Describing Oscillations of High Energy Neutrinos in Matter Precisely

YITP-SB-05-16 Oscillations of high energy neutrinos in matter: Precise formalism and parametric resonance E. Kh. Akhmedova,b , M. Maltonic and A. Yu. Smirnovd,e arXiv:hep-ph/0506064v2 8 Jul 2005 a Physik Department T30, Technische Universität München James-Franck Straße, D-85748 Garching, Germany b Kurchatov Institute, Moscow, Russia c C.N. Yang Institute for Theoretical Physics SUNY at Stony Brook, Stony Brook, NY 11794-3840, USA d ICTP, Strada Costiera 11, 34014 Trieste, Italy e Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia We present a formalism for precise description of oscillation phenomena in matter at high energies or high densities, V > ∆m2 /2E, where V is the matter-induced potential of neutrinos. The accuracy of the approximation is determined by the quantity sin2 2θm ∆V /2πV , where θm is the mixing angle in matter and ∆V is a typical change of the potential over the oscillation length (l ∼ 2π/V ). We derive simple and physically transparent formulas for the oscillation probabilities, which are valid for arbitrary matter density profiles. They can be applied to oscillations of high energy (E > 10 GeV) accelerator, atmospheric and cosmic neutrinos in the matter of the Earth, substantially simplifying numerical calculations and providing an insight into the physics of neutrino oscillations in matter. The effect of parametric enhancement of the oscillations of high energy neutrinos is considered. Future high statistics experiments can provide an unambiguous evidence for this effect. PACS numbers: 14.60.Pq, 14.60.Lm 1. Introduction. Neutrino physics enters a new phase now, where the objectives are precision measurements of the parameters, studies of subleading oscillation effects and searches for new physics beyond the already standard picture, which includes non-zero neutrino masses and mixing. Detection of neutrinos from new sources, in particular, of cosmic neutrinos, is in the agenda. Substantial new information is expected from the studies of high energy (E > 1 GeV) neutrinos. This includes investigations of atmospheric neutrinos with new large volume detectors [1], long baseline accelerator experiments [2] and detection of cosmic neutrinos from galactic and extragalactic sources, as well as of neutrinos produced in interactions of cosmic rays with the solar atmosphere [3]. Another possible source of neutrinos is annihilation of hypothetical WIMP’s in the center of the Earth and the Sun [4]. In all these cases beams of high energy neutrinos can propagate significant distances in the matter of the Earth (or of the Sun) and therefore undergo oscillations/conversions in matter. Increased accuracy and reach of neutrino experiments put forward new and more challenging demands to the theoretical description of neutrino oscillations. In the present Letter we derive very simple and accurate analytic formulas describing oscillations of neutrinos in matter. Our primary goal is to study oscillations of high energy neutrinos [5], but the formulas we obtain are actually applicable in a wide range of neutrino energies. They simplify substantially numerical calculations and allow a deep insight into the physics of neutrino conversions in matter. In particular, they provide a useful tool for studying parametric resonance enhancement of neutrino oscillations. The parametric resonance can occur in oscillating systems with varying parameters due to a specific correlation between the rate of the change of these parameters and the values of the parameters themselves. In the case of neutrino oscillations in matter, the parametric enhancement is realized when the variation of the matter density along the neutrino trajectory is in a certain way correlated with the change of the oscillation phase [6, 7]. This effect is different from the MSW effect [8, 9], and yet can result in a strong amplification of the oscillations. We apply the formalism developed here to study the parametric resonance in oscillations of high energy neutrinos in the Earth. 2. Formalism. We consider oscillations in the 3-flavor neutrino system (νe , νµ , ντ ), with the mass squared differences ∆m231 and ∆m221 responsible for the oscillations of atmospheric and solar neutrinos, respectively. In certain situations, the full three-flavor problem is approximately reduced to the effective two-flavor ones, in which the electron neutrino νe oscillates into a combination of νµ and ντ . We shall be mainly interested in oscillations of neutrinos with energies E > ∆m231 /2V , where the matter-induced potential of neutrinos V (x) ≡ √ 2GF Ne (x), with Ne (x) the electron number density in matter and GF the Fermi constant. Numerically, this corresponds to E > 8 − 10 GeV for the matter of the Earth. In this case the 1-2 mixing is strongly suppressed by matter, and the problem is reduced to an effective two-flavor one, described by the potential V , mass 2 squared difference ∆m2 ≡ ∆m231 and vacuum mixing angle θ ≡ θ13 (which is assumed to be non-zero) [10]. In particular, the oscillations of electron neutrinos are determined by the transition probability P2 ≡ P (νe ↔ νa ), where νa = sin θ23 νµ + cos θ23 ντ is the state which mixes with νe in the third mass eigenstate. In terms of P2 the flavor transition probabilities are P (νe → νµ ) = P (νµ → νe ) = sin2 θ23 P2 , P (νe → ντ ) = P (ντ → νe ) = cos2 θ23 P2 [10]. The third state (orthogonal to νa ) decouples from the rest of the system and evolves independently. In the (νe , νa ) basis, the evolution matrix S(x) describing neutrino oscillations satisfies the equation i dS = H(x)S , dx with the Hamiltonian    V − cos 2θ 1 0 H(x) = +δ sin 2θ 2 0 −1 (1)  sin 2θ . cos 2θ (2) Here δ ≡ ∆m2 /4E , (3) and the first (potential) term dominates in the high energy limit. However, in most situations of interest the neutrino path length in matter L satisfies δ · L & 1, therefore we cannot consider the whole second term as a small perturbation, and the effect of δ on the neutrino energy level splitting should be taken into account. For this reason we split the Hamiltonian as H = H 0 + HI , (4) where S0 is the solution of the evolution equation with H replaced by H0 . From (5) we find  −iφ(x)  e 0 S0 (x) = , (10) 0 eiφ(x) where φ(x) ≡ Z x dx′ ω(x′ ) is the adiabatic phase. Then, according to (1), (4) and (9), the matrix SI satisfies the equation i dSI = S0−1 HI S0 SI = H̃I SI , dx H0 = ω  1 0 , 0 −1   −ǫ 1 . HI = sin 2θ δ 1 ǫ (5) Here q ω(x) ≡ (V /2 − δ cos 2θ)2 + δ 2 sin2 2θ , (6) cos 2θ δ − V /2 + ω δ ≈ sin 2θ ≪ 1. sin 2θ δ V (7) 2ω being the difference of the eigenvalues of H(x); ǫ≡ The ratio of the second and the first terms in the Hamiltonian in (5) is determined by the mixing angle in matter θm : sin 2θ δ = sin 2θm . ω (8) Therefore for sin 2θm ≪ 1 the term HI can be considered as a perturbation. Furthermore, according to (7), ǫ ∼ sin 2θm , so that the diagonal terms in HI can be neglected in the lowest approximation. We seek the solution of eq. (1) in the form S = S0 · SI , (9) (12) where H̃I ≡ S0−1 HI S0 is the perturbation Hamiltonian in the “interaction” representation. Eq. (12) can be solved (1) by iterations: SI = 1 + SI + ..., which leads to the standard perturbation series for the S matrix. For neutrino propagation between x = 0 and x = L we have Z L Z x Z L dx′ H̃I (x′ )... dxH̃I (x)− dxH̃I (x) SI (L) = 1−i 0 0 0 (13) Taking SI to the first order, we obtain from (9) the evolution matrix " # Z L  0 ei2φ(x) S(L) = S0 (L) 1 − iδ sin 2θ . dx e−i2φ(x) 0 0 (14) The νe ↔ νa transition probability P2 (L) is given by the squared modulus of the off-diagonal element [S(L)]ae : with  (11) 0 P2 = δ 2 sin2 2θ Z L 2 dx e−i2φ(x) . (15) 0 For density profiles that are symmetric with respect to the center of the neutrino trajectory, V (x) = V (L − x), eq. (15) gives "Z #2 2  L/2 ∆m2 2 sin 2θ P2 = 4 dz cos 2φ(z) , (16) 4E 0 where z = x − L/2 is the distance from the midpoint of the trajectory and φ(z) is the phase acquired between this midpoint and the point z. The transition probability P2 scales with neutrino energy essentially as E −2 . The accuracy of eq. (15) also improves with energy as E −2 . This is illustrated by panels (a) and (b) of fig. 1, which show P2 as a function of neutrino energy for several trajectories inside the Earth and as a function of the zenith angle of the neutrino trajectory Θ for several neutrino energies. One can see that already for E & 8 MeV the accuracy of our analytic formula is extremely good. Note that when neutrinos do not cross the Earth’s core (cos Θ > −0.837) and so experience a slowly changing potential V (x), the accuracy 3 of the approximation (15) is very good even in the MSW resonance region E ∼5 – 8 GeV. The accuracy of eq. (15) is also good for energies below ∼ 2 GeV (not shown in the figure); however, in this region the domain of the applicability of (15) is relatively narrow, since for E . 0.5 GeV the oscillations driven by the “solar” parameters (∆m221 , θ12 ) can no longer be neglected. To understand the remarkable accuracy of eq. (15), we find the correction ∆P2 to the transition probability in (15) emerging in the next nontrivial order in HI . Note that from the above considerations one can expect ∆P2 /P2 to be proportional to sin 2θm . Furthermore, as can be easily seen, for uniform matter eq. (15) reproduces the exact transition probability; therefore one expects ∆P2 /P2 ∝ V ′ . A straightforward calculation indeed gives ∆V ∆V ∆P2 ≃ sin2 2θm ≃ sin2 2θm , P2 4πω 2πV (17) where ∆V is the change of the potential over the oscillation length π/ω, and the last equality holds in the high energy regime. For slowly changing density this is equivalent to ∆P2 V′ ≃ sin2 2θm 2 . P2 4ω (18) Introducing the adiabaticity parameter γ = 4πω/(sin 2θm ∆V ), we find that ∆P2 /P2 ≃ sin 2θm γ −1 , and therefore for small mixing in matter our approximation is better than the adiabatic one. At the same time, for ∆V /4πω < 1 it is better than the simple expansion in powers of sin2 2θm . The matter density profile of the Earth satisfies V ′ /V 2 . 0.5, and therefore for neutrino oscillations in the Earth our approximation is expected to work well when sin2 2θm ≪ 1. This is fulfilled in the high energy (or, equivalently, high density) limit EV /∆m2 ≫ cos 2θ, i.e. above the MSW resonance. If the vacuum mixing angle is small (i.e. θ = θ13 ), our expansion parameter is also small below the resonance. The above formalism applies in this low energy case as well, with only minor modifications: the sign of H0 in (5) has to be flipped, and correspondingly one has to replace ω → −ω in eq. (7). These modifications are necessary because of the interchange of the eigenvalues of the Hamiltonian upon crossing the MSW resonance. Expressions for the transition probability in eqs. (15), (16) remain unchanged. Thus, our results are in general valid outside the MSW resonance region, which for small θ is very narrow. For the non-resonant channels (ν̄ channels for ∆m2 > 0 or ν channels for ∆m2 < 0) and small vacuum mixing our formulas are valid in the whole diapason of energies because sin 2θm is always small. If θ13 is very small or vanishes, νe ↔ νµ,τ oscillations are driven by ∆m2 = ∆m221 and the large mixing angle θ = θ12 . The oscillation probabilities can then be expressed through another effective two-flavor probability, P̃2 ≡ P̃2 (∆m221 , θ12 , V (x)), in terms of which the flavor transition probabilities are P (νe ↔ νµ ) = cos2 θ23 P̃2 , P (νe ↔ ντ ) = sin2 θ23 P̃2 [11]. For EV /∆m221 ≫ cos 2θ12 (which for the typical densities inside the Earth corresponds to E & 0.5 GeV), the probability P̃2 is very well approximated by eq. (15). Let us consider the case of symmetric matter. Integrating (16) by parts, one finds " #2 Z L/2 dω 1 0 P2 = sin2 2θm sin φL + ω0 dz sin 2φ(z) , dz ω 2 0 (19) 0 where θm ≡ θm (V0 ), ω0 ≡ ω(V0 ), V0 being the potential at the initial and final points of the neutrino trajectory, and φL is the adiabatic phase acquired along the entire neutrino path. If the potential changes slowly with distance, so that ω −2 dω/dz ≪ 1, the second term in (19) can be neglected, and P2 reduces to the usual adiabatic probability in symmetric matter: Padiab = 0 sin2 2θm sin2 φL . The second term in (19) describes the effects of violation of adiabaticity. Let us apply the above results to neutrino beams (atmospheric, accelerator, cosmic neutrinos) crossing the Earth. According to the PREM model [12], the Earth density profile can be described as several spherical shells of radii Ri with rather smooth density change within the shells and sharp change at the borders between them. Then, along a direction from the center of the Earth outwards, ω(z) decreases abruptly from ωi+ to ωi− in very narrow regions around Ri . Therefore dω/dz is large in these narrow regions and small outside them. The integration in (19) can then be easily done, leading to 2  ω+ − ω− 0 sin φL − ω0 Σi i + −i sin 2φi . (20) P2 = sin2 2θm ωi ωi Here φi is the adiabatic phase acquired by neutrinos between the points z = 0 and z = Ri . We will use eq. (20) for interpreting the results of our calculations. 3. Parametric enhancement of oscillations. In the PREM model of the Earth density all the density jumps between different shells except those between the mantle and core are relatively small [12]. Therefore the density profile felt by neutrinos crossing the core of the Earth can be approximated by three layers (mantle - core - mantle) with slow density changes within the layers. Eq. (20) then gives  2   ω0 ωm 0 sin(φc + 2φm ) − P2 = sin2 2θm 1− sin φc , ωm ωc (21) where ωm and ωc are the values of ω(x) in the mantle and core on the respective sides of their border, and φm and φc are the phases acquired in the mantle (one layer) and core. Eq. (21) corresponds to the adiabatic neutrino propagation inside the core and mantle and violation of the adiabaticity at the borders between them. In the approximation of constant effective densities in the mantle 4 2 sin 2θ13 = 0.15 P2 0.4 10 0.2 E = 12 GeV 0.1 cos Θ 6 8 (c) 0.5 E = 10 GeV 0.2 0 (b) 0.3 -0.8 Θ= cos 0.6 (b’) 3 0.5 -0.9 Θ= cos 0.8 (a) 10 10 12 14 10 E = 15 GeV = -1 2 E [GeV] 1 1 E = 21 GeV 0 -1 -0.8 E -0.6 -0.4 cos Θ 1.5 0.5 -0.2 0 -1 -0.95 -0.9 -0.85 -0.8 cos Θ FIG. 1: Panels (a) and (b): transition probability P2 vs neutrino energy for different trajectories inside the Earth [panel (a)] and vs cos Θ for different neutrino energies [panel (b)]. Solid curves are the results of exact numerical calculations, dashed curves are obtained using formula (15). Small window (b’) in panel (b) shows the values of the parameter X3 [eq. (25)] calculated in the three-layer model of the Earth’s density. Panel (c): contours of constant phases φm (solid curves), φc (dashed curves) and of constant X3 = 0 (dotted curves). The numbers at the curves are the values of the phases in units of π. The shaded areas in panesls (b) and (c) correspond to the Earth’s core. For all panels we take sin2 2θ13 = 0.15 and ∆m2 = 2 · 10−3 eV2 . eff and core we have ω0 = ωm , and the non-adiabatic term eff eff is proportional to (1 − ωc /ωm ). For neutrino trajectories that cross the mantle only (φc = 0), eq. (21) reduces to the adiabatic probability. The passage of neutrinos through the core can lead to an enhancement of the oscillations. As follows from (21), the maximum enhancement of the probability can be achieved when sin(φc +2φm ) and sin φc are of opposite sign and maximal amplitude: sin φc = − sin(φc + 2φm ) = ±1, (22) that is, when φc = ± π + 2πn, 2 φm = ± π + 2πk . 2 (23) Here n and k are integers, and the signs in front of π/2 are correlated. In this case the enhancement factor is  2  2  P2max ωm Vm ω0 1− ≈ 2− ≈ 2.5 , = 1+ ωm ωc Vc sin2 2θm (24) where sin2 2θm in the denominator corresponds to the maximum possible transition probability for neutrinos crossing only the mantle, and we have taken into account that at high energies ωm /ωc ≈ Vm /Vc . The condition in eq. (23) and the enhancement described by eq. (24) are the particular cases of the parametric resonance condition and the parametric enhancement of neutrino oscillations [6, 7]. In [13] it was shown that in the case of matter consisting of alternating layers of two different constant densities and (in general) different widths the parametric resonance condition is X3 ≡ −(sin φm cos φc cos 2θm +cos φm sin φc cos 2θc ) = 0 , (25) where θm,c and φm,c are the mixing angles and the acquired oscillations phases in the layers m and c. This condition can also be used as an approximate one when matter density inside the layers is not constant but varies sufficiently slowly. For neutrino oscillations in the Earth we identify the layers m and c with the mantle and core. Since in the energy region sin 2θ δ ≪ V one has θm ≈ θc ≈ π/2, condition (25) reduces to X3 ≃ sin(φm + φc ) = 0 . (26) Eq. (23) is a particular realization of this condition. In the high energy limit the parametric resonance condition (23) was previously considered in the framework of active-sterile atmospheric neutrino oscillations in [14]. It should be noted that, while the realization (23) of condition (25) corresponds to the maximal possible parametric enhancement of oscillations of high energy neutrinos in the Earth, a sizable amplification is also possible if the equality X3 = 0 is realized differently, i.e. when the two terms in (25) do not separately vanish but cancel each other [15]. The parametric resonance conditions in eq. (25) or (23) require a subtle correlation between the matter density of the Earth, the distances that neutrinos travel in the Earth’s mantle and core (which are not independent), and also in general neutrino energy and oscillation parameters. Therefore it is far from obvious that these 5 conditions can actually be satisfied. Amazingly, this is indeed the case [14],[16],[13]. Our present analysis shows that, for νe ↔ νµ,τ oscillations of high energy neutrinos in the Earth, there are two regions of the zenith angles of neutrino trajectories in which (25) is satisfied: cos Θ = −1 ÷ −0.93 and cos Θ = −0.88 ÷ −0.84. This is illustrated in panel (b) of fig. 1. Since at high energies matter suppresses neutrino mixing, one could expect that for the trajectories crossing the core (cos Θ < −0.837), where the densities are higher, the probability would be suppressed or at least would not change. Instead, we see two prominent peaks there, exceeding maximal allowed by the MSW effect value of probability sin2 2θm by up to a factor of 2. This is the result of the parametric enhancement of neutrino oscillations. As can be seen from the figure, for core-crossing trajectories the positions of the peaks of P2 essentially coincide with zeros of X3 . For neutrino energies E ≃ 10 – 15 GeV, the oscillation phases corresponding to the peak with cos Θ . −0.93 (the inner peak) are φm ≃ π/4, φc ≃ 7π/4, while for the peak with cos Θ = −0.88 ÷ −0.84 (the outer peak), they are φm ≃ 0.35π, φc ≃ 0.65π. In both peaks to a good accuracy φm + φc = nπ, so that eq. (26) is satisfied. The phases in the outer peak are closer to the realization (23) of the parametric resonance condition, and therefore in this peak the parametric enhancement of oscillations is closer to the maximal possible one. From panel (c) of fig. 1 one can see that at E ≃ 21 GeV the maximum enhancement condition (23) can be exactly realized in the outer peak. For neutrinos of very high energies (E & 100 GeV), it can be realized nearly exactly in the inner peak, whereas the outer peak becomes slimmer and lower, and at energies E & 300 GeV virtually disappears. [1] H. Back et al., hep-ex/0412016. [2] C. Albright et al. [Neutrino Factory/Muon Collider Collaboration], physics/0411123. [3] J. G. Learned and K. Mannheim, Ann. Rev. Nucl. Part. Sci. 50, 679 (2000). [4] G. Bertone, D. Hooper and J. Silk, Phys. Rept. 405, 279 (2005). [5] For other studies of oscillations of high energy neutrinos in matter see, e.g., S. P. Mikheyev and A. Yu. Smirnov, ’86 Massive Neutrinos in Astrophysics and in Particle Physics, proceedings of the Sixth Moriond Workshop, edited by O. Fackler and J. Trân Thanh Vân (Editions Frontières, Gif-sur-Yvette, 1986), p. 355; A. Cervera et al., Nucl. Phys. B 579, 17 (2000) [Erratum-ibid. B 593, 731 (2001)] [hep-ph/0002108]; M. Freund, P. Huber and M. Lindner, Nucl. Phys. B 615, 331 (2001) [hep-ph/0105071]; I. Mocioiu and R. Shrock, JHEP 0111, 050 (2001) [hep-ph/0106139]; M. Blennow and T. Ohlsson, Phys. Lett. B 609, 330 (2005) [hep-ph/0409061], and also refs. [10, 11, 13, 14] below. [6] V. K. Ermilova, V. A. Tsarev and V. A. Chechin, Kr. Soob. Fiz. [Short Notices of the Lebedev Institute] 5, 26 (1986). [7] E. Kh. Akhmedov, preprint IAE-4470/1, (1987); Sov. J. Nucl. Phys. 47, 301 (1988) [Yad. Fiz. 47, 475 (1988)]. [8] L. Wolfenstein, Phys. Rev. D 17, 2369 (1978). [9] S. P. Mikheev and A. Yu. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985) [Yad. Fiz. 42, 1441 (1985)]. [10] E. Kh. Akhmedov, A. Dighe, P. Lipari and A. Yu. Smirnov, Nucl. Phys. B 542, 3 (1999) [hep-ph/9808270]. [11] O. L. G. Peres and A. Yu. Smirnov, Phys. Lett. B 456, 204 (1999) [hep-ph/9902312]; Nucl. Phys. B 680, 479 (2004) [hep-ph/0309312]. [12] A. M. Dziewonski and D. L. Anderson, Phys. Earth Planet Interiors 25, 297 (1981). [13] E. Kh. Akhmedov, Nucl. Phys. B 538, 25 (1999) [hep-ph/9805272]. [14] Q. Y. Liu and A. Yu. Smirnov, Nucl. Phys. B 524, 505 (1998) [hep-ph/9712493]: Q. Y. Liu, S. P. Mikheyev and A. Yu. Smirnov, Phys. Lett. B 440, 319 (1998) [hep-ph/9803415]. [15] In principle, the parametric resonance can also lead to a parametric suppression of oscillations; however, this does not happen for oscillations of high energy neutrinos in the Earth. [16] S. T. Petcov, Phys. Lett. B 434, 321 (1998) [hep-ph/9805262]. 4. In conclusion – We have derived very simple and accurate integral formulas describing neutrino oscillations in matter with arbitrary density profiles. They can be applied to all possible situations where the effective mixing in matter is small. In particular, our results can be applied to atmospheric, accelerator and cosmic neutrinos crossing the Earth as well as to cosmic neutrinos crossing the Sun. They can be used for probing the effects of small jumps in the density profile of the Earth as well as evaluating the effects of uncertainties in this profile on the interpretation of the results of future accelerator experiments. They can also be employed for studying the effects of proper averaging over the energy spectrum of the neutrino beam. We used the obtained formulas to study the parametric enhancement of oscillations of high energy neutrinos in the Earth and identified two peaks in the zenith angle distribution of core-crossing neutrinos, which are due to the parametric effects. Observation of these peaks in future high statistics experiments with high energy neutrinos would provide an unambiguous evidence for the parametric resonance effects in neutrino oscillations in matter. This work was supported in part by SFB-375 für Astro-Teilchenphysik der Deutschen Forschungsgemeinschaft (EA), the National Science Foundation grant PHY0354776 (MM) and by the Alexander von Humboldt Foundation (AS).