Higgs Naturalness and Dark Matter Stability by Scale Invariance

Extending the spacetime symmetries of standard model (SM) by scale invariance (SI) may address the Higgs naturalness problem. In this article we attempt to embed accidental dark matter (DM) into SISM, requiring that the symmetry protecting DM stability is accidental due to the model structure rather than imposed by hand. In this framework, if the light SM-like Higgs boson is the pseudo Goldstone boson of SI spontaneously breaking, we can even pine down the model, two-Higgs-doublets plus a real singlet: The singlet is the DM candidate and the extra Higgs doublet triggers electroweak symmetry breaking via the Coleman-Weinberg mechanism; Moreover, it dominates DM dynamics. We study spontaneously breaking of SI using the Gillard-Weinberg approach and find that the second doublet should acquire vacuum expectation value near the weak scale. Moreover, its components should acquire masses around 380 GeV except for a light CP-odd Higgs boson. Based on these features, we explore viable ways to achieve the correct relic density of DM, facing stringent constraints from direct detections of DM. For instance, DM annihilates into $b\bar b$ near the SM-like Higgs boson pole, or into a pair of CP-odd Higgs boson with mass above that pole.


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Scale invariance (SI) "removes" quardratic divergency (QD) Recently, the LHC discovered a new resonance, which is putative the Higgs boson predicted by the SM [1]. Its couplings are well consistent with the SM predictions and give no illustrative hints for new physics beyond the SM. However, this by no means implies the end of SM. We have a variety of reasons to go beyond the SM, e.g., its lack of DM candidates. The most robust one is theoretical: If the Higgs boson is fundamental, it will suffer QD which renders the weak scale instable under quantum corrections. Removing QD guides the direction of new physics in decades.
A new symmetry may be indispensable to protect this particle, e.g., the well known supersymmetry. In 1995, classical SI was proposed by Bardeen as a candidate [2]. Here we present the argument. QD may be exaggerated by the choice of an improper regularization method like the cut-off regularization. While in the more elaborate dimension regularization (DR) [3] it is not worse than the logarithmic divergency, both as a pole 1/ǫ. People may still insist on the right to use cut-off regularization for the Higgs sector, thus facing the QD problem again. But SI rejects that right, otherwise we have to introduce a Higgs mass counter term δm 2 Φ |Φ| 2 , which however does not appear in the original Lagrangian. Thus this regularization is not consistent with SI. By contrast, using DR the QD automatically disappears. So, DR is the self-consistent regulation method in the the SM with SI (SISM), where we may do not suffer the QD problem 1 . SI anomaly and the origin of EW scale The realistic SM is not SI since it involves a characteristic scale, the EW scale. Therefore SI should be broken somehow. This surprisingly accords with the fact that at quantum level 1 Here argument of SI getting rid of QD is a little bit different to the original paper [2], which did not stress the necessity of DR.
SI is anomalous, manifested in the running of coupling constants. Actually, symmetry spontaneously breaking in the scaless theory was explored long ago by Coleman and Weinberg (CW) in the classical paper [4], and they found that indeed it can happen through dimension transition. So hopefully we can understand electroweak symmetry breaking (EWSB) and the origin of the EW scale in the SISM by means of SI anomaly. Let us briefly review how the CW mechanism works and its generic features. Assuming that the vacuum of a model is determined by the minimum of the effective potential V eff (φ cl ) which contains a single classical field φ cl . In the scaless theory, at one-loop level V eff (φ cl ) can be generically written as with Q the renormalization scale. A and B are functions of dimensionless constants involving the couplings of φ cl . In the MS scheme, they are given by with λ the tree-level quartic coupling constant in the potential. P sums over particles which have internal degrees of freedom n P and field-dependent masses m P = g P Φ cl . The factor A P = 3/2, 3/2, 5/6 for the spin 0, 1/2 and 1 particles, respectively. Due to the field-dependent logarithmic term, an extreme is created given ln(Q/ φ cl ) = 1 4 + A/2B. To avoid a large logarithmic term such that the loop-expansion of V eff is invalid, one may want to choose Q = φ cl and then A/B| Q= φ cl = −1/2. Expanding around φ cl , it is not difficulty to get the cur-vature of V eff at φ cl : Thus, given B < 0 the extreme is a maximum. By contrast, if B > 0 we obtain a local minimum and m 2 φ is the mass square of the corresponding quantum in this vacuum. As expected, its mass is loop suppressed since it is the pseudo Goldstone boson of SI anomaly. This may interpret the lightness of the SM-like Higgs boson.
Applying the above results to the ISSM without any extension, one soon finds that B < 0 owing to the heaviness of top quark and its large internal degrees of freedom n t = −12. To overcome it, naturally we introduce scalar or vector bosons which have large couplings to the SM Higgs doublet Φ [5]. A variant is the Higgs portal λ X (Φ † Φ)|X| 2 , which produces the ordinary Higgs mass term m 2 Φ = λ X v 2 X < 0 with v X = X via a hidden CW mechanism [6] (or hidden confining gauge dynamics [7]). In this letter, it is found that the original version provides a predictive framework for DM.
Accidental dark matter (aDM) from SI DM guides us to new physics, nevertheless to too many directions thus lacking unambiguous predictions. What will happen when we embed DM into SISM?
Let us start from a toy SISM, that only includes singlet scalars S i to implement the CW mechanism. It has local symmetries G SM = SU (3) C × SU (2) L × U (1) Y , Poincare and SI spacetime symmetries. They restrict the most general renormalizalbe potential to be with i, ... = 1, 2, ...N . The singlets obtain masses through the second term, and thus it is convenient to work in the basis where they are diagonal, i.e, λ ij = λ i δ ij > 0.
Remarkably, an accidental Z 2 −symmetry, only S i odd under it, emerges. If only Higgs doublet acquires VEV, this Z 2 survives after EWSB. Consequently the lightest singlet, denoted by S, will be stable and service as a DM candidate. Such accidental DM stability is a reminiscence of proton stability in the SM, where the baryon number is accidentally conserved and protects proton stability due to its field content and symmetries. Stabilizing DM similarly is elegant, since we do not have to impose any ad hoc symmetry by hand.
On top of stability, other DM particle properties are largely specified. The single term λS 2 |Φ| 2 /2 accounts for that. First, just like other massive members in the SM, DM acquires mass m DM = λ/2v through EWSB. Next, interactions between DM and the visible particles are via the Higgs portal. The Higgs mediated DM-nucleon spinindependent cross section σ SI = 4f 2 p µ 2 p /π, with µ p the reduced mass of the proton-DM system and where m h = 125 GeV and the values of the nucleon form factors f (p) Tq can be found in Ref. [8] (The updated data favors a smaller f (p) Ts , but it does not affect our ensuing qualitative conclusions). f p has an unique dependence on λ. We can get a conservative upper bound λ 0.03 from XENON100 [9]. It means that m DM 30 GeV and then the Higgs invisible decay into a pair of S kinematically opens and has a width The SM Higgs width at 125 GeV is about 4.1 MeV. Thus, even if the branching ratio of invisible decay is allowed to be as large as 20%, we still get a more stringent upper bounds on λ in turn DM mass: That small coupling causes S to annihilate ineffectively, and then fail to get correct relic density Ω DM h 2 ≃ 0.1 (see a relevant discussion [10]). We argue that a singlet scalar is the unique candidate of aDM. Consider a fermionic singlet ψ. It fails because SI can not forbid the couplinglΦψ which spoils the accidental Z 2 . We can exhaust all possible DM candidate dwelling in a multiplet (2j + 1, Q Y ) with j an integer or half integer, which forms a representation of SU (2) L × U (1) Y . There are some arguments against DM from j ≥ 1/2 multiplets, in particular one is that the Z−boson mediated DM-nucleon SI scattering is too large. But it has a loophole, multiplets with Q Y = 0 possessing no DM-DM-Z coupling. A rather strict no-go may be established as the following: (I) The Higgs doublet is the unique, at least dominant, source of DM mass via The fermionic DM is a Dirac particle because it must carry hypercharge, otherwise it can not couple to Φ; (II) As in the toy model, the constraint from XENON100 and then the Higgs invisible decay forces m DM to lie well below m Z /2; (III) Consequently, the invisible decay widths of Z−boson into the charged partners of DM are too large 2 . In conclusion, only a real scalar singlet is the viable aDM candidate.
Pin down aDM model We have shown that for aDM by SI, its quantum number, mass origin and interactions are almost fixed, leading to an unsuccessful toy model. We can cure it by considering non-singlet EWSB triggers. With them, DM gains effective annihilation channels. We have two choices, into a pair of photons/gluons at looplevel, or into the SM fermions at tree-level. The former requires a quite large quartic coupling between DM and triggers, as jeopardizes perterbility of the model around the weak scale. What is more, such DM is excluded by the search of gamma-ray line or colored relics (the triggers must carry color thus being stable). Aside from these, they are likely to affect the Higgs production/decay rates too significantly. The (2j +1, Q Y ) scalar trigger shifts the amplitude of Higgs to di-photon by an amount (normalized to the W −loop amplitude) [11] 1 24 Thus, only these with j ≤ 1 can change the Higgs to diphoton rate by less than 40%. In particular, for (2, ±1/2) the corresponding change is about 7%. So, we have to rely on the latter. By virtue of Eq. (9), three kinds of candidates are left, (3,0), (3, ±1) and (2, ±1/2). Triplets allow non-accidental Z 2 couplings and hence the doublets are unique candidates. Further using the economic criteria, we eventually pine down the model: two-Higgs-doublets plus a real singlet (2HDM+S). There are different versions of 2HDM classified by the pattern of couplings between the extra doublet and SM fermions [12]. For each version, DM has proper annihilating channels into the SM fermions. For definiteness, here we focus on type-II, where Φ 1,2 couple to the up-type and down-type quarks (and leptons), respectively. Its general tree-level potential is given by Here λ ij = λ ji and we have assumed real parameters. Φ 1,2 develop VEVs to account for the fermion masses. As usual, we define tan Viewing from the current Higgs data, which favors a quite SM-like Higgs boson, it is reasonable to consider the decoupling limit of the 2HDM. Then we have a large tan β 10 thus a small v 2 20 GeV. Later it is found that a large tan β is also favored to enhance the DM annihilation rate.
which however produce mass splittings ∝ Q 3 H v 2 /m DM , with Q 3 H the charge of H−components under T 3 H . Thus some components will become even lighter.
Before heading towards the details of EWSB through the CW mechanism, we first make some reasonable simplifications. The singlet S can not acquire a VEV. For the doublets, Φ 1 is the dominant source of EWSB while Φ 2 , the unique trigger, its VEV v 2 should be sufficiently small for the sake of DM and Higgs phenomenologies. Such vacuum configuration can be arranged by a small λ 12 , which drives Φ 2 away from the origin via a tadpole term of Φ 2 . In that limit Φ 2 locates at With them, it is justified to approximately study EWSB similar to the original CW analysis based on the SM, i.e., with one Higgs doublet Φ 1 , and Φ 2 is merely a trigger. Then the (Higgs) field-dependent masses of Φ 2 are: We have decomposed fields as Φ T with G 0,− the Goldstones bosons. λ 2 −dominance renders either H or A tachyonic. λ 1 −dominance causes large mass splitting between the neutral and charged components, which implies a large violation of EW precise test. Therefore, λ 12 −dominance is the favored case where λ 12 and λ 12 play the main roles in determining the vacuum. The results The model mainly is described by only four parameters, λ 12 , λ 12 (traded with tan β using Eq. (11)), η 1 and η 12 . Higgs and DM phenomenologies tightly constrains them, leading to a quite predictive aDM.
In the RGE-improved effective potential, the curvature becomes larger and accordingly the needed λ 12 becomes smaller. The concrete value depends on the input λ 12 via Eq. (14), but typically is small, ∼ 5%. Note that at Q = v we have β λ12 approaching 1, which means that at the TeV scale β λ12 will exceed 1 and then perturbativity breaks down there.  (17) taking m DM = 10.0 GeV. The large tan β enhancement is manifested, and it is just the reason why aDM succeeds here. On the other hand, σ SI from the H contribution is suppressed. Actually, due to the cross symmetry their ratio is fixed up to DM mass: For m b m DM 10 GeV, typically σ SI should be larger than a few 10 −5 σv bb . It is on the edge of the exclusion lines of XENON10 and in particular LUX [14], however interestingly close to the value to explain the CoGeNT anomaly, which hinted a 8 GeV DM with σ SI ∼ 10 −5 pb [15] 3 . Further experimental progress is expected to clear the confusing picture in the low DM mass region. In Fig. 1 we display the status of aDM, using micrOMEGAs 3.1 [16] for complete numerical analysis.
Comments on the heavy Higgs states at LHC The 2HDM in this paper has two features, a large tan β and heavy Higgs states around (λ 12 /2) 1/2 v ≃ 370 GeV. They can be searched at LHC. Due to large tan β, the production cross section of H, associated with bb, is enhanced and moreover H dominantly decays into bb. Thus, multib jet is a promising signature to hunt such a Higgs. The present 7 TeV CMS data already has sensitivity to it [17] and the full 8 TeV data set of 25 fb −1 is able to exclude tan β 20 [18]. In addition, a large λ 12 enhances the branching ratio of H decay into a pair of SM-like Higgs bosons: which can be 10% for tan β ≃ 10, giving rise to a remarkable 6b−jets signature.
Conclusions SI may provide a simple way to address the Higgs naturalness problem. We explore the idea of accidental DM in the SISM to find that 2HDM+S is the unique model that can give rise to acceptable DM phenomenologies. We study the type-II, which gives two clear predictions: A scalar DM about 5 GeV and heavy Higgs states about 370 GeV. They can be examined soon both from DM detections and LHC searches. Actually, the real singlet scalar DM, because of its simplicity, is extensively studied based on SM [19] or 2HDM [20]. But here we reveal that this somewhat trivial particle has depth, associated with naturalness of Higgs.