Dark Matter in a Simplest Little Higgs with T-Parity Model

Home , Little Higgs

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T-parity model. Here η is a real scalar field and h = (h0,h−)T is an SU(2) doublet which can be identified as the SM Higgs. The remaining 5 degrees of freedom of the symme- II. SIMPLEST LITTLE HIGGS WITH T-PARITY try breaking are “eaten” by the additional heavy gauge bosons and therefore are omitted from Eq.(2). Scalar field interactions with the gauge boson sector Little Higgs models [7, 8, 9, 10, 11, 12] offer an inter- and the masses of gauge bosons are determined by the esting alternative to supersymmetry in solving various standard kinetic term for the Φ [18]: problems of the SM. For instance, the hierarchy prob- i lem is resolved by introducing new heavy particles at the 2 a a igx x TeV scale that cancel quadratic divergences of the Higgs Φ = ∂ + igA T B Φ , (3) L µ µ − 3 µ i boson mass arising in the SM from one-loop radiative corrections. This cancellation occurs between particles where the SU(3) gauge coupling g is equal to the SM with the same statistics in Little Higgs models, as op- SU(2) gauge coupling and the U(1) gauge coupling g posed to the case of supersymmetry where divergences L X x is fixed in terms of g and the weak mixing angle tW are canceled by opposite statistics partners of the SM tan θ by ≡ particles. The original little Higgs models [10, 13, 14], W however, suffered from stringent constraints from elec- gtW gx = . (4) troweak precision tests (EWPT) which required the in- 1 t2 /3 troduction of fine tuning. An elegant solution to this − W problem is to introduce a discrete symmetry (like the R- The gauge bosons correspondingp to the broken gener- parity in supersymmetry) called T-parity [15]. T-parity ators get masses of order f TeV and consist of a Zh not only remedies the problem of precision electroweak boson, which is a linear combination∼ of A8 and Bx constraints, but also leads to the appearance of promising dark matter candidates. We must note that, recently, 1 2 8 x Zh = 3 t A + tW B , (5) it has been pointed out that T-parity in general may be √ − W 3 violated by anomalies [16], making the lightest T-odd q 0 − particle unstable. However, such violation strongly de- and a complex SU(2)L doublet (Y ,X ) pends on the UV completion of the Little Higgs model A6 iA7 A4 iA5 and may not be realized in nature [17]. Consequently, we X− = − , Y 0 = − . (6) consider models respecting T-parity here. √2 √2 The Simplest Little Higgs with T-parity model [6] is 0 The SM mass eigenstates W ±, Z , and A are also ex- constructed by enlarging the SM SU(2) U(1) gauge L ⊗ Y pressed in terms of the gauge fields: group to SU(3)W U(1)X in a minimal way. This en- tails enlarging the⊗ SU(2) doublets of the SM to SU(3) 1 2 ± A iA triplets, including additional SU(3) gauge bosons, and W = ∓ , (7) √2 writing SU(3) invariant interactions in a way that re- produces all the SM couplings when restricted to SM fields. The SU(3)W U(1)X gauge symmetry is broken 0 3 tW 8 x 2 ⊗ Z = A cW + (A sW B cW 3 tW ), (8) down to the SM electroweak gauge group by two com- √3 − − q plex scalar fields Φ1,2, which are triplets under the SU(3) with aligned vevs f1,2. When the scalar fields acquire 2 3 8 sW x cW 2 vevs, the initial global symmetry [SU(3)W U(1)X ] is A = A sW A + B 3 tW . (9) 2 √ √ spontaneously broken to [SU(2) U(1) ] .⊗ At the same − 3 3 − ⊗ X q time, the global symmetry is explicitly broken to its di- As in the SM, masses of the heavy gauge bosons are deter- agonal subgroup SU(3) U(1) by the gauge interactions. ⊗ mined by the symmetry breaking vev and can be written The scalar fields are conveniently parameterized by a as: 2 nonlinear sigma model of the [SU(3) U(1)X ] /[SU(2) U(1) ]2 symmetry breaking as ⊗ ⊗ √2gf gf X ± 0 MZh = ,MX = MY = . (10) 3 t2 √2 − W f 0 f 0 iΘ 2 −iΘ 1 Φ1 = e f1 0 , Φ2 = e f2 0 (1) The symmetryp breaking SU(2) U(1)/U(1)em by the     × f1 f2 vev of the SM Higgs introduces corrections to the masses     of the gauge bosons. For example, the charged heavy where bosons X± become slightly lighter than the neutral gauge boson Y 0. However, for our studies of dark matter 0 0 physics, we are interested only in the first order effects 1 η 2 2 2 2 Θ= + 0 0 and f = f1 + f2 . (2) and therefore will disregard (v/f) corrections to masses f √2  h† 0  and to the couplings in the Feynman rules.    3

Anomaly-free embedding which immediately requires f1 = f2 = f/√2. This def- inition of T-parity also determines the following trans- fermion (SU(3)c × SU(3)w)U(1)X formation rules for the gauge boson sector, which follow Q1,2 (3, 3¯)0 from the requirement of the T-parity invariance for the Q3 (3, 3) 1 3 kinetic term (3), c c ua,T (3¯, 1) −2 3 c c c A ΩˆA Ωˆ. (14) da, D ,S (3¯, 1) 1 µ µ 3 → Lm (1, 3) −1 3 It is important to note that, under this transformation, c Nm (1, 1)0 the new neutral gauge boson Zh remains even, and hence c its mass is required to be relatively heavy in order to em (1, 1)1 avoid the precision electroweak constraints.

TABLE I: The SU(3)c ⊗ SU(3)W ⊗ U(1)X representations T-parity invariance requires that both scalar triplets of the fermions in the anomaly-free embedding of the Sim- Φ1 and Φ2 couple to the fermion fields with equal plest Little Higgs with T-parity model. Index a runs over the strength. This introduces major modifications to the first two generations of quarks, while m runs over all three standard Simplest Little Higgs model. Yukawa interac- generations of leptons. tions in the leptonic sector can be written as

c † † l = iλNm Nm(Φ1 +Φ2)Lm + (15) There are two possible gauge charge assignments for L m iλe c i j k fermions in the Simplest Little Higgs model. The first + emǫijkΦ1Φ2Lm + h.c., (universal embedding) yields heavy partners of the top, Λ charm and up quarks, with all generations carrying iden- where m =1, 2, 3 is a generation index and i, j, k =1, 2, 3 tical quantum numbers. However, this embedding is are SU(3) indices. Yukawa Lagrangians for the third gen- known to be anomalous and also appears to be ruled out eration of quarks and for the first two generations are, by precision electroweak constraints [14]. In the second respectively, scenario (anomaly-free embedding) the top, strange and c + + down quarks have heavy partners. The first two gener- 3 = iλtQ3[tL Φ2 Φ1 + (16) ¯ L − ations of quarks transform under the antifundamental 3 λ +tc Φ+ +Φ+ ]+ b ibcǫ Φi Φj Qk + h.c., representation of the SU(3), while the third generation H 2 1 Λ ijk 1 2 3 3 quarks and leptons transform as ’s [18]: T c c n T nc Lm = (ν,e,iN)m, iem,iNm, n = iλDQn (Φ1 +Φ2)dH + (17) T c c c L n Q1 = (d, u,iD), id ,iu ,iD , n T nc λu c ∗i ∗j k − +iλd Qn (Φ1 Φ2)dL + i unǫijkΦ1 Φ2 Qn + h.c., QT = (s, c,iS), isc,icc,iSc, − Λ 2 − T c c c c Q3 = (t,b,iT ), it ,ib ,iT , (11) where n =1, 2; i, j, k =1, 2, 3 are SU(3) indexes; tL and c c tH are T-even and T-odd linear combinations of t and where m is the generation index, and uc, dc, ... are c nc nc c T , while dH and dL are linear combinations of d and the right-handed Weyl fermions invariant under SU(3)W c c c c D for n = 1 and of s and S for n = 2; um runs over that marry corresponding components of the SU(3)W all the up-type conjugate quarks (uc,cc,tc). Also, we are triplets to produce fermion masses. Hypercharge assign- neglecting flavor mixing here. ments and representations of the fundamental fields in The requirement of T-invariance leads to a simple re- the anomaly-free embedding are summarized in Table I. lation for the heavy top mass: We consider only this case. Following the initial investigation [6], we further intro- √2f M = M . (18) duce a discrete symmetry (T-parity) into the model to T t v avoid precision electroweak constraints and to produce a heavy stable particle as a dark matter candidate. A The masses of the remaining new fermions come from straightforward way to implement T-parity into the Sim- the Yukawa interaction in the form Mf = √2λf f. We plest Little Higgs model is to start with the fermionic sec- choose a universal value of the Yukawa coupling λ for all tor. The action of the T-parity operator on the fermion new fermions, except for the first generation heavy neu- triplets can be defined as [6] trino. In this work we choose λ to be large and hence additional fermions decouple and have negligible influ-

ΨQ,L ΩΨˆ Q,L, (12) ence on the results considered. We also note that a more →− general study without decoupling of the fermions should where Ωˆ = diag( 1, 1, 1). For the scalar fields the be undertaken. transformation is − − Introduction of T-parity makes the lightest T-odd particle (LTP) stable. There are two neutral heavy particles Φ1 ΩΦˆ 2 (13) that should be considered as the potential LPT: a heavy → 4 neutrino N and a new scalar η. The mass of η depends on where c is a constant of order one determined by match- the mass of the SM Higgs boson as well as on the masses ing the late-time and early-time solutions. of other heavy particles through the Coleman-Weinberg Major annihilation reactions that kept the heavy neu- potential and is generally of order Mη > 700 GeV [18]. trino coupled to the primordial plasma before decoupling Although it is possible to make η a LTP it∼ requires signif- are shown in Fig. 1. It is important to emphasize that in icant fine tuning in the parameter space. Moreover, the this paper we consider additional quarks as well as the coupling of η to other particles in the primordial plasma second and third generation heavy neutrinos to be much is usually f 2 suppressed which makes it practically im- heavier than the lightest T-odd particle and, hence, the possible to satisfy relic density constraints [6]. On the effects of coannihilations are ignored. Without coanni- other hand, the mass of a heavy neutrino remains a free hilations the dark matter relic density is determined by ± parameter in the model and can be chosen to be signif- the Zh s-channel exchange as well as by the t-channel X icantly lighter than the masses of other particles in the and Y processes. The t-channel processes are especially model. A heavy neutrino is weakly coupled to the rest important for the heavy LTP. The increase in the cross ± of the particles through Zh and X , Y exchanges and section caused by the t-channel processes and its impact it decouples at the right time to produce the dark mat- on the dark matter freeze-out is depicted in Fig. 2. ter density observed today. In light of this, we choose to For the calculation of the relic density constraints we consider a heavy neutrino as a dark matter candidate of wrote a Mathematica code using the Feyncalc package the Simplest Little Higgs with T-parity Model. to calculate invariant amplitudes of the annihilation processes shown in Fig. 1. The constraints that arise are provided in Fig. 3. We plot allowed regions that fall III. RELIC ABUNDANCE CONSTRAINTS within 2σ and 5σ around the result obtained by the 2 WMAP team 0.094 < ΩDM h < 0.126. The allowed The amount of dark matter today, the relic density, region can be conveniently parametrized by a linear rela- has been measured with good precision by WMAP as tion that links the dark matter mass, MN , and the sym- 2 0.094 < ΩDM h < 0.126 [2]. Here ΩDM is the ratio of metry breaking scale, f, of the initial global symmetry dark matter density to the critical density of the universe in the SLHT: and h parametrizes the Hubble constant today. From a f =3.2M + 600, (21) theoretical side we can calculate the relic density of dark N matter in terms of the major parameters of any models where f and MN are in GeV. that produce a stable WIMP and hence impose stringent constraints on the allowed parameter space of the models. In the early universe, dark matter particles were IV. COLLIDER SIGNATURES present in large numbers in thermal equilibrium with the rest of the cosmic plasma; as the universe cooled, they As we have shown in the previous sections, astrophys- could reduce their density through pair annihilation. As ical observations provide stringent constraints on the their density decreases, however, it becomes more and properties of dark matter particles and, hence, can di- more difficult for particles to find partners to annihilate rect the search for dark matter at existing and future with, and the comoving density becomes constant, known colliders. Astrophysical probes, however, are unable to as “freeze-out”. Therefore, the present day density of determine and study the exact properties of a dark mat- dark matter particles is completely determined at the ter particle and it largely remains for collider physics to time of freeze-out. It can be conveniently expressed in unravel the mystery surrounding dark matter. terms of the thermally averaged product of the annihila- Regardless of its origin, if cold dark matter is composed tion cross velocity σv at the time of freeze-out [19]: h i of WIMPs, then it may be possible to produce and study 9 −1 the dark matter particle(s) directly at the LHC [21] . In 2 1.07 10 GeV xF 1 any collider experiment, WIMPs would be like neutrinos Ωχh × . (19) ≈ MPl √g∗ σv in that they would escape the detector without depositing h i any energy in the experimental apparatus, resulting in an Here g∗ counts the number of relativistic degrees of free- apparent imbalance of energy and momentum in collider dom and is approximately equal to the number of bosonic events. While WIMPs would manifest themselves only relativistic degrees of freedom plus 7/8 of the fermionic as missing (transverse) energy at collider experiments, it relativistic degrees of freedom evaluated at the freeze- should nevertheless be possible to analyze the visible par- out temperature (a plot of g∗ as a function of time ticles produced in association to study the new physics can be found for instance in [20]). Also x m/T F ≡ F associated with the WIMP sector. parametrizes freeze-out temperature TF and can be esti- We identify two promising channels of dark matter pro- mated through iterative solution of the equation duction for the model under consideration: ¯ 45 g m M σv pp jet + NN, x = ln c(c + 2) Plh i , (20) → F 3 1/2 1/2 + − ¯ " 8 2π g∗ x # pp l l + NN, r F → 5

N¯ f¯ N¯ f N ν

X±(Y ) Y Zh

N f N f¯ N ν

FIG. 1: Dominant channels for dark matter annihilation in the Simplest Little Higgs with T-parity model. These diagrams provide the largest contributions to the heavy neutrino annihilation cross section for the range of SLHT parameters examined in this paper.

106 3000

105

, fb 2500

> 4 v 10 <Σ

1000 2000 f, GeV

100 200 300 400 500 600 M , GeV N 1500 FIG. 2: t-channel contribution (short-dash line) to the thermally averaged annihilation cross section hσvi (dashed line) in the Simplest Little Higgs with T-parity model as a function of dark matter mass. The WMAP allowed region (gray) is also 1000 shown for reference. This plot emphasizes the contribution of 100 200 300 400 500 600 the t-channel diagrams to the annihilation cross section. MN , GeV where l+l− represents an e+e− or µ+µ− pair. Feynman FIG. 3: Relic abundance constraints on the parameter space of the Simplest Little Higgs with T-parity model. On this diagrams contributing to these channels are presented in plot the dark region corresponds to 2σ variation about the Figs. 4 and 5, respectively. Cross sections at LHC en- WMAP observed relic density, while the gray region covers ergy as a function of dark matter mass for these two 5σ confidence level. channels are shown in Fig. 7. Monojet associated dark matter production is enhanced by the QCD coupling and we can expect a large number of events for this channel. A large cross section of the monojet plus missing en- be around 10 fb−1 after the first year of operation and ergy channel makes it an extremely important signature more than 300 fb−1 over the full period of operation. of the Simplest Little Higgs with T-parity model. This From Fig. 7 we can see that, if the Simplest Little Higgs signature may help to distinguish this model from both mechanism is realized in nature, and the heavy neutrino supersymmetry, in which dark matter is predominantly is light enough, then dark matter indeed will be copi- produced in multijet events, and the Littlest Higgs with ously produced at the LHC. However, the SM model T-parity model, in which monojet dark matter produc- background is also significant. Some Feynman diagrams tion is highly suppressed. In fact we identify monojet contributing to the SM background of the monojet and associated dark matter production as a smoking-gun sig- dilepton channels are presented in Fig. 6. The total cross nature of the Simplest Little Higgs with T-parity. If the section at LHC for the monojet plus missing energy pro- SLHT is realized in nature then this process should be duction in the SM is σ 2 103 pb, while for the dilepton observed at a very early stage of the LHC operation and plus missing energy channel≈ · σ 2 pb. In both cases we the model can be confirmed with the dilepton channel expect many more background≈ events than DM events. which has a distinct signature, but a lower cross section. However, because of the large mass of the dark matter The integrated luminosity of the LHC is expected to particle, signal events appear in the kinematic regions 6

g (jet) q (jet) q¯ g q q N N¯ Zh q Zh q

N¯ N

FIG. 4: Sample Feynman diagrams for the monojet associated dark matter production at the LHC in the Simplest Little Higgs with T-parity model.

l− N Zh − q q¯ X N¯ ¯ N¯ q Zh − X+ N l q q Z l+ l+

FIG. 5: Sample Feynman diagrams for the dileptonic dark matter production at the LHC in the Simplest Little Higgs with T-parity model. where the SM background is relatively small. To study the LHC signal of dark matter production in the SLHT we implemented the model into the Mad- Parameter in GeV Set 1 Set 2 Set 3 Graph package [22]. Based on the relic density con- f 1000 1560 2520 straint we have chosen three sets of parameters that MN 125 300 600 cover most of the allowed parameter space and hence MZh 560 872 1409 are representative of the model in general.Our param- Mh 120 120 120 eters are given in Table II. Also, for all collider cal- ΓZ 4.0 5.8 8.6 culations we use the standard set of major SM param- h −5 −2 MX 460 717 1158 eters [23]: GF = 1.16637(1) 10 GeV – Fermi constant, α = 1/137.036(1) –× fine-structure constant, ΓX 1.1 1.5 2.0 2 MT 983 1533 2477 sinθW (MZ )=0.23152(14) – square of the electroweak mixing angle, and αs(MZ )=0.1176(20) – strong cou- MD, MS, MN2 , MN3 707 1103 1782 pling constant. In Fig. 8 and 9 we show the predicted signal of the two TABLE II: Parameters used for the analysis of dark matter production at the LHC in the Simplest Little Higgs with T- missing energy channels compared to the SM background parity model. Values of the tested parameters are chosen to for various values of the dark matter mass. A bin-by-bin 2 be consistent with the relic density constraints from Fig. 3 χ analysis of the data clearly shows the possibility of and to cover most of the allowed parameter space. 5σ detection of dark for the considered region of SLHT parameter space. In order to increase the signal to background ratio we can impose kinematic cuts on the observed particles as well as on the missing energy. To determine optimal cuts variety of kinematic cuts depending on the specific objec- we considered predicted distributions for the observed tives. In this paper, we considered a cut on the charged particles as well as for missing energy in both the SLHT lepton momentum pT which allows us to significantly in- and the SM. In this study we analyze cuts only for the crease the threshold on the dark matter mass that can be dilepton channel, since the monojet signal is very well observed in the dilepton channel. In Fig. 10 we show the distinguished over the SM background without introduc- dilepton plus missing energy signal for the heavy neutrino ing any cuts over all values of the considered parameters. mass of MN = 300 GeV both before and after kinematic Leptons produced in the dark matter related processes cut pT > 200 GeV. This cut reduces the cross section of are more energetic and mainly appear in the central re- the SM background by more than 3 orders in magnitude, gion of the detector. This suggests the introduction of a while the signal is decreased only 2 orders. Remarkably, 7

l− q (jet) g − q q¯ W ν¯ ν¯ Z Z ν W + q q

l+ ν

FIG. 6: Sample Feynman diagrams for the Standard Model background to the dileptonic (left) and monojet (right) associated dark matter production at the LHC.

f (TeV) 1 1.2 1.4 1.6 1.8 2 2.2 2.4 -1 8 10 MN = 125 GeV 105 LHC, 14 TeV MN = 300 GeV 10 → pp j + DM MN = 600 GeV → + - pp l l + DM 4 Background 107 10

1 -1 103 106 Events / 20 GeV 300 fb (pb) σ 10-1 102 Events/300 fb 105 10 10-2

104 1 500 1000 1500 -3 Missing E [GeV] 10100 200 300 400 500 600 T MN (GeV) FIG. 8: Missing energy distribution expected in the SLHT in the dilepton plus missing energy channel. The SM back- FIG. 7: Cross section of dark matter production at the LHC ground is also plotted for comparison. is plotted as a function of dark matter mass. On the top scale we identify the SLHT symmetry breaking scale f that corresponds to the dark matter mass via the relic density constraints of Eq. 21. The number of events for an integrated constraints on the model’s parameter space are imposed. 1 luminosity of 300 fb− is plotted on the second y-axis. It has been shown that, if the heavy neutrino is respon- sible for the observed dark matter in the universe, then its mass is approximately linearly related to the value of this cut also allows us to study a different aspect of the the SLH symmetry breaking scale f as f 3.2M +600, ≃ N SLHT model, namely the mass of the new heavy neutral where f and MN are in GeV. gauge boson Zh, with the observation of the invariant We have analyzed couplings specific to the Simplest mass distribution for the charged leptons. In Fig. 10, we Little Higgs with T-parity (SLHT) model and introduced clearly see that the distribution in the SLHT model has the model into the MadGraph package. Collider phe- a distinctive peak around the mass of Zh that is strongly nomenology of SLHT dark matter was discussed and we emphasized by the introduction of the cut. have shown that, for a large region of the allowed parameter space, dark matter particles could be detected at the LHC. Problems of the dark matter analysis at the V. SUMMARY LHC were outlined. We showed that the SLHT has an important monojet plus missing energy signature that In this paper, we have investigated dark matter can- clearly distinguishes it from both the Littlest Higgs with didates that arise in the Simplest Little Higgs model af- T-parity model and from supersymmetry. A dileptonic ter the introduction of a discrete Z2 symmetry called T- channel of dark matter production was shown to be a parity. A heavy neutrino is identified as the most promis- promising independent channel of heavy neutrino pro- ing dark matter candidate of this model and relic density duction that can be used to confirm the SLHT mecha- 8 -1 -1 M = 125 GeV N Signal MN = 300 GeV 6 M = 600 GeV 10 N Background Background 102 105

104 Events / 20 GeV 300 fb Events / 15 GeV 10 fb 3 10 10

102

10 1 200 400 600 800 1000 1200 1400 1600 1800 1 m [GeV] 200 400 600 800 1000 1200 1400 ij Missing E [GeV] T -1 FIG. 9: Missing energy distribution expected in the SLHT in Signal the one jet plus missing energy channel. The SM background is also plotted for comparison. Background 104 nism if the larger monojet dark matter signal is detected. We also presented various kinematic distributions for the 103 leptons of the dilepton associated dark matter production and discussed kinematic cuts that provide signiﬁcant en- Events / 20 GeV 300 fb hancement of the signal to background ratio and thus 102 permit observation of dark matter particles with higher mass as well as allow to study various properties of the gauge sector of the SLHT. 10

200 400 600 800 1000 1200 1400 1600 1800 Acknowledgments mij [GeV]

D.T. would like to thank Aliaksei Charnukha for valu- FIG. 10: Invariant mass distribution of electrons in dilepton able discussions and technical support throughout this plus missing energy production for MN = 300 GeV with pT > research. D.T. is grateful to Dr. Michael Santos for his 200 GeV cut imposed(top) and without cuts (bottom). SM support and advice during this work. The authors are expectation for this channel is also presented. also grateful to Ken Moats for insightful discussions during the implementation of the SLHT into MadGraph and subsequent analysis of the data. This research was sup- symmetry factors equal, f1 = f2. In the notation of [18] ported in part by the Natural Sciences and Engineering this requirement leads to tanβ = f2/f1 = 1. We summa- Research Council of Canada. rize the triple and quartic gauge boson vertex factors in Tables III and IV, respectively. The triple and quartic couplings are given in the form APPENDIX: FEYNMAN RULES FOR THE µ ν ρ µν ρ SIMPLEST LITTLE HIGGS WITH T-PARITY V1 (k1)V2 (k2)V3 (k3): i[g (k1 k2) + MODEL νρ µ ρµ− ν− +g (k2 k3) + g (k3 k1) ]g , − − V1V2V3 In this appendix we provide a list of Feynman rules µ ν ρ σ specific to the SLHT model. As discussed in Section 2, V1 V2 V3 V4 : µν ρσ µρ νσ νρ µσ implementing T-parity in the gauge and scalar sectors of i(2g g g g g g )gV1V2V3V4 , the Simplest Little Higgs model does not require signifi- − − − cant change to the structure of the original Simplest Lit- where gV1V2V3 and gV1V2V3V4 are vertex factors provided tle Higgs model. The major effect in the gauge sector is to in the tables. We also must note that the W −Z0X+Y 0 set the aligned vevs of the two global [SU(3) U(1) ] and W +Z0X−Y¯ 0 couplings can not be written in the W ⊗ X i 9 standard form due to the approximate nature of the should be replaced by T a∗ – generators of the antifun- ¯ − gauge eigenstates. The appropriate couplings have the damental 3 representation of the SU(3)W . Fermion cou- form: plings to the gauge bosons in the SLHT are identical to those of the original Simplest Little Higgs model and are provided in Table VI. We note, however, that our value 2 − 0 + 0 isW for the coupling of D and S quarks to the heavy Zh is Wµ Zν Xρ Yσ : − [2gµν gρσ 2√2cW − slightly different from the coupling calculated in [18]. 2 Finally, in Table VII, we provide the couplings of the gµσgνρ gµρgνσ] 3c (gµσgνρ gµρgνσ), − − − W − new heavy gauge bosons and the scalar Higgs boson. We also note that, because the new electroweak singlet η is a real scalar field, it does not couple directly to the 2 gauge bosons, while its quartic couplings are f 2 sup- + 0 − ¯ 0 isW Wµ Zν Xρ Yσ : − [2gµν gρσ pressed. Therefore, for the investigation of dark mat- 2√2cW − ter phenomenology, effects of the new heavy scalar were g g g g ] 3c2 (g g g g ). − µσ νρ − µρ νσ − W µσ νρ − µρ νσ completely negligible.

Particles gV V V Particles gV V V + + W W −A −gsW X X−A −gsW 2 + 0 + 0 g 1 2sW W W −Z −gcW X X−Z − − 2 cW + + g 2 W W −Zh 0 X X−Zh 2 3 − tW Y 0Y¯ 0Z0 − 1 g Y 0Y¯ 0A 0 2cW p 0 0 g 2 + 0 g Y Y¯ Zh − 3 − t X W −Y − 2 W √2

TABLE III: Feynmanp rules for the triple gauge boson interactions in the Simplest Little Higgs with T-parity model. All momenta are deﬁned as outgoing. The SM couplings of W ± to Z0 and photon are also presented for reference. In the table sW , cW , and tW stand for the sine, cosine and tangent of the electroweak mixing angle respectively.

The fermion sector of the Simplest Little Higgs with T- parity model is very different from the original Simplest Little Higgs. The modifications are especially significant in the Yukawa interactions. Things are also complicated by the fact that, choosing the anomaly-free implementation of fermion assignments, the third generation quarks transform differently from the quarks of the first two generations under the global symmetry of the model. For the purposes of dark matter study we ignore mixings in the leptonic and quark sectors. This leads to a consider- able simplification in the Feynman rules, and allows us to concentrate on the most significant “smoking-gun” signatures of the model. The more complete set of couplings including v2/f 2 corrections and various mixing can be derived from generic Simplest Little Higgs Lagrangians provided in [18] after imposing the T-parity requirements described in Section 2. In Table V we show scalar-fermion couplings that arise from the Yukawa Lagrangians introduced in Section 2. The fermion couplings to the gauge bosons are determined by the standard fermion kinetic term:

= ψi¯ γµψ, = ∂ + igAaT a + ig Q Bx, L Dµ D x x a with the Qx charges given in Table I. T are the generators of the fundamental 3 representation of the SU(3)W . In the case of the ﬁrst and second generation quarks T a 10

Particles gVVVV Particles gVVVV + 2 2 + + 2 W W −AA g sW W W W −W − −g 2 + 2 2 + 0 0 g 2 2 − − 2 X X AA g sW X X Z Z c (1 − 2sW ) 4 W 2 2 + g 2 + 0 g sW 2 X X−ZhZh (3 − t ) X X−AZ (1 − 2s ) 4 W 2cW W 2 2 + g sW 2 + 0 g 2 2 X X−AZh − 3 − t X X−Z Zh − (1 − 2s ) 3 − t 2 W 4cW W W 2 0 ¯ 0 g 2 0 ¯ 0 0 0 1 2 Y Y ZhZh 4 (3 − tW ) Y Y Z Z 4c2 g p W p 2 0 0 0 g 2 0 0 0 0 2 Y Y¯ ZhZ 3 − t Y Y¯ Y Y¯ g 4cW W 2 + 0 sW 2 + 0 g 2 W −X AY g W −ZhX Y 3 − t √2 2√2 W p 2 + 0 sW 2 + 0 g 2 W X−AY¯ g W ZhX−Y¯ 3 − t √2 2√2 p W 2 + ¯ 0 0 g + + 2 X Y X−Y − 2 X X−X X− g 2 p 2 + 0 ¯ 0 g + + g W Y W −Y − 2 W X W −X− − 2 2 0 + 0 isW 2 Wµ−Zν Xρ Yσ − (2gµν gρσ − gµσgνρ − gµρgνσ) − 3cW (gµσgνρ − gµρgνσ) 2√2cW 2 is + 0 ¯ 0 W 2 Wµ Zν Xρ−Yσ − (2gµν gρσ − gµσgνρ − gµρgνσ) − 3cW (gµσgνρ − gµρgνσ) 2√2cW

TABLE IV: Feynman rules for the quartic gauge boson interactions in the Simplest Little Higgs with T-parity model. Sample SM couplings are also provided for reference.

Particles Vertices Particles Vertices Particles Vertices vMNm MT vMD ηN¯mνm −i 2 PL ηTt¯ −i PL ηDd¯ −i 2 PL 2f √2f 2f M ¯ Nm ¯ MT ¯ MD ηηNmNm 2f 2 ηηTT 2f 2 ηηDD 2f 2 vM ¯ Nm ¯ vMT ¯ vMD hNmNm 2f 2 hTT 2f 2 hDD 2f 2 M ¯ Nm ¯ MT ¯ MD hhNmNm 2f 2 hhTT 2f 2 hhDD 2f 2

5 TABLE V: Feynman rules for scalar-fermion couplings in the Simplest Little Higgs with T-parity model. PL = (1 − γ )/2 is a projection operator. Only the leading terms are given.

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Particles Vertices Particles Vertices 0µ g µ 2 2 2 µ g µ 1 2 Z ee : 2c γ [(cW − sW )PL − 2sW PR] Zh νν : − 2 γ ( 2 − sW )PL W c 3 4s W − W µ g µ 1 1 2 2 2 µ g µ 2 Zh tt : − 2 γ [( 2 − 3 sW )PL + 3 sW PR] Zh NN : − 2 γ (−1+ sW )PL c 3 4s c p3 4s W − W W − W µ g µ 1 1 2 1 2 ig − Zh bb : − 2 γ [( 2 − 3 sW )PL − 3 sW PR] Xµ Du : √ γµPL c p3 4s p 2 W − W µ g µ 1 2 2 2 2 0 ig Zh uu : − 2 γ [(− 2 + 3 sW )PL + 3 sW PR] Yµ Dd : − √ γµPL c p3 4s 2 W − W µ g µ 1 2 2 1 2 ig − Zh dd : − 2 γ [(− 2 + 3 sW )PL − 3 sW PR] Xµ eN : − √ γµPL c p3 4s 2 W − W µ g µ 1 2 2 0 ig Zh ee : − 2 γ [( 2 − sW )PL − sW PR] Yµ νN : − √ γµPL cp 3 4s 2 W − W µ g µ 5 2 2 2 ig − Zh TT : − 2 γ [(−1+ 3 sW )PL + 3 sW PR] Xµ bT : − √ γµPL c p3 4s 2 W − W µ g µ 4 2 1 2 0 ig Zh DD : − 2 γ [(1 − 3 sW )PL − 3 sW PR] Yµ tT : − √ γµPL c p 3 4s 2 W − W

p 5 TABLE VI: Feynman rules for gauge boson-fermion couplings in the Simplest Little Higgs with T-parity model. PL = (1−γ )/2 5 and PR = (1+ γ )/2 are projection operators. Couplings for the second generation quarks are identical to those of the ﬁrst generation. The SM lepton Z0-boson coupling is also provided to clarify our phase choice. Only the leading terms are provided.

Particles Vertices + 1 2 W W −h : 2 g v + 1 2 X X−h : − 2 g v + 1 2 X X−hh : − 2 g 2 2 0 ¯ 0 g v v Y Y h : 8 f 2 2 2 0 ¯ 0 g v Y Y hh : 8 f 2 2 g v ZhZhh : − 2c2 W 2 g ZhZhhh : − 2c2 W 2 (1 t ) 0 1 2 − W Z Zhh : 2 g v 2 c t W 3 W 2 − (1 t ) 0 1 2 − W Z Zhhh : 2 g 2 c p3 t W − W

TABLE VII: Feynman rules for gauge boson-scalar couplings in the Simplestp Little Higgs with T-parity model. The SM coupling of the W ± to the Higgs ﬁeld h is also provided for reference. Factor of gµν is implied in each vertex.