he neutrinocase oss-induce

Physics Letters A 351 (2006) 373–378

www.elsevier.com/locate/pla

Neutrino wave packet propagation in gravitational fields

Dinesh Singha, Nader Mobeda,∗, Giorgio Papinia,b,c

a Department of Physics, University of Regina, Regina, SK, S4S 0A2, Canadab Prairie Particle Physics Institute, Regina, SK, S4S 0A2, Canada

c International Institute for Advanced Scientific Studies, 89019 Vietri sul Mare (SA), Italy

Received 1 May 2005; received in revised form 13 November 2005; accepted 14 November 2005

Available online 21 November 2005

Communicated by P.R. Holland

Abstract

We discuss the propagation of neutrino wave packets in a Lense–Thirring metric using a gravitational phase approach. We show that toscillation length is altered by gravitational corrections and that neutrinos are subject to helicity flip induced by stellar rotation. For thef arapidly rotating neutron star, we show that absolute neutrino masses can be derived, in principle, from rotational contributions to the madenergy shift, without recourse to mass generation models presently discussed in the literature. 2005 Elsevier B.V. All rights reserved.

PACS: 04.90.+e; 14.60.Pq; 04.80.+z; 97.60.Bw

Keywords: Neutrino wave packets; Gravitational phase; Helicity flip; Absolute neutrino mass

dif-aks os oillaactrinhe

ofem

tes-

ofthe

nsity,singinsi-Theforran-

in-

n.

1. Introduction

Recent experimental evidence from Super-Kamiokande[1]and SNO[2] has significantly endorsed the claim[3] that neu-trinos undergo flavour oscillations in vacuum due to theference of their rest masses. This discovery, however, mno pronouncements about the intrinsic physical propertieneutrinos, particularly whether they exist as plane wavewave packets. In addition, the actual calculation of the osction length for solar neutrinos assumes a flat space–time bground. While this assumption seems reasonable for neupropagation in our solar system, it may be unwarranted wthe gravitational source is a neutron star or a black hole.

The purpose of this work is to investigate the contributiongravitation to the flavour oscillations of a two-neutrino systin a background described by the Lense–Thirring metric[4],assuming a wave packet description of the mass eigenstamomentum space. This approach[5], so far untested in gravi

* Corresponding author.E-mail addresses: singhd@uregina.ca(D. Singh),

nader.mobed@uregina.ca(N. Mobed),papini@uregina.ca(G. Papini).

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2005.11.032

esfr-k-on

in

tational problems, offers an intuitively satisfying descriptionneutrinos. It relates directly the physical characteristics ofgravitational source, such as its stellar temperature and deto the size of the wave packet. It also lends itself well to sendistortions in space–time because of the common and intrcally non-local nature of both curvature and wave packets.choice of the Lense–Thirring metric is particularly relevantthis purpose, since rotational effects may induce a helicity tsition while the neutrino is in transit.

2. Gravitational phase and spin-gravity coupling

A particularly interesting feature of our approach is thetroduction of the gravitational phase that leads to adirect spin-gravity coupling interaction within the neutrino’s wavefunctioIt was shown[6–9] that the gravitational phase

ΦG ≡ 1

2

x∫x0

dzλ γαλ(z)pα

(1)+ 1

4

x∫dzλ

[γβλ,α(z) − γαλ,β(z)

]Lαβ(z),

x0

374 D. Singh et al. / Physics Letters A 351 (2006) 373–378

avt o,n-thede

ns-

by

n e-

m,

ds oed

s

estor

b

rtaklypat

oachtatesof

t isrip-

n

or.

n-erm

and

de-and

er-d tonon-

enters the description of quantum particles in external grtational fields in a way that is essential and independentheir spin. In (1), γµν = gµν − ηµν is the metric deviationandpµ andLαβ are momentum and orbital angular mometum operators of the free particle. It leads to a solution ofcovariant Klein–Gordon equation that is exact to first orin γµν and invariant under the gauge transformationsγµν →γµν − (ξµ,ν + ξν,µ) induced by the co-ordinate transformatioxµ → xµ + ξµ. Application of (1) to closed space–time integration paths gives rise to a covariant Berry’s phase[10] withconsequences for particle interferometry[11]. Detailed com-parisons of the results obtained from(1) with those given byDeWitt [12], Papini [13], Anandan[14], Stodolsky[15], andother authors are available[11] for further reference.

Equation(1) also yields the particle deflection predictedgeneral relativity in the geometrical optics approximation[16].When applied to fermions,ΦG and the spin connectionΓµ re-produce all those gravitational-inertial effects that have beether observed[17,18]or predicted[19,20], and predicts, in particular, the non-conservation of helicity[21] for strictly mass-less fermions.

To show howΦG acts on a neutrino propagating in vacuuwe first start with the covariant Dirac equation[iγ µ(x)(∇µ +iΓµ) − m/h]ψ(x) = 0, wherem is the neutrino rest mass, an∇µ is the usual covariant derivative. We use geometric unitG = c = 1 [22], so that all physical quantities can be describin units of length. Then for the Lense–Thirring metric in−2signature given by

g =(

1− 2M

r

)dt ⊗ dt

−(

1+ 2M

r

)(dx ⊗ dx + dy ⊗ dy + dz ⊗ dz)

+ 4

5

MΩR2

r3

[x(dy ⊗ dt + dt ⊗ dy)

(2)− y(dx ⊗ dt + dt ⊗ dx)],

wherer = √x2 + y2 + z2, M andR are the mass and radiu

of the gravitational source, andΩ is its angular velocity, thecorresponding Dirac Hamiltonian is

H0 ≈(

1− 2M

r

)α · p + m

(1− M

r

)β + ih

M

2r3(α · r)

(3)+ 4

5

MΩR2

r3Lz + 1

5

hMΩR2

r3

[3z

r2(σ · r) − σ z

],

whereα andβ are the usual four-dimensional Dirac matricandLz = xpy − ypx is the orbital angular momentum operain thez-direction. The gravitational phase can be introducedmeans of the transformationψ(x) → exp(iΦG/h)ψ(x) and thenew Hamiltonian takes the formH = H0 + HΦG, where

(4)HΦG = α · (∇ΦG) + (∇tΦG)

is a first-order correction. We then treat(4) as a perturbation foa two mass-species neutrino system. While the approachis based on standard quantum mechanics, in order to app(1)it is necessary to assume that the average of all possible

i-f

r

i-

f

y

en

hs

reduces to the integration path ofΦG. Because some terms ofH

are non-diagonal with respect to spin, the wave packet apprshould also make known whether the neutrino mass eigensare subject to a helicity transition in the gravitational fielda rotating source.

3. Neutrino wave packets

Adopting the Dirac representation, the matrix elemen〈ψ(r)|HΦG|ψ(r)〉, and we assume for the wave packet desction

(5)∣∣ψ(r)

⟩ = 1

(2π)3/2

∫d3k ξ(k)eik·r ∣∣U(k)

⟩±,

where ξ(k) = ξ(kx)ξ(ky)ξ(kz) is the normalized Gaussiawavefunction of widthσp and centroid〈k〉, and

(6)ξ(kj

) = 1

(√

2π σp)1/2exp

[− (kj − 〈k〉j )2

4σ 2p

].

The normalized four-spinor|U(k)〉± is

∣∣U(k)⟩± =

√E + m

2E

(1

hσ ·kE+m

)⊗ |±〉,

(7)|+〉 =(

10

), |−〉 =

(01

),

whereE = √p2 + m2 is the energy associated with the spin

By symmetrizing over the exchange betweenk andk′ to clearlyidentify all non-zero terms in the matrix element, explicit costruction shows contributions due to both a spin diagonal tand a helicity transition, in the form⟨ψ(r)

∣∣HΦG

∣∣ψ(r)⟩

= h3/2

(2π)3V

∫d3r d3k d3k′ ξ(k)ξ(k′)

√E + m

2E

√E′ + m

2E′

×

cos[(k − k′) · r][

h(∇ΦG)S ·(

k

E + m+ k′

E′ + m

)

+ (∇tΦG)S

(1+ h2(k · k′)

(E + m)(E′ + m)

)]〈±|±〉

− sin[(k − k′) · r][

h(∇ΦG)S ×(

k

E + m− k′

E′ + m

)

(8)− h2(∇tΦG)S(k × k′)(E + m)(E′ + m)

]· 〈∓|σ |±〉

,

where (∇µΦG)S = (h/2)[(∇µΦG)(k) + (∇µΦG)(k′)] is thesymmetrized form of the gravitational phase gradients,V = (4π/3)(r3 − R3) is the volume of spatial integration fromthe star’s surface. In principle, for an integration measurescribed in terms of spherical co-ordinates for both positionmomentum space, the radial integration limits go fromr = R tor = ∞ and the wave number integration limits go from|k| = 0to |k| = ∞. Because of numerical limitations inherent in pforming the computation, the integration limits are truncatethe region of phase space where the wave packet is clearlyzero. It is clear from the last line of(8) that there would be no

D. Singh et al. / Physics Letters A 351 (2006) 373–378 375

ar og

nerenwit

tho

e tin

thear

n atith

heve

l tesinents

f

ff-tion

tion

-s-

Fig. 1. Orientation of the neutrino beam in relation to a rotating neutron stmassM , radiusR, and angular velocityΩ which describes the Lense–Thirrinmetric.

helicity transition contribution if we considered strictly plawaves, since a non-zero transition amplitude requires diffemomentum components within the wave packet to interactthe Pauli spin matrices to yield a non-zero result. Choosingspin quantization axis to be along the neutrinos’ directionpropagation, the helicity transition element is

〈∓|σ |±〉 = [cosθ cosϕ ∓ i sinϕ]x(9)+ [cosθ sinϕ ± i cosϕ]y − sinθ z,

where the upper sign refers to the transition from negativpositive helicity.Fig. 1shows the neutrino beam orientationterms of the angular co-ordinates adopted here.

4. Matrix element due to gravitational phase

The explicit calculation of the integrals of(8) in sphericalco-ordinates is prohibitively complicated by the coupling ofamplitudes to the oscillatory functions, whose argumentsthemselves dependent on sinusoidal functions. However, aproximate, but analytic expression for(8) can be found by firsperforming a 2nd-order Taylor expansion of the amplitude wrespect tom, along with an 11th-order Taylor expansion of toscillatory functions, and then integrating term by term oboth position and momentum space. Then(8) takes the form

⟨ψ(r)

∣∣HΦG

∣∣ψ(r)⟩

= h〈k〉

M

r

[C0(q, r) + C1(q, r)m + C2(q, r)m2]

(10)

+ MΩR2

r2sinθ

[D0(q, r) + D1(q, r)m + D2(q, r)m2],

where m = m/〈p〉 = m/(h〈k〉), q = 〈k〉/σp, and 〈k〉 ≡ |〈k〉|.The dimensionless functionsCj (q, r) andDj(q, r) that arisefrom the integration are very lengthy in form, each one equathe sum of dozens of terms. An analytic description of thfunctions will be presented in a future paper. The first lof (10) is the contribution due to the diagonal compone

f

thef

o

ep-

r

oe

Fig. 2. Dimensionless functions for Eq.(10) due to diagonal elements o〈ψ(r)|HΦG|ψ(r)〉 assuming 10 MeV neutrinos, given ar = 10 kpc neutronstar source ofM = 1.5M andR = 10 km.

of 〈ψ(r)|HΦG|ψ(r)〉, while the second line is due to the odiagonal components and refers to the helicity flip contribuof the perturbation.

For this Letter, the gravitational source under considerais a rapidly rotating 1.5M neutron star withR = 10 km andΩ = 1 kHz, at a distance ofr = 10 kpc from the neutrino detector. Figs. 2 and 3contain a list of plots for the functiondescribed in the matrix element(10). For this choice of pa

376 D. Singh et al. / Physics Letters A 351 (2006) 373–378

o

torala

con

1thwitht si-t in

d re-% ins no

das

cetialirelynsi-gre-thattiontricrti-histua-

eu-ntsentlues

en

il-

Fig. 3. Dimensionless functions due to non-diagonal elements〈ψ(r)|HΦG|ψ(r)〉 in Eq. (10) for 10 MeV neutrinos, given ar = 10 kpcneutron star source ofM = 1.5M andR = 10 km.

rameters, it is clear from a comparison ofFigs. 2(a)–(c) andFigs. 3(a)–(c) that |C2(q, r)| |C1(q, r)| |C0(q, r)| and|D2(q, r)| |D1(q, r)| |D0(q, r)| for all choices ofq. Thissuggests the trend towards convergence of the series duepansion with respect tom. To justify the truncation of the Tayloexpansion of(8), we note from analyzing a one-dimensionanalogue of the problem that the Gaussian functions whichpresent in the integrand have the effect of damping out the

f

ex-

re-

tribution of the higher-order expansion terms beyond the 1order. A plot of this series expansion matches preciselythat due to the corresponding product function of the exacnusoidal function with a one-dimensional Gaussian, excepthe tail regions of the Gaussian envelope. In those isolategions of parameter space, we estimate an error of 10–15the amplitude and are confident that this degree of error hasignificant bearing on our results.

We stress some interesting features of(10). The most obvi-ous one is the presence of termslinear in m, due to the Taylorexpansion of〈ψ(r)|HΦG|ψ(r)〉 for both the spin diagonal anoff-diagonal terms. This fact has interesting implications,shown below, in the calculation of the energy shifts. Given(9),we know that the helicity transition contribution to(10) is dueentirely to thez-component of the transition amplitude, sinthe x- and y-components average out to zero over all spaangles. Furthermore, the non-zero contribution is due entto the rotation of the source, which induces the helicity tration. Because of the sinθ term, propagation of a neutrino alonthe±z-axis suggests that its initially prepared helicity statemains fixed throughout its motion. This is a sensible resultis consistent with our present understanding of spin-rotacoupling, since the rotational term in the Lense–Thirring metends to act like a uniform magnetic field that orients the pacle spin either parallel or antiparallel to the field strength. Tresult also agrees with the fact that, barring quantum fluctions about the classical integration path of(1), ∇tΦG = 0 in theLense–Thirring field, which implies helicity conservation[21].

5. Gravitational corrections to neutrino oscillation length

To determine the mass-induced energy shift for the ntrino oscillation length, we note that the off-diagonal elemeof (10) contribute to a second-order effect in time-independperturbation theory, where the unperturbed energy eigenvaE±

0 come fromH0ψ(x) = E±0 ψ(x) for E±

0 ≈ √〈p〉2 + m2 −2Mr

〈p〉 + 45

MΩR2

r3 (Lz ± h2), and

(11)E+0 − E−

0 = 4

5

hMΩR2

r3.

By virtue of (11), we can calculate the energy shift for a givneutrino, leading to a new energy ofE±

m ≈ E±0 +(E)±m, where

(12)(E)±m = 〈±|HΦG|±〉 ±∣∣〈−|HΦG|+〉∣∣2

E+0 − E−

0

.

This leads[23,24] to the final expression for the neutrino osclation lengthLosc. = 2π/(E±

m2− E±

m1), where

E±m2

− E±m1

= h〈k〉

1

2

(m2

2 − m21

)

+[M

rC1(q, r) ± MΩR2

r2sin2 θF1(q, r)

](m2 − m1)

+[M

rC2(q, r) ± MΩR2

r2sin2 θF2(q, r)

](m2

2 − m21

)

(13)

D. Singh et al. / Physics Letters A 351 (2006) 373–378 377

e-rg

netiae tnelein il

ondseg-

d-liondif-the

-r

dthn

r’sshowp-

ee

ons,isms

uslytis-are

uire-ing

n in-rino

nces

. 86

t. 86

t. 82

Fig. 4. Dimensionless functions for helicity flip terms in Eq.(13), assuming thesame source and neutrino energy conditions as listed inFigs. 2 and 3.

and

(14)F0(q, r) = 5

4r〈k〉D2

0(q, r),

(15)F1(q, r) = 5

2r〈k〉D0(q, r)D1(q, r),

(16)F2(q, r) = 5

4r〈k〉[D2

1(q, r) + 2D0(q, r)D2(q, r)].

The plots of(15)and(16)are listed inFig. 4.It is clear from Figs. 2 and 4that the functions becom

extremely large forq → 0, corresponding to very large momentum spread in the wave packet for a given neutrino enewhich then rapidly decay to zero for largeq. In particular, theplots show that the helicity transition terms in(13) will dom-inate asq → 0, which is consistent with(8), since a largemomentum spread in the wave packet is required to sense thedifferential rotational effects within a localized region of space–time near the star’s surface.

One important question concerns the coherence of thetrino wave packet while propagating outwards, and its potenimpact on the quantitative results presented here. Becauseffects of gravitational phase are most dominant when thetrino is near the surface of the central mass, it is reasonabsurmise that the wave packet remains coherent at this pointtime evolution. Referring to(8), the portion of the radial integra

y,

u-lheu-tots

for distances much larger than the wave packet size correspto a very rapid oscillation of the integrand, which yields a nligible contribution to the matrix element.

6. Conclusions

For solar neutrinos and those due to supernovae[24], theexpected wave packet widths in momentum space arehσp ≈10−5 MeV and 2× 10−2 MeV, respectively. The corresponing values forq ≈ 106 andq ≈ 50 indicate that gravitationacorrections have negligible effect on the neutrino oscillatlengths for these scenarios. However, the situation is quiteferent when applied to rapidly rotating neutron stars. Forneutron star parameters we consider,M/r ≈ 7.175×10−18 andMΩR2/r2 ≈ 7.751× 10−36. In order to predict a 1% correction to the value ofm2

21 = m22 − m2

1 as determined by solaneutrino experiments, we require|F2(q, r)| ≈ 1.3667× 1033,which suggests a choice ofq ≈ 2.1 × 10−5 from Fig. 4(b),and implies a wave packet width ofhσp ≈ 4.762× 105 MeV.This prediction corresponds very well to the calculated wiof hσp ≈ 3.260× 105 MeV for neutrinos emitted from neutrostars, as determined by a mean-free-path calculation[24], as-suming a stellar temperature of 3×106 K [25], and an effectivestellar density of 1011 g/cm3 averaged over the neutron staexpected core and surface densities. Our results thereforethat helicity flip likely plays a role in the case of neutrinos proagating in the field of rotating neutron stars.

From Fig. 4(a), it also follows that|F1(q, r)| ≈ 1.2150×1038 for the same choice ofq as when applied toFig. 4(b),and so the contribution in(13) due to linear mass differencm21 = m2 − m1 is not negligible. This result suggests thpossibility of performing, in principle, a parameter fit of(13)for suitable choices ofq, m2

21, and m21 so that we caninfer the absolute neutrino masses entirely from observatiand without any reference to the mass generation mechanpresently under consideration in the literature[23]. In practice,however, such an undertaking would require an enormolarge neutrino flux and large counting rates to obtain statically significant measurements. To our knowledge, thereno experimental facilities available that can meet these reqments. Nonetheless, the theoretical possibility of determinabsolute neutrino masses by this technique makes for ateresting consideration in the development of future neutobservatories.

Acknowledgements

This research was supported, in part, by the Natural Scieand Engineering Research Council of Canada (NSERC).

References

[1] Y. Fukuda, et al., Super-Kamiokande Collaboration, Phys. Rev. Lett(2001) 5656;Y. Fukuda, et al., Super-Kamiokande Collaboration, Phys. Rev. Let(2001) 5651;Y. Fukuda, et al., Super-Kamiokande Collaboration, Phys. Rev. Let(1999) 2644;

378 D. Singh et al. / Physics Letters A 351 (2006) 373–378

433

436

467

02)

02)

01)

on:84)

ntemic

gnce

005)

979)

9..As-

ood

ron

Y. Fukuda, et al., Super-Kamiokande Collaboration, Phys. Lett. B(1998) 9;Y. Fukuda, et al., Super-Kamiokande Collaboration, Phys. Lett. B(1998) 33;Y. Fukuda, et al., Super-Kamiokande Collaboration, Phys. Lett. B(1999) 185.

[2] Q.R. Ahmad, et al., SNO Collaboration, Phys. Rev. Lett. 89 (20011302;Q.R. Ahmad, et al., SNO Collaboration, Phys. Rev. Lett. 89 (20011301;Q.R. Ahmad, et al., SNO Collaboration, Phys. Rev. Lett. 87 (20071301.

[3] V. Gribov, B. Pontecorvo, Phys. Lett. B 28 (1969) 493.[4] J. Lense, H. Thirring, Z. Phys. 19 (1918) 156 (English translati

B. Mashhoon, F.W. Hehl, D.S. Theiss, Gen. Relativ. Gravit. 16 (19711).

[5] C. Giunti, Found. Phys. Lett. 17 (2004) 103, hep-ph/0302026.[6] Y.Q. Cai, G. Papini, Phys. Rev. Lett. 66 (1991) 1259;

Y.Q. Cai, G. Papini, Phys. Rev. Lett. 68 (1992) 3811.[7] D. Singh, G. Papini, Nuovo Cimento B 115 (2000) 223.[8] G. Papini, in: P.G. Bergmann, V. de Sabbata (Eds.), Advances in the I

play between Quantum and Gravity Physics, vol. 317, Kluwer AcadeDordrecht, 2002, gr-qc/0110056.

[9] G. Papini, in: G. Rizzi, M.L. Ruggiero (Eds.), Relativity in RotatinFrames, Kluwer Academic, Dordrecht, 2004, Chapter 16 and referetherein, gr-qc/0304082.

r-,

s

[10] Y.Q. Cai, G. Papini, Mod. Phys. Lett. A 4 (1989) 1143;Y.Q. Cai, G. Papini, Class. Quantum Grav. 7 (1990) 269;Y.Q. Cai, G. Papini, Gen. Relativ. Gravit. 22 (1990) 259.

[11] Y.Q. Cai, G. Papini, Class. Quantum Grav. 6 (1989) 407.[12] B.S. DeWitt, Phys. Rev. Lett. 16 (1966) 1092.[13] G. Papini, Nuovo Cimento B 45 (1966) 66;

G. Papini, Phys. Lett. A 23 (1966) 418;G. Papini, Phys. Lett. A 24 (1967) 32.

[14] J. Anandan, Phys. Rev. D 15 (1977) 1448.[15] L. Stodolsky, Gen. Relativ. Gravit. 11 (1979) 391.[16] G. Lambiase, G. Papini, R. Punzi, G. Scarpetta, Phys. Rev. D 71 (2

073011.[17] S.A. Werner, J.-L. Staudenmann, R. Colella, Phys. Rev. Lett. 42 (1

1103;L.A. Page, Phys. Rev. Lett. 35 (1975) 543.

[18] U. Bonse, T. Wroblewski, Phys. Rev. Lett. 51 (1983) 1401.[19] F.W. Hehl, W.-T. Ni, Phys. Rev. D 42 (1990) 2045.[20] B. Mashhoon, Phys. Rev. Lett. 61 (1988) 2639.[21] D. Singh, N. Mobed, G. Papini, J. Phys. A: Math. Gen. 37 (2004) 832[22] C.S. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, Freeman, 1973[23] M. Fukugita, T. Yanagida, Physics of Neutrinos and Applications to

trophysics, Springer-Verlag, 2003.[24] C.W. Kim, A. Pevsner, Neutrinos in Physics and Astrophysics, Harw

Academic Press, 1993.[25] S.L. Shapiro, S.A. Teukolsky, Black Holes, White Dwarfs, and Neut

Stars: The Physics of Compact Objects, John Wiley & Sons, 1983.