OPERA, SN1987a and energy dependence of superluminal neutrino velocity
aa r X i v : . [ h e p - ph ] O c t OPERA, SN1987a and energy dependence of superluminal neutrino velocity
N.D. Hari Dass ∗ Chennai Mathematical Institute, Chennai & CQIQC, Indian Institute of Science, Bangalore, India.
This is a brief note discussing the energy dependence of superluminal neutrino velocities recentlyclaimed by OPERA [1, 2]. The analysis is based on the data provided there on this issue, as well as onconsistency with neutrino data from SN1987a as recorded by the Kamioka detector [4]. It is seen thatit is quite difficult to reconcile OPERA with SN1987a. The so called Coleman-Glashow dispersionrelations do not do that well, if applied at all neutrino energies. The so called quantum gravityinspired dispersion relations perform far worse. Near OPERA energies both a energy-independentvelocity, as well as a linear energy dependence with an offset that is comparable in value to theobserved δv by OPERA at 28.1 GeV work very well. Our analysis shows that precision arrivaltime data from SN1987a still allow for superluminal behaviour for supernova neutrinos. A smoothinterpolation is given that reconciles OPERA and SN1987a quite well. It suggests a fourth powerenergy dependence for δvc of supernova neutrinos. This behaviour is insensitive to whether thevelocities are energy-independent, or linearly dependent on energy, near OPERA scale of energies.Suggestions are made for experimental checks for these relations. PACS numbers:
INTRODUCTION
The recent announcement by the OPERA collabora-tion [1, 2] that they have measured superluminal neutrinovelocities has stunned both the scientific community, aswell as public at large. The announced magnitudes of theeffects are large and statistically very significant. The fullimplications, should the experiment be vindicated in fu-ture, are mind-boggling. Everything in modern physics,as we understand it today, depends one way or the otheron the correctness of the Special Theory of Relativity. In-terestingly, speeds greater than c for photons in the sameenergy range as OPERA neutrinos, had been reported byGharibyan [3] before.Because of the enormous stakes involved, every ef-fort, both experimental and theoretical, must be madeto make sure that the claims are correct. On the ex-perimentalists side, one needs to thoroughly check anddouble check all sources of errors. One needs to repeatthe experiment, both the original version, as well as vari-ants of it like at different energies, different path lengthsetc.. On the theoretical side efforts have to be madeto check the consistency of the OPERA data both con-ceptually, as well as with earlier precision data. Indeed,after the OPERA announcement, a number of interestingideas have been put forward(see [7]-[34]). Some of theseideas even predated the OPERA results [10–12, 70, 71].But if the OPERA results are really right, all of physicshas to be radically revised and there may not exist, atpresent, any sensible way of analysing it. While one maycome up with some parametrizations, or some pertur-bative effective treatments like the one offered by Kost-elecky and coworkers [35–38] or by Coleman and Glashow[48], the inner selfconsistency of such approaches is neverguaranteed till another succesful theory replaces SpecialTheory of Relativity. However, theories based on Spon- taneous Breaking of Lorentz Invariance [50, 51] may bebetter in this regard. That will certainly entail some rad-ical revisions of our notions of space and time, to say theleast. That way we can liken the situation to the days ofthe Bohr atom, where one could parametrize some smallcorrections to observables in the large quantum numberstates, without having the slightest idea of the deep con-ceptual changes that the full quantum theory eventuallybrought about. So here too one may have to develop in-termediate crutches like the correspondence principle etcto make progress.To illustrate this point, consider a number of very in-teresting papers have appeared which have shown thatthe OPERA result in conjunction with reasonable as-sumptions like energy-momentum conservation, can leadto contradictory results [52–56]. On the one hand it isshown that the kinematics of pion decay is constrainedto the extent of not permitting neutrinos of the en-ergy ’seen’ at OPERA, and on the other hand that thehigh energy neutrino looses much of its energy throughbremsstahlung before it reaches the OPERA detector. Inthese works, some ’reasonable’ assumptions like energy-momentum conservation, or that charged mesons obeynormal dispersion relations (based on limits on Cerenkovradiation in vacuum), are made. The point is, that inthe new theory all these notions might themselves un-dergo radical revisions. Even if some sort of momentumis conserved, it may not be of the form used currently.If some sort of relativity principle still survives, that willautomatically take care of the issue of Cerenkov radia-tion in vacuum. In fact such a relativity principle wouldforbid the bremsstrahlung process considered by Cohenand Glashow [52]. According to [57–59] there are indeedLorentz violating theories that nevertheless are compat-ible with a relativity principle.In this context, it is worth remembering that theOPERA experiment is really probing space-time struc- ture . Though they talk of neutrino energies and suchother notions, these are not central to an analysis of theirdata.Even concepts in which we have great faith like tim-ing need to be reassessed. This is essentially a so called
Time of Flight (TOF) method (is it time to rename itas
Time of Fright method?), though it is not the tradi-tional ’start time - stop time’ TOF method, but is insteada variant with a start time distribution and a stop timedistribution . TOF data is mostly analysed using classi-cal methods. But the situation here is highly quantummechanical, and it is known that time in quantum me-chanics can be a very delicate issue (for various perspec-tives on this, see [40]). There are various characterstictimes like a rrival time, passage time etc whose meaningin the quantum context can be ambiguous. In the presentcircumstance, neutrinos being very weakly interacting, detection time treated quantum mechanically adds fur-ther complications. In the end it may well turn out thatthese nuances are not particularly relevant for interpret-ing OPERA data, but it is imperative to get a soundunderstanding of these issues given the earth shatteringnature of the results.In a most interesting paper, perhaps the most credibleinterpretation of the OPERA events, Naumov and Nau-mov [41] have argued that neutrino wavepackets becomeeffectively oblate spheroids due to highly relativistic ef-fects, and that can lead to a systematic early arrival timesfor neutrinos that are not exactly aligned, at the source,with the detector.Other subtleties to worry about, regarding the timingof neutrinos at OPERA, are conceptual issues tied upwith the correct way of synchronizing clocks [72, 73]. Aspointed out by Contaldi [72], OPERA uses the so called one way synchronisation of clocks, which can be par-ticularly problematic when synchronizing clocks in non-inertial frames such as the earth (which is non-inertialboth because it is a rotating frame [76], and also be-cause of the earth’s gravitational field). According to thisanalysis, the lack of synchronization due to time dilationhas the correct sign to explain the OPERA results. Butbuilding the lack of synchronization to 60 ns from timedilation, in the manner described, seems unrealistic, buta careful appraisal is certainly called for.Before turning to our article, two other proposals, bothhaving to do with the beam structure, that could seri-ously impact on the interpretation of the OPERA data,are worth discussing. The first one, due to Henri [74],claims that a fluctuation in neutrino energies of the or-der of 10% during the roughly 500 ns. of the leading andtrailing edges of the PDF can mimic the alleged superlu-minal effects claimed by OPERA. While the number ofprotons increase and decrease during these phases, theproton energy itself remains at 400 GeV. Thus it is hardto see the source of this variation in the neutrino energy.Besides, this has to be systematic for the nearly million extractions, which is hard to accept.The other, potentially more damaging to the OPERAinterpretation, is the claim by Knobloch [75] that the Beam Current Tracker (BCT) at the CERN end has notregistered the true PDF, and that even after several sta-tistical averaging, 30 and 60 ns. structures are still seenin the PDF as measured by the beam current tracker(see also [2]). If true, this will cast doubts about thesystematics as claimed by OPERA, and could well nul-lify their claims. It is beyond our expertise to assess thiscriticism adequately, and we eagerly await clarificationby OPERA.The plan of this article is as follows: in the next sec-tion we recapitulate the most relevant part of the dataon the supernova 1987a. Here we have only looked atthe data recorded by the Kamioka detector [4]. We thenanalyse a power law dispersion relation for neutrinos, fix-ing the parameters by requiring compatibility with theobserved differential superluminal neutrino velocity of2 . · − c at an average energy of 17 GeV and the Kamioka data. We then compute the expecteddifferentials at the other energies 13 . GeV , 28 . GeV and 42 . GeV where OPERA has data. We compare thecalculated values with the measured values though thestatistics of these measurements are not as good as at17
GeV . We also repeat this exercise by first demandingcompatibility between OPERA data at 28 . GeV , andthen comparing the calculated values with the data at13 . GeV and 42 . GeV .In the section after this, we analyse both a flat, energy-independent velocity, as well as a linear energy depen-dence taking into account a possible constant shift term.In this section we also analyse a linear energy dependencewith a different shift term which is taken to be valid onlynear OPERA energies. In the subsequent section we dis-cuss the difficulties of a treatment that is good both atthe OPERA energy scales as well as at the SN1987a en-ergy scales. We propose, as an illustrative example, aform that works at both ends. In the last section wemake some concluding remarks.
ESSENTIAL 1987A DATA
There is a truly enormous amount of literaturediscussing various aspects of the supernova explosionSN1987a in the large magellanic cloud in February 1987.Fortunately for this analysis we need to concentrate onlyon a few essentials of the detection by Kamioka [4]. Webriefly summarize them in the next para.The distance to the event, L SN , is 1,68,000 light years.A neutrino cluster of 12 events was observed 2 hours be-fore the optical sighting [5] of the supernova. The neu-trino energies spread over a range 7 . − . low energy event of6 . ± . .
686 s. after the burst neu-trino, and three rather high energy events with energies35 . ± . , . ± . , . ± . . , . , .
915 s. after the burst event. Theexclusion of these anomalous events [42] does not affectthe reliability of the analysis given here. The first eventwas a ν e event with 20 . ± . burst neutrino. The subsequent neutrinos, the so called ther-mal neutrinos comprise of neutrinos and antineutrinos ofall flavours. The arrival times of the cluster was spreadaround 12 seconds, and this accurately established datais crucial to this analysis. Leaving out the anomalousevents, the highest neutrino energy is around 20 MeV,and the lowest around 7.5 MeV. The average neutrinoenergy is around 15 MeV.One should not interpret the early arrival of neutrinosby two hours compared to photons as evidence for anysuperluminal neutrino velocity as the photons take longerto escape the opaque stellar medium. This is similar tolight taking a million years longer to escape the solarinterrior.The SN1987a neutrinos are of electron flavour, whilethe OPERA neutrinos are predominantly ν µ . Thus onecould take the attitude that there is no common ground,and hence no real conflict between the two. But os-cillations make the situation more complicated, and areasonable attitude to take is that on the average everyflavour of neutrino would have velocities behaving simi-larly in both set ups. There are also indications that theOPERA effect has to be by and large flavour indepen-dent if serious troubles with oscillation phenomena areto be avoided [68]. This provides a justification for theanalyses to be discussed next. Analysis based on photon arrival time
As mentioned before the first optical sighting of thesupernova was 2 hrs after the neutrino burst [5]. It ispossible to use this information to already draw variousconclusions of interest.The analysis proceeds as follows: let t γem , t νem be theemission times, at the supernova, of photons and neu-trinos, respectively. Let us for the moment assume thatphotons travel with velocity c . The fact that they actu-ally travel slower because of interstellar dispersion, anddue to a possibly very small mass, will only strengthenthe conclusions drawn here, not weaken them. Let t γarr , t νarr be the respective arrival times.If the neutrinos also travel luminally or subluminally,no interesting conclusions can be drawn. However, if theSN1987a neutrinos also travelled superluminally, and thiscan not be ruled out, one can use the optical sightingdata to constrain the superluminality. This will be a cruder bound compared to the one to be given in thenext subsection, but still useful.With these assumptions, and an optical sighting 7200 seconds later, one gets t γem + L SN c = 7200 + t νem + L SN c − δt SN (1)which can be rearranged as δt SN = 7200 + t νem − t γem (2)Since under no circumstances can the photons leave thesupernova earlier than the neutrinos, one gets δt SN ≤ δt SN ≤ δv SN c ≤ s. y. = 6 . · − (4) Analysis based on neutrino arrival times
The analysis now proceeds as follows: let t highem , t lowem bethe emission times, at the supernova, of neutrinos of thehighest and lowest energies, respectively. Let t higharr , t lowarr be the respective arrival times. One now gets t lowarr = t lowem + L SN c − L SN δv lowSN c t higharr = t highem + L SN c − L SN δv highSN c (5)leading to L SN c { δv highSN − δv lowSN } ≤ [12 + ( t highem − t lowem )] (6)Unlike the photon arrival case, no simple arguments existfor ( t highem − t lowem ), and it would require details of neutrinoemission mechanisms from supernovae(to get some ideaof these issues see [43–47]). In most scenarios consideredin these works, high energy neutrinos precede low energyones. It seems very reasonable to take the quantity [ .. ]in eqn.(6) to be about 10 seconds. This leads to { δv highSN − δv lowSN } ≤ . · − c (7)It is also important to consider the IMB data [6] in amore careful analysis. The energies of the detected neu-trinos are higher in IMB. If very accurate timing wasavailable for all detectors, that also could have been usedto constrain the energy dependence of δv SN . But unfor-tunately, the Kamioka timing had an uncertainty of ± ±
50 milliseconds. A determination of theso called offset time from fitting data to models of neu-trino emission seems to indicate that the first IMB neu-trino arrived 0 . δv SN . They donot imply that supernova neutrinos are superluminal, butthat the available data are indeed consistent with thembeing so. They may in the end turn out to be subluminal,or just luminal too. We have left out any discussion of theeffect of neutrino masses. The current thinking on thesemasses is that they can at most be of a few eV with oneof them even being massless. For 20 MeV neutrinos, thecorrection to δvc is ≃ · − , if the mass is taken tobe 2 eV . GENERAL POWER LAW
Now we consider a possible power law dependence for δv . This analysis should be taken as an exercise in phe-nomenology with no pretenses to any deeper theoreticalconsiderations. For integer powers, these belong to theclass of so called Distorted Special Relativistic theoriesintroduced in [57–59]. OPERA data has been analysedin the context of these models in [49]. An interestingpoint stressed in [49] is that these DSR type theories stillallow for a relativity principle.The simplest candidate for a power law behaviour is δvc = α E m (8)where α is a constant with dimensions of E − m . We shallrequire eqn.(8) to be valid at all energies. Therefore δv SN δv OP = ( E SN E OP ) m (9)The data from OPERA for the advancementtimes are: 60 . , . , . , . . , . , . , . δvc these correspond to (2 . , . , . , . · − respectively. The statistical errors are the leastat 17 GeV, getting progressively higher at theother energies. The quoted statistical errors are :( ± . , ± . , ± . , ± . · − respectively, andthe systematic errors are ± . · − [1]. Estimating α , m from photon arrival time data We shall estimate the parameters α , m first usingthe photon arrival time data. Taking average energy ofSN1987a at 15 Mev and the OPERA energy at 17 GeV,where the statistics are the best, the energy ratio comesout to be 8 . · − . Taking the estimate of eqn.(4) for δv SN one gets, on using eqn.(9), m = 1 . α = (20 T eV ) − . (10)Using eqn.(10), one can estimate δβ = δvc , in units of10 − , at the energies of 13 . , . , . δβ (13 .
9) = 1 . δβ (28 .
1) = 5 . δβ (42 .
9) = 10 . . , ,
10 standard deviations(statistical)respectively. Thus this power law fit works rather poorly.This can be understood from the fact that the val-ues at 17
GeV and 28 . GeV are very close. In fact, theyare consistent with no energy dependence at all. It istherefore instructive to make an analysis with the dataat 13 . , . , . m = 1 . α = (50 T eV ) − . (12)Using eqn.(12), one can estimate δβ = δvc , in units of10 − , at the energies of 13 . , . δβ (13 .
9) = . δβ (42 .
9) = 4 .
46 (13)respectively. Now the estimates based on the powerlawfit of eqn.(8) are off the experimentally measured cen-tral values by 1 . , . δβ at 17 GeV that eqn.(12) would give: δβ (17) = 1 . · − ,differing by the measured value by 3 . Estimating α , m from neutrino arrival time data In a similar manner we can estimate the parameters α , m by using the estimates based on neutrino arrivaltimes as given in eqn.(7). As input, we need δv SN at thehighest energy, 20 MeV. Getting it from eqn.(7) requiresan estimate for δv SN at the lowest energy, taken hereto be 7.5 MeV, which can be obtained as a fraction ofthe former if we already know the exponent m . We shallcircumvent this in a practical manner by simply takingthe rhs of eqn.(7) to be the estimate for δβ SN at 20 MeV.As before, using this estimate for δβ SN in eqn.(9) gives m = 2 . α = (1 . T eV ) − . (14)Using eqn.(14), one can estimate δβ = δvc , in units of10 − , at the energies of 13 . , . , . δβ (13 .
9) = 1 . δβ (28 .
1) = 8 . δβ (42 .
9) = 23 . . , . , . m = 2 . α = (4 . T eV ) − . (16)Using eqn.(16), one can estimate δβ = δvc , in units of10 − , at the energies of 13 . , . δβ (13 .
9) = . δβ (42 .
9) = 6 .
00 (17)Quadratic corrections of the type m = 2 have been sug-gested countless number of times in the so called Quan-tum Gravity inspired models of dispersion relations. IfOPERA data even at high energies is taken seriously,these are seen to perform rather poorly. Furthermore,the intrinsic energy scales in these models is the
PlanckMass , but in these fits that scale is turning out to bearound TeV. But then, there are also models where the’Planck Mass’ can be much lower.The lesson one learns from the analysis so far is thatwhile fitting SN1987a requires steeper energy depen-dence, the OPERA data on energy dependence prefers flatter dependence. Striking a balance between these op-posing trends is the key to reconciling the OPERA andSN1987a data.
FLATTER ENERGY DEPENDENCES.Energy Independent Superluminality.
If one takes both the data of 60.7 nanoseconds at 17GeV, as well as the 60.3 nanoseconds at 28.1 GeV, onehas to more or less conclude that the observed superlu-minality is energy independent : δβ OP ≃ . · − (18)This is what the dispersion relations from the socalled Standard Model Extensions [35–37] as well as theColeman-Glashow analysis [48] would give. These pro-posals were made in the context of the so called
HighEnergy Violations of Lorentz invariance. But for neutri-nos, these analyses would seem to imply their validity as long as neutrino energies are much higher than neutrinomasses. Hence they should be applicable to the SN1987aneutrinos too.A flat velocity surplus of eqn.(18) would imply thatthe supernova neutrinos preceded the photons by about4 years! While neutrinos hitting the earth 4 years beforethe photons can not be ruled out [68], the fact that theneutrinos observed by Kamioka almost fully accountedfor the energy output in neutrinos that is expected fromthe standard supernova models, would cause some prob-lems of energetics. But it is fair to say that one needsa complete analysis of neutrino oscillations with the su-perluminality of eqn.(18) built in before one can makeconfident statements. There are already such analyses ofneutrino oscillations available in the literature [37, 60] .But this line of thinking requires the above equation tobe valid at all energies. But we shall argue that it is es-tablished, if at all, only at OPERA energies, and we shallpropose interpolations that can comfortably sit with theSN1987a data presented earlier.
Linear Energy Dependences.
It is interesting that the central values of δt quotedby OPERA at 13.9 and 42.9 GeV, namely, 53.1 ns and67.1 ns, when combined with 60.3 ns at 28.1 GeV, sug-gest a rather good linear fit to the data. But from ouranalysis till now, a flatter energy dependence at OPERAenergy scales does not extrapolate well to the supernovascales. In fact the linear dependence suggested by theabove mentioned data is roughly 1 ns increase for every2 GeV. Extrapolating 53.1 ns at 13.9 GeV to supernovaenergy scales (nearly zero on OPERA energy scales), onestill gets 46 ns which translates to a δvc of ≃ · − .This is unacceptably large, leading to neutrinos arrivingnearly 3.4 years before photons(see, however, [68])!A simple way of circumventing this is to consider alinear dependence with a suitable offset δβ OP = α E + δβ (19)where δβ is so adjusted as to cancel most of the largevalue mentioned before. In eqn.(19), α is roughly 4 . · − per 2 GeV. Even without working out δβ , we cansee that this too ends up in trouble. This is because δβ highSN − δβ lowSN ≃ α · . M eV ≃ . · − (20)is still very large, predicting a differential time of arrivalof nearly two hours for these neutrinos, whereas Kamiokadata constrains them to be at most 12 seconds.Eqn.(19), with just the offset term, is what theColeman-Glashow like Lorentz violating dispersion rela-tion [35, 36, 48] would give. Thus unless one interpretsthe Coleman-Glashow form to be valid only for energieswell above a GeV scale, we see that it does not work forneutrinos. Eqn.(19) is an example of the so called DSRtype theories discussed by [57–59]. Among the manyLorentz violating theories, these may have the advantagethat they may still accommodate a relativity principle.As it appears to be very difficult to reconcile bothOPERA and SN1987a data with a single scale energydependence, let us abandon that approach and first seekan energy dependence valid close to the OPERA ener-gies (namely 10 to 40 GeV) only. Then, a very nicefit to the available data is provided by eqn.(19) with α = (5 · T eV ) − and δv c = 1 . · − , that is δβ OP = 2 · − EGeV + 1 . · − (21)A comparison of OPERA data with various forms ofdispersion relations has also been done in [49]. Theircomparison with SN1987a is rather preliminary, and notas detailed as the one given here. They even hint at somesort of a metatheorem (!) claiming that it would not bepossible to achieve consistency with both OPERA andSN1987a data. We do not agree with that. Many pa-pers have appeared that have analyzed the compatibilityissues between OPERA and SN1987a [61–69]. INTERPOLATING OPERA AND SN1987A
Thus, if we want to give an energy dependence formula,with a single intrinsic scale , that fits both OPERA andSN1987a data, it is almost impossible. If , however, oneallows another intrinsic scale, many possibilities open up.We do not wish to explore all possible such forms, as suchan exercise is somewhat pointless at the present juncture.We give one of the simplest forms: δβ = δβ OP ( E GeV E GeV + M L ) n (22)where δβ OP is from either eqn.(18) or eqn.(21), and M L in GeV is a scale which is much larger than supernovaenergies, but much smaller than OPERA energies. Theparameters M L , n are constrained by SN1987a data ac-cording to δβ SN ≃ . · − ( E GeV M L ) n (23)We have used ≃ here because the prefactor in eqn.(23)is 2.48 if we had used eqn.(18), and 1.91 if we had usedeqn.(21). This hardly makes much difference to the esti-mates of M L , n . We will see in the next subsections thatestimates of eqn.(4) and eqn.(7) quite severely constrainthe parameters in eqn.(23). Photon arrival constraints
Let us begin with the estimate as given in eqn.(4) andtake it to be for 15 MeV neutrinos. It is straightforward to work out that for n = 1 , M L works outto 2 . , . Neutrino arrival constraints
If instead of eqn.(4) we use eqn.(7), we get that for n = 1 , M L is 50 , M L = 50 GeV is unacceptable, while M L = 1 GeV iscertainly acceptable. In any case, one should be using themore accurate and cleaner estimates provided by eqn.(7),and rule out n = 1 in eqn.(23). DISCUSSION AND CONCLUSIONS
Our aim was to discuss possible energy dependenceof the superluminal neutrino velocities reported by theOPERA collaboration. Our motivations are three fold:i. an understanding of energy dependence will providean important key to unraveling the mysteries, ii. withoutany energy dependence it is impossible to reconcile thefindings of OPERA with the accurate timing data fromSN1987a, and finally, iii. if OPERA data at both 17 and28.1 GeV are taken seriously, then, near OPERA energiesthe velocities are energy-independent. But the mean val-ues they quote for 13.9, 28.1 and 42.9 GeV hint at a clearenergy dependence. Though the statistical errors for themeasurements at 13.9 and 42.9 GeV are much poorer,as the sample sizes were drastically reduced in numberdue to the reliance on only CC events, we have takenthe central values, which are significant at three stan-dard deviations, to parametrize the energy dependenceon the one hand, and to reconcile them with SN1987a onthe other.For the SN1987a data, we have used both the timingof the first optical sighting, as well as the time spread inthe cluster of 12 events observed at the Kamioka detec-tor. We have first attempted a power law fit to boththe OPERA data at 17 GeV as well as the SN1987aestimates. Fitting the light arrival time gives an in-trinsic mass scale of 20
T eV and an energy exponentof 1.50. The calculated values at 13.9, 28.1 and 42.9GeV from this fit yield values that deviate from thequoted mean values by .5, 5.2 and 10 standard devia-tions(statistical). On the other hand, fitting photon ar-rival times to OPERA data at 28.1 GeV yields a scale of50 TeV, and an exponent of 1.39.A fit based on the 12 seconds clustering of the neutrinoevents at Kamioka iand 17 GeV data from OPERA gives,on the other hand, an intrinsic mass scale of 1 . T eV andan exponent of 2 .
42, which could be interpreted as really2 . δβ to be 1.52, 8.4 and 23.3. Suchdependences with m = 2 are suggested by many quan-tum gravity inspired models, but the scales found hereare very different.Motivated by the near linear increase of δt reportedby OPERA for the observations at 13.9, 28.1 and 42.9,we next attempted a linear fit but with an offset so thatthe linear behaviour extrapolated to supernova energieswould reconcile with the supernova arrival limits. If theslope is determined by the OPERA data, it is found thatirrespective of the offset, one gets a differential arrivaltime for 7.5 MeV and 20.0 MeV supernova neutrinos thatis very clearly in conflict with the arrival times.Interpreting all this as the inability of any single-scaleenergy dependence to account for both OPERA andSN1987a data, we decided to check how well a linearfit will work only when restricted to OPERA energies.A mass scale of 5 · TeV and an intercept(offset) of2 . · − was obtained. This fit works very well. Thiscan be tested by measuring δt at other, but comparable,energies as OPERA.Finally, an interpolation was sought between this lin-ear behaviour valid near OPERA energies, and the up-per limits on neutrino velocities obtained from supernovadata. The important point to notice is that these super-nova arrival time limits are still consistent with superlu-minal velocities for supernova neutrinos. This is an ex-citing possibility that needs to probed further. There areclearly many such interpolations possible. One of themis given in eqn.(23). The additional scale M L and theexponent n are again constrained by arrival time dataand the following, δβ = δβ OP ( E GeV E GeV + 1 . (24)seems most probable. Thus, this analysis anticipates an E - dependence for δv near supernova energies. Even at1 GeV , this relation predicts a δvc which is only th ofthat reported by OPERA at 28.1 GeV! Further Work
The OPERA collaboration should increase the statis-tics of the measurements done with the CC events. De-termining reliably the energy dependence of the effect,if true, is one of the most important things to be done.Efforts should also be made to measure the effect at dif-ferent energies. An accurate determination of the energydistribution of the neutrino flux is also important.It is important also to perform this experiment at dif-ferent baseline lengths because it is crucial to separateeffects of production and detection from those of propa-gation(the effects discussed by [74] are an example of the kind that have no dependence on the baseline distance).The claim of 18cm accuracy in the baseline length is cen-tral to the establishment of the superluminality. It istherefore critical to find several independent checks onwhat could easily become the
Achilles heel for the exper-iment.As already emphasized by Coleman and Glashow [48],the superluminal velocity effects and the energy depen-dence of them can have novel manifestations in oscilla-tion phenomena. It is therefore of utmost importanceto remodel oscillation calculations by including the en-ergy dependences considered here, and then subject themto experimental tests. In principle, the various param-eters that entered the interpolation formula in eqn.(23)could be flavour dependent. Even the TOF analysis itselfshould be remodelled taking into account possible flavourdependences, as well as effects of oscillations.The supernova neutrino data, from all the detectorsthat observed them, also needs to be thoroughly reexam-ined, particularly from the points of view of energeticsand timing.The most challenging task, should OPERA results befound to stand further scrutiny, will be in arriving at areappraisal of our notions of space and time. The entireedifice of theoretical physics stands on these pillars.
Acknowledgements
The author thanks Bala Sathiapalan, Kalyana Rama,M.V.N. Murthy, R. Shankar, H.S. Mani and Kamal Lo-daya for many useful discussions. He is thankful toKalyana Rama for stressing the importance of energy de-pendences. He acknowledges support from Departmentof Science and Technology to the project IR/S2/PU-001/2008. ∗ Electronic address: [email protected][1] OPERA, arXiv:1109.4897v1[2] G. Brunetti, Neutrino velocity measurement with theOPERA experiment in the CNGS beams, PhD thesis, injoint supervision from Universite Claude Bernard Lyon-I and Universita di Bologna, 2011.[3] V. Gharibyan, Phys. Lett. B611, 2005, p.231-238.[4] IAU Circular No 4338; K. Hirata et al [Kamiokande-IICollaboration], Observa- tion of a Neutrino Burst fromthe Supernova SN1987A, Phys. Rev. Lett. 58 (1987) 1490.[5] W. Kunkel and B. Madore, IAU Circular No 4316.[6] IAU Circular No 4340; R.M. Bionta et al, ” Observa-tion of a Neutrino Burst in Coincidence with Supernova1987A in the Large Magellanic Cloud”, Phys. Rev. Lett.58 (1987) 1494.[7] Emmanuel Saridakis,
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