PHYSICAL REVIEW D 101, 075009 (2020)

Composite Higgs bosons from neutrino condensates in an inverted seesaw scenario

† ‡ Leonardo Coito,* Carlos Faubel, and Arcadi Santamaria Departament de Física Teòrica, Universitat de Val`encia and IFIC, Universitat de Val`encia-CSIC, Doctor Moliner 50, E-46100 Burjassot (Val`encia), Spain

(Received 8 January 2020; accepted 11 March 2020; published 7 April 2020)

We present a realization of the idea that the is mainly a bound state of neutrinos induced by strong four-fermion interactions. The conflicts of this idea with the measured values of the and Higgs boson masses are overcome by introducing, in addition to the right-handed neutrino, a new fermion singlet, which, at low energies, implements the inverse seesaw mechanism. The singlet fermions also develop a scalar bound state that mixes with the Higgs boson. This allows us to obtain a small Higgs boson mass even if the couplings are large, as required in composite scalar scenarios. The model gives the correct masses for the top quark and Higgs boson for compositeness scales below the Planck scale and masses of the new particles above the electroweak scale, so that we obtain naturally a low-scale seesaw scenario for neutrino masses. The theory contains additional scalar particles coupled to the neutral fermions, which could be tested in present and near future experiments.

DOI: 10.1103/PhysRevD.101.075009

I. INTRODUCTION are included). Since then, many authors have tried to generalize the mechanism to give predictions in In 1989 Bardeen et al. [1] (BHL) put forward the agreement with experiment (for a review see, for idea that the Higgs boson could be a bound state of instance, [14,15]). top quarks by using an adapted Nambu-Jona-Lasinio Among the different ideas, we find particularly inter- (NJL) model [2,3] (see also [4–9] for similar esting the possibility that the Higgs boson is, mainly, a approaches1). The mechanism is very attractive because bound state of neutrinos [16–19] because, after all, it gives a prediction for the top quark mass and for the neutrinos are already present in the Higgs boson mass, which can be compared with (SM) and should have some non-SM interactions in experiment. These predictions are based on two main order to explain the observed neutrino masses and ingredients: (i) The existence of a Landau pole in the mixings. In particular, if neutrino masses come from top quark Yukawa and Higgs boson self-couplings at the type I seesaw model, neutrino Yukawa couplings the compositeness scale and (ii) the existence of could be large enough to implement the BHL mecha- infrared fixed points in the nism. This approach has two important problems: (i) In equations (RGEs), which make the low-energy predic- the type I seesaw, the Majorana masses of right-handed tions stable [12,13]. Unfortunately, the minimal version 13 neutrinos should be quite large (at least ∼10 GeV) for predicts a too heavy top quark (mass above 200 GeV) Yukawa couplings of order one, which are needed and an extremely heavy Higgs boson (m ∼ 2m at h t to generate the bound state. This means that there are leading order and above 300 GeV once corrections just a few orders of magnitude of running to reach the Landau pole before the Plank scale. (ii) It is very *[email protected] † difficult to obtain the 125 GeV Higgs boson mass [email protected] ‡ because it tends to be too heavy. In Ref. [16] problem [email protected] 1NJL interactions among “subquarks” were also used in (i) was circumvented by adding three families of [10,11] to give predictions for the top quark and Higgs boson neutrinos with identical couplings and problem (ii) by masses. adding, by hand, a fundamental scalar singlet that mixes with the Higgs doublet. This produces a shift in the Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Higgs boson mass and allows one to accommodate the Further distribution of this work must maintain attribution to measured value. the author(s) and the published article’s title, journal citation, Here, we propose a quite different approach: (i) will be and DOI. Funded by SCOAP3. solved by lowering the right-handed neutrino mass. This

2470-0010=2020=101(7)=075009(10) 075009-1 Published by the American Physical Society COITO, FAUBEL, and SANTAMARIA PHYS. REV. D 101, 075009 (2020) can be implemented naturally in inverse seesaw2 type transparently by using the “bosonized” version [1]; namely, scenarios [25,26] (see also [27,28]). To solve (ii) we will the four-fermion interaction can be written as also introduce a new scalar, however, this scalar will be a L ¼ − 2 j j2 þ ¯ þ ð Þ composite of the new fermions required in the inverse Λ m0H H y0tTLtRH H:c:; 2 seesaw scenario, and therefore, its couplings will be fixed by the compositeness condition. where H is a scalar doublet. On can easily check the Thus, in Sec. II we briefly describe the BHL mechanism. equivalence of Eqs. (1) and (2) by using the equations of 2 ¼ 2 In Sec. II A we sketch the minimal version as applied to the motion to remove the scalar field H, which gives ht y0t. pure SM, and in Sec. II B we present the case in which the This equivalence is exact, at some scale Λ, because the Higgs boson is mainly a bound state of neutrinos within field H has no kinetic term and, at this point, it must be seen the type I seesaw scenario according to Ref. [16]. In Sec. III as an auxiliary field. However, quantum corrections involv- we discuss our implementation of the BHL mechanism. ing only fermion loops will necessarily generate a scalar First, in Sec. III A, we briefly review the inverse seesaw kinetic term and scalar self-interactions. Thus, at a scale μ model. Then, in Sec. III B we give the high-energy just below the Λ scale, one generates kinetic terms for the Lagrangian, which only contains fermions and three four- H, a renormalization of the mass and quartic terms fermion interactions, and derive the low-energy Lagrangian, 1 which contains the SM Higgs doublet as a bound state 2 2 2 ˜ 4 Lμ ¼ Z ðμÞjDμHj − m˜ ðμÞjHj − λðμÞjHj of the fermions plus and additional composite scalar singlet. H H 2 We also obtain the matching conditions for the couplings of ˜ ¯ þ ytðμÞTLtRH þ H:c: ð3Þ the two Lagrangians and show that all the dimensionless couplings of the low-energy Lagrangian (three Yukawa and Calculation of the corresponding fermion loops with a cutoff three quartic couplings) are written in terms of two param- Λ and imposing the “compositeness” boundary conditions eters at the compositeness scale. Last, in Sec. III C we run all couplings down to the electroweak scale and compute the ˜ 2 2 ZHðΛÞ¼0; λðΛÞ¼0; m˜ ðΛÞ¼m0 ; y˜tðΛÞ¼y0t; top quark and Higgs boson masses, which are compared H H with the experimental values. Finally, in Sec. IV we discuss ð4Þ the main results of this work. one obtains II. THE BHL MECHANISM ðμÞ¼ 2 ðμÞ λ˜ðμÞ¼2 4 ðμÞ ZH Ncy0tL ; Ncy0tL ; A. The SM case N m˜ 2 ðμÞ¼m2 − 2y2 c ðΛ2 − μ2Þ; ð Þ In the BHL approach, one considers a SM without the H 0H 0t 16π2 5 scalar doublet and, instead, one introduces a four-fermion interaction among top quarks where 1 Λ2 h2 ðμÞ ≡ ð Þ L ¼ t ð ¯ Þð¯ Þþ ð Þ L 2 log 2 : 6 4f 2 TLtR tRTL H:c:; 1 16π μ m0H Notice that Yukawa couplings y˜t do not receive one-loop where T is the SM left-handed third generation quark ˜ ðμÞ¼˜ ðΛÞ¼ L corrections from fermions, therefore, yt pyt ffiffiffiffiffiffiffiffiffiffiffiffiffi y0t. doublet and tR is the top quark right-handed singlet. By Then, one rescales the field H → H= Z ðμÞ to iterating this interaction one can show [1,3] that if it is H obtain the SM Lagrangian (with only top quark Yukawa strong enough it will induce spontaneous symmetry break- ¯ couplings), ing (SSB), htLtRi ≠ 0, and the presence of a scalar bound state of top quarks. For our purposes, this can be seen more 1 L ¼j j2 − 2 ðμÞj j2 − λðμÞj j4 μR DμH mH H 2 H 2 There is some recent work in which low-scale seesaw models þ y ðμÞT¯ t H þ H:c:; ð7Þ are used to explain small neutrino masses. In particular, in [20] t L R they are used in a composite scalar scenario. However, in this work the Higgs boson doublet is a fundamental Higgs and no with attempt is made to explain the observed Higgs boson and top 2 ðμÞ¼ ˜ 2 ðμÞ ðμÞ quark masses. Rather, the NJL framework is used to justify lepton mH mH =ZH ; number violation and provide solutions to the cosmological 1 baryon asymmetry and dark matter problems (for the use of 2ðμÞ¼ 2 ðμÞ¼ ð Þ yt y0t=ZH ðμÞ ; 8 right-handed neutrino condensates to solve these problems, see NcL also [21–23]). On the other hand, in [24] it is shown that the 2 λðμÞ¼λ˜ðμÞ 2 ðμÞ¼ ¼ 2 2ðμÞ ð Þ inverse seesaw scenario is the most natural way to implement =ZH ðμÞ yt : 9 neutrino masses in the littlest Higgs model with T parity. NcL

075009-2 COMPOSITE HIGGS BOSONS FROM NEUTRINO CONDENSATES … PHYS. REV. D 101, 075009 (2020)  We see that the two couplings ytðμÞ and λðμÞ diverge 2 9 4 3 2 2 3 4 2 βλ ¼ 12 λ þ g þ g g þ g together when μ ¼ Λ, λðμÞ¼2y ðμÞ. The last equation 400 1 40 1 2 16 2 t    is very important since it gives the relation between the 3 3 þ λ 2 − 2 − 2 − 4 ð Þ Higgs bosonp andffiffiffi the top quark masses. In fact, if yt 20 g1 4 g2 yt ; 14 H0 ¼ðv þ hÞ= 2, one finds

v 2 2 2 2 2 where g3, g2, g1 are the SM SU(3), SU(2), and U(1) SM m ¼ y ðm Þpffiffiffi;m¼ λðm Þv ¼ 2y ðm Þv ¼ 4m ; t t t 2 h t t t t gauge couplings [normalized with the SU(5) prescription, 2 θ ¼ ð Þ such that the weak mixing angle is given by tan W 10 2 2 ð3=5Þg1=g2 and, for a generic coupling g, we are using the β ¼ 16π2μ μ which is the standard compositeness result. Moreover, once convention g dg=d ]. Λ is given, we have a prediction for the top quark and Higgs One can check that the couplings in Eq. (8) satisfy these ¼ 3 boson masses (for simplicity we take m ≈ λðm Þv2; the equations once one takes Nc , removes gauge terms, h t and includes only contributions from fermion loops. small running from mt to mh and finite corrections can also be included), To impose the boundary conditions in Eq. (8) we cannot take directly μ ¼ Λ since then the couplings diverge. We 2 will take the boundary conditions slightly below Λ,atμ ¼ 2 1 16π ð Þ¼ ¼ Λκ ≡ Λ=κ with κ ≳ 1, yt mt ð Þ 2 2 NcL mt Nc logðΛ =mt Þ 8π2 2 2 v 8π2 16π2 → m ¼ : ð11Þ 2 2 t 2 2 y ðΛκÞ¼ ; λðΛκÞ¼2y ðΛ=κÞ¼ ; Nc logðΛ =mt Þ t t Nc logðκÞ Nc logðκÞ ð Þ The solution can be written in terms of the Lambert 15 function W−1ðxÞ    which can be seen as matching conditions between the 1 8π2 2 ¼ Λ − v ð Þ SM and the theory at the compositeness scale. Thus, we mt exp 2 W−1 2 12 NcΛ are assuming that we have the complete SM below Λκ ≲ Λ,whilefromΛκ to Λ we have an effective theory 15 in which the Higgs boson is not a dynamical degree of and gives mt ¼ 164 GeV for Λ ¼ 10 GeV, which is freedom (does not run in loops) and gauge interactions reasonable, while the prediction mh ∼ 2mt is quite wrong. are neglected. This setup eventually leads to a Landau For lower Λ, mt (and so mh) is larger. For instance, if 10 Λ Λ ¼ 10 GeV, Eq. (12) gives m ¼ 210 GeV. However, pole for all couplings at the scale , but introduces a t κ this calculation is not complete. Equations (8) should be dependence on the parameter , which parametrizes Λ understood as boundary conditions for scales close to the possible matching uncertainties at the scale . Most compositeness scale Λ, where the Higgs boson is not a of these uncertainties will be erased in the running from Λ Λ ≫ dynamical field (therefore, cannot appear in loops), and to the electroweak scale, if mt, because of the gauge corrections are presumably small (the main contri- infrared fixed point structure of Eqs. (13) and (14), butions come from QCD, which are small at large scales). which drive the couplings at low energies toward values Thus, below the compositeness scale, the Higgs boson at which the corresponding beta functions cancel. For contributions (fermion self-energies, vertex corrections, instance, if in Eq. (13) weak couplings are neglected, ∼ 4 and scalar self-interactions) should be included. Also, low-energy values of yt are attracted toward yt 3 g3 strong interactions could become important at lower (see, for instance, [12,13]). Notice that the details of the energies. Therefore, to give accurate predictions, one complete theory (in this case the use of four-fermion should use the complete RGEs of the SM with the interactions to obtain the bound states) are encapsulated boundary conditions Eqs. (8) and (9). Still, the calculation in the boundary conditions (15). above illustrates the main consequences of the approach; Thus, one takes the gauge couplings measured at the Λ once Λ is given, everything is fixed, in particular mt Z-boson mass scale mZ, runs them up to the scale k, and and mh. then, using Eq. (15) and Eqs. (13) and (14), one obtains ð Þ λð Þ 3 Let us now see how the full predictions can be obtained. yt mt and mh , and therefore, mt and mh. ¼ 223 3 ¼ The complete SM RGE beta functions are With this procedure, one gets mt and mh 17 246 4 GeV for Λ ¼ 10 GeV, and mt ¼ 455 45 and   9 9 17 2 2 2 2 3 β ¼ y y − 8g3 − g2 − g1 ; ð13Þ Well-known SM finite corrections at the weak scale can also yt t 2 t 4 20 be included, if necessary.

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¼ 605 142 Λ ¼ 104 2 2 mh GeV for GeV, where the uncer- L ¼ hν ð ¯ ν Þðν¯ Þþ htν ð ¯ ν Þð¯ Þ ð Þ 4f 2 LL R RLL 2 LL R tRTL tainties come from the input parameters [basically g3 mZ ] m m κ κ 0H 0H and , which we vary from 2 to 10. These values are 1 compatible with the results of Ref. [1] when their input þ νc ν þ ð Þ 2 RMR R H:c:; 16 parameters are used. This has to be compared with the ∼ 173 ∼ 125 measured values mt and mh GeV. Quarks, where L are the left-handed lepton doublets, ν are the however, are not observed as free particles and there are L R right-handed neutrinos, and MR is a 3 × 3 right-handed several possible definitions for their masses. The most neutrino Majorana mass matrix necessary to implement the precisely measured value for the top quark mass is type I seesaw mechanism for neutrino masses. If hν ≫ hν , obtained by kinematic reconstruction and yields t the NJL interaction can be written in terms of an auxiliary m ∼ 173 GeV. Its connection with the parameters of t scalar doublet H (to be identified as the Higgs doublet) the Lagrangian is not clear, although it is believed to be related, up to corrections of the order of the QCD scale L ¼ − 2 † þ ¯ þ ¯ ν Λ m0HH H y0tTLtRH y0νLL RH Λ with the so-called pole mass, denoted here by mt QCD 1 and defined as the position of the pole of the propagator þ νc M ν þ H:c:; ð17Þ computed perturbatively. Running Yukawa couplings are 2 R R R defined in the MS scheme and are more closely related to the running mass m¯ ðm¯ Þ. It is known that the relation as can be checked by removing the field H using the t t equations of motion. Notice that in this procedure one between these two quantities, pole and running masses, 2 is affected by large QCD radiative corrections, which neglects terms of order y0t, which would induce a pure top produces a shift of the order of 10 GeV between quark four-fermion interaction. This means that H will be the two definitions (see, for instance, Ref. [29]). This mainly a bound state of neutrinos with a small contribution would lower the mass from 173 to 163 GeV. The situation from top quarks. is even more complicated if one also includes electroweak Below the scale Λ, a Higgs kinetic term and a potential corrections, which can be large because of the presence of are generated and, after Higgs wave function renormaliza- tadpole contributions [29]. Fortunately, one can show that, tion, one recovers a SM Lagrangian, Eq. (7), but including at least at one loop, the connection between the pole mass neutrino Yukawa couplings and right-handed neutrino and the Yukawa coupling is free from these tadpole Majorana masses, written in terms of renormalized cou- ðμÞ ðμÞ contributions, rendering the electroweak corrections plings yt and yν . Then, one runs this Lagrangian Λ small. Therefore, we use the known expressions that from to the scale MR, where right-handed neutrinos connect the pole quark mass with the Yukawa coupling decouple and generate active neutrino masses given by the 2 [29,30]. Anyway, even taking into account all these standard seesaw formula mν ∼ yνðMRÞhHi=MR. Since corrections, it is clear that the top quark mass prediction yνðMRÞ are expected to be Oð1Þ in order to drive the is off by more than 50 GeV. The Higgs mass prediction is NJL and mν < 1 eV, MR is expected to be larger than ¼ 125 1 13 even worse since the measured value is mh . 10 GeV. Below MR one has the SM with tiny neutrino 0 14 . GeV and its connection to quartic couplings is only Majorana masses. The point is that, above the scale MR, the affected by small weak corrections. neutrino Yukawa coupling yν contributes to the running of Clearly the minimal version of the mechanism is off even the top quark Yukawa coupling for scales close to the Planck scale. The Higgs boson mass   prediction, above the top quark mass, seems particularly 9 2 2 β ¼ y y þ 3yν þ ; μ >M ; ð18Þ difficult to reconcile with experiment. Moreover, the infra- yt t 2 t R red fixed point structure of the RGEs suggests that, to reconcile the mechanism with experiment, it is not enough where the ellipsis represents SM gauge terms. to modify the theory at high scales, but it is also necessary Then, above MR, the neutrino Yukawa couplings drive to introduce new light degrees of freedom that modify the top quark Yukawa coupling to diverge at some scale the RGEs. 20 Λ ∼ 10 GeV if the ratio yν=yt is large enough. What about the Higgs boson mass? In Sec. II A we have B. The Higgs as a neutrino bound state seen that, in the NJL scheme, quartic couplings should also Here we briefly review the scenario in which the Higgs diverge at the scale Λ. However, in the pure SM this leads to boson is a bound state of neutrinos [16–19], specifically, we a too heavy Higgs boson. Unfortunately, the introduction of follow more closely the Krog and Hill (KH) approach [16]. the neutrino Yukawa couplings does not help here, in fact, it KH introduce the following four-fermion interactions (they even worsens the situation because Yukawa couplings give assume three families of leptons with a common coupling always a negative contribution to the running of quartic and three colors for the quarks, whose indices will not be couplings. KH solve this problem by introducing a funda- displayed explicitly): mental neutral scalar singlet S at the electroweak scale, as in

075009-4 COMPOSITE HIGGS BOSONS FROM NEUTRINO CONDENSATES … PHYS. REV. D 101, 075009 (2020) scalar Higgs portal models [31–33]. In these models, if the Neutrino masses are given by a similar expression but singlet scalar develops a vacuum expectation value (VEV) with an extra loop suppression factor, which allows for hSi ≫ hHi, the mass of the lighter scalar is given by larger values of μν.   λ2 2 ≃ 2 λ − HS h i2 ð Þ mh H H ; 19 B. The inverse seesaw model with composite scalars λS In the following, we will embed this mechanism in the BHL λ λ λ where H, S and HS are the quartic couplings of H, S, and scheme, and the interesting thing is that, since the masses of the mixed H, S respectively. From Eq. (19) it is clear that mh new neutral heavy leptons could be naturally at the electro- can be relatively small even if the quartic couplings are weak scale, they can be obtained through SSB of a composite order one. Moreover, in the KH setup, only λH is fixed by λ λ singlet scalar at low scales and implement the Higgs portal the compositeness boundary conditions, while HS and S mechanism to accommodate the Higgs boson mass. are arbitrary and can be adjusted at will. We will consider a Lagrangian with only fermions and the following interactions and Majorana mass terms4: III. THE INVERSE SEESAW MODEL WITH 2 2 COMPOSITE SCALARS hν ¯ hs L4 ¼ ðL ν Þðν¯ L Þþ ðn¯ ν Þðν¯ n Þ f m2 L R R L m2 L R R L A. The inverse seesaw model of neutrino masses 0H 0H  2 h ν 1 A nice way to lower the seesaw scale at arbitrary scales is þ t ð ¯ ν Þð¯ Þþ c μ þ ð Þ 2 LL R tRTL 2nL nnL H:c: ; 23 provided by the so-called inverse seesaw (ISS) mechanism. m0H In this mechanism [25,26], one introduces, in addition to ν three right-handed neutrinos R, three new singlet fermions where the Majorana mass term for nL, μn, can be included nL, with Lagrangian because nL is a singlet and it is necessary to obtain masses for active neutrinos (as discussed before, an alternative would be 1 ¯c L ¼ ¯ ν þ ν¯ þ c μ þ ð Þ to add a right-handed neutrino Majorana mass term ν μννR). ISS LLyν RH RMνnnL 2 nL nnL H:c:; 20 R Notice that this Lagrangian, when μn ¼ 0, preserves two iα iα global phase symmetries Lν ∶ ν → e ν , L → e L , μ 3 3 R R R L L where yν, Mνn, and n are × matrices. Notice that, if iβ and Ln ∶ nL → e nL.Ifμn ≠ 0, Ln is explicitly broken but μn ¼ 0, the lepton number can be assigned in such a way L L Lν is preserved; if μν ≠ 0, Lν wouldbebrokenbutL that it is conserved. After SSB the Lagrangian (20) leads to R R nL ν¯ the following Majorana mass term would be preserved and, finally, a term RnL would break the two but keep Lν þ L . This Lagrangian can be obtained (in 0 10 1 L nL 0 h i 0 ≫ yν H νL the limit in which hν htν) from 1 B CB C L ¼ ð νc ν c Þ@ †h i 0 A@ νc A ISS L R nL yν H Mνn R 2 † ¯ ¯ 2 LΛ ¼ −m0 H H þ y0 T t H þ y0νL ν H 0 T μ n H t L R L R Mνn n L 1 − 2 † þ ν¯ þ c μ þ ð Þ þ H:c: ð21Þ m0SS S y0sS RnL 2 nL nnL H:c:; 24

If μn ¼ 0, this can be diagonalized exactly and leads to where S is a singlet scalar field, which will be interpreted as a three Dirac neutrinos, whose masses squared are the n¯ LνR bound state. Fermion loops will induce a scalar 2 2 † ¼ h i þ ν eigenvalues of the matrix MνH yνyν H M nMνn, potential and kinetic terms for the scalars and three exactly massless Weyl neutrinos. If μn ≠ 0, the L ¼ ðμÞj j2 − ˜ 2 ðμÞj j2 þ ðμÞj∂ j2 − ˜ 2ðμÞj j2 lepton number is explicitly broken and the would-be μ ZH DμH mH H ZS μS mS S massless neutrinos acquire a mass matrix given by (in 1 1 1 ≫ h i − λ˜ ðμÞjHj4 − λ˜ ðμÞjSj4 − λ˜ ðμÞjHj2jSj2 the limit Mνn yν H ) 2 H 2 S 2 HS  h i h i ¯ ¯ H H † þ y˜ ðμÞT t H þy˜νðμÞL ν H þy˜ ðμÞSν¯ n ≃ μ ν ð Þ t L R L R s R L mν yν T n y ; 22 Mνn Mνn  1 c þ n μnnL þH:c: : ð25Þ so that if μn is small, mν can be below 1 eV even if yν is 2 L order one and Mνn about 1 TeV. An interesting variation consists in taking μ ¼ 0 and n 4 adding a Majorana mass term for right-handed neutrinos, For simplicity, we use only one family of leptons and nL, but ala` νc μ ν the mechanism can be generalized easily to three families R ν R. In that case, active neutrino masses are not KH. Moreover, to generate masses for the other quarks and generated at tree level (the determinant of the mass leptons, one should also introduce additional four-fermion matrix remains zero), but are generated at one loop [34]. interactions, which will be neglected here.

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Calculation of the corresponding fermion loops and impos- where we have defined p ≡ y0t=y0ν, which characterizes ing the compositeness boundary conditions the relative strength of top quark to neutrino interactions and must be small. ˜ ˜ ˜ ZHðΛÞ¼ZSðΛÞ¼0; λHðΛÞ¼λSðΛÞ¼λHSðΛÞ¼0; If the two scalar fields develop a VEV, the model 2 2 2 2 specified by the Lagrangian in Eq. (28) implements the m˜ ðΛÞ¼m ; m˜ ðΛÞ¼m ; H 0H S 0S ISS mechanism described in Sec. III with a mass ˜ ðΛÞ¼ ˜ ðΛÞ¼ ˜ ðΛÞ¼ ð Þ yt y0t; yν y0ν; ys y0s; 26 Mνn ¼ yshSi. Therefore, if μn ≪ hHi ≪ hSi, one can explain small neutrino masses. Moreover, with this hier- gives archy of scales, one can also implement the Higgs portal model [31–33], in which the effective low-energy Higgs ðμÞ¼ð 2 þ 2 Þ ðμÞ ðμÞ¼ 2 ðμÞ ZH y0ν Ncy0t L ;ZS y0sL ; quartic coupling λ can be small even if the complete theory ˜ 4 4 quartic couplings are large, as usually required in NJL λ ðμÞ¼ð2y0ν þ 2N y0 ÞLðμÞ; H c t scenarios (see Sec. II B). This leads to the following λ˜ ðμÞ¼2y4 LðμÞ; λ˜ ðμÞ¼2y2 y2 LðμÞ 2 2 S 0s HS 0ν 0s hierarchy of masses: mν ∼μnhHi =hSi ≪mt, mh ∝ hHi ≪ 1 M ∼ m , mν ∝ hSi ≪ Λ, where we have denoted by M, ˜ 2 ðμÞ¼ 2 − ð2 2 þ 2 2 Þ ðΛ2 − μ2Þ s H mH m0H y0ν Ncy0t 2 ; 16π generically, the scale of new particles, the scalar S, ms, and 2 the neutral heavy leptons νH, mν . Notice that, since the y0 H m˜ 2ðμÞ¼m2 − s ðΛ2 − μ2ÞðÞ 5 → iα S 0S 8π2 27 scalar potential has an extra global symmetry, S e S, broken spontaneously, the low-energy spectrum contains, in and, as in the SM case, y˜tðμÞ¼y˜tðΛÞ¼y0t, y˜νðμÞ¼y˜νðΛÞ¼ addition to the SM fields, a Goldstone boson coupled mainly ˜ ðμÞ¼˜ ðΛÞ¼ y0ν, ys ys y0s. pffiffiffiffiffiffiffiffiffiffiffiffiffi to the neutral heavy particles. Then, it is a kind of singlet → ðμÞ Majoron [35] (triplet and doublet Majorons [36,37] are now Nowp oneffiffiffiffiffiffiffiffiffiffiffiffi rescales the scalar fields H H= ZH and excluded because of the well-measured invisible decay S → S= Z ðμÞ to obtain S width of the Z boson). The phenomenology of this type 2 2 2 2 2 2 of model is very interesting and one can usually cope with it Lμ ¼jDμHj − m ðμÞjHj þj∂μSj − m ðμÞjSj R H S (a detailed phenomenological study of the model and some 1 1 1 − λ ðμÞj j4 − λ ðμÞj j4 − λ ðμÞj j2j j2 of its variations will be given elsewhere). Just note that the 2 H H 2 S S 2 HS H S  mixing of the singlet scalar with the doublet will induce modifications of the Higgs boson couplings, which are þ ðμÞ ¯ þ ðμÞ ¯ ν þ ðμÞ ν¯ yt TLtRH yν LL RH ys S RnL experimentally constrained, and an invisible decay of the  1 Higgs boson to Majorons, which is also constrained. These þ c μ þ ð Þ constraints can be satisfied by taking hSi large enough. Here 2 nL nnL H:c: ; 28 we are more interested in the possibility of obtaining the observed top quark and the Higgs boson masses in this NJL with scenario. Since the Majorana mass for the new fermion nL must be much below the electroweak scale μ ≪ hHi, it will m2 ðμÞ¼m˜ 2 ðμÞ=Z ðμÞ;m2ðμÞ¼m˜ 2ðμÞ=Z ðμÞ; n H H H S S S not affect this calculation and can be safely neglected. We 2 2 1 will reintroduce it at the end when we discuss neutrino y ðμÞ¼y =Z ðμÞ¼ ; ð29Þ s 0s S LðμÞ masses.

p2 y2ðμÞ¼y2 =Z ðμÞ¼ ; C. The top quark and Higgs boson masses t 0t H ð1 þ N p2ÞLðμÞ c To obtain the value of m , we take the measured values of 1 t 2ðμÞ¼ 2 ðμÞ¼ ð Þ the gauge couplings at mZ and run them up to Λκ with SM yν y0ν=ZH 2 ; 30 ð1 þ Ncp ÞLðμÞ RGEs. Since the new particles are all singlets, at one loop, they do not affect the running of gauge couplings. At the 2ð1 þ 4Þ Λ – λ ðμÞ¼λ˜ ðμÞ 2 ðμÞ¼ Ncp scale κ, we impose the boundary conditions (29) (32) and H H =ZH 2 2 ; ð1 þ Ncp Þ LðμÞ obtain all Yukawa couplings, yt, yν, ys, and quartic 2 couplings λH, λS, λHS, as functions of κ and p. Then we λ ðμÞ¼λ˜ ðμÞ=Z2ðμÞ¼ ; ð31Þ S S S LðμÞ run them with the RGEs of the complete model (see the Appendix for the beta functions) up to the scale M, which 2 we fix at some value above the electroweak scale. At the λ ðμÞ¼λ˜ ðμÞ ð ðμÞ ðμÞÞ ¼ HS HS = ZH ZS 2 ; ð1 þ Ncp ÞLðμÞ 5 ≡ ð Þ This is just a consequence of the global symmetries of the new p y0t=y0ν; 32 four-fermion interactions we have introduced.

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6 scale M we assume that S develops a VEV, giving a ν¯RnL mass term so that the fermions νR and nL combine to form a 5 Dirac fermion (if μn ¼ 0,ifμn ≠ 0 a pseudo-Dirac fermion) ∼ ð Þh i ∼ with mass mνH ys M S M. Then, to obtain the top quark mass, we decouple the heavy particles and run the top quark Yukawa coupling with the SM RGEs from M to 2 μ ¼ mt, which at this point is still unknown, but can easily be computed by using the SM relation7 pffiffiffi m y ðm Þ¼ 2 t ð1 þ δ Þ; δ ≈−0.059: ð33Þ 1 t t v t t

Here δt represents the well-known SM corrections to the relation between the top quark pole mass and the Yukawa coupling [29,30]. δt includes QCD corrections that, as commented on in Sec. II, are very large and some small electroweak corrections. For masses mt ∼ 173 and FIG. 1. Evolution of the Yukawa couplings, as explained in the ¼ 0 1 Λ ¼ 1017 κ ¼ 2 mh ∼ 125 GeV, δt can be well approximated by the number text, for p . , GeV, . The new heavy particles given above, which we use in the following calculations, are assumed to have a mass M ¼ 10 TeV where they decouple. Then, from M to the electroweak scale, the top quark Yukawa but it can be computed for arbitrary values of mt and mh. We represent in Fig. 1 an example of the running of all coupling yt is run according to SM RGEs. For the chosen values, this procedure gives finally m ¼ 173 GeV. Yukawa couplings for p ¼ 0.1, Λ ¼ 1017 GeV, κ ¼ 2, and t M ¼ 10 TeV that reproduces the correct value of ≫ mt ∼ 173 GeV. The SM RGEs running is shown in the will give the mass squared of the new scalar (for u v, 2 2 dashed blue line, while the running with the new particles is ms ∼ λSu ). It is easy to check that these two eigenvalues shown in the solid blue line (from M to Λκ). Above Λκ all are related by Yukawa couplings run only with fermion loops (dotted   line) and meet the Landau pole at μ ¼ Λ. We see how the λ2 1 − λ v2=m2 m2 ¼ v2 λ − HS H s Yukawa couplings yν and ys pull the top quark Yukawa h H λ 1 − ðλ − λ2 λ Þ 2 2 S H HS= S v =ms coupling yt toward the Landau pole.    ≫ λ2 λ2 2 To obtain the Higgs boson mass we have to study the m⟶s v 2 λ − HS 1 − HS v þ ð Þ v H λ λ 2 : 36 Higgs potential. As commented before, we assume that the S S ms two scalars obtain a VEV, then we write Then if ms ≫ v, the effect of the new scalar on the Higgs ð0Þ 1 ð0Þ 1 boson mass is just a redefinition of the SM quartic coupling H ¼ pffiffiffiðv þ h þ iω Þ;S¼ pffiffiffiðu þ s þ iθÞ: ð34Þ 2 2 λ, in terms of the couplings of the complete theory8   Since the potential has an extra global symmetry S → eiαS, λ2 ð Þ λð Þ¼ λ ð Þ − HS M ð Þ broken spontaneously, the low-energy spectrum (below M H M : 37 λSðMÞ μ ¼ M) contains, in addition to the SM fields, a Goldstone θ boson, which is given by the imaginary part of S, . On the Since we know, from the measured value of the Higgs other hand, the real part of S mixes with the Higgs doublet boson mass, that λ is small, and the λ are naturally ð Þ H;HS;S with a mass matrix squared given by [in the h; s basis] large in composite scenarios, Eq. (37) requires a cancella-   tion. It is interesting to note9 that in our model this λ v2 λ vu pffiffiffi pffiffiffi M2 ¼ H HS ;v≡ 2hHi;u≡ 2hSi: cancellation naturally occurs for p ≪ 1 because of the scalars λ λ 2 HSvu Su boundary conditions in Eqs. (31) and (32). ð35Þ 8We perform the matching at a scale M of the order of the mass 2 of the new particles, the fermions, and the scalars, which we The smallest of the eigenvalues mh can be identified with pffiffiffi assume are of the same order. Since for u ≫ v, mν ∼ ysu= 2 the observed Higgs boson mass squared, while the largest 2 2 2 H and ms ∼ λSu , one needs λS and ys to be of the same order. This is guaranteed by the boundary conditions at the Λκ scale. To be 6 We assume that the parameters of the model are adjusted, definite, in our calculations we take M ¼ ms, but we have such as both the doublet H and the singlet S develop a vacuum checked that this condition is not strongly modified by the 2 ð Þ 0 2ð Þ 0 Λ expectation value, i.e., mH M < and mS M < . running from κ to M. 7Therefore, we stop the running when this equation is satisfied. 9We thank the referee for remarking this important point to us.

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δ 5 where h is given by a complicated expression that depends on the masses of SM particles [38], but for mh ∼ 125 and mt ∼ 173 GeV, it is well approximated by the value above, 2 while δhs is obtained by comparing the first line in Eq. (36) with Eq. (38) when one-loop corrections are set to zero, that δ ¼ 0 λð Þ¼λð Þ 1 is, when h and mt M ,     λ2 ðMÞ v2 λ2 ðMÞ v2 −1 0.5 δ ≃− HS 1− λ ð Þ− HS ð Þ hs λ ð Þ 2 H M λ ð Þ 2 : 39 S M ms S M ms λ 0.2 To evaluate these expressions we need the H;S;SH couplings at the scale M and λ at the scale mt. For that we run them using the beta functions given in the Appendix. In Fig. 2 we give an example with the same values of p, M, Λ, and κ as in Fig. 1, where we see how λ FIG. 2. Evolution of the scalar quartic couplings, as explained H;S;SH evolve to lower energies from the Landau pole. in the text, for the same values as in Fig. 1. At the scale M ¼ Since at μ ¼ M ¼ 10 TeV all the heavy particles decouple, 10 TeV the new particles decouple, leaving a SM quartic the couplings λH;S;SH do not run anymore but leave a SM- coupling, given by Eq. (37), which runs up to the weak scale like theory with an effective coupling λðMÞ given by according to the SM RGEs. For the chosen values, this procedure Eq. (37). Then, from M to the electroweak scale λ runs ¼ 125 gives finally mh GeV. according the SM RGEs. This procedure gives finally (for the chosen values of p, M, Λ, and κ) mh ¼ 125 GeV ¼ 173 The connection between m and λ will have tree-level (and mt GeV). h Λ corrections δhs, which vanish for ms ≫ v, as shown in We can repeat this procedure for different values of p, Eq. (36). Moreover, one-loop logarithmic electroweak (and κ, M) and check if they are able to reproduce the ∼ 173 ∼ 125 corrections can be incorporated by running λ from M to measured values of mt and mh GeV. In Fig. 3 we depict the region of p; Λ that can reproduce mt according to the SM. Finally, to connect λðmtÞ with the values of m in a region of 1 GeV around m ¼ 173 GeV physical Higgs boson mass mh, one should also take into t t (band with green-pink colors) and m in a region of 1 GeV account the well-known one-loop SM corrections at the mt h around m ¼ 125 GeV (gray band). We do this for two scale [38], δh. Thus, one has h values of κ in each plot (for fixed M ¼ 1 TeV on the right and M ¼ 1000 TeV on the left). We see that, indeed, there 1 þ δ m2 ¼ λðm Þv2 hs ; δ ∼−0 011; ð Þ is an overlapping region where one can reproduce both the h t 1 þ δ h . 38 h Higgs boson and the top quark masses. For M ¼ 1 TeV

174.0 126.0

173.5 125.5

173.0 125.0

172.5 124.5

172.0 124.0

FIG. 3. Region of p, Λ that can reproduce values of mt in a region of 1 GeV around mt ¼ 173 GeV (bands with green-pink colors, lighter grays if seen as grayscale) and mh in a region of 1 GeV around mh ¼ 125 GeV (darker thinner gray bands). On the left for M ¼ 1000 TeV and on the right for M ¼ 1 TeV. In each plot, we present results for two different values of κ.

075009-8 COMPOSITE HIGGS BOSONS FROM NEUTRINO CONDENSATES … PHYS. REV. D 101, 075009 (2020) this is found around Λ ∼ 1019 GeV, while for M ¼ neutrino masses with the presence of additional neutral 1000 TeV the overlapping region occurs around scalars coupled to the neutral fermions. If the scale of the Λ ∼ 1012 GeV. Larger values of M lead to lower values new particles is not much larger than 1 TeV, the model of Λ, but for M ≳ 108 GeV there are no solutions. The exhibits a very rich phenomenology that could be tested in effect of the exact scale at which we perform the matching present and near future experiments and will be studied in between the complete model and the model with static another publication. Further extensions in which the ν scalars, which is parametrized by κ ¼ Λ=Λκ, only changed Majorana mass terms for R and/or nL are also generated the preferred value of p, which is always small, as required by dynamical symmetry breaking might also be interesting. for consistency. We only represent values of Λ a couple of orders of magnitude above M, since for Λ close to M the ACKNOWLEDGMENTS range of running is very small and the results are com- This work is partially supported by the FEDER/MCIyU- pletely dominated by the matching, which cannot reliably AEI Grant No. FPA2017-84543-P, by the “Severo Ochoa” be computed without knowing the details of the complete Excellence Program under Grant No. SEV-2014-0398 and by theory behind the four-fermion interactions. the “Generalitat Valenciana” under Grant No. PROMETEO/ Once mt and mh are obtained with the correct values, all 2019/087. L. C. and C. F. are also supported by the the couplings and scales are quite constrained. However, “ ” “ ” μ Generalitat Valenciana under the GRISOLIA and the Majorana mass terms of the heavy fermions ν and/or “ACIF” fellowship programs, respectively. μn are completely free and can be adjusted to obtain neutrino masses below 1 eV using the inverse seesaw formula, Eq. (22). A complete analysis of neutrino masses, APPENDIX: RGEs OF THE MODEL as for the rest of the fermions, requires a three family Here we give the RGE beta functions of the model, analysis, but it is clear that given the freedom in μν;n there which have been computed with the help of SARAH [40] 2 02 should be no problem for adjusting neutrino masses and [we use the SU(5) convention 3g1 ¼ 5g for the U(1) mixings. Alternatively, one could also try to generate the factor]. Majorana mass terms by using composite scalars breaking Gauge couplings (same as in the SM): lepton number as done in [39] with the interesting conse- quences discussed there. 41 19 β ¼ g3; β ¼ − g3; β ¼ −7g3: ðA1Þ g1 10 1 g2 6 2 g3 3 IV. CONCLUSIONS Yukawas: Following previous works [16,18,19], we have explored   the possibility that the observed Higgs boson is mainly a 9 2 2 9 2 17 2 2 β ¼ y y − 8g3 − g2 − g1 þ yν bound state of neutrinos formed because a strong four- yt t 2 t 4 20 fermion interaction between neutrinos appears at high   5 1 9 scales. The minimal version of this scenario has problems 2 2 2 2 2 β ¼ yν yν þ y þ 3y − ð5g2 þ g1Þ to reproduce the observed top quark mass and, especially, yν 2 2 s t 20 the Higgs boson mass. We have overcome these problems β ¼ ð2 2 þ 2Þ ð Þ ys ys ys yν : A2 by introducing, in addition to right-handed neutrinos νR,a new singlet fermion nL, with four-fermion interactions Quartic couplings: ðν¯RnLÞðn¯ LνRÞ, which gives rise to a new scalar bound state. This singlet scalar develops a VEV and, therefore, mixes 2 27 4 9 2 2 9 4 βλ ¼ 12λ þ g1 þ g1g2 þ g2 with the Higgs doublet, allowing us to obtain a small Higgs H H 100 10 4   mass even if the couplings are large, as required in 9 composite scalar models. þ λ 12 2 − 2 −9 2 þ 4 2 þ 2λ2 − 4 4 − 12 4 H yt 5g1 g2 yν HS yν yt The compositeness condition basically fixes all Yukawa and quartic couplings at the compositeness scale; therefore, β ¼ 10λ2 þ 4λ 2 þ 4λ2 − 4 4 λS S Sys HS ys the parameters of the model are very constrained. In spite of  that, this setup can accommodate the correct masses for the β ¼ λ 4λ þ 6λ þ 4λ þ 2 2 þ 2 2 þ 6 2 λHS HS HS H S yν ys yt top quark and Higgs boson for compositeness scales below  the Planck scale and masses of the new particles above the 9 2 9 2 2 2 8 − g − g − 4y yν: ðA3Þ electroweak scale but below ∼10 GeV. 10 1 2 2 s If small Majorana masses are allowed for νR and/or nL, we naturally obtain a low-scale seesaw scenario for

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