January 6, 2012 1:12 WSPC/INSTRUCTION FILE wfj2010 2 NEUTRINO OSCILLATIONS 1 IN STRONG GRAVITATIONAL FIELDS 0 2 n a MAREKGO´Z´DZ´1 andMAREKROGATKO2 J 1Department of Informatics, 2Department of Physics, 5 Maria Curie-Sk lodowska University, pl. Marii Curie–Sk lodowskiej 5, 20-031 Lublin, Poland ] h [email protected], [email protected] p - p e h Neutrinos do oscillate, which up to our best knowledge implies that they are massive [ particles. As such, neutrinos should interact with gravitational fields. As their masses are tiny, the gravitational fields must be extremely strong. In this paper we study the 1 influenceofblackholesdescribedbynon-trivialtopologies ontheneutrinooscillations. v Wepresentapproximateanalytical andnumericalsolutionsofcertainspecificcases. 9 4 2 1. Introduction 1 . 1 Neutrinooscillationsareoneofthemostinterestingphenomenainparticlephysics. 0 They were anticipated long time ago,1 but their detection was complicated due 2 to very weak interaction of these particles. Nowadays,these phenomena have been 1 : detected and studied in the case of solar neutrinos, reactor neutrinos, and atmo- v spheric interaction of cosmic rays.2,3 Up to our best knowledge massless particles i X cannotoscillate,andsoneutrinos(atleasttwooutofthree)musthavemass,which r isestimatedfromthesupernovaandotherastrophysicalobservationstoberoughly a 4 0.3 eV. Inthispaperwearegoingtodiscussthe(veryweak)interactionbetweenmassive neutrinos and a gravitational field. We focus here on the change of oscillation rate forneutrinospropagatingclosetoblackholes.Inprinciple,the stronggravitational field should modify the vacuum oscillation results, introducing additional phase shift. The quantum mechanical phase of neutrinos in the Schwarzschild spacetime 5 was presented in Ref. , where the authors discussed propagation including the possibleeffectofinteractionwithmatter(MSWeffect).Thenon-radialpropagation 6 of neutrinos in the aforementioned background was elaborated in Ref. , while the critical examination of the gravitationally induced quantum mechanical phases in 7 neutrino oscillations was given in Ref. . 6 WehavefoundtheresultsofRef. particularlyinteresting,astheexpressionsfor the oscillationphase Φ ,gainedby neutrinos during propagationina gravitational k field, contained terms proportional to M, the mass of the gravitating source. This observationsuggeststhatforcertainastrophysicalobjects,likesuper-massiveblack 1 January 6, 2012 1:12 WSPC/INSTRUCTION FILE wfj2010 2 M. G´o´zd´z and M. Rogatko holesforexample,thecontributiontoΦ maybesubstantial.Byperformingamore k carefull analysis we show that unfortunatelly this is not the case. In our work we use a general form of the metric, which allows us to discuss notonly the flatSchwarzschildbackground,butalsothe caseofcertaintopological defects(likeablackholepiercedbyacosmicstringandablackhole-globalmonopole system). 2. Neutrino oscillations in the vicinity of a black hole In this section we derive the formula for the quantum mechanical phase, acquired by a neutrino which propagates in a strong gravitational field, like in the vicinity of a black hole. Let us start with the following line element, dr2 ds2 = B(r)dt2+ +C˜(r)r2dθ2+C(r)r2sin2θdφ2, (1) − B(r) where B(r), C˜(r), and C(r) are functions of the radial coordinate r. In our case we areinterestedinthe motionofa neutrino inthe space-timearounda blackhole described by Eq. (1). Because of the spherical symmetry we may always confine this motion to the plane θ =π/2. This simplifies the line element to the following form: dr2 ds2 = B(r)dt2+ +C(r)r2dφ2. (2) − B(r) Moreover,weassumethatthecoefficientfunctionsB(r)andC(r)donotdependon (k) (k) time nor on the φ coordinate. It follows that the canonical momenta p and p t φ willplay the roleofconstantsofmotion:the energyE andangularmomentumJ k k of the neutrino k-th mass eigenstate, as seen by an observer at infinity. The phase,gainedby a neutrino during propagationfrom a space-time point A to a space-time point B, written in a covariant form, reads6,8 B Φ = p(k)dxµ, (3) k µ ZA where p(k) is the four-momentumofthe k-thmasseigenstateofthe neutrino,char- µ acterized by the mass m , p(k) = m g dxν. These quantities are related to each k µ k µν ds other and to the mass m by the mass-shell relation m2 =gµνp(k)p(k). k k µ ν 6 Following closely the method presented in Ref. (a detailed presentation is be- yond the scope of this contribution) we find, that the metric parameter C(r), de- scribingthepossibletopologicalnon-triviality,willcontributeonlyiftherelativistic expansion in the neutrino mass to energy ratio squared parameter, (m2/E2), will k k be performed up to the second order. Consequently, after a rather lengthy compu- tation, we finish with Φ = rB Ekdr m2k 1 m2k 2 C(dr2)r2(1+2B(r))+B(r) ,(4) k −ZrA 1−B(r)C(dr2)r2 2Ek2 − 2(cid:18)2Ek2(cid:19) 1−B(r)C(dr2)r2  q   January 6, 2012 1:12 WSPC/INSTRUCTION FILE wfj2010 Neutrino Oscillationsin Strong Gravitational Fields 3 where d denotes the impact parameter. We notice that when the (m2/E2)2 is ne- k k 6 glected, the result of Ref. is reproduced.Also, the minus sign is irrelevantand we will drop it for simplicity. 3. Flavour changing probability in neutrino oscillations Knowingthephasegainedbyneutrinosduringtheirpropagationonemayaskabout theprobabilityofaneutrinotochangeitsflavourasafunctionofthedistancefrom the source. Let us for simplicity limit ourselves to the two-neutrino case. Then the flavour changing probability in its text-book form is ∆E =sin2(2θ)sin2 t , (5) P 2 (cid:18) (cid:19) θbeingthemixingangle.Inourcase,thetime-dependentphaseshavetobemodified by the phase coming from the gravitational field. The latter, however, depends on theimpactparameterd,whichrepresentsthedistanceinwhichtheneutrinopasses the black hole. Therefore we recognize two sources of possible interference among 6 neutrinostates: betweendifferentmasseigenstatesgoingalongthesamepath,and between mass eigenstates going along slightly different paths. We call the different paths “long” (L) and “short” (S) and rewrite the standard definitions of the time- dependent fields as νe(t) = cosθ(e−i(E1t+ΦL1)+e−i(E1t+ΦS1))ν1 | i 2 | i + sinθ(e−i(E2t+ΦL2)+e−i(E2t+ΦS2))ν , (6) 2 2 | i νµ(t) = sinθ(e−i(E1t+ΦL1)+e−i(E1t+ΦS1))ν1 | i − 2 | i + cosθ(e−i(E2t+ΦL2)+e−i(E2t+ΦS2))ν . (7) 2 2 | i The usual expression for the probability now turns into sin2(2θ) = 2+cos(ΦL ΦS)+cos(ΦL ΦS) Pgrav 8 1 − 1 2 − 2 cos(ΦL hΦL+∆Et) cos(ΦS ΦS +∆Et) − 1 − 2 − 1 − 2 cos(ΦS ΦL+∆Et) cos(ΦL ΦS +∆Et) , (8) − 1 − 2 − 1 − 2 which represents the probability of ν ν transition in the bacikground of the e µ → gravitational field. As a check one notices, that for Φ = 0 the usual probability Eq.(5)isrecovered.Inwhatfollowswewillattempttoestimatethefullprobability as well as the difference for some special cases. grav grav P P −P 4. Approximate solution for the phase Φk The integral Eq. (4) is not solvable exactly in terms of elementary or special func- tions.Wewillattempt,however,togiveanestimateofthesolutionfortwoextreme cases: a super-massive and a micro black hole. January 6, 2012 1:12 WSPC/INSTRUCTION FILE wfj2010 4 M. G´o´zd´z and M. Rogatko 6 An approach that has been used in Ref. was the so-called weak field approxi- mation,inwhichitisgloballyassumedthatGM r.Thishasleadtothesolution Φ m2GM/E ,cf.Eq.(59)inRef.6,whichincr≪easeswiththemassofthe source k ∼ k 0 of the gravitational field. As this approximation is valid in certain cases, we will notuseithere.Oneexampleforwhichitcannotbeusedisthesuper-massiveblack hole which is believed to reside in the center of our galaxy. Its mass is estimated to be around 1037 kg.9 One may easily check that neither for r close to its event horizon 1010 m, nor for r being approximately the distance between the Earth ∼ and the center of the Milky Way 1020 m, this approach is not justified. ∼ A few words about possible metrics describing topological defects are in order. For example, a black hole pierced by a cosmic string is described by the metric (2) with B(r)=1 R, C(r) =1 4µ, while a black hole with a global monopole has − r − B(r)=1 8πGη2 R,C(r)=1 8πGη2.Bothofthese cases,althoughphysically − − r − different, are mathematically equivalent, with B(r) being a function of the black hole’s mass M and the distance r, and C(r) = const. The parameters µ and η are purelytheoreticalandonlyroughbounds forthemcanbe formulated,buttheyare genericallyverysmall.Inordernottoviolateexistingobservations,C(r)isbelieved to be of the order 1 10−α with α=6 15.10 − − Forthe asymptoticcaser onemayreplacebothB(r) andC(r) by1.This →∞ yields in the leading order m2 rB dr m2 d2 rB m2 Φfar = k = k r 1 k (r r ). (9) k 2Ek ZrA 1− dr22 2Ek " r − r2#rA ≈ 2Ek B − A q For the close limit, r d R, we approximate B(r) 1 R. This simplification ≈ ≥ ≈ − d results in d′ m2 R d2 Φclose = k r 1 1 . (10) k 2Ek " s −(cid:18) − d(cid:19)Cr2#d Onemaycheckbysolvingtheinequalityr3 d2r/C+d2R/C >0,thatifonlyr >R − thereisalwaysaC suchthatΦ isreal.AnexampleispresentedinFig.1,inwhich k theSchwarzchildradiushasbeentakentobe1010 m.Thisvaluecorrespondstothe super-heavy (M 1037 kg) black hole that is anticipated to reside in the center of our galaxy. On t∼he other hand, for a micro black hole (M 10−27 kg) that may appear in the LHC experiments, Fig. 1 has to be rescaled su∼ch that R 10−54 m. ∼ Thegenericapproximationpresentedabovemaybereformulatedinsomespecial numerical cases by taking the leading terms, which dominate significantly over the others. For instance, in the super massive case, the close limit up to the second order in relativistic expansion is given by Φclose Ek m2k 2 1+ m2k r52 + 1 m2k r92 d′. (11) k ≈ √2GMd22E2 5 4E2 9E2d2 k (cid:20) (cid:18) k(cid:19) k (cid:21)d January 6, 2012 1:12 WSPC/INSTRUCTION FILE wfj2010 Neutrino Oscillationsin Strong Gravitational Fields 5 ´1010m 4 Fig.1. Distancer as afunctionof the met- 2 ric parameter C, that corresponds to real R~d C phases Φk in the case of a super massive 0.2 0.4 0.6 0.8 1.0 blackholefromthecenteroftheMilkyWay. -2 Theshadedregionrepresentsphysicallyac- -4 ceptablesolutions. -6 On the other hand, the close limit for a micro black hole takes the form d′ m2 1 m2 r2 5d2 Φclose E k r2 d2 k − . (12) k ≈ k2Ek2" − − 22Ek2√r2 d2# p − d The far limits are basically unaffected, as all metrics we may be interested in are asymptotically flat. The oscillation probabilities for the super-massive black hole are depicted in Fig. 2. We recall here, that the Earth, thus our would-be observation point, is roughlyatthedistancer 2.5 1020m.Theactualnumbersusedinthecalculations ≈ × were d =R, d =100R, m =0.30 eV, m =0.29 eV, and ∆E =1 eV. In Fig. 3 S L 1 2 we have collected the functions calculated for different energies of the grav P −P neutrinos. In all the cases the oscillatory behaviour with interference patterns is eminent. Another possible case of interest is a micro black hole. Given that certain the- ories which assume the existence of additional spatial dimensions are true, at the energyscaleswhichwillbereachedintheLargeHadronColliderinCERNsomeex- tradimensionalblackholesshouldappear.Theirmassisexpectedtobeoftheorder of 10−27 kg, which corresponds to the Schwarzschild radius R 10−54 m. These ∼ extremely tiny objects would almost instantly evaporate, however, some models predictsimilarobjectstobe createdandtravelalmostfreelythroughthe Universe. As such, they may act as gravitational lenses (in the same way as does regular black holes and other massive dark objects). The results are presented in Fig. 4 fromwhichoneseesthatthedifferenceintransitionprobabilitiesmayreachtheor- der of10−6 andmore onthe distance ofatleast1500km.This mayhypothetically 1.2 1.2 1 1 0.8 0.8 grav 00..46 P 00..46 P 0.2 0.2 0 0 -0.2 -0.2 2.0 2.2 2.4 2.6 2.8 3.0 2.0 2.2 2.4 2.6 2.8 3.0 r [1020 m] r [1020 m] Fig. 2. Flavour changing probability Pgrav in neutrino oscillations as a function of the distance totheMilkyWayblackhole.TheresultbasedonthestandardformulaP isshownforreference (rightpanel).NeutrinoenergyEk=1GeV.SeealsoFig.3. January 6, 2012 1:12 WSPC/INSTRUCTION FILE wfj2010 6 M. G´o´zd´z and M. Rogatko 1 1 Pgrav - 00 ..055 E=10 MeV - 00 ..055 E=100 MeV -1 -1 2.0 2.2 2.4 2.6 2.8 3.0 2.0 2.2 2.4 2.6 2.8 3.0 1 1 Pgrav - 00 ..055 E=1 GeV - 00 ..055 E=10 GeV -1 -1 2.0 2.2 2.4 2.6 2.8 3.0 2.0 2.2 2.4 2.6 2.8 3.0 1 V 1 Pgrav - 00 ..055 100 Ge - 00 ..055 =1 TeV = E -1 E -1 2.0 2.2 2.4 2.6 2.8 3.0 2.0 2.2 2.4 2.6 2.8 3.0 r [1020 m] r [1020 m] Fig.3.DifferencePgrav−P forvariousenergiesEk oftheneutrinos. givesomechancesforthefutureverylongbaselineneutrinooscillationexperiments to detect the presence of such an object in the area of the beam. 5. Conclusions 6 Basing on the results presented in Ref. , in which the phase Φ is proportional k to M, one may expect that the black holes should generate a gravitational field strong enough, to give a substantial contribution to the neutrino oscillations. The conclusion from our calculations is quite contrary. Even though the general formula is known, see Eq. (4), its solution depends on theactualmetricparameters,andinmostcasesrequirespurenumericaltreatment. We have managed to formulate approximate analytical solutions for two distinct 3e-06 1e-11 8e-12 2e-06 6e-12 P 4e-12 1e-06 P-grav --- 2642eeee----1111 02222 -1e-0 06 -8e-12 -2e-06 -1e-11 -3e-06 1 2 3 4 5 6 7 8 9 10 0 250 500 750100012501500 r [m] r [km] Fig.4.DifferencePgrav−P asafunctionofthedistancetothemicroblackhole.Neutrinoenergy Ek =1GeV. January 6, 2012 1:12 WSPC/INSTRUCTION FILE wfj2010 Neutrino Oscillationsin Strong Gravitational Fields 7 examples of a super-massive and a micro black hole with topological defects. 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