Dipartimento di Scienze Fisiche e Chimiche, Università di L’Aquila, 67100
L’Aquila (AQ), Italy INFN, Laboratori Nazionali del Gran Sasso (LNGS), 67100
Assergi (AQ), Italy. 14.60.PqNeutrino mass and mixing 14.60.StNon-standard-
model neutrinos, right-handed neutrinos, etc.

# Dark Matter and IceCube Neutrinos

R. Biondi

###### Abstract

We show that the excess of high energy neutrinos observed by the IceCube
collaboration at energies above 100 TeV might originate from baryon number
violating decays of heavy shadow baryons from mirror sector, which in turn
constitute Dark Matter. Due to tiny mixing between mirror and ordinary
neutrinos, it is possible to explain the specific features of the IceCube
events spectrum.

Discovery of high-energy neutrino events by the IceCube Collaboration [1]
opened a new era of experimental high-energy neutrino astrophysics. Recently
the collaboration published the three year data collected between 2010 and
2013, containing 37 candidate in the energy range from 30 TeV to 2 PeV. While
the number of events with energies below 100 TeV or so is compatible with the
background expectations, there is an evident excess of the events with $E>100$
TeV or so. On the other hand, no events were observed in the gap between 400
TeV and 1 PeV, but three most energetic shower events emerged at the end of
the spectrum with energies 1.04 PeV, 1.14 PeV and 2.0 PeV where the
atmospheric background is practically vanishing. On the other hand, the
spectrum is apparently cut off at energies larger than about 2 PeV. The gap in
the energy spectrum is difficult to explain in known models of high-energy
neutrinos of astrophysical origin.

The model that may explain such a spectrum was suggested in ref. [2]. Here we
present main guidelines of this model. Nowadays it becomes popular the idea
that dark matter of the universe can emerge from a parallel gauge sector, with
particles and interactions sharing many similarities with ordinary particle
sector. Such a shadow sector would contain particles like quarks which form
composite baryons, as well as leptons and neutrinos which are all sterile for
ordinary gauge interactions. In particularly interesting example is
represented by so-called mirror world [3], which has the content of particles
and interactions exactly identical to that of ordinary sector, with the same
gauge and Yukawa coupling constants. Taking into consideration also attractive
possibilities for particle physics beyond the Standard Model related to
supersymmetric grand unified theory (SUSY GUT), one can consider that at
higher energies both ordinary and mirror sectors are presented by identical
GUTs, e.g. $SU(5)$ and $SU(5)^{\prime}$ or $SU(6)$ and $SU(6)^{\prime}$, which
in both sectors break down to their Standard Model subgroups $SU(3)\times
SU(2)\times U(1)$ at the scale $M_{G}\sim 10^{16}$ GeV. However, following
refs. [4, 5], one can assume that later on the symmetry between two sectors is
broken so that the electroweak symmetry breaking scale $V$ in mirror sector is
much larger than than in the Standard Model, $v=174$ GeV. Namely, if $V\sim
10^{11}$ GeV, then the lightest shadow baryons should have mass order few PeV.
Due to baryon violating GUT gauge bosons, they decay with $\tau_{\rm dec}\sim
t_{U}$ producing energetic shadow neutrinos which oscillate into active
neutrinos (with oscillation probabilities $\sim 10^{-10}$ or so) transferring
their spectrum to the latter.

Let us consider, for simplicity, a supersymmetric grand unification theory
$SU(5)\times SU(5)^{\prime}$. 111 As discussed in ref. [2], our proposal can
be more nicely realized in SUSY $SU(6)$ theory [6] which gives natural
solution to the so called hierarchy and doublet-triplet splitting problems by
relating the electroweak symmetry breaking scale to the supersymmetry breaking
scale $M_{S}\sim 1$ TeV and naturally explains the fermion mass spectrum.
Consider a situation when in both sectors the GUT symmetries are broken at the
scale $M_{G}\simeq 2\times 10^{16}$ GeV, $SU(5)\rightarrow SU(3)\times
SU(2)\times U(1)$ and $SU(5)^{\prime}\rightarrow SU(3)^{\prime}\times
SU(2)^{\prime}\times U(1)^{\prime}$. Below this scale our sector is
represented by MSSM with chiral superfields of quarks $q_{i}=(u,d)_{i}$,
$u^{c}_{i},d^{c}_{i}$ and leptons $l_{i}=(\nu,e)_{i}$, $e^{c}_{i}$, $i=1,2,3$
being family index, and two Higgs superfields $h_{1},h_{2}$. As for parallel
sector we have the similar particle content, quarks $Q_{i}=(U,D)_{i}$,
$U^{c}_{i},D^{c}_{i}$, leptons $L_{i}=(N,E)_{i}$, $E^{c}_{i}$, and two Higgses
$H_{1},H_{2}$. At the scale $\mu=M_{G}$ all gauge coupling constants are equal
and the Yukawa couplings have the same pattern for ordinary and shadow
fermions:

$(Y^{e}_{ij}l_{i}e^{c}_{j}h_{1}+Y^{d}_{ij}q_{i}d^{c}h_{1}+Y^{u}_{ij}q_{i}u^{c}_{j}h_{2})\leavevmode\nobreak\
+\leavevmode\nobreak\
(Y^{e}_{ij}L_{i}E^{c}_{j}H_{1}+Y^{d}_{ij}Q_{i}D^{c}H_{1}+Y^{u}_{ij}Q_{i}U^{c}_{j}H_{2})$
(1)

As for the neutrinos, their masses emerge from the following $D=5$ operators
[4]:

$\frac{\alpha_{ij}}{M}(lh_{2})(lh_{2})+\frac{\alpha_{ij}}{M}(LH_{2})(LH_{2})+\frac{\beta_{ij}}{M}(lh_{2})(LH_{2})$
(2)

where $M$ is an effective scale of the order of Planck scale $M_{Pl}$ [5]. The
first two terms give the Majorana masses respectively to ordinary neutrinos
$\nu_{e,\mu,\tau}$ and their shadow (sterile) partners while third term
induces the mixing between ordinary and shadow neutrinos.

Below GUT scale, the gauge coupling constants of two sectors, respectively
$g_{3,2,1}$ and $g^{\prime}_{3,2,1}$ evolve down in energies in both sectors
in the same way (see Fig. 1). For ordinary sector we follow to the standard
paradigm. Supersymmetry is broken at the scale $M_{S}\sim 1$ TeV, the Higgses
are not protected anymore to be massless and they get VEVs $v_{1}=v\cos\beta$,
$v_{2}=v\sin\beta$, where $v=174$ GeV, which induce the electroweak symmetry
breaking and generate the fermion masses.

Figure 1: Running of gauge couplings below the GUT scale,
$\alpha_{i}=g_{i}^{2}/4\pi$ and $\alpha^{\prime}_{i}=g_{i}^{\prime 2}/4\pi$.

As for parallel sector, let us assume now that mirror symmetry between two
sectors is spontaneously broken in the following way. Imagine that
supersymmetry is broken in shadow sector at the scale $M^{\prime}_{S}\sim
10^{11}$ GeV, supposedly due to non-zero $F$ or $D$ terms of some auxiliary
fields, so that shadow scalars, including squarks and leptons as well as Higgs
doublets $H_{1}$ and $H_{2}$ acquire soft masses order $M_{S}$.222
Interestingly, if supersymmetry breaking is transferred to our sector via
gravity or other Planck scale mediators, this would explain ordinary soft
masses order $M_{S}\sim M^{\prime 2}_{S}/M_{Pl}\sim 1$ TeV.  Respectively
shadow Higgses Get VEVs $V_{1}=V\cos\beta^{\prime}$ and
$V_{2}=V\sin\beta^{\prime}$, and electroweak symmetry is broken at the scale
$V\leq M^{\prime}_{S}$. Therefore, the masses of shadow fermions are rescaled,
modulo renormalization factors order 1, by a factor $\xi=V/v$ with respect to
ordinary fermion masses. Namely, taking $V/v=10^{9}$ and assuming
$\tan\beta^{\prime}=\tan\beta$, by the RG running of gauge and Yukawa
constants from the GUT scale down in energies, one obtains for the shadow
electron mass $M_{E}\simeq 0.4$ PeV, $M_{D}\simeq 1.1$ PeV and $M_{U}\simeq
1.9$ PeV, against $m_{e}=0.5$ MeV, $m_{d}\simeq 5$ MeV and $m_{u}\simeq 3$ MeV
for ordinary electron and quarks. One the other hand, the RG evolution shows
that the shadow QCD scale becomes $\Lambda^{\prime}\sim 100$ TeV (c.f.
$\Lambda\simeq 200$ MeV in ordinary QCD). Therefore, $\Lambda^{\prime}\ll
M_{U},M_{D}$ and the shadow QCD looks like a rescaled version of our QCD
without light quarks, but only with heavy quarks like $c$ and $b$. Notice also
that in shadow sector up quark $U$ becomes heavier than the down quark $D$
[2], and $M_{U,D}/\Lambda^{\prime}$ is of the same order as $m_{b,c}/\Lambda$
in ordinary QCD.

The lightest state would appear to be a shadow $\Delta^{-}$ baryon of spin
$3/2$, consisting of three down quarks $D$ and having mass $M_{\Delta}\approx
3M_{D}=3.3$ PeV. All states, containing up quark $U$, will be unstable against
weak decays, $U\rightarrow D\bar{E}N$. As for mesons, the lightest
pseudoscalar is shadow neutral pion $\pi^{0}$ consisting of $D\bar{D}$, with
mass $M_{0}\approx 2M_{D}=2.2$ PeV, while the lightest vector meson
$\rho^{0}(D\bar{D})$ is slightly heavier than $\pi^{0}$. Charged Pion
$\pi^{-}$ as well as $\rho^{-}$-meson consisting of $D\bar{U}$ will have mass
$M\simeq M_{U}+M_{D}=3$ PeV, with $\rho^{-}$ a bit heavier than $\pi^{-}$. All
pseudoscalar and vector mesons have excited states with mass gap order
$\Lambda^{\prime}$ between the levels, just like $c\bar{c}$ or $b\bar{b}$
states in ordinary QCD.

Now we come to the role of baryon violation and proton decay which is
fundamental prediction of the GUTs. The heavy gauge bosons of $SU(5)$ with
baryon violating couplings between quarks and leptons induce the decay of the
lightest ordinary baryons (proton, or neutron bound in nuclei), with lifetime
$\tau_{p}\sim M_{G}^{4}(\alpha_{5}^{2}m_{p}^{5})^{-1}\sim 10^{31}$ Gyr or so,
where $\alpha_{5}$ is $SU(5)$ coupling constant at the GUT scale $M_{G}\sim
2\times 10^{16}$ GeV [7].

In the shadow sector, the similar couplings of GUT gauge bosons should
destabilize the shadow baryon. However, taking into account that shadow
$\Delta$ baryon is much heavier than the ordinary proton,
$M_{\Delta}/m_{p}\sim 10^{6}$, its lifetime must be about 30 orders of
magnitude smaller than the proton lifetime. Hence we get $\tau_{\Delta}\sim
M_{G}^{4}(\alpha_{5}^{2}m_{\Delta}^{5})^{-1}\sim 10$ Gyr or so, with lifetime
comparable to the age of the Universe $t_{U}=14$ Gyr.

Figure 2: (a) Shadow neutrino spectrum produced by $\Delta$-barion decay and
subsequent decay of shadow pions (b) Spectrum of VHE neutrino events as
predicted in our model.

The principal decay mode of $\Delta$ baryon is in vector mesons,
$\Delta^{-}\rightarrow\rho^{-}_{a}+\bar{N}_{x}$, where the neutrino state
$N_{x}$ is generically a superposition of shadow neutrino flavor eigenstates.
However, taking into account the active sterile mixing $\sim\beta v/\alpha V$
due to operators (2), it contains also small admixture of ordinary neutrinos.
Each decay produces monoenergetic neutrinos $N_{x}$ with energies
$E_{a}=\frac{1}{2}M_{\Delta}(1-M_{a}^{2}/M_{\Delta}^{2})$, where $M_{0}$ is
the mass of $\rho^{-}$ meson and $M_{1,2,...}$ are the masses of its
excitations. In Fig. 2(a) the spectrum of neutrinos produced by decay of
galactic dark matter is shown by sharp peaks (solid blue) for
$M_{0},M_{1},M_{2}$ respectively being $3.0,3.1$ and 3.2 PeV. Due to close
degeneracy between the masses of $\Delta$-baryon and $\rho^{-}_{a}$ mesons,
the neutrino energies are $E_{a}\ll 1$ PeV while the most of initial energy
$=M_{\Delta}$ is taken away by mesons $\rho^{-}_{a}$.

Vector mesons promptly decay into mirror pion and photon,
$\rho^{-}_{a}\rightarrow\pi^{-}+\gamma^{\prime}$, and subsequent decay of the
pion produces the neutrino once again (solid red curves in Fig. 2(a)). Shadow
$\pi^{-}$ has two decay modes, two body $\pi^{-}\rightarrow E\bar{N}_{E}$ and
three body $\pi^{-}\rightarrow\pi^{0}E\bar{N}_{E}$. Interestingly, their
branching ratios are comparable which fact is intimately related to the value
$\Lambda^{\prime}\sim 100$ TeV [2]. Two body decay produces neutrinos with a
narrow energy spectrum concentrated around $M_{\Delta}/2\simeq 1.6$ PeV, while
the three body decay, due to smaller phase space, produces less energetic
neutrinos with a wide spectrum extending up to the value $E_{\rm
max}=M_{-}-M_{0}-M_{E}\simeq M_{U}-M_{D}-M_{E}\simeq 0.4$ PeV.

Fig. 2(a) shows the final spectrum including the neutrinos produced by the
decay of dark matter in the galactic halo, and extragalactic neutrinos
produced by the decay of cosmological dark matter at large redshifts. The
fraction of extragalactic neutrinos strongly depends on the decay time
$\tau_{\Delta}$ (in Fig. 2 we take $\tau_{\Delta}=10t_{U}$). Fig. 2(b) shows
how such a spectrum will be seen by the IceCube provided that active-shadow
neutrino mixing angles are order $10^{-5}$. Here the effective areas for the
neutrino detection by IceCube [1] and characteristic error bars in estimation
of neutrino energies (of about 13$\%$) are taken into account. Needless to
say, that the obtained spectrum of events look very much like the spectrum
observed by the IceCube [1]. The validity of our model will be tested with
increasing statistics by the IceCube collaboration.

## References

  * [1] M. G. Aartsen et al. [IceCube Collaboration], Science 342, no. 6161, 1242856 (2013); Phys. Rev. Lett. 113, 101101 (2014).
  * [2] Z. Berezhiani, R. Biondi, G. Di Panfilo and A. Gazizov, in preparation; Z. Berezhiani, talk at NOW 2014 [arXiv:1506.09040 [astro-ph.HE]]
  * [3] For a review, see e.g. Z. Berezhiani, Int. J. Mod. Phys. A 19, 3775 (2004); hep-ph/0508233.
  * [4] Z. Berezhiani and R. N. Mohapatra, Phys. Rev. D 52, 6607 (1995); Z. Berezhiani, A. D. Dolgov and R. N. Mohapatra, Phys. Lett. B 375, 26 (1996).
  * [5] E. K. Akhmedov, Z. Berezhiani and G. Senjanovic, Phys. Rev. Lett. 69, 3013 (1992).
  * [6] Z. Berezhiani and G. Dvali, Bull. Lebedev Phys. Inst. 5, 55 (1989); R. Barbieri et al., Nucl. Phys. B 432, 49 (1994). Z. Berezhiani et al., Nucl. Phys. B 444, 61 (1995).
  * [7] For a review, see e.g. P. Nath and P. Fileviez Perez, Phys. Rept. 441, 191 (2007).