Neutrino oscillations and Lorentz invariance breakdown

Physics Letters B 560 (2003) 1–6 www.elsevier.com/locate/npe

Neutrino oscillations and Lorentz invariance breakdown G. Lambiase a,b a Dipartimento di Fisica “E.R. Caianiello” Universitá di Salerno, 84081 Baronissi (Sa), Italy b INFN, Gruppo Collegato di Salerno, Italy

Received 27 January 2003; received in revised form 24 February 2003; accepted 6 March 2003 Editor: G.F. Giudice

Abstract The breakdown of Lorentz’s and CPT invariance gives rise to the occurrence of helicity terms in the dispersion relation of fermions. Consequences of these terms are reviewed in the framework of spin flavor oscillations of neutrinos, and resonance effects are discussed. Bounds on the 0-component of the pseudo-vector potential bµ are derived for solar and atmospheric neutrinos.  2003 Published by Elsevier Science B.V. PACS: 11.30.Cp; 14.60.Pq

The Lorentz invariance breakdown and its consequences in physics are nowadays a very active area of research. Theoretical evidence of a possible violation of the Lorentz invariance has been suggested by Kostelecký and Samuel [1] in the framework of string theory. More recently a deviation from the Lorentz symmetry has been also suggested in [2], as well as in the context of D-branes [3], and loop quantum gravity [4]. Lorentz’s invariance violation due to non-trivial solution of (open) string field theory, as proposed in Ref. [1], follows from the observation that even though the underlying theory is invariant under Lorentz and CPT transformations, the vacuum solution of the theory could spontaneously violate these symmetries. The breakdown of these fundamental symmetries occurs in the extension of the standard model, and for preserve the standard model power-counting renormalizability, one has to take into account only terms involving operators of mass dimension four or less [5]. The fermion Lagrangian density is generalized as [5–7] L=

i i ¯ ← →ν →ν → ¯ µ ψ − bµ ψγ ¯ 5 γ µ ψ + i cµν ψγ ¯ µ← ¯ 5γ µ← ∂ ψ + dµν ψγ ∂ ψ ψγµ ∂ µ ψ − aµ ψγ 2 2 2 ¯ ¯ µν ψ − mψψ, − Hµν ψσ

(1)

where aµ , bµ , . . . are constants. As shown by Bertolami and Carvalho, the dispersion relation of particles turns out to be [7] (we shall use natural units c = 1 = h¯ ) pµ pµ = −2c00E 2 − 2a0E ± (b0 + d00E)p. E-mail address: [email protected] (G. Lambiase). 0370-2693/03/$ – see front matter  2003 Published by Elsevier Science B.V. doi:10.1016/S0370-2693(03)00379-4

(2)

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Eq. (2) is a special case of Eq. (14) derived in Ref. [6]. The ± signs stand for the helicity of propagating fermions. In what follows, we drop the a0 -term since it may lead to changing flavor neutral currents when more than one flavor is involved [7]. Besides, we take c00  1 and d00  1. The aim of this Letter is to investigate the consequence of fermion helicity terms appearing in the dispersion relation (2) and determine an upper bound on b0 . The analysis is carried out in the framework of spin flip conversion (ν ) of neutrino flavors by assuming that b0 is the same for all species of neutrinos (b0(νe ) = b0 µ = b0(ντ ) = b0 ). Since in the relativistic regime the helicity operator is equal, up to the factor (m/E)2  1, to the chiral operator [8], Eq. (2) can be cast in the form EL,R p +

m2 ± b0 , 2p

(3)

where not essential terms have been omitted. The terms ±b0 change the effective energy of neutrinos depending if they are left- or right-handed. If neutrinos propagate in a medium, then their energy is shifted owing to the weak interaction with the background matter (MSW √[9]. The elastic scattering through charged (neutral) current √ effect) interaction gives the energy contribution 2 GF ne ( 2 GF nn /2), where GF is the Fermi constant, and ne (nn ) is −10.5r/R , the electron (neutron) density. In what follows we shall take nn ∼ ne /2 [10]. For the 0e √ Sun, ne (r) = n−12 −3 eV [10]. where n0 = 85N √ A cm , NA is the Avogadro number [11,12]. At r = 0, one gets 2 GF ne (0) ∼ 10 For the Earth, 2 GF ne (0) ∼ 10−14 eV [10]. Besides, neutrinos interact with the magnetic field [13] µν ¯ Fµν ψ, Lint = ψµσ

(4) 1 µ ν 4 [γ , γ ].

where µ is the magnetic momentum of the neutrino, Fµν = The equation of evolution describing the conversion between two neutrino flavors is, therefore, [12,14,15]     νeL νeL d  ν¯   ν¯  i  eR  = H  µR  , (5) νµL dr νµL ν¯ µR ν¯ µR is the electro-magnetic field tensor, and σ µν

where, in the chiral base, the matrix H is the effective Hamiltonian [9,10,12,14,16]  m2 m2 − 4p cos 2θ + b0 + 3A 0 4p sin 2θ  2 m  0 −µB− 4p cos 2θ − b0 − 3A H= 2 2  m −µB+ − m  4p sin 2θ 4p cos 2θ + b0 − A µB−

m2 4p

sin 2θ

0

µB+ m2 4p

sin 2θ 0

m2 4p

     

cos 2θ − b0 + A (6)

√ up to terms proportional to identity matrix. Here A ≡ 2 GF ne /4, m2 ≡ m22 − m21 (we assume the mass hierarchy m2 > m1 ), and B± = Bx ± iBy is the magnetic field [14]. The effective Hamiltonian H does contain also a gravitational field term, but it vanishes for Schwarzschild-like geometry due to spherical symmetry. We restrict to flavors νe − νµ , but obviously the analysis works also for different neutrino flavors (νµ − ντ , νe − ντ ). The resonant conditions follow from the diagonal elements of the effective Hamiltonian (6) √ 2 m2 cos 2θ − GF ne (rres ) = 0, νeL → ν¯ µR , (7) −2b0 + 2p 2 √ 2 m2 cos 2θ + GF ne (rres ) = 0. −2b0 − νµL → ν¯ eR , (8) 2p 2 In both cases, we only have transitions from left- to right-handed neutrinos, the latter being sterile neutrinos do not have electroweak interaction with matter. Clearly, the resonant transitions do not occur simultaneously. Since

G. Lambiase / Physics Letters B 560 (2003) 1–6

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the spin flip transition into muon antineutrinos is a viable solution to solar neutrino problem [17], we analyze the νeL → ν¯µR transition. From Eq. (7), one gets (E ∼ p) √ 2E[2b0 + 2 GF ne (rres )/2]
1. | cos 2θ | ∼ (9) |m2 | The transition between the two flavors νeL → ν¯ µR is determined by defining the effective mixing angle θ˜ (r) which diagonalizes the corresponding submatrix in (6) (B = Bx ) 4µB(r)E

˜ = tan 2θ(r)

|m2 | cos 2θ

− 4Eb0 −

√

(10)

2 GF Ene (r)

and relates flavor and mass eigenstates |νe  = cos θ˜ |ν1  + sin θ˜ |ν2 ,

|νµ  = − sin θ˜ |ν1  + cos θ˜ |ν2 .

For homogeneous (constant) magnetic field, the conversion probability is [13] PνeL →νµR =

(2µB)2

+

|m2 | 2E

(2µB)2 cos 2θ − 2b0 −

√

2 2 2 GF ne

sin2 α,

(11)

where  α=



√ 2 |m2| 2 cos 2θ − 2b0 − GF ne + (2µB)2 r 2E 2

and r is the distance travelled by neutrinos. At the resonance it becomes1 PνeL →¯νµR = sin2 µBr.

(13)

The neutrino magnetic momentum is µ ∼ 10−11 µB ∼ 6 × 10−16 eVT−1 [12]. Taking the magnetic field of Earth B⊕ ≈ 5 × 10−5 T, constant over the region r ∼ 2R⊕ ∼ 1.2 × 107 m, it follows PνeL →¯νµR ∼ 3 × 10−4 . For the Sun, the superficial magnetic field B ∼ 10−1 T [12], and r ∼ R ∼ 7 × 108 m give PνeL →¯νµR ∼ 0.04. On the other hand, in the convective zone B ∼ 10 T [12], thus PνeL →¯νµR ∼ 0.65. Let us now use Eq. (9) to determine the bounds on the parameter b0 . Oscillations in matter. Due to the fact that the matter densities ne varies with the distance r, as well as the magnetic field B(r), the mass eigenstates evolve adiabatically only if the adiabaticity condition is satisfied at the resonance [10,12,16], that is    a1 − a2    1,  γ ≡ (14)  ˜ d θ/dr res 1 Unfortunately, the profile of magnetic fields in the core, radiation zone or convection zone of Sun is little known, except that they may be quite large. Similarly for Earth. For adiabatic variation of the magnetic field and electron density, the conversion probability is given by [18,19]

PνeL →ν¯ µR =

1 − 2



 1 − Pc cos 2θ˜ cos 2θ, 2

(12)

where Pc = exp[−2π γ sin2 θ˜ cos θ˜ / sin2 2θ˜ ] and γ is defined in Eq. (14) [10]. At the resonance (θ˜ = π/4) and when γ  1, the formula (12) assumes the maximum value PνeL →ν¯ µR = 1/2.

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G. Lambiase / Physics Letters B 560 (2003) 1–6

where the effective mixing angle is define by (10) and ai , i = 1, 2, are the eigenvalues of the submatrix of evolution Eq. (6) corresponding to νeL –νµR transition   √ √  2 |m2| 2 2 1 2 2 GF ne ± − 2b0 − GF ne + 4µ |B | . a1,2 = (15) 2 2 2E 2 ˜ Eq. (10) allows to calculate (d θ/dr) res , and by using (15), the adiabaticity condition (14) assumes the form    √ 2 GF  dne  γ −1 =  1. 16µ2 |B |2  dr  res

(16)

Being B  10 T in the solar convective zone [12], Eq. (16) reduces to e−10.5rres/R  0.1, which is fulfilled by rres > 0.4R . From Eq. (9) it follows the upper bound on b0   1 |m2 | b0
−10−12e−10.5rres/R eV + . 2 2E

(17)

(18)

Notice that the bound (18) is not affected by magnetic field; the latter, in fact, is related to the transition probability (see Eq. (13)) but it does not enter in the resonance condition from which Eq. (18) is inferred. We shall use the following values of the m2 − sin2 2θ parameters [10]   m2  ∼ (3–10) × 10−6 eV2 , sin2 2θ ∼ (0.6–1.3) × 10−2 , SMA,   m2  ∼ (1–20) × 10−5 eV2 , sin2 2θ ∼ 0.5–0.9, LMA, where SMA and LMA stand for Small and Large Mixing Angle solution, respectively. Estimations on b0 are derived for solar neutrinos with the maximum energy E ∼ 15 MeV, produced by 8 B and hep reactions [10]. Notice that the flux of 8 B neutrinos produced in the Sun is much smaller than the fluxes of pp, 7 Be, and pep neutrinos, but 8 B neutrinos give the major contribution to the event rates of (Kamiokande and SuperKamiokande (SK)) experiments with a high energy detection threshold [10]. The resonance occurring at rres > 0.4R implies that the background matter density reduces and the dominant term in (18) is the massive one. It then follows that the upper bound on b0 is b0
10−14 –10−12 eV.

(19)

Atmospheric neutrinos. Evidence in favor of atmospheric neutrino oscillations come from SK experiment [20], and the relevant values of |m2 | − sin2 2θ parameters are (νµ − ντ ) [10,21]   m2  ∼ 5 × 10−3 eV2 , (20) sin2 2θ ≈ 1. Neglecting the interaction with matter background (ne ≈ 0), Eq. (18) gives (E ∼ GeV) b0  10−13 eV.

(21)

Bounds on the pseudo-vector bµ = (b0 , b), can be put from tests proposed in [22,23]. In Ref. [22], the EotWash II experiment [24] is discussed, leading to a possible measure up to |b| ∼ 10−19 eV. The method discussed in Ref. [23] invokes a squid to measure |b| via the magnetisation in a paramagnetic material; the field b plays the role of an effective magnetic field, and it can be measured at the level of 10−20 eV. CPT and Lorentz tests in hydrogen and antihydrogen atoms confined in magnetic trap, and anomalous magnetic moments give |b|
10−18 eV [25] and |b|
10−15 eV [26], respectively. The pseudo-vector bµ arises as curvature coupling of fermion in a rotating frame [27], in such a case |b| ∼ 10−28 eV.

G. Lambiase / Physics Letters B 560 (2003) 1–6

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As shown in this Letter, (upper) bounds on the 0-component of the pseudo-vector bµ may be inferred from neutrino oscillations physics. Our analysis agrees with Refs. [22–27] only if b0 is assumed to be universal (in these papers, experiments are performed considering not neutrinos, but electrons, protons, etc.). The helicity terms ±b0 (see Eq. (3)) give rise to the occurrence of resonance conditions in the spin-flip conversion of neutrino flavor. Thus, in addition to the tests above discussed, neutrino oscillations might provide another framework to probe the Lorentz invariance breakdown, as suggested by Kostelecký and collaborators.

Acknowledgement The author thanks the referee for comments.

References [1] V.A. Kostelecký, S. Samuel, Phys. Rev. D 39 (1989) 683; V.A. Kostelecký, S. Samuel, Phys. Rev. Lett. 63 (1989) 224. [2] G. Amelino-Camelia, J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, S. Sarkar, Nature 393 (1998) 763. [3] J. Ellis, K. Farakos, N.E. Mavromatos, V.A. Mitsou, D.V. Nanopoulos, Astrophys. J. 535 (2000) 139; J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, G. Volkov, Gen. Relativ. Gravit. 32 (2000) 1777. [4] R. Gambini, J. Pullin, Phys. Rev. D 59 (1999) 124021; J. Alfaro, H.A. Morales-Técolt, L.F. Urrutia, Phys. Rev. Lett. 84 (2000) 2318; J. Alfaro, H.A. Morales-Técolt, L.F. Urrutia, Phys. Rev. D 65 (2002) 103509. [5] D. Colladay, V.A. Kostelecký, Phys. Rev. D 58 (1998) 116002; D. Colladay, V.A. Kostelecký, Phys. Rev. D 55 (1997) 6760. [6] V.A. Kostelecký, R. Lehnert, Phys. Rev. D 63 (2001) 065088. [7] O. Bertolami, C.S. Carvalho, Phys. Rev. D 61 (2000) 103002. [8] C.I. Itzykson, J.-B. Zuber, Quantum Field Theory, McGraw–Hill, New York, 1980. [9] L. Wolfenstein, Phys. Rev. D 17 (1978) 2369; L. Wolfenstein, Phys. Rev. D 20 (1979) 2634; S.P. Mikheyev, A.Yu. Smirnov, Yad. Fiz. 42 (1985) 1441, Sov. J. Nucl. Phys. 42 (1985) 913; S.P. Mikheyev, A.Yu. Smirnov, Nuovo Cimento C 9 (1986) 17. [10] S.M. Bilenky, C. Giunti, W. Grimus, Prog. Part. Nucl. Phys. 43 (1999) 1. [11] J.N. Bahcall, Neutrino Astrophysics, Cambridge Univ. Press, Cambridge, 1989. [12] G.G. Raffelt, Stars as Laboratories for Fundamental Physics, Chicago Univ. Press, IL, 1996. [13] L.B. Okun, Yad. Fiz. 44 (1986) 847, Sov. J. Nucl. Phys. 44 (1986) 546; L.B. Okun, M.B. Voloshin, M.I. Vyotsky, Yad. Fiz. 91 (1986) 754, Sov. J. Nucl. Phys. 44 (1986) 440; C.S. Lim, W.J. Marciano, Phys. Rev. D 37 (1988) 1368; E. Akhmedov, M. Khlopov, Yad. Fiz. 47 (1988) 1079, Sov. J. Nucl. Phys. 47 (1986) 689. [14] J. Schechter, J.W.F. Valle, Phys. Rev. D 24 (1981) 1883. [15] D. Piriz, M. Roy, J. Wudka, Phys. Rev. D 54 (1996) 1587. [16] R.N. Mohapatra, P.B. Pal, Massive Neutrinos in Physics and Astrophysics, World Scientific, Singapore, 1991. [17] O.G. Miranda, C. Pena-Garay, T.I. Rashba, V.B. Semikoz, J.W.F. Valle, Phys. Lett. B 521 (2001) 299; J. Barranco, O.G. Miranda, T.I. Rashba, V.B. Semikoz, J.W.F. Valle, Phys. Rev. D 66 (2002) 093009. [18] S.J. Parke, Phys. Rev. Lett. 57 (1986) 1275. [19] W.C. Haxton, Annu. Rev. Astron. Astrophys. 33 (1995) 459. [20] Y. Fukuda, et al., Phys. Rev. Lett. 81 (1998) 1562; Y. Fukuda, et al., Phys. Lett. B 433 (1998) 9; Y. Fukuda, et al., Phys. Lett. B 436 (1998) 33. [21] G.L. Fogli, E. Lisi, A. Marrone, G. Scioscia, Phys. Rev. D 60 (1999) 053006. [22] R. Bluhm, V.A. Kostelecký, Phys. Rev. Lett. 84 (2000) 1381. [23] W.-T. Ni, et al., Phys. Rev. Lett. 82 (1999) 2439. [24] E.G. Adelberger, et al., in: P. Herczag, et al. (Eds.), Physics Beyond the Standard Model, World Scientific, Singapore, 1999, p. 717. [25] R. Bluhm, V.A. Kostelecký, N. Russell, Phys. Rev. Lett. 82 (1999) 2254.

6

G. Lambiase / Physics Letters B 560 (2003) 1–6

[26] R. Bluhm, V.A. Kostelecký, N. Russell, Phys. Rev. Lett. 79 (1997) 1432; R. Bluhm, V.A. Kostelecký, C.D. Lane, Phys. Rev. Lett. 84 (2000) 1098; H. Dehmelt, et al., Phys. Rev. Lett. 83 (1999) 4694; G. Gabrielse, et al., Phys. Rev. Lett. 82 (1999) 3198. [27] S. Mohanty, B. Mukhopadhyay, A.R. Prasanna, Phys. Rev. D 65 (2002) 122001.