Physics Letters B 727 (2013) 443–447 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Darkening the little Higgs ∗ Travis A.W. Martin , Alejandro de la Puente TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada article info abstract Article history: We present a novel new method for incorporating dark matter into little Higgs models in a way that Received 7 August 2013 can be applied to many existing models without introducing T -parity, while simultaneously alleviating Received in revised form 15 October 2013 precision constraints arising from heavy gauge bosons. The low energy scalar potential of these dark Accepted 28 October 2013 little Higgs models is similar to, and can draw upon existing phenomenological studies of, inert doublet Available online 4 November 2013 models. Furthermore, we apply this method to modify the littlest Higgs model to create the next to Editor: G.F. Giudice littlest Higgs model, and describe details of the dark matter candidate and its contribution to the relic Keywords: density. Dark matter © 2013 Elsevier B.V. All rights reserved. Little Higgs Two Higgs doublet model Inert doublet model Naturalness Collective symmetry breaking 1. Introduction It has been noted that certain classes of little Higgs models may contain discrete symmetries that can be used to introduce a viable Little Higgs (LH) models [1–3] are extensions of the Standard dark matter candidate. In particular, three such classes of models Model (SM) that stabilize the electroweak scale with a light Higgs have been studied: theory space models [13], T -parity models [14] boson and weakly coupled new physics. These models resolve the and skyrmion models [15,16].Inthelatter,T -parity [14] requires fine-tuning problem within the SM by embedding the Higgs bo- () = () new fermions, forces the gauge couplings to be equal, g1 g2 , son within a non-linear sigma field, and by introducing new gauge forces conservation of a T -charge for all interactions (therefore, the and fermion states that result in a collective breaking of the scalar lightest T -odd state is stable), and results in an elimination of the Higgs potential. This collective symmetry breaking ensures can- triplet vacuum expectation value (vev). Theory space models [13] cellation of the quadratic divergences that result from radiative contain a Z4 symmetry that can be used to interchange the non- corrections from gauge boson and top quark loops that plague the linear sigma model fields amongst themselves. Within this class of SM Higgs boson. models, the scalar identified with the SM-like Higgs boson breaks The challenges in constructing a modern little Higgs model in- the Z4 symmetry down to a Z2 symmetry after electroweak sym- clude: generating a natural mass hierarchy between the heavy top metry breaking (EWSB), and the lightest particle charged under the partner(s) and heavy gauge bosons that fits within precision elec- Z2 may become a viable dark matter candidate. Additionally, dark troweak constraints; avoiding the generation of a dangerous singlet matter can arise in some little Higgs models from topological con- in the scalar potential [4]; and, in light of the mounting evidence siderations [15,16]. In these models, skyrmions take the form of for dark matter, the inclusion of a dark matter candidate. For ex- topological solitons. ample, the littlest Higgs model [5–7] and simplest little Higgs In this Letter, we explore an alternative method of introduc- model [8] do not include a dark matter candidate, and are largely ing dark matter to little Higgs models by incorporating a second constrained by precision measurements [9–11].Whilethebestest non-linear sigma field, . This expands upon the concept intro- little Higgs (BLH) model [12] resolves these precision constraint duced in the bestest little Higgs model [12] and in another T-parity issues by including a custodial SU(2) symmetry and introducing model [17], and provides a relatively simple means of implement- a second non-linear sigma field that couples only to the gauge ing an inert doublet potential [18–20] – in effect, we prescribe a bosons, it does not include a dark matter candidate. means of little Higgs-ing the inert doublet models. It should be noted that this is not the only implementation of an inert dou- blet potential in little Higgs models (see [21]). This presents a new * Corresponding author. E-mail addresses: [email protected] (T.A.W. Martin), [email protected] class of little Higgs models, dark little Higgs (DLH) models, which (A. de la Puente). follow the general structure: 0370-2693/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physletb.2013.10.060 444 T.A.W. Martin, A. de la Puente / Physics Letters B 727 (2013) 443–447 √ • duplicate global symmetry (G /H duplicates group structure + 5(Y − Y )σ , ⎛ 1 2 √ ⎞ of GΣ /HΣ ) that breaks at scale F > f ; † † • 0√ ξ / 2 χ√ G gauged in the same way as GΣ ; ⎝ ∗ ⎠ a a a Π = ξ/ 2 0 ξ / 2 + Q − Q α • and, fermions transform only under GΣ . √ 1 2 χ ξ T / 20 √ Since fermions do not transform under the second global sym- + 5(Y − Y )β, (4) metry, the complex doublet embedded in does not develop a 1 2 non-zero vev, and thus remains as a possible dark matter can- where ξ and χ are the analogous fields to the h and φ from didate. Additionally, by following this prescription, the heavy top the Σ sector, and the real triplet (ηa, αa) and singlet (σ , β) partner masses are disconnected from the mass of the heavy gauge representations of the two non-linear sigma fields mix to form bosons, which relaxes electroweak precision constraints on the a combination that becomes the longitudinal components of the models without reintroducing fine-tuning constraints. a = a + a 2 + 2 = + heavy gauge bosons (αe ( f η F α )/ f F and βe ( f σ In this Letter, we describe the details of a simplistic version of F β)/ f 2 + F 2), and an orthogonal combination that is physical. this by modifying the littlest Higgs model into the next to littlest These new fields couple to the gauge bosons in the normal way, Higgs model, a DLH class model, and explore the relic abundance via the kinetic term of the Lagrangian, such that, generated by the inert doublet. 2 2 f μ † F μ † 2. The model LK = Tr (DμΣ) D Σ + Tr (Dμ) D . (5) 8 8 The littlest Higgs is based on a non-linear sigma field (Σ) The covariant derivative is given as that parametrizes an SU(5) /SO(5) coset space. We introduce a Σ Σ second non-linear sigma field, , parametrizing a separate coset = − a a + aT DμΣ() ∂μΣ() i g j W j Q j Σ() Σ()Q j space, SU(5)/SO(5), but require that both the SU(5)Σ and j SU(5) global symmetries contain the same gauged [SU(2) × ]2 − + U (1) subgroup. Fermions transform only under the SO(5)Σ sym- i g j B j Y jΣ() Σ()Yi , (6) metry, and so the scalar doublet embedded in does not acquire j a radiatively generated negative mass squared. As with other little = × Higgs models, this description does not explain the physics origin where the sum is over j 1, 2 for each of the two SU(2) U (1). of the non-linear sigma model, which is relevant only at or above The heavy gauge boson masses pick up an extra contribution pro- ∼ portional to F 2, such that M2 = 1 (g2 + g2)( f 2 + F 2) and M2 = the “compositeness” scale λ 4π f . W H 4 1 2 B H The SU 5 symmetry is broken to SO 5 at a scale f ,asin 1 2 + 2 2 + 2 ( )Σ ( )Σ 20 (g1 g2 )( f F ). the littlest Higgs, while SU(5) is broken to SO(5) at a scale F The Coleman–Weinberg (CW) derived couplings (λ’s) for the h (> f ). The vacuum expectation values that generate this breaking and φ in the scalar potential remain predominantly unchanged at are the same as in the littlest Higgs model, given by: leading order, as factors of F cancel out, leaving a dependence only ⎛ ⎞ ⎛ ⎞ on the scale Λ.FactorsofF still contribute in the μ2 term, which 001 × 001 × 2 2 2 2 contains logarithmic divergences, through the masses of the heavy Σ = ⎝ 010⎠ ,= ⎝ 010⎠ . (1) 0 0 gauge bosons. The negative contribution from the heavy quark sec- × 00 × 00 12 2 12 2 tor is still dominant in the μ2 term in the potential, and induces The non-linear sigma fields are then parameterized as: spontaneous symmetry breaking. We can examine the degree of fine tuning in the model as 2iΠΣ / f 2iΠ/F Σ(x) = e Σ0,(x) = e 0 (2) in [22] by examining the logarithmically divergent contributions 2 2 2 where Π = a Xa and Π = a Xa, summing over the 14 to the μ term in the scalar potential. Examining δT μ , δW μ , Σ a πΣ a π 2 2 2 a a δB μ and δφμ , we similarly find that δT μ is responsible for the Goldstone bosons (πΣ,) corresponding to the 14 generators (X ) in each sector. In the littlest Higgs model, four fields correspond- largest degree of fine tuning of the μ parameter. For a Higgs boson = = ing to four of the broken generators are eaten to give mass to the mass of 125 GeV, and scale parameters f 1 TeV and F 5TeV, 2 2 2 2 heavy gauge bosons, and three are eaten to give mass to the SM we find δW μ /mh < 11, as compared with δT μ /mh < 180.