Flavor ratios of extragalactical neutrinos and neutrino shortcuts in extra dimensions

The recent measurement of high energy extragalactic neutrinos by the IceCube Collaboration has opened a new window to probe non-standard neutrino properties. Among other effects, sterile neutrino altered dispersion relations (ADRs) due to shortcuts in an extra dimension can significantly affect astrophysical flavor ratios. We discuss two limiting cases of this effect, first active-sterile neutrino oscillations with a constant ADR potential and second an MSW-like resonant conversion arising from geodesics oscillating around the brane in an asymmetrically warped extra dimension. We demonstrate that the second case has the potential to suppress significantly the flux of specific flavors such as $\nu_\mu$ or $\nu_\tau$ at high energies.

Recently the IceCube Collaboration has reported the detection of 28 neutrino events in the energy range of 30 TeV up to 2 PeV. Atmospheric neutrinos as the source of this signal are ruled out at more than 4 sigma [1] and it has been argued that these neutrinos are extra-galactic in origin [2]. Consequently this result allows for the first time to study the flavor composition of high energy extra-galactic neutrinos and thus opens up a window onto neutrino properties exhibiting themselves at extreme energies and propagation distances.
An apparent deficit of muon track events in the IceCube data has recently been discussed as a possible indication for new physics beyond the Standard Model [3][4][5], albeit this anomaly is so far not statistically significant [6]. In spite of the seeming deficit of muons, the flavor analysis of IceCube events finds that the data are consistent with the expected canonical mix of 1:1:1 [7]. On the other hand a careful best fit analysis finds that the best fit is 1:0:0 [5] which is very hard to understand without some new physics.
In this paper we discuss how scenarios in which the simple dispersion relation E 2 = p 2 + m 2 is altered due to sterile neutrino shortcuts in extra dimensions [8] can affect these flavor ratios. For example, sterile neutrinos propagating in spacetimes with an asymmetrically warped extra dimension u [9][10][11] propagate on geodesics oscillating around the brane [12] and experience a shorter propagation time [8] which can be parametrized by a shortcut parameter defined as the normalized difference of propagation times on the brane and in the bulk, ≡ δt/t.
As a consequence, the effective neutrino masses and mixings are altered in a way similar to what happens when neutrinos propagate inside matter [8], so that the effective two-flavor oscillation amplitude becomes and the Hamiltonian is altered by the additional term where E Res denotes the resonance energy Further details of such models have been worked out in [13][14][15][16] and a similar effect due to the interaction with the matter potential of intergalactic dark matter has been discussed in [17].
Astrophysical neutrinos are typically assumed to originate from a pion source where the dominant number of neutrinos is produced in the decays of charged pions and kaons resulting from high energy proton-proton and few proton-photon collisions [18,19], ν e + ν µ +ν µ . In this case the initial flavor ratio at the source is where Φ α denotes the integrated flux of both neutrinos and anti-neutrinos. To estimate the flavor ratios after propagation from the source to the detector we follow [20].
As neutrino oscillations decohere over the large propagation distances the flavor conversion is properly described by the mean of the oscillation probability so that the oscillatory term averages out to sin 2 (x/L osz ) ∼ 1/2 and the conversion probability P αβ and consequently the normalized neutrino fluxes depend only on the mixing matrix elements, Adopting a 3+1 scenario, U αi denotes a 4 × 4 mixing matrix including a sterile neutrino mixing with ν µ and ν τ . The flavor ratios at the detection point are then given by (Φ νe : Φ νµ : Φ ντ : Φ νs ), with the neutrino fluxes Φ β and β ∈ {ν e , ν µ , ν τ , ν s }.
In order to illustrate the effect of neutrino shortcuts we concentrate on two concrete scenarios for sterile neutrinos: • A scenario with light sterile neutrinos fitting the LSND, MiniBoone and Gallium anomalies with an δm 2 of order 1 eV 2 . Here we use sin 2 (2θ) 0.12 (sin 2 θ 0.03) and δm 2 1 eV 2 according to the global analysis presented in [28].
For an estimation how large the portion of the neutrino spectrum affected by the resonance is it is helpful to calculate the Full Width in energy at a fraction f of Maximum (FWfM) as which, for small θ reduces to 2θ 1−f f . Thus, the resonance is very narrow for a small mixing angle. For example, Full Width at Half Max is δE(FWHM) = 2 θ E Res for a small angle.
Assuming a resonance energy of 300 TeV yields δE(FWHM) = 104 TeV (35%), 95 MeV (3 · 10 −2 %) for the light sterile and νMSM scenarios above, respectively. This demonstrates that the affected portion of the neutrino spectrum depends crucially on the active-sterile neutrino mixing Figure 1: Neutrino flavor rates for a pion source as a function of E/E Res : Φ e (red), Φ µ (green), In Fig. 1 the resonance peaks are shown as a function of E/E Res in the light sterile neutrino (left panel) and νMSM scenario (right panel). Depending on the active-sterile mixing angle a significant portion of the spectrum can be affected. In Table 1 we summarize the change in flavor ratios at resonance energies in the light sterile neutrino for different astrophysical sources and mixing scenarios in addition to the one presented above, the pion source with (ν µ , ν τ ) − ν s mixing. The sources are considered to be ideal, based on processes described in [20]. Also given are the flavor The breaking of the original µ − τ symmetry in the fluxes obtained results from the additional mixing of both flavors with the sterile neutrino. A similar effect due to a non-vanishing sin θ 13 = 0 has been found in [21].
A particularly interesting limit exists in the case where the change in the dispersion relation occurs adiabatically so that an MSW-like [30][31][32] resonant conversion can occur (for details see [33]). In the following we assume a single non-zero mixing angle between ν µ / ν τ and the sterile neutrino ν s and concentrate on a 1+1 active-sterile neutrino pattern. Consequently,   P νa→νa (x) = cos 2 θ · cos 2θ + sin 2 θ · sin 2θ (14) P νa→νs (x) = sin 2 θ · cos 2θ + cos 2 θ · sin 2θ (15) Here the effective mixing angleθ is given by The sterile neutrino propagates on the geodesic see [12,33] for details. The periodic length of the geodesic l is a function of the initial conditions for the velocities parallel and perpendicular to the brane and of the warp factor k [12]: whereẋ 0 andu 0 denote the initial speed of the sterile neutrino along and perpendicular to the brane, respectively. We assume that l is not much larger than typical travel distances of the high energetic neutrinos from extragalactic sources.
The limit of resonant conversion applies as long as the change in the dispersion relation is adiabatic, corresponding to the following condition: where the shortcut parameter is given by This is the condition for adiabatic transition at resonance. If the condition is fulfilled at resonance, it is fulfilled everywhere else. As the derivative of the shortcut parameter with respect to the baseline x at resonance becomes rather complicated, for sake of simplicity we adopt the maximum value of d dx at x = 0 as an upper bound: Consequently, if the condition is fulfilled, the adiabatic condition is fulfilled as well.
In order to calculate the survival probability we just average over the periodic length l, which with the help of (16) reduces tō P = 1 2 (I cos 2θ + 1) where This expression yields an energy dependent transition probability. As can seen in fig. 2, the averaged survival probability breaks down at the resonance energy E Res and is extremely low for higher energies. This corresponds to the fact, that level crossing occurs only at higher energies. Assuming an adiabatic change in the dispersion relation the high and low energy limits for the transition and survival probabilities can be calculated as follows, The survival probabilityP only depends on the product of the geometric parameters k and l.
As can be seen the fall-off of the survival probability is shifted to higher energies for smaller factors kl. This corresponds to a smaller ratio of the initial velocities |u 0 | x 0 . For smaller kl the sterile neutrino thus plunges less deeply into the bulk and thus is less affected by the shortcut, therefore it needs a have higher energy to experience level-crossing.
Fixing the product kl and varying the vacuum mixing angle does not affect the position of the tilt in the conversion probability but makes the function steeper for smaller vacuum mixing angles.
For a given set of mixing parameters δm 2 , sin 2 (2θ) and a given energy E we obtain the following constraints for the geometric parameters k and l:  If the product kl is too small, no level crossing will occur, and consequently no resonant conversion will emerge. On the other hand, if the product k 2 l is too large, the adiabatic condition will not be fulfilled and the level crossing will not result in resonant conversion.
Adopting again the parameter sets for light sterile and νMSM neutrinos discussed above, and requiring a transition probability > 75 % we arrive at the conditions under which the major part of muon and tau neutrinos are converted into sterile neutrinos. Numerical values for various parameter sets and neutrino energies are given in table II. Fig. 3 displays the bounds obtained on k and l. As can be seen the periodic length l has to be very large to fulfill both conditions. Also taken into account is the constraint resulting from the fact that atmospheric and solar neutrino data are in conflict with significant oscillation or conversion into sterile neutrinos. In order to have atmospheric and solar neutrinos unaffected by the change in the dispersion relation we assume that the resonance length, is much larger than the distance from the Earth to the sun, but much smaller than the diameter of the Milky Way, 10 6 ly = 5 · 10 25 1 eV > x res > 8 · 10 17 1 eV = 1.6 · 10 −5 ly (29) which leads to 1 64 10 −17 1 eV This ensures that neutrinos from extra-terrestrial astrophysical sources indeed pass through the resonance length, even if the periodic length of the geodesic can assume beyond horizon values. As can be seen in Fig. 3 large parts of the parameter space are consistent with all constraints and allow for a significant reduction of the rates of astrophysical muon and tau neutrinos.
In summary, we have discussed the effect of altered dispersion relations for sterile neutrinos propagating in extra dimensions on the flavor ratios of astrophysical neutrinos. We have confined our discussion to two important limits: the case where the extra-dimensional shortcut can be described in terms of a constant effective potential and the case where the potential changes adiabatically giving rise to an MSW-like resonant flavor conversion. In the first case one gets significant distortions of the expected flavor ratio around the resonance region whose width depends strongly on the active-sterile neutrino mixing. In the second case strong deviations, including a flavor ratio of 1:0:0 are possible over a large range of energies and baselines. The flavor ratios of high energy astrophysical neutrinos can thus qualify as a prime probe for non-standard dispersion relations originating from exotic physics such as shortcuts in extra dimensions.

I. ACKNOWLEDGEMENTS
We thank Danny Marfatia for useful discussions. HP was supported by the 'Helmholtz Alliance for Astroparticle Physics HAP' funded by the Initiative and Networking Fund of