Darkening the Little Higgs ∗ Travis A.W

Physics Letters B 727 (2013) 443–447

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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 doublet 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. Thus gauge bosons, leaving seven observable scalar states. In our model, it is clear that the degree of fine tuning in the model is controlled there are 14 broken generators for each of the Σ and sectors by the heavy quark sector, and larger values of MW that result in (total of 28), and a total of seven are eaten to give mass to the a relaxation of electroweak (EW) precision constraints are viable gauge bosons, leaving 21 observable scalars. without significantly increasing the degree of fine tuning. [ × ]2 Other EW precision constraints arise in the model as a result of Both SU(5) symmetries are gauged by the same SU(2) U (1) = − − = the triplet vev, v . The scalar potential for φ is unchanged from the subgroups, with generators Y1 diag( 3, 3, 2, 2, 2)/10 and Y2 diag −2 −2 −2 3 3 10 for the two U 1 groups, and littlest Higgs model, which provides the relation v <(v/4 f )v [23]. ( , , , , )/ ( ) ⎛ ⎞ ⎛ ⎞ Since the v contributions to the EW precision observables are sub- a 2 2 2 2 σ /200 00 0 dominant over those proportional to v / f (or M /M ) [23] for a = ⎝ ⎠ a = ⎝ ⎠ W W Q 1 000, Q 2 00 0 (3) most of the parameter space, the overall constraints on the scales a∗ 000 00−σ /2 f and F arising from EW precision observables will be improved a over the original littlest Higgs model. In [6], it was argued that v for the two SU(2) groups. In this notation, σ are the Pauli matri- passes the constraints on g Z for values of v < 10%v, which is ces. 1 easily satisfied within the NLH model. The new fields of the non-linear sigma field are embedded The masses of the and h fields in the sector are similar to in the Pion matrix as: φ Σ ⎛ √ ⎞ those found in the littlest Higgs. The χ triplet obtains a quadrati- † † 0√ h / 2 φ√ cally divergent mass from the one loop CW potential, while the ξ = ⎝ ∗ ⎠ + a − a a ΠΣ ξ/ 2 0√ h / 2 Q 1 Q 2 η doublet only obtains a logarithmically divergent mass. The domi- φ hT / 20 nant terms in the masses of these states are given by: T.A.W. Martin, A. de la Puente / Physics Letters B 727 (2013) 443–447 445 2 2 2 † 2 3 Λ 2 2 V =−λ f F Tr T (Σ − Σ )T ( − ) + h.c. (11) M = M + M , (7) Σ Σ Σ 0 Σ 0 χ 16π 2 f 2 + F 2 W H B H The operator TΣ has a certain amount of flexibility, but must be chosen to prevent mixing between the ξ and h fields. An oper- 3 Λ2 M2 = f 2 + F 2 g2 g2 log ator constructed from a linear combination of the generators of ξ 128 2 1 2 2 the [SU(2) × U (1)]2 symmetry is well motivated for this, since π MW H 1 2 these operators already preserve gauge invariance. In particular, 3 2 2 2 2 Λ + f + F g g log . (8) TΣ = n1Diag[1, 1, 0, 0, 0]+n2Diag[0, 0, 0, 1, 1] will resolve the 1280π 2 1 2 M2 B H mass splitting, and respects a Z2 symmetry that protects the ξ field from decaying. The following contribution to the masses of After electroweak symmetry breaking, contributions to the mass the neutral ξ states are generated with such an operator: of the ξ doublet proportional to the square of the vev, v2,arising from the one loop logarithmic terms create a small mass splitting 2 2 2 2 δmξ ≡Re[ξ] =−(n1 + n2) λΣv − 4 n − n fv, between the neutral and charged states, and the neutral compo- 0 1 2 = − 2 2 + 2 − 2 nent becomes the lightest state. δmΞ0≡Im[ξ] (n1 n2) λΣv 4 n1 n2 fv. (12) The CW potential also generates a small negative mass for As described in [24], the mass splitting needs only to be of the σ field, which would necessarily induce spontaneous symme- O (100 keV), indicating that small values of λ are acceptable. try breaking. To avoid this, we introduce a small, positive, explicit Σ Of note, a value of λ ∼ 0.01 will produce a mass splitting of mass term for the σ field of the form: Σ O (1GeV). Additional contributions to the masses of the χ0, η0, ± ± 4 † σ0, χ and η will also arise, but can be eliminated by setting ei- V = λ F Tr T( − 0)T( − 0) . (9) ther n1 = 0orn2 = 0. However, we use n1 = n2 = 1, as a simple The operator T has a certain amount of flexibility, so long as it implementation, since the contribution to the masses of the other does not violate gauge invariance. We take T = Diag[0, 0, 1, 0, 0], states will be of a similarly small nature, and thus unimportant to which is a minimal solution that avoids contributions to the the overall phenomenology of the model. masses of other fields. The value of λ is restricted by perturba- One interesting aspect of the NLH model is that, after decou- − tivity constraints only, but is taken to be O (10 1). The masses of pling the electroweak scalar triplets, the heavy top and the heavy the real singlet and triplet are then given by: electroweak gauge bosons, the scalar potential reduces to the inert doublet potential: 16 f 2 F 2 f 2 F 2 λ4 Λ2 M2 ≈ λ − t log , σ 2 2 2 2 2 2 2 2 2 2 2 2 4 4 2 2 5( f + F ) 40π ( f + F ) s c M V = μ |H1| + μ |H2| + λ1|H1| + λ2|H2| + λ3|H1| |H2| t t T 1 2 2 2 † 2 † 2 3( f 2 + F 2)g g Λ2 + H H + Re H H2 (13) 2 ≈ 1 2 λ4 1 2 λ5 1 . Mη log , 0 128π 2 M2 W H Within this framework, H1 ≡ h is the SM Higgs doublet that is ≡ 9( f 2 + F 2)g2 g2 Λ2 spontaneously broken; H2 ξ is the doublet from the sector; 2 ≈ 1 2 2 M ± log . (10) and the λΣ terms contribute to λ4 and λ5, where λ4 v contri- η 128π 2 M2 W H butions to the neutral component of H2 preserve a mass degener- 2 The parameters in the model are thus limited to the two acy and λ5 v contributions generate a mass splitting between the symmetry breaking scales, f and F ; three mixing angles (s = scalar and pseudoscalar states. = = = = = Many of the studies of inert doublet dark matter can thus sin θg g/g1, s sin θg g /g1, st sin θt λt /λ1) identical to those defined in the littlest Higgs model; the explicit scalar cou- be applied to the NLH model. For example, by comparing super- symmetry search results from LEPI and LEPII to the inert doublet pling λ; and two parameters which characterize the higher scale physics that is responsible for cancelling the divergences in the model, the authors in [26] found mass constraints of: Coleman–Weinberg potential, a and a .Theparametersa and a m 80 GeV, are the same as those defined in the littlest Higgs model; for a ξ0 more detailed discussion, see [6,22]. mΞ0 100 GeV, The heavy gauge bosons (W H , Z H , and A H ) and complex scalar m − m 8GeV (14) triplet states (φ and χ ) are typically quite heavy in the NLH model, Ξ0 ξ0 . at least in the several TeV range, while the heavy top partner can These constraints are easily satisfied within the NLH model. easily be lighter than a TeV. The real scalar triplet (η) and singlet The Planck Collaboration [27] has recently published updated states (σ ) masses typically vary between a few hundred GeV and results on the relative relic abundance of dark matter, giving a best the low TeV range. The mass of the complex doublet, ξ ,typically fit value of Ωh2 = 0.11889. Using FeynRules [28],wehaveimple- takes values below 1 TeV, which will be discussed in the following mented the model in the software package MicrOMEGAs [29] and section. calculated the relic density arising from the lightest stable state in our model, assuming a Higgs boson mass of 125 GeV. Using a 3. Dark Matter Monte Carlo method to select parameter values (0.05 < s < 0.95, 0.05 < s < 0.95, 600 < f < 2000, 0.2 < f /F < 0.8, 0.05 < st < As defined, the ξ is a degenerate, two component dark mat- 0.95, 0 <λΣ < 0.5, 0 <λ < 0.5, a = 1, a = 1), ∼ 130k mod- ter candidate (scalar and pseudoscalar), with a mass in the els were generated and the relic density calculated. Of the models O (100 GeV) range. Such degenerate, complex DM candidates nec- that produced viable masses for the DM candidate, 65.4% of the essarily generate a large direct detection signal through a vector parameter sets could only account for less than half of the relic coupling to nuclei [24]. A recent study of this phenomenon is density. Only 1.2% of the parameter space explored could account present in [25]. This can be resolved by introducing an explicit for 75–100% of the relic density, while 2.6% of the parameter space symmetry breaking term into the Lagrangian that breaks the acci- dental symmetry that maintains the mass degeneracy between the 1 Re[ξ] and Im[ξ] fields, such as: A similar argument motivates the operator T from the previous section. 446 T.A.W. Martin, A. de la Puente / Physics Letters B 727 (2013) 443–447

The parameter λΣ plays two roles in this, which accounts for the contours observed in Fig. 2. The ﬁrst is that λΣ directly controls the coupling between ξ and h: increasing the value of λΣ results in a larger ξξ → hh annihilation rate. This is the dominant annihilation mode in the region where the relic abundance predicted in the model is in the < 25% range. The second role is that λΣ controls the mass separation between ξ and = 2 + 2 − Ξ (M Mξ λΣv Mξ ), which affects the co-annihilation rates [31]. The relic abundance is otherwise understood as a man- ifestation of the inert doublet models [19,20,26,30].

4. Summary

We have presented a new class of little Higgs models, called dark little Higgs models, that employ the little Higgs method of re- solving the large quadratically divergent Higgs boson mass present in the standard model, and generate an inert doublet model that can simultaneously account for dark matter. In addition, we have presented a simple implementation of this class of models in the form of the next to littlest Higgs model – a modiﬁcation of the littlest Higgs model – and explored the relic abundance predicted in the model. We found that a heavy dark matter candidate with a mass on the order of 500 GeV can be generated for regions of ∼ Fig. 1. Density plot of the 130k parameter points examined, where darker shaded the parameter space that can account for the observed relic abun- regions correspond to a greater number of parameter points resulting in the value dance, in agreement with existing studies of inert doublet models. of mξ for a given value of f that account for the given relic abundance relative to the measured value. Histogram density scales are unique to each subplot. Acknowledgements

The authors would like to thank Heather Logan, Thomas Gré- goire, and David Morrissey for guidance and assistance. This work was supported by the National Science and Engineering Research Council of Canada (NSERC).

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