Higgs vacuum stability from the dark matter portal
Abstract
We consider classically scaleinvariant extensions of the Standard Model (CSI ESM) which stabilise the Higgs potential and have good dark matter candidates. In this framework all mass scales, including electroweak and dark matter masses, are generated dynamically and have a common origin. We consider Abelian and nonAbelian hidden sectors portally coupled to the SM with and without a real singlet scalar. We perform a careful analysis of RG running to determine regions in the parameter space where the SM Higgs vacuum is stabilised. After combining this with the LHC Higgs constraints, in models without a singlet, none of the regained parameter space in Abelian ESMs, and only a small section in the nonAbelian ESM survives. However, in all singletextended models we find that the Higgs vacuum can be stabilised in all of the parameter space consistent with the LHC constraints. These models naturally contain two dark matter candidates: the real singlet and the dark gauge boson in nonAbelian models. We determine the viable range of parameters in the CSI ESM framework by computing the relic abundance, imposing direct detection constraints and combining with the LHC Higgs constraints. In addition to being instrumental in Higgs stabilisation, we find that the singlet component is required to explain the observed dark matter density.
JHEP 1408 (2014) 026 IPPP/14/22, DCPT/14/44
Higgs vacuum stability from the dark matter portal

Institute for Particle Physics Phenomenology, Durham University,
South Road, Durham, DH1 3LE, U.K.
ArXiv ePrint: 1403.4953
Contents
1 Introduction
Following the discovery of the Higgs boson by the ATLAS and CMS experiments [1, 2] particle physics has entered a new epoch. The particle spectrum of the Standard Model is now complete yet nevertheless, we know that the Standard Model cannot be a complete theory of particle interactions, even if we do not worry about gravity. The more fundamental theory should be able to address and predict the matterantimatter asymmetry of the universe, the observed dark matter abundance, and it should stabilise the Standard Model Higgs potential. It should also incorporate neutrino masses and mixings. In addition it is desirable to have a particle physics implementation of cosmological inflation and possibly a solution to the strong CP problem. Finally there is still a question of the naturalness of the electroweak scale; the Standard Model accommodates and provides the description of the Higgs mechanism, but it does not, and of course was not meant to, explain the origin of the electroweak scale and why it is so much lighter than the UV cutoff scale.
In this paper we concentrate on a particular approach of exploring Beyond the Standard Model (BSM) physics, based on the fact that the Standard Model contains a single mass scale, the negative Higgs mass squared parameter, , in the SM Higgs potential,
(1.1) 
In the unitary gauge, the vacuum expectation value (vev) and the mass of the physical SM Higgs field are triggered by the scale,
(1.2) 
If this single mass scale is generated dynamically in some appropriate extension of the SM, the resulting theory will be manifestly classically scaleinvariant. Such theories contain no explicit massscales (all masses have to be generated dynamically), but allow for nonvanishing beta functions of their dimensionless coupling constants. In section 2 we employ the seminal mechanism of massscale generation due to Coleman and Weinberg (CW) [3] and show how the electroweak scale emerges in the Standard Model coupled to the CW sector.
Classically ScaleInvariant Extensions of the Standard Model – CSI ESM – amount to a highly predictive model building framework. The high degree of predictivity/falsifiability of CSI ESM arises from the fact that one cannot start extending or repairing a CSI model by introducing new mass thresholds where new physics might enter [4, 5]. All masses have to be generated dynamically and, at least in the simple models studied in this paper, they are all related to the same dynamical scale, which is not far above the electroweak scale. This is consistent with the manifest CSI and as the result protects the electroweak scale itself by ensuring that there are no heavy massscales contributing radiatively to the Higgs mass. Furthermore, in the CSI ESM approach one naturally expects the common origin of all mass scales, i.e. the EW scale relevant to the SM, and the scales of new physics. In other words the CSI ESM framework, if it works, realises the Occam’s razor succinctness.
The CSI ESM theory is a minimal extension of the SM which should address all the subPlanckian shortcomings of the SM, such as the generation of matterantimatter asymmetry, dark matter, stabilisation of the SM Higgs potential, neutrino masses, inflation, without introducing scales much higher the electroweak scale. It was shown recently in Ref. [6] that the CSI U(1) SM theory where the ColemanWeinberg U(1) sector is reinterpreted as the gauged U(1) symmetry of the SM, can generate the observed value of the matterantimatter asymmetry of the Universe without introducing additional mass scales nor requiring a resonant finetuning. This CSI U(1) SM theory also generates Majorana masses for the righthanded sterile neutrinos in the range between 200 MeV and 500 GeV and leads to visible neutrino masses and mixings via the standard seasaw mechanism [7, 6].
It follows that not only the baryonic matterantimatter asymmetry, but also the origin of dark matter must be related in the CSI ESM to the origin of the electroweak scale and the Higgs vacuum stability. Papers [8, 9] have shown that in the nonAbelian CSI SU(2) SM theory there is a common origin of the vector dark matter and the electroweak scale. It was also pointed out in [10] that a CSI ESM theory with an additional singlet that is coupled nonminimally to gravity, provides a viable particle theory implementation of the slowroll inflation. Furthermore, the singlet responsible for inflation also provides an automatic scalar dark matter candidate.
The main motivation of the present paper is to study in detail the link between the stability of the electroweak vacuum and the properties of multicomponent (vector and scalar) dark matter in the context of CSI ESM theory. Our main phenomenological results are described in sections 4 and 5. There, in a model by model basis we determine regions on the CSI ESM parameter space where the SM Higgs vacuum is stabilised and the extended Higgs sector phenomenology is consistent with the LHC exclusion limits. We then investigate the dark matter phenomenology, compute the relic abundance and impose constraints from direct detection for vector and scalar components of dark matter from current and future experiments.
2 CSI ESM building generation of the EW scale
In the minimal Standard Model classical scale invariance is broken by the Higgs mass parameter in eq. (1.1). Scale invariance is easily restored by reinterpreting this scale in terms of a vacuum expectation value (vev) of a new scalar , coupled to the SM via the Higgs portal interaction, Now, as soon as an appropriate nonvanishing value for can emerge dynamically, we get in (1.1) which triggers electroweak symmetry breaking.
In order to generate the required vev of we shall follow the approach reviewed in [5, 10] and employ the seminal mechanism of the mass gap generation due to Coleman and Weinberg [3]. In order for the CW approach to be operational, the classical theory should be massless and the scalar field should be charged under a gauge group . The vev of the CW scalar appears via the dimensional transmutation from the running couplings, leading to spontaneous breaking of and ultimately to EWSB in the SM.
The CSI realisations of the Standard Model which we will concentrate on in this paper are thus characterised by the gauge group where the first factor plays the role of the hidden sector. The requirement of classical scale invariance implies that the theory has no input mass scales in its classical Lagrangian; as we already mentioned, all masses have to be generated dynamically via dimensional transmutation. The basic treelevel scalar potential is
(2.1) 
The matter content of the hidden sector gauge group can vary: in the minimal case it consists only of the CW scalar ; more generally it can contain additional matter fields, including for example the SM fermions. We will discuss a few representative examples involving Abelian and nonAbelian gauge groups with with a more and a lessminimal matter content.
The minimal U(1) theory coupled to the SM via the Higgs portal with the scalar potential (2.1) was first considered in [11]. The phenomenology of this model was analysed more recently in the context of the LHC, future colliders and low energy measurements in [5]. Classical scale invariance is not an exact symmetry of the quantum theory, but neither is it broken by an arbitrary amount. The violation of scale invariance is controlled by the anomaly in the trace of the energymomentum tensor, or equivalently, by the logarithmic running of dimensionless coupling constants and their dimensional transmutation scales. In weakly coupled perturbation theory, these are much smaller than the UV cutoff. Therefore, in order to maintain anomalously broken scale invariance, one should select a regularisation scheme that does not introduce explicit powers of the UV cutoff scale [12]. In the present paper we use dimensional regularisation with the scheme. In dimensional regularisation, and in theories like ours that contain no explicit mass scales at the outset, no large corrections to mass terms can appear. In this regularisation, which preserves classical scale invariance, the CSI ESM theory is not finetuned in the technical sense [10, 5].
Other related studies of CSI ESModels can be found in [13, 14, 15, 16, 17, 18, 19]. We would also like to briefly comment on two scaleinvariancedriven approaches which are different from ours. The authors of Refs. [4, 20, 21, 22, 23] envision CSI models with dimensional transmutation, which are not based on the CW gaugesectorextension of the SM, but rather appeal to an extended matter content within the SM, or to a strongly coupled hidden sector. One can also consider model building based on the approach with an exact quantum scale invariance of the UV theory, as discussed recently in [24] and [25]. It is important to keep in mind that classical scale invariance of the effective theory below the Planck scale does not necessarily assume or is directly related to the hypothesised conformal invariance of the UV embedding of the SM.
2.1 Csi U(1) Sm
This is the minimal classically scaleinvariant extension of the SM. The SM Higgs doublet is coupled via the Higgsportal interactions to the complex scalar
(2.2) 
where is a Standard Model singlet, but charged under the U(1)ColemanWeinberg gauge group. The hidden sector consists of this U(1) with plus nothing else. In the unitary gauge one is left with two real scalars,
(2.3) 
and the treelevel scalar potential (2.1) reads
(2.4) 
where the superscripts indicate that the corresponding coupling constants are the treelevel quantities.
We now proceed to include radiative corrections to the classically scaleinvariant potential above. Our primary goal in this section is to show how quantum effects generate the nontrivial vacuum with nonvanishing vevs and , to derive the matching condition between coupling constants in the vacuum and to compute the scalar mass eigenstates, and of the mixed scalar fields and . We then determine the SM Higgs selfcoupling in terms of and other parameters of the model. The fact that is not identified with will be of importance later when we discuss the stability of the SM Higgs potential in our model(s).
For most of this section we will follow closely the analysis of Ref. [5], but with a special emphasis on two aspects of the derivation. First, is that the effective potential and the running couplings need to be computed in the scheme, which is the scheme we will also use later on for writing down and solving the RG equations.
Following the approach outlined in [5] one can simplify the derivation considerably by first concentrating primarily on the CW sector and singling out the 1loop contributions arising from the hidden U(1) gauge field.^{1}^{1}1Radiative corrections due to the CW scalar selfcoupling will be subleading in this approach cf. eq. (2.8) below. Perturbative corrections arising from the SM sector will then be added later. Effective potentials and running couplings in this paper will always be computed in the scheme. In this scheme the 1loop effective potential for reads, cf. [26],
(2.5) 
which depends on the RG scale that appears both in the logarithm and also in the 1loop running CW gauge coupling constant . The running (or renormalised) selfcoupling at the RG scale is defined via
(2.6) 
We can now express the effective potential in terms of this renormalised coupling constant by substituting into eq. (2.5), obtaining
(2.7) 
The vacuum of the effective potential above occurs at Minimising the potential (2.7) with respect to at gives the characteristic ColemanWeinbergtype relation between the scalar and the gauge couplings,
(2.8) 
It is pleasing to note that this matching relation between the couplings takes exactly the same form as the one obtained in the CW paper in the cutoff scheme – i.e. accounting for the 3! mismatch in the definition of the coupling in [3] we have where is the coupling appearing in [3], .
Shifting the CW scalar by its vev and expanding the effective potential in (2.7), we find the mass of ,
(2.9) 
and the mass of the U(1) vector boson,
(2.10) 
The expressions above are once again identical to those derived in the cutoff scheme in [3, 5].
We now turn to the SM part of the scalar potential (2.4), specifically
(2.11) 
where we have dropped the superscript for the portal coupling, as it will turn out that does not run much. The SM scale is generated by the CW vev in the second term,
(2.12) 
and this triggers in turn the appearance of the Higgs vev as in the first equation in (1.2).
The presence of the portal coupling in the potential (2.11) (or more generally (2.4)) provides a correction to the CW matching condition (2.8) and the CW mass (2.9). By including the last term on the r.h.s of (2.4) to the effective potential in (2.5) and (2.7), we find the induced correction to the equations (2.8)(2.9) which now read
(2.13)  
(2.14) 
in full agreement with the results of [5]. In this paper, we consider small values of so that these corrections are negligible, since
Our next task is to compute the Higgs mass including the SM radiative corrections. To proceed we perform the usual shift, , and represent the SM scalar potential (2.11) as follows,
(2.15) 
where for overall consistency we have also included oneloop corrections to the Higgs mass arising in the Standard Model,
(2.16) 
These corrections are dominated by the topquark loop and are therefore negative. The appearance of in the denominator of is slightly misleading, and it is better to recast it as,
(2.17) 
The vev is determined from (2.15) by minimisation and setting , and thus the last term in (2.15) does not affect the value of , however it does contribute to the oneloop corrected value of the Higgs mass. We have,
(2.18) 
where is the oneloop corrected value of the selfcoupling.
The two scalars, and , both have vevs and hence mix via the mass matrix,
(2.19) 
where is given in (2.14) (and already includes the correction).^{2}^{2}2The mass mixing matrix (2.19) is equivalent to the mass matrix derived in [5] which was of the form: in terms of and , with The mass eigenstates are the two Higgs fields, and with the mass eigenvalues,
(2.20) 
It is easy to see that in the limit where the portal coupling is set to zero, the mixing between the two scalars and disappears resulting in and mass eigenvalues, as one would expect. However, for nonvanishing , the mass eigenstates and are given by
(2.21) 
with a nontrivial mixing angle . Which of these two mass eigenstates should be identified with the SM Higgs of eq. (1.2)?
The answer is obvious, the SM Higgs is the eigenstate which is ‘mostly’ the scalar (i.e. the scalar coupled to the SM electroweak sector) for small values of the mixing angle,
(2.22) 
The SM Higgs selfcoupling constant appearing in the SM Higgs potential (1.1) can be inferred from , but it is not the relevant or primary parameter in our model ( is).
In our computations for the RG evolution of couplings and the analysis of Higgs potential stabilisation carried out in this paper, we solve the initial condition (2.22) for the eigenvalue problem of (2.19) numerically without making analytical approximations. However, we show some simple analytic expressions to illuminate our approach.
In the approximation where is small we can expand the square root in (2.20) and obtain:
(2.23)  
(2.24) 
Note that our requirement of assigning the SM Higgs mass value of 126 GeV to the ‘mostly state’ selects two different roots of (2.20) in the equations above, depending on whether the state or the state is lighter. As the result, there is a ‘discontinuity of the SM Higgs identification’ with in the first equation, while in the second equation. Similarly, the value of is smaller or greater than the perceived value of in the SM, in particular,
(2.25) 
One concludes that in the case of the CW scalar being heavier than the SM Higgs, it should be easier to stabilise the SM Higgs potential, since the initial value of here is larger than the initial value of the coupling and as such, it should be useful in preventing from going negative at high values of the RG scale.^{3}^{3}3This point has been noted earlier in the literature in [27, 28], [8] in the context of assisting the stabilisation of the SM Higgs by integrating out a heavy scalar. In our case the second scalar does not have to be integrated out. In fact, the required stabilising effect arises when the second scalar is not much heavier than the SM Higgs, which manifests itself in keeping the denominator in (2.25) not much greater than the square of the EW scale.
On a more technical note, in our computations we also take into account the fact that the requirement of stability of the Higgs potential at high scales goes beyond the simple condition at all values of , but should be supplemented by the slightly stronger requirement emerging from the treelevel stability of the potential (2.4), which requires that
In the following sections 2.22.4, we extend the construction above to models with more general hidden sectors. First of all, the G ColemanWeinberg sector can be extended so that SM fermions are charged under G, and, secondly, G can also be nonAbelian. In addition, these CSI ESM models can include a gauge singlet with portal couplings to the Higgs and the CW scalar field. In sections 4 and 5 we will explain how the combination of constraints arising from the Higgs vacuum stability, collider exclusions, and dark matter searches and phenomenology will apply to and discriminate between these varieties of CSI SM extensions.
2.2 Csi U(1) Sm
The theory was originally introduced in [29], and in the context of the CW classically scaleinvariant extension of the SM this theory was recently studied in [17] and by the two of the present authors in [6]. In the latter reference it was shown that this model can explain the matterantimatter asymmetry of the universe by adopting the ‘Leptogenesis due to neutrino oscillations’ mechanism of [30] in a way which is consistent with the CSI requirement that there are no large mass scales present in the theory.
The U(1) SM theory is a particularly appealing CSI ESM realisation, since the gauge anomaly of U(1) cancellation requires that the matter content of the model automatically includes three generations of righthanded Majorana neutrinos. All SM matter fields are charged under the U(1) gauge group with charges equal to their Baryon minus Lepton number. In addition, the CW field carries the charge 2 and its vev generates the Majorana neutrino masses and the mass of the U(1) boson. The standard seesaw mechanism generates masses of visible neutrinos and also leads to neutrino oscillations.
The scalar field content of the model is the same as before, with being the complex doublet and , the complex singlet under the SM. The treelevel scalar potential is given by (2.1) which in the unitary gauge takes the form (2.4). Our earlier discussion of the mass gap generation in the CW sector, the EWSB and the mass spectrum structure, proceeds precisely as in the previous sections, with the substitution . The oneloop corrected potential (2.7) becomes:
(2.26) 
Minimising it at gives the matching condition for the couplings and the expansion around the vacuum at determines the mass of the CW scalar field (cf. (2.13)(2.14)),
(2.27)  
(2.28) 
in agreement with [6]. The expressions for the Higgs field vev, , and the Higgs mass, , are unchanged and given by (2.18). The mass mixing matrix is the same as in (2.19) with given by (2.28).
2.3 Csi Su(2) Sm
One can also use a nonAbelian extension of the SM in the CSI ESM general framework. In this section we concentrate on the simple case where the CW group is SU and for simplicity we assume that there are no additional matter fields (apart from the CW scalar ) charged under this hidden sector gauge group. This model was previously considered in [8] and subsequently in [9]. The novel feature of this model is the presence of the vector dark matter candidate – the SU ColemanWeinberg gauge fields [8].
The classical scalar potential is the same as before,
(2.29) 
where as well as the Higgs field are the complex doublets of the SU(2) and the SU(2) respectively. In the unitary gauge for both of the SU(2) factors we have,
(2.30) 
The analogue of the oneloop corrected scalar potential (2.7) now becomes:
(2.31) 
where is the coupling of the SU(2) CW gauge sector. Minimising at gives:
(2.32)  
(2.33) 
2.4 Csi Esm singlet
All Abelian and nonAbelian CSI extensions of the SM introduced above can be easily extended further by adding a singlet degree of freedom, a onecomponent real scalar field . Such extensions by a real scalar were recently shown in Ref. [10] to be instrumental in generating the slowroll potential for cosmological inflation when the scalar is coupled nonminimally to gravity. The two additional features of models with the singlet, which are particularly important for the purposes of this paper, are that (1) the singlet portal coupling to the Higgs will provide an additional (and powerful) potential for the Higgs stabilisation, and (2) that the singlet is also a natural candidate for scalar dark matter.
The gauge singlet field is coupled to the ESM models of sections 2.12.3 via the scalar portal interactions with the Higgs and the CW field ,
(2.34) 
Equations (2.1), (2.34) describe the general renormalisable gaugeinvariant scalar potential for the three classically massless scalars as required by classical scale invariance. The coupling constants in the potential (2.34) are taken to be all positive, thus the potential is stable and the positivity of and ensure that no vev is generated for the singlet . Instead the CW vev generates the mass term for the singlet,
(2.35) 
in the vacuum , , .
3 RG Evolution
In this section our aim is to put together a tool kit which will be necessary to determine regions of the parameter spaces of CSI ESModels where the Higgs vacuum is stable. To do this we first need to specify the RG equations for all CSI ESM theories of interest, with and without the additional singlet. We will also fix the initial conditions for the RG evolution.
Following this more technical build up in the present section, the Higgs vacuum stability and collider constraints on the Higgssector phenomenology will be analysed in section 4.
3.1 Standard Model U(1)
This is the simplest scaleinvariant extension of the SM. The hidden sector is an Abelian U(1) which couples only to the CW scalar (of charge 1) and no other matter fields. We now proceed to write down the renormalisation group equations for this model.
The scalar couplings , and are governed by:
(3.2)  
The RG equation for the top Yukawa coupling is,
(3.4) 
Finally, , and denote the gauge couplings of the U(1), which obey,
(3.5)  
(3.6)  
(3.7) 
A characteristic feature of the Abelian ESM theory is , the kinetic mixing of the two Abelian factors, U(1). For a generic matter field transforming under both U(1)’s with the charges and , the kinetic mixing is defined as the coupling constant appearing in the the covariant derivative,
(3.8) 
Kinetic mixing is induced radiatively in so far as there are matter fields transforming under both Abelian factors. In the present model it is induced by the mass eigenstates of the scalar fields. In what follows for simplicity we will choose at the top mass.
3.2 Standard Model U(1)
The RG equations in the theory are the appropriate generalisation of the equations above. These equations were first derived in [31] and were also discussed recently in [17]. In our conventions the RG evolution in the CSI U(1) SM theory with the classical scalar potential (2.1) is determined by the set of RG equations below:
(3.9)  
(3.10)  
(3.11)  
The Yukawas for the top quark and for 3 Majorana neutrinos are determined via
(3.12)  
(3.13) 
and the gauge couplings are given by eqs. (3.7) together with
(3.14)  
(3.15) 
3.3 Standard Model U(1) real scalar
When discussing the Higgs vacuum stability we will soon find out that the size of the available region on the CSI ESM parameter space will be significantly dependent on whether or not the theory includes an additional singlet field. We are thus led to extend the RG equations above to the case with the singlet.
The scalar selfcouplings and portal couplings in this model are governed by the following equations,
(3.16)  
(3.17)  
(3.18)  
(3.19)  
(3.20)  
(3.21) 
The rest of the RG equations are the same as before. Equations for Yukawa couplings are (3.12)(3.13), and the gauge couplings are given by eqs. (3.7) together with (3.14)(3.15). As always, we set .
Note that it is easy to derive a simple formula, eq. (3.24) below, which computes the coefficients in front of scalar couplings on the right hand sides of the RG equations. First, let us write the classical scalar potential in the form,
(3.22) 
where in our case , and the second sum is understood as over the three pairs of indices, , and . The notation denotes the canonically normalised real components of the Higgs, , the complex doublet and the real singlet . In general we denote the number of real components of each of the species of and . It is then easy to derive the expressions for scalarcoupling contributions to all the selfinteractions, by counting the contributing 4point 1PI diagrams involving 2 scalar vertices. For the beta functions of the selfcouplings we get,
(3.23) 
and the portal couplings are governed by,
(3.24) 
This formula is valid for all of the CSI ESM examples considered in this paper.
3.4 Standard Model Su(2)
We can also write down the relevant renormalisation group equations for the classically scaleinvariant Standard Model SU(2) theory with the scalar potential given by eq. (2.29). These RG equations were first derived in Refs. [8, 9]. For scalar selfcouplings and , and the portal coupling we have:
(3.25)  
(3.26)  
(3.27) 
where the top Yukawa coupling obeys
(3.28) 
and , are the gauge couplings of the SU(2) SU(3) SU(2) U(1),