this minimum, the mass matrix takes the form √ 2 √2λH −2 −λH λHS v λH . (28) −2 −λH λHS λHS − βλ S λHS B. Communicated CW symmetry breaking Assuming that v2 u2 which is to say −λHS 1, we λH may expand the eigenvalues to the leading order in λHS λH Common to the scalar sectors discussed so far has been and obtain the feature that a mass scale has been inserted by hand into the potential, either for both scalars, or for one of 2 2 2 βλS λH 2 m1 = 2λ , m2 = − v , (29) them. This enables the generation of a vast interval of λHS possible scalar masses, but intrinsically means that these where the indices 1 and 2 relate to the state composed of are very sensitive to the input parameters. An alterna- 2 λHS mostly H and S respectively, and λ = λH − . This tive way to generate mass scales is the dynamical one, βλS where the mass scales arise directly from the RG evolution. We will show in the following that Landau poles in the quartic couplings, in accordance with a composite picture, may also accommodate spontaneous symmetry 1 Similar models with a portal coupling to another scalar sector are often used to probe dark matter phenomenology. breaking due to the CW mechanism as demonstrated for 2 elementary scalars in [17]. The critical property of the gauge group is asymptotic freedom, so any other gauge group with this property could have been It is central to the success of this model, that we now used. 7

14 naturally resembles (16), and we see once again, how M = 4.110 GeV , mΝ = 0.3 eV, gx HMZ L = 5.5 the effective quartic coupling is smaller than the true 5.0 coupling for the Higgs. gX yΝ So far, the setup seems to resemble the simple one 2.0 given in the previous chapter. The key difference is that while a high degree of tuning was needed for the initial 1.0 Λ conditions in the simple setup to guarantee the correct S 0.5 hierarchy at smaller scales, this is no longer the case, ΛH since the dynamics at these scales are controlled mainly by the evolution of the new gauge coupling. 0.2 ÈΛHSÈ Our probes of the parameter space for this theory will follow the lines of logic from the previous section: Assum- 100 5 8 11 14 ing a certain neutrino mass m and Majorana mass M, 10 10 10 10 ν Μ the scale of compositeness scale Λ is determined uniquely. GeV We will then impose the constraint (22), which fixes the FIG. 4: The RG evolution for the quartic couplings in quartic couplings at this scale3. The last remaining free the communicated CW setup. Choosing the active neu- parameter is the new gauge coupling gX , which will be trino mass and Majorana mass, determines the composite- fixed at the mass of the Z boson. The only free parame- ness scale, where the quartics are given values such that 2 ters in our analysis is thus the two masses associated to λH = |λHS | = λS ≈ yν at this scale. The final assump- the neutrino sector and the value gx(mZ ). tion is that gX (mZ ) = 5.5, which determines the IR behavior The RG equations for the remaining quartic coupling and symmetry breaking pattern. The evolution shown above and the new gauge coupling is given to one loop by yields v ' 174 GeV and mH ' 126 GeV.

2 2 2 4 4 = HL L = βλH = (12yt + 12 θM yν − 3g1 − 9g2)λH − 12(yt + θM yν ) mΝ 0.3 eV, gx EW 5.5 3 3 9 4 2 2 2 2 2 300 + g1 + g1g2 + g2 + 12λH + 2 θuλHS, (30) 4 2 4 MX 2 3 2 9 2 β = 6y + 6 θ y − g − g + 6λ λ (31) m λHS t M ν 2 1 2 2 H HS 200 s

9 GeV Υ + 6λ − g2 λ + 4λ2 , 150 S 2 X HS HS 43 mh β = − g3 . (32) 100 gX 6 X A numerical evaluation of the running of the couplings 2.5 ´ 1014 3.0 ´ 1014 3.5 ´ 1014 4.0 ´ 1014 4.5 ´ 1014 5.0 ´ 1014 as described above will yield the vevs of H and S as M GeV well as the masses of the respective eigenstates, through (26),(27), and (29), when the couplings are evaluated at FIG. 5: Values of the Higgs vev (v) and mass (mh) along with the scale of symmetry breaking s∗. the mass of the extra scalar (ms) and the dark gauge bosons (M ) as the Majorana mass M is varied, while the active A sample RG evolution yielding v ' 175 GeV and X neutrino mass mν = 0.3 eV and gX (mZ ) = 5.5. The grey mH ' 125 GeV is shown in Fig. 4, where the increase of dashes indicate the point where correct Higgs phenomenology the gauge coupling gX in the IR is displayed alongside is realized. the decrease of the dark quartic λS, which is the source of the symmetry breaking. We warn the reader that the value for hSi = u ' 227 GeV, such that v2/u2 ∼ 0.6 such that the approximation used in (29) may be invalid the tuning between is no longer needed. Instead, having and a more complete analysis should be performed. Once settled on a specific neutrino mass, only the Majorana again, we postpone this for other work, while aiming for mass M and gX (mZ ) require balancing in order to get a qualitative description for now. the correct phenomenology in the Higgs sector. Keep- For the RG evolution shown above all quartic values ing gX (mZ ) fixed while varying M with respect to the are fixed to be equal at the compositeness scale, and sample calculation above, yields the Higgs vev and mass depicted in Fig. 5. Interestingly, the Higgs mass seems to be stabilized around ∼ 130 GeV for a range of different Majorana masses, while the vev has a stronger depen- 3 We will assume that all quartic couplings are large at this scale dence on M. which would be true in a theory where all scalars are composite in the sense we have described here. This is not a necessary Varying gX (mZ ), one sees that in order to get val- assumption, and it may be relaxed if one wishes to consider ele- ues of v and mH close to the correct values, one has mentary scalar dark matter extensions. to remain within the interval gX (mZ ) ∈ [5; 6] with 8

14 M ∼ 4 × 10 GeV for the chosen value of mν = 0.3 Presently, however, we essentially abandon the top eV. Thus the tuning problems within the parameters of quark as the constituent of the Higgs, and have followed the theory have been greatly reduced, and the interesting Martin [9] to adopt the neutrinos as the Higgs boson con- region of parameter space has been discovered. For the stituents. In the neutrino see-saw model [11] the Higgs higher neutrino mass mν = 1 eV, the relevant values of will necessarily have Yukawa couplings to the conven- M are centered at M ∼ 1.2 × 1014 GeV, while for the tional left-handed lepton doublets, and right-handed neu- lower mass mν = 0.1 eV, realistic Higgs phenomenology trino singlets, ∼ yν ψLνR · H + h.c. . These Yukawa cou- 15 requires M ∼ 1.2 × 10 GeV, while the value of gX (mZ ) plings are not seen as d = 4 operators in the low energy is kept constant. theory below the scale M of the right-handed neutrino Along with the values for the Higgs mass and vev, masses, rather we observe only the d = 5 “Weinberg op- 2 c we obtain values for the mass of the other scalar state erators,” ∼ yν (ψLH) HψL/M + h.c.. ms = m2 from (29) along with the mass for the dark Above the scale of the Majorana mass, M, the d = 4 matter candidate MX = gX ∗ u/2, which are also shown operators materialize and the Yukawa couplings, yν , be- in Fig. 5. For the choice of parameters corresponding to gin to run. We assume that neutrino mixing is driven by the values for the Higgs observables, marked with a grey M, and assume degeneracy of three large Higgs Yukawas, i line, we obtain mS ∼ 190 GeV and MX ∼ 300 GeV. yν . Thus our Higgs boson is engineered in a “large The predictions for these dark matter observables are Nflavor = 3 fermion bubble approximation.” The yν fairly independent on the choice of neutrino and Majo- have a Landau pole and can match to the compositeness rana mass in the setup. conditions for the Higgs. The top quark Yukawa is also The phenomenology of the model presented here is by pulled up to the Landau pole, but remains a spectator to construction virtually identical to the one of its elemen- the new dynamics that forms the Higgs boundstate. tary counterpart, as reviewed in [17]. The main eﬀect of The model has some nice features, lending a physical imposing the composite picture is that the absolute value role to the right-handed neutrinos and demanding some for the portal coupling λHS is larger in our setup. new strong dynamics at Λ (e.g., a gauged SU(3) neutrino ﬂavor?). New dynamics near the weak scale is relevant for this. As a proof of principle, there remains much to do VI. SUMMARY AND CONCLUSIONS to survey viable schemes and explore their phenomenological consequences.

Our main goal in the present paper was to see how Acknowledgements difficult it is to maintain the idea of a composite Higgs boson in the sense of ref.[3–5], in light of modern standard This work was done at Fermilab, operated by Fermi model constraints. While the composite models fine-tune Research Alliance, LLC under Contract No. DE-AC02- the scale vweak, they are in rough concordance with the 07CH11359 with the United States Department of En- values of the Higgs boson and top quark masses as seen ergy. J.K is partially supported by the Danish National in nature, and offer potential predictivity. Research Foundation under grant number DNRF:90. Nature appears at face value to resist the idea of a strong, dynamical fermion condensate as the origin of the Higgs mechanism, given the apparent highly perturbative and critical behavior of the quartic coupling λ. It is, nonetheless, readily possible to construct a model that can yield the compositeness conditions at large scale Λ ∼ 1015 − 1019 GeV. Our main ingredient is the portal interaction that demotes λ to an effective low energy coupling, while the high energy theory is controlled by λH . We find a typical result that λH ∼ 2 × λ. This is sufficient to completely redefine the UV behavior of the theory. λH can easily have a Landau pole and satisfy the Higgs boson compositeness conditions [5]. Here we use a portal interaction near the TeV scale, which is popular in a large number of scale invariant Higgs theories [13]. The constituents of the composite Higgs boson must couple with large Yukawa interactions to the Higgs doublet, and these couplings must also have a Landau pole at the scale Λ. The top quark in the large Nc fermion loop approximation in the standard model has too weak a Higgs Yukawa coupling to produce the Landau pole. This is easier to solve than the λ problem, and one can imagine a number of alternative theoretical fixes for it. 9

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