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In the framework of quantum field theory, a graviton interacts locally with a quantum state having definite mass, that is, the gravitational mass eigenstate, while a weak boson interacts with a state having definite flavor, that is, the flavor eigenstate. An interaction of a neutrino with an energetic graviton may trigger the collapse of the neutrino to a definite mass eigenstate with probability expressed in terms of PMNS mixing matrix elements. Thus, gravitons would induce quantum decoherence of a coherent neutrino flavor state similarly to how weak bosons induce quantum decoherence of a neutrino in a definite mass state. We demonstrate that such an essentially quantum gravity effect may have strong consequences for neutrino oscillation phenomena in astrophysics due to relatively large scattering cross sections of relativistic neutrinos undergoing large angle radiation of energetic gravitons in gravitational field of a classical massive source (i.e., the quasi-classical case of gravitational Bethe-Heitler scattering). This graviton-induced

A theoretical extrapolation of the fundamental quantum mechanics concepts to Einstein’s gravity suffers from major difficulties with quantization of space-time, ultraviolet behavior and nonrenormalizability of the resulting theory (for more details, see [

Typically, in the standard quantum field theory framework which unifies three of four basic forces of nature, the quantum gravity effects are disregarded as being phenomenologically irrelevant at energy scales much smaller than the Planck scale,

In this paper, we propose a new approach for indirect experimental studies of (local) quantum gravity interactions based upon an effect of the large angle energetic gravitational Bremsstrahlung (or Gravi-strahlung, in short) off an astrophysical neutrino passing through an external classical gravitational potential on neutrino oscillation observables. This process, known as the gravitational Bethe-Heitler (GBH) process, can be considered in the quasi-classical approximation for large angle and/or large energy graviton emission; that is, the Born approximation is sufficient. Such a process may happen with a rather high probability, such as in the case of an astrophysical neutrino scattering off a massive source of classical gravitational field (like a star, black hole, dark matter distribution, or galaxy). In quantum mechanics, the latter process may serve as a direct

A graviton couples to the full energy-momentum tensor. From the quantum mechanical point of view, we work in the mass eigenstate basis where the Hamiltonian of local quantum gravitational interactions has a diagonal form and identify the particle mass eigenstates with gravitational eigenstates (due to equivalence of gravitational and inertial mass). In this approach, higher Fock states are created by the graviton creation operator acting on a particle mass eigenstate. By measuring the quasi-classical graviton cross section and deviations from it, we would be engaging in the first investigations of the deeper quantum gravity theory similar to how electroweak

Generically, weakly interacting neutrinos can be considered as an efficient carrier of information across the universe as they are not absorbed or scattered by interstellar mediums. In practice, this unique property of neutrinos enables us to utilize them for large-scale astrophysical “experiments,” such as searching for possible tiny signatures of Lorentz invariance violation [

The traditional source of decoherence typically referred to in astrophysical neutrino oscillations studies can be called

The role of classical Einstein’s gravity in quantum mechanics is under extensive consideration in the literature and may be sizeable under certain conditions. As was claimed in [

We expect elementary particles in the mass basis to be gravitational eigenstates of the Hamiltonian of quantum gravitational interactions in the same way as leptons and quarks are weak eigenstates in the flavor and CKM basis, respectively. The advantage of the neutrinos which we exploit here is that they interact via the weak force and that neutrino mass and flavor eigenstates are not the same and that they propagate at cosmological distances/times. For particles whose flavor and mass eigenstates are identical this technique would not work to identify that a graviton induced quantum mechanical interaction had happened, which means that the neutrino is a unique carrier of astrophysical quantum gravity interactions.

Consider first a relativistic neutrino state propagating in the gravitational potential of a supermassive black hole, dark matter halo, or another massive system. These not only are sources of strong gravitational fields but could also be significant sources of astrophysical neutrinos. Suppose now that at the quantum level a graviton interacts only with a definite mass state (or gravitational mass eigenstate)

The neutrino is “converted” to mass state with a probability

The amplitudes of typical quasi-classical gravity scattering processes which may lead to the quantum decoherence effect under certain conditions can be represented as follows:

In a sense, the quantum gravity-induced decoherence of a definite flavor state described above is in close analogy to the weak-induced decoherence of a definite mass state. For example,

In the case of vacuum neutrino oscillations, the traveling neutrino is not in a definitive mass eigenstate but is rather in a superposition of mass eigenstates which evolves when the neutrino travels in space-time. Then, with respect to the weak interactions, the nondiagonal

Contrary to the Penrose-Diósi effect of classical decoherence [

The proposed effect is also different from the standard propagation decoherence (see Figure

We would like to note that while the flux from quantum (gravitational) decoherence is a flux of pure mass eigenstates as noted, the important difference is that in the propagation decoherence case the flux is not of pure mass eigenstates, but rather decoherent (spatially separated) mass eigenstates. No quantum measurement of the state of these neutrinos has taken place, and the neutrino still exists as a superposition of mass states (just no longer with off diagonal elements in the density matrix). While these two situations are exactly the same when detected in the case where the flux is detected without passing through matter, in the case where the flux passes through matter, the regeneration which the neutrino flux experiences is different for the two cases. In the quantum gravitational decoherence case, the neutrino flux experiences regeneration as fluxes of neutrinos in pure mass eigenstates. However, in the propagation decoherence case, the neutrino flux experiences regeneration as a superposition of mass eigenstates; individual actual neutrinos continue to exist in a spatially separated quantum superposition of mass eigenstates. These spatially separated quantum superpositions experience the potential of the Earth. Simulation was done to demonstrate the possible size of this effect due to the difference in regeneration in the two cases (details below). In the simulation the neutrino is considered to have experienced propagation decoherence and the exact distance the neutrino travelled is not important (as long as it fulfils the conditions in [

The simulation presented here operates in the

The theory of neutrino propagation, including neutrino propagation in medium and neutrino propagation when the neutrino experiences propagation decoherence, is well presented in the papers by Beuthe [

For ease of discussion we will consider just two regimes, the vacuum and the Earth (with constant density) and two neutrino flavors. Due to the discontinuity at the Earth’s surface, the adiabatic formulas do not describe the neutrino propagation. However, the solution is to match the flavor conditions between the two regimes. The flavor at the point before the density jump is used to determine the initial state after the jump [

Propagation decoherence was studied in detail by Beuthe [

The condition for the wave packet separation to be complete is given explicitly by Farzan and Smirnov [

To determine the proper state we must consider the proper normalisation and phase for the states. In the two-flavor approximation, the probability is given by

This allows us to give a clear description of a produced

In the case of quantum decoherence we have the emission of a graviton off a neutrino mass state in the vacuum. An interaction of a neutrino with the graviton serves essentially as a measurement of the neutrino state, both its detection and production in quantum mechanical language. This tells us the condition on the graviton which must be true for the effect presented in this work; it is the condition of coherent production/detection of a neutrino as presented in [

The ratio of neutrinos which have undergone

Analogically, the ratio of neutrinos which have undergone

Now consider which quantum gravity processes the neutrino could possibly experience so as to experience the quantum decoherence effect in the astrophysical medium. As mentioned we will be considering quasi-classical gravity processes.

As is known the Coulomb field is measured by inserting a charged probe into it. From the quantum electrodynamics (QED) point of view, an electromagnetic scattering of a charged particle off the Coulomb field is due to an exchange of virtual photons (with small negative momentum transfer squared

Generically, in quantum electrodynamics (QED) the virtual photons may become real (produced on mass-shell) if one disturbs the field pumping energy into it. This is the physical reason for photon Bremsstrahlung in QED. Specifically, the standard Bethe-Heitler scattering in electrodynamics demonstrates that only an accelerated charge emits real photons (corresponding to electromagnetic wave in the classical limit of multiple soft photon radiation). Likewise, in the quasi-classical gravity framework, the virtual graviton, as a quantum of the gravitational field of a static massive object, may turn into the real one (corresponding to gravitational wave in the classical limit of multiple soft graviton radiation) if the source of the gravitational field is accelerated or, in general, when the energy-momentum tensor experiences disturbances.

Possible sources of real gravitons in the universe include active galactic nuclei (AGN), binary systems, supernova explosions (SNe), primordial black holes collisions, compact star/black holes binaries, quantum bremsstrahlung of gravitons of particles scattering off a massive object, black hole (BH) evaporation, relic isotropic gravitational background from the early universe, inflation, phase transitions in the primordial plasma, and the decay or interaction of topological defects (e.g., cosmic strings). For details and references, see [

Consequently, in the cosmological medium a neutrino can scatter either off a classical gravitational potential with accompanying radiation of an energetic real graviton off the scattered neutrino (e.g., Bethe-Heitler-type scattering) or off real graviton in the astrophysical medium (e.g., Compton-type scattering). Let us consider both cases and conditions for initiation of the quantum neutrino decoherence in more detail.

In fact, all elementary particles, including neutrinos, when traveling in the vicinity of massive objects (sources of classical gravitational field) can emit real gravitons with a certain energy spectrum. This process has a straightforward QED analog of a photon emission in relativistic electron scattering off the Coulomb field of a heavy nucleus mentioned above, the Bethe-Heitler process at the Born level. Even though the energy spectrum of radiated real gravitons is peaked in the forward direction and in the infrared limit (corresponding to forward radiation of classical gravitational waves), there is a nonnegligible probability to radiate

The decoherence of the neutrino at the quantum level can only be initiated by hard energetic interactions with relatively hard gravitons whose energies exceed the mass difference between different mass states

A soft graviton with energy lower than the difference between mass states will be unable to resolve the individual mass eigenstates in this superposition and will instead couple to the whole energy-momentum tensor of the flavor state, nonlocally, which is the classical general relativity limit. In the latter case quantum decoherence is not triggered, and the effect will be as discussed in [

The Born-level calculation is good first order approximation in the case of off-forward hard graviton emissions at large angles relevant for the quantum decoherence effect; this is the reason why one can disregard higher-order radiative corrections which are highly suppressed (by extra powers of the Planck mass) as long as one cuts off the problematic but uninteresting infrared/collinear parts of phase space. As was previously shown in [

In the considered GBH case, shown in Figure

The quasi-classical gravity processes which destroy the coherence of the neutrino flavor eigenstate (

Shown are diagrams of neutrino propagation in the quantum field theoretical description (such as found in [

Diagram for neutrino propagation with quantum decoherence

Diagram for neutrino propagation

The GBH cross section has initially been calculated for the gravitational scattering of scalar particles with ^{−4}, so there is no significant suppression of the cross section for relativistic neutrinos. This is a particle physics magnitude cross section which naively implies particle physics size impact parameters. Larger impact parameters would exist for larger masses, such as the dark matter halo. The above cross section is integrated over impact parameter, and it may be instructive to look into differential cross section in order to find the probability of this process as a function of distance to the massive astrophysical body. For the current study it suffices to note that such probability can potentially be significant.

It is worth noticing that the Bethe-Heitler calculation in QED to first order gives the correct cross section for the photon Bremsstrahlung for extended objects such as a nucleus as shown in [

In Figure

Differential cross section of the gravitational Bethe-Heitler scattering of neutrino off a massive object, for example, a black hole (BH) in radiated graviton energy

This observation strongly suggesting the importance of the quantum decoherence initiated by interactions with such energetic real gravitons. The latter source of decoherence does not have a classical interpretation. As we have already mentioned, due to a quantum mechanical nature of a

Of course, the

The cross section of the considered GBH process can be enhanced for the galactic center (~10^{6}–10^{9} solar masses) or the dark matter halo (~10^{20}–10^{24} solar masses). It can also be enhanced for ultra-relativistic neutrinos which are potentially detectable at neutrino observatories such as IceCube and Super-K. As is our main result, we notice that the GBH scattering may cause the quantum decoherence of astrophysical neutrinos and this effect can be measured via neutrino flavor composition measurements. A massive classical source of the gravitational field may not necessarily be a black hole, but any compact star or, in general, any bound gravitational potential induced by continuous matter distribution in the galactic disk and halo.

Due to rather large cross sections it can be that most of the astrophysical neutrinos which are observed at the Earth from a given direction and have passed in close vicinity of a massive object would have experienced the quantum decoherence due to a graviton-induced scattering. In other words, the probability for a given neutrino in a superposition of mass states

A precise theoretical calculation for

Another possibility for quasi-classical gravity induced interactions with neutrino participation is shown in Figure

As presented above, the neutrino in a mass eigenstate does not oscillate unless it scatters off ordinary matter via a weak channel which will cause it to be in a flavor eigenstate. It is likely that the

Previously, in [

Here we consider a very massive source of strong gravitational fields like cluster of stars (e.g., the center of our Galaxy) or a dark matter halo as a good example of a graviton “detector.” This section provides predictions for such an extreme large-scale quasi-classical gravity measurement.

As we have demonstrated above, the probability of an individual (elementary) act of the “quantum gravity measurement” defined by the graviton-neutrino cross section can be rather large due to a large GBH cross section and there may be scenarios where it should not be neglected.In particular, utilizing the dense region of stars and black holes in the galactic center (GC) as our “graviton detector” in the above sense, one could expect that a significant fraction of neutrinos passing by the dense region would have experienced the GBH scattering. Then since many of the neutrinos are now in a mass eigenstate, they will no longer undergo flavor oscillation. Due to the neutrino existing in a mass eigenstate during propagation, further graviton rescattering would not constitute additional quantum measurements of an “undetermined” quantum state. Depending on the astrophysical process, one might favor energies of neutrinos where the neutrino oscillation may not be suppressed due to the MSW effect where the neutrino exists in a single mass eigenstate, so that the graviton-induced effect would be cleaner. We suggest that this effect could be tested in neutrino telescopes and observatories by looking at the galactic center neutrino flavor composition and comparing it to the composition expected without quantum gravitational decoherence. It might be possible that close, “standard candle,” neutrino emitters in other parts of the sky provide a flux of neutrinos which have not undergone quantum gravitational decoherence (note: a very similar effect should take place in flavor oscillations in the neutral kaons

The general formula for the number of electron type neutrinos observed from an electron type source in the vacuum is

If all neutrinos have interacted with at least one graviton, that is, fixing

Here the difference due to the Earth matter effect in the observed number of electrons is plotted (assuming an initial flux that only contains electron neutrinos). The difference is defined to be the ratio of the observed number of electron type neutrinos for the case where there is a maximal Earth matter effect (impact parameter is

As one notices in Figure

For illustration in this calculation we use a constant, maximal probability for quantum decoherence case,

The neutrino flux spectrum from astrophysical sources is still being modeled [

This is our prediction for the quantum gravity-induced effect on the detected flavor composition in the “maximal quantum decoherence” scenario. While numerically the effect can be very large, one would certainly need to have a good understanding of all the other statistical and systematical uncertainties for a possible measurement of the dependence of

In conclusion, we have considered a quasi-classical gravity process, the gravitational Bethe-Heitler scattering of a neutrino off a massive object accompanied by an energetic graviton radiation, which can have a rather large cross section proportional to the mass squared of the classical source. Due to hard gravitons interacting with a neutrino mass eigenstate only, opposite to weak bosons which interact with a flavor eigenstate only, the considered process is a measurement of the incoming neutrino flavor state at the quantum level, causing its decoherence in a different manner than the process which can be called propagational decoherence or other sources of decoherence. This quantum decoherence affects astrophysical neutrino behavior. Namely, quantum decoherence can be considered as a specific quasi-classical gravitational measurement of the neutrino propagating state, which changes the behavior of the neutrino in the presence of a potential (such as the Earth matter effect) compared to the traditional source of decoherence known as propagation decoherence, which can be observed in the neutrino flavor composition in an Earth based detector (see the Appendix).

This enables the utilization of neutrinos traveling across the Galaxy as a source of information about the graviton-induced interactions they might have experienced on their journey to Earth. Specifically, the measured probability to find a given flavor component in the neutrino flux coming from a vicinity of a super massive black hole or another super massive object (galactic center or dark matter halo) will be different from the corresponding probability measured from a source of neutrinos where the neutrinos never pass near a massive system. In the case where no astrophysical neutrinos can be identified which have not interacted with a gravitational potential, the flavor composition can be compared to the expectation for the Earth matter effect which can be determined using reactor, atmospheric, and accelerator neutrinos. We have explicitly demonstrated that the maximal difference corresponding to an assumption that all of the detected neutrinos have experienced an interaction with a graviton, that is,

Thus, the probability for a neutrino state to interact with at least one energetic graviton,

The difference between propagation decoherence and quantum gravitational decoherence is a crucial component in our study and so we provide short summary. In the classical case [

In the quasi-classical gravity case (this study), the neutrino is produced and detected in distinct flavor (at the astrophysical source and the earth detector); however, the neutrino exists in a single mass state due to being “observed” by the emitted graviton which distinguishes which mass state the neutrino exists in. We describe this effect as

While we give an explicit calculation of the GBH process to demonstrate that the emission of a hard graviton via gravitational Bremsstrahlung is relatively large and used this fact to motivate discussion of a maximal possible signature, that is,

Having all that in mind, as a natural starting point in this very first paper we would like to present the basic concept/idea of quantum decoherence due to large angle neutrino-graviton interactions (Gravi-strahlung) in strong gravitational fields and its possible effect on neutrino flavor observables. In this paper we report on our preliminary study of such a graviton-induced effect on neutrino oscillations and motivate future studies in this direction. We plan to improve our simulation with fluxes and the astrophysical medium in a future study. The possibility that

The program used to produce Figure

In the review process, [

The central effect of the graviton observation of the neutrino which is utilised in this proposed measurement is that a superposition is different than a classical ensemble of states. Distinguishing these two things is of key interest to the Quantum Information and Quantum Foundations communities and they have been shown to be different in experiments which investigate Bell Inequalities. To quote a member of the Quantum Foundations community who are also interested in distinguishing the situation where the particle is in a superposition (neutrino which has not undergone an interaction with a graviton, in our case) and those where the wave function has collapsed (in the mass basis in our case, where the neutrino has undergone an interaction with a graviton) [

What is required to distinguish these two cases is for the phase between the states to not be rapidly varying. In the case where interference phenomena may be observed (the phenomena of neutrino oscillation for neutrinos) this is obviously the case. For the case where the particle is still coherent but the phase difference between states is rapidly varying, it is obvious that it is impossible to differentiate a classical ensemble from a superposition. For neutrinos this is the situation where there is still overlap between the states but the energy resolution of the detector is not good enough to observe the oscillation and has been talked about in [

However, there is an additional case where the neutrino in the flavor basis ceases to oscillate. The states no longer overlap. This is the case of propagation decoherence and generally the case for astrophysical neutrinos. In this case, in the flavor basis, the neutrino has a constant phase difference between (matter) states. If we measure this state without making any changes to it based on the phase difference we get the same result as if we measure a classical ensemble of states (the quantum gravity decoherence case). However, if we modify this (flavor) state by sending it through matter, the constant phase difference is changed differently than the classical ensemble of states and the neutrino can be distinguished as being in a classical ensemble of states (or having undergone quantum gravity decoherence) rather than a separated superposition. The boundary between matter regimes has a finite width and so the (flavor) state is going to have a constant phase difference (for large enough separations), independent of the energy resolution of the final (flavor) detector which collapses the wave function.

This can be clearly described in the density matrix formalism. In the formalism, the evolution of the density matrix is given by

For the case of 2 neutrinos flavours, the constant phase difference for a neutrino in a separated superposition is obvious and is given by

The authors declare that there is no conflict of interests regarding the publication of this paper.

Stimulating discussions and helpful correspondence with Sabine Hossenfelder and Alexei Vladimirov are gratefully acknowledged. Jonathan Miller was supported in part by PROYECTO BASAL FB 0821 CCTVal and Fondecyt (Grant no. 11130133). Roman Pasechnik was supported in part by Fondecyt (Grant no. 1090291) and the Crafoord Foundation (Grant no. 20120520). Roman Pasechnik is grateful to the “Beyond the LHC” Program at Nordita (Stockholm) for support and hospitality during a portion of this work. This research was supported in part by the National Science Foundation under Grant no. NSF PHY11-25915.