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ZU-TH 10/09 A Little Higgs Model with Exact Dark Matter Parity 0 A. Freitas1, P. Schwaller2, D. Wyler2 1 0 2 n 1 Department of Physics & Astronomy, University of Pittsburgh, a J 3941 O’Hara St, Pittsburgh, PA 15260, USA 6 2 2 Institut fu¨r Theoretische Physik, Universität Zu¨rich, Winterthurerstrasse 190, CH-8057 Zu¨rich, Switzerland ] h p - p e h [ 4 v 6 Abstract 1 8 1 Based on a recent idea by Krohn and Yavin, we construct a little Higgs model . with an internal parity that is not broken by anomalous Wess-Zumino-Witten terms. 6 0 The model is a modification of the “minimal moose” models by Arkani-Hamed et al. 9 and Cheng and Low. The new parity prevents large corrections to oblique electroweak 0 parameters and leads to a viable dark matter candidate. It is shown how the complete : v Standard Model particle content, including quarks and leptons together with their i X Yukawacouplings,canbeimplemented. Successfulelectroweak symmetrybreakingand r consistency with electroweak precision constraints is achieved for natural parameters a choices. A rich spectrum of new particles is predicted at the TeV scale, some of which have sizable production cross sections and striking decay signatures at the LHC. 1 Introduction Little Higgs models are effective non-supersymmetric theories with a natural cutoff scale at about 10 TeV, where the Higgs scalar is a pseudo-Goldstone boson of a global symmetry, whichisspontaneouslybrokenatascalef 1TeV.Thesymmetry breakingpatternprotects ∼ the Higgs mass from quadratically divergent one-loop corrections, which are cancelled by new gauge bosons and fermions with masses near f. Therefore the hierarchy of scales can be realized without fine-tuning the parameters in the Higgs potential. A simple implementation of this mechanism is given by the “minimal moose” model of Ref. [1]. This model has two copies of the Standard Model (SM) gauge group, which are broken to the diagonal group at the scale f, reminiscent of chiral symmetry breaking in QCD. However, tree-level mixing between the gauge bosons introduces large corrections to the oblique electroweak parameters for f 1 TeV, unless the gauge couplings of the two gauge ∼ sectors are almost equal [2]. This equality of couplings can be explained by a discrete symmetry called T-parity [3,4], under which the SM fields are T-even and the new TeV- scale particles are odd1. As a result, all tree-level interactions between T-even and T-odd particles are forbidden, so that corrections to the electroweak precision observables occur only at one-loop level and thus are sufficiently small to allow values of f of 1 TeV and below. Furthermore, the lightest T-odd particle is stable and, if neutral, can be a good dark matter candidate. Often it is assumed that the new physics entering near the scale of 10 TeV are some strong dynamics similar to technicolor theories2. In this case, however, the fundamental theory can induce a Wess-Zumino-Witten (WZW) term [8], which is T-odd [9] if T-parity is implemented as in Ref. [4]. The breaking of T-parity by the WZW term, though suppressed by the large symmetry breaking scale, rules out the lightest T-odd particle as a dark matter candidate, since this particle would decay promptly into gauge bosons [10]. On the other hand, it was recently shown that a different construction of the parity in moose models leads to a parity-even WZW term [11]. The authors present a simple toy model that shows the relevant features. In this article we adopt the idea of Ref. [11] for the “minimal moose” model in order to constructafullyrealisticmodelwhichreproducestheStandardModelasalow-energytheory, admits electroweak symmetry breaking (EWSB), is consistent with electroweak precision constraints, and has a viable dark matter candidate. In the following section, the model and theimplementation of thenew X-parityis described explicitly. In section 3 thephysical mass spectrum of the model is analyzed, and it is shown that successful electroweak symmetry breaking can be achieved. Finally, section 4 discusses electroweak precision constraints and gives a brief overview of the collider phenomenology, before the conclusions are presented in section 5. 1A different discrete symmetry, which does not lead to a complete doubling of the SM particle content, has been proposed in Ref. [5,6]. 2An alternative approachinvolving a weakly coupled symmetry breaking sector can be found in Ref. [7]. 1 2 The model The model is based on a large SU(3)8 = [SU(3) SU(3) ]4 global symmetry group that is L R × spontaneously broken to the diagonal vector group SU(3)4 at a scale f, giving rise to four V sets of SU(3) valued nonlinear sigma model fields X = e2ixi/f, i = 1,...,4. (1) i Under the global symmetry group they transform as X1,3 → L1,3X1,3R1†,3 and X2,4 → R2,4X2,4L†2,4. The axial components of the global symmetries shift the Goldstone fields, x x +ǫ , thereby forbidding any nonderivative couplings for the Goldstone fields. In par- i i i → ticular, as long as these symmetries are not explicitly broken, a mass term can’t be generated for the Goldstone fields at any loop order. Adding gauge and Yukawa interactions will in general break some of the global symmetries and therefore generate (f) mass terms for the corresponding Goldstone bosons. The O idea of collective symmetry breaking is to implement the required interactions in such a way that each interaction respects parts of the global symmetry and therefore keeps the corresponding Goldstone bosons massless. Only the simultaneous presence of different symmetry breaking interactions can then generate a mass for those Goldstone bosons. Since appropri- ate diagrams only appear at the two-loop level, the generated masses are suppressed by an additional loop factor and can be significantly below the scale f. Our goal is to have at least one light electroweak doublet that we can identify with the SM Higgs boson. Under the SM gauge interactions, the Goldstone fields x decompose as i follows φ +η /√12 h /2 x = i i i , (2) i (cid:18) h†i/2 −ηi/√3(cid:19) where φ = φaσ2/2 are triplets under the SU(2) gauge group, h are complex doublets, and i i i η are real singlets. We further demand that the physical Higgs boson is even under the dark i matter parity that acts as x x and x x on the Goldstone fields. This leaves us with 1 2 3 4 ↔ ↔ two candidates for the SM Higgs doublets, h 1 (h +h ), h 1 (h +h ). (3) a ≡ √2 3 4 b ≡ √2 1 2 ThephysicalHiggsfieldwilllaterbeidentifiedash andisprotectedbytheglobalsymmetries a SU(3) = SU(3) SU(3) /SU(3) and SU(3) = SU(3) SU(3) /SU(3) , L,a L,3 L,4 DL R,a R,3 R,4 DR × × where SU(3) denotes the diagonal subgroups of these product groups. As long as no Di single interaction breaks both SU(3) and SU(3) at the same time, the mass of the Higgs L,a R,a will be sufficiently small. For models based on the symmetry structure used here, possibilities to introduce interactions that preserve enough global symmetries are discussed in [1]. We found that we could adopt their rules to introduce scalar self-interactions as well as gauge interactions, but that some modifications are required in the Yukawa sector in order to maintain the parity symmetry. In particular partners for the standard model fermions must be introduced so that the dark matter parity can be implemented in a linear way. 2 x 1 SU(3) SU(3) x 1) 2 S ( SU(3) SU(3) U U ( x 2) ) x x (2 3 U U SU(3) SU(3) ( S 1 ) x 4 SU(3) SU(3) Figure 1: Illustration of the global and gauge symmetry structure of the model. 2.1 Scalar and gauge sector The global symmetry structure of the model is is depicted in Fig. 1. On each site, a SU(2) U(1) subgroup is gauged, with equal strength for both sites. The gauge group × generators are given by σa/2 0 1 0 Qa = , Y = 1 , (4) L,R 0 0 L,R √12 0 2 (cid:18) (cid:19) (cid:18) − (cid:19) written in terms of 2 2 and 1 1 blocks. Here σa denote the Pauli matrices. The kinetic × × term of the sigma fields reads f2 4 = tr[(D X )(DµX ) ], with D X = ∂ X iA X +iX A , (5) G µ i i † µ 1,3 µ 1,3 Lµ 1,3 1,3 Rµ L 4 − Xi=1 DµX2,4 = ∂µX2,4 iARµX2,4 +iX2,4ALµ, − and A g Wa Qa +g y B Y , (6) Lµ ≡ L Lµ L L′ LX Lµ L A g Wa Qa +g y B Y , (7) Rµ ≡ R Rµ R R′ RX Rµ R where the gauge couplings at the two sites are chosen to be equal, g = g = √2g and L R g = g = √2g , and g,g are the SM gauge couplings. Furthermore, y denote the L′ R′ ′ ′ LX,RX U(1) charges of the fields X . The choice y = y = 1/√3 ensures the correct values i LX RX for the Higgs doublet hypercharge and Weinberg angle. Note that the definition (5) of the covariant derivatives corresponds to assigning opposite directions for the link fields 1,3 and 2,4, which is important for the definition of the X-parity below. Each gauge interaction separately only break either SU(3) or SU(3) and therefore L,a R,a respects collective symmetry breaking. Actually since the gauge interactions are either on the left or on the right side of the moose diagram, no large mass is generated for any of the Goldstone fields from these interactions. The kinetic term (5) has a symmetry, called X-parity, defined by 2 Z X-parity: A A , X X , X X . (8) L R 1 2 3 4 ↔ ↔ ↔ 3 This definition is a straightforward generalization of the parity of the two-link model in Ref. [11]. Under this parity, the WZW terms [8] for the four link fields transform as Γ (x ,A ,A ) Γ (x ,A ,A ), Γ (x ,A ,A ) Γ (x ,A ,A ), (9) WZW 1 L R WZW 2 R L WZW 3 L R WZW 4 R L ↔ ↔ so that the combined term = Γ (x ,A ,A )+Γ (x ,A ,A )+Γ (x ,A ,A )+Γ (x ,A ,A ) WZW WZW 1 L R WZW 2 R L WZW 3 L R WZW 4 R L L (10) remains invariant. As a result, X-parity is an exact symmetry of the model and the lightest X-odd particle is stable. In addition to the X-parity in eq. (8) a second symmetry, called T-parity, is imposed, 2 Z under which T-parity: AL ↔ AR, Xi → ΩXi†Ω, (11) where Ω diag(1,1, 1). Our T-parity is identical to the original version in Ref. [4], and it ≡ − ensures that the triplet and singlet scalar do not receive any vacuum expectation values. In our implementation, T-parity is respected by the model at the classical level, but broken by . However, since the stability of the dark matter candidate is already guaranteed by WZW L X-parity (8), this does not lead to any problems. In the gauge sector, the X-odd linear combinations of gauge bosons, Wa = 1 (Wa Wa), B = 1 (B B ), (12) H √2 L − R H √2 L − R acquire masses of order f from the kinetic term (5), while the X-even combinations Wa = 1 (Wa +Wa), B = 1 (B +B ), (13) √2 L R √2 L R remain massless before EWSB and are identified with the SM gauge bosons. The scalar fields form the following X-even and X-odd combinations: 1 1 w = (x x +x x ) x = ( x +x +x x ) (X-odd), (14) 1 2 3 4 1 2 3 4 2 − − 2 − − 1 1 y = ( x x +x +x ) z = (x +x +x +x ) (X-even). (15) 1 2 3 4 1 2 3 4 2 − − 2 The triplet φ and the singlet η are eaten to form the longitudinal components of Wa and w w H B . H A large Higgs quartic coupling, required for electroweak symmetry breaking, is generated by the following X-invariant plaquette operator: κ LP = 8f4tr X1X3†X2†X4 +X2X4†X1†X3 +h.c. (16) h i This operator contains an explicit (f) mass term for the scalar fields in x, but preserves O enough global symmetries so it does not generate large masses for any other Goldstone bosons at the one loop level, in particular not for h , h . a b 4 Successful electroweak symmetry also requires the introduction of a second plaquette term [1], which breaks a different subset of the global symmetry: ǫ L′P = 8f4tr T8X1X3†X2†X4 +T8X2X4†X1†X3 +X1X3†T8X2†X4 +X2X4†T8X1†X3 +h.c. (17) (cid:16) (cid:17) whereT = diag(1,1, 2)/√12, andǫisacomplexconstant. Asexplained inRef.[1], eq. (17) 8 − can be generated radiatively by two-loop diagrams involving the top quark, and therefore it is natural to assume that ǫ κ /10. We can assume ǫ to be purely imaginary, since the | | ∼ | | real part only gives small corrections to the scalar potential. 2.2 Fermion sector For the construction of the kinetic and Yukawa terms of the fermions, several conditions need to be considered. First, one has to make sure that these terms do not break too many of the global symmetries, so that the mass of the little Higgs doublet remains protected from quadratic corrections. Secondly, the minimal construction using only X-even fermions [4] leads to unsuppressed four-fermion operators at one-loop level, thus forcing the scale f be about 10 TeV or larger [12]. The second problem can be solved by introducing “mirror” fermions[12], i.e. twosetsoffermionsthatarepartnersunder X-parity. Ourimplementation closely resembles the setup in the appendix of Ref. [13]. For each SM flavor two doublets of left-handed fermions are introduced, located at the two sites of the moose diagram. With the exception of the top quark, they are embedded into incomplete representations of SU(3) as follows Q = (d ,u ,0) , Q = (d ,u ,0) . (18) a a a ⊤ b b b ⊤ Under the global SU(3) SU(3) group they transform as Q L Q and Q R Q , L R a i a b i b × → → while X- and T-parity interchange the two fields, Q Q . a b ↔ Since (18) are incomplete multiplets, their interaction terms break the global symmetries that protect the Higgs mass and lead to quadratically divergent contributions from one-loop diagrams involving the Yukawa couplings. For the first two generations this is not a problem since the Yukawa couplings are very small, but for the third generation we need to introduce complete multiplets Q = (d ,u ,U ) , Q = (d ,u ,U ) . (19) 3a 3a 3a a ⊤ 3b 3b 3b b ⊤ Here the additional singlets U cancel the quadratically divergent Higgs mass contributions a,b induced by the large top Yukawa coupling. The X- and T-invariant fermion kinetic terms have the standard form = iQ σ¯µDaQ +iQ σ¯µDbQ , with Da = ∂ +ig Wa (Qa) ig y B , (20) LF a µ a b µ b µ µ L Lµ L ⊤ − L′ LQ Lµ Db = ∂ +ig Wa (Qa) ig y B , µ µ R Rµ R ⊤ − R′ RQ Rµ where σ¯µ (1, ~σ), and y and y are diagonal matrices composed of the U(1) charges LQ RQ ≡ − in Table 1. The SM fermions emerge from the X-even linear combination Q = 1 (Q +Q ). √2 a b 5 To give mass to the X-odd combination Q = 1 (Q Q ), we need to introduce conjugate H √2 a− b Dirac partners Qc = (dc,uc,0) , Qc = (dc ,uc ,Uc) , (21) c c c ⊤ 3c 3c 3c c ⊤ Under SU(3) SU(3) they transform as Qc U Qc, where U (i = 1,...,4) belongs to L × R c → i c i the unbroken diagonal subgroup of SU(3) SU(3) and is a non-linear function of L and L R i × R . Furthermore, the effect of X- and T-parity is defined as Qc ΩQc. Then a X- and i c → − c T-invariant mass term for the X-odd fermions is given by λ LM = −√c2f Qaξ1Qcc −QbΩξ1†Qcc −Qbξ2ΩQcc +QaΩξ2†ΩQcc +h.c., (22) (cid:16) (cid:17) where ξi = eixi/f. Under global SU(3)L ×SU(3)R rotations ξi transforms as ξi → LiξiUi† = UiξiRi† for i = 1,3 and analogous for i = 2,4, so that eq. (22) is evidently gauge invariant. In general, λ is a 3 3 matrix in flavor space. Since it can contribute to flavor-changing c × neutral currents (FCNCs) at one-loop level, it is constrained by data on heavy-flavor decays and oscillations. Such effects are studied for example in [14] for the case of the littlest Higgs model with T-parity. For the analyses in section 3 and 4 we assume a flavor diagonal λ for c simplicity. Since the Qc transform non-linearly, one must make use of the ξ fields to construct a c i gauge- and X- and T-invariant kinetic term. Following the formalism of Callan, Coleman, Wess, and Zumino [15], it can be written as 1 Lc = iQccσ¯µ ∂µ + 4(ξ1†Dµξ1 +ξ1Dµξ1† +ξ2†Dµξ2 +ξ2Dµξ2†)−ig′(yQc + √13YV)Bµ Qcc, (cid:18) (cid:19) (23) where ξi†Dµξi = ξi†(∂µ +igWaQaV +igWHaQaA +ig′√13BYV +ig′√13BHYA)ξi, (24) ξiDµξi† = ξi(∂µ +igWaQaV −igWHaQaA +ig′√13BYV −ig′√13BHYA)ξi†, (25) and Qa,Y and Qa,Y are the unbroken and broken gauge generators, respectively. Both V V A A equations (22) and (23) do not involve the x and x Goldstone fields and therefore do not 3 4 break the global symmetries that protect the Higgs mass. They do however generate masses for some of the other Goldstone bosons that will be explicitly calculated in section 3.2. Now Yukawa couplings can be constructed for the X-even massless combinations of the fermions. For the up-type quarks of the first two generations they read 0 0 Lu = −λufQa(X3 +ΩX4†Ω)0−λufQb(ΩX3†Ω+X4)0+h.c., (26) uc uc     where uc is are the right-handed quarks (one for each flavor), which are X- and T-even. As already mentioned above, the presence of incomplete multiplets in the Yukawa couplings 6 leads to quadratically divergent contribution to the Higgs mass. Therefore the top Yukawa coupling has a slightly different form [13], 0 0 Lt = −λfQ3a(X3 +ΩX4†Ω) 0 −λfQ3b(ΩX3†Ω+X4) 0 +h.c.. (27) Uc Uc b a     Here the two singlets Uc and Uc transform under X- and T-parity as Uc Uc. Their X-even a b a ↔ b combination Uc +Uc emerges in the right-handed top quark, while the X-odd combination a b Uc Uc forms the right-handed partner of the X-odd U U . In addition there are one a − b a − b more X-even and X-odd fermion in the top sector, which receive masses from eq. (22). This will be explained in more detail in section 3.1. The use of complete multiplets Q , Q in (27) makes sure that each term preserves one 3a 3b of the global SU(3) symmetries that protect the Higgs mass. Finally, the down-type Yukawa couplings are given by 0 0 Ld = −λdfQa(X3 +ΩX4†Ω)∗0−λdfQb(ΩX3†Ω+X4)∗0+h.c., (28) dc dc e   e   where σ2/2 0 Q = 2iT Q = ( u ,d ,0) , T = . (29) a,b − 2 a,b − a,b a,b ⊤ 2 0 0 (cid:18) (cid:19) The lepton Yuekawa interactions are defined similarly. In contrast to the up-type Yukawa couplings, the all three generations of down-type fermions generate quadratically divergent contributions to the Higgs doublet masses from eq. (28), which is permissible since the bottom Yukawa coupling is much smaller than the top Yukawa coupling. The kinetic term for the singlet conjugate fields ψc uc,dc,Uc,Uc simply reads ≡ a b R = iψcσµ(∂µ ig′yψcBµ)ψc = iψcσµ ∂µ i√2g′(yLψcBLµ +yRψcBRµ) ψc, (30) L − − where σµ (1,~σ) and yψc = 2yLψc = 2yRψc is t(cid:0)he fermion hypercharge. (cid:1) ≡ Table 1 summarizes the fermion contained in the model and their transformation properties. Note that the model is non-renormalizable and considered to be a low-energy effective theory of some fundamental dynamics associated with the UV cutoff scale Λ 10f ∼ ∼ 10 TeV. This UV completion could, but does not need to, consist of some strongly coupled gauge interaction, which breaks the global symmetry through the formation of a fermion condensate, similar to technicolor. 7 SU(2) SU(2) U(1) U(1) X T L R L R q 2 1 1 1 q q a 12 12 b b U 1 1 7 1 U U a 12 12 b b q 1 2 1 1 q q b 12 12 a a U 1 1 1 7 U U b 12 12 a a Qc nonlinear ΩQc ΩQc c − c − c dc 1 1 1 1 dc dc 6 6 uc 1 1 1 1 uc uc −3 −3 Uc 1 1 7 1 Uc Uc a −12 −12 b b Uc 1 1 1 7 Uc Uc b −12 −12 a a Table 1: Quantumnumbers of thefermion multiplets under the [SU(2) U(1)]2 gaugesym- × metry, and their transformation properties under X and T. The physical U(1) hypercharge Y is the sum of both U(1) +U(1) charges. There is some freedom in the assignment of U(1) 1 2 1 (c) (c) and U(1) charges to U , U . Here the conventions of [13] have been adapted. 2 a b 3 Mass spectrum 3.1 Top quark sector Expanding the Yukawa couplings (22) and (27) in the top quark sector in powers of 1/f yields = √2λ f(u u )uc √2λ f(U +U )Uc 2λf (U Uc +U Uc) Lt − c 3a − 3b 3c − c a b c − a b b a λ(q (h +h )Uc +q (h +h )Uc) − 3a y z b 3b y z a + 2√12λc (q3a +q3b)(hy −hz)Ucc +(Ua −Ub)(h†y −h†z)qcc +···+h.c., (31) where q = (d ,u )(cid:2), q = (d ,u ) , and the dots indicate ((cid:3)f 1) terms and (f0) 3a 3a 3a ⊤ 3b 3b 3b ⊤ − O O terms that do not involve Higgs doublets. With suitable phase redefinitions of the fields, both λ and λ can be chosen to be real3. Introducing the X-even and -odd combinations c 1 1 U (U U ), Uc (Uc Uc), (32) ± ≡ √2 a ± b ± ≡ √2 a ± b 1 1 q (q q ), u (u u ), (33) 3 3a 3b 3 3a 3b ± ≡ √2 ± ± ≡ √2 ± one obtains = 2λ fu uc 2λ fU Uc 2λf U Uc +U Uc Lt − c 3− 3c − c + c − + + − − −λ q3+(hy +hz)U+c +q3−(hy +h(cid:0)z)U−c + 21λcq3+((cid:1)hy −hz)Ucc +h.c. (34) 3A relative factor i(cid:0)between the second line of (31) and (27(cid:1)) has been absorbed by this same procedure. 8 Neglecting contributionsoforderv2/f2, theX-oddmasseigenstates inthetopsector, written in terms of left- and right-handed components, are (T ,Tc) (u ,uc ), (T ,T c) (U ,Uc), (35) H H ≡ 3− 3c ′ ′ ≡ − − with masses 2λ f and 2λf, respectively. In the X-even top sector, the following Dirac c fermions are formed: λ Uc +λUc λ Uc λUc (T,Tc) U , c c + , (t,tc) u , c + − c . (36) + 3+ ≡ λ2 +λ2 ! ≡ λ2 +λ2 ! c c p p The T obtains a mass m = 2 λ2 +λ2f, while the SM-like top quark t remains massless T c before EWSB and has a Yukawa coupling given by p √2λλ λ q htc +h.c., λ = c . (37) t 3 t − λ2 +λ2 c Note that the X-odd top partner T is responsible for the canpcellation of the quadratically ′ divergent contribution to the Higgs mass. Therefore the X-even T as well as the X-odd T H can be given masses of several TeV by increasing λ , thus effectively decoupling them from c The remaining fermion masses can be found in table 2. Once electroweak symmetry is broken mixing of the top quark with the T quark is reintroduced. The resulting mass matrix can be diagonalized by redefining the t and T quark as follows: t c t s T, T c T +s t, (38) L L L L → − → tc c tc s Tc, Tc c Tc +s tc, (39) R R R R → − → where s sinα , c cosα are the sine and cosine of the left-handed mixing angle and L L L L ≡ ≡ similarly for s , c . To leading order in an expansion in (v/f), these mixing angles are given R R by λ m m2 sinα α = t + t , (40) L ≈ L λ m O m2 c T (cid:18) T(cid:19) m2 sinα α = 0+ t , (41) R ≈ R O m2 (cid:18) T(cid:19) while the mass eigenvalues remain unperturbed at this order. 3.2 Scalar masses Since the non-linear sigma model breaks the complete symmetry down to its diagonal vector group, the X-odd SU(2) and U(1) gauge bosons, which are associated with the broken generators, become massive by eating the triplet φ and singlet η in the scalar w multiplet, w w respectively. The other scalars are pseudo-Goldstone bosons that receive masses from all 9