# Recent Paper: Dark Matter in Leptophilic Higgs Models After the LHC Run-I

In this post, I'll talk about my recent paper, written with my graduate student, David Feld.

This paper is interested in leptophilic Higgs models, and their possible connection to dark matter. I'll explain what those are in a bit. Such models have been considered before, but looking around at the literature, we didn't see a lot that had been updated after the discovery in 2012 of the Higgs boson at 125 GeV. We wanted to see what changed once we folded these new results in to the mix.

So what is a leptophilic Higgs model? The Higgs boson was predicted from our understanding of the Standard Model of particle physics, but really you should think of the theory that Peter Higgs and others wrote down as just the simpliest version of many possible theories for a "Higgs sector." All these theories need to share a number of characteristics, because they all need to do two things: they need to "break electroweak symmetry" to give mass to the $W$ and $Z$ bosons and they need to give masses to the Standard Model quarks and leptons.

In any Higgs theory, both these objectives are achieved via a common mechanism. Some number of scalar fields (Higgs fields) with the right quantum numbers get non-zero values (vacuum expectation values, or vevs) everywhere in the Universe. The interaction of these vevs with the gauge bosons on the weak nuclear force and the "hypercharge" force mixes up these two forces and results in the massive $W$ and $Z$ bosons and massless photon. At the same time, the interaction of the vevs with the quarks and leptons allows them to obtain mass as well.

In the simpliest Higgs theory, you have just one Higgs scalar. Such a field turns out to have four "degrees of freedom" that ordinarily would be seen as particles (two neutral particles and the positively and negatively parts of an electrically charged particle). However, unlike the other particle fields we know about, the Higgs field gets a vev, $v$, which we know experimentally is $v =$ 246 GeV. As a result three of those degrees of freedom get "eaten" by the $W$ and $Z$ particles (to use the technical term), giving these particles masses and leaving one remaining particle which we see as "The Higgs Boson." That single Higgs field also interacts with all the quarks and leptons, and that interaction results in the quarks and leptons having non-zero masses once the Higgs obtains its vev.

So that in itself is pretty complicated. However, what if you had more than one Higgs scalar? Its certainly possible; while we might want to resort to Occam's Razor here, we also know that the Standard Model is not as simple as it apparently could be (after all, why do we need the muon anyway?).

So we can consider the next-simpliest theory, two Higgs scalars, each with vevs $v_1$ and $v_2$. Then the experimental constraint (coming from the masses of the $W$ and $Z$ bosons) is that $$v_1^2+v_2^2 = v^2 = (246~\mbox{GeV})^2.$$ You might recognize this as the equation for a circle, so we can parameterize "how much" of the 246 GeV of vev lies in $v1$ and $v2$ with an angle $\beta$.

However, with two scalars to play with, some of the quarks and leptons can get masses from their interaction with one Higgs scalar, and the rest can get masses with their interaction with the other. The way you as a theorist decide to link up Higgs scalars with the fermions determines what sort of 2 Higgs Model you are working with. A "leptophilic" model has one Higgs giving mass to the up-type and down-type quarks, and the other Higgs giving mass to all the charged leptons. Thus it is "leptophilic" since one Higgs "likes" to talk only to leptons and ignores the quarks.

This is in contrast to the most popular Two Higgs Model, known as the Type-II model, where one Higgs gives mass to the up-type quarks and the other gives mass to both the down-type and the leptons. It might seem odd that the most commonly discussed Two Higgs model splits up the interactions this way; the reason is supersymmetry. Technically, in a leptophilic model, one Higgs (let's call it $H_Q$) interacts with the up-type quarks to give them mass, while the antimatter version of $H_Q$ gives mass to the down-type quark. Another Higgs $H_\ell$ gives mass to the leptons. In Type-II models, none of the quarks or leptons needs to interact with an "antimatter" version of the Higgs field to get mass. Why is this good? Because in theories of supersymmetry, such antimatter interactions are forbidden. We like supersymmetry in theoretical physics, so we spend a lot of time thinking about Type-II models.

Example Higgs production made possible by Large couplings to top quarks

However, this concentration on Type-II models might be a bit problematic, even though this is the Higgs sector the "Minimal Supersymmetric Standard Model" would have to have. In Type-II models, both Higgs doublets talk to quarks. The LHC collides protons, and so makes a lot of strongly interacting particles - that is, the LHC produces a lot of quarks. It is therefore somewhat easier to look for Type-II Higgs models at the LHC than a leptophilic Higgs model.

In particular, the mass of each quark or lepton indicates the size of the coupling that particle has with the Higgs field(s). In the Type-II model, both Higgs must talk to heavy quarks: the bottom quark and the top quark. These particles, being heavy, have large interactions with the Higgs fields, and being quarks are produced in large numbers at the LHC. Then, you can look for new Higgs particles being "radiated" off of top or bottom quarks. These processes are closely related to the sort of thing I discussed in my previous research paper write-up. An example Feynman diagram is reproduced here.

In a model with one Higgs field, after the "eating" of some of the fields by the $W$ and $Z$ bosons, there is one particle remaining (which turns out to be CP-even). With two Higgs fields, there would initially be eight particles; but after the fields gain vevs, some combination of three get "eaten," leaving 5 remaining. Two become a charged Higgs $H^\pm$, one becomes a CP-odd "pseudoscalar" $a$, and two become CP-even scalars $h$ and $H$. One of these last two particles must be what we discovered at the LHC, with a mass of 125 GeV.

But what about the remaining particles? The charge Higgs, the pseudoscalar, and the other CP-even Higgs? Where are they? Should we have seen them yet? And is the Higgs we found at 125 GeV the lighter or heavier of the two CP-even Higgses? Generically, we assume that the Higgs we found is the lightest, and the other possible new particles which are part of the "Higgs sector" are significatly heavier.

In contrast, what we found in the paper is that, adding in the new information about the 125 GeV Higgs we've discovered, we could have new leptophilic Higgs particles lurking well below 125 GeV. Indeed, given the properties of the Higgs we have discovered, direct discovery of these leptophilic particles at the LHC would be extremely difficult. However, these new particles fit nicely into some possible models of dark matter, possibly explaining some odd observations from the Galactic Center. Thus, the best place to look for Leptophilic Higgses might be through their interaction with dark matter.

Experimental constraints on Higgs parameters in Leptophilic Model.

The first thing we looked at in the paper were the measured properties of the 125 GeV Higgs. Generically, when you have a model with two Higgs fields, the physical states, the particles you can produce at a collider, are a mixture of the two primordial Higgses. So the 125 GeV Higgs we've discovered would have to be a combination of parts of the $H_Q$ and $H_\ell$. The fact that it is a combination of two particles means that it's interactions with other particles could differ from the simpliest Single-Higgs model.

One of those properties we can measure is split between the vacuum expectation values of the two Higgses. We define this as the angle $\beta$, and like any angle we can take sines, cosines, and tangents of it. The value of $\tan\beta$ is especially useful: as $\tan\beta \to \infty$, more and more of the vev is in the $H_Q$ field, as $\tan\beta \to 0$, most of the vev is in $H_\ell$. We can't directly measure this quantity, but we can get indirect handles on it.

We can also look at the interaction of the 125 GeV Higgs with the $W$ and $Z$ bosons. In a certain sense, there's a maximum "amount" of coupling between these particles and all the Higgses; if the Higgs we found saturates that limit, the other possible Higgs particles will have very small couplings with these bosons. The way we like to look at the amount of coupling is through a combination of $\beta$ and another angle $\alpha$: in particular when the quantity $\cos(\beta-\alpha) = 0$, the Higgs we discovered has the maximum amount of coupling to the $W$ and $Z$ bosons.

I reproduce here our Figure 1, where I show our best fit to the ATLAS and CMS data in a leptophilic model in terms of $\tan\beta$ and $\cos(\beta-\alpha)$. The green region is the area that fits the data best, up to 68% confidence. The yellow area is the 90% confidence region. Other than a little banana-shaped region, most of the parameter space compatible with data is close to $\cos(\beta-\alpha)=0$.

This means that the Higgs we discovered is aligned (and this is true for all Two-Higgs models, not just leptophilic ones). That means that the Higgs we found looks a lot like the simpliest Higgs possible. It also means any additional Higgses that could be out there have very suppressed couplings to the $W$ and $Z$ bosons.

This makes discovery difficult for leptophilic Higgses. Such extra Higgs particles would have small couplings to quarks due to the lepton-specific interactions, and we now know also have small couplings to the $W$ and $Z$ bosons due to this alignment. Thus, most of the search channels at the LHC, considered in previous leptophilic Higgs papers written prior to 2012, and relying on large couplings to $W$ and $Z$ particles, become inoperative. This means that there could be new Higgs particles lurking at very low masses at the LHC, and we would not know it. Not only could the 125 GeV Higgs not be the lightest Higgs particle, there could be new Higgs particles as low as a few 10's of GeV in mass.

Unfortunately, as far as we could tell in our paper, there is no obvious search strategy we can rely on at the LHC to find such light Higgses. The problem comes down to the fact that leptophilic Higgses are produced at very low rates (due to the newly-discovered alignment) and will decay into the heaviest lepton available: the tau lepton. Taus decay very messily in the experimental detectors, and are difficult to pick out over the many backgrounds. New ideas would be needed to boost our ability to pick out tau-rich signals.

Given the difficulty in finding light leptophilic Higgs particles, what can we do? Well, in addition to the unknown Higgs sector, we know that there is something else new in the Universe: dark matter. What if the dark matter was connected to the Higgs particles? Such ideas are not new to our paper, of course, and have been considered in leptophilic, Type-II, and other Higgs models.

Take our leptophilic Higgs model and add on dark matter inspired from supersymmetry. Though the minimal supersymmetric model cannot have a leptophilic Higgs, you can imagine expanding supersymmetry far beyond the minimal case. And indeed, maybe the reason we haven't found supersymmetry yet means that it isn't the minimal version we spent so much time thinking about? Indeed, the last part of the paper looks at a full supersymmetric model that incorporates the leptophilic Higgs. It's rather more complicated than the Minimal Supersymmetric Standard Model, with four Higgs fields, rather than two, but like the non-supersymmetric leptophilic Higgs model, allows very light new Higgs fields without running into serious collider constraints.

Annihilation cross section for dark matter in leptophilic Higgs model. Galactic Center region of interest from various studies in yellow, purple, and red

Working in our "supersymmetric-like" dark matter model, interacting via the Higgs fields, we find some interesting things. For dark matter model-building, you actually need your dark matter to interact relatively significantly with something. Otherwise, when you look at the evolution of dark matter in the early Universe, it tends to freeze-out too early, and you end up with too much of the dark matter left around today. I talk about this in some detail here.

In Type-II models, you often have to play a bit of delicate game with this requirement, if you want to have the interaction go through the Higgs. To have sufficiently large interaction in the early Universe, you tend to need some light Higgs particles. But if you have light Higgses, then you often should either have seen something at the LHC, or you should expect to see evidence of that interaction at direct detection experiments. These latter experiments link dark matter to the quarks inside protons and neutrons, and since Type-II models have Higgses interacting with quarks, you can't suppress that interaction too much.

Direct detection constraints for leptophilic Higgs dark matter as parameter $tan\beta$ varied.

In leptophilic models, however, don't have this problem. Leptophilic Higgses don't talk much to quarks, so the direct detection experiments are much less important. In addition, we can generally get away with much lighter particles in leptophilic models at colliders.

What we found was that we would pretty easily accommodate sufficiently light particles to make the early Universe story make sense without running into any problems with existing experimental constraints from direct detection or colliders. In addition, we found reasonable parameter space where some unusual signals of gamma rays from the Galactic Center could be interpreted as dark matter annihilating through leptophilic Higgses into tau leptons. This is actually very interesting (and secretly why I was interested in this model to begin with). Explaining the Galactic Center anomaly is very difficult in Type-II models, as usually the parameters you need are ruled out by the direct detection bounds. Leptophilic models can do it easily.

So, if this is the sort of Higgs sector the Universe chose to realize, and if this is the sort of dark matter we have, the place we'd discover the new physics would be in our dark matter experiments. We'd see signals of indirect detection of dark matter, and suppressed (but non-zero) signals of dark matter in our direct detection experiments. If we start seeing that sort of result, we may need to look more carefully at methods to tease out the small, tau-rich signals of leptophilic Higgses at colliders. Given the measured properties of the bit of the Higgs sector we now know about, these leptophilic particles could be lurking at surprisingly low masses.