Is the Higgs Boson Composed of Neutrinos? Jens Krog1, ∗ and Christopher T. Hill2, y 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 re- quires (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 yElectronic 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 y (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 y λeH y 2 by the small value of λH in the standard model, and L = ZH jDHj − 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 ob- tainable [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 = p y = p 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.