This report presents a brief review on the experimental measurements of the muon neutrino velocities from the OPERA, Fermilab and MINOS experiments and that of the (anti)-electron neutrino velocities from the supernova SN1987A, and consequently on the theoretical attempts to attribute the data as signals for superluminality of neutrinos. Different scenarios on how to understand and treat the background fields in the effective field theory frameworks are pointed out. Challenges on interpreting the OPERA result as a signal of neutrino superluminality are briefly reviewed and discussed. It is also pointed out that a covariant picture of Lorentz violation may avoid the refutation on the OPERA experiment.
The OPERA collaboration recently reported that muon neutrinos could be superluminal. More recently, Cohen and Glashow pointed that such superluminal neutrinos would be suppressed since they lose their energies rapidly via bremsstrahlung. In this Letter, we propose that Finslerian nature of spacetime could account for the superluminal phenomena of particles. The Finsler spacetime permits the existence of superluminal behavior of particles while the casuality still holds. A new dispersion relation is obtained in a class of Finsler spacetime. It is shown that the superluminal speed is linearly de…
We consider Lorentz- and CPT-violating dimension-5 operators to address the issue of superluminal neutrinos recently pointed out in OPERA experiments. We assume these operators in the photon and neutrino sectors coupled to Lorentz-violating backgrounds in a preferred frame defined by a time-like direction. We show that such operators can produce a curve with OPERA's slope that fits OPERA, MINOS and supernova SN1987a data.
OPERA has claimed the discovery of superluminal propagation of neutrinos. We analyze the consistency of this claim with previous tests of special relativity. We find that reconciling the OPERA measurement with information from SN1987a and from neutrino oscillations requires stringent conditions. The superluminal limit velocity of neutrinos must be nearly flavor independent, must decrease steeply in the low-energy domain, and its energy dependence must depart from a simple power law. We construct illustrative models that satisfy these conditions, by introducing Lorentz violation in a sector wit…
This talk opens the CosPA2011 session on OPERA's superluminal neutrino claim. I summarize relevant observations and constraints from OPERA, MINOS, ICARUS, KamLAND, IceCube and LEP as well as observations of SN1987A. I selectively review some models of neutrino superluminality which have been proposed since OPERA's announcement, focusing on a neutrino dark energy model. Powerful theoretical constraints on these models arise from Cohen-Glashow bremsstrahlung and from phase space requirements for the initial neutrino production. I discuss these constraints and how they might be evaded in models i…
The recent OPERA experiment of superluminal neutrinos has deep consequences in cosmology. In cosmology a fundamental constant is the cosmological constant. From observations one can estimate the effective cosmological constant $Λ_{eff}$ which is the sum of the quantum zero point energy $Λ_{dark energy}$ and the geometric cosmological constant $Λ$. The OPERA experiment can be applied to determine the geometric cosmological constant $Λ$. It is the first time to distinguish the contributions of $Λ$ and $Λ_{dark energy}$ from each other by experiment. The determination is based on an explanation o…
The new results by OPERA collaboration claim the discovery of superluminal neutrinos. Superluminal particles have to break Lorentzian symmetry or causality principle. The method discussed gives us the possibility to reintroduce Lorentzian symmetry without breaking of causality.
It is suggested that recent superluminal neutrinos from the OPERA collaboration might indicate that there are other ultimate speeds than usual speed of light in our universe. The leptonic sector of the standard model (SM) is reformulated incorporating with new ultimate speeds. In minimal cases where there is another maximum speed $c'$ which equals to the speed of the OPERA neutrinos, new effects would appear at the level of ${\cal O}(10^{-5})$. Improved precisions for the electroweak observables would check the validity of this scenario.
Recently the OPERA collaboration {\cite{opera}} has reported the observation of superluminal neutrinos traveling a distance of 730 km from Grand Sasso Laboratory to CERN.These results contradict the basic tenet of the Theory of Special Relativity: No particle can travel faster than light in vacuum. Moreover they seem to be in conflict with the speed of (anti)neutrinos detected from the explosion of the SP1987A Super Nova{\cite{SP1987A}}.Here we show that the relative velocity between neutrino and photon has the following property: It depends weakly on the energy of the particle excepts in cert…
We consider a possibility that the recent OPERA results on neutrinos' superluminality could be caused by a local effect of a new gauge field sourced by the earth. If neutrinos couple to this gauge field via a gauge-invariant but non-renormalizable interaction, the coupling effectively changes a background metric, thereby leading to superluminality of neutrinos. This possibility naturally might explain why neutrinos from CERN CNGS beam to Gran Sasso Laboratory become superluminal while those from SN1987A to Earth become subluminal.
The OPERA collaboration has claimed the discovery of supeluminal neutrino propagation. However the superluminal interpretation of the OPERA result was refuted by Cohen and Glashow because it was shown that such superluminal neutrinos would lose energy rapidly via the bremsstrahlung of electron-positron pairs (arXiv:1109.6562). We note that the superluminal interpretation is still possible if there exists a new (dark) neutrino which can propagate with a superluminal velocity and which couples with usual neutrinos only via the mass mixing leading to neutrino oscillations. It is supposed that the…
In this short note we remark that the OPERA measurement could be reconciled with information from SN1987a in the context of deformed special relativity without the loss of energy through Cherenkov-like process reported by Cohen and Glashow.
We show that in the presence of a distance dependent Lorentz-violating time-like background one may find superluminal neutrinos at some high energy such as OPERA's scale. The similar behavior but approaching the subluminal branch is found to appear for inflaton field in the low energy scale. Thus this scenario seems to be suitable to acomplish both superluminal neutrinos and dark energy.
Quantum trajectories are used to study OPERA findings regarding superluminal neutrinos. As the applicable stationary quantum Klein-Gordon equation is real, real quantum reduced actions and subsequent real quantum trajectories follow. The requirements for superluminal neutrinos are examined. A neutrino that is self-entangled by its own backscatter is shown to have a nonlocal quantum trajectory that may generate a superluminal transit time. Various cases are shown to produce theoretical superluminal neutrinos consistent with OPERA neutrinos. Quantum trajectories are also shown to provide insight…
An interpretation of the superluminal velocity observed by OPERA collaboration is given in terms wave packet distortion of ultrarelativistic massive neutrinos. Standard quantum-mechanical time evolution of physical states can explain the observed Delta x/Delta t > c without violating the fundamental principles of special relativity.
In this letter, we show that superluminal neutrinos announced by OPERA could be explained by the existence of a monopole, which is left behind after the spontaneous symmetry breaking (SSB) phase transition of some scalar fields in the universe. We assume the 't Hooft-Polyakov monopole couples to the neutrinos but not photon fields. The monopole introduces a different effective metric to the neutrinos from the one experienced by photons. We find that the superluminal propagation only exists in a very short distance from the monopole, about $10^3$ cm in OPERA. No matter how far they travel, neut…
Most recently, the measurement of the neutrino velocity with the OPERA detector in the CNGS beam shows unexpected indication, that the muon neutrino velocity, $v_ν$, exceeds the velocity of light in the vacuum, $c$, which is obviously in contradiction with the most basic hypothesis of modern physics. Within the framework of minimal Standard Model Extension, we discuss the modified dispersion relation and consequently the velocity-energy relation of muon neutrinos. The simplified models are fit to the OPERA data, Fermilab experiment and MINO data. We find that minimal Standard Extension can des…
The superluminal is reported in OPERA neutrino experiment (arXiv:1109.4897v2), this result is consistent with previous MINOS experiment about neutrino velocity (Phys. Rev. D 76, 072005 (2007)). Here we propose a Dirac equation of tachyon for superluminal neutrino in which the mass matrix is anti-hermitian. Additionally we find that this anti-hermitian mass matrix which is responsible for superluminal can only be for neutrinos, while other results of Standard Model are unchanged. In case that Lorentz invariant is remained, the masses of neutrinos are estimated based on available data of experim…
We argue that the recent measurement of the neutrino velocity to be higher than the velocity of light could be due to violation of Lorentz invariance by the muon neutrinos. This result need not undermine special-relativistic foundational notions of causality.
The splitting of neutrinos, a viable reaction for superluminal neutrinos, is shown to have phenomenologically relevant consequences if one accepts the recent OPERA results for the velocity of neutrinos. Neutrino splitting can be used to put strong constraints on the energy dependence of the velocity of propagation of neutrinos in a general analysis of modifications of relativistic kinematics and to propose observable effects due to the departures from special relativity in neutrino physics.
Motivated by the tentative observation of superluminal neutrinos by the OPERA experiment, we present a model of active-sterile neutrino oscillations in which sterile neutrinos are superluminal and active neutrinos appear superluminal by virtue of neutrino mixing. The model demonstrates some interesting possibilities and challenges that apply to a large class of models aiming to explain the OPERA result.
The superluminal propagation of neutrinos observed by OPERA collaboration can be interpreted as neutrinos traveling in a pseudoscalar potential which may be generated by a medium. The OPERA differential arrival time data set a constraint on the form of the pseudoscalar potential.
The OPERA collaboration has reported the observation of superluminal muon neutrinos, whose speed $v_ν$ exceeds that of light $c$, with $(v_ν- c)/c \simeq 2.5 \times 10^{-5}$. In a recent work, Cohen and Glashow (CG) have refuted this claim by noting that such neutrinos will lose energy, by pair-emission of particles, at unacceptable rates. Following the CG arguments, we point out that pair-emissions consistent with the OPERA anomaly can lead to detectable signals from decays of highly boosted top quarks at the LHC, allowing an independent test of the superluminal neutrino hypothesis.
The superluminal behaviors of neutrinos were reported by the OPERA collaboration recently. It was also noticed by Cohen and Glashow that, in standard quantum field theory, the superluminal neutrinos would lose their energy via the Cherenkov-like process rapidly. Finslerian special relativity may provide a framework to cooperate with the OPERA neutrino superluminality without Cherenkov-like process. We present clearly the symmetry, causal structure and superluminality in Finsler spacetime. The principle of relativity and the causal law are preserved. The energy and momentum are well defined and…
The result of the OPERA experiment revealed that the velocity of muon-neutrinos was larger than the speed of light. We argue that this apparent superluminal velocity can be interpreted as a weak value, which is a new concept recently studied in the context of quantum physics. The OPERA experiment setup forms a scheme that manifests the neutrino velocity as a weak value. The velocity defined in the scheme of weak measurement can exceed the speed of light. The weak velocity is not a concept associated to a single phenomenon but it is a statistical concept defined by accumulating data at separate…
A simple discussion of the recent OPERA result on the apparent critical speed of the muon neutrino is presented. We point out in particular some of the possible consistency problems of such an interpretation of the OPERA data with respect to well-established astrophysical observations.
The effect of nonlinear self-interaction can be associated with superluminal velocity of neutrino. The power energy spectrum E=p+Cp^a is derived from the nonlinear Dirac equation when interaction term V=λ(ψγ_μψψγ^μψ)^a is added to the Lagrangian of a free spin-1/2 particle. The superluminal velocity recorded by the OPERA and MINOS collaborations is achieved when the coupling constants are taken in the range a=0.4-1.18 and λ=-(0.5-1.6)x10^-4. The self-interaction Lagrangian V=λ(ψγ_μψψγ^μψ) with the coupling constant λ=-(0.7-0.9)x10^-4 yields the same result. Scalar interaction V=λ(ψψ)^b and sca…
In the context of theories where particles can have different limiting velocities, we review the running of particle speeds towards a common limiting velocity at low energy. Motivated by the recent OPERA experimental results, we describe a model where the neutrinos would deviate from the common velocity by more than do other particles in the theory, because their running is slower due to weaker interactions.
We revisit the bremsstrahlung process of a superluminal neutrino motivated by OPERA results. From a careful analysis of the plane wave solutions of the superluminal neutrino, we find that the squared matrix elements contain additional terms from Lorentz violation due to the modified spin sum for the neutrino. We point out that the coefficients of the decay rate and the energy loss rate significantly depend on the details of the model, although the results are parametrically similar to the ones obtained by Cohen and Glashow [1]. We illustrate this from the modified neutral current interaction o…
OPERA collaboration has reported evidence of superluminal phenomenon for neutrinos. One of the possible ways to explain the superluminality is to have Lorentz symmetry violated. It has been shown that dispersion relations put forwards has the problem of physics laws vary in different inertial frames leading to conflicting results on Cherenkov-like radiation, pion decay and high energy neutrino cosmic ray. For theories with deformed Lorentz symmetry, by modifying conservation laws corresponding to energy and momentum in the usual Lorentz invariant theory, it is possible to have superluminal eff…
Violation of Lorentz invariance (VLI) has been suggested as an explanation of the superluminal velocities of muon neutrinos reported by OPERA. In this note we show that the amount of VLI required to explain this result poses severe difficulties with the kinematics of the pion decay, extending its lifetime and reducing the momentum carried away by the neutrinos. We show that the OPERA experiment limits $α= (v_ν- c)/c < 4 \times 10^{-6}$. We then take recourse to cosmic ray data on the spectrum of muons and neutrinos generated in the earth's atmosphere to provide a stronger bound on VLI: $(v-c)/…
Contrary to a widespread belief, measures of velocity can yield a value larger than $c$, the instantaneous light speed in vacuum, without contradicting Einstein's relativity. Nevertheless, the effect turns out to be too small to explain the recently claimed superluminal velocity by the OPERA collaboration. Several other general relativistic effects acting on the OPERA neutrinos are also analyzed. All of them are unable to explain the OPERA result.
In the context of the sensational results concerning superluminal velocities, announced recently by the OPERA Collaboration, we have proposed a classical model yielding a statistically calculated measured velocity of a beam, higher than the velocity of the particles constituting the beam. The two key elements of our model, necessary and sufficient to obtain this curious result, are a time-dependent "transmission" function and statistical method of the maximum-likelihood estimation.
We analyze the velocity dependence on energy of superluminal neutrino recorded by the OPERA and MINOS collaborations and manage to approximate the energy spectrum by a power law E=p+Cp^a where parameters must be taken in the range a=0.40--1.18 and C=1.5x10^{-5}--4.15x10^{-4} (momentum and energy are expressed in GeV). This rough estimation is constrained by the errors of measurements, and new experimental data are requested.
The OPERA collabotation has reported evidence of superluminal neutrinos with a mean energy 17.5 GeV ranging up to 50 GeV. However, the superluminal interpretation of the OPERA results has been recently refuted theoretically by Cherenkov-like radiation. We discuss a loophole of this argument from the kinematical viewpoint and find it possible to avoid the Cherenkov-like radiation of the OPERA neutrinos. The key idea of our argument is to admit the fact that the neutrinos travel faster than the observed speed of light while they do slower than the true speed of light in vacuum so strictly speaki…
In this work it is shown, that for short 3ns neutrino pulses reported by OPERA, a relativistic shape deforming effect of the neutrino distribution function due to spontaneous emission, produces an earlier arrival of 65.8ns in agreement with the reported 62.1ns\pm 3.7ns, with a RMS of 16.4ns explaining the apparent superluminal effect. It is also shown, that early arrival of long 10500ns neutrinos pulse to Gran Sasso, by 57.8ns with respect to the speed of light, could be explained by a shape deforming effect due to a combination of stimulated absorption and spontaneous emission, while travelin…
We show that the superluminal speeds of the muon neutrinos observed in the OPERA experiment can be explained within a relativity theory with extra time like dimensions. In addition, such theory predicts, the existence of dark matter.
We suggest a possible interpretation of the recent observation by the OPERA collaboration of superluminal propagation of neutrinos. We show that it is in principle possible that the group velocity of neutrinos exceeds the speed of light without violating special relativity.
Recently, Cohen and Glashow [A.G. Cohen, S.L. Glashow, Phys. Rev. Lett. {\bf 107}, 181803 (2011)] pointed out that the superluminal neutrinos reported by the OPERA would lose their energy rapidly via the Cherenkov-like process. The Cherenkov-like process for the superluminal particles would be forbidden if the principle of special relativity holds in any frame instead violated with a preferred frame. We have proposed that the Finslerian special relativity could account for the data of the neutrino superluminality (arXiv:1110.6673[hep-ph]). The Finslerian special relativity preserves the princi…
According to the measurement of muon-neutrino experiment done by the OPERA collaboration, the speed of high-energy neutrino exceeds that of light in vacuum by 25ppm. Assuming that this result is correct, a possible resolution of the dilemma between it and the validity of relativity is proposed without changing the framework of the Standard Model of elementary particles. The essential idea is based on a possible resolution, proposed previously, of the color confinement problem of quantum chromodynamics.
The OPERA collaboration has announced to have observed superluminal neutrinos with a mean energy 17.5 GeV, but afterward the superluminal interpretation of the OPERA results has been refuted theoretically by Cherenkov-like radiation and pion decay. In a recent work, we have proposed a kinematical resolution to this problem. A key idea in our resolution is that the OPERA neutrinos are not superluminal but subluminal since they travel faster than the observed speed of light in vacuum on the earth while they do slower than the true speed of light in vacuum determining the causal structure of even…
Recently it was suggested that the observation of superluminal neutrinos by the OPERA collaboration may be due to group velocity effects resulting from close-to-maximal oscillation between neutrino mass eigenstates, in analogy to known effects in optics. We show that superluminal propagation does occur through this effect for a series of very narrow energy ranges, but this phenomenum cannot explain the OPERA measurement.
The OPERA Collaboration reported evidence for muonic neutrinos traveling slightly faster than light in vacuum. While waiting further checks from the experimental community, here we aim at exploring some theoretical consequences of the hypothesis that muonic neutrinos are superluminal, considering in particular the tachyonic and the Coleman-Glashow cases. We show that a tachyonic interpretation is not only hardly reconciled with OPERA data on energy dependence, but that it clashes with neutrino production from pion and with neutrino oscillations. A Coleman-Glashow superluminal neutrino beam wou…
In recent experiments conducted by the OPERA collaboration, researchers claimed the observation of neutrinos propagating faster than the light speed in vacuum. If correct, their results raise several issues concerning the special theory of relativity and the standard model of fundamental particles. Here, the physical consequences of superluminal neutrinos described by a tachyonic Dirac lagrangian, are explored within the standard model of electroweak interactions. If neutrino tachyonic behavior is allowed, it could provide a simple explanation for the parity violation in weak interactions and …
In this letter, we propose that the recent measurement of superluminal neutrinos in OPERA could be explained by the existence of a domain wall which is left behind after the phase transition of some scalar field in the universe. The scalar field couples to the neutrino and photon field with different effective couplings. It causes different effective metrics and the emergence of superluminal neutrinos. Moreover, if the supernova and the earth are in the same plane parallel to the wall, or the thickness of the wall is much smaller than the distance from the supernova to the earth, the contradic…
Various approaches aim to describe the recent analysis by the OPERA experiment, which indicates that neutrinos travel faster than the speed of light. We demonstrate that any such theoretical or experimental explanation must not destroy the complicated (nonlinear) structure of the proton waveform recovered in the neutrino signal. As one example, consider that only a fraction of the neutrinos travel faster than the speed of light, such as sterile neutrinos. We fit the OPERA data including this fraction as a free variable, assuming that the OPERA result is correct. In our analysis, the best-fit v…
Quantum coherence can significantly increase the strength of the forward scattering of neutrinos propagating through the Earth and interacting with matter. The index of refraction of the neutrinos propagating in a medium and hence their phase velocity is determined by the forward scattering. So, depending on the nature of the interaction of neutrinos with matter, their phase velocity can be larger than the speed of light in vacuum. We show that such effects can explain the apparent superluminal propagation of muon neutrinos found recently by the OPERA experiment. Our proposal explains why the …
We suggest that the recently observed superluminal neutrino propagation observed by OPERA may have an environmental origin that can simultaneously explain the DAMA/LIBRA annual modulation. Our proposal can be unambiguously tested by the prediction that OPERA will also observe an annual modulation in the speed of neutrinos. Experiments geographically distant from Gran Sasso will however not be able to observe these effects. We do not rely upon any violation of Lorentz symmetry except for spontaneous breaking of translational invariance. We discuss implications for the CRESST experiment, also lo…
This is a brief note discussing the energy dependence of superluminal neutrino velocities recently claimed by OPERA [1,2]. The analysis is based on the data provided there on this issue, as well as on consistency with neutrino data from SN1987a as recorded by the Kamioka detector [3]. It is seen that it is quite difficult to reconcile OPERA with SN1987a. The so called Coleman- Glashow dispersion relations do not do that well, if applied at all neutrino energies. The so called quantum gravity inspired dispersion relations perform far worse. Near OPERA energies both an energy-independent velocit…
The superluminal propagation of neutrinos observed by the OPERA collaboration can be explained by an energy dependent potential for the neutrino beam in passage through the Earth.
We digress on the implications of recent claims of superluminal neutrino propagation. No matter how we turn it around such behaviour is very odd and sits uncomfortably even within "far-fetched" theories. In the context of non-linear realizations of the Lorentz group (where superluminal misbehaviour is run of the mill) one has to accept rather contrived constructions to predict superluminal properties for the neutrino. The simplest explanation is to require that at least one of the mass states be tachyonic. We show that due to neutrino mixing, the flavor energy does not suffer from the usual ru…
Non-dispersive wave packet for massless fermions is formulated on the basis of squeezed coherent states that are put in a form of common eigenfunction for the Hamiltonian and the helicity operator, starting from the Dirac equation. The wave packet thus constructed is demonstrated to propagate at a constant velocity as that of light. This explicit expression of wave packet for the massless fermions can facilitate theoretical analysis of problems where a wave packet is of formal significance. Furthermore, extensive wave packet may result in a superluminal velocity statistics if determined from t…
We test several phenomenological scenarios of faster-than-light neutrinos, by fitting to the experimental data from OPERA, MINOS and Fermilab79. Our purpose is to see, from the perspective of the current data, whether or not the speed of the superluminal neutrino depends on its energy. We show that the Coleman-Glashow scenario in which the velocity of the neutrino is free of the energy fits the data best. However, the result of SN1987A cannot be explained by this model. We find that a power-law model with the power close to zero can simultaneously explain the results of SN1987A and OPERA+MINOS…
We revisit the model building challenges that one faces when trying to reconcile the OPERA claim of neutrino superluminality with other observational constraints. The severity of the supernova bound and of the kinematical constraints of Cohen-Glashow type lead us to focus on scenarios where all types of particles are superluminal inside matter. In contrast to the Dvali-Vikman proposal, this matter effect needs to be very short-ranged to avoid constraints from experiments on the Earth's surface in low-density environments. Due to this short range, the interaction underlying such a matter effect…
From the data release of OPERA - CNGS experiment, and publicly announced on 23 September 2011, we cast a phenomenological model based on a Majorana neutrino state carrying a fictitious imaginary mass term, already discussed by Majorana in 1932. This mass term can be induced by the interaction with the matter of the Earth's crust during the 735 Km travel. Within the experimental errors, we prove that the model fits with OPERA, MINOS and supernova SN1987a data. Possible violations to Lorentz invariance due to quantum gravity effects have been considered.
We present a possible solution to the reported OPERA anomaly for the speed of neutrinos, based on the idea that it is a local effect caused by a scalar field sourced by the earth. The coupling of the scalar to neutrinos effectively changes the background metric where neutrinos propagate, leading to superluminality. The strength of the coupling is set by a new mass scale, which should be at $1\, {\rm TeV}$ to account for the OPERA anomaly. Moreover, if this scenario is valid, the neutrino velocity depends on the baseline distance between the emission and detection points in such a way that supe…
The superluminal neutrinos detected by OPERA indicates Lorentz invariance violation (LIV) of the neutrino sector at the order of $10^{-5}$. We study the implications of the result in this work. We find that such a large LIV implied by OPERA data will make the neutrino production process $π\to μ+ ν_μ$ kinematically forbidden for neutrino energy greater than about 5 GeV. The OPERA detection of neutrinos at 40 GeV can constrain the LIV parameter to be smaller than $3\times 10^{-7}$. Furthermore the neutrino decay in the LIV framework will modify the neutrino spectrum greatly. The atmospheric neut…
Modified neutrino dispersion relations, which still obey the relativity principle, can have both a superluminal (muon-type) neutrino and a luminal (electron-type) neutrino, as long as neutrino-mass effects can be neglected. The idea is to allow for flavor-dependent deformed Lorentz transformations and an appropriate hierarchy of energy scales. If OPERA's result on a superluminal velocity of the muon-neutrino is confirmed, the model has a matching superluminal velocity of the corresponding charged lepton, the muon, at equal particle energy. Assuming that this model is not already ruled out, new…
By analyzing the structure of the Weyl spinor field in the Clifford bundle formalism we show that in each spinorial frame it is represented by F\insec(\doublebarwedge^0 T^\starM + \doublebarwedge^2 T^\star M + \doublebarwedge^{4} T^\star M)\hookrightarrowsecC\ell(M,g) satisfying the equation \partialF=0, where \partial is the Dirac operator acting on sections of the Clifford bundle C\ell(M,g). With this result we show that introducing a generalized potential A=(A + γ_5 B)\insec(\doublebarwedge^{1}T^{\star}M + \doublebarwedge^3 T^\star M)\hookrightarrowsecC\ell(M,g) for the Weyl field such that…
