Dynamical alignment of visible and dark sector gauge groups

TL;DR

This paper proposes a model where dark leptons and gauge bosons form a hidden sector that interacts with the Standard Model via a Higgs portal, enabling dark atom formation and influencing dark matter properties.

Contribution

It introduces a dynamically broken dark gauge symmetry with a novel Higgs portal, allowing dark leptons to contribute to dark matter and form bound states, expanding dark sector models.

Findings

Dark matter mass predicted around 96 GeV.

Single-component model excluded by experiments.

Multi-component or alternative models remain viable.

Abstract

We discuss a dark family of lepton-like particles with their own "private" gauge bosons under a local SU'(2)xU'(1) symmetry. The product of dark and visible gauge groups SU'(2)xU'(1)xSU_w(2)xU_Y(1) is broken dynamically to the diagonal (vector-like) subgroup SU(2)xU(1) through the coupling of two scalar fields M_i to the Higgs field and the dark lepton-like particles. After substituting vacuum expectation values for the fields M_i, the Higgs doublet couples in the standard way to the left-handed SU'(2) doublet and right-handed singlets of the dark gauge group, but not to the extra gauge bosons. This defines a new Higgs portal, where the "dark leptons" can contribute to the dark matter and interact with Standard Model matter through Higgs exchange. It also defines a dark matter model with internal interactions. At low energies, the Standard Model Higgs boson aligns the two electroweak-type symmetry groups in the visible and dark sectors and generates the masses in both sectors. We also identify charge assignments in the dark sector which allow for the formation of dark atoms as bound states of dark lepton-like particles. The simplest single-component dark matter version of the model predicts a dark matter mass around 96 GeV, but the corresponding nucleon recoil cross section is ruled out by the xenon based experiments. However, multi-component models or models with a dark SU'(2) doublet mediator instead of the Higgs portal would still be viable.

Equations

$${\mathcal{L}}_{H-DM}=-\frac{\sqrt{2}}{v_h}\left(m_2{\overline{\Psi }}_L\cdot H\cdot {\psi }_{2,R}+m_1{\overline{\Psi }}_L\cdot \underline{ϵ}\cdot H^{\ast }\cdot {\psi }_{1,R}\right)+h.c.$$

$$D_{\mu }H(x)$$

$$D_{\mu }H(x)={\partial }_{\mu }H(x)-i{g}_w{\mathbf{W}}_{\mu }(x)\cdot \frac{\mathbf{\sigma }}{2}H(x)-i\frac{g_Y}{2}{B}_{\mu }(x)H(x).$$

$$D_{\mu }{\underline{M}}_i$$

$${\mathcal{L}}_{MH-DM}=-\frac{\sqrt{2}}{v_h}\left({\overline{\Psi }}_L\cdot {\underline{M}}_2\cdot H\cdot {\psi }_{2,R}+{\overline{\Psi }}_L\cdot {\underline{M}}_1\cdot \underline{ϵ}\cdot H^{\ast }\cdot {\psi }_{1,R}\right)+h.c.$$

$$V({\underline{M}}_1,{\underline{M}}_2)=\frac{1}{4}\sum _{i=1}^2{\lambda }_i{\left[\mathrm{Tr}\left({\underline{M}}_i\cdot {\underline{M}}_i^{+}\right)-2\mathrm{Det}{\underline{M}}_i\right]}^2.$$

$$\mathrm{Tr}\left({\underline{M}}_i\cdot {\underline{M}}_i^{+}\right)=2\mathrm{Det}{\underline{M}}_i$$

$${\underline{M}}_i=m_i\underline{1},$$

$$S{U}^{\prime }(2)\times U^{\prime }(1)\times S{U}_w(2)\times U_Y(1)\to S{U}^{\prime }(2)\times U^{\prime }(1)=S{U}_w(2)\times U_Y(1).$$

$$Y_L^{\prime }-Y_{1,R}^{\prime }=-Y_h=-1,Y_L^{\prime }-Y_{2,R}^{\prime }=Y_h=1,$$

$$H^{\prime }(x)=\exp [i\alpha (x)/2]U(x)\cdot H(x),$$

$$\left(\begin{array}[]{c}\nu^{\prime}_{L}(x)\\ e^{\prime}_{L}(x)\\ \end{array}\right)=\exp[-i\alpha(x)/2]U(x)\cdot\left(\begin{array}[]{c}\nu_{L}(x)\\ e_{L}(x)\\ \end{array}\right),$$

$${\nu }_R^{\prime }(x)={\nu }_R(x),e_R^{\prime }(x)=\exp [-i\alpha (x)]e_R(x),\dots$$

$${\mathbf{W}}_{\mu }^{\prime }(x)\cdot \mathbf{\sigma }=U(x)\cdot ({\mathbf{W}}_{\mu }(x)\cdot \mathbf{\sigma })\cdot U^{+}(x)+\frac{2i}{g_w}U(x)\cdot {\partial }_{\mu }{U}^{+}(x),$$

$$B_{\mu }^{\prime }(x)=B_{\mu }(x)+\frac{1}{g_Y}{\partial }_{\mu }\alpha (x),$$

$$\Psi^{\prime}_{L}(x)\equiv\left(\begin{array}[]{c}\psi^{\prime}_{1,L}(x)\\ \psi^{\prime}_{2,L}(x)\\ \end{array}\right)=\exp[iY^{\prime}_{L}\alpha(x)/2]U(x)\cdot\left(\begin{array}[]{c}\psi_{1,L}(x)\\ \psi_{2,L}(x)\\ \end{array}\right),$$

$${\psi }_{1,R}^{\prime }(x)=\exp [i(Y_L^{\prime }+1)\alpha (x)/2]{\psi }_{1,R}(x),{\psi }_{2,R}^{\prime }(x)=\exp [i(Y_L^{\prime }-1)\alpha (x)/2]{\psi }_{2,R}(x),$$

$${\mathbf{X}}_{\mu }^{\prime }(x)\cdot \mathbf{\sigma }=U(x)\cdot ({\mathbf{X}}_{\mu }(x)\cdot \mathbf{\sigma })\cdot U^{+}(x)+\frac{2i}{q_2}U(x)\cdot {\partial }_{\mu }{U}^{+}(x),$$

$$C_{\mu }^{\prime }(x)=C_{\mu }(x)+\frac{1}{q_1}{\partial }_{\mu }\alpha (x).$$

$$D_{\mu }{\Psi }_L(x)={\partial }_{\mu }{\Psi }_L(x)-i{q}_2{\mathbf{X}}_{\mu }(x)\cdot \frac{\mathbf{\sigma }}{2}{\Psi }_L(x)-i\frac{q_1}{2}{Y}_L^{\prime }{C}_{\mu }(x){\Psi }_L(x),$$

$$D_{\mu }{\psi }_{i,R}(x)=\left({\partial }_{\mu }-i\frac{q_1}{2}{Y}_{i,R}^{\prime }{C}_{\mu }(x)\right){\psi }_{i,R}(x),$$

$$Y_{i,R}^{\prime }=Y_L^{\prime }-(-{)}^i.$$

$$H_C$$

$${\partial }_{\mu }{F}^{\mu 0}=e\left({\psi }_e^{+}{\psi }_e-{\psi }_p^{+}{\psi }_p\right)$$

$${\mathbf{E}}_{\Vert }(\mathbf{x})=\nabla \int d^3{\mathbf{x}}^{\prime }\frac{e}{4\pi |\mathbf{x}-{\mathbf{x}}^{\prime }|}\left({\psi }_e^{+}({\mathbf{x}}^{\prime }){\psi }_e({\mathbf{x}}^{\prime })-{\psi }_p^{+}({\mathbf{x}}^{\prime }){\psi }_p({\mathbf{x}}^{\prime })\right)$$

$$H_{EM}=\frac{1}{2}\int d^3\mathbf{x}\left({\mathbf{E}}_{\Vert }^2(\mathbf{x})+{\mathbf{E}}_{\perp }^2(\mathbf{x})+{\mathbf{B}}^2(\mathbf{x})\right)=H_C+\frac{1}{2}\int d^3\mathbf{x}\left({\mathbf{E}}_{\perp }^2(\mathbf{x})+{\mathbf{B}}^2(\mathbf{x})\right),$$

$$\bm{E}_{\perp}(\bm{x})=-\partial\bm{A}(\bm{x},t)/\partial t\Big{|}_{t=0},\quad\bm{B}(\bm{x})=\bm{\nabla}\times\bm{A}(\bm{x}).$$

$$\partial_{\mu}C^{\mu 0}=-\varrho_{C}=-\frac{q_{1}}{2}\Big{[}Y^{\prime}_{L}\left(\psi^{+}_{1}\psi_{1}+\psi^{+}_{2}\psi_{2}\right)+\psi^{+}_{1,R}\psi_{1,R}-\psi^{+}_{2,R}\psi_{2,R}\Big{]},$$

$$D_{\mu }{X}_a^{\mu 0}=-{\varrho }_a=-\frac{q_2}{2}{\Psi }_L^{+}\cdot {\sigma }_a\cdot {\Psi }_L,$$

$$H_{dC}=\int d^3\mathbf{x}\int d^3{\mathbf{x}}^{\prime }\frac{{\varrho }_d^2(\mathbf{x},{\mathbf{x}}^{\prime })}{8\pi |\mathbf{x}-{\mathbf{x}}^{\prime }|}$$

Full text

1 Introduction

The identification of the dark matter which dominates the large scale structure in the universe remains an open problem. Higgs exchange had been proposed as an option for non-gravitational interactions between dark matter and ordinary matter [1, 2, 3, 4, 5, 6, 7, 8], and Higgs portal models with couplings to the scalar product $H^{+}H$ of the Higgs doublet $H=(H^{+},H^{0})$ have been discussed extensively for bosonic [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] and fermionic [19, 27, 28, 29] dark matter, but have meanwhile been ruled out in the lower region of the preferred WIMP mass range between about 100 GeV and about 1 TeV [30, 31, 32]. So far the traditional Higgs portal couplings still remain an option for light dark matter, or for WIMPs heavier than 1 TeV.

In the present paper we suggest studies of another kind of Higgs portal which does not involve the scalar product $H^{+}H$ of the Higgs doublet, but may contribute to the masses of dark fermions, although the Higgs field and the dark fermions transform under a priori separate $SU(2)$ tranformations at high energies.

Mass terms in the dark sector are usually inserted by hand or generated by a separate symmetry breaking mechanism. However, it is an intriguing question whether $SU(2)$ breaking by the Standard Model (SM) Higgs boson could also generate the dark matter masses without violating the Standard Model gauge symmetries, and yet be safe from constraints arising through couplings to the electroweak gauge bosons. We propose a mechanism to achieve this. The key idea is to have an a priori separate $SU^{\prime}(2)\times U^{\prime}(1)$ gauge symmetry in the dark sector with its own gauge bosons $\bm{X}_{\mu}$ and $C_{\mu}$. Dynamical breaking of the dark symmetry, e.g. through scalar fields $\underline{M}_{i}\equiv\{M_{i,ab}\}$, which are charged both under the dark and visible gauge groups, then induces standard chiral Yukawa couplings of dark left-handed $SU^{\prime}(2)$ lepton-like doublets and right-handed $SU^{\prime}(2)$ singlets with the Standard Model Higgs doublet,

[TABLE]

These Yukawa couplings in turn break the gauge symmetry of the coupled SM+dark matter model to $SU_{c}(3)\times SU_{w}(2)\times U_{Y}(1)$, because the Higgs couplings align the local $SU(2)\times U(1)$ gauge transformations in the visible and dark sectors.

Irrespective of whether a variant of the model from Sec. 2 turns out to generate the low energy Higgs alignment (1) of visible and dark sector gauge groups, or whether some other mechanism is at work, the dark gauge groups should also be broken up to a possible remnant $U(1)$, in order not to generate too many dark radiation degrees of freedom. This leaves a remnant Yukawa potential interaction in the low energy sector if the remnant dark $U(1)$ is broken at low energy, or a dark photon. Dark photons comply with Planck’s CMB constraints on dark radiation if they decouple early enough [33, 34], and additional relativistic degrees of freedom would also favor a higher value of $H_{0}$, thus reducing the tension between Planck and the astronomical measurements of the Hubble constant. The dark leptons, on the other hand, acquire mass terms through their SM-like Yukawa couplings to the Higgs (and possibly also due to internal symmetry breaking in the dark sector) and can constitute the dark matter, assuming that leptogenesis also had an analog in the dark sector. Neutrality under the internal gauge interactions of the dark sector then implies that the dark matter will consist of dark atoms. The resulting dark matter model is therefore similar to the dark atom models, see [35, 36, 37, 38] and references there. The primary new idea is a dynamically generated Higgs alignment between visible and dark gauge groups.

Breaking of dark gauge symmetries occurs in particular if the Higgs boson couples both to the visible and dark electroweak type gauge bosons,

[TABLE]

Combined with Eq. (1), this defines a class of renormalizable Higgs portal models for dark matter with $SU_{w}(2)\times U_{y}(1)\times SU^{\prime}(2)\times U^{\prime}(1)$ gauge symmetry and spontaneously broken $SU(2)$ factors. This model would not be ruled out (yet) through bounds on the invisible Higgs decay width if all massive particles in the dark sector are heavier than $m_{h}/2$, and if all dark particles with masses below $m_{h}/2$ have small masses, $m\ll m_{h}/2$.

However, as outlined above, in the present investigation we will focus on the case that the Higgs boson does not directly couple to the $SU^{\prime}(2)\times U^{\prime}(1)$ gauge bosons, but is only charged under the Standard Model gauge group,

[TABLE]

Section 2 provides a dynamical toy model which generates the Higgs portal coupling (1), although the covariant derivative on the Higgs field is only given by (3). Section 3 then provides a more fulsome discussion of the low energy formulation of the theory, and the formation of dark atoms in this theory is discussed in Sec. 4. Annihilation cross sections and the constraints from thermal dark matter creation on the simplest version of dark matter in this framework are discussed in Section 5. Section 6 summarizes our conclusions.

2 A dynamical mechanism for alignment of visible

and dark gauge symmetries

A dynamical mechanism to generate the coupling (1) with a Higgs field which is not charged under the dark gauge groups, cf. (3), can be constructed with scalar fields $\underline{M}_{i}\equiv\{M_{i,ab}\}$ which are charged both under the dark gauge group and the Standard Model gauge group. The first index $a$ refers to the fundamental representation of $SU^{\prime}(2)$ from the left, while the second index $b$ refers to $SU_{w}(2)$ acting through the adjoint $SU_{w}(2)$-matrices from the right. The $U^{\prime}(1)$ charges are given in terms of the dark fermion charges by $Y^{\prime(M)}_{i}=Y^{\prime}_{L}-Y^{\prime}_{i,R}$, and the $U_{Y}(1)$ charges are $Y^{(M)}_{i}=-Y_{h}=-1$. The covariant derivatives of the $\underline{M}_{i}$-fields are therefore

[TABLE]

The following coupling term between the visible and the dark sector is then invariant under the full gauge group $SU^{\prime}(2)\times U^{\prime}(1)\times SU_{w}(2)\times U_{Y}(1)$,

[TABLE]

This yields the new Higgs portal coupling (1) in the low energy sector through the potential

[TABLE]

A $2\times 2$ matrix $\underline{M}_{i}$ satisfies the ground state condition

[TABLE]

if and only if the matrix is proportional to the unit matrix,

[TABLE]

up to an additional possible unitary factor $\underline{V}_{i}$. The parameter $m_{i}$ in Eq. (8) can be chosen to satisfy $m_{i}\geq 0$. These results can easily be proved using the polar decomposition $\underline{M}_{i}=\underline{H}_{i}\cdot\underline{V}_{i}$ of the matrix $\underline{M}_{i}$ into a positive semidefinite hermitian factor $\underline{H}_{i}$ and a unitary factor $\underline{V}_{i}$.

This simple dynamical model is only a toy model for the demonstration that the new Higgs portal (1) can arise dynamically, if we are primarily interested in the traditional WIMP mass range, which will be the focus of the remainder of this paper. Any actual phenomenological implementation of the dynamical model (4-6) could only work if dark matter is extremely light.

3 Dark matter masses from the Standard Model Higgs boson

For all we know, the Higgs field is the only dynamical field (besides gravity) which couples to all massive fields, and indeed generates the masses of those fields through Yukawa couplings. It is therefore compelling to assume that the Higgs field also generates the masses in the dark sector. The Yukawa couplings (1) to fermionic dark matter $\Psi_{L}=(\psi_{1,L},\psi_{2,L})$, $\psi_{1,R}$ and $\psi_{2,R}$ provide a way to achieve this, and the mechanism outlined in Sec. 2 provides a way to generate these couplings in the low energy sector of a renormalizable gauge theory.

We assume a single generation of dark lepton-like particles in (1), but the generalization to more generations is straightforward through promotion of the masses $m_{1}$ and $m_{2}$ to mixing matrices.

A priori the dark gauge group $SU^{\prime}(2)\times U^{\prime}(1)$ acting on the fields $\Psi_{L}$ and $\psi_{i,R}$ and the electroweak gauge group are different symmetries, with $SU^{\prime}(2)\times U^{\prime}(1)$ acting only in the dark sector while the electroweak symmetry only acts in the visible sector. However, the Yukawa couplings (1) of the Higgs doublet align the transformations in both symmetry groups at low energies, thus breaking the direct product of symmetry groups to its diagonal component,

[TABLE]

The corresponding dark hypercharge ($U^{\prime}(1)$) assignments have to satisfy

[TABLE]

where $Y_{h}=1$ is the weak hypercharge of the Higgs doublet.

The gauge symmetry $SU_{w}(2)\times U_{Y}(1)$ is therefore implemented for low energy in the visible and dark sectors through the $SU(2)$ transformations $U(x)=\exp[i\bm{\varphi}(x)\cdot\bm{\sigma}/2]$ and the $U(1)$ transformations $\exp[i\alpha(x)Y/2]$, $\exp[i\alpha(x)Y^{\prime}/2]$. The action of the SM fields is as usual,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and the corresponding action in the dark sector is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here $q_{2}$ and $q_{1}$ are the gauge couplings of the dark $SU^{\prime}(2)\times U^{\prime}(1)$ symmetry group, and the covariant derivatives on the dark matter fields are

[TABLE]

[TABLE]

[TABLE]

The low energy sector of the theory outlined in Sec. 2 therefore looks like the Standard Model augmented with a fourth lepton-like family with its own “private gauge bosons” $\bm{X}_{\mu}$ and $C_{\mu}$. Since we assume $m_{i}>m_{h}/2$, this “fourth family” does not contribute to $Z$ nor Higgs decays, nor does it produce a fourth low mass thermalized particle.

The reader will certainly have noticed that the assumed preference for left-chirality also in the dark sector is not determined by any experimental observation and therefore ambiguous with our current knowledge. Instead, we could just as well use right-handed $SU^{\prime}(2)$ doublets and left-handed singlets in the dark sector. This would be compelling from an enhanced symmetry point of view. Opposite chirality in the dark sector could restore $CP$ and time-reversal symmetry in particle physics through mappings between visible sector and dark sector fields. $CP$ and time reversal symmetry might then only be hidden from us through the very weak Higgs-coupling between the visible sector and the dark sector. The model would then provide a Higgs portal realization of the old idea of a $CP$ mirror of the Standard Model [39]. However, currently we can constrain the dark sector only through gravitational effects and direct and indirect search limits. We cannot currently distinguish between a $V-A$ or $V+A$ preference in the dark sector, and at this stage a $V-A$ formulation is as good as a $V+A$ formulation.

There are obviously infinitely many possible ramifications of this model by also including corresponding families of dark quarks with dark symmetry group $SU^{\prime}_{c}(N)$ and corresponding gauge bosons $F_{\mu}^{a}$. The Yukawa couplings of the Higgs boson to the dark quark-like multiplets would obviously not align the $SU^{\prime}_{c}(N)$ with the Standard Model $SU_{c}(3)$. In such a model it would be natural to expect the dark matter to consist of dark atoms formed from dark leptons and dark nucleons, with the atoms formed due to the $SU^{\prime}(2)\times U^{\prime}(1)$ interactions in the dark sector.

However, we will focus on the simplest model of private gauge bosons with one dark family of lepton-like particles and their private $SU(2)\times U(1)$ gauge bosons (1,13,14,15). This is practically equivalent to a dark multi-family model with one family being lighter than the other families, thus being the relevant family for dark matter. Because we do not need to exclude the presence of heavier lepton-like or quark-like $SU^{\prime}(2)\times U^{\prime}(1)$ doublets, we also do not have to worry about anomaly cancellation within the lightest lepton-like dark family. However, we will later find that the requirement for dark matter in the single lepton-like dark family implies $Y^{\prime}_{L}=0$, such that anomaly cancellation is actually fulfilled already with the single dark doublet $\Psi_{L}$.

Without dark nucleons from a dark quark sector, Coulomb-type repulsion between the dark leptons could counteract the pull of gravity. Generically this would seem to imply that the $SU^{\prime}(2)\times U^{\prime}(1)$ couplings $q_{2}$ and $q_{1}$ should only be of gravitational strength for not actually preventing dark halo formation. However, we can choose the $SU^{\prime}(2)\times U^{\prime}(1)$ charges of the dark leptons in a way which allows for the formation of $SU^{\prime}(2)\times U^{\prime}(1)$ neutral dark atoms as bound states of the dark leptons, thus alleviating the contraints on the magnitudes of $q_{1}$ and $q_{2}$.

4 Dark atoms

Recall that the Coulomb potential for electrons and protons in quantum optics (expressed in terms of Schrödinger picture operators)

[TABLE]

arises from

[TABLE]

in Coulomb gauge, and substitution of the solution

[TABLE]

of Eq. (17) into the contribution from $\bm{E}_{\|}$ to the electromagnetic field Hamiltonian

[TABLE]

[TABLE]

Substitution of the $\bm{k}$-space expansions of the electron and proton operators and normal-ordering then yields the usual electron-electron, electron-(anti-proton), proton-proton, positron-proton etc. repulsion terms, and the electron-positron, electron-proton etc. attraction terms.

A corresponding analysis helps to identify any attractive particle-particle combinations in the dark sector, although for non-abelian dark interactions the Coulomb kernel will only yield the perturbative short-distance part of the actual potential if the interaction is asymptotically free, or the Coulomb kernel needs to be replaced by a Yukawa kernel for massive gauge fields.

In the non-relativistic limit, the equations (with $\psi^{+}_{i}\psi_{i}\equiv\psi^{+}_{i,L}\psi_{i,L}+\psi^{+}_{i,R}\psi_{i,R}$)

[TABLE]

[TABLE]

yield the dark sector Coulomb operator

[TABLE]

with the charged density-density correlation operator

[TABLE]

Most of the terms in Eqs. (19,20) are repulsive between pairs of particles and attractive between particles and anti-particles in the dark sector. The attractive channels between particles and anti-particles allow for the formation of dark $SU^{\prime}(2)$ mesons which will decay fast through the Higgs portal (1) and accelerate annihilation of any remnant dark anti-leptons. We are therefore interested in attractive terms between pairs of particles for the formation of dark atoms.

The terms in Eq. (20) with coupling constants $q_{1}^{2}Y^{\prime}_{L}(Y^{\prime}_{L}\pm 1)$ yield attractive interactions in the two-particle states $\psi^{+}_{1,L}(\bm{x})\psi^{+}_{1,R}(\bm{x}^{\prime})\bm{|}0\bm{\rangle}$ and $\psi^{+}_{2,L}(\bm{x})\psi^{+}_{1,R}(\bm{x}^{\prime})\bm{|}0\bm{\rangle}$ if $Y^{\prime}_{L}(Y^{\prime}_{L}+1)<0$, and attractive interactions in the two-particle states $\psi^{+}_{1,L}(\bm{x})\psi^{+}_{2,R}(\bm{x}^{\prime})\bm{|}0\bm{\rangle}$ and $\psi^{+}_{2,L}(\bm{x})\psi^{+}_{2,R}(\bm{x}^{\prime})\bm{|}0\bm{\rangle}$ if $Y^{\prime}_{L}(Y^{\prime}_{L}-1)<0$. However, the resulting bound states would have residual charges $q_{2}/2$ under $SU^{\prime}(2)$ and $(2Y^{\prime}_{L}\pm 1)q_{1}/2$ under $U^{\prime}(1)$, respectively. This would yield repulsive Coulomb-type atom-atom interactions and again prevent dark halo collapse unless $q_{1}$ and $q_{2}$ would be tuned below gravitational strength.

However, the Coulomb term for the dark two-lepton states $\psi^{+}_{1,R}(\bm{x})\psi^{+}_{2,R}(\bm{x}^{\prime})\bm{|}0\bm{\rangle}$,

[TABLE]

is attractive if $Y^{\prime 2}_{L}<1$. It has no $SU^{\prime}(2)$ charge, but a resulting atomic $U^{\prime}(1)$ charge $q_{1}Y^{\prime}_{L}$. Setting $Y^{\prime}_{L}=0$ therefore yields uncharged dark atoms consisting of the two right-handed dark leptons. This implies overall charge neutrality of the universe also under the dark gauge groups, in the same way as anomaly cancellation in the Standard Model ensures charge neutrality in the visible sector. At temperatures above 1 TeV there are equal abundances of particle species, and the vanishing sum of visible $U_{Y}(1)$ hypercharges over SM particle species and dark $U^{\prime}(1)$ hypercharges over dark sector species ensures overall charge neutralities in both sectors.

The potential in the dark sector therefore splits into the left-handed and right-handed parts, $H_{dC}=H_{LL}+H_{RR}$, with fine structure constants $\alpha_{L}=q_{2}^{2}/16\pi$ and $\alpha_{R}=q_{1}^{2}/16\pi$, and the only attractive particle-particle channel for neutral atoms consisting of the two right-handed dark leptons, which are bound due to the $U^{\prime}(1)$ interaction. The states of the dark atoms are therefore for separation $\bm{r}=\bm{x}_{1}-\bm{x}_{2}$ of the two dark leptons and total atomic momentum $\bm{K}$ given in the standard way by direct transcription of the corresponding results of non-relativistic QFT,

[TABLE]

with hydrogen type wave functions $\Psi_{n,\ell,m_{\ell}}(\bm{r})$ for coupling constant $\alpha_{R}/4$ and reduced mass $m_{12}=m_{1}m_{2}/M$, $M=m_{1}+m_{2}$. The energy eigenvalues in the non-relativistic regime are

[TABLE]

Spin singlet or triplet factors were apparently omitted, since we are not interested in the fine structure of the dark atoms. The chiral projectors in the potential (21) reduce the effective dark sector fine structure constant to $\alpha_{R}/4$. This is explained in detail in the Appendix.

We also assume $\alpha_{R}<\alpha_{S}$ since dark $U(1)$ interactions must be weaker than electromagnetism. The separation of gas and dark matter in bullet-type clusters warrants this conclusion [40, 41]. The fact that dark matter halos are much more extended and form less concentrated substructures than baryonic matter also tells us that dark matter cannot cool down as efficiently as baryons.

5 Dark matter annihilation and thermal creation

We are interested in the non-relativistic thermal freeze-out of WIMP scale dark leptons. The annihilation of a heavy dark lepton species is then dominated by branching ratios into Standard Model particles through the Higgs portal (1). The $t$ and $u$ channel annihilations $\psi_{i}\overline{\psi}_{i}\to CC$ and $\psi_{i}\overline{\psi}_{i}\to\bm{X}\bm{X}$ are suppressed with $\alpha_{R}^{2}\bm{p}^{2}/m_{i}^{2}$ due to the chirality factors in the amplitudes, which come from the chirality factors $(1\pm\gamma_{5})/2$ in the vertices $q_{1}Y^{\prime}_{i,R}\overline{\psi}_{i}\gamma^{\mu}C_{\mu}(1+\gamma_{5})\psi_{i}/4$ and $q_{2}\overline{\psi}_{i}\gamma^{\mu}\bm{X}_{\mu}\cdot\bm{\sigma}(1-\gamma_{5})\psi_{i}/4$ from Eqs. (13,14).

The leading order cross sections into the Standard Model states are (with $VV=W^{+}W^{-}$ or $VV=ZZ$ and $\delta_{W}=1$ for annihilation into $W^{+}W^{-}$ or $\delta_{W}=0$ otherwise)

[TABLE]

[TABLE]

[TABLE]

The total annihilation cross section increases with mass $m_{i}$ for masses above 80 GeV, and therefore the heavier dark lepton species will determine both the mass $M$ of the dark atoms and the freeze out temperature. We will assume $m_{1}\lesssim 0.01m_{2}$ and therefore $M\simeq m_{2}$.

The requirement of thermal freeze out then determines $M\simeq 96$ GeV, see Fig. 1, where the thermally averaged [42] annihilation cross for a particle with mass $m_{2}\simeq M$ is compared to the required value from thermal dark matter creation. The logarithmically varying required value of $\langle v\sigma\rangle$ for thermal creation varies from $3.38\times 10^{-26}\,\mathrm{cm}^{3}/\mathrm{s}$ for $M=65$ GeV to $3.45\times 10^{-26}\,\mathrm{cm}^{3}/\mathrm{s}$ for $M=100$ GeV.

A low mass value $m_{1}\lesssim 1$ GeV implies an invisible Higgs decay width which is well below the current limits [43, 44], $\Gamma_{h\to\psi_{1}\overline{\psi}_{1}}\lesssim 82\,\mathrm{keV}\simeq 0.018\Gamma_{h\to\mathrm{SM}}$.

This dark matter model is even more predictive than the standard Higgs portal dark matter models because the coupling to the Higgs field is already determined in terms of the mass, $g=m_{2}/v_{h}\simeq M/v_{h}$. The requirement of thermal dark matter creation therefore does not yield a parametrization $g(M)$ of the Higgs portal coupling as a function of the dark matter mass, but determines $M$. However, the corresponding nucleon recoil cross section

[TABLE]

is about $1.2\times 10^{-44}\,\mathrm{cm}^{2}$ for $M=96$ GeV, $m_{N}=930.6$ MeV (the weighted average of the nucleon masses in long lived xenon isotopes), and the SVZ value $gv_{h}=210$ MeV [45] for the effective Higgs-nucleon coupling. This is in conflict with the exclusion limits from the xenon based direct search experiments [46, 47, 30, 31, 32].

6 Conclusions

Alignment of gauge symmetries in the visible and dark sectors through the new Higgs portal is an interesting new tool for dark matter model building. It can arise as a consequence of dynamical symmetry breaking in gauge theories, and it opens a door to fermionic Higgs portal models without the need of higher mass-dimension effective vertices.

Apparently, the construction presented here opens the Higgs portal into much more complicated and rich dark sectors, even with the possibility of $CP$ and time-reversal reciprocity between the visible and dark sectors, which would be broken through the different mass spectra in the two sectors. Furthermore, the construction also generalizes to alignment of gauge groups through other fields. Every field which transforms in a faithful representation of a symmetry group $G$ can align different copies $G_{1},G_{2},\ldots$ of the symmetry group, each with their own gauge bosons $A^{a}_{i,\mu}$ and coupling constants $g_{i}$, through Yukawa couplings to fields transforming under the different symmetries $G_{i}$. In particular, it is conceivable that the dark leptons may not couple to the Standard Model through the Higgs portal, but through a dark scalar $SU^{\prime}(2)$ doublet $H^{\prime}$, which couples to the dark sector gauge bosons $\bm{X}_{\mu}$ and $C_{\mu}$. This would also align the electroweak-type gauge symmetries in the dark and visible sectors through the Yukawa couplings of the scalar $SU^{\prime}(2)$ doublet. However, it would not be constrained by the current limits from the direct search experiments, since the Yukawa couplings of $H^{\prime}$ in the visible sector must be weaker than the Higgs couplings, thus also implying a weaker effective $H^{\prime}$-nucleon coupling.

\ack

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada through a subatomic physics grant. I very much enjoyed the hospitality of the Kavli Institute for Cosmological Physics during my sabbatical, and I benefitted greatly from discussions with KICP members and the particle cosmology group, especially Rocky Kolb, Lian-Tao Wang, Andrew Long, Michael Fedderke, and Dan Hooper. I also wish to acknowledge Valeri Galtsev for his excellent computer support at KICP.

Appendix: The Schrödinger equation in the dark sector

To understand the impact of the chiral projectors in the dark sector Coulomb potential (21), we follow the procedure which yields the corresponding Schrödinger equation for the hydrogen atom in the baryonic sector while keeping track of the chiral projectors.

The relevant states for the dark atoms are two-particle states

[TABLE]

The states are written in the Schrödinger picture, and the indices $\alpha,\beta$ are Dirac indices.

The relevant part of the Hamiltonian of the theory in the sector of Fock space wich is spanned by the states (27) (i.e. suppressing all parts of the Hamiltonian which map into different sectors of Fock space) is with $P_{R}=(1+\gamma_{5})/2$,

[TABLE]

The corresponding Hamiltonian without the chiral projectors $P_{R}$ arises for the electron-proton system from the energy-momentum tensor of QED in Coulomb gauge, see e.g. Sec. 21.4 in Ref. [48].

The relativistic Schrödinger equation for the two-particle system follows from $id\bm{|}\Psi(t)\bm{\rangle}/dt=H\bm{|}\Psi(t)\bm{\rangle}$ and after decomposition in the basis (27) in the form

[TABLE]

with the Hamilton operator

[TABLE]

To derive the nonrelativistic limit, we use the Dirac representation of $\gamma$ matrices and write the $4\times 4$ matrix $\Psi_{\alpha\beta}(\bm{x},\bm{x}^{\prime},t)$ in terms of $2\times 2$ matrices in the form

[TABLE]

Substitution into Eq. (29) then yields in leading order of $m_{i}^{-1}$ and $\alpha_{R}$ for the “small” components the equations

[TABLE]

[TABLE]

and $\underline{\chi}(\bm{x},\bm{x}^{\prime},t)=0$, and substitution into the equation following for $\underline{\psi}(\bm{x},\bm{x}^{\prime},t)$ from Eq. (29),

[TABLE]

yields the standard two-particle Schrödinger equation up to an extra factor of $1/4$ in the Coulomb potential,

[TABLE]

Separation in center of mass and relative coordinates for the mapping into the effective single-particle equations with masses $M=m_{1}+m_{2}$ and $m_{12}=m_{1}m_{2}/M$ then proceeds as usual.