- Email: [email protected]

Quantum decoherence and neutrino data Gabriela Barenboim a , Nick E. Mavromatos b,∗ , Sarben Sarkar b , Alison Waldron-Lauda b a Departamento de Física Teórica and IFIC, Centro Mixto, Universidad de Valencia-CSIC,

E-46100 Burjassot (Valencia), Spain b King’s College London, University of London, Department of Physics, Strand WC2R 2LS, London, UK

Received 7 June 2006; received in revised form 9 August 2006; accepted 13 September 2006 Available online 10 October 2006

Abstract Microscopic decoherence models for neutrinos are presented. The decoherence in these models has contributions not only from stochastic quantum gravity vacua, operating as an effective medium as well as from conventional uncertainties in the energy of the (anti)neutrino beam. All these contributions lead to damping factors modulating the oscillatory terms. On fitting all available current data including LSND and KamLAND spectral distortion results, we find that some (but not all) of the oscillatory terms are damped over an oscillation length L by a factor Γ which is independent of L. The order of magnitude of Γ favours the interpretation of decoherence due to conventional physics. However, there is still the puzzling aspect of the undamped terms, which currently remains unexplained. The microscopic nature of the models is key to reaching this conclusion. © 2006 Elsevier B.V. All rights reserved. PACS: 14.60.Pq; 14.60.St; 03.65.Yz; 04.60.-m

1. Introduction The theory of quantum gravity (QG) is still elusive. In some theoretical models, the phenomenon of space–time ‘foam’, invoked by Wheeler [1], may be incorporated; according to this picture, the singular microscopic fluctuations of the metric give the ground state of QG the structure of a ‘stochastic medium’. The medium has the profound effect of CPT-violating [2] * Corresponding author.

E-mail address: [email protected] (N.E. Mavromatos). 0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.09.012

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decoherence of quantum matter as it propagates. This may have experimentally observable consequences in principle [3]. One of the basic effects of gravitational decoherence is the presence of factors which damp oscillatory terms or coherence. However, it is necessary to be very careful when interpreting decoherence effects, if observed in an experiment, because interactions with standard matter can easily ‘fake’ decoherence effects, especially the damping factors and their associated exponents [4,5]. For instance, uncertainties in the energy of a neutrino beam [4], which are associated with conventional physics can reproduce a damping exponent similar to that encountered in Lindblad decoherence models [6] of QG. Of course, stochastic quantum gravity effects can induce such uncertainties in the energy of the beam, and hence contribute to the damping exponent themselves [3], but such effects are usually subleading. Thus, one should know the energy of the beam with high precision in order to eliminate ‘fake’ decoherence effects and probe quantum gravity effects with sufficient precision. The models used recently [7] to fit available neutrino data, including LSND results [8], use phenomenological models of decoherence with mixing in all three generations of neutrinos. They are in the same spirit as earlier studies of decoherence with two-generation neutrino models [9]. Such models are based on a mathematical parametrisation of quantum dynamical semi-groups (describing time evolution) which map positive operators into positive operators; positivity guarantees that calculated probabilities remain positive at all times [6]. Even if such a parametrisation is successful in terms of data fits, it is difficult to rule in or out space–time foam as the origin of decoherence since no knowledge of any microscopic mechanism is incorporated in the model. Hence our primary aim is to give a microscopic and physically motivated model which would fit into the general scheme of the linear Lindblad decoherence with specific forms for the coupling. Hence the parameters of the model will have a clear microscopic interpretation together with well-defined relations among themselves. Furthermore by relaxing these relations a wider class of models is considered which has the property of positivity. Such models fill an important void in previous analyses [10] which constructed three-generation Lindblad models without any physical guidance and consequently failed to find parameter values which satisfied the criterion for positive dynamical semi-groups. Consequently in certain regions in phase space the difficulties of negative probabilities arose [7] (for two neutrino generations, Lindblad models respecting positivity have been discussed in [9,11]). The parameter space is formidably large a priori, and so searching for parameters to satisfy such criteria is very difficult. Our microscopic model presented here, thus, has the pleasing bonus of leading to the first mathematically satisfactory Lindblad approach to decoherence for the case of three generations of neutrinos. In the present work, the decoherence parameters in the model are assumed to be the same in both neutrino and antineutrino sectors, consistent with the expected universal property of a quantumgravity environment. Moreover for the case of decoherence for other sensitive particle probes, like neutral mesons [12], the oscillations between particle and antiparticle sectors, necessitate a common decoherence environment between mesons and antimesons. Consequently we will take the point of view that the LSND result is correct in both channels, although the observed excess of ν¯ e events is not corroborated (at the same level at least) in the neutrino channel. This is to be contrasted with the approach of [7], where following [13], only the evidence in the antineutrino sector was considered. On considering space–time foam as a stochastic medium it is necessary to consider the propagation of neutrinos through a medium with a fluctuating density. The ground state of the medium is taken to be a quantum space–time foam [1], characterised by fluctuating densities of charged black-hole/anti-black-hole pairs produced and absorbed by the vacuum within Planckian time

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scales [14,15]. Such a case, will not produce any vacuum charge on average, but the associated density fluctuations will produce vacuum fluctuations in electron currents with which electron neutrinos will interact preferentially. Given the underlying microscopic picture and effective medium description we can proceed using the formalism of quantum mechanics in stochastic media [16] in order to obtain the microscopic model for decoherence. The structure of this article is the following: in Section 2 we review the basic theory of Lindblad decoherence, and specify the conditions for complete positivity. We then define the microscopic model within this framework. In Section 3 we extract the salient features of the microscopic model to obtain a more general but simple Lindblad decoherence model in which positive definite transition probabilities are guaranteed for the entire regime of the experimental parameter space. The decoherence implies exponential damping with time (oscillation length), which violates microscopic time irreversibility, irrespective of the CP properties of neutrinos, and hence CPT violation within a quantum gravity context [2,3]. In order to agree with the experimental results of KamLAND on spectral distortions [17] we require such exponential damping factors to imply a modulation of the oscillatory terms in the (survival) transition probabilities of order per mil. We obtain stringent constraints on the exponents of the damping factors. The sample point that fits all available data, including LSND, is discussed in detail in Section 4. The statistical analysis leads to the nonzero exponents of the decoherent damping factors over an oscillation length L being independent of L. An attempt to explain such a result in terms of microscopic models of stochastic space–time foam is given in Section 5. However, as we show there, explanations based on conventional physics such as the uncertainties in the (anti)neutrino beam energy, are definitely much more plausible, as far as the order of magnitude is concerned. The present data when interpreted in terms of our microscopic model [15] make quantum gravity not a dominant candidate for the origin of decoherence (claimed to be observed in our fits). Nevertheless, the aspect of selective damping, suggested by the fit, remains a puzzle, and, hence one cannot exclude the possibility that in other models the situation could be different. Finally, conclusions and outlook are presented in Section 6. Some technical details of our formalism are presented in Appendix A. 2. Lindblad decoherence and microscopic model In this section we will present the details of a microscopic model within a Lindblad framework [6] and the calculation for transition probabilities of three generations of neutrinos. In this framework the general evolution equation of the ρ density matrix, representing a (spinless) neutrino state reads ∂t ρ = L[ρ],

(2.1)

where there are conditions on the decoherence contribution to L which guarantee complete positivity of the probabilities as they evolve in time. The spin of the neutrino will not play an important rôle in constraining the decoherence sector by comparing with experimental data, and hence we shall present a formalism based on scalar particles. Detailed studies of Dirac and spinless neutrinos have been performed in [15]; as explained there the inclusion of spin does not affect qualitatively the main decoherence effects which is the damping of oscillation probabilities. In Section 4, where we attempt to interpret the time (i.e. oscillation length) dependence and order of magnitude of the decoherence parameters, we shall present a more detailed discussion based on the results of [15].

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With the above in mind, we commence our analysis with a theorem due to Gorini, Kossakowski and Sudarshan [18] on the structure of L, the generator of a quantum dynamical semi-group [6,18]. For a nonnegative matrix ckl (i.e. a matrix with nonnegative eigenvalues) such a generator is given by dρ 1 ckl Fk ρ, Fl† + Fk , ρFl† , = L[ρ] = −i[H, ρ] + dt 2

(2.2)

k,l

where H = H † is a Hermitian Hamiltonian, {Fk , k = 0, . . . , n2 − 1} is a basis in Mn (C) such that F0 = √1n In , Tr(Fk ) = 0 ∀k = 0 and Tr(Fi† Fj ) = δij . In our application we can take Fi = Λ2i (where Λi are the Gell-Mann matrices) and satisfy the Lie algebra [Fi , Fj ] = i k fij k Fk (i = 1, . . . , 8), fij k being the standard structure constants, antisymmetric in all indices. It can always be arranged that the sum over k and l run over 1, . . . , 8. Without a microscopic model, in the three generation case, the precise physical significance of the matrix ckl cannot be understood. Moreover a general parametrisation of ckl is too complicated to have any predictive power. The effective Heff for the interaction of neutrinos with the space–time foam medium in our model can be written as Heff = H + ncbh (t)HI ,

(2.3)

where nbh c is the number of virtual particles emitted from the foam and can be taken to be a Gaussian random variable [14]. We take ncbh (t) = n0 and ncbh (t)ncbh (t ) ∼ Ω 2 (ncbh )2 δ(t −t ). Note that in the present work we consider media, including the quantum-gravity foam, which are homogeneous and isotropic, and hence the density ncbh (t) depends only on time t . For light particles, such as neutrinos, t r (c = 1). In more complicated situations of space–time foam one may also have nfoam (r, t), a case which, as far as we are aware of, has not been analysed in the literature as yet. HI is a 3 × 3 matrix which in flavour space will be taken, in analogy with the MSW effect [19], to have the form aν e (2.4) aν μ aν τ and H is the free Hamiltonian. For our qualitative purposes in this section we have taken aντ aνμ . The master equation in (2.2) is best expressed in the basis of mass eigenstates for the neutrinos since oscillation phenomena are clearest in this basis. Consequently we need the three generation mixing matrix U relating flavour eigenstates to mass eigenstates. Our discussion is simplified (but phenomenologically realistic) if we consider the dominant mixing to be between νe (family 1) and νμ (family 2), and between νμ and ντ (family 3) [15]; the mixing angles, for simplicity, are chosen so that θ12 = θ23 = θ , and θ13 = 0 and lead to ⎞ ⎛ cos(θ ) sin(θ ) 0 U = ⎝ − sin(θ ) cos(θ ) (2.5) sin(θ ) ⎠ . cos2 (θ ) 2 sin (θ ) − sin(θ ) cos(θ ) cos(θ ) Simulations indicate that the three generation case with full mixing does not affect qualitatively the form of the fit for damping exponents to the oscillation experiments. Following standard analysis [16] we find that the evolution equation of ρ of the neutrino in this microscopic model involves a time-reversal (actually [2,3] CPT) breaking decoherence term

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in the form of a double commutator ∂t ρ = i[ρ, Heff ] − Ω 2 HI , [HI , ρ] .

(2.6)

Furthermore this double-commutator decoherence is a specific case of Lindblad evolution since HI , [HI , ρ] = − [HI ρ, HI ] + [HI , ρHI ] . (2.7) The resulting C-matrix has the form: ⎛ h2 0 h h 0 0 0 1

⎜ 0 ⎜ ⎜ h1 h3 ⎜ ⎜ C=⎜ 0 ⎜ 0 ⎜ ⎜ 0 ⎝ 0 h1 h8

1 3

0 0 0 h23 0 0 0 0 0 0 0 0 0 h3 h8

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 h1 h8 ⎞ 0 0 ⎟ ⎟ 0 h3 h8 ⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎠ 0 0 0 h28

(2.8)

(aν −aν )

with h1 = (aνe − aνμ ) sin(2θ ), h3 = (aνe − aνμ ) cos(2θ ), and h8 = e√ μ . This matrix indeed 3 has positive eigenvalues for real hi . The difference aνe −aνμ is proportional to the average density of the medium n0 . It is interesting to note the largely diagonal form of the symmetric matrix C. We note at this stage that for the case of interest, the gravitationally-induced MSW [14], we may write aeμ ≡ aνe − aνμ ∝ GN n0

(2.9)

with GN = 1/MP2 , MP ∼ 1019 GeV, the four-dimensional Planck scale. This gravitational coupling replaces the weak interaction Fermi coupling constant GF in the conventional MSW effect. In such a case the density fluctuations Ω 2 can be assumed small compared to other quantities present in the formulae, and an expansion to leading order in Ω 2 is sufficient. Following the standard analysis given in Appendix A, to leading order in the small parameter Ω 2 1 we obtain the following expression for the neutrino transition probability νe ↔ νμ in this case: 212 √ 2 2 Pνe →νμ = e−aeμ Ω t 1+ 4G (cos(4θ)−1) sin(t G) cos2 (θ ) sin2 (2θ ) 3 sin2 (2θ )Δ212 1 2 × aeμ Ω 2 Δ212 − 4G 5/2 G 3/2 212 √ Δ2 2 2 − e−aeμ Ω t 1+ 4G (cos(4θ)−1) cos(t G) cos2 (θ ) sin2 (2θ ) 12 2G 2 Ω 2 2 t sin2 (2θ) 2 aeμ (a + cos(2θ )Δ ) 1 12 eμ 12 G − e− (2.10) cos2 (θ ) + cos2 (θ ), 2G 2 m2

2 sin2 (2θ ), Δ = 12 where G = (aeμ cos(2θ ) + Δ12 )2 + aeμ 12 2p . From (2.10) we easily conclude that the exponents of the damping factors due to the stochastic-medium-induced decoherence, are therefore of the generic form, for t = L, the oscillation length (in units of c = 1): Δ212 (cos(4θ ) − 1) 2 2 . exponent ∼ aeμ Ω t 1 + (2.11) 4G

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We should note that, in the limit Δ12 → 0, which could characterise the situation in [7], where the space–time foam effects on the induced neutrino mass difference are the dominant ones, the damping factor is of the form: exponentgravitational MSW ∼ Ω 2 (aeμ )2 L, with the precise value of the mixing angle θ not affecting the leading order of the various exponents. However, in that case, as follows from (2.10), the overall oscillation probability is suppressed by factors proportional to Δ212 , and, hence, the stochastic gravitational MSW effect [14,15], although in principle capable of inducing mass differences for neutrinos, however does not suffice to produce the bulk of the oscillation probability, which is thus attributed to conventional flavour physics. The implications for this model of the fit to data obtained in the next section will be given later. The presence of several off-diagonal terms in the decoherence matrix (2.8) leads to considerable algebraic complexity for the expressions of the flavour-oscillation probabilities. However, one may consider other simplified scenarios of space–time foam, which can lead to algebraically less complicated expressions, while maintaining the basic qualitative features of the stochastic quantum gravitational environments. Such scenarios are those in which the entanglement with the space–time foam is, for all practical purposes, insensitive to mixing, in other words the interaction Hamiltonian ncbh HI is diagonal in the mass eigenstate space instead of the flavour space, aν1 HI = (2.12) . aν 2 aν 3 Under the same simplifications of the interaction couplings aνi as in the flavour favouring MSW case above, that is assuming that the couplings are aν1 = aν2 = aν3 , and assuming again stochastic fluctuations of the foam density ncbh , it is easy to see that the relevant decoherence c-matrix is formally obtained from (2.8) upon the substitutions aνα (α = e, μ) → aνi (i = 1, 2) and θ → θQG 0. This leaves only the corresponding h3 and h8 nonzero in (2.8). Models of this type account for direct quantum-gravitational effects on the mass eigenstates of the system, through—for instance—dynamically-generated mass contributions for neutrinos, as a result of their interaction with the foam. One may think of this situation more precisely as follows: the quantum gravitational interactions generate a very weak mixing θQG 0, as a primordial contribution, which suffices to generate mass eigenstate differences. Then, ordinary physics is responsible for the bulk of the mixing θ θQG , which thus affects primarily the Hamiltonian terms of the evolution of the neutrino density matrix, leading to flavour oscillation probabilities Pνα →νβ . The latter occur in a Lindblad-type quantum-decohering-foam environment with a decoherence C-matrix of the form ⎛ ⎞ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎟ ⎜0 0 ⎜ ⎟ 0 0 0 0 h˜ 3 h˜ 8 ⎟ ⎜ 0 0 h˜ 23 ⎜ ⎟ 0 0 0 0 0 0 ⎟ ⎜0 0 Csimple = ⎜ (2.13) ⎟ 0 0 0 0 0 0 ⎟ ⎜0 0 ⎜ ⎟ 0 0 0 0 0 0 ⎟ ⎜0 0 ⎝ ⎠ 0 0 0 0 0 0 0 0 0 0 h˜ 3 h˜ 8 0 0 0 0 h˜ 28 √ with h˜ 3 = 3h˜ 8 = aν1 − aν2 . As we shall discuss below, such a situation leads to considerable algebraic simplifications for three-generation models, as compared to the flavour-favouring

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MSW-like quantum gravity scenarios. Nevertheless, all the important qualitative features of the stochastic-quantum-foam-induced decoherence are maintained, and thus from now on we shall restrict ourselves to such simplified models. 3. General Lindblad decoherence model The matrix C ≡ (ckl ) (2.13) of the microscopic model (of the previous section) is symmetric and is suggestive of the following general ansatz: ⎞ ⎛ 0 0 0 0 0 0 c11 0 0 0 0 0 0 ⎟ ⎜ 0 c22 0 ⎟ ⎜ 0 0 0 0 c 0 0 c ⎜ 33 38 ⎟ ⎟ ⎜ 0 0 c44 0 0 0 0 ⎟ ⎜ 0 C=⎜ (3.1) ⎟, 0 0 ⎟ 0 0 0 c55 0 ⎜ 0 ⎟ ⎜ 0 ⎟ 0 0 0 0 c66 0 ⎜ 0 ⎠ ⎝ 0 0 0 0 0 0 c77 0 0 0 0 c88 0 0 c38 0 where, for the sake of generality within the framework of our pilot study in this section, we have included additional diagonal terms (phenomenological in origin) diagonal terms. Such terms might owe their existence to contributions of space–time foam different from the simple stochastic fluctuations of the density of the foam medium considered in Section 2. Such extra diagonal contributions may be associated, for instance, with (flavour-insensitive) quantum fluctuations (uncertainties) of the neutrino energy as a result of stochastic fluctuations of the space–time metric itself [15] (see also Section 5), or other back reaction effects, such as recoil contributions to the space–time metric in some stringy quantum-gravity models, e.g., those pertaining to “D-particle” foam [3], as a consequence of space–time distortions induced by the recoil of the D-particle space–time defects (playing the rôle of the quantum-gravity “environmental” degrees of freedom) during their scattering with the matter (neutrino) probe. For our phenomenological purposes in this work we shall not attempt to specify further the microscopic origin of the elements of the decoherence matrix (3.1), but instead proceed to examine its consequences on the form of the flavour-oscillation probabilities, to be used in the experimental fit below. As stated in Section 2, positivity can be guaranteed if and only if the matrix C is positive and hence has nonnegative eigenvalues. The simplification in [10] (and subsequent works relying on [10]) was based on the L matrix and is the main reason why that approach was not successful as regards maintaining positivity. In Section 4, we shall discuss the viability of the interpretation of the fit in terms of the microscopic models of space–time foam. The simplified form of the cij matrix given in (3.1) implies a matrix Lij of the form (see Appendix A) ⎞ ⎛ 0 0 0 0 0 0 D11 −Δ12 0 0 0 0 0 0 ⎟ ⎜ Δ12 D22 ⎟ ⎜ 0 D33 0 0 0 0 D38 ⎟ ⎜ 0 ⎟ ⎜ 0 0 D44 −Δ13 0 0 0 ⎟ ⎜ 0 L=⎜ (3.2) ⎟, 0 0 0 ⎟ 0 0 Δ13 D55 ⎜ 0 ⎟ ⎜ 0 ⎟ 0 0 0 0 D66 −Δ23 ⎜ 0 ⎠ ⎝ 0 0 0 0 0 0 Δ23 D77 0 0 0 0 D88 0 0 D83

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97

m2 −m2

where we have used the notation Δij = i 2p j . The corresponding eigenvalues are: 1 1 λ1 = (D11 + D22 ) − (D22 − D11 )2 − 4Δ212 ≡ (D11 + D22 ) − Ω12 , 2 2 1 1 λ2 = (D11 + D22 ) + (D22 − D11 )2 − 4Δ212 ≡ (D11 + D22 ) + Ω12 , 2 2 1 1 2 2 λ3 = (D33 + D88 ) − (D33 − D88 ) + 4D38 ≡ (D11 + D22 ) − Ω38 , 2 2 1 1 2 2 λ4 = (D44 + D55 ) − (D44 − D55 ) − 4Δ13 ≡ (D44 + D55 ) − Ω13 , 2 2 1 1 2 λ5 = (D44 + D55 ) + (D44 − D55 )2 − 4Δ13 ≡ (D44 + D55 ) + Ω13 , 2 2 1 1 λ6 = (D66 + D77 ) − (D66 − D77 )2 − 4Δ223 ≡ (D66 + D77 ) − Ω23 , 2 2 1 1 λ7 = (D66 + D77 ) + (D66 − D77 )2 − 4Δ223 ≡ (D66 + D77 ) + Ω23 , 2 2 1 1 2 2 λ8 = (D33 + D88 ) + (D33 − D88 ) + 4D38 ≡ (D11 + D22 ) + Ω38 , 2 2 where the matrix D and its inverse are ⎛ λ1 −D22 λ2 −D22 0 0 0 0 0 0 Δ12 Δ12 ⎜ 1 1 0 0 0 0 0 0 ⎜ λ3 −D33 λ8 −D33 ⎜ 0 0 0 0 0 0 ⎜ D38 D38 ⎜ ⎜ 0 λ4 −D55 λ5 −D55 0 0 0 0 0 Δ13 Δ13 D=⎜ ⎜ ⎜ 0 0 0 1 1 0 0 0 ⎜ λ6 −D77 λ7 −D77 ⎜ 0 0 0 0 0 0 ⎜ Δ23 Δ23 ⎝ 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1

(3.3) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (3.4)

and

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ −1 D =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

Δ12 Ω12

λ2 −D22 Ω12 λ1 −D22 − Ω12

0

0

38 −D Ω38

0

0

0

0

0

Δ13 −Ω 13

λ5 −D55 Ω13 −D55 − λ4Ω 13

0

Δ23 −Ω 23

λ7 −D77 Ω23 −D77 − λ6Ω 23

0

Δ12 −Ω 12

0

0

0

0

0

0

0

0

0

0

0

0

0

0

λ8 −D33 Ω38

0

0

0

0

0

0

0

0

0

Δ13 Ω13

0

0

0

0

0

0

0

0

0

0

0

D38 Ω38

Δ23 Ω23

0

0

0

0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

−D33 − λ3Ω 38

(3.5) It will be sufficient to look explicitly at the k = 1 and k = 2 terms in the sum of the right-hand side of (A.8) in Appendix A, since by block symmetry the other terms will be of the same form.

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For the k = 1 term we have β −1 α eλ1 t Di1 D1j ρj (0)ρi β β β β −1 −1 −1 −1 + ρ2α ρ2 D21 D12 + ρ2α ρ1 D11 D12 + ρ1α ρ2 D21 D11 = eλ1 t ρ1α ρ1 D11 D11 (D11 +D22 )t −Ω12 t β −D11 + D22 + Ω12 β −D22 + D11 + Ω12 2 =e e 2 ρ1α ρ1 + ρ2α ρ2 2Ω12 2Ω12 β Δ12 β −Δ12 + ρ2α ρ1 + ρ1α ρ2 . Ω12 Ω12 Likewise for k = 2 we have β −1 α ρj (0)ρi eλ2 t Di2 D2j β β β β −1 −1 −1 −1 = eλ2 t ρ1α ρ1 D12 D21 + ρ2α ρ2 D22 D22 + ρ2α ρ1 D12 D22 + ρ1α ρ2 D22 D21 (D11 +D22 )t Ω12 t β −D22 + D11 + Ω12 β (−D22 + D11 − Ω12 ) 2 2 =e e + ρ2α ρ2 ρ1α ρ1 2Ω12 2Ω12 β −Δ12 β Δ12 . + ρ2α ρ1 + ρ1α ρ2 Ω12 Ω12 We thus obtain for the relevant probability: t t e−Ω12 2 + eΩ12 2 1 1 α β α β ρ1 ρ1 + ρ2 ρ2 Pνα →νβ (t) = + 3 2 2 β β β β α α 2Δ12 (ρ1 ρ2 − ρ2 ρ1 ) + D21 (ρ1α ρ1 − ρ2α ρ2 ) + Ω12 −Ω12 t t Ω 12 2 −e 2 t e e(D11 +D22 ) 2 × 2 t t e−Ω13 2 + eΩ13 2 α β α β + ρ4 ρ4 + ρ5 ρ5 2 β β β β 2Δ13 (ρ4α ρ5 − ρ5α ρ4 ) + D54 (ρ4α ρ4 − ρ5α ρ5 ) + Ω13 −Ω13 t t 2 − e Ω13 2 t e e(D44 +D55 ) 2 × 2 −Ω23 t t 2 + e Ω23 2 β β e + ρ6α ρ6 + ρ7α ρ7 2 β β β β α α 2Δ23 (ρ6 ρ7 − ρ7 ρ6 ) + D76 (ρ6α ρ6 − ρ7α ρ7 ) + Ω23 −Ω23 t t Ω 2 − e 23 2 t e e(D66 +D77 ) 2 × 2 t t e−Ω38 2 + eΩ38 2 α β α β + ρ3 ρ3 + ρ8 ρ8 2

(3.6)

(3.7)

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β β β β 2D38 (ρ3α ρ8 − ρ8α ρ3 ) + D83 ρ3α ρ3 − ρ8α ρ8 + Ω38 −Ω38 t Ω38 2t 2 −e e (D33 +D88 ) 2t . e × 2

Above we have used the notation that Dij = Dii − Djj . We have assumed that 2|Δij | > |Dij | with the consequence that Ωij , ij = 12, 13, 23 is imaginary. However, 2 Ω38 = (D33 − D88 )2 + 4D38 will be real. Thus, the final expression for the probability reads β D21 ρ1α ρ1 t |Ω12 |t 1 1 |Ω12 |t α β ρ1 ρ1 cos + sin e(D11 +D22 ) 2 Pνα →νβ (t) = + 3 2 2 |Ω12 | 2 β D54 ρ4α ρ4 t |Ω13 |t |Ω13 |t α β + sin e(D44 +D55 ) 2 + ρ4 ρ4 cos 2 |Ω13 | 2 β D76 ρ6α ρ6 t |Ω23 |t |Ω23 |t β + sin e(D66 +D77 ) 2 + ρ6α ρ6 cos 2 |Ω23 | 2 α β Ω38 t α β + ρ3 ρ3 + ρ8 ρ8 cosh 2 β β β β 2D38 (ρ3α ρ8 − ρ8α ρ3 ) + D83 (ρ3α ρ3 − ρ8α ρ8 ) + Ω38 t Ω38 t e(D33 +D88 ) 2 . × sinh (3.8) 2 On using the relations 2 α ρ0 = , 3 ∗ ρ1α = 2 Re Uα1 Uα2 , ∗ ρ2α = −2 Im Uα1 Uα2 , ρ3α = |Uα1 |2 − |Uα2 |2 , ∗ ρ4α = 2 Re Uα1 Uα3 , ∗ ρ5α = −2 Im Uα1 Uα3 , ∗ ρ6α = 2 Re Uα2 Uα3 , ∗ ρ7α = −2 Im Uα2 Uα3 , 1 α |Uα1 |2 + |Uα2 |2 − 2|Uα3 |2 ρ8 = 3 we can readily see that for a real U matrix (i.e. no CP violating phases) the relevant probabilities are bounded. Indeed, there is no danger of the terms involving cosh and sinh becoming unbounded with time t, since we always have Ω38 < D33 + D88 . We are now ready to discuss the fit to the experimental data. This is done in the next section.

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4. Fitting the experimental data In order to check the viability of our simplified scenario, we have performed a χ 2 comparison (as opposed to a χ 2 fit) to SuperKamiokande sub-GeV and multi-GeV data (the 40 data points that are shown in Fig. 1), CHOOZ data (15 data points), KamLAND (13 data points, shown in Fig. 2) and LSND (1 datum), for a sample point in the vast parameter space of our extremely simplified version of decoherence models. Rather than performing a χ 2 -fit (understood as a run over all the parameter space to find the global minimum of the χ 2 function) we have selected (by “eye” and not by χ ) a point which is not optimised to give the best fit to the existing data. Instead, our sample point must be regarded as a local minimum around a starting point chosen by an educated guess. It follows then that it may be quite possible to find a better fitting point through a complete (and highly time consuming) scan over the whole parameter space. To simplify the analysis and gain intuition concerning the rather cumbersome expressions for the transition probabilities, we have imposed, D11 = D22 ,

D44 = D55 ,

D66 = D77 ,

D33 = D88 ,

D38 = D83 = 0

(4.1)

implying a diagonal D-matrix. Such conditions correspond, for instance, to the simplified microscopic model of stochastic foam, discussed in Section 2, with a decoherence C-matrix (2.13), as one can readily see from the relations (A.6) (in Appendix A) connecting the Dij coefficients to the elements of the decoherence C-matrix. As stated previously, such simplified microscopic models of space–time foam lead to considerable simplifications in the algebraic expressions pertaining to the oscillation probabilities, while maintaining the majority of the most important qualitative features, and associated effects, of Lindblad-type foam decoherence that we wish to test experimentally. As a concrete example, we mention that, if we were to use decoherence matrices of the form (2.8), with c18 terms present, then such terms would imply nontrivial offdiagonal terms D75 , D57 , D46 , D64 , D38 , D83 which would in turn lead to cumbersome and non-illuminating expressions for the oscillation probabilities. To avoid such situations we shall therefore stick for the remainder of this section to the conditions (4.1). As we shall demonstrate below, the excellence of the experimental fit justifies a posteriori such choices. Later on we shall set some of the Dii to zero. Furthermore, we have also set the CP violating phase of the KMS matrix to zero, so that all the mixing matrix elements become real. With these assumptions, the complicated expression for the transition probability (3.8) simplifies to 1 1 α β |Ω12 |t D22 t ρ1 ρ1 cos e Pνα →νβ (t) = + 3 2 2 |Ω13 |t D55 t |Ω23 |t D66 t β β + ρ4α ρ4 cos + ρ6α ρ6 cos e e 2 2 Ω38 t D33 t β β + ρ3α ρ3 + ρ8α ρ8 cosh (4.2) e 2 for both, neutrino and antineutrino sectors. As indicated by the state-of-the-art analysis, masses and mixing angles are selected to have the values m212 = 7 × 10−5 eV2 ,

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m223 = 2.5 × 10−3 eV2 , θ23 = π/4,

θ12 = 0.45,

θ13 = 0.05.

For the decoherence parameters we find 1.3 × 10−2 , (4.3) L in units of 1/km with L = t the oscillation length. The 1/L-behaviour of D11 , implies oscillationlength independent Lindblad exponents. We shall attempt to interpret this result in the next section. The complete positivity of the case defined by (4.1) and (4.3) is guaranteed; this follows from the fact that the√solutions for cij in terms of Dk (cf. (A.7)) is such that the only nonzero elements are c88 = c38 / 3. Such a C-matrix has only nonnegative eigenvalues D33 = D66 = 0,

D11 = D22 = D44 = D55 = −

C-matrix eigenvalues = (0, 0, 0, 0, 0, 0, 0, −8D11 /3).

(4.4)

In summary we have introduced only one new parameter, a new degree of freedom, by means of which we shall try to explain all the available experimental data. It is important to stress that the inclusion of one new degree of freedom by itself does not guarantee a priori that all the experimental observations can be accounted for. Indeed for situations without decoherence the addition of a sterile neutrino (which introduces four new degrees of freedom—excluding CP violating phases) seemed to be insufficient for matching all available experimental data, at least in CPT conserving situations [20]. In order to test our model with this decoherence parameter, we have calculated the zenith angle dependence of the ratio “observed-events/(expected-events in the no oscillation case)”, for muon and electron atmospheric neutrinos, for the sub-GeV and multi-GeV energy ranges, when mixing is taken into account. Since matter effects are important for atmospheric neutrinos, we have implemented them through a two-shell model, where the density in the mantle (core) is taken to be roughly 3.35 (8.44) g/cm3 , and the core radius is taken to be 2887 km. The results are shown in Fig. 1, where, for the sake of comparison, we have also included the experimental data. As readily seen, the agreement is remarkable. As eyeball comparisons can be misleading, we have also calculated the χ 2 value for each of the cases, defining the atmospheric χ 2 as 2 χatm =

exp 10 th )2 (Rα,i − Rα,i M,S α=e,μ i=1

2 σαi

.

(4.5)

Here σα,i are the statistical errors, the ratios Rα,i between the observed and predicted signal can be written as exp

exp

MC Rα,i = Nα,i /Nα,i

(4.6)

(with α indicating the lepton flavor and i counting the different bins, ten in total) and M, S stand for the multi-GeV and sub-GeV data respectively. For the CHOOZ experiment we used the 15 data points with their statistical errors, where in each bin we averaged the probability over energy. For the KamLAND experiment, their 13 data points have been used for a fixed distance of L0 = 180 km, as if all antineutrinos detected in KamLAND were due to a single reactor at this distance and plotted in Fig. 2 while for LSND one datum has been included.

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Fig. 1. Decoherence fit. The dots correspond to SK data.

The results are summarised in Table 1, where we present the χ 2 comparison for the model in question and the standard scenario (calculated with the same program). From the table it becomes clear that our simplified version of decoherence in both neutrino and antineutrino sectors can easily account for all the available experimental information, including LSND. It is important to stress once more that our sample point was not obtained through a scan over all the parameter space, but by an educated guess, and therefore plenty of room is left for improvements. On the other hand, for the mixing-only/no-decoherence scenario, we have taken the best fit values of the state of the art analysis and therefore no significant improvements are expected. As we have seen, the decoherence effects suffered by our model, are just an overall suppression at the per mil level (to account precisely for LSND, a per mil evidence) and therefore, no effect is expected (or found) in the oscillation dominated physics, where the level of precision is at the percent level, at most. We are guaranteed then to have an excellent agreement with solar data, as long as we keep the relevant mass difference and mixing angle within the LMA-I region, something which we certainly did. Thus, there is no need to include these data on our fit.

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Fig. 2. Ratio of the observed ν e spectrum to the expectation versus L0 /E for our decoherence model. The dots correspond to KamLAND data. Table 1 χ 2 obtained for our model and the one obtained in the standard scenario for the different experiments calculated with the same program χ2

Decoherence

Standard scenario

SK sub-GeV SK multi-GeV CHOOZ KamLAND LSND

38.0 11.7 4.5 16.7 0

38.2 11.2 4.5 16.6 6.8

Total

70.9

77.3

At this point, a word of warning is in order. From the table, it may seem that the decoherence model that we are presenting here and the standard scenario (with no decoherence) provide equally good fits, i.e. the former has a χ 2 /DOF = 70.9/63 and the latter has a χ 2 /DOF = 77.3/64; although both are quite “acceptable” from the statistical point of view, one must remember that only the decoherence model can explain the LSND result. This fact, however, gets blurred in the total χ 2 because LSND is represented by only one experimental point with a poor precision. Before closing this section, it is worth revisiting the models of [7], in order to understand, in the above context, the failure of complete positivity in certain regions of the parameter space. In that case, the following restrictions on the antineutrino-sector decoherence matrix (which was also diagonal, as in the case (4.1)) had been imposed [7]: D11 = D22 = D44 = D55 = −2 × 10−18 · E = −A,

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D66 = D77 = D33 = D88 = −10−24 /E = −B, D38 = D83 = 0,

(4.7) √ √ leading to a solution for the c-matrix c38 = 23 3A − 23 3B, c55 = 23 B, c44 = 23 B, c88 = 23 A, c66 = 23 B, c22 = 23 B, c33 = −4/3B + 2A, c77 = 23 B, c11 = 23 B (all other matrix entries zero) such that the pertinent eigenvalues (−2B + 83 A, 23 B, 23 B, 23 B, 23 B, 23 B, 23 B, 23 B) which are not positive for arbitrary values of A and B. The condition for positivity can be obtained by demanding positivity of the first eigenvalue (since the others are trivially so), and so −2B + 83 A 0, −18 · E, where E is in units of GeV. This leads to the conwhich implies 2 × 10−24 /E < 16 3 × 10 dition E > O(1 MeV), which was also the condition found in [7]. This result was interpreted in that work as implying that, outside the regime of parameters where positivity holds, that particular linear Lindblad model was not valid. However, we have shown here that another simple linear Lindblad model, with a different energy dependence, does not have problems with positivity. 5. Attempt at interpreting the fit To understand the results (4.3), (4.4), in connection with either the microscopic model [3, 15], or energy-uncertainty driven decoherence [4], it suffices to consider the simplified case with dominant mixing only between neutrino families 12, and 23 [15], i.e. with mixing angles θ12 = θ23 = θ , and θ13 = 0. 5.1. Stochastic quantum-gravity models We commence our analysis with the microscopic model of Section 2. We assume the case where Δ12 aeμ required from our earlier discussion, and this is the case we shall compare with the results of our experimental fit above. The result of the fit (4.3), (4.4), then, implies that the above decoherence-induced damping exponent (2.11) is independent of L and actually we have, to leading order in aeμ /Δ12 1 (reinstating dimensions of h¯ , c): cos(4θ ) − 1 Ω 2 (aeμ )2 1 + (5.1) · L ∼ 2.56 × 10−19 GeV km. 4 This in turn implies that in this specific model of foam, the density fluctuations of the space–time charged black holes is such that, for maximal mixing, say, θ = π/4 assumed for concreteness, and for L ∼ 180 Km, as appropriate for the KamLAND experiment, the decoherence damping factor is D = Ω 2 G2N n20 ∼ 2.84 × 10−21 GeV, if the result of the fit is due exclusively to this effect (note that the mixing angle part does not affect the order of the exponent). Smaller values are found for longer L, such as in atmospheric neutrino experiments. The independence of the relevant damping exponent from the oscillation length, then, implied by our fit above, may be understood as follows in this context: in the spirit of [14], the quantity 2 GN n0 = ξ m E , where ξ 1 parametrises the contributions of the foam to the induced neutrino mass differences, according to our discussion above. Hence, the damping exponent becomes in this case ξ 2 Ω 2 (m2 )2 · L/E 2 . Thus, for oscillation lengths L we have L−1 ∼ m2 /E, and one is left with the following estimate for the dimensionless quantity ξ 2 m2 Ω 2 /E ∼ 1.3 × 10−2 . This implies that the quantity Ω 2 is proportional to the probe energy E. In principle, this is not an unreasonable result, and it is in the spirit of [14], since back reaction effects onto space–time, which affect the stochastic fluctuations Ω 2 , are expected to increase with the probe energy E.

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However, due to the smallness of the quantity m2 /E, for energies of the order of GeV, and m2 ∼ 10−3 eV2 , we conclude (taking into account that ξ 1) that Ω 2 in this case would be unrealistically large for a quantum-gravity effect in the model. We remark at this point that, in such a model, we can in principle bound independently the Ω and n0 parameters by looking at the modifications induced by the medium in the arguments of the oscillatory functions of the probability (2.10), that is the period of oscillation. Unfortunately this is too small to be detected in the above example, for which hi Δ12 . The result of the fit, however, may be interpreted more generally, as implying oscillation length L independent exponents in the decoherence exponential suppression factors in front of the oscillatory terms in the transition probabilities. In this sense, the bound (4.3), (4.4) determined by the fit above, can be applied to other stochastic decoherence models, for instance the one discussed in [15], in which one averages over random (Gaussian) fluctuations of the background space–time metric over which the neutrino propagates. In such an approach, one considers merely the Hamiltonian of the neutrino in a stochastic metric background. This is one contribution to decoherence, since other possible non-Hamiltonian interactions of the neutrino with the foam (such as the Lindblad terms considered earlier) are ignored. In this case, one obtains transition probabilities with exponential damping factors in front of the oscillatory terms, but now the scaling with the oscillation length (time) is quadratic [15], consistent with time reversal invariance of the neutrino Hamiltonian. For instance, for the two generation case, which suffices for our qualitative purposes in this work, we have: + 2 i(ω −ω )t (z0 − z0− )t (m1 − m22 ) 1 1 2 = exp i exp − −iσ1 t + V cos 2θ e k 2 k iσ3 t 1 iσ2 t (m21 − m22 ) + V cos 2θ − V cos 2θ × exp − 2 2 k 2 2 2 2 (m1 − m2 ) (9σ1 + σ2 + σ3 + σ4 ) × exp − 2k 2 2V cos 2θ (m21 − m22 ) (12σ1 + 2σ2 − 2σ3 ) t 2 , + (5.2) k where k is the neutrino energy, σi , i = 1, . . . , 4, parametrise appropriately the stochastic fluctuations of the metric in the model of [15], Υ = 2V k 2 , |Υ | 1, and k 2 m21 , m22 , and m1 −m2

1 2 m1 + Υ (1 + cos 2θ ) m21 − m22 + Υ 2 m21 − m22 sin2 2θ , 2 1 z0− = m22 + Υ (1 − cos 2θ ) m21 − m22 − Υ 2 m21 − m22 sin2 2θ . 2 z0+ =

(5.3)

In principle, effects from all the terms in (2.2) have to be simultaneously considered when fitting data. In the present approach the fit has indicated that the non-Hamiltonian decoherence term dominates, for strength of fluctuations consistent with quantum gravity. Note that the metric fluctuations-σi induced modifications of the oscillation period, as well 2 as exponential e−(···)t time-reversal invariant damping factors [15]. The latter is attributed to the fact that in this approach, only the Hamiltonian terms are taken into account (in a stochastically fluctuating metric background), and as such time reversal invariance t → −t is not broken explicitly. But there is of course decoherence, and the associated damping.

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We, then, observe that the result of the fit above, (4.3), (4.4), implying L-independent exponents in the associated damping factors due to decoherence, may also apply to this case, implying for the damping exponent: 2 2V cos 2θ (m21 − m22 ) (m1 − m22 )2 (9σ1 + σ2 + σ3 + σ4 ) + (12σ1 + 2σ2 − 2σ3 ) t 2 k 2k 2 ∼ 1.3 × 10−2 .

(5.4)

Ignoring subleading MSW effects V , for simplicity, and considering oscillation lengths t = L ∼ 2k 2 2 , we then observe that the independence of the length L result of the experimental fit, (m1 −m2 )

found above, may be interpreted, in this case, as bounding the stochastic fluctuations of the metric to 9σ1 + σ2 + σ3 + σ4 ∼ 1.3 × 10−2 . This is too large to be a quantum gravity effect, which means that the L2 contributions to the damping due to stochastic fluctuations of the metric, found in the model of [15] above, cannot be the explanation of the fit. 5.2. Conventional explanation: energy uncertainties The reader’s attention is drawn at this point to the fact that such time-reversal invariance decoherence may also be due to ordinary uncertainties [4] in energies and/or oscillation lengths, which are unrelated to quantum gravity effects. Such “ordinary” effects would be beyond those accounted for by the statistical analysis of systematics in the experiments. For instance, consider the ordinary oscillation formula for neutrinos, with a mixing matrix U , Pαβ = Pαβ (L, E) n = δαβ − 4

2 m2ab L ∗ ∗ Re Uαa Uβa Uαb Uβb sin 4E a=1 b=1,a

(5.5)

a=1 b=1,a

where α, β = e, μ, τ, . . . , a, b = 1, 2, . . . , n, m2ab = m2a − m2b . In general there are uncertainties in the energy E in the production of a ν (and/or ν¯ )-wave, and also in the oscillation length. As a result of these uncertainties one has to average the oscillation probability (5.5) over the L/E dependence. Considering a Gaussian average [4], ∞ P = −∞

≡ x, σ =

2 1 − (x−) dx P (x) √ e 2σ 2 , σ 2π

(x − x)2 , x = L/4E, and approximating L/E L/E we obtain

Pαβ = δαβ − 2 −2

n

n

a=1 b=1,a

∗ 2 2 2 ∗ 1 − cos 2m2ab e−2σ (mab ) Re Uαa Uβa Uαb Uβb

∗ 2 2 2 ∗ sin 2m2ab e−2σ (mab ) . Im Uαa Uβa Uαb Uβb

a=1 b=1,a

(5.6)

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Notice the exponential damping factors due to the fluctuations σ . In fact, as discussed in [4], there are two kinds of bounds for σ : a pessimistic: one, according L L L L L 2 E 2 1/2 4E ( L + E . to which σ x 4E E ) and an optimistic: σ 4E ([ L ] + [ E ] ) In our case, where we consider long baseline experiments, the uncertainties in the oscillation length L are negligible, and hence the two cases degenerate to a single expression for L E σ = 4E E . The damping exponent, then, in (5.6), arising from the uncertainties in the energy of the (anti)neutrino beam, becomes 2 L2 E 2 2 2 2 m2 . 2σ m = 2 (5.7) 2 E (4E) As mentioned above, for oscillation lengths we have Lm2 /2E ∼ O(1), and hence, the result of the best fit (4.3), (4.4), implying independence of the damping exponent on L (irrespective of the power of L), yields an uncertainty in energy of order E ∼ 1.6 × 10−1 (5.8) E if one assumes that this is the principal reason for the result of the fit. This is not an unreasonable result, thus implying, at first sight, that the result of the fit may be interpreted as being due to ordinary physics associated with uncertainties in the energy of the neutrino beam. However, the fact that the fit indicates a selective damping among the various oscillatory terms in the expression for the pertinent probability, with some of the terms being not damped at all, poses a serious puzzle, which is still not resolved. Hence one cannot yet exclude the possibility that the fundamental physics of quantum gravity may play a significant rôle in some models other than the ones examined in this work, thereby providing an explanation for the results of the current fit. We next notice that the important difference of the effects due to ordinary uncertainties in energy or oscillation length from the stochastic fluctuations of gravity medium, discussed above, lies on the fact that the period of oscillation is not affected by the averaging procedure (5.6), in contrast to the stochastic gravity cases (2.10) and (5.2), and thus in principle the effects can be disentangled. However, in general these latter corrections are small, and beyond the sensitivity of the current experiments. Nevertheless, as we have seen, some effects, such as the time-reversal symmetric stochastic fluctuations of the background metric (5.2), can be already disentangled straightforwardly by their order as compared with the above energy-uncertainty effects due to ordinary physics. The precise energy and length dependence of the damping factors is an essential step in order to determine the microscopic origin of the induced decoherence and disentangle genuine new physics effects [3] from conventional effects, which as we have seen above may also contribute to decoherence-like damping [5]. For instance, as we discussed above, some genuine quantumgravity effects, such as the stochastic fluctuations of the space–time, are expected to increase in general with the energy of the probe [3], as a result of back reaction effects on space–time geometry, in contrast to ordinary-matter-induced ‘fake’ CPT violation and ‘decoherence-looking’ effects, which decrease with the energy of the probe [5]. At present, the sensitivity of the experiments is not sufficient to unambiguously determine the microscopic origin of the decoherence effects, as we have seen above, but we think that in the near future, when experiments involving both higher energy and precision become available, one would be able to arrive at definite conclusions on this important issue. Thus, phenomenological analyses like ours are of value and

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should, in our opinion, be actively pursued, in our opinion, in the future, not only in neutrino physics but also in other sensitive probes of quantum mechanics, such as neutral mesons. 6. Conclusions and outlook In this work we have presented a microscopic model and a complete analysis of a related three-generation neutrino transition probabilities, which include decoherence effects with guaranteed positivity. In our opinion this is the first complete, mathematically consistent, example of a Lindblad-decoherence model for neutrinos with full three generation mixing with a plausible microscopic interpretation. We have shown that decoherence effects can account for all available neutrino data, including LSND results even in a minimalistic scenario (with only one new parameter, which parametrises all the decoherence effects). Contrary to other approaches in the literature, using sterile neutrinos [20], and following the spirit of our earlier work [7,14], we have attempted to interpret the LSND results not by means of oscillations, but as a decoherence effect inducing damping in the oscillatory terms, which is also present (as a permilish additional suppression) in other neutrino experiments as well. The specific oscillation length L dependence of the single decoherence parameter, implying L-independence of the corresponding exponents of damping over an oscillation length, could in principle find a natural explanation in some theoretical models of stochastic quantum gravity. However, its order of magnitude seems incompatible with this possibility, at least in the concrete space–time foam microscopic models considered here, since it would imply quantum-gravity effects to be unrealistically large. On the other hand, the result of our fit can find a natural explanation in terms of ordinary physics. It could be due, for instance, to uncertainties in the energy beam of (anti)neutrinos, and indeed this scenario seems to provide the most natural explanation of our fit. However, the selective damping of oscillation terms remains a serious unresolved issue. We now remark that quantum-gravity contributions could indeed be present, and lead to similar damping in oscillatory terms, but their suppressed order of magnitude would imply that they could only be probed at higher energies. For instance, high-energy neutrinos detected from distant supernovae, may probe these issues further, since they will increase significantly the sensitivity to genuine quantum gravity effects [21], and thus may probe the induced changes in the damping as well as the oscillation period, as discussed in this work. It goes without saying that there is much more work to be done, both theoretical and experimental, before definite conclusions are reached on this important issue, but we believe that neutrino (astro)physics will provide a very sensitive probe of new physics, including quantum gravity, in the not-so-distant future. Acknowledgements The work of N.E.M. is partially supported by funds made available by the European Social Fund (75%) and National Resources (25%), EPEAEK II, PYTHAGORAS (Greece). G.B. and N.E.M. would like to thank M. Baldo-Ceolin for the invitation to the III International Workshop on Neutrino Oscillations in Venice, 7–10 February 2006, Venice (Italy), where preliminary results of this work have been presented (N.E.M).

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Appendix A. Calculation of the transition probability We will now outline calculation of the neutrino transition probability νe ↔ νμ which is quite general. Since the Fi are Hermitian, we can rewrite the expression for L[ρ] as 1 L[ρ] ≡ −i[H, ρ] + D[ρ] = −i[H, ρ] + (A.1) ckl [Fk ρ, Fl ] + [Fk , ρFl ] . 2 k,l

After standard manipulations, we may write the non-Hamiltonian decoherence part D[ρ] as 1 ckl Fk , [ρ, Fl ] + Fk , [ρ, Fl ] − Fl , [Fk , ρ] + Fl , [Fk , ρ] D[ρ] = 4 k,l 1 + ρ, [Fk , Fl ] . (A.2) 2 On using the expansion ρ = i ρi Fi , this expression can be written ρi ckl 1 1 −film fkmj Fj + ifilm δkm + dkmj Fj + fkim flmj Fj D[ρ] = 4 3 2 i,j,k,l,m 1 1 1 1 + ifkim δlm + dlmj Fj + i2fklm δim + dimj Fj , (A.3) 3 2 3 2 where dij k is the symmetric rank 3 tensor associated with the anticommutator relations of SU(3) and the summation symbol is understood to refer to the appropriate repeated indices, as usual. We note that the only terms which contribute are ρi4ckl (−film fkmj + fkim flmj )Fj . We follow the basic notation [6,10] and express the time evolution of the density matrix as ρ˙k = (A.4) hi fij k + Dkj ρj = Lkj ρj , j

i

j

where we have ckl (−film fkmj + fkim flmj ). Dij = 4

(A.5)

k,l,m

Using the values of the structure constants fij k of the SU(3) group (appropriate to the three generation case being examined here) we arrive at 1 1 D11 = − c22 + c33 + (c44 + c55 + c66 + c77 ) , 2 4 1 1 D22 = − c11 + c33 + (c44 + c55 + c66 + c77 ) , 2 4 1 1 D33 = − c11 + c22 + (c44 + c55 + c66 + c77 ) , 2 4 √ 3 1 1 D44 = − c55 + (c11 + c22 + c33 + c66 + c77 + 3c88 ) + c38 , 2 4 2 √ 3 1 1 c38 , D55 = − c44 + (c11 + c22 + c33 + c66 + c77 + 3c88 ) + 2 4 2

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1 1 c77 + (c11 + c22 + c33 + c44 + c55 + 3c88 ) − 2 4 1 1 D77 = − c66 + (c11 + c22 + c33 + c44 + c55 + 3c88 ) − 2 4 3 D88 = − (c44 + c55 + c66 + c77 ), 8 √ 3 (c44 + c55 − c66 − c77 ), D83 = D38 = − 8 or conversely, D66 = −

√ 3 c38 , 2 √ 3 c38 , 2

1 c11 = D88 + D11 − D22 − D33 , 3 1 c22 = −D11 + D88 + D22 − D33 , 3 1 c33 = D88 − D11 − D22 + D33 , 3 2 2 c44 = −D55 + D44 − √ D38 − D88 , 3 3 2√ 2 3D38 − D88 , c55 = D55 − D44 − 3 3 2 2 c66 = −D77 + D66 + √ D38 − D88 , 3 3 2 2 c77 = D77 − D66 + √ D38 − D88 , 3 3 2 2 2 2 1 1 1 c88 = − D55 − D77 − D66 − D44 + D88 + D11 + D22 + D33 , 3 3 3 3 3 3 3 1 1 1 1 2 c38 = − √ D55 + √ D77 + √ D66 − √ D44 + D38 . 3 3 3 3 3

(A.6)

(A.7)

In terms of the eigenvalues λi of Lkj the probability of a neutrino of flavor να , created at time t = 0, being converted to a flavor νβ at a later time t, is calculated in this framework [6,10] to be 1 1 λt β −1 α e k Dik Dkj ρj (0)ρi . Pνα →νβ (t) = Tr ρ α (t)ρ β = + 3 2

(A.8)

i,j,k

References [1] See for instance: J.A. Wheeler, K. Ford, Geons, Black Holes, and Quantum Foam: A Life in Physics, 1998, and references therein. [2] R.M. Wald, Phys. Rev. D 21 (1980) 2742. [3] N.E. Mavromatos, in: Lecture Notes in Physics, vol. 669, Springer-Verlag, Berlin, 2005, p. 245, gr-qc/0407005, and references therein. [4] T. Ohlsson, Phys. Lett. B 502 (2001) 159, hep-ph/0012272. [5] M. Blennow, T. Ohlsson, W. Winter, JHEP 0506 (2005) 049, hep-ph/0502147; M. Jacobson, T. Ohlsson, Phys. Rev. D 69 (2004) 013003, hep-ph/0305064. [6] G. Lindblad, Commun. Math. Phys. 48 (1976) 119; R. Alicki, K. Lendi, in: Lecture Notes in Physics, vol. 286, Springer-Verlag, Berlin, 1987.

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