Tadpole-Induced Electroweak Symmetry Breaking and pNGB Higgs Models

We investigate induced electroweak symmetry breaking (EWSB) in models in which the Higgs is a pseudo-Nambu-Goldstone boson (pNGB). In pNGB Higgs models, Higgs properties and precision electroweak measurements imply a hierarchy between the EWSB and global symmetry-breaking scales, $v_H \ll f_H$. When the pNGB potential is generated radiatively, this hierarchy requires fine-tuning to a degree of at least $\sim v_H^2/f_H^2$. We show that if Higgs EWSB is induced by a tadpole arising from an auxiliary sector at scale $f_\Sigma \ll v_H$, this tuning is significantly ameliorated or can even be removed. We present explicit examples both in Twin Higgs models and in Composite Higgs models based on $SO(5)/SO(4)$. For the Twin case, the result is a fully natural model with $f_H \sim 1$ TeV and the lightest colored top partners at 2 TeV. These models also have an appealing mechanism to generate the scales of the auxiliary sector and Higgs EWSB directly from the scale $f_H$, with a natural hierarchy $f_\Sigma \ll v_H \ll f_H \sim{\rm TeV}$. The framework predicts modified Higgs coupling as well as new Higgs and vector states at LHC13.


I. INTRODUCTION
The discovery of the Higgs boson has sharpened the problem of the naturalness of the electroweak (EW) scale. An attractive solution is that the Higgs boson is a composite pseudo-Nambu-Goldstone Boson (pNGB) of a global symmetry that is spontaneously broken at a scale f H not far above the electroweak scale v H = 246 GeV [1,2]. More modern realizations of this idea include Composite Higgs (CH) models (with partial compositeness) [3][4][5], as well as Twin Higgs (TH) [6,7] and Little Higgs [8][9][10][11].  [12][13][14] give lower bounds on the tuning of such theories, but current bounds can allow a totally natural mass scale for the Higgs when colored top partner decays are hidden [15,16] or the global symmetry is partially restored by neutral particles, as in Twin Higgs models [6,7].
However, observations of Higgs properties [17][18][19] re- quire v H f H so that the curvature of the pNGB manifold does not induce significant Higgs coupling deviations from the SM values (see, e.g., [20,21]). SM-like Higgs measurements at the level of ∼ 10% constrain and future measurements will reach the ∼ 1% level [22][23][24]. This makes realizing a natural model much more difficult. Minimal versions of 3rd generation partners can only obtain m h = 125 GeV when v H f H with severe radiative tuning [20,21]. More elaborate/extended fermionic sectors can improve the situation, but the structure of radiative contributions to the pNGB potential still leads to an 'irreducible' tuning ∆ ∼ These obstacles motivate studying pNGB Higgs models with a combination of additional tree-level contributions to the potential and top sectors that minimize radiative contributions, as such models stand the best chance to be 'maximally natural.' One well-known strategy, used in Little Higgs (as well as some TH models [25]), arXiv:1603.03772v1 [hep-ph] 11 Mar 2016 is to introduce additional dynamics generating a treelevel quartic without a significant contribution to the Higgs mass-squared parameter. The quartic is the dominant term in the potential, stabilizing the vacuum at v H = 0, and the radiative potential, which generates a negative mass-squared parameter, is a small perturbation moving the vacuum to a non-zero vev with a natural hi- Here, we study an alternative approach. The pNGB potential will naturally be of the size of the radiative contributions, but with a positive mass-squared stabilizing the vacuum at v H = 0. An auxiliary decoupling EWSB sector Σ is then introduced to trigger Higgs EWSB through a linear coupling to the Higgs sector, perturbing the Higgs vacuum to a non-zero vev with a nat- . This is an application of Bosonic Technicolor (BTC) or, as it is more recently dubbed, induced EWSB [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] to a pNGB Higgs. A schematic comparison of this approach to the tuned minimal radiative approach is shown in Fig. 1.
The tuning problem in pNGB models in many ways resembles the little hierarchy problem of the minimal supersymmetric standard model (MSSM), where obtaining m h = 125 GeV radiatively requires stop masses mt TeV and/or large A-terms, both of which directly contribute to the tuning [41]. It is not surprising then that parallels can be drawn between proposed solutions in the two frameworks. For example, the addition of radiatively-safe tree-level quartics is commonplace both in supersymmetric models (as in, e.g., the NMSSM) [42][43][44][45] and in Little Higgs. Indeed, the approach we take here to reconciling the Higgs mass with naturalness has been considered previously in the context of supersymmetric models [27,28,[36][37][38][39][40], but has not yet been employed in composite/pNGB Higgs models.
In the subsequent sections, we will explore the details of tadpole-induced EWSB in two concrete realizations of composite Higgs models-'conventional' Minimal Composite Higgs models (MCHM) based on SO(5)/SO (4) [5,20,21] and composite Twin Higgs models [6,7,46,47] based on SO(8)/SO(7) (or SU (4)/SU (3) for weakly-coupled UV completions). The ability of tadpole-induced EWSB to improve naturalness differs for the two frameworks:   In Sec. II, we discuss in more detail the structure of these models in the presence of a non-dynamical tadpole term. In Sec. III, we give concrete examples of top sectors and discuss the advantages of the additional tadpole contribution to the potential, including when the tuning can be substantially improved by the induced EWSB structure. In Sec. IV, we discuss the dynamics of the Σ sector, demonstrating that a realistic strongly-coupled auxiliary sector preserves the improved tuning and that the dynamical scale of the Σ sector may even arise from the Higgs sector itself. In Sec. V we discuss phenomenolog- A schematic depiction of "regular" radiative EWSB (left) versus induced EWSB (right) in a pNGB Higgs model. In this figure we take the Twin Higgs as an example where HA is the SM Higgs doublet and HB is its mirror partner (but the mechanism applies more broadly). In both cases the non-linear sigma model constrains the vev to live on a "pNGB manifold" (dotted circle). In the radiative EWSB the generic, untuned, EW vev is tuned down from fH to vH using a mass term. In the induced case an untuned EW vev of zero is brought up to vH without tuning by a tadpole.
while the physical mass-squared is and m h = 125 GeV is realized for β = β SM 1/32.
A key point is that radiative contributions to the po- However, this tuning is not 'irreducible'-it can be avoided by including an additional tadpole-like contribution to the potential. The structure of the low-energy theory is that of 'induced' EWSB [39,40]. In induced EWSB, the Higgs vev arises as a result of a coupling linear in the Higgs to another sector, Σ, that breaks the with |Σ| = fΣ √ 2 . If this additional sector were not present or did not acquire a vev, Higgs EWSB would not occur. In the limit that the extra modes of the additional sector are decoupled, the dominant component of   EWSB can be viewed as arising from an effective tadpole   for the Higgs; we first focus on this case before returning   to the dynamics of the Σ sector in Sec. IV. For a composite Higgs model, we can parameterize the tadpole by a term γ = κ 2 f Σ /f 3 H in the non-linear realization, This mechanism requires α > 0, such that v H = 0 for γ = 0. The tadpole perturbs the vacuum from v H = 0 and a small value of γ naturally leads to v H f H . As such, the correct Higgs mass and vev can be achieved even with β β SM . Moreover, since γ explicitly breaks SU (2) L , a hierarchy γ α is naturally preserved by radiative corrections. As long as radiative contributions to the mass-squared can be made naturally small, δαf 2 H ∼ < m 2 h , the overall naturalness of the model can be improved.

B. Twin Higgs Models
Twin Higgs models extend the coset and low-energy content of the theory to preserve a spontaneously broken The original twin Higgs model [6,7] consisted of an where H A,B are doublets of weakly-gauged SU (2) A,B ⊂ SU (4), with a small SU (4)-violating but Z 2 -preserving The parity exchanges A and B. In strongly-coupled realizations a larger SO(8) symmetry should be considered [7,[46][47][48]. 1 When the approximate SU (4) is spontaneously broken by a large vev f H v H , there is an uneaten pNGB that is associated with the Higgs, which develops a potential proportional to explicit SU (4) breaking. Parameterizing one finds a potential for the light Higgs mode of the form of Eq. (2) with β = −α = δ 2 . The Z 2 symmetry ensures that quadratically-divergent radiative contributions take Below the scale f H , the Higgs potential takes the form such that the Higgs vev is determined by Just as before, the tadpole allows the vev to be continu- to realize EWSB with the observed Higgs mass. We can therefore estimate the tuning of the tadpole model The radiative contribution from the top sector is often negative in the concrete models of the top sector we study. Sources of α 0 from outside the top sector that can be used to tune against this negative contribution and achieve the α = α obs > 0 required for induced EWSB are discussed in Sec. IV C.
In practice, we find that the tadpole mechanism in the This is an interesting case to apply the tadpole mechanism of EWSB as |α| > |β| implies that, if the top sector is responsible for radiatively generating the observed value of β = β SM , the tuning is considerably worse than the minimal tuning, ∆ The Yukawa coupling is to leading order, which requires m 4 ∼ > f H , and gives a lower bound M4 f H ∼ > 2 sin θ R for the top partner mixing with the elementary t L . For numerical results, we use y t = The full definition of the two-site model and the radiative Higgs potential is given in App. A. In the limit of a fully composite t R , sin θ R = 1 and The one-loop quadratic divergences are cut-off, but a residual logarithmic scale-dependence remains associated with the scale µ of the next set of top partner resonances [21]. For concreteness, we set µ = 3M 4 .
We will study two concrete models of Twin top sectors to determine the degree to which light colored top partners can lower the radiative tuning of the tadpole potential with respect to the minimal The results are summarized in Fig. 3 Ref. [6] proposed completing the top sector by extend- ). We will refer to this as the '6 × 4' model.
there is a minimal value for the colored top partner mass We evaluate Eq. (23) using the SM MS value of the top mass at µ = m t B 700 GeV. Refs. [46,47] studied pNGB Twin Higgs models based on an SO(8)/SO(7) coset with a partially composite top sector, similar to those studied in the MCHM [5,20,21] and above. In particular we focus on the model studied in Ref. [47] with q L embedded in an 8 = 7+1, t R in a singlet, The Yukawa coupling is to leading order, which requires m 7 ∼ > f H .
The full definition of the two-site model and expressions for the radiative corrections are described in App. A following Ref. [47]. In the Twin model the contributions to α are only logarithmically sensitive to the colored top partner masses, and therefore the residual scale dependence found in the two-site 5+1 model is absent. First, experimental constraints on Higgs couplings re- This is very similar to the size κ 2 ∼ 4πf 2 Σ suggested by naïve dimensional analysis for a strongly-coupled aux- In this section, we shall explore the structure of the auxiliary sector, beginning first with a linear model. As the above constraints likely imply strong coupling, this model is more useful for developing intuition (i.e., in the large self-coupling limit) than it is realistic. We shall subsequently discuss strongly-coupled auxiliary sectors, focusing on the additional higher-order operators between the Higgs and Σ sectors we expect in this scenario. While these operators may have interesting implications, the qualitative features of the model remain unchanged. Finally, we will highlight some additional UV considerations relevant for models that attempt to address the origin of the two sectors.

A. Linearly-Realized Auxiliary Sectors
An effective theory analysis has previously been carried out in the context of a simplified model of induced EWSB with a single Higgs doublet coupled to a linearlyrealized Σ doublet in [39]. They confirmed that it was possible to achieve a stable vacuum with f Σ < v H and, as the tadpole limit is approached, tuning does indeed become small. Here, we extend this analysis to the case of the MCHM and Twin Higgs scenarios. While the requirement of strong coupling limits the validity of a linear description of the auxiliary sector, this approach allows us to investigate the back-reaction, tuning and impact of Σ-sector G-breaking described above, as well as the form of the tadpoles generated. We will discuss in more generality the strongly-coupled case in the following subsection.

Composite Higgs
Starting with the SO(5)/SO(4) case, we take the Σ sector to be a simple linear model, which only realizes the custodial SO(4) symmetry. In the absence of a coupling to the Higgs, SO(4) is spontaneously broken at scale f 2 Σ = Λ 2 Σ δΣ . The Higgs and auxiliary sectors are linked by a Bµ-type term, producing the necessary EWSB tadpole. In addition, this term explicitly breaks SO (5) The effective tadpole for σ shifts the Σ-sector EWSB vev The auxiliary sector minimization condition combined with Eq. (25) implies So, the EWSB vev in the Σ sector does receive a correction due to back-reaction from the Higgs vev, but this effect is suppressed in the strong coupling regime when δ Σ is large. 5 In particular, that the shift in Σ is relatively small in this regime indicates that back-reaction does not result in additional tuning.
Meanwhile, the Higgs experiences explicit SO(5)breaking in addition to the tadpole through its interactions with σ. In this simplified picture, this breaking can be viewed as communicated via mixing of the CP-even states, which induces higher-order operators in the pNGB potential. It is useful to define = κ 2 2Λ 2 Σ to parameterize the mixing angle of the Higgs pNGB and σ, Again, these effects are suppressed in the large-coupling limit. Integrating out σ gives rise to new terms in the pNGB potential, including corresponding to a contribution to α In the strong-coupling limit, this effect is of similar size to the experimentally-required value of α, and therefore does not induce additional tuning. Higher-order terms are suppressed by powers of mixing between the Higgs and Σ sector, but can be relevant for the phenomenology of the extra Higgs states, as will be discussed in Sec. IV B.
This analysis indicates that the dynamics of the auxiliary sector do not disrupt the leading-order description of a Higgs pNGB with positive mass term (α > 0) and EWSB induced by a tadpole as in Sec. II, particularly in the strong-coupling limit required by experimental constraints. Back-reaction and explicit SO(5)-breaking lead to at most O(1) shifts to (f Σ , α), and so for strongcoupling induce no additional tuning in either sector.

Twin Higgs
The Twin Higgs case is similar to the SO(5)/SO(4) The Higgs sector is of the same form as given in Eqs. (9) and (10) which is an explicit soft breaking of the SU (2) Σ A × SU (2) Σ B × SU (4) H global symmetry to the gauge and discrete symmetry SU (2) A × SU (2) B × Z 2 .
Following the same strategy of integrating out the Σ sector, we have the leading quadratic terms where we have elided terms proportional to λ Σ that do not couple the Σ A and Σ B sectors. The B-sector vev is shifted by   The linear sigma model nicely captures the backreaction on the Σ sector and its effects on the pNGB potential, as well as elucidating the possibility of cogenerating the Higgs and Σ sector scales. However, because the Σ sector must be near strong coupling and its interactions with the Higgs sector can be a strong perturbation, there may be important higher-order effects neglected in this description. In the following section, we give an effective description of strongly-coupled UV completions and argue that the qualitative features remain the same.

B. Strongly-Coupled Auxiliary Sectors
We now focus on the case that H and Σ emerge from independent strongly-coupled sectors with compositeness scales Λ Σ < Λ H .

Composite Higgs -SO(5)/SO(4)
The global symmetries of the two sectors are SO(5) H and SO(4) Σ . At scales above Λ H , the two sectors are weakly coupled by an operator explicitly breaking The spurion κ 2 Ij parameterizes the breaking, We normalize these operators so that, in terms of the The fields Π a h and Π a Σ correspond to the pNGBs of the broken SU (2) H and SU (2) Σ , with a linear combination absorbed by the gauge bosons and the remaining The effects on the pNGB Higgs potential are determined by treating H as a background field and integrating out the Σ sector at Λ Σ to obtain the full Goldstone potential, This term fully describes the IR contributions from the Σ sector, and connects the size of the tadpole to the higherorder terms. For instance, these terms will generate a contribution to α, again consistent with the results for a linearly-realized auxiliary sector, although with undetermined coefficient.
Higher-order terms in Eq. (45) can also give O(1) shifts in the masses of the extra Higgs sector states Π A . For example, the tadpole and first leading contribution to the masses of the Π A have the form Integrating out the Σ sector also generates terms of the Following the same analysis as for the SO(5)/SO(4) model, the IR contribution to the Higgs potential has the form with the structure enforced by the Z 2 symmetry. We choose to express the potential in terms of a redefined parameter κ 2 ∼κ 2 to normalize the tadpole term as As for the composite example, the higher-order terms are parametrically the same size as calculated in the linear realization for Λ Σ 4πf Σ , such that the tadpole due to f H can readily constitute a significant perturbation on the Σ B sector. In addition, we expect the operators in the pNGB potential to be generated with O(1) coefficients, permitting the possibility that these terms can generate additional positive contributions to α, perhaps alleviating the need for additional UV contributions required to overcome the δα < 0 from the SM top sector.
Another notable detail is that non-negligible higherorder terms coupling H and Σ should be generated. Depending on their sign and size, these terms may lead to complete breaking of SU (2) B × U (1) B (in the event that Twin hypercharge is gauged). In particular, as  fΣΛΣ ψ ΣψΣ . Contributions to the potential for H are cut off at Λ 2 H and give a leading one-loop UV contribution This exceeds the IR-generated quadratic term by a factor ∝ be a concern [46], but such contributions are forbidden if the global symmetry is expanded to SO(8) [7,46]. This indicates small explicit breakings of SO (8) to SU (4) may also be useful to obtain α > 0.
A UV completion should also address the potentially The allowed values of f Σ are thus constrained by the combined ATLAS and CMS Higgs measurements [17][18][19]-for a strongly-coupled auxiliary sector, f Σ ∼ < 0.3v [40]. Motivated by the discussion of Sec. IV, we focus on strongly-coupled auxiliary sectors here. However we do note that, if the auxiliary sector is at least somewhat weakly-coupled, the constraints vary due to the mixing between the Higgs and the radial mode of the auxiliary sector. This mode couples to gauge bosons but not to fermions, so mixing partially restores the depletion of κ V while also reducing the enhancement of κ f .
In pNGB Higgs models, there is additional universal suppression of Higgs couplings due to where Br (SM) (h → bb) = 0.577 for m h = 125 GeV. However, depending on the exact details of the quark couplings, this decay may be suppressed and a variety of Higgs decays to Twin sector states, including displaced decays, may be possible (see, e.g., [59][60][61]).
In Fig. 4, we plot the (κ V , κ f ) that can occur in induced EWSB models with a pNGB Higgs and a stronglycoupled auxiliary sector, as well as the combined ATLAS and CMS measurements [17]. We consider both a general MCHM model (i.e., with additional suppression κ (pNGB) h relative to Eqs. (51) and (52) and, correspondingly, In our case this relationship is modified by higher-order terms. First, in the pNGB Higgs potential, α > 0 yields a negative quartic, which would tend to enhance m A relative to the above estimate, but we also expect higher-order terms including β = 0 to be generated. A second set of constraints comes from vector resonances. If the auxiliary sector is indeed strongly-coupled, we expect vector resonances with masses m ρ ∼ 4πf Σ associated with the strong dynamics [63]. These "technirhos" are constrained both by direct searches (notably, ρ ± → W ± Z [64]) and by electroweak precision measurements [32]. The exact constraints depend on the properties of the technirhos, which depend on the details of the unknown strong dynamics. However, for lighter technirhos (such as those predicted by a QCD-like auxiliary sector), these can be the dominant constraints, eliminating the majority of the allowed parameter space [40]. Thus, for a truly strongly-coupled auxiliary sector, the strong dynamics must be such that the vector resonances are at least somewhat heavy. For instance, the (non-excluded) strongly-coupled benchmarks considered in [40] [12][13][14]. A top partner of this variety is expected to be somewhat light as it is responsible for cutting off quadratic divergences due to the SM top quark.
However, in 'maximally natural' models, the full global symmetry is likely restored not too far above m T (see Sec. III). As a result, searches for other states implied by the global symmetry, such as heavy charge-1/3 B-quarks [65,66] or exotic charge-5/3 quarks [65,67] (present in complete multiplets of custodial SO(4)) may also be relevant [68,69]. In particular, for Twin Higgs models, the lightest top partner responsible for regulating the quadratic divergences is uncolored, leading to weak constraints from the LHC. But natural models likely exhibit colored top partners not too much heavier than the uncolored twin top (as in Sec. III B), which may be probed up to m * ∼ 2.5 TeV at the LHC [70].

VI. CONCLUSION
Tadpole-induced electroweak symmetry breaking gives an alternative structure for the low-energy potential of a pNGB Higgs model. This structure allows the desired EWSB pattern with m h = 125 GeV and v H f H to be achieved in Composite Higgs models that could not otherwise realize a large enough quartic term β without excessive tuning. Unlike other tree-level modifications to the pNGB Higgs potential, which focus on increasing the quartic term β (e.g., Little Higgs), the tadpole structure simply makes β irrelevant in the limit v H f H .
In SO (5) Higgs decays, may be observable. It has not escaped our attention that the auxiliary sector generically contains composite singlet pseudoscalars at the scale Λ Σ ∼ 4πf Σ ∼ 750 GeV with large branching ratios to diphotons [71][72][73][74][75][76][77], which may be able to explain recent hints for a resonance at LHC13 [78,79]. In particular, small mixings between the auxiliary sector and singlet pseudoscalars in the composite Higgs sector [80][81][82][83] can lead to an appreciable gluon fusion production cross section even if the auxiliary sector contains no colored states.
Not only can tadpole-induced models feature a pNGB potential with a fully natural scale for EWSB, but in fact searches at LHC13 and future colliders will likely be able to probe the entire remaining range of viable models independent of any naturalness arguments.
Acknowledgements: We thank Nathaniel Craig for useful discussions at various stages of this work and also wish to congratulate him for his productivity. We also .
(A2) ∆β is obtained in the same fashion.