Advances in High Energy Physics

Volume 2015, Article ID 381569, 15 pages

http://dx.doi.org/10.1155/2015/381569

## Quasi-Classical Gravity Effect on Neutrino Oscillations in a Gravitational Field of a Heavy Astrophysical Object

^{1}Departamento de Física, Universidad Técnica Federico Santa María, Casilla 110-V, 2930123 Valparaíso, Chile^{2}Theoretical High Energy Physics, Department of Astronomy and Theoretical Physics, Lund University, Sölvegatan 14A, 223-62 Lund, Sweden

Received 1 December 2014; Revised 4 March 2015; Accepted 1 May 2015

Academic Editor: Srubabati Goswami

Copyright © 2015 Jonathan Miller and Roman Pasechnik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

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 *decoherence* is compared to *decoherence* due to propagation in the presence of the Earth matter effect. Based on this study, we propose a new technique for the indirect detection of energetic gravitons by measuring the flavor composition of astrophysical neutrinos.

#### 1. Introduction

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 [1, 2] and references therein). A wealth of theoretical studies have been presented in the literature and many different quantum gravity models have been developed. However, no conclusive statement about the true quantum nature of gravity has been made. Only a real experiment can settle the longstanding confusion between the different approaches and provide guidance in developing the correct underlying theory.

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, GeV. Moreover, due to enormous suppression, quantum gravity effects are often referred to as nearly unobservable [3–5]. While observing a single graviton directly may be impossible, it is not impossible to find an indirect evidence for quantum gravity. For an overview of potential phenomenological opportunities for indirect signatures of quantum gravity, see [6–9]. Nevertheless, our understanding of the quantum nature of gravity suffers from the lack of accessible sources of information.

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* quantum measurement* of the microscopic properties of the gravitational field at astrophysical scales.

*Quasi-Classical Gravity*. In the limit of weak gravity, the quasi-classical approximation to quantum gravity is a valid framework. In this case, the graviton field is a correction determined on the flat Minkowskian background and the metric operator in the Heisenberg representation is given by . Here, the -number part is the Minkowski metric and is the graviton arising after the quantization procedure. The Einstein-Hilbert action provides the mechanism for* virtual gravitons* to propagate in the flat space-time and to interact with one another in the quantum case as an analog of the standard QED picture of the Coulomb field around an electric charge. These virtual gravitons should be distinguished from* real gravitons* which are radiated off an accelerated massive body and their coherent wave packets correspond to gravitational waves in the classical limit. A “cloud” of virtual gravitons around a static massive body can be reinterpreted geometrically in terms of a deviation from the flat metric (or curvature) in Einstein’s classical relativity [10, 11] (the background must be chosen to be flat since only in this case is it possible to use the Casimir operators of the Poincaré group and show that the quanta have spin two and rest mass zero, thus being identified as gravitons).

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 - measurements provided the first investigation of the deeper quantum Weinberg-Salaam theory.

#### 2. Decoherence of Neutrino State

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 [12], testing general relativity [13] and quantum mechanics [14–16], testing the equivalence principle [17, 18], and testing minimal length models [19, 20]. Ultimately, it is possible to identify an* extraterrestrial large-scale quantum experiment* where neutrinos “change” their quantum state due to a local quantum gravity process (in terms of local graviton coupling to a fundamental matter particle) and further convey information about such a process unchanged through the cosmological medium to the Earth.

##### 2.1. Propagation Decoherence

The traditional source of decoherence typically referred to in astrophysical neutrino oscillations studies can be called* propagation decoherence*. This is when the distance that a neutrino travels exceeds the neutrino oscillation length. In this case, the neutrino mass states have separated so that they no longer interfere at large distances from the production point. This source of decoherence depends on the energy resolution of the detection process, the energy of the neutrino, the masses of the neutrino mass states, and other details of the production and detection processes. In neutrino experiments, the time between neutrino production and detection is normally not measured. In a real experiment, this means that beyond the neutrino oscillation length the propagating neutrino mass states no longer interfere during the interaction process in a detector [21, 22]. For cosmic/astrophysical neutrinos, in some cases and for some processes, this decoherence effect is irrelevant [23, 24].

##### 2.2. Classical Diósi-Penrose Decoherence

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 [25], the gravity-induced quantum state reduction can be tested by observing the neutrino flavor oscillations at cosmological distances, while in [26] it was regarded as practically undetectable. This classical gravity effect on real-time evolution of a quantum state composed of several mass eigenstates was initially considered by Diósi [27–29] and Penrose [30]. In the classical gravity limit, the latter can be approximated by a change in the phase of the flavor wave function which appears mainly due to a nondegeneracy of neutrino mass eigenstates; that is, , where is the mass of the mass eigenstate . This is caused by different mass states traveling along different geodesics in curved space-time and the whole effect gradually accumulates over large cosmological distances [31]. This is the essence of* classical decoherence* of a neutrino flavor state which is typically regarded as a probe for neutrino wave function collapse models and, more generally, alternatives to conventional (linear) quantum mechanics [32]. Instead, we consider another possible decoherence mechanism of a neutrino flavor state triggered* at the quantum level* by a single local graviton-neutrino interaction. Let us discuss this phenomenon in detail.

##### 2.3. Quantum Decoherence

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) or . This is equivalent to saying that definite mass eigenstates (the propagating states) are conserved by the quantum gravity Hamiltonian while superpositions, such as the flavor eigenstates, are not [33]. Note, the astrophysical neutrinos are initially produced in electroweak processes (e.g., in SNe processes) in a definitive flavor state, or , which are coherent superpositions of mass eigenstates. In an astrophysical environment, a high-energy graviton can interact only with a definite mass component of the neutrino wave function, thus causing* quantum decoherence* of the neutrino which is in a superposition of mass states, effectively “converting” it into a definitive mass eigenstate. This neutrino is quantum mechanically observed as being in a definite mass state. This means that between the production in an AGN or supernova (SNe) or other astrophysics source and the detection in an Earth based detector, the neutrino which was observed by the graviton exists in a definite mass state. This is independent from propagation decoherence.

The neutrino is “converted” to mass state with a probability , given in terms of the corresponding wave function which projects out a flavor state onto a mass state and is typically expressed in terms of the corresponding PMNS mixing matrix element, . The considered effect is different from other known classical decoherence sources emerging due to a mere propagation (without a hard graviton radiation) in classical gravitational potential and/or neutrino propagation in flat space-time. The effect under consideration is a straightforward consequence of fundamental time-energy uncertainty relation for the real hard Gravi-strahlung and should be taken into account in studies of astrophysical neutrino oscillations.

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: Here, is a source of strong classical gravitational fields, such as a massive star or a black hole. The first amplitude corresponds to the gravitational Compton scattering of a neutrino mass state off a real graviton in the medium, the second amplitude represents the GBH scattering of a neutrino in gravitational mass state off a classical heavy source (with energetic graviton radiation). Clearly, a mass eigenstate “produced” in this interaction due to decoherence does not undergo oscillation until it interacts weakly with normal matter (e.g., in an Earth detector) by means of , -exchange. Therefore, quantum decoherence may have an nonnegligible effect on neutrino oscillation observables, along with other existing sources of classical decoherence and medium matter effects [34, 35]. Explicitly, oscillation characteristics of neutrinos coming from, for example, a vicinity of the Galactic Center, may differ from vacuum oscillations. The latter case could be where a source of neutrinos is “nearby” but where there is no massive objects between the source and the Earth (nor significant variations in dark matter density). Such neutrinos, if identified, could be used as a control sample.

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, , bosons interact only with a coherent flavor state inducing a “conversion” of a definite mass state into a definite flavor state. Namely, a neutrino in a mass eigenstate turns into a flavor eigenstate through an interaction with the virtual , -bosons propagating in the -channel, that is, four different reactions, are possible as follows: such that . Here, a definitive mass state which may exist due to previous hard neutrino-graviton interaction or due to the resonance MSW effect [34, 35] is “converted” back into a flavor state which may undergo oscillation. It is important to note that because the neutrino is not likely to interact weakly between the source and the Earth, if the neutrino is in a definitive mass state induced by the hard neutrino-graviton scattering event which occurred long before it arrives at the Earth it will still be in the definitive mass state at the Earth. The distance between a hard neutrino-graviton scattering event and detection event is not important.

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 transition amplitude between two flavor states and is given by [36] here is the distance from where the neutrino was created in a definite flavor eigenstate , and is the energy of the neutrino. Analogically, for neutrino-graviton interactions the transition amplitude between a flavor state and a mass state can be written as which means that the probability for a given flavor neutrino state to decohere by transforming into a mass state due to a hard graviton-neutrino interaction, given by , is independent of the neutrino mass, , the mass splitting, , and the distance from the neutrino source, . The dependence on the relativistic neutrino energy, , for a given scattering comes from the neutrino mass state scattering amplitude squared, (for more details, see the next section).

###### 2.3.1. Differences from Other Sources of Decoherence

Contrary to the Penrose-Diósi effect of classical decoherence [27–30], the quantum decoherence of a neutrino flavor state happens at small space-time scales, , which are much smaller than the neutrino wave length scale: , due to the quantum nature of the tree-level graviton-neutrino interaction. As an additional significant difference, the quantum decoherence effect is not sensitive to the mass differences of the mass eigenstates, or to , while they are crucial for and determine the classical decoherence of the neutrino flavor state at large separations, . Most importantly, quantum decoherence provides us with a key for phenomenological verification of quantum gravity models with possible deviations from quasi-classical gravity through measurement of neutrino oscillation characteristics.

The proposed effect is also different from the standard propagation decoherence (see Figure 2). In propagation decoherence, the neutrino mass states are separated in time and/or space and so the local weak interaction (the detection process)* observes* an incoherent sum of the propagating mass states in a given space-time point. In quantum decoherence, the neutrino exists only within a given mass eigenstate after being “observed” by the hard graviton (e.g., in the quantum processes of hard GBH or Compton scattering; see below). This difference is important. Indeed, while a flux of neutrinos which have undergone quantum decoherence is* observed* by a weak interaction in an Earth-based detector as an incoherent sum of the mass states, they do not experience a change of potential induced by matter (e.g., the MSW effect) as an incoherent sum of mass states. Namely, the neutrino which has not undergone quantum decoherence experiences matter as a superposition of mass states, while the neutrino which has undergone quantum decoherence would not experience matter as a superposition of mass states. Also, it is possible that a neutrino passing through densities which change nonadiabatically might demonstrate interference phenomena as presented in [23]. As we will explicitly demonstrate below, such a difference between the quantum and propagational decoherence in the presence of the Earth matter effect may be observable and is important in studies of astrophysical neutrino oscillations.

##### 2.4. Decoherence in the Presence of the Earth Matter Effect

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 [24]); in this work we consider relatively small distances (1000 kpc) since we do not explicitly consider differentiating quantum decoherence from potentially very long distance effects, for example, classical decoherence.

The simulation presented here operates in the -matrix oscillations formalism [37] and was realised in Python, but for the neutrino propagation the code gives the same results as the Fortran simulation found in [38]. In the case of quantum decoherence where the neutrino exists as a single mass eigenstate, the neutrino is not coherent and the Earth matter effect causes no significant change to the measured flavor composition at low energies. For the case of propagational decoherence, the neutrino is a decoherent superposition of mass states where the description of the neutrino in the flavor basis is given by constant phase differences between the mass eigenstates and an effect may be seen. While many models for supernova neutrino spectra, see, for example, [39, 40], and other astrophysical neutrino sources may be interesting, for simplicity, we adopt an initial uniform electron neutrino flux with energies between 0.5 and 20 MeV. We are interested in a significant measurement so incorporated propagation of the neutrino directly through the Earth and have obtained a maximal difference between propagation decoherence and the quantum decoherence effect at greater than 100%. The latter is important for next generation neutrino measurements.

##### 2.5. Analytic Illustration: Two-Flavour Case without Core

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 [22] and Akhmedov and collaborators [41, 42] and Blennow and Smirnov [43]. These papers give the essential understanding of neutrino propagation in matter and propagation decoherence, but no explicit formula is given for a neutrino which undergoes propagation decoherence and then experiences the Earth matter effect.

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 [43, 44].

Propagation decoherence was studied in detail by Beuthe [22] but unfortunately only in the vacuum case. Akhmedov and Wilhelm [41] explicitly consider decoherence due to production or detection conditions but only note that finite coherence length is recovered during the integration over energy. Beuthe [22] goes into great detail about the physics, which is that the wave packets separate or that the wave packet spatial spread is so large that the phase varies over the wave packet and the information is lost. These are changes of the relationship between the states which make up the neutrino and the changes can be accounted for by integrating the phase in the vacuum transition probability. In the case of two regimes with a sharp transition, as we consider here, it is necessary to find the flavor states at the transition.

The condition for the wave packet separation to be complete is given explicitly by Farzan and Smirnov [24]. They explicitly note that this is different than the effect due to averaging (Section 2.1 in [24]) over the energy, despite the effect being computationally the same for the vacuum (and adiabatic) case [22]. As noted by [24], once the phase difference becomes large, the phase difference between the mass eigenstates can be expressed as a constant. This happens once [24]; considerwhich is achieved for both eV and eV for kpc since we expect to be less than .

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 where is the neutrino two-flavor mixing angle in vacuum and and are the two neutrino flavors. The amplitude can then be given by These give the correct flavor amplitudes of a neutrino produced in a state which has travelled through vacuum and experienced wave packet separation when it reaches the earth-to-vacuum transition. There is an overall phase, but for length scales (such as the earth-to-vacuum transition) much smaller than the the distance between the wave packets (which can be 1 km or more) the phase difference between the wave packets is a constant as expressed above. An amplitude where the phase between the wave packets changes with distance would be incorrect for large wave packet separations.

This allows us to give a clear description of a produced which experiences propagation decoherence, travels through the mantle of the earth, and then is detected as a . This is which for the standard description in terms of a matter mixing angle and the matter phase is where [37]and is a constant density. This formula is different from that which is given for solar neutrinos and which is presented in the paper by Dighe et al. [45]. They give the calculation for an incoherent mixture of mass eigenstates originating in the sun, but this important paper does not explicitly consider propagation decoherence (decoherence due to wave packet separation) but rather the effects of coherent neutrino propagation at long baselines in addition to the known solar MSW resonance effect where neutrinos which leave the sun exist in only a single mass eigenstate. Farzan and Smirnov [24] explicitly give the condition to have the phase change for decoherent, and also for detection to restore coherence. The condition is that the measurement takes place over large enough time scales (or small enough energy resolutions) for both wave packets to be measurable [46].

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 [22, 41]. The formulation can be considered in the framework of [41] but where the matrix elements represent the mass eigenstates and not the matter eigenstates (so the identity since we are assuming propagation in the vacuum). Then the relationship for neutrinos which have undergone quantum gravity decoherence is simply (for two flavours) This is equivalent, as discussed in [45], to a coherent neutrino being observed a long distance from the neutrino source by an experiment without arbitrary energy resolution. The probability is then given by where and are as mentioned before.

The ratio of neutrinos which have undergone* propagation decoherence* and at the same time propagated through a region of constant density to those which have only propagated through the vacuum is given by the following expression:

Analogically, the ratio of neutrinos which have undergone* quantum decoherence* in the presence of matter effect in a constant density medium to those which have propagated through the vacuum takes a different form: A difference between the ratios and , in principle, could be measurable and indicates the principal difference between propagation and quantum decoherence emerging in the presence of matter effect. Measurement of such a difference could therefore serve as a clear example of graviton detection.

#### 3. Graviton-Neutrino Scattering

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 in the -channel) between the probe and the source. Analogically, it is correct to discuss multiple exchange of virtual -channel gravitons in a scattering event as a signature of non-zeroth curvature itself (for more detailed discussions of the principles, see, for example, [47]).

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 [48].

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.

##### 3.1. Gravitational Bethe-Heitler Scattering

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* hard or energetic gravitons*, namely, with energies comparable to the incoming relativistic neutrino energy. Due to the quantum nature of the neutrino and graviton, the latter process can trigger a dramatic decoherence of an incoming neutrino flavor state at the quantum level during a very short time scale (inversely proportional to the energy of the radiated hard graviton).

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 and therefore requires hard real graviton emission. In this case, the hard graviton probe has a small wave length and thus can resolve separate mass states in a coherent or incoherent neutrino flavor state in the quantum mechanical sense (likewise a hard enough photon can resolve an internal substructure of the proton wave function and interacts with the separate quarks it is composed of while a soft one “sees” a proton as a whole only).

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 [31].

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 [49], the radiative corrections can only be relevant in the deep infrared limit of soft real gravitons emitted in the forward direction where they will cancel the soft/collinear divergences. The latter classical limit represents classical gravitational waves emitted off a neutrino state with very small or no impact on it.

In the considered GBH case, shown in Figure 1(a), one deals with the graviton exchange with negative momentum transfer squared in the -channel with the propagator stretched between the relativistic neutrino of mass and energy and a massive classical gravitational field source with mass . The wave function, , describes a projection of a given flavor state onto a fixed mass state denoted as a dark ellipse, while the heavy classical source of the gravitational field is shown by a shaded circle.