Probes of Lorentz violation in neutrino propagation

It has been suggested that the interactions of energetic particles with the foamy structure of space-time thought to be generated by quantum-gravitational (QG) effects might violate Lorentz invariance, so that they do not propagate at a universal speed of light. We consider the limits that may be set on a linear or quadratic violation of Lorentz invariance in the propagation of energetic neutrinos, $v/c=[1\pm (E/M_{\nu \mathrm{QG}1})]$ or $[1\pm (E/M_{\nu \mathrm{QG}2}{)}^2]$, using data from supernova explosions and the OPERA long-baseline neutrino experiment. Using the SN1987a neutrino data from the Kamioka II, IMB, and Baksan experiments, we set the limits $M_{\nu \mathrm{QG}1}>2.7(2.5)\times {10}^{10}\mathrm{GeV}$ for subluminal (superluminal) propagation and $M_{\nu \mathrm{QG}2}>4.6(4.1)\times {10}^4\mathrm{GeV}$ at the 95% confidence level. A future galactic supernova at a distance of 10 kpc would have sensitivity to $M_{\nu \mathrm{QG}1}>2(4)\times {10}^{11}\mathrm{GeV}$ for subluminal (superluminal) propagation and $M_{\nu \mathrm{QG}2}>2(4)\times {10}^5\mathrm{GeV}$. With the current CERN neutrinos to Gran Sasso extraction spill length of $10.5\mu \mathrm{s}$ and with standard clock synchronization techniques, the sensitivity of the OPERA experiment would reach $M_{\nu \mathrm{QG}1}\sim 7\times {10}^5\mathrm{GeV}$ ($M_{\nu \mathrm{QG}2}\sim 8\times {10}^3\mathrm{GeV}$) after 5 years of nominal running. If the time structure of the super proton synchrotron radio frequency bunches within the extracted CERN neutrinos to Gran Sasso spills could be exploited, these figures would be significantly improved to $M_{\nu \mathrm{QG}1}\sim 5\times {10}^7\mathrm{GeV}$ ($M_{\nu \mathrm{QG}2}\sim 4\times {10}^4\mathrm{GeV}$). These results can be improved further if a similar time resolution can be achieved with neutrino events occurring in the rock upstream of the OPERA detector: we find potential sensitivities to $M_{\nu \mathrm{QG}1}\sim 4\times {10}^8\mathrm{GeV}$ and $M_{\nu \mathrm{QG}2}\sim 7\times {10}^5\mathrm{GeV}$.

Figure 1

The distribution ${\tau }_{\min }$ of 1000 Monte Carlo simulations of the KII data on neutrinos from SN1987a, including the smearing due to energy uncertainties.

Figure 2

The regions of parameter space excluded by SN1987a, for subluminal (dashed lines) and superluminal (solid lines) linear (left panel) and quadratic (right panel) models.

Figure 3

The time distribution of events predicted by our Monte Carlo simulation for the case of subluminal Lorentz violation at the mass scales $M={10}^{10}\mathrm{GeV}$ and $M={10}^{11}\mathrm{GeV}$.

Figure 4

The neutrino energy spectra from the Livermore simulation [22].

Figure 5

The ECF linearly weighted with energy from one realization of the simulated time profile of Fig. 3 with neutrino energies smeared by MC applying to the expected energy resolution of SK, for the case of linear energy depending neutrino velocity.

Figure 6

The distribution of the positions of the maximums from fits of ECFs like in Fig. 5 of 1000 realizations of the simulated time profile of Fig. 3 with neutrino energy smeared by MC.

Figure 7

The time profile of 600 events, which could be detected by SK from a future extra galactic supernovae, occurred at the distance 40 kpc from the Earth. The events are simulated with respect to the energy spectrum given in Fig. 4 with linear energy depending propagation effect, encoded at the level ${\tau }_1=55\mathrm{s}·{\mathrm{MeV}}^{-1}$.

Figure 8

The dispersion (2) versus $\tau$ from one realization of the simulated time profile of Fig. 7 with neutrino energies smeared by MC applying to the expected energy resolution of SK, for the case of linear energy depending neutrino velocity.

Figure 9

The expected CNGS neutrino beam energy spectrum [20].

Figure 10

A superposition of the production times of neutrinos in CNGS spills reflects the bunch structure of the CNGS beam [20].

Figure 11

The time structure of events in the CNGS beam, including a 100 ns timing uncertainty without (upper panel) Lorentz violation in neutrino propagation, and with (lower panel) a linearly energy-dependent time delay during neutrino propagation at the level of $\tau =5\mathrm{ns}/\mathrm{GeV}$.

Figure 12

The mean arrival times of 1000-event slices with increasing energies without Lorentz violation in the neutrino propagation (asterisks) and with the effect of a time delay during neutrino propagation at the level of $\tau =5\mathrm{ns}/\mathrm{GeV}$ (triangles). The latter corresponds to $M_{\nu \mathrm{QG}1}=4.8\times {10}^5\mathrm{GeV}$. One particular simulation of the OPERA experiment is shown: others are similar, exhibiting the expected statistical fluctuations.

Figure 13

The measured shifts in the average arrival times of neutrinos in 1000-event slices with increasing energies, assuming a time delay during neutrino propagation at the level of $\tau =5\mathrm{ns}/\mathrm{GeV}$.

Figure 14

The 68% (dashed-dotted line), 95% (dashed line), and 99% (solid line) sensitivity contours for the case of a linear energy-dependent fit (8).

Figure 15

The same as in Fig. 14 calculated for the sensitivity of the quadratically energy-dependent fit (10).

Figure 16

The left (right) spill edges fitted using 20 000 detector events for scenarios with a linear energy dependence of the neutrino velocity. The solid gray (red) line is the reference histogram, while the points represent the shifted data. The solid black line represents the probability distribution function.

Figure 17

A particular realization of the bunch structure with $\approx 1\mathrm{ns}$ relative time uncertainty incorporated. The histogram is binned with a resolution suitable for resolving the bunch structure.

Figure 18

The Gaussian fit to the CCF calculated for the case of a linear energy dependence with time smearing $\approx 1\mathrm{ns}$.

Figure 19

The same as in Fig. 18 for the quadratic energy dependence.

Figure 20

The profile of the CCF calculated with a 2 ns time resolution for the case of linear energy dependence in neutrino propagation.

Figure 21

A simulated realization of the bunch structure for rock events, incorporating a timing uncertainty $\approx 1\mathrm{ns}$. The histogram is binned with a resolution suitable for resolving the bunch structure.

Figure 22

The CCF for rock events with time resolution $\approx 1\mathrm{ns}$ in the case of linear energy dependence, compared with a Gaussian fit.