Neutrino Spin Oscillations in a Black Hole Background in Noncommutative Spaces

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1 1 Neutrino Spin Oscillations in a Black Hole Background in Noncommutative Spaces S. A. Alavi; S. Nodeh Department of Physics, Hakim Sabzevari University, P. O. Box 397, Sabzevar, Iran. s.alavi@hsu.ac.ir; alialavi@fastmail.us; somayenode@yahoo.com. We study neutrino spin oscillations in a black hole background in noncommutative spaces. In the case of a charged black hole, the maximum frequency of oscillation is a monotonically increasing function of the noncommutativity parameter. For a neutral black hole the maximum frequency decreases with increasing the noncommutativity parameter. In both cases, the frequency of spin oscillations decreases as the distance from the black hole grows. We present a phenomenological application of our results. It is also shown that the noncommutativity parameter is bounded as Key words: Neutrino spin oscillation, Noncommutative spaces, Schwarzschild metric, Reissner-Nordstrom (RN) metric, Minimal length. Introduction Recently there have been notable studies on the formulation and possible experimental consequences of extensions of theories in the noncommutative spaces.there has been also a growing interest in possible cosmological consequences of space noncommutativity. In field theories the noncommutativity is introduced by replacing the standard product by the star product. For a manifold parametrized by the coordinates the noncommutative relations can be written as:,, where is an antisymmetric tensor which can be defined as, see [1] for a review. Coordinate noncommutativity implies the existence of a finite minimal length, below which the concept of distance becomes physically meaningless so in noncommutative spaces there can t be point like objects. To study black holes in the noncommutative spaces, it is not

2 necessary to change the Einstein s tensor part of the field equations and noncommutative effects act only on the matter source. So one can modify the distribution of point like source in favor of smeared objects. The effect of smearing is mathematically implemented as follows: position Dirac- delta function is replaced everywhere with a Gaussian distribution of minimum width. So we choose the mass density of a static, spherically symmetric, smeared, particle- like gravitational source as [2,3] : (1) A particle of mass M, instead of being perfectly localized at a point is diffused throughout a region of line size. On the other hand the neutrino physics is an active area of research with important implications for particle physics, cosmology and astrophysics. Cosmological implications of neutrinos include lepto/ baryogenesis and possible connections to the dark sector of the universe [4]. The phenomenon of neutrino oscillations can explain solar and atmospheric neutrino problems. It also provides the first experimental evidence for physics beyond the standard model since it requires nonzero mass for neutrinos. Interaction of neutrinos with an external field provides one of the factors required for a transition between helicity states. In this paper, we study neutrino spin oscillations in Reissner-Nordstrom (RN) and Schwarzschild backgrounds in noncommutative spaces. We recall that the effect of a quantum gravity-induced minimal length on neutrino oscillations has been studied in [5]. 2 Neutrino spin oscillations in noncommutative Schwarzschild metric By solving the Einstein s equations with as a matter source and setting ħ, we have [6]: (2) where : (3)

3 3 Here M is the mass of the black hole. The components of vierbein four velocity are as follows [7]: where (4) (5) is the four velocity of a particle in its geodesic path, which is related to vierbein four velocity through. The four velocity of a particle in the relevant metric is related to the world velocity of the particle through where and is the proper time. The non-zero vierbein vectors are as follows: (6) where X= To study the spin evolution of a particle in a gravitational field, we calculate which is the analogue of the electromagnetic field tensor. It is defined as follows: (7) where are the covariant derivatives of vierbein vectors which are defined through the following expression: (8) We use Eqs.(6) and (8) to calculate the covariant derivatives of vierbein vectors :

4 4 (9) Using the fact that any antisymmetric tensor in four-dimensional Minkowskian space-time can be stated in terms of two three dimensional vectors (such as electric and magnetic fields), we have: (10) Using Eqs. (5), (7) and (10) we have the following forms for the analogues of the electric and magnetic fields: (11) where. Geodesic equation of a particle in a gravitational field is as follows [8]:. (12) where the variable parameterizes the particle's world line. We assume that the motion is in a stable circular orbit with constant radius ( ). From Eqs. (2) and (12), we can calculate the values of and :

5 5, (14) (13) where =. Neutrino spin precession is given by the expression Ω where vector is defined as follows [7]: (15) By substituting (4), (11), (13), (14) and (15) in Ω, the only non-zero component of frequency i.e. Ω is obtained: Ω If we recover the results of the commutative case [7]: (16) Using Eq. (16), we have plotted Ω versus for different values of. (a) (b) Figure 1: Neutrino spin oscillations frequency versus radius of the neutrino orbit for different values of (a), and the commutative case (b).

6 6 We observe that the maximum frequency decreases with increasing the noncommutativity parameter. Transition probability versus time for different values of η has been plotted in Fig. (2). (a) (b) Figure 2: The transition probability versus time for different values of (a), and the commutative case (b). Neutrino spin oscillations in noncommutative Reissner-Nordstrom (RN) metric. The RN metric in a noncommutative space is given by [9,10]: where: (17) Here and M are the charge and the mass of the black hole respectively. The non-zero vierbein vectors are as follows:

7 7 (18) We use Eqs. (8) and (18) to calculate the non-zero covariant derivatives of the vierbein vectors, we have: (19) Using Eqs.(5), (7) and (10) we have the following forms for the electric and magnetic fields:., (20) From Eqs. (12) and (17) we can calculate the values of and :

8 8 (21) where. By substituting (4), (15), (20) and (21) in Ω, the only non-zero component of frequency i.e. Ω is obtained: Ω Using Eq. (22), we can plot Ω versus for different values of. One can recover the results of Schwarzschild metric (Eq. (16)) as a special case by taking can be also shown that in the limit, we recover the results of commutative case presented in [11].. It (22) (a) (b) Figure 3: Neutrino spin oscillation frequency versus radius of the neutrino orbit for different values of the commutative case (b). (a), and

9 9 It is seen that the maximum frequency of oscillation is a monotonically increasing function of the noncommutativity parameter. In Figs. (4a),(4b) we show the neutrino transition probability Ω Figure 4a: The transition probability versus time for different values of η. Figure 4b: The transition probability versus time for the commutative case X.

10 11 Phenomenological application We consider the simple bipolar neutrino system which helps us to understand many qualitative features of collective neutrino oscillations in supernovae. The system composed of a homogeneous and isotropic gas that initially consists of mono-energetic and and is described by the flavor pendulum [12]. We introduce as the fractional excess of neutrino over antineutrinos,. Fig. (2) shows that the period of transition probability is higher for the case of a neutral black hole in noncommutative spaces: (23) So at a later time t > 0, we have: (24) This implies that the precession frequency Ω of the flavor pendulum as a function of the neutrino number density for the case of a neutral black hole will be higher in commutative spaces. We have interpreted the neutrino spin precession as neutrino-antineutrino oscillations which is true for Majorana neutrinos. Similarly, it is seen from Figs. (4a), (4b) for a charged black hole that: (25) Which results in: (26) This implies that for a charged black hole the precession frequency Ω of the flavor pendulum as a function of the neutrino number density will be higher in noncommutativity spaces.

11 11 Bound on Noncommutativity Parameter. An interesting point is that by analyzing the diagrams of neutrino spin oscillations in noncommutative RN metric, we obtain the following bound for the parameter : (27) This is because the diagrams corresponding to are not acceptable. For instance we have plotted the diagram for which in Fig.(5). Figure 5. Neutrino spin oscillation frequency versus radius of the neutrino orbit for Using the fact that, where is the Planck mass, we arrive at the following bound for the noncommutativity parameter : (28)

12 12 which is consistent with the results reported in [6, 9, 13]. Here is the planck length. Conclusion In this paper we have studied neutrino spin oscillations in gravitational fields created by a charged and a neutral black hole in noncommutative spaces. We have also analyzed the dependence of the neutrino spin oscillations frequency on the radius of the orbit. For the case of a charged black hole, the maximum frequency of oscillation is a monotically increasing function of the noncommutativity parameter. For a neutral black hole, the maximum frequency decreases with increasing the noncommutativity parameter. We have also briefly studied the effects of noncommutativity of space on a bipolar neutrino system. Finally we have obtained the estimation for the noncommutativity parameter. If there exists any noncommutativity in nature, as seems to emerge from different theories and arguments, its implications should appear in neutrino oscillations in gravitational fields and in collective neutrino oscillation in supernovae. References [1] R. J. Szabo, Quantum gravity, field theory and signatures of noncommutative space time, arxiv: ; Quantum Field Theory on Noncommutative Spaces Phys. Rept. 378 (2003) ; arxiv : hep-th/ [2]. A. Smailagic, E. Spallucci, Feynman Path Integral on the Noncommutative Plane, J. Phys. A36 L467 (2003); UV divergence-free QFT on noncommutative plane, J. Phys. A36 L517 (2003). [3]. M. Rinaldi, A New approach to non-commutative inflation, Class.Quant.Grav. 28 (2011) [4]. N. E. Mavromatos, CERN-PH-TH/ , KCL-PH-TH/ , LCTS/ ; arxiv : hep-ph/

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