January 2014

Mirror dark matter:

Cosmology, galaxy structure and direct detection

R. Foot^{1}^{1}1
E-mail address: rfoot@unimelb.edu.au

ARC Centre of Excellence for Particle Physics at the Terascale,

School of Physics, University of Melbourne,

Victoria 3010 Australia

A simple way to accommodate dark matter is to postulate the existence of a hidden sector.
That is, a set of new particles and forces interacting with the known particles predominantly via gravity.
In general this leads to a large set of unknown parameters, however if the hidden sector
is an exact copy of the standard model sector, then an enhanced symmetry arises.
This symmetry, which can be interpreted as space-time parity, connects each ordinary particle
( with a mirror partner (.
If this symmetry is completely unbroken, then the mirror particles are degenerate with
their ordinary particle counterparts, and would interact amongst themselves with exactly
the same dynamics that govern ordinary particle interactions.
The only new interaction postulated is photon - mirror photon kinetic mixing, whose strength ,
is the sole new fundamental (Lagrangian) parameter relevant for astrophysics and cosmology.
It turns out that such a theory, with suitably chosen initial conditions effective in the very early Universe,
can provide an adequate description of dark matter phenomena provided that .
This review focusses on three main developments of this mirror dark matter theory during the last decade:
Early universe cosmology, galaxy structure and the application to direct detection experiments.

## 1 Introduction and Overview

### 1.1 Introduction

Astronomical observations provide a strong case for the existence of non-baryonic dark matter in the Universe. The first evidence arose in the 1930’s from observations of galaxies in clusters which showed unexpectedly high velocity dispersion [1, 2]. Further evidence followed from measurements of optical and radio emissions in spiral galaxies [3, 4, 5, 7, 6, 8, 9] (for a review, and more detailed bibliography, see Ref.[10]). These observations allowed galactic rotation curves to be obtained, which greatly strengthened the case for dark matter. It was found that rotation curves in spiral galaxies were roughly flat near the observed edge of the galaxy, in sharp contrast to expectations from Newton’s law of gravity applied to the inferred baryonic mass.

Dark matter is also needed to explain the observed Large-Scale Structure (LSS) of the Universe [11, 12, 13] and also the anisotropies of the Cosmic Microwave Background radiation (CMB) [14]. Such cosmological observations have provided, perhaps, the strongest evidence yet for dark matter in the Universe. These, and other measurements, can be explained within the Friedmann Robertson Walker (FRW) cosmological model. This model, with significant developments over the years, has a small number of parameters, among which are , , and [ = baryon, = dark matter, = cosmological constant]. Here , with critical density and with all densities evaluated today (for a review, see e.g. [15]). Comparison of the model with observations allows these parameters to be determined:

(1) |

That is, the energy density of dark matter in the Universe is currently around five times larger than that of ordinary matter.

Most recently, evidence of dark matter direct detection in underground experiments has emerged [16, 17, 18, 19, 20, 21]. The strongest such dark matter signal is the measurement of an annually modulated event rate by the DAMA collaboration. Due to the Earth’s motion around the Sun, the dark matter interaction rate in an Earth based detector should modulate with a period of one year and have a maximum near June [22]. Such a modulation was observed by the DAMA/NaI [16] and DAMA/Libra [17, 18] experiments and provides tantalizing evidence that dark matter particles may have been detected in the laboratory.

The non-gravitational interactions of the known elementary particles are well represented by the standard model. This theory exhibits gauge symmetry, along with a host of space-time symmetries, and can be described by a Lagrangian:

(2) |

This model together with Einstein’s General Relativity theory provides an excellent description of the elementary particles and their interactions. Although the standard model is very successful, it contains no suitable dark matter candidate, so one is naturally led to consider new particle physics.

Perhaps the simplest way to accommodate dark matter is via a hidden sector. This entails extending the standard model to include an additional set of matter particles and gauge fields (associated with a gauge group ) so that:

(3) |

If the new particles do not interact with any of the standard gauge fields then their properties are experimentally unconstrained. They cannot be produced in colliders, unless some additional interactions, , are assumed. It is for this reason that such an additional set of particles is called a hidden sector. The hidden sector can have accidental global or discrete symmetries stabilizing one or more of the lightest particles: , , … In this case these stable particles can potentially constitute the inferred dark matter in the Universe.

From this perspective, the astrophysics of dark matter can be
simple or complex depending on the properties of the hidden sector. For example, if all the new gauge bosons
are heavy, like the and gauge bosons, then , , … are essentially collisionless
particles, also called WIMPs in the literature ^{2}^{2}2
Collisionless dark matter (WIMPs) can be motivated in other frameworks, such as in models with large extra dimensions
and supersymmetry. For a review, see for example [23] or [24]..
On the other hand, if one or more of the new gauge fields are light or massless
then the , ,… can have significant self interactions which are also dissipative.
The case of collisionless dark matter has been very well studied in the literature, in part because the astrophysics is particularly
simple. However the alternative possibility, where dark matter has more complex astrophysical properties is equally simple from
a particle physics standpoint, and is also worth investigating. Naturally, the astrophysical implications of such
complex dark matter depends, to a significant extent, on the details of the particular model.

This review will focus on a very special hidden sector model - mirror dark matter, which I will argue is exceptionally simple and well motivated from a particle physics perspective. However, even if the reader does not share my enthusiasm for this particular model, this study at least serves to illustrate some of the rich dark matter phenomenology that is possible in generic hidden sector models. It may thus (hopefully) provide useful insight if the astrophysical properties of dark matter are, in fact, non-trivial.

Mirror dark matter corresponds to the theoretically unique case where the hidden sector is an exact copy of the standard model sector (up to an ambiguity concerning whether or not chirality is flipped in the hidden sector, to be discussed in a moment). This means that each of the known particles has a mirror partner, denoted with a prime (). The mirror particles interact amongst themselves with exactly the same dynamics that govern the ordinary-particle interactions. That is, is just the standard model Lagrangian:

(4) |

where contains possible non-gravitational interactions coupling ordinary and mirror particles.
The mirror particles don’t interact with any of the known particles except
via gravity and the terms in .
This particular hidden sector theory is also tightly constrained: The only new parameters are
those in (to be discussed shortly).
The vacuum structure of the mirror sector is presumed identical to the ordinary
sector so that the mass and the lifetime
of each mirror particle is exactly identical to the corresponding ordinary matter
particle. That is etc.
^{3}^{3}3
In the presence of mass-mixing terms arising from
massive neutral particles, such as the neutrinos and the Higgs boson,
can have their particle - mirror particle degeneracy broken.
and the mirror electrons and mirror protons (mirror neutrons in mirror nuclei) are stable and can constitute
the non-baryonic dark matter in the Universe.

Having the hidden sector isomorphic to the standard model sector is a sensible thing to do, not just because it reduces the number of parameters, but also because it increases the symmetry of the theory. There is an unbroken discrete symmetry interchanging each ordinary particle with its mirror partner. To appreciate the significance of this particular discrete symmetry we need to remember some basic particle physics. Firstly, recall that ordinary (Dirac) fermion fields are a combination of two chiralities: left and right-handed chiral fields. These chiral states describe the two helicity states of the spin 1/2 fermion in a Lorentz covariant manner. The two chiral states, together with the antiparticles are described by two complex fields, . That is, we have four degenerate physical states.

With the presumption of a hidden sector isomorphic to the standard model sector,
each fermion and its mirror partner forms now
eight degenerate chiral states.
Considering for example the electron and its degenerate partners, we have four complex fields: .
Let us examine all possible discrete transformations of these degenerate states.
A discrete symmetry which interchanges states of opposite chirality is possible if it
also maps .
Thus, for example, the conventional parity transformation maps and ^{4}^{4}4
Technically, the transformation is , where is a Dirac gamma matrix.
As this detail is inessential for the purposes of this introductory discussion, the required matrices
are taken as implicit..
However it is known that only
the left-handed ordinary fermion fields couple to the (charged current) weak interactions ( bosons).
Thus a discrete symmetry which maps cannot be an invariance of the full theory.
Since the fields are complex, the transformations: ,
or , known as the CP and C transformation respectively, are both
possible,
but again experiments have shown that these cannot lead to an invariance
of the full theory.
This leaves two possibilities.
Either or
. The latter case requires also because
it interchanges chiralities.
In both cases, the symmetry also interchanges the gauge bosons ( etc.) with
their mirror partners ( etc.) and can be a full invariance of the theory ^{5}^{5}5
The cases: and
do not lead to any new theories, beyond the ones considered..

The conclusion is that, if the standard model is extended with an isomorphic hidden sector then there are actually two (almost) phenomenologically equivalent theories, depending on whether the chirality of the fermions are swapped in the mirror sector. If the left and right-handed chiral fermion fields are in fact interchanged, then the discrete symmetry can be interpreted as space-time parity symmetry as it also maps . The theory also exhibits an exact time reversal invariance, which means that the full Poincar group becomes an unbroken symmetry of the theory [25].

Particle physics considerations have often been guided by symmetry principles, and space-time parity appears to be as good a candidate as any for a fundamental symmetry of nature. This was well recognized by our pre-1956 ancestors, who generally assumed fundamental interactions were invariant under space inversion. Things changed in 1956: Some experimental anomalies led Lee and Yang to suggest that space-time parity might be broken in nature [26]. Lee and Yang also pointed out that even if the interactions of the known particles were to violate parity, the symmetry could be restored if a set of mirror particles existed. [Although at that time it wasn’t clear if every known particle had a mirror partner, or just some of them.] Shortly thereafter it was realized by Landau that the CP transformation could play the role of space-time parity [27], and thereby argued that a mirror sector was not necessary. Space inversion accompanied by particles swapping with antiparticles might be the mirror symmetry chosen by nature. However, following experiments in 1964 which showed that CP was in fact violated [28], Landau’s former student Pomeranchuk and collaborators, influenced by Landau’s strong belief in parity symmetry, reconsidered Lee and Yang’s original idea [29]. There they argued that a complete doubling of the known particles and forces (except gravity) was necessary to realize Lee and Yang’s vision. Related ideas were also discussed around a decade later by Pavsic [30].

The potential application to dark matter
was suggested in 1982 [31] and also independently in 1985 [32] (the latter motivated not by space-time parity
but by anomaly free superstring theories [33]). However, with the exception of two significant papers
in 1986, 1987 [34, 35], the idea was not actively pursued.
So, perhaps surprisingly, the extension of the standard model
extended with such a mirror sector was not written down until 1991 [25].
The 1991 work was independent of the earlier developments, and arose out of
studies investigating a gauge model with non-standard parity symmetry which
interchanged quarks with leptons
as well as [36]. ^{6}^{6}6
The gauge model referred to here, called the quark-lepton symmetric model in [36],
extended the gauge symmetry of the standard model to include a gauged
color symmetry for leptons.
This means that the leptons interact with an octet of leptonic gluons ,
in the same way in which the quarks interact with the familiar gluons, .
With an leptonic color, the Lagrangian can possess a discrete parity
symmetry which not only maps but additionally interchanges
left-handed (right-handed) leptons with right-handed (left-handed) quarks and also with .
Consistency with experiments requires the
vacuum such that is spontaneously broken, which means that the parity symmetry
is also spontaneously broken. The non-degeneracy of the quarks and leptons can thereby be explained,
and a phenomenologically consistent model results.
With gauged, the leptons appear as a parity double of the quarks.
The jump from this model, to the model where the parity symmetry was completely unbroken followed
by assuming that all of the particles in the standard model have a parity double.
This review will mainly be concerned with the post-1991 evolution of the theory and its application to
dark matter. Readers wishing to know more about pre-1991 work on the subject might
consult Okun’s articles [37, 38].

Returning to the Lagrangian of Eq.(4), we have yet to define the term. This piece describes possible non-gravitational interactions coupling ordinary and mirror particles together. It turns out that there are just two mixing terms consistent with the symmetries of the minimal theory and which are also renormalizable [25, 39]:

(5) |

where () is the ordinary (mirror) gauge boson field strength tensor and () is the ordinary Higgs (mirror Higgs) field. The two Lagrangian terms above involve two dimensionless parameters: , both of which are not determined by the symmetries of the theory. Of these two terms, only the first term, the kinetic mixing term will be important for the astrophysical and cosmological applications discussed in this review. The relevant particle physics thus involves only one additional fundamental parameter, .

The physical effect of the kinetic mixing interaction is to induce a tiny ordinary electric charge for the mirror proton and mirror electron of [25, 40]. Kinetic mixing can thereby lead to electromagnetic interactions of the form: . Although the cross-section for such processes is suppressed by , these kinetic mixing induced interactions can still have extremely important astrophysical and cosmological implications. In particular, such kinetic mixing can make supernovae - both ordinary and mirror varieties - play critical roles in astrophysics and cosmology. Recall that in standard theory, ordinary supernovae release almost all of their core collapse energy into neutrinos since these particles can escape from the core due to their extremely weak interactions. If photon - mirror photon kinetic mixing exists with strength , then around half of this energy can instead be released into light mirror particles: , produced through processes such as in the hot supernova core [41, 42]. These light mirror particles, once produced, escape from the core and are injected into the region around the supernova. Ultimately this energy is expected to be converted into mirror photons. These mirror photons, as we will see, provide an excellent candidate for the heat source responsible for stabilizing mirror-particle halos hosting spiral galaxies. At an earlier epoch, mirror supernovae might also have played an important role. These supernovae can release a large fraction of their core collapse energy into ordinary photons. A rapid period of mirror star formation at an early epoch: , might have been responsible for the reionization of ordinary matter - inferred from CMB and other observations.

In the mirror dark matter scenario, it is supposed that all of the inferred non-baryonic dark matter in the Universe, on both large and small scales, is comprised of mirror particles, in one form or another. At the particle level, dark matter consists of a spectrum of stable massive mirror particles which are not only self interacting but also dissipative. It turns out that this dark matter picture gives consistent early Universe cosmology, and predicts large-scale structure and CMB anisotropies which are compatible with observations. Furthermore, on smaller scales the dissipative interactions lead to non-trivial halo dynamics. The picture is that dark matter halos hosting spiral galaxies are composed predominately of a mirror-particle plasma containing: [43]. The loss of energy due to dissipative processes, such as thermal bremsstrahlung, is (currently) being replaced by a heat source, with ordinary core-collapse supernovae, as briefly described above, posing the best available candidate. It turns out that this dynamics leads to a satisfactory explanation of the inferred dark matter properties of spiral galaxies, i.e. asymptotically flat rotation curves, cored density profile, empirical scaling relations and so on [43, 44, 45, 46].

A key test of this dark matter theory comes from direct detection experiments. Ordinary supernovae can only stabilize dark matter halos if the kinetic mixing interaction exists, with . Such an interaction also implies that mirror particles can elastically scatter off ordinary nuclei and thereby be observed in direct detection experiments. The impressive annual modulation signal recorded by the DAMA collaboration [16, 17, 18], and the low energy excesses observed by CoGeNT [19], CRESST-II [20] and CDMS/Si [21] can all be simultaneously explained in this framework [47]. However the dust has not completely settled; some tension with the null results of the XENON-100 [48] and LUX [49] experiments remain.

The purpose of this article is to review these developments, in a (hopefully) coherent and pedagogical manner. This review is structured as follows: In the remainder of this section a qualitative overview of the mirror dark matter picture is provided. Section 2 reviews the relevant particle physics of mirror matter. Section 3 discusses early Universe cosmology: Big Bang Nucleosynthesis (BBN), mirror BBN, CMB and LSS. Section 4 looks at small-scale structure, reviewing recent work on the nontrivial halo dynamics suggested by this dark matter candidate. Section 5 examines the mirror dark matter interpretation of the direct detection experiments, especially DAMA, CoGeNT, CRESST-II and CDMS/Si. Finally, some concluding remarks are given in section 6.

### 1.2 Overview

Cosmological observations indicate that the energy in the Universe consists of ordinary matter,
non-baryonic dark matter and the cosmological constant. This review is concerned with a particular
dark matter theory - mirror dark matter.
The mirror dark matter hypothesis contains three main ingredients.
First, the particle physics Lagrangian is extended to include a hidden sector
exactly isomorphic to the ordinary matter sector.
This provides stable massive particles which make up the presumed dark matter
in the Universe.
Second, we assume ordinary and mirror matter interact with each other via
gravity and also the photon - mirror photon kinetic mixing interaction, with
.
This assumption is required to account for small-scale structure and also direct detection
experiments (as we will see).
Third, we need appropriate initial conditions arising in the very early Universe.
In addition to the usual assumptions of tiny adiabatic scalar perturbations which seed the structure
in the Universe,
we also have: .^{7}^{7}7
The precise value for is set by fits to the CMB anisotropy spectrum, in the same
way in which cold dark matter density is determined in the CDM model.
These initial conditions are required to explain large-scale structure and CMB anisotropies.

It is perhaps useful to first give a qualitative discussion of how these three ingredients might combine to provide an adequate description of dark matter phenomena. The subsequent sections will review what is known quantitatively about the various parts of this picture. Our starting point is the early Universe, around the time of the BBN epoch, second. By then, any antibaryons created in the early Universe have efficiently annihilated with baryons. It follows that our existence today requires the generation of a baryon - antibaryon asymmetry in the Universe. In a similar manner, any mirror antibaryons created in the early Universe would have efficiently annihilated with mirror baryons, so it is safe to assume that dark matter is composed of mirror baryons, with a negligible mirror antibaryon component (or vice versa). The origin of the mirror-baryon asymmetry of the Universe is unknown, although several mechanisms have been discussed, e.g. [50, 51, 52, 53]. Clearly, the result that does suggest that these asymmetries might be connected in some way [54]. This kind of asymmetric dark matter has also been examined in the context of more generic hidden sector models. See the recent reviews [55, 56] and references therein for relevant discussions.

Of course radiation - not baryons - dominated the energy density during the BBN epoch. Since BBN arguments constrain the energy density of the Universe to be less than around one additional neutrino at that time, the mirror particles and ordinary counterparts did not have the same temperature. The mirror particles must have been cooler than the ordinary particles. This is possible, if the interactions in which couple the two sectors together, are small enough. In fact, we make the simple assumption that holds at some early time before the BBN epoch (our notation is that [] without subscript is the photon [mirror photon] temperature). We take a similarly pragmatic approach to . CMB observations (and others) constrain . We call these effective initial conditions since it is certainly possible that they might have arisen from symmetric ones at an even earlier time. [This occurs, for instance, in chaotic inflation models where the reheating of the ordinary and mirror sectors can be asymmetric [32, 50, 57].]

Even if the Universe started with , entropy in the mirror sector can be generated via kinetic mixing induced interactions: [35]. For , the asymptotic value (i.e. ) of the ratio is [58, 59]. Since , mirror nucleosynthesis would have occurred somewhat earlier than ordinary nucleosynthesis. To understand what this means, let us first recall what happens in ordinary BBN. The nucleon number densities are determined by the two-body and three-body reactions:

(6) |

Initially these reactions drive the neutron to proton ratio to unity but as the temperature drops to around 1 MeV, the neutron - proton mass difference leads to a larger proportion of protons. Eventually the rate of these reactions became less frequent than the expansion rate of the Universe. When this happens the two-body reactions become infrequent enough to effectively freeze the neutron/proton ratio. The temperature where this occurs is MeV. This ratio is then only further modified by neutron decays which occur until deuterium formation at MeV. The end product is that around 25% of the baryons end up in helium and 75% of the baryons in hydrogen, with trace amounts of other light elements. Mirror nucleosynthesis is qualitatively similar, but since it occurs earlier, the expansion rate is greater so that the mirror-neutron/mirror-proton ratio freezes out at a higher temperature: . For this reason, and also because there is less time for mirror neutrons to decay, the mirror-neutron/mirror-proton ratio remains close to unity. This means that there is a high proportion of mirror helium in the mirror sector [51]. For , around 90% of mirror baryons are synthesized into mirror helium, with 10% into mirror hydrogen [60].

At these early times the Universe is remarkably isotropic and homogeneous. The Universe is not completely smooth though, tiny perturbations, possibly seeded by quantum fluctuations and amplified by inflation, are present. Consider a perturbation to the matter density: . In Fourier space such a perturbation is described by a wavevector :

(7) |

While these perturbations are small: , modes with different wavevector evolve in time independently and linearly. This is the so-called linear regime. The linear evolution of such modes is described by linearlized Boltzmann-Einstein equations. Qualitatively, the evolution of these modes depends on their scale relative to the comoving horizon size at the time under consideration. Large-scale modes with much larger than the horizon are not influenced by causal physics; they remain unchanged. Small-scale modes with less than the horizon can be influenced by the physical processes of gravity and potentially also pressure. As the Universe expands, the comoving horizon increases; large-scale modes enter the horizon and are processed by causal physics (the comoving wavelength remains constant).

Matter density perturbations can be divided into baryonic perturbations and mirror-baryonic ones. For baryonic perturbations prior to hydrogen recombination, the photons are tighly coupled to electrons via Compton scattering and electrons to protons via Coulomb scattering. At this time, the particles: can be treated as a tightly coupled fluid. The effects of gravity and pressure are well understood for this system: acoustic oscillations occur and are responsible for the peaks in the CMB anisotropy spectrum. The physics of mirror baryonic perturbations is very similar. Prior to mirror-hydrogen recombination, i.e. when eV, the mirror particles: also form a tightly coupled fluid. Fourier modes which are small enough to have entered the horizon at this epoch undergo acoustic oscillations due to the () radiation pressure; this suppresses perturbations on scales smaller than the horizon at this time. Only after mirror-hydrogen recombination can matter density perturbations grow.

Perhaps it is useful to pause here and compare this picture with that of collisionless dark matter. Collisionless dark matter by definition has no pressure and therefore no acoustic oscillations. Mirror dark matter might therefore appear to be very different, however this need not be the case. In the limit , equivalently given the assumed initial condition , mirror nuclei were always in neutral atoms. Mirror-baryonic acoustic oscillations would not then occur. In this limit therefore, mirror dark matter would be indistinguishable from collisionless cold dark matter during the linear regime [51, 61]. Clearly, for nonzero departures from collisionless cold dark matter would be expected on small scales, smaller than a characteristic scale . Observations can then be used to yield an upper limit on , and hence also on .

Within the mirror dark matter context, the formation and evolution of structure on scales larger than should be similar to collisionless cold dark matter, at least in the linear regime. If is small enough, then linear evolution of structures on galactic scales and larger can therefore be very similar to collisionless cold dark matter. What about the early evolution of small-scale structure in the nonlinear regime? Consider first collisionless cold dark matter. In that model, halos hosting galaxies such as the Milky Way are believed to have formed hierarchically from the merging of smaller structures [62] (see also [63] for an up-to-date review and more detailed bibliography). This picture would presumably need some revision if acoustic oscillations were effective at suppressing small-scale inhomogenities in the linear regime. It could happen for instance that structure evolves hierarchically above a certain scale and top down below this scale [and some mixture of both mechanisms on scales near ].

It is tempting to speculate that the suppression of small-scale structure below
might be connected with the surprisingly small number of satellite galaxies that have been observed in the local group.
This “missing satellites problem” is considered to be a serious issue for the
collisionless cold dark matter model (for a review and references to the original literature
see for example [64]).
Mirror dark matter appears to have the potential to address this and other small-scale
shortcomings of collisionless cold dark matter, however much more work is needed ^{8}^{8}8
Another small-scale puzzle of collisionless cold dark matter is the observed large proportion of bulgeless disk galaxies.
That is, pure disk galaxies with no evidence for merger-built bulges.
This is surprising given the level of hierarchical clustering anticipated if dark matter were collisionless. For relevant
discussions see [65, 66] and references therein.
In fact [66] describes this as the biggest problem in the theory of galaxy formation.
The suppression of small scale structure below and also the early heating of
ordinary matter from mirror supernovae (to be discussed) may help address this issue.
Qualitatively, mergers should be less frequent, and importantly, the formation of the baryonic disk might be delayed due to the
early heating..
Suffice to say that the formation and early evolution of structure on galaxy scales
is a complex issue and is, at present, poorly understood in the mirror dark matter framework.
Ideally, hydrodynamical simulations taking into account mirror dark matter interactions,
both dissipative and non-dissipative, along with heating from supernovae
in the presence of kinetic mixing (see below) could be attempted.
Alternatively, analytic or semi-analytic techniques could conceivably be developed using
the Press-Schechter formalism as a starting point [67].
At the present time though, such work has not yet been done.
In the absence of such computations or analytic studies, any discussion is
certainly speculative. Nevertheless, a self-consistent if not quantitative picture appears to be emerging.

Initially, mirror density perturbations evolve linearly and grow in both density and size as the Universe expands.
Consider now a particular galaxy-scale perturbation.
When the matter overdensity reaches the evolution starts to become nonlinear.
Around this time the perturbation breaks away from the expansion and can begin to collapse.
Mirror dark matter is collisional, however it is also dissipative, and if the
cooling time scale is faster than the free-fall time scale then the collapse of mirror-particle
perturbations are not impeded [68]. The perturbation will collapse into a disk-like system on the free-fall time scale
(the size of the disk depending on details such as the amount of angular momentum)
^{9}^{9}9
The disk is not expected to be completely uniform and smaller scale perturbations on the edge of the disk might break away
from the main perturbation and collapse. Such perturbations might seed satellite galaxies and could potentially
explain why the bulk of the dwarf satellite galaxies of the Milky Way and M31 in the local group are aligned in a plane [69, 70].
Alternatively [68] the dwarf satellite galaxies might have originated much later as tidal dwarf galaxies formed during
a merger event [71]..
Mirror star formation
can occur during the free-fall phase and/or later in the collapsed disk.
Mirror supernovae are also expected to be occurring during this early time.
This is especially important assuming photon - mirror photon kinetic mixing interaction exists with
. As briefly mentioned in subsection 1.1, mirror
supernovae would then influence ordinary matter by providing a huge heat source.
Basic particle processes such as in the mirror supernova’s core
would convert about half of the mirror supernova’s core collapse energy into creation
of light ordinary particles [43, 42]. In the region around
each supernova () this energy is converted (via complex and poorly understood
processes, e.g. generation of shocks etc.) into ordinary photons which are anticipated
to have an energy spectrum in the X-ray region.
These
photons would not only heat ordinary matter but might have been responsible for its reionization - inferred from observations to
have occurred at early times at redshift: .

Once the ordinary matter is ionized it can no longer efficiently absorb radiation.
This is because ordinary matter has very little metal content at this early time, and the Thomson scattering
cross-section is so small. [We adopt the astrophysics convention of describing every element heavier than helium as a metal.]
Ordinary matter can now start to cool and accumulate in these mirror dark matter structures.
One expects, therefore, that the ordinary baryons will ultimately collapse potentially forming a separate disk ^{10}^{10}10
Gravitational interactions between the baryonic disk and mirror baryonic one, should both form, could lead to their alignment cf. [72]..
Ordinary star formation can now begin and is expected to proceed extremely rapidly.
In fact, the density of the baryonic gas () in these collapsed structures would be very high, which
is known to be directly correlated with the star formation rate:

(8) |

where [73, 74]. Thus leads inevitably to the production of ordinary supernovae. Now, the physics of ordinary supernovae, like mirror supernovae as we briefly described above, is extremely interesting if the kinetic mixing interaction exists. Ordinary supernovae will produce a huge flux of mirror photons in the presence of the kinetic mixing interaction of strength . These mirror photons can heat the mirror disk, which by now has a substantial mirror metal fraction. [This energy is absorbed very efficiently because of the large photoionization cross-section of the mirror metal atoms.] This huge energy input can potentially expand the gas in the mirror disk out into an approximately spherically distributed plasma. This, it is presumed, is the origin of the roughly spherical halos inferred to exist around spiral galaxies today. Naturally, much work needs to be done in order to check this qualitative picture of the early period of galaxy evolution.

The (current) structure of galactic halos appears to be a more tractable problem [43, 44, 45, 46]. As described above, the dark matter distribution in galaxies was once very compact until heating by ordinary supernovae occurred. If the rate of supernovae became large enough, then the heating rate of the mirror-particle plasma could exceed its cooling rate (due to processes such as thermal bremsstrahlung) in which case this plasma component will expand. The mirror star formation rate falls drastically at this time as the gas component heats up and its mass density falls. As the mirror dark matter expands, the ordinary star formation rate (and hence supernova rate) also falls as the ordinary matter densities drop in the weakening gravitational potential. The halo will continue to expand until the heating is balanced by cooling. The end result is that at the current epoch the halo should have evolved to a quasi-static equilibrium configuration where the energy being absorbed in each halo volume element is balanced by the energy being emitted in the same volume element:

(9) |

Under the simplifying assumption of spherical symmetry, the above dynamical condition, along with the hydrostatic equilibrium equation,

(10) |

can be used to determine the dark matter density and temperature profiles: . That is, the current bulk properties of the dark matter halo around spiral galaxies can be derived from this assumed model of halo dynamics.

Numerically it has been shown that this dynamics requires dark matter to have an approximate quasi-isothermal distribution [45, 46]:

(11) |

where is the dark matter central density and core radius. Numerically it is also found that the core radius, , scales with disk scale length, , via and that the product is roughly , i.e. independent of galaxy size (the is set by the parameters of the model).

Dark matter with this constrained distribution is known to provide an excellent description of galactic rotation curves in spiral galaxies [75]. Indeed, a result of all these scaling relations, together with baryonic relations connecting the disk mass with the disk scale length, is that the ordinary and mirror dark matter content of spiral galaxies are, roughly, specified by a single parameter. This parameter can be taken to be or the galaxy’s luminosity in some band, . This has important implications for the galaxy’s rotation curve. It should be roughly universal, i.e. completely fixed once is specified. This is consistent with observations, which show just this behaviour [76, 77, 78]. As should be clear from the above scaling relations, the agreement with observations is not just qualitative, the dynamics allows quantitative predictions to be made, all of which appear to be consistent with the observations.

Another result of numerical solution to Eqs.(9), (10) is that the halo is approximately isothermal. Numerical work and also some analytic arguments [43] indicate that the average halo temperature is approximately:

(12) |

where is the galactic rotational velocity (for the Milky Way km/s)
and is the mean mass of the particles, , constituting the plasma.
Arguments from early Universe cosmology (mirror BBN) indicate that GeV [60].
This means that for the Milky Way the halo temperature is roughly: eV, i.e. a few million
degrees kelvin ^{11}^{11}11Unless otherwise indicated,
we use natural units with throughout..

The end result of all this, is that at the present time, spiral galaxies such as the Milky Way are at the center of an extended dark matter halo. This halo is predominately in the form of a hot spherical plasma, which is composed of an array of mirror particles: . These particles are continuously undergoing both dissipative and non-dissipative self interactions, with the energy dissipated from the halo being replaced by energy produced from ordinary supernovae, made possible if kinetic mixing with strength exists. The current mirror-star formation rate in such a plasma is expected to be very low; the plasma cannot locally cool and condense into stars. The vast bulk of mirror star formation is therefore expected to have occurred at very early times - in the first billion years or so. As discussed above, this is the presumed origin of the mirror metal component of the halo plasma. Although very rare, mirror supernovae might still occur today. Observationally, a mirror supernova might appear to be something like a Gamma Ray Burst given the assumed kinetic mixing interaction, and indeed it has been proposed as a candidate for the central engine powering (at a class of) such objects [42, 79].

The halo dynamics described above requires
the kinetic mixing interaction to not only exist but have strength .
This interaction induces also an interaction between charged mirror particles and ordinary nuclei.
This enables halo mirror particles to thereby scatter off ordinary nuclei, essentially
a Rutherford-type (spin independent) elastic scattering process.
Hence, mirror particle interactions might potentially be seen in direct detection experiments searching
for halo dark matter
^{12}^{12}12
Given that mirror dark matter arises from a particle-antiparticle asymmetry
in the Universe, signals from the annihilation of mirror baryons
with mirror antibaryons are not anticipated. There would be far too few mirror antibaryons in the halo for
such annihilation to be detected.
Thus, observable indirect detection
signatures of mirror dark matter are expected to be very limited; possibly only an excess of positions produced via kinetic
mixing induced processes in mirror supernovae, should such supernovae occur at a sufficient rate [42].
.

Consider a mirror nuclei of type of atomic number (e.g. ) that is moving with velocity . If this mirror nuclei passes close to an ordinary nucleus of atomic number (presumed at rest), then it can scatter leaving with a recoil energy . The differential cross-section for this process has a characteristic dependence:

(13) |

where , are the relevant form factors. If this kinetic mixing induced interaction does indeed exist, then halo mirror dark matter can be probed in direct detection experiments. In fact, a kinetic mixing strength happens to be just the right magnitude for the current generation of direct detection experiments to be sensitive to this interaction [80, 81, 82, 83, 47].

The rates in such an experiment depend not just on the cross-section but also on the halo velocity distribution of the mirror particles. The self interactions of the mirror particles in the halo plasma should help keep these particles in thermal equilibrium. Their velocity distribution is therefore expected to be Maxwellian:

(14) |

The quantity , which characterizes the velocity dispersion, evidently depends on the mass , of the particular component:

(15) |

where Eq.(12) has been used. Observe that such a mass dependent velocity dispersion is very different from the distribution expected for collisionless cold dark matter, where is anticipated [84].

Clearly, mirror dark matter has a number of distinctive features: It is (a) multi-component, with a spectrum of particles with known masses (b) interacts with ordinary matter via kinetic mixing induced interactions, leading to Rutherford-type (spin independent) elastic scattering and (c) heavy mirror particles, , have small velocity dispersion (). These features, it turns out, combine to provide a consistent explanation of the DAMA annual modulation signal [16, 17, 18] and also the low energy excesses found by CoGeNT [19], CRESST-II [20] and CDMS/Si [21]. In this interpretation, these experiments have detected the kinetic mixing induced interactions of halo mirror metal components, [47]. While these developments appear to be very encouraging, the experimental situation is still not completely settled. Significant tension with the null results of XENON100 [48] and LUX [49] exists. Also, important and necessary checks have yet to be made such as an experiment located in the southern hemisphere. Such an experiment is important, not just as a check of the DAMA annual modulation signal, but also to search for the expected diurnal modulation [85].

## 2 The particle physics

The standard model of particle physics is a highly predictive gauge theory, based on the gauge symmetry: . This theory has been extremely successful in accounting for the electromagnetic, weak, and strong interactions of the known particles [86, 87]. For a review, see for instance [88]. The electromagnetic and strong interactions are associated with unbroken gauge symmetries and , while weak interactions arise from the spontaneous breaking of . The recent discovery of a Higgs-like resonance at the Large Hadron Collider (LHC) [89, 90] appears to confirm that this symmetry breaking is due to the nonzero vacuum expectation value of an elementary Higgs doublet field, [91]. Indeed, the measured properties of the Higgs-like resonance are (currently) consistent with those expected for the standard model Higgs scalar [92, 93, 94].

The standard model can be described by a renormalizable Lagrangian:

(16) |

This Lagrangian respects an array of symmetries including proper orthochronous Lorentz transformations, space-time translations and gauge symmetries, as discussed above. Notably, the standard model Lagrangian does not respect improper Lorentz symmetries, such as parity and time reversal. Parity, in particular, is violated maximally by the weak interactions as the gauge bosons couple only to the left-handed chiral fermion fields. However improper space-time symmetries, appropriately defined, can be exact and unbroken symmetries of nature if a set of mirror particles exist. The simplest such model has been called the exact parity symmetric model [25].

### 2.1 Exact parity symmetric model

Mirror particles are defined as follows. For every known particle, a mirror partner is hypothesized, which we shall denote with a prime (). The interactions of these duplicate set of particles are described by a Lagrangian of exactly the same form as that of the standard model. That is, the ordinary particles and mirror particles are described by the Lagrangian:

(17) |

where accounts for possible non-gravitational interactions connecting
ordinary and mirror particles, which we set aside for the moment.
That is, the ordinary and mirror particles form parallel sectors, each
respecting independent gauge symmetries . This means that the gauge symmetry of the full Lagrangian, , is .
As defined in Eq.(17) above, the Lagrangian has a discrete symmetry which swaps each ordinary particle
with its partner. If we make a slight adjustment, and interchange left and right-handed chiral fields
in the mirror sector so that mirror weak interactions couple to right-handed chiral fermion fields (instead of left-handed
fields) then
the Lagrangian, , respects an exact parity symmetry, which we also refer to as mirror symmetry [25]:
^{13}^{13}13
Technically, there are two possible theories depending on whether or not we flip the left
and right chiralities in the mirror sector.
Although these two theories are formally distinct, they
are phenomenologically
almost indistinguishable. Certainly,
for the applications to dark matter phenomena, this distinction is unimportant.
See also section 6 of [95] for further discussions about this dichotomy.

(18) |

Here are the spin-one gauge bosons, the fermion fields represent the leptons and quarks, is the generation index and is a Dirac gamma matrix. Also included is the Higgs doublet along with its mirror partner, . This review discusses the parameter region (to be defined in section 2.3) where , so that the mirror symmetry is not spontaneously broken by the vacuum; mirror symmetry is an exact, unbroken symmetry of the theory.

The parity transformation as given in Eq.(18), which we here define as , involves swapping ordinary particles with mirror particles in addition to . Although this is non-standard, and is perhaps subtle, it is of course a perfectly acceptable definition of space-time parity in the presence of degenerate partners [25, 96, 95] (see also [26, 97] for early related discussions). This theory also exhibits an exact time reversal symmetry , defined by CPT where CPT is the conventionally defined CPT transformation (the CPT transformation is an invariance of itself and so is also an invariance of ). The and transformations do not separately commute with proper Lorentz transformations (which is, of course, a general property of space and time inversion transformations) but together with space-time translations close to form the Poincar group - the group of isometries of Minkowski space-time.

Figure 2.1: The process and the mirror particle analogue: . Mirror symmetry implies that the cross-section for both processes is exactly the same.

Mirror symmetry, so long as it is not spontaneously, or otherwise broken, ensures that the masses and couplings of the particles in the mirror sector are exactly identical to the corresponding ones in the ordinary sector. The only new parameters are those in , which by hypothesis conserve mirror symmetry. An important, but trivial consequence of mirror symmetry is that every ordinary particle process has a mirror particle analogue. Take elastic scattering as an example (figure 2.1). Mirror symmetry implies that can also occur, and since the symmetry is exact and unbroken, the cross-section for each process is exactly the same. In the Thomson limit, for instance, the (Born) cross-section for both processes is . The same thing happens, of course, for every other ordinary particle process.

Mirror symmetry does not exclude the possible existence of new interactions coupling ordinary and mirror particles together. However, with the minimal particle content, the (mirror, gauge, Lorentz) symmetries of the theory restrict such renormalizable interactions to just two terms [25]:

(19) |

where () is the ordinary (mirror) gauge boson field strength tensor. The first interaction is a mixing of the kinetic terms for the and gauge bosons, while the second interaction is a Higgs - mirror Higgs quartic coupling which forms part of the full Higgs potential. We now discuss each of these terms in more detail.

### 2.2 Photon - mirror photon kinetic mixing

The kinetic mixing term in Eq.(19) is gauge invariant, since itself is gauge invariant under the gauge transformation, . Kinetic mixing respects mirror symmetry Eq.(18), and all the other known symmetries of the theory. Furthermore, since kinetic mixing is a renormalizable interaction, can be viewed as a fundamental parameter of the theory [39].

In standard electroweak theory, the gauge boson , is a linear combination of the photon and the Z-boson :

(20) |

It follows that there is both and kinetic mixing. However, experiments and observations are much more sensitive to kinetic mixing interaction so we need not discuss mixing any further.

What is the physical effect of photon - mirror photon kinetic mixing? Consider quantum electrodynamics of the electron , and photon , mirror electron , and mirror photon :

(21) | |||||

where we have adopted the convenient notation: , , and so on. The kinetic mixing can be removed with a non-orthogonal transformation: , . One then has two massless (i.e. degenerate) and kinetically unmixed states; any orthogonal transformation of which will leave the kinetic terms invariant. One can transform to a basis where only one of these states couples to electrons. The state coupling to the electrons is the physical photon , appropriate for an ordinary matter dominated environment, such as the Earth [40] (see also [98]). The orthogonal state we call the sterile photon :

(22) |

In this physical basis for an ordinary matter environment, the Lagrangian is (to leading order in ):

(23) | |||||

where (). Evidently, the physical photon couples to mirror electrons with electric charge , while the mirror photon doesn’t couple to ordinary matter at all. The mirror symmetry appears to be broken, but it is not of course; it is simply the result of a mirror asymmetric environment consisting of ordinary matter.

For completeness, let us briefly digress to discuss the physical states appropriate for a mirror matter environment, such as a star composed of mirror baryons. These are the physical mirror photon , and the sterile mirror photon :

(24) |

In this physical basis for a mirror matter environment, the Lagrangian is (to leading order in ):

(25) | |||||

Evidently, a mirror star would emit the state . In terms of the ordinary matter physical states,
(to leading order in ). Thus, the flux of mirror photons detectable in
an ordinary matter telescope is reduced by a factor . This makes such radiation undetectable with current
technology. Also note that such radiation would decohere into ordinary matter eigenstates on passing through
ordinary matter and thus could not even be detected in underground experiments
^{14}^{14}14For a mixed ordinary/mirror matter environment oscillations between ordinary and mirror photons are possible in principle..

Generalization of this quantum electrodynamics to the exact parity symmetric model is straightforward.
The physical photon , is the photon. It couples to the known particles in the usual
way and additionally couples to mirror charged particles with coupling suppressed by .
That is, the photon couples to mirror protons with ordinary electric charge ,
mirror electrons with ordinary electric charge etc. As discussed above,
the orthogonal state doesn’t couple to
ordinary matter at all ^{15}^{15}15
In principle, the sign of can be either positive or negative.
Although these two cases are physically inequivalent, this detail is unimportant for the
kinetic mixing applications discussed in this review. For this reason, subsequent reference to the parameter
are statements about its magnitude only..
The small induced electric charge means that mirror particles can elastically scatter off ordinary
nuclei and can thereby be directly detected in experiments such as DAMA, CoGeNT, CDMS etc.
Another consequence of the small induced electric charge is that mirror electron - mirror positron pairs
can be produced from processes such as in the core of ordinary supernovae and in the early
Universe. The cross-section for this process is proportional to .

The magnitude of the kinetic mixing parameter of astrophysical interest and also of interest for dark matter direct detection experiments turns out to be very small: . This is nearly two orders of magnitude smaller than the direct laboratory upper limit of (90% C.L.) which arises from the orthopositronium system [99]. The kinetic mixing interaction induces orthopositronium - mirror orthopositronium mass mixing which leads to oscillations of orthopositronium into mirror orthopositronium [34] (see also [100, 101]). There are important proposals to improve the precision of orthopositronium experiments to directly explore the parameter region [102].

As a final comment, the approach taken here is to consider kinetic mixing as a fundamental interaction in the Lagrangian [39, 25]. An alternative possibility is that kinetic mixing is radiatively generated [40]. In particular, in Grand Unified models, such as those based on gauge symmetry, the is embedded in a non-abelian gauge symmetry. This additional symmetry prevents kinetic mixing from arising at tree-level (i.e. in the classical limit). However if there exists particles that are charged under both ordinary and mirror electromagnetism, e.g. under , then kinetic mixing can be radiatively generated at 1-loop level. Such induced kinetic mixing is typically around [40]. However if kinetic mixing cancels at 1-loop, as happens if the particles are degenerate in mass, then it can be shown to cancel also at 2-loop level [103]. At three loops, kinetic mixing might conceivably be of order , although this has yet to be demonstrated in an actual calculation.

The kinetic mixing interaction is the only term in [Eq.(19)] which is used in the applications to the astrophysical and cosmological problems discussed in subsequent sections of this review. Nevertheless, for completeness we now briefly consider other possible non-gravitational interactions connecting ordinary and mirror particles discussed in the literature.

### 2.3 Higgs portal coupling

In addition to kinetic mixing, there is only one other renormalizable term (in the minimal model) which can couple the known particles with the mirror particles. This is the Higgs - mirror Higgs quartic interaction, also called Higgs portal coupling:

(26) |

The possible effects of this interaction have been discussed in a number
of papers [25, 96, 104, 105, 106, 107, 108, 109]
^{16}^{16}16The Higgs portal coupling and kinetic mixing interaction have also
been discussed in the context of more general hidden sector dark matter models, for a flavour of such work
see for example [110, 111, 112]..
We shall summarize some of the main results here.

The complete Higgs potential, including the above portal coupling, is:

(27) | |||||

This potential can be minimized to obtain the non-trivial vacuum:

(28) |

where GeV.
There is a second possible vacuum, one with ,
in which the mirror symmetry is spontaneously broken ^{17}^{17}17
When QCD effects are taken into account is perturbed away
from zero, but is still very small:
where MeV [113]..
This broken phase occurs for
a distinct region of parameter space, namely , .
The phenomenology of this second solution is clearly quite different,
and has been discussed in several papers [113]. Next to minimal models, with additional singlet scalar(s) and/or soft
breaking terms have also been considered in the literature. Such models can accommodate
[114], or
[106, 115].
The mirror dark matter discussed in this review refers to
the theoretically unique case where mirror symmetry is completely unbroken.
As discussed
above, this assumes the minimal scalar content with
so that the parity conserving vacuum, Eq.(28), results.

Expanding the potential around the parity conserving vacuum allows one to identify the two mass eigenstate Higgs fields: and . These states are maximal combinations of the weak eigenstates:

(29) |

where and are the real parts of the neutral components of and , respectively. The two states, and , have definite exact mirror parity, with being even while is odd. When the Lagrangian, Eq.(17), is rewritten in terms of and , one finds that and each couple to ordinary fermions and gauge bosons similar to the standard model Higgs, but with coupling reduced by [96]. Whether or not this is observable depends on the mass difference between and . The masses of are:

(30) |

We see that the effect of the Higgs portal coupling [Eq.(26)] is to break the mass degeneracy. The mass difference , is given by

(31) | |||||

The rough consistency of the Higgs-like resonance discovered at the
LHC [89, 90] with standard model expectations already
puts restrictions on . This mass difference must be less than the Higgs decay width otherwise
the two states will be produced incoherently
^{18}^{18}18
For the Higgs mass difference
to be less than the Higgs decay width requires
small values of . Small values of (and also kinetic mixing, )
are technically natural as
the limit , corresponds to the decoupling of the ordinary and mirror
sectors. There is consequent increase in symmetry in this limit (cf. [116]) as one can perform independent
Poincar
symmetry transformations on the ordinary sector and mirror sector separately..
Incoherent production
leads to a large deviation from standard Higgs physics [96, 108, 109], which
is already excluded.

Coherent production occurs when , where MeV is the standard model Higgs decay width [117]. In this parameter region, the weak eigenstate, , is produced and starts to oscillate into the mirror state (the discussion below closely follows the treatment of [109]). The oscillation probability is then:

(32) |

where is the oscillation time in the non-relativistic limit. The average oscillation probability to the mirror state, which determines the invisible decay width, is given by

(33) | |||||

Evidently in this coherent production regime the branching fraction to invisible channels is always less than . The oscillations also modify the cross-sections into visible channels. These cross-sections are reduced by the factor , where

(34) | |||||

Observe that the Higgs physics becomes indistinguishable from that of the standard model in the limit where . This occurs when , or equivalently when .

What is the experimental limit on from collider data? Ref. [118] studied the standard model Higgs augmented with invisible decay modes. There, they found that LHC and Tevatron data implied a limit on the branching ratio: at C.L. Setting , and using Eq.(33), it follows that:

(35) |

Massaging this expression, using Eq.(31), leads to the limit:

(36) |

This experimental limit can be compared with the cosmological bound [104, 107]. This bound arises by demanding that scattering be small enough so that the mirror sector is not thermalized with the ordinary matter sector in the early Universe. Note however that the cosmological limit can be evaded in inflationary scenarios with low reheating temperature, GeV [107].

### 2.4 Neutrino - mirror neutrino mass mixing

Neutrino oscillations have been observed in a variety of experiments which indicates that neutrinos have nonzero masses. For a review see for example [119]. This means that the standard model will have to be extended in some way to accommodate massive neutrinos. Although the neutrinos have mass, their overall mass scale is sub eV, which is much smaller than the other fermions in the standard model. If mirror symmetry is unbroken, then we expect a set of mirror neutrinos, also sub eV mass scale. They need not be exactly degenerate with their ordinary matter counterparts if there is mass mixing between ordinary and mirror neutrinos. Such mass mixing is possible since it does not violate any of the fundamental unbroken symmetries of the theory such as of electromagnetism or mirror symmetry. Neutrino mass mixing, if it exists, would lead to oscillations between the ordinary and mirror neutrinos [96, 120, 95, 121, 122].

Whether or not neutrino - mirror neutrino mass mixing is expected to occur depends on the mechanism by which neutrinos gain their masses. Here we consider the three simplest seesaw neutrino mass generating models. These are now called type-I, type-II and type-III seesaw models [123]. In principle the following analysis could be repeated for any other model generating nonzero neutrino masses.

Type-I seesaw In this model, three gauge singlet right-handed neutrinos, , , are added to the standard model [124]. The coupling of these neutrinos to is described by the following Lagrangian, restricting here to the first generation for simplicity:

(37) |

where is the standard CP transformation. [In the Dirac-Pauli representation of the matrices, .] In the Lagrangian above, and is the Higgs doublet field whose neutral component develops a non-zero vacuum expectation value, . If , then diagonalization of the resulting neutrino mass matrix yields two Majorana states: and , with masses, and . Type-II seesaw For the type-II seesaw model, an electroweak triplet scalar , is introduced instead of the [125]. In this case a Yukawa term:

(38) |

generates a Majorana mass for when the neutral component of gains a nonzero vacuum expectation value: . Type-III seesaw In the type-III seesaw option three fermionic triplets , are introduced (instead of ). These states couple to in the following way, again restricting to one generation for simplicity [126]:

(39) |

The resulting neutrino mass matrix has the same form as for the type-I seesaw.

For each of these three models we can easily add an isomorphic Lagrangian (with ordinary fields replaced by mirror fields) to the mirror sector. As before, there is an exact parity symmetry again swapping each ordinary particle with its mirror partner. An important question arises: Are masses that mix ordinary and mirror neutrinos allowed in any of these three models? With the particle content described above, only the type-I seesaw model can have mass mixing between ordinary and mirror neutrinos. [For the type-II and type-III seesaw, mass mixing between ordinary and mirror neutrinos is forbidden by the gauge symmetry of the Lagrangian.] This arises through terms such as:

(40) |

The above term leads to off-diagonal contributions to the mass matrix describing neutrinos and their mirror partners.

The effect of mass mixing is to induce oscillations between ordinary and mirror neutrinos. At one time it was suggested that such ordinary - mirror neutrino oscillations might be implicated in the atmospheric and solar neutrino observations [96, 120, 95]. Experiments have shown that this is not the case; the solar, atmospheric, and long-baseline neutrino experiments can all be accounted for with just the three ordinary neutrinos (see for example the review [127]). Some anomalies remain, but it seems unlikely that they could be explained with mirror neutrino oscillations, unless the mirror symmetry was broken in some way (see [128] for some recent work in this direction). The conclusion is that on length scales probed by the solar, atmospheric, and long-baseline neutrino experiments, there is no convincing evidence for any oscillations into mirror neutrinos. Thus, either the mass mixing between ordinary and mirror neutrinos is zero, as occurs in e.g. the type-II and type-III seesaw models, or it is small. Small mass mixing is possible in the type-I seesaw model, and the experiments could be used to place an upper limit on the parameters in Eq.(40). A more sensitive probe of neutrino - mirror neutrino mass mixing could come from measurements of energetic neutrinos of astrophysical origin by experiments such as IceCube and ANTARES. Indeed, ref.[129] points out that such oscillations would modify the flavour ratios of the observed neutrinos away from standard expectations of . Experiments could thereby find evidence for, or against, ordinary - mirror neutrino oscillations, which might tell us something about the neutrino mass generation mechanism.

Oscillations of ordinary neutrinos into mirror neutrinos can also be important for cosmology, modify BBN etc., and potentially have also astrophysical implications. Some cosmological effects of neutrino oscillations were discussed in the context of the now disfavored solutions to the atmospheric and solar neutrino anomalies involving oscillations of ordinary neutrinos into mirror neutrinos [130]. Cosmological effects of oscillations of the heavy Majorana fermions were also considered in [131] and some astrophysical applications of neutrino oscillations into mirror neutrinos were discussed in [79].

### 2.5 Higher dimensional effective operators in ?

So far, it must be said, the discussion has been conservative. We have only examined the consequences of the two renormalizable interactions given in Eq.(19), and briefly considered also neutrino mass mixing. It is possible that we might be lucky. If nature is more liberal there could be interesting TEV scale physics connecting ordinary particles with their mirror counterparts. A common gauge interaction perhaps, coupling equally to both ordinary and mirror particles. The LHC signatures of this type of interaction has been discussed in the context of more generic hidden sector dark matter models in [132]. Even if such interactions are not (yet) directly observable at the LHC, they might be probed indirectly through effective interactions inducing mixing of some of the known neutral particles with their mirror counterparts. We consider here two examples that have been discussed in the literature.

Neutron oscillations into mirror neutrons

Effective interactions leading to
neutron - mirror neutron () mass mixing and thus to oscillations are an interesting
possibility [133].
Such oscillations could have important implications for cosmic ray physics [134] and big
bang nucleosynthesis [135].
Neutron - mirror neutron mass mixing doesn’t violate any of the mirror, gauge or Lorentz unbroken symmetries,
however it does require the generation of a dimension nine operator: ^{19}^{19}19
The neutron - mirror neutron mass mixing operator [Eq.(41)]
does violate the separate global symmetries generated by baryon number ,
and mirror-baryon number , but conserves a diagonal subgroup.
Baryon number is usually considered to be an accidental symmetry of the standard model, so
its violation in models beyond the standard model is possible, and of no cause for concern. Naturally, one must check that the
underlying particle physics model which produces the effective operator in Eq.(41) is
consistent with constraints such as proton lifetime bounds.