Implications of unitarity and gauge invariance for simplified dark matter models
Felix Kahlhoefer, Kai Schmidt-Hoberg, Thomas Schwetz, Stefan Vogl
PPrepared for submission to JHEP
DESY 15-182
Implications of unitarity and gauge invariance forsimplified dark matter models
Felix Kahlhoefer, a Kai Schmidt-Hoberg, a Thomas Schwetz b and Stefan Vogl b,c a DESY, Notkestraße 85, D-22607 Hamburg, Germany b Institut f¨ur Kernphysik, Karlsruher Institut f¨ur Technologie (KIT), D-76021 Karlsruhe, Germany c Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University,SE-10691 Stockholm, Sweden
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We show that simplified models used to describe the interactions of dark mat-ter with Standard Model particles do not in general respect gauge invariance and thatperturbative unitarity may be violated in large regions of the parameter space. The mod-ifications necessary to cure these inconsistencies may imply a much richer phenomenologyand lead to stringent constraints on the model. We illustrate these observations by con-sidering the simplified model of a fermionic dark matter particle and a vector mediator.Imposing gauge invariance then leads to strong constraints from dilepton resonance searchesand electroweak precision tests. Furthermore, the new states required to restore pertur-bative unitarity can mix with Standard Model states and mediate interactions betweenthe dark and the visible sector, leading to new experimental signatures such as invisibleHiggs decays. The resulting constraints are typically stronger than the ‘classic’ constraintson DM simplified models such as monojet searches and make it difficult to avoid thermaloverproduction of dark matter.
Keywords:
Mostly Weak Interactions: Beyond Standard Model; Astroparticles: Cosmol-ogy of Theories beyond the SM a r X i v : . [ h e p - ph ] J a n ontents S matrix unitarity constraints 32.2 Application to a simplified model with a Z (cid:48) mediator 4 A.1 Gauge boson mixing 26A.2 Scalar mixing 27
After the successful discovery of a Higgs Boson consistent with the predictions of theStandard Model (SM), the focus of the current and upcoming runs of the Large HadronCollider (LHC) at 13 TeV will be to discover evidence for physics beyond the SM. Amongthe prime targets of this search is dark matter (DM), which has so far only been observedvia its gravitational interactions at astrophysical and cosmological scales. Since no particlewithin the SM has the required properties to explain these observations, DM searches atthe LHC are necessarily searches for new particles.In fact, LHC DM searches are also likely to be searches for new interactions. Giventhe severe experimental constraints on the interactions between DM and SM particles, itis a plausible and intriguing possibility that the DM particle is part of a (potentially rich)hidden sector, which does not couple directly to SM particles or participate in the knowngauge interactions. In this setup, the visible sector interacts with the hidden sector onlyvia one or several new mediators, which have couplings to both sectors.In the simplest case the mass of these mediators is large enough that they can beintegrated out and interactions between DM particles and the SM can be described by– 1 –igher-dimensional contact interactions [1, 2]. This effective field theory (EFT) approachhas been very popular for the analysis and interpretation of DM searches at the LHC [3–5].Nevertheless, as any effective theory it suffers from the problem that unitarity breaks downif the relevant energy scales become comparable to the cut-off scale of the theory [6–10](for other examples of applying unitarity arguments in the DM context see refs. [11–14]).The easiest way to avoid this problem appears to be to explicitly retain the (lightest)mediator in the theory. The resulting models are referred to as simplified DM models, inwhich couplings are only specified after electroweak symmetry breaking (EWSB) and noultraviolet (UV) completion is provided [15]. Compared to the EFT approach, simplifiedmodels have a richer phenomenology [8, 16–23], including explicit searches for the mediatoritself [24–26]. Moreover, it is possible to achieve the DM relic abundance in large regionsof parameter space [26–28]. Constraining the parameter space of simplified DM models istherefore a central objective of experimental collaborations [29–31].In the present work we focus on the case of a spin-1 s -channel mediator [24, 28, 32–46].Our central observation is that the simplified model approach is not generally sufficient toavoid the problem of unitarity violation at high energies and that further amendments arerequired if the model is to be both simple and realistic. In particular, a spin-1 mediatorwith axial couplings violates perturbative unitarity at large energies, pointing towards thepresence of additional new physics to restore unitarity.Indeed, the simplest way to restore unitarity is to assume that the spin-1 mediator isthe gauge boson of an additional U (1) (cid:48) gauge symmetry [47, 48] and that its mass as wellas the DM mass are generated by a new Higgs field in the hidden sector. The famous Lee-Quigg-Thacker bound [49] implies that the additional Higgs boson cannot be arbitrarilyheavy and may therefore play an important role for LHC and DM phenomenology. Inparticular, it can mix with the SM-like Higgs boson and mediate interactions between DMparticles and quarks.Furthermore, we require for a consistent simplified DM model that the coupling struc-ture respects gauge invariance of the full SM gauge group before EWSB (see [50] for asimilar discussion in the EFT context). If the mediator has axial couplings to quarks,this requirement implies that the new mediator will also have couplings to leptons andmixing with the SM Z boson, both of which are tightly constrained by experiments. Muchweaker constraints are obtained for the simplified DM model containing a spin-1 mediatorwith vectorial couplings to quarks. Constraints from direct detection can be evaded if themediator has only axial couplings to DM, which naturally arises in the case that the DMparticle is a Majorana fermion. We discuss the importance of loop-induced mixing effectsin this context, which can play a crucial role for both direct detection experiments andLHC phenomenology.The outline of the paper is as follows. Starting from a simplified model for a spin-1 s -channel mediator, we explore in section 2 the implications of perturbative unitarity,deriving a number of constraints on the model parameters and in particular an upperbound on the scale of additional new physics. In section 3 we then consider the case wherethis additional new physics is a Higgs field in the hidden sector and derive an upper boundon the mass of the extra Higgs boson. We then discuss additional constraints on the SM– 2 –ouplings implied by gauge invariance. Section 4 focuses on the case of non-zero axialcouplings between SM fermions and the mediator, whereas in section 5 we assume that theSM couplings of the mediator are purely vectorial. Finally, we discuss the experimentalimplications of a possible mixing between the SM Higgs and the hidden sector Higgs insection 6. A discussion of our results and our conclusions are presented in section 7. S matrix unitarity constraints Consider the scattering matrix element M if ( s, cos θ ) between 2-particle initial and finalstates ( i, f ), with √ s and θ being the centre of mass energy and scattering angle, respec-tively. We define the helicity matrix element for the J th partial wave by M Jif ( s ) = 132 π β if (cid:90) − d cos θ d Jµµ (cid:48) ( θ ) M if ( s, cos θ ) , (2.1)where d Jµµ (cid:48) is the J th Wigner d-function, µ and µ (cid:48) denote the total spin of the initial and thefinal state (see e.g. [51]), and β if is a kinematical factor. In the high-energy limit s → ∞ ,which we are going to consider below, β if →
1. The right-hand side of eq. (2.1) is to bemultiplied with a factor of 1 / √ S matrix impliesIm( M Jii ) = (cid:88) f |M Jif | = |M Jii | + (cid:88) f (cid:54) = i |M Jif | ≥ |M Jii | (2.2)for all J and all s . The sum over f in the first line runs over all possible final states.Restricting these to be all possible 2-particle states leads to a conservative bound. Ifthe relation (2.2) is strongly violated for matrix elements calculated at leading order inperturbation theory one can conclude that either higher-order terms in perturbation theoryrestore unitarity (i.e. break-down of perturbativity) or that the theory is not complete andadditional contributions to the matrix element are needed.From eq. (2.2) one obtains the necessary conditions0 ≤ Im( M Jii ) ≤ , (cid:12)(cid:12) Re( M Jii ) (cid:12)(cid:12) ≤ . (2.3)In the following we will apply these inequalities to leading-order matrix elements in orderto identify regions in parameter space where perturbative unitarity is violated. Since thesematrix elements are always real in the present context, only the second constraint will berelevant.If the matrix M Jif is diagonalized the inequality in eq. (2.2) becomes an equality. Hence,stronger constraints can be obtained by considering the full transition matrix connectingall possible 2-particle states with each other (or some submatrix thereof) and calculating– 3 –he eigenvalues of that matrix. Then the bounds from eq. (2.3) have to hold for each ofthe eigenvalues [52].We note that d J ( θ ) = P J (cos θ ), where P J are the Legendre polynomials. If initial andfinal state both have zero total spin, eq. (2.1) therefore becomes identical to the familiarpartial wave expansion of the matrix element. In the following we will focus on the J = 0partial wave, which typically provides the strongest constraint. Since d µµ (cid:48) is non-zero onlyfor µ = µ (cid:48) = 0, we then obtain from eq. (2.1) M if ( s ) = 164 π β if δ µ δ µ (cid:48) (cid:90) − d cos θ M if ( s, cos θ ) . (2.4) Z (cid:48) mediator Let us consider a simplified model for a spin-1 mediator Z (cid:48) µ with mass m Z (cid:48) and a DiracDM particle ψ with mass m DM . The most general coupling structure is captured by thefollowing Lagrangian: L = − (cid:88) f = q,l,ν Z (cid:48) µ ¯ f (cid:2) g Vf γ µ + g Af γ µ γ (cid:3) f − Z (cid:48) µ ¯ ψ (cid:2) g V DM γ µ + g A DM γ µ γ (cid:3) ψ . (2.5)Although these interactions appear renormalisable, the presence of a massive vector bosonimplies that perturbative unitarity may be violated at large energies. In the following, wewill study this issue in detail and derive constraints on the parameter space of the model.Let us first consider diagrams between 2-fermion states with the Z (cid:48) as mediator. Theappropriate propagator for the mediator is (cid:104) Z (cid:48) µ ( k ) Z (cid:48) ν ( − k ) (cid:105) = 1 k − m Z (cid:48) (cid:18) g µν − k µ k ν m Z (cid:48) (cid:19) , (2.6)where k µ is the momentum of the mediator. For the case of a gauge boson this correspondsto unitary gauge in which the Goldstone boson has been absorbed. Since we are interestedin the high-energy behaviour of the theory we concentrate on the second term, whichdoes not vanish in the limit k → ∞ . This corresponds to restricting to the longitudinalcomponent of the mediator, Z (cid:48) L , which dominates at high energy [51]. For instance,considering DM annihilations, we can contract the longitudinal part of the propagatorwith the DM current. Making use of k = p + p , where p and p are the momenta of thetwo DM particles in the initial state, leads to a factor k µ ¯ v ( p ) (cid:0) g V DM γ µ + g A DM γ µ γ (cid:1) u ( p ) = ¯ v ( p ) (cid:104) g V DM ( /p + /p ) + g A DM ( /p γ − γ /p ) (cid:105) u ( p )= − g A DM m DM ¯ v ( p ) γ u ( p ) . (2.7) In the case of Majorana DM the vector current vanishes and hence there can only be an axial couplingon the DM side. We will come back to this case shortly but will consider Dirac DM here to allow for bothvectorial and axial couplings. It turns out that for certain processes the transversal part of the propagator leads to a logarithmicdivergence for m Z (cid:48) (cid:28) s . This divergence is not related to the UV completeness of the theory, but signalsbreakdown of perturbativity in the IR, see also [14]. By restricting to the longitudinal components of the Z (cid:48) [51] we can avoid the occurence of those IR divergences. – 4 –ence, the second term in the propagator behaves exactly like a pseudoscalar with mass m Z (cid:48) and couplings to DM equal to 2 g A DM m DM /m Z (cid:48) , just like the Goldstone boson presentin Feynman gauge. Note that the term is independent of the vector couplings. The sameargument holds for the quark couplings, which are found to be given by 2 g Af m f /m Z (cid:48) . Thisconsideration suggests that perturbative unitarity will not only lead to bounds on g V,A ,but also on the combination g Af m f /m Z (cid:48) .We can make this statement more precise by applying the methods outlined in theprevious subsection to the self-scattering of two DM particles or two SM fermions. Weobtain for any fermion f with axial couplings g Af (cid:54) = 0 that the fermion mass must satisfythe bound m f (cid:46) (cid:114) π m Z (cid:48) g Af . (2.8)Here f can be any fermion, including SM fermions and the DM particle. As suggested bythe above discussion we do not obtain any bound on the masses of fermions with purelyvectorial couplings, nor on the scale of new physics.Let us now turn to the discussion of processes involving Z (cid:48) in the external state,in particular Z (cid:48) with longitudinal polarisation. For concreteness, we study the process ψ ¯ ψ → Z (cid:48) L Z (cid:48) L . At large momenta, k (cid:29) m Z (cid:48) , the polarisation vectors of the gauge bosonscan be replaced by (cid:15) µL ( k ) = k µ /m Z (cid:48) . One might therefore expect the matrix element forthis process to grow proportional to s/m Z (cid:48) . However, such a term is absent due to acancellation between the t - and u -channel diagram. To obtain a non-zero contribution,one needs to include a mass insertion along the fermion line [53]. It turns out that thecontribution proportional to g V DM still cancels in this case and that the leading contributionat high energies becomes proportional to ( g A DM ) √ s m DM /m Z (cid:48) . As a result, perturbativeunitarity is violated unless [53–55] √ s < π m Z (cid:48) ( g A DM ) m DM . (2.9)For larger energies new physics must appear to restore unitarity. This can be accomplishedby including an additional diagram with an s -channel Higgs boson, since both contributionshave the same high-energy behaviour. The consideration above implies an upper bound onthe mass of the Higgs that breaks the U (1) (cid:48) and gives mass to the Z (cid:48) : m s < π m Z (cid:48) ( g A DM ) m DM . (2.10)We will discus the consequences of such an extension of the minimal model in section 3.In summary we have found that there are two different types of constraints on theparameters of this simplified model, even for perturbative couplings. For non-vanishing Note that this process corresponds to an off-diagonal element of M if , with i (cid:54) = f , whereas the boundsfrom eq. (2.3) apply for diagonal elements. In order to apply the unitarity constraint we consider the2 × M if spanned by the states ψ ¯ ψ and Z (cid:48) L Z (cid:48) L . For s → ∞ only the off-diagonal elementsurvives, and hence the eigenvalues of the matrix become equal to the off-diagonal element, and we canapply eq. (2.3). Our result differs from the one in [53] by a factor 1 / √ – 5 – s / [ GeV ] m Z ' [ G e V ] m DM =
500 GeV g DM A = g DM A = Figure 1 . Parameter space forbidden by the requirement of perturbative unitarity in the √ s − m Z (cid:48) plane for m DM = 500 GeV. The constraint resulting from DM scattering is shown in grey (solid),the constraint resulting from DM annihilation into Z (cid:48) s is shown in blue (dashed). Thick (thin)lines correspond to g A DM = 1 ( g A DM = 0 . Z (cid:48) can never be lighter than about400 GeV (40 GeV) irrespective of the UV completion. axial couplings there is an energy scale for which the theory violates perturbative unitarityand needs to be UV completed, see eq. (2.9). In addition, imposing that the couplingbetween the longitudinal component of the vector mediator and the DM particle remainperturbative, we find that the vector mediator cannot be much lighter than the DM, seeeq. (2.8). This constraint is not related to missing degrees of freedom and is thereforecompletely independent of the UV completion. We illustrate both constraints in figure 1for different axial couplings and a DM mass m DM = 500 GeV.To conclude this section, we emphasise that for pure vector couplings of the Z (cid:48) ( g A DM = g Af = 0) the simplified model considered in this section is well-behaved in the UV in thesense that there is no problem with perturbative unitarity. Indeed in this specific casea bare mass term for the dark matter is allowed such that it is sufficient to generate thevector boson mass via a Stueckelberg mechanism without the need for additional degrees offreedom [56, 57]. However, this specific coupling configuration is highly constrained, sinceit is very difficult to evade bounds from direct detection experiments and still reproduce theobserved DM relic abundance. This is illustrated in figure 2 where we show the parameterregion excluded by the bound on the spin-independent DM-nucleon scattering cross sectionfrom LUX [58] and the parameter region where the DM annihilation cross section becomesso small that DM is overproduced in the early Universe. One can clearly see that only afinely-tuned region of parameter space close to the resonance m DM = m Z (cid:48) / As discussed below there can be anomalies which require additional fermions. – 6 – m DM [ GeV ] m Z ' [ G e V ] g DM V = g qV = Figure 2 . Vector(SM)–Vector(DM): Parameter space excluded by the bound on the spin-independent DM-nucleon scattering cross section from LUX (green, dashed) and the parameterregion where the DM annihilation cross section becomes so small that DM is overproduced in theearly Universe (red, solid).
As we have seen in the previous section, for non-zero axial couplings the simplified modelviolates perturbative unitarity at high energies, implying that additional new physics mustappear below these scales. This observation motivates a detailed discussion of how togenerate the vector boson mass from an additional Higgs mechanism. To restore unitaritylet us therefore now consider the case that the Z (cid:48) is the gauge boson of a new U (1) (cid:48) gaugegroup. To break this gauge group and give a mass to the Z (cid:48) , we introduce a dark Higgssinglet S , which needs to be complex in order to allow for a U (1) (cid:48) charge. We then obtainthe following Lagrangian L = L SM + L DM + L (cid:48) SM + L S , (3.1)where the first term is the usual SM Lagrangian and the second term describes the inter-actions of DM. The third term contains the interactions between SM states and the new Z (cid:48) gauge boson while the fourth term contains the extended Higgs sector. As mentioned above, it is well-motivated from a phenomenological perspective to considerthe case that vector couplings to the Z (cid:48) mediator vanish in at least one of the two sectors,so that direct detection is suppressed. On the DM side this is naturally achieved for aMajorana fermion, which we will focus on from now. We therefore write ψ = (cid:32) χ(cid:15)χ ∗ (cid:33) , (3.2)– 7 –here χ is a Weyl spinor. We assume that χ carries a charge q DM under the new U (1) (cid:48) gauge group, such that under a gauge transformation ψ → exp (cid:2) i g (cid:48) q DM α ( x ) γ (cid:3) ψ , (3.3)where g (cid:48) is the gauge coupling of the new U (1) (cid:48) . The kinetic term for ψ can hence bewritten as L kin = 12 ¯ ψ ( i /∂ − g (cid:48) q DM γ /Z (cid:48) ) ψ = i ψ /∂ψ − g A DM Z (cid:48) µ ¯ ψγ γ µ ψ , (3.4)with g A DM ≡ g (cid:48) q DM . The U (1) (cid:48) charge forbids a Majorana mass term. Nevertheless, ifthe Higgs field S carries charge q S = − q DM , we can write down the gauge-invariantcombination L mass = − y DM ¯ ψ ( P L S + P R S ∗ ) ψ . (3.5)Including the kinetic and potential terms for the Higgs singlet, the full dark Lagrangiantherefore reads L DM = i ψ /∂ψ − g A DM Z (cid:48) µ ¯ ψγ γ µ ψ − y DM ¯ ψ ( P L S + P R S ∗ ) ψ , L S = (cid:2) ( ∂ µ + i g S Z (cid:48) µ ) S (cid:3) † (cid:2) ( ∂ µ + i g S Z (cid:48) µ ) S (cid:3) + µ s S † S − λ s (cid:16) S † S (cid:17) . (3.6)Once the Higgs singlet aquires a vacuum expectation value (vev), it will spontaneouslybreak the U (1) (cid:48) symmetry, thus giving mass to the Z (cid:48) gauge boson and the DM particle.After symmetry breaking, we obtain the following Lagrangian (defining S = 1 / √ s + w )and using g S ≡ g (cid:48) q S = − g A DM ) L = i ψ /∂ψ − F (cid:48) µν F (cid:48) µν − g A DM Z (cid:48) µ ¯ ψγ γ µ ψ − m DM ψψ − y DM √ s ¯ ψψ + 12 m Z (cid:48) Z (cid:48) µ Z (cid:48) µ + 12 ∂ µ s∂ µ s + 2( g A DM ) Z (cid:48) µ Z (cid:48) µ ( s + 2 s w ) + µ s s + w ) − λ s s + w ) , (3.7)with F (cid:48) µν = ∂ µ Z (cid:48) ν − ∂ ν Z (cid:48) µ and m DM = 1 √ y DM w , m Z (cid:48) ≈ g A DM w . (3.8)If the SM Higgs is charged under the U (1) (cid:48) the Z (cid:48) mass will receive an additional contri-bution from the SM Higgs vev, see eq. (3.19) below. Electroweak precisison data requiresthat this contribution is small, and therefore we neglect this term in eq. (3.8) and for therest of this subsection. Note that without loss of generality we can choose w and y DM tobe real (ensuring real masses) by absorbing complex phases in the field definitions for S and ψ . This will no longer be true if we allow for an explicit mass term for ψ . In this case the relative phasebetween y DM and the mass term is physical (see e.g. [59]). Here we do not allow for an explicit mass termand we assume that the vev of the singlet is the only source of U (1) (cid:48) symmetry breaking. – 8 –s discussed above, the mass of the additional Higgs particle must satisfy m s < π m Z (cid:48) ( g A DM ) m DM (3.9)in order for perturbative unitarity to be satisfied, which when substituting the masses ofthe Z (cid:48) and DM becomes m s < √ πwy DM . (3.10)Once we include such a new particle coupling to the Z (cid:48) , however, there are additionalscattering processes such as ss → ss that need to be taken into account when check-ing perturbative unitarity [60]. Here we consider the scattering of the states ss/ √ Z (cid:48) L Z (cid:48) L / √
2. In the limit √ s (cid:29) m s (cid:29) m Z (cid:48) , the J = 0 partial wave of the scattering matrixtakes the form [49] lim √ s →∞ M if = − ( g A DM ) m s πm Z (cid:48) (cid:32) (cid:33) . (3.11)Partial wave unitarity requires the real part of the largest eigenvalue, which correspondsto the eigenvector ( ss + Z (cid:48) L Z (cid:48) L ) /
2, to be smaller than 1 /
2. We hence obtain the inequality m s ≤ √ π m Z (cid:48) g A DM = √ πw . (3.12)This inequality together with eq. (2.8) gives a stronger bound on the Higgs mass thanthe one obtained in eq. (2.10). In other words, the bound in (2.10) can never actually besaturated in this UV completion. We note that eqs. (2.8) and (3.12) can be unified to √ π m Z (cid:48) g A DM ≥ max (cid:104) m s , √ m DM (cid:105) . (3.13) For the discussion above we only needed to consider the DM part of the Lagrangian. Letus now also look at the coupling to the SM, see e.g. [61]. The interactions between SMstates and the new Z (cid:48) gauge boson can be written as L (cid:48) SM = (cid:104) ( D µ H ) † ( − i g (cid:48) q H Z (cid:48) µ H ) + h.c. (cid:105) + g (cid:48) q H Z (cid:48) µ Z (cid:48) µ H † H − (cid:88) f = q,(cid:96),ν g (cid:48) Z (cid:48) µ (cid:2) q f L ¯ f L γ µ f L + q f R ¯ f R γ µ f R (cid:3) , (3.14)where D µ denotes the SM covariant derivative. We can now immediately write down a listof relations between the different charges q required by gauge invariance of the SM Yukawaterms: q H = q q L − q u R = q d R − q q L = q e R − q (cid:96) L . (3.15) If right-handed neutrinos exist their charge q ν R would be constrained by q H = q (cid:96) L − q ν R to allow for aYukawa term with the lepton doublet. In the following we assume that if right-handed neutrinos exist theyare heavy enough to decouple from all relevant phenomenology. – 9 –fter electroweak symmetry breaking, we obtain L (cid:48) SM = 12 e g (cid:48) q H s W c W ( h + v ) Z µ Z (cid:48) µ + 12 g (cid:48) q H ( h + v ) Z (cid:48) µ Z (cid:48) µ − (cid:88) f = q,l,ν g (cid:48) Z (cid:48) µ ¯ f (cid:2) ( q f R + q f L ) γ µ + ( q f R − q f L ) γ µ γ (cid:3) f . (3.16)Comparing the second line of eq. (3.16) with eq. (2.5) we can read off the vector and axialvector couplings of the fermions: g Vf = 12 g (cid:48) ( q f R + q f L ) , g Af = 12 g (cid:48) ( q f R − q f L ) . (3.17)It is well known that a U (1) (cid:48) under which only SM fields are charged is in generalanomalous, unless the SM fields have very specific charges (e.g. U (1) B − L is anomaly free).The relevant anomaly coefficients can e.g. be found in [61]. The presence of these anomaliesimplies that the theory has to include new fermions to cancel the anomalies. While thesefermions can be vectorlike with respect to the SM, they will then need to be chiral withrespect to the U (1) (cid:48) . The mass of the additional fermions is therefore constrained by thebreaking scale of the U (1) (cid:48) . In particular, the bound from eq. (2.8) applies to these fermionsas well and therefore they cannot be decoupled from the low-energy theory.It is however interesting to note that the anomaly involving two gluons and a Z (cid:48) isproportional to A ggZ (cid:48) = 3 (2 q q L − q u R − q d R ) , (3.18)which always vanishes if we restrict the charges based on gauge invariance of the Yukawacouplings (see eq. (3.15)). This implies that no new coloured states are needed to cancelthe anomalies, greatly reducing the sensitivity of colliders to these new states. In any case,there are many different possibilities for cancelling the anomalies via new fermions. Whilethe existence of additional fermions will lead to new signatures, a detailed investigation ofthese is beyond the scope of this work.If the SM Higgs is charged under U (1) (cid:48) ( q H (cid:54) = 0) the mass of the Z (cid:48) receives a contri-bution from both Higgses: m Z (cid:48) = ( g (cid:48) q H v ) + 4( g A DM w ) , (3.19)and we obtain a mass mixing term of the form δm Z µ Z (cid:48) µ with δm = 12 e g (cid:48) q H s W c W v , (3.20)where s W ( c W ) is the sine (cosine) of the Weinberg angle.As we are going to discuss below, electroweak precision data requires | δm | (cid:28) | m Z − m Z (cid:48) | (see also App. A.1). Using m Z = ev/ (2 s W c W ), g (cid:48) q s = − g A DM , and neglecting orderone factors this requirement implies either g (cid:48) q H (cid:28) e or q s w (cid:29) v . In the parameter regionsof interest it follows from those conditions that the first term in eq. (3.19) is small and This conclusion is in disagreement with the observations made in [41]. – 10 –ence the mass of the Z (cid:48) is dominated by the vev of the dark Higgs. Taking into accounteqs. (3.15) and (3.17), the condition | δm | (cid:28) | m Z − m Z (cid:48) | then implies either small axialcouplings ( g Af (cid:28)
1) or m Z (cid:48) (cid:29) m Z . We are going to present more quantitative results inthe next section and discuss a number of interesting experimental signatures resulting fromthe new interactions due to eq. (3.16).To conclude this section, it should be noted, that the Lagrangian introduced above isUV-complete (up to anomalies) and gauge invariant but does not correspond to the mostgeneral realization of this model. In particular the term L ⊃ − λ hs ( S ∗ S )( H † H ) , (3.21)which will lead to mixing between the SM Higgs h and the dark Higgs s can be expectedto be present at tree level. Furthermore, the term L ⊃ −
12 sin (cid:15)F (cid:48) µν B µν , (3.22)which generates kinetic mixing between the Z (cid:48) and the Z -boson, respects all symmetriesof the Lagrangian. It can be argued that (cid:15) might vanish at high scales in certain UV-completions, but even in this case kinetic mixing is necessarily generated at the one-looplevel and can have a substantial impact on EWPT. We will return to these issues and theresulting phenomenology of the model in sections 5 and 6. For the moment, however, weare going to neglect these additional effects and focus on the impact of δm (cid:54) = 0, whichnecessarily leads to mass mixing between the neutral gauge bosons in the case of non-vanishing axial couplings. Let us start with the case that axial couplings on the SM side are non-vanishing. Animmediate consequence is that the SM Higgs is charged under the U (1) (cid:48) , which followsfrom eqs. (3.15) and (3.17) for g Af (cid:54) = 0. Note that these equations also imply that it isinconsistent to set the vectorial couplings for all quarks equal to zero. For example, if weimpose that the vectorial couplings of up quarks vanish, i.e. g Vu = 0, eq. (3.17) implies q u R = − q q L , which using eq. (3.15) leads to g Vd = 2 g (cid:48) q q L . In the following, whenever g Af (cid:54) = 0,we always fix g Vf = g Af , which corresponds to setting q q L = q (cid:96) L = 0.Furthermore, eq. (3.15) requires that Z (cid:48) couplings are flavour universal and leptonscouple with the same strength to the Z (cid:48) as quarks. This conclusion could potentially bemodified by considering an extended Higgs sector, e.g. a two-Higgs-doublet model. Here wefocus on the simplest case where a single Higgs doublet generates all SM fermion masses.This implies that the leading search channel at the LHC will be dilepton resonances, whichgive severe constraints. In principle also electron-positron colliders can constrain this sce-nario efficiently. Limits on a Z (cid:48) lighter than 209 GeV derived from LEP data imply g (cid:46) − [62] (see also [63, 64]). We do not include LEP constraints here since other constraints willturn out to be at least equally strong. – 11 –or general couplings, the partial decay width of the mediator into SM fermions isgiven byΓ( Z (cid:48) → f ¯ f ) = m Z (cid:48) N c π (cid:115) − m f m Z (cid:48) (cid:34) ( g Vf ) + ( g Af ) + m f m Z (cid:48) (cid:0) g Vf ) − g Af ) (cid:1)(cid:35) , (4.1)where N c = 3 (1) for quarks (leptons). The decay width into DM pairs isΓ( Z (cid:48) → ψψ ) = m Z (cid:48) π ( g A DM ) (cid:18) − m m Z (cid:48) (cid:19) / . (4.2)Consequently, for m DM (cid:28) m Z (cid:48) and g A(cid:96) = g Aq (cid:28) g A DM the branching ratio into (cid:96) = e, µ is given by BR( R → (cid:96)(cid:96) ) ≈ g A(cid:96) ) / ( g A DM ) . For m DM > m Z (cid:48) /
2, on the other hand, thebranching ratio is given by BR( R → (cid:96)(cid:96) ) ≈ . .
10 depending on the ratio m Z (cid:48) /m t .We implement the latest ATLAS dilepton search [65], complemented by a Tevatrondilepton search [66] for the low mass region, and show the resulting bounds in figure 3.One can see that the bounds strongly depend on the assumed branching ratio of the Z (cid:48) .As a conservative limiting case we show g A DM = 1 and m DM = 100 GeV, which leadsto a rather large branching fraction into DM and hence suppressed bounds. The secondbenchmark, m DM = 500 GeV, allows for Z (cid:48) decays to DM only for rather heavy Z (cid:48) s, leadingto correspondingly more restrictive dilepton constraints. Overall the bounds turn out to bevery stringent and the Z (cid:48) coupling to leptons and quarks needs to be significantly smallerthan unity for 100 GeV (cid:46) m Z (cid:48) (cid:46) g q = g l .The fact that the SM Higgs is charged also implies potentially large corrections toelectroweak precision observables. In particular we obtain the non-diagonal mass term δm Z µ Z (cid:48) µ leading to mass mixing between the SM Z and the new Z (cid:48) . The diagonalisationrequired to obtain mass eigenstates is discussed in the appendix. In the absence of kineticmixing between the U (1) (cid:48) and the SM U (1) gauge bosons ( (cid:15) = 0), the resulting effectscan be expressed in terms of the mixing parameter ξ = δm / ( m Z − m Z (cid:48) ) (see eq. (A.5) inthe appendix with (cid:15) = 0). In particular, we can calculate the constraints from electroweakprecision measurements, which are encoded in the S and T parameters. To quadratic orderin ξ we find [67] αS = − c s ξ ,αT = ξ (cid:18) m Z (cid:48) m Z − (cid:19) , (4.3)where α = e / π . The resulting bounds are shown in figure 3. To infer our bounds we usethe 90% CL limit on the S and T parameters as given in [62].Note that the bound from electroweak precision data is completely independent of the Z (cid:48) couplings to the DM as well as the DM mass. Hence, the same bound would apply alsoin the case of Dirac DM with vector couplings to the Z (cid:48) . Note however that since vectorialcouplings to quarks are necessarily non-zero if g Aq (cid:54) = 0 (see eqs. (3.15), (3.17)), there will be– 12 – - - - m Z ' [ GeV ] g l A = g q A m DM =
100 GeV g DM A = P e r t u r ba t i v eun i t a r i t yv i o l a t i on L HC d il ep t on r e s onan c e s T e v a t r ond il ep t on r e s onan c e s DM overproduction E l e c t r o w e a k p r e c i s i o n t e s t s - - - m Z ' [ GeV ] g l A = g q A m DM =
500 GeV g DM A = Figure 3 . Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dileptonresonance searches from ATLAS (light green, dashed) and Tevatron (dark green, dashed), elec-troweak precision observables (blue, dotted) and DM overproduction (red, solid) in the m Z (cid:48) − g Aq,l parameter plane for two exemplary DM masses 100 GeV (left) and 500 GeV (right). In the shadedregion to the left of the vertical grey line the Z (cid:48) -mass violates the bound from perturbative unitarityfrom eq. (2.8). very stringent bounds from direct detection experiments on any model with g V DM (cid:54) = 0 dueto unsuppressed spin-independent scattering. For Majorana DM, the vectorial couplingalways vanishes and the constraints from direct detection are much weaker.In figure 4 we show the constraints from electroweak precision data as well as LHCdilepton searches in the m DM − m Z (cid:48) plane for different values of the axial vector couplingto fermions. In the lower right corner of the plots (grey area) the perturbative unitaritycondition from eq. (2.8) is violated. We also show the region excluded by direct detectionsearches (dark region in the lower left corners). For the axial-axial couplings DM-nucleusscattering proceeds through spin-dependent interactions, with a scattering cross sectiongiven by σ SD N = 3 a N ( g A DM ) ( g Aq ) π µ m Z (cid:48) , (4.4)where µ is the DM-nucleon reduced mass, N = p, n and a p = − a n = 1 .
18 is the effectivenucleon coupling [62]. This is the dominant contribution in this case as the vector-axialcoupling combination is even further suppressed. In the plots we show the bound on thespin-dependent scattering cross section that can be calculated from the published LUXresults [58], following the method described in [68]. We observe that in this case directdetection is never competitive with other constraints.The red solid curves in Figs. 3 and 4 show the parameter values that lead to thecorrect relic abundance. In order to calculate the relic abundance we have implementedthe model in micrOMEGAs v4 [69], assuming that the mass of the Higgs singlet saturates– 13 – m DM [ GeV ] m Z ' [ G e V ] g DM A = g q , lA = m DM [ GeV ] m Z ' [ G e V ] g DM A = g q , lA = P e r t u r ba t i v e un i t a r i t y v i o l a t i on Dilepton resonances D M o v e r p r o d u c t i o n Electroweak precision tests D i r e c t de t e c t i on M ono j e t s m DM [ GeV ] m Z ' [ G e V ] g DM A = g q , lA = m DM [ GeV ] m Z ' [ G e V ] g DM A = g q , lA = Figure 4 . Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dileptonresonance searches (green, dashed) and electroweak precision observables (blue, dotted) in the m DM − m Z (cid:48) plane for four different sets of couplings. We also show the regions excluded by DMoverproduction (red), direct detection bounds (purple, dot-dashed) and the parameter space whereperturbative unitarity is violated (grey). For the relic density calculation we have assumed that themass of the hidden sector Higgs saturates the unitarity bound. the unitarity bound and setting the mixing with the SM Higgs to zero. In the regionsshaded in red (to the right/above the solid curve) there is overproduction of DM. In thisregion additional annihilation channels are required to avoid overclosure of the Universe,since the interactions provided by the Z (cid:48) are insufficient to keep DM in thermal equilibriumlong enough. Such additional interactions could be obtained for instance from the scalarmixing discussed in section 6. Conversely, to the left/below the red solid curve the model Note that since ξ can be large in some regions of parameter space, it is not a good approximation toexpand the annihilation cross section in ξ . We therefore use the exact expression for the mixing betweenthe neutral gauge bosons in terms of (cid:15) and δm as derived in the appendix. – 14 –oes not provide all of the DM matter in the Universe, since the annihilation rate is toohigh.Let us briefly discuss the various features that can be observed in the relic abundancecurve. First there is a significant decrease of the predicted abundance as the DM masscrosses the top-quark threshold, m χ > m t , resulting from the fact that the s -wave con-tribution to the annihilation cross section is helicity suppressed and hence annihilationinto top-quarks becomes the dominant annihilation channel as soon as it is kinematicallyallowed. The second feature occurs at m χ ∼ m Z (cid:48) and reflects the resonant enhancementof the annihilation process χχ → q ¯ q as the mediator can be produced on-shell. A thirdvisible feature is a very narrow resonance at 2 m χ ∼ m s = √ π m Z (cid:48) /g A DM due to a resonantenhancement of the process χχ → s → Z (cid:48) Z . The position and magnitude of this effectdepends on the mass of the dark Higgs, which has been (arbitrarily) fixed to saturate theunitarity bound. However, even for this extreme choice, it turns out to give a non-negligiblecontribution to the relic abundance. For m χ > m Z (cid:48) direct annihilation into two media-tors becomes possible, leading to a significant decrease of the predicted relic abundance.Finally, the fact that the relic abundance curve in figure 4 touches the unitarity boundfor high DM masses reflects the well-known unitarity bound on the mass of a thermallyproduced DM particle [11].All in all we find the case with non-vanishing axial couplings on the SM side to bestrongly constrained by dilepton searches as well as electroweak precision observables, im-plying that in a UV complete model this is where a signal should first be seen. For com-parison, we show recent bounds from LUX as well as from the CMS monojet search [29]. We find that these searches, as well as searches for dijet resonances, are not competitive.Note that in figure 4 we assume g A DM = 1. We comment on smaller couplings on the DMside later in the context of figure 7. Let us now look at the case where axial couplings toquarks are taken to be zero, which will turn out to be somewhat less constrained. Let us now consider the case with purely vectorial couplings on the SM side, i.e. g Aq = g A(cid:96) = g Aν = 0. In this case the SM Higgs does not carry a U (1) (cid:48) charge and thereforethe charges of quarks and leptons are independent. In particular, it is conceivable that g Vq (cid:29) g V(cid:96) , so that constraints from dilepton resonance searches can be evaded. Also therecan in principle be a flavour dependence of the Z (cid:48) couplings to quarks. Nevertheless, toavoid large flavour-changing neutral currents, we will always assume the same coupling forall quark families in what follows [15]. Finally, in contrast to the case discussed above,tree-level Z − Z (cid:48) mass mixing is absent. It therefore seems plausible that the Z (cid:48) is theonly state coupling to both the visible and the dark sector. Nevertheless, as mentioned To interpret the CMS results in the context of our model, we implement our model in
Feynrules v2 [70]and simulate the monojet signal with
MadGraph v5 [71] and
Pythia v6 [72]. Imposing a cut on the missingtransverse energy of /E T >
450 GeV, we exclude all parameter points that predict a contribution to themonojet cross section larger than 7 . CalcHEP v3 [73] and
DELPHES v3 [74]. – 15 – - - - m Z ' [ GeV ] ϵ m DM =
100 GeV g DM A = g qV = P e r t u r ba t i v eun i t a r i t yv i o l a t i on L HC d il ep t on r e s onan c e s T e v a t r ond il ep t on r e s onan c e s DM overproduction E l e c t r o w e a k p r e c i s i o n t e s t s m DM [ GeV ] m Z ' [ G e V ] ϵ = g DM A = g qV = Figure 5 . Vector(SM)–Axial(DM): Parameter space forbidden by constraints from ATLAS andTevatron dileptons (green, dashed), electroweak precision observables (blue, dotted) and relic DMoverproduction (red, solid) in the m Z (cid:48) - (cid:15) parameter plane (left) and the m DM - m Z (cid:48) parameter plane(right). In both panels we show the parameter space where perturbative unitarity is violated (grey).For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturatesthe unitarity bound. above, potentially important effects in this scenario can be kinetic mixing of the U (1) gaugebosons as well as effects induced by the dark Higgs, which we are going to discuss below.Let us just mention that all these effects will also be present in the scenario discussed in theprevious section. They are, however, typically less important than the effects of tree-level Z - Z (cid:48) mixing.We first consider the effects of kinetic mixing between the Z (cid:48) and the SM hyperchargegauge boson B : L ⊃ −
12 sin (cid:15) F (cid:48) µν B µν , (5.1)where F (cid:48) µν = ∂ µ Z (cid:48) ν − ∂ ν Z (cid:48) µ and B µν = ∂ µ B ν − ∂ ν B µ . A non-zero value of (cid:15) leads tomixing between the Z (cid:48) and the neutral gauge bosons of the SM (see App. A.1). As in thecase of mass mixing discussed above, there are strong constraints on kinetic mixing fromsearches for dilepton resonances and electroweak precision observables.The dilepton couplings induced via the kinetic mixing parameter (cid:15) can be inferred fromthe mixing matrices and are given in the appendix, cf. eq. (A.10). The S and T parametersare given by αS =4 c s W ξ ( (cid:15) − s W ξ ) ,αT = ξ (cid:18) m Z (cid:48) m Z − (cid:19) + 2 s W ξ(cid:15) , (5.2)where for δm = 0 the mixing parameter ξ is given by ξ = m Z s W (cid:15)/ ( m Z − m Z (cid:48) ) at leadingorder. If (cid:15) is sizeable, i.e. if mixing is present at tree level, the resulting bounds can be– 16 – - - - m Z ' [ GeV ] g q V m DM =
100 GeV g DM A = m DM [ GeV ] m Z ' [ G e V ] g DM A = g qV = L HC m ono j e t s P e r t u r ba t i v e un i t a r i t y v i o l a t i on D il ep t on r e s onan c e s DM overproductionElectroweak precision tests
Figure 6 . Vector(SM)–Axial(DM): Parameter space forbidden by constraints from ATLAS andTevatron dileptons (green, dashed) and electroweak precision observables (blue, dotted) in the m Z (cid:48) - g Vq parameter plane (left) and the m DM - m Z (cid:48) parameter plane (right), assuming that (cid:15) = 0at Λ = 10 TeV so that kinetic mixing is only induced at the one-loop level. In the right panelwe show also the region excluded by LHC monojet (orange, dashed) and dijet (violet, dot-dashed)searches due to tree-level Z (cid:48) exchange for the adopted coupling choice. In both panels we show theparameter space where perturbative unitarity is violated (grey). For the relic density calculationwe have assumed that the mass of the hidden sector Higgs saturates the unitarity bound. quite strong. This expectation is confirmed in figure 5. Note that the relic density curvesshown in figure 5 are basically independent of (cid:15) , because freeze-out is dominated by direct Z (cid:48) exchange for the adopted choice of couplings.While tree-level mixing is tightly constrained, it is reasonable to expect that (cid:15) vanishesat high scales, for example if both U (1)s originate from the same underlying non-Abeliangauge group, as in Grand Unified Theories. Since quarks carry charge under both U (1) (cid:48) and U (1) Y , quark loops will still induce kinetic mixing at lower scales [47], but the magnitudeof (cid:15) can be much smaller than what we considered above. The precise magnitude of thekinetic mixing depends on the underlying theory, but if we assume that (cid:15) (Λ) = 0 at somescale Λ (cid:29) µ > m t will be given by [75] (cid:15) ( µ ) = e g Vq π cos θ W log Λ µ (cid:39) . g Vq log Λ µ . (5.3)We can use this equation (setting µ = m Z (cid:48) ) to translate the bounds from figure 5 intoconstraints on g Vq . The results of such an analysis are shown in figure 6 assuming Λ =10 TeV. As can be seen in figure 6 (left), searches for dilepton resonances give againstringent constraints, implying g Vq < . m Z (cid:48) = 200 GeV and g Vq < m Z (cid:48) = 1 TeV.In the right panel of figure 6 we also show the constraints coming from LHC searchesfor monojets (i.e. jets in association with large amounts of missing transverse energy) andfor dijet resonances, adopted from ref. [26]. These limits are independent of the kinetic Note that as long as the mediator is produced on-shell, the production cross section is proportional to – 17 –ixing (cid:15) since they originate from the tree-level Z (cid:48) exchange and probe larger values of m Z (cid:48) and smaller values of m DM . Nevertheless, dilepton resonance searches and EWPTgive relevant constraints for small m Z (cid:48) and large m DM , which are difficult to probe withmonojet and dijet searches.We conclude from figure 6 that the combination of constraints due to loop-inducedkinetic mixing and bounds from LHC DM searches leave only a small region in parame-ter space (a small strip close to the Z (cid:48) resonance), where DM overabundance is avoided.While this result depends somewhat on our choice Λ = 10 TeV and µ = m Z (cid:48) , it is onlylogarithmically sensitive to these choices.It is worth emphasising that the unitarity constraints shown in figures 4– 6 dependsensitively on the choice of g A DM (cf. figure 1). We therefore show in figure 7 how theseconstraints change if we take g A DM = 0 . g A DM = 1. In this case both m DM and m s can be much larger than m Z (cid:48) . At the same time, however, relic density constraintsbecome significantly more severe, excluding almost the entire parameter space with m DM
Figure 7 . Constraints for small DM couplings ( g A DM = 0 . (cid:15) = 0 at Λ = 10 TeV(cf. figure 6). q χ qχs qZ ′ Z ′ qχ qχZ ′ qχ Z ′ qχ qχq Z ′ Z ′ χ Figure 8 . Contributions to spin-independent scattering. an effective coupling between DM and nucleons of the form
L ∝ f N m N m DM ¯ N N ¯ ψψ ,where m N is the nucleon mass, N = p, n and f N ≈ . L eff ⊃ ( g A DM ) ( g Vq ) π m s m Z (cid:48) × m DM f N m N ¯ N N ¯ ψψ . (5.4)The corresponding spin-independent scattering cross section is given by σ SI N = m f N m N µ π ( g A DM ) ( g Vq ) π m s m Z (cid:48) , (5.5)where µ is the DM-nucleon reduced mass. For masses of order 300 GeV and couplings oforder unity this expression yields σ SI N ∼ − cm , which is below the current bounds fromLUX but well within the potential sensitivity of XENON1T.We note that there are two additional diagrams (shown in the second and third panelof figure 8) that also lead to unsuppressed spin-independent scattering of DM particles [76].– 19 –or m DM (cid:29) m N , the resulting contribution is given by L eff ⊃ ( g A DM ) ( g Vq ) π m − m Z (cid:48) + m Z (cid:48) log( m Z (cid:48) /m ) m Z (cid:48) ( m Z (cid:48) − m ) × m DM f N m N ¯ N N ¯ ψψ . (5.6)If there is no large hierarchy between m DM , m Z (cid:48) and m s , this contribution is of comparablemagnitude to the one from dark Higgs exchange and interference effects can be important.Moreover, there may be a relevant contribution from loop-induced spin-dependent scatter-ing. We leave a detailed study of these effects to future work. In addition to the loop-induced couplings of the dark Higgs to SM fermions discussed inthe previous section, such couplings can also arise at tree-level from mixing. In fact, animportant implication of the presence of a second Higgs field is that the two Higgs fields willin general mix, thus modifying the properties of the mostly SM-like Higgs. Furthermore,the mixing opens up the so-called Higgs portal between the DM and SM particles, leadingto a much richer DM phenomenology than in the case of DM-SM interactions only via thevector mediator.The mixing between the scalars is due to an additional term in the scalar potential: V ( S, H ) ⊃ λ hs ( S ∗ S )( H † H ) . (6.1)The coupling λ hs is a free parameter, independent of the vector mediator. For non-zero λ hs , the scalar mass eigenstates H , are given by H = s sin θ + h cos θH = s cos θ − h sin θ (6.2)where, as shown in App. A.2, θ ≈ − λ hs v wm s − m h + O ( λ hs ) . (6.3)We emphasise that perturbative unitarity implies that m s cannot be arbitrarily large (forgiven m Z (cid:48) and g A DM ) and hence it is impossible to completely decouple the dark Higgs.The resulting Higgs mixing leads to three important consequences. First, the (mostly)dark Higgs obtains couplings to SM particles, enabling us to produce it at hadron collidersand to search for its decay products (or monojet signals). Second, the properties of the(mostly) SM-like Higgs, in particular its total production cross section and potentiallyalso its branching ratios, are modified. And finally, both Higgs particles can mediateinteractions between DM and nuclei, leading to potentially observable signals at directdetection experiments.Higgs portal DM has been extensively studied, see for instance [12, 59, 77–81] for anincomplete selection of references. A full analysis of Higgs mixing effects is beyond the– 20 – m DM [ GeV ] m s [ G e V ] λ hs = D i r e c t d e t e c t i o n P e r t u r ba t i v eun i t a r i t yv i o l a t i on - - m DM [ GeV ] λ h s m s =
250 GeV
Figure 9 . Constraints on m s and λ hs from bounds on the Higgs invisible branching ratio (blue,dotted) and from bounds on the spin-independent DM-nucleon scattering cross section (green,dashed). In the grey parameter region unitarity constraints are in conflict with the stability of thepotential. scope of the present paper. Nevertheless, to illustrate the magnitude of potential effects,let us consider the induced coupling of the SM-like Higgs H ≈ h to DM particles L ⊃ − m DM sin θ w h ¯ ψψ (cid:39) m DM λ hs v m s − m h ) h ¯ ψψ . (6.4)For small λ hs , the resulting direct detection cross section is given by [77] σ SI N (cid:39) µ π m h f N m N m λ hs ( m s − m h ) , (6.5)where we can neglect an additional contribution from the exchange of a dark Higgs provided m s (cid:29) m h . The parameter regions excluded by the LUX results [58] are shown in figure 9(green regions).We note that (in the linear approximation) the direct detection cross section is in-dependent of w and does therefore not depend on m Z (cid:48) or g A DM . Nevertheless w is notarbitrary, because unitarity gives a lower bound √ πw > max (cid:2) √ m DM , m s (cid:3) . At the sametime, stability of the Higgs potential requires 4 λ s λ h > λ hs . These two inequalities canonly be satisfied at the same time if m DM < √ π m s m h v . (6.6)In figure 9, we show the parameter region where unitarity and stability are in conflict ingrey.Finally, if the DM mass is sufficiently small, the SM-like Higgs can decay into pairs ofDM particles, with a partial width given by [77]Γ inv = 18 π m λ hs ( m s − m h ) v m h (cid:18) − m m h (cid:19) / . (6.7)– 21 – m DM [ GeV ] m Z ' [ G e V ] g DM A = g qV = λ hs = D i r e c t d e t e c t i o n P e r t u r ba t i v eun i t a r i t yv i o l a t i on DijetresonancesDijetresonances L HC m ono j e t s D M o v e r p r odu c t i on m DM [ GeV ] m Z ' [ G e V ] g DM A = g qV = λ hs = Figure 10 . Constraints in the m Z (cid:48) - m DM plane for different values of λ hs , taking the mass ofthe hidden sector Higgs to saturate the unitarity bound. The blue (dotted) region is excluded bybounds on the Higgs invisible branching ratio and the green (dashed) region is in conflict withbounds on the spin-independent DM-nucleon scattering cross section. The orange (dashed) regionshows constraints from the CMS monojet search, the purple (dot-dashed) region is excluded by acombination of dijet searches from the LHC, Tevatron and UA2 (adopted from ref. [26]). In thegrey parameter region unitarity constraints are in conflict with the stability of the potential, thered region corresponds to DM overproduction. Note the change of scale in these figures. The invisible branching fraction is tightly constrained by LHC measurements: BR( h → inv) < .
27 [82]. Furthermore, a combined fit from ATLAS and CMS yields µ = 1 . +0 . − . for the total Higgs signal strength [83], which can be used to deduce BR( h → inv) < . σ SI N from LUX, are shownin figure 9 (blue regions).The crucial observation is that the necessary presence of a dark Higgs will in generalinduce additional signatures and therefore lead to new ways to constrain models with a Z (cid:48) mediator using both direct detection experiments and Higgs measurements. However,since λ hs and m s are effectively free parameters, it is difficult to directly compare theconstraints shown in figure 9 to the ones obtained from monojet and dijet searches at theLHC. Nevertheless, we can conservatively estimate the relevance of these effects by fixingthe dark Higgs mass m s to the largest value consistent with perturbative unitarity.The resulting constraints in the conventional m Z (cid:48) - m DM parameter plane with fixedcouplings are shown in figure 10. For comparison we show the constraints from the CMSmonojet search [29] and a combination of dijet searches from the LHC, Tevatron and UA2(adopted from ref. [26]). We find that the additional constraints due to Higgs mixingprovide valuable complementary information in the parameter region with small m Z (cid:48) andlarge m DM , which is difficult to probe with monojet or dijet searches. Note that for m Z (cid:48) > m DM ≈ m Z (cid:48) / m DM > m Z (cid:48) (cf. figure 6). Furthermore, it is worth emphasising that– 22 –or smaller values of m s significantly stronger constraints are expected from Higgs mixing.Moreover, these constraints are independent of the SM couplings of the Z (cid:48) and will thereforebecome increasingly important in the case of small g q . In this paper we have studied the so-called simplified model approach to DM used toparametrise the interactions of a DM particle with the SM via one or several new mediators.It should be clear that simplified models are considered merely as an effective description,used as a tool to combine different DM search strategies. Nevertheless, it is importantthat such models fulfil basic requirements, such as gauge invariance and that perturbativeunitarity is guaranteed in the regions of the parameter space where the model is used todescribe data. To ensure gauge invariance, one needs to impose certain relations betweenthe different couplings, whereas it is necessary to introduce additional states in order torestore perturbative unitarity.We have illustrated these issues by considering a simplified model consisting of afermionic DM particle and a vector mediator, which may for example be the Z (cid:48) gaugeboson of a new U (1) (cid:48) gauge symmetry in the hidden sector. The phenomenology of thismodel depends decisively on whether the couplings of the mediator are purely vectorial orwhether there are non-zero axial couplings (implying that left- and right-handed fields arecharged differently under the new U (1) (cid:48) ). Since the coupling structure on the SM side maybe different from the one of the DM side, there are four different cases of interest: purelyvectorial couplings on both sides, non-zero axial couplings on either the SM side or theDM side, and non-zero axial couplings in both sectors. Our results can be summarized asfollows:1. Vector(SM)–Vector(DM):
In this case no additional new physics is needed toguarantee perturbative unitarity and the mass of the Z (cid:48) can be generated via theStueckelberg mechanism. This model is however highly constrained phenomenolog-ically and a thermal DM is excluded for large parts of the parameter space due tostrong limits on the spin-independent DM-nucleon scattering cross section.Generally, if at least one of the axial couplings is non-zero one needs new physics to unitarizethe longitudinal component of the Z (cid:48) . As a simple example we consider a SM-singletHiggs breaking the dark U (1) (cid:48) . Unitarity then requires the mass of the new Higgs to becomparable to the Z (cid:48) mass. Models with non-zero axial couplings are therefore expected tohave a rich phenomenology with promising experimental signatures in DM direct detectionexperiments and invisible Higgs decays as well as additional DM annihilation channels.2. Axial(SM)–Axial(DM):
The crucial observation in this case is that gauge invari-ance of the SM Yukawa terms requires that the SM Higgs has to be charged underthe U (1) (cid:48) . This requirement has important implications for phenomenology:(a) Electroweak symmetry breaking leads to mass mixing between the Z (cid:48) and theSM Z -boson, which is strongly constrained by EWPT.– 23 –b) The axial couplings of SM fermions to the Z (cid:48) are necessarily flavour universal andequal for quarks and leptons. Hence, it is not possible to couple the DM particleto quarks without also inducing couplings of the Z (cid:48) to leptons. Since the LHCis very sensitive to dilepton resonances, the resulting bounds severely constrainthe model (dominating over constraints from monojet and dijet searches).3. Axial(SM)–Vector(DM):
The constraints from EWPT and dilepton resonancesearches are largely independent of the coupling between the Z (cid:48) and DM and there-fore also apply in the case of purely vectorial couplings on the DM side. However,gauge invariance of the Yukawa couplings implies that it is impossible for the Z (cid:48) tohave purely axial couplings to quarks. Consequently, as soon as there is a vectorialcoupling on the DM side, one necessarily obtains a vector-vector component inducingunsuppressed spin-independent DM-nucleus scattering, which is strongly constrainedby direct detection (see item 1 above).4. Vector(SM)–Axial(DM):
In contrast to the couplings between the Z (cid:48) and quarksit is possible for the DM- Z (cid:48) coupling to be purely axial. Indeed, this situation arisesnaturally in the case that the DM particle is a Majorana fermion such that the vectorcurrent vanishes. If the couplings on the SM side are purely vectorial (i.e. left- andright-handed SM fields have the same charge), the SM Higgs is uncharged under the U (1) (cid:48) and consequently the constraints discussed in item 2 do not apply. Furthermore,the tree-level direct detection cross section is velocity suppressed, leading to muchweaker constraints on this particular scenario.Nevertheless, sizeable spin-independent DM-nucleus scattering can be induced atloop level. In addition, kinetic mixing between the Z (cid:48) and SM gauge bosons (attree level or loop-induced) can be potentially important for EWPT and dileptonsignatures. Assuming (cid:15) = 0 at Λ = 10 TeV, we find that bounds from searches fordilepton resonances due to loop-induced kinetic mixing can still be relevant and giveconstraints that are complementary to the ones obtained from monojet and dijetsearches.All in all we find that imposing gauge invariance and conservation of perturbativeunitarity has important implications for the phenomenology of DM interacting via a vectormediator and that relevant experimental signatures are not captured by considering onlythe interactions of the vector mediator with DM and quarks. This observation is relevantfor the interpretation of various recent analyses of Z (cid:48) -based simplified models, e.g. [22, 26,28, 40, 41, 46]. Indeed, the Z (cid:48) model considered here is severely constrained by EWPT anddilepton resonance searches, either due to tree-level effects or loop-induced kinetic mixing.Moreover, the general expectation is that the mixing between the dark Higgs and the SMHiggs is sizeable and that as a result Higgs portal interactions are present in addition tothe interactions mediated by the Z (cid:48) .The weakest constraints are obtained in the case of purely vectorial couplings on theSM side and purely axial couplings on the DM side. Indeed, this is the only case whereLHC monojet and dijet searches are potentially competitive with other kinds of constraints.– 24 –n all cases that we have considered we find the hypothesis of thermal DM production tobe under significant pressure. In large regions of the parameter space which is still allowedby experiments additional annihilation channels (beyond the Z (cid:48) -mediated interactions) arenecessary to avoid DM overabundance. A more systematic parameter scan of the modelwill be performed in a forthcoming publication [84].Two final comments are in order. First, in this work we have not taken into accountgauge anomalies. In general new fermions are needed to cancel the anomalous trianglediagrams, potentially leading to additional signatures and further constraints on the model.However, due to the constraints implied by gauge invariance of the SM Yukawa terms, thegluon-gluon- Z (cid:48) anomaly vanishes automatically, so that the new fermions need not becharged under colour, making them difficult to probe at the LHC.Second, the requirement of universality of all axial fermion charges (including leptons)follows from the gauge invariance of the SM Yukawa term. It relies on the fact that in theSM all fermion masses are generated by the same Higgs doublet. If the Higgs sector is morecomplicated, for example in a two-Higgs-doublet model, this condition is relaxed and it ispossible to have different axial couplings to up- and down-type quarks or different axialcouplings to quarks and to leptons. In any case, such extensions of the SM go significantlybeyond the simplified model approach and would most likely have a number of implicationsfor Higgs physics, EWPT and other searches for new physics.In conclusion we would like to emphasise that one of the most intriguing implicationsof our study is that a model with a vector mediator should generically also contain a scalarmediator, corresponding to the dark Higgs that generates the vector mass. In the limitthat the mass of the vector mediator is much larger than the mass of the scalar, our Z (cid:48) model can also be used to study simplified models with a scalar mediator, where gaugeinvariance and perturbative unitarity can be similarly problematic. Indeed, our findingssuggest that a strict distinction between simplified models with scalar and vector mediatorsis unnatural and many of the issues with these models may be best addressed in a morerealistic set-up combining the two. Future direct detection experiments together with theupcoming runs of the LHC will be able to thoroughly explore the parameter space of sucha realistic simplified model and test the hypothesis of DM as a thermal relic. Acknowledgements
We thank Sonia El Hedri, Juan Herrero-Garcia, Matthew McCullough and Oscar St˚alfor helpful discussions, and Michael Duerr and Jure Zupan for valuable comments on themanuscript. FK would like to thank the Oskar Klein Center and Stockholm University forhospitality. This work is supported by the German Science Foundation (DFG) under theCollaborative Research Center (SFB) 676 Particles, Strings and the Early Universe as wellas the ERC Starting Grant ‘NewAve’ (638528).– 25 –
Coupling structure from mixing
A.1 Gauge boson mixing
In this appendix we discuss the mixing of a gauge boson ˆ Z (cid:48) of a new U (1) (cid:48) gauge groupwith the SM U (1) Y gauge field ˆ B and the neutral component ˆ W of the SU (2) L weakfields, where we use hats to denote the interaction eigenstates in the original basis. Themixing will then lead to the mass eigenstates Z (cid:48) , Z and A . Following the discussion in [67],we consider an effective Lagrangian including both kinetic mixing and mass mixing (seealso [48]) L = L SM −
14 ˆ X µν ˆ X µν + 12 m Z (cid:48) ˆ Z (cid:48) µ ˆ Z (cid:48) µ −
12 sin (cid:15) ˆ B µν ˆ X µν + δm ˆ Z µ ˆ Z (cid:48) µ , (A.1)where ˆ X µν ≡ ∂ µ ˆ Z (cid:48) ν − ∂ ν ˆ Z (cid:48) µ . Furthermore, we have defined ˆ Z ≡ ˆ c W ˆ W − ˆ s W ˆ B , whereˆ s W (ˆ c W ) is the sine (cosine) of the Weinberg angle and ˆ g (cid:48) , ˆ g are the corresponding gaugecouplings.The field strengths are diagonalised and canonically normalised via the following twoconsecutive transformations [48, 67, 85] ˆ B µ ˆ W µ ˆ Z (cid:48) µ = − t (cid:15) /c (cid:15) B µ W µ Z (cid:48) µ , (A.2) B µ W µ Z (cid:48) µ = ˆ c W − ˆ s W c ξ ˆ s W s ξ ˆ s W ˆ c W c ξ − ˆ c W s ξ s ξ c ξ A µ Z µ R µ , (A.3)where t ξ = − c (cid:15) ( δm + m Z ˆ s W s (cid:15) ) m Z (cid:48) − m Z c (cid:15) + m Z ˆ s s (cid:15) + 2 δm ˆ s W s (cid:15) . (A.4)For (cid:15) (cid:28) δm (cid:28) m Z , m Z (cid:48) , this equation can be approximated by ξ = δm + m Z ˆ s W (cid:15)m Z − m Z (cid:48) . (A.5)The mass eigenvalues m Z and m Z (cid:48) are given by m Z = m Z (1 + ˆ s W t ξ t (cid:15) ) + δm t ξ c (cid:15) ≈ m Z + ( m Z − m Z (cid:48) ) ξ , (A.6) m Z (cid:48) = m Z (cid:48) + δm (ˆ s W s (cid:15) − c (cid:15) t ξ ) c (cid:15) (1 + ˆ s W t ξ t (cid:15) ) ≈ m Z (cid:48) + m Z (cid:48) ξ ( ξ − ˆ s W (cid:15) ) − m Z ( ξ − ˆ s W (cid:15) ) . (A.7)– 26 –e define the ‘physical’ weak angle via s c = π α ( m Z ) √ G F m Z , (A.8)where α = e / (4 π ). Eq. (A.8) also holds with the replacements s W → ˆ s W , c W → ˆ c W and m Z → m ˆ Z , leading to the identity s W c W m Z = ˆ s W ˆ c W m ˆ Z . This equation implies s = ˆ s − ˆ s ˆ c ˆ c − ˆ s (cid:32) − m Z (cid:48) m Z (cid:33) ξ . (A.9)These equations allow us to fix ˆ s W and m ˆ Z in such a way that we reproduce the experi-mentally well-measured quantities s W and m Z .The couplings of the Z (cid:48) to SM fermions induced via mixing can e.g. be found in [24].Of particular interest to our current analysis are the couplings to leptons which are stronglyconstrained. In terms of the mixing parameters they can be written as g V (cid:96) = 14 (3ˆ g (cid:48) (ˆ s W s ξ − c ξ t (cid:15) ) − ˆ g ˆ c W s ξ ) , g A (cid:96) = −
14 (ˆ g (cid:48) (ˆ s W s ξ − c ξ t (cid:15) ) + ˆ g ˆ c W s ξ ) , (A.10)with ˆ g and ˆ g (cid:48) the fundamental gauge couplings of SU (2) L and U (1) Y . A.2 Scalar mixing
Considering the SM Higgs h plus the dark Higgs s , the most general scalar potential afterelectroweak and dark symmetry breaking can be written as V ( s, h ) = − µ s s + w ) − µ h h + v ) + λ h h + v ) + λ s s + w ) + λ hs h + v ) ( s + w ) . (A.11)For λ hs = 0, we obtain the usual formulas v = µ h λ h , m h = 2 λ h v , (A.12) w = µ s λ s , m s = 2 λ s w . (A.13)In this case, there is no mixing between the two Higgs fields even at one-loop level. Nev-ertheless, there is no reason why λ hs should be negligible and therefore the two fields willin general mix. One then obtains for the minimum (assuming 4 λ h λ s > λ hs ) v = 2 2 λ s µ h − λ hs µ s λ s λ h − λ hs , (A.14) w = 2 2 λ h µ s − λ hs µ h λ s λ h − λ hs , (A.15)and for the mass squared eigenvalues m , = λ h v + λ s w ∓ (cid:113) ( λ s w − λ h v ) + λ hs w v . (A.16)– 27 –he corresponding mass eigenstates are H = s sin θ + h cos θH = s cos θ − h sin θ (A.17)with tan 2 θ = λ hs v wλ h v − λ s w . (A.18)For small λ hs we find m ≈ λ h v ≡ m h and m ≈ λ s w ≡ m s . This yields θ ≈ − λ hs v wm s − m h + O ( λ hs ) . (A.19) References [1] M. Beltran, D. Hooper, E. W. Kolb, and Z. C. Krusberg,
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