Table Of ContentJanuary 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. Nu-
mericalillustrationsoftheflavourchangingprobabilityasafunctionofthedistance
and neutrino energies have also been presented. In the case of the super-massive
black hole, the interference patterns are quite different for different energies. It is,
however,difficult to imagine how one would be able to use this knowledge in prac-
tice. A more promising case is the micro black hole. Firstly, using accelerators like
the LHC or even more powerful which will be built in the future, one may pretty
well localize the spot in which such a (hypothetical) black hole will be produced.
Thedifferenceinflavourchangingprobabilities,10−6for1500kmuptoalmost10−5
fortheEarthdiameter,arestillnotpossibletodetectnow,butsuchsensitivitymay
be probably reached in the future.
References
1. B. Pontecorvo, Sov. Phys. JETP 26, 984 (1968); V.N. Gribov, B. Pontecorvo, Phys.
Lett. B28, 493 (1969).
2. S.Fukudaetal.(Super-KamiokandeCollaboration),Phys.Rev.Lett.81,1562(1998);
Y. Ashie et al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 93, 101801
(2004);Phys.Rev.D71,112005(2005);K.Abeetal.(Super-KamiokandeCollabora-
tion),Phys.Rev.Lett.97, 171801 (2006); Phys.Rev.D77, 052001 (2008); T. Araki
et al. (KamLAND Collaboration), Phys. Rev. Lett. 94, 081801 (2005); S. Abe et al.
(KamLANDCollaboration), Phys.Rev.Lett.100,221803(2008);Q.R.Ahmedet al.
(SNOCollaboration),Phys.Rev.Lett.87,071301(2001);Phys.Rev.Lett.89,011301
(2002); Phys. Rev. Lett. 89, 011302 (2002); B. Aharmin et al. (SNO Collaboration),
Phys. Rev. C 72 055502 (2005); Phys. Rev. C 75 045502 (2007); M. Apollonio et
al. (CHOOZ Collaboration), Phys. Lett. B 466, 415 (1999); Eur. Phys.J. C 27, 331
(2003); G.L. Fogli et al.,Phys. Rev.D 66, 093008 (2002).
3. S.M.Bilenky,Phys.Scr.T121,17(2005);R.N.Mohapatra,A.Y.Smirnov,Ann.Rev.
Nucl. Part. Sci. 56, 569 (2006); D.P. Roy,Phys. News 39, 51 (2009).
4. G.L. Fogli et al.,Phys.Rev.D 78, 033010 (2008).
5. C.Y. Cardall, G.M. Fuller, Phys.Rev. D 55, 7960 (1997).
6. N. Fornengo, C. Giunti, C.W. Kim, J. Song, Phys. Rev.D 56, 1895 (1997).
7. T. Bhattacharya, S.Habib, E. Mottola, Phys.Rev.D 59, 067301 (1999).
8. L. Stodolsky,Gen. Relativ. Gravit. 11, 391 (1979).
9. A.M. Ghez et al., Astrophys.J. 620, 744 (2005).
10. A. Villenkin, E.P.S. Shellard, Cosmic Strings and other Topological Defects, Cam-
bridge (1994); W.H.C. Freire, V.B. Bezerra, J.A.S. Lima, Braz. J. Phys. 30, 398
(2000).
