Origins of Inert Higgs Doublets

We consider beyond the standard model embedding of inert Higgs doublet fields. We argue that inert Higgs doublets can arise naturally in grand unified theories where the necessary associated $Z_2$ symmetry can occur automatically. Several examples are discussed.


I. INTRODUCTION
With the discovery [1,2] of a Higgs-like boson at about 125 GeV at the Large Hadron Collider (LHC), the standard model (SM) of particle physics comes close to its completion in terms of particle spectrum. While many of the detailed Higgs properties, uncannily dictated by spontaneously symmetry breaking, still needed to be pinned down at the LHC or perhaps by the International Linear Collider (ILC) for Higgs precision measurements, there are existing phenomena indicating that we must extend the SM. Among these are the neutrino masses, dark matter (DM), and baryo-leptogenesis which might be related to TeV scale physics. On the other hand, not a single clue for new physics signal has been found in existing LHC data.
Extensions of the scalar sector beyond the lone doublet in SM is quite common in the literature for various reasons. Perhaps the most studied are the two Higgs doublet models (2HDM) [3] since a second doublet is required in the popular minimal supersymmetric standard model (MSSM) [4], where, with a discrete symmetry imposed upon it, a scalar field component can play the role of dark matter in the inert Higgs doublet model [5,6].
Since the 125 GeV boson behaves very much like the SM Higgs, this indicates that maybe the SM doublet will play the dominant role in spontaneous electroweak symmetry breaking.
In other words, if there are other Higgs multiplets present in the extended scalar sector at the TeV scale, their vacuum expectation values (VEVs) must be minuscule or even vanish. Thus an inert Higgs doublet model (IHDM), in which the second doublet has neither a VEV nor couplings with the quarks or leptons, may be a very realistic extension of the scalar sector of the SM. With the upgrade of the LHC coming this year, more data will be accumulated that could easily reveal this exciting possibility, or put stringent constraints on this simple extension. For detailed studies of phenomenological constraints on IHDM, see for example Refs. [7,8]. In this letter we study the rationale for the presence of an inert Higgs doublet at low energy.
The paper is organized as follows. In section II, we discuss in general how an inert Higgs doublet embedded in grand unified theories (GUTs). In section III, we classify all the inert Higgs doublet possibilities for low lying irreducible representations (irreps) of frequently studied GUT gauge groups. This is done by constructing concrete examples using SU(5), SO(10) and E 6 as our GUT gauge groups. In section IV, we discuss some explicit models.

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In section V, some phenomenological implications are discussed.

II. EMBEDDING THE INERT HIGGS DOUBLET IN A GUT
It is interesting to explore how an inert Higgs models embedded in more fundamental theories. Let us consider grand unification theories and show that inert Higgses and their concomitant Z 2 symmetry can arise naturally. We note that there are other means for an inert Higgs doublet embedded in a higher theory, for example in a composite dark sector [9] or in a scale invariance extension of IHDM [10]. We could also avoid the 24 and break to SU(3) × SU(2) × U(1) with something else like a 75 to change things. As we will see below, SUSY can also be used to help with some of the issues.

III. CLASSIFICATION
It is useful to classify all the inert Higgs possibilities for the popular GUT groups like SU(5), SO(10) and E 6 . The criteria for the inert Higgs are the following: (1) It has no VEV; (2) It does not couple to SM fermions; (3) It is odd under a Z 2 symmetry under which all the SM particles transform trivially.
A number of questions arise as to the nature of inert Higgses. Could EW scalar doublets with non standard U(1) charges be of interest? Could they also play the part of an inert Higgs? I.e., are they close enough to being inert Higgses that they can deliver the same or similar phenomenology? Are there other ways the idea of inert Higgs can be generalized?
We will address some of these questions below.
Examples of EW scalar doublets in SU (5)  A. SU (5) There are 51 irreps of SU(5) with dimensions less than or equal to 1000, but there are not that many (only 12) that contain EW doublets. A systematic collection of these results is given in Table I. Note that none of these irreps contain more than one doublet. As we see, only 6 of the 12 have the standard EW hypercharge. (These results and those given below are all easily checked using the software package LieART [11].) B. SO (10) For SO(10) all the fermions are in 16s, where 16 × 16 = 10 + 120 + 126.
To couple to fermions, a Higgs must be in a 10, 120 or 126 (both the 10 and 120 are real irreps).
Again the SM Higgs can only be in the 5 or 45 of SU(5) since only they contain the (1, 2) −3 that couples to fermions.
To find other doublets in SO(10) irreps with SM charges, we can just find those SU (5) irreps where the SM doublets can live -i.e., those on the list above in Table I (10) was studied previously in [12].
Non standard hypercharged doublets live in the following SO (10)  models (For a summary see e.g., [13] and references therein.), but we will resist the urge for sake of brevity.

V. DISCUSSIONS AND CONCLUSIONS
Here we will discuss some possible phenomenological implications based on our findings in the previous section, continuing to focus on the special case of SU(5).
• In the SU(5) examples above, if the 280 H (or the 480 H ) does not get a VEV, then the Z 2 never gets broken, so the lightest component of the associated inert Higgs doublet will be a DM candidate.
• All the doublets in SU(5) irreps in Table I  • For n > 0 the inert Higgses have to appear in pairs and the lightest component would be stable. They can annihilate pair-wise into photons, but after freeze-out they could form neutral "atoms" and be part of the dark matter. For instance the charge 2 component φ ++ of Φ 2 could bind two electrons to form a helium-like atom (dark helium atom). The energy levels would be only slightly shifted from true helium since the nucleus would weigh a few TeV instead of 4 GeV. These particles could be easily hidden from observations. The φ −− component may be harder to hide since it would need to bind either to positrons, which are probably not available, or protons which would have helium like energy levels but shifted into the X-ray spectrum.
• One could also think of having all the DM in the lightest stable inert Higgs state, say φ ++ and look for an "apparent" excess of helium from standard BBN predictions.
(Here, to simplify the discussion, we assume a φ ++ /φ −− asymmetry.) If the φ mass is 1 TeV, then a pseudo-He atom is 250 times as heavier as normal He. From BBN we know 25% of the baryons are in He. We have about 5 times as much energy density in DM as baryons. So 1 TeV pseudo-He DM would contribute what appears like a 0.5% excess in He in the Universe, which is probably close to the detectable range. If the lightest stable inert Higgs state is say φ +3 , then we'd get an apparent excess of pseudo-Lithium. Since they are predicted to be more rare, Li and other heavy elements would give much stronger limits than H or He, and so we could expect to get strong bounds on the φ +n masses, for n > 2.
• Free stable φ ±n s could potentially be primary cosmic ray components. They would be charged (i.e. charge n) heavy particles without strong interactions. They would be highly penetrating like muon but difficult to accelerate to relativistic velocities because of their small charge-to-mass ratio. However, we note that cosmic charged stable particles are usually considered to be excluded by cosmological arguments coupled with terrestrial searches for anomalously heavy water molecules [14].
• The renormalization group (RG) running of the hypercharge U(1) Y coupling would be faster when we include an extra inert Higgs, so we would need to add color thresholds to compensate in order to preserve unification. To be more specific, first we note that by adding particles with large hypercharges the U(1) Y coupling grows even faster with mass scale. Secondly, we also change the SU(2) L running since we are adding EW doublets. For the RG trajectories to unify at the same place (say around 10 16 GeV for MSSM), we then would need to change the SU(3) C running (and probably adjust the SU(2) a bit too) so that it bends in the same direction as the U(1) Y . This requires particle with color charges, i.e., quarks or maybe exotics. Thus finding a highly charged doublet could indicate a 4 th family at a fairly low scale (ignoring other problems with having a 4 th family).
A Z 2 discrete symmetry, unbroken at tree level in a renormalizable model, never gets broken by loop diagrams (higher order operators). All terms in the Lagrangian are even in R H , and since R H gets no VEV, all higher order operators are also even in R H and hence conserve the Z 2 symmetry.
We can consider these models as UV completion of the inert Higgs doublet -at least up to near the GUT scale ∼ 10 16 GeV. Beyond that we need to worry about the fact that quantum gravity effects can violate any global discrete symmetry [15]. The way to avoid this problem is to gauge the Z 2 , promoting it into local discrete symmetry [15], but that is beyond the scope of our present analysis.
Besides decoupling an inert Higgs H ′ doublet from SM fermions we have seen that we can also decouple it from the SM Higgs H. (We call these cases 'strongly inert'.) Examples include the MSSM extended by either 280 H or a 480 H , both of which deliver automatic Z 2 s. For this reason we have been lead to broaden our definition of what we mean by an inert Higgs. We can also generalize Z 2 to any discrete group [16][17][18], either abelian or nonabelian that accommodates either one of these decouplings. One could even have multiple inert and/or strongly inert Higgses. These alternative systems will have phenomenology that differs from the standard IHDM. In particular since the H ′ in the strongly inert case only couples to gauge bosons, the global fit results found in [8] will require modification.
Phenomenology of the generalized inert Higgs explored in this work is quite rich and further study is deserved.