Very special relativity induced phase in neutrino oscillation
VVery special relativity induced phase in neutrino oscillation
Alekha C. Nayak ∗ Physical Research Laboratory, Ahmedabad 380009, India
Abstract
In Very Special Relativity (VSR), the neutrino mass term is coupled with the VSR preferred axis, andhence Lorentz violating in nature. Beyond standard model physics predicts neutrino magnetic momentwhich is linearly proportional to the mass eigenstates of the neutrinos. We report an additive kinematicphase in the neutrino flavor oscillation due to the neutrino magnetic moment in the VSR framework. Thisphase is proportional to the coupling between the VSR preferred axis with the external magnetic field aswell as the spin of the neutrino. Furthermore, we predict time variation in the neutrino oscillation with aperiod of one sidereal day. ∗ [email protected] a r X i v : . [ h e p - ph ] J a n . THEORY The particle interactions in the Standard Model (SM) are based on the assumption that it pre-serves Lorentz and CPT symmetry. However, SM is a low energy limit of a unified theory at Planckscale. Local Lorentz and CPT symmetry violation are severely constrained by the experiments[1–4].To test these assumptions in experiments, an alternate theory of Lorentz violation have proposedby Cohen and Glashow[5]. The null result of Michelson-Morley experiment does not necessarilyrequire full Lorentz group, it only demands particular subgroup of the Lorentz group such as T(2),E(2), HOM(2) and SIM(2)[5]. Adjoining space-time translation with these Lorentz subgroups inknown as very special relativity. The generators for T(2) group are T = K x + J y and T = K y − J x , generator of T(2) along with J z forms E(2) subgroup, generator of T(2) along with K z formsHOM(2) subgroup, generator of T(2) along with K z and J z forms SIM(2) subgroup[5]. Here J and K represent rotation and boost respectively.The VSR invariant Lagrangian for neutrino is given by [5, 6] L = ¯ ν L (cid:0) i /∂ − m ν /nin.∂ (cid:1) ν L (1)where the null vector: n µ = (1 , , ,
1) defines the preferred axis. n µ is invariant under T , T and J z , but transforms under K z as : n µ → n µ e φ , where e φ = ( E − p z ) /m ν [5, 6]. The presence ofpreferred axis n µ in the second term of Eq.(1) violates Lorentz symmetry. However, the numeratorand denominator are homogeneous in n µ , hence it becomes invariant under HOM(2) and SIM(2)subgroup [5, 6]. In principle, we can add such VSR mass term for all the SM leptons by replacingthe neutrino field with SM doublet and gauge it by replacing ∂ µ with covariant derivative D µ . Di-agonalization of the Yukawa mass term and VSR mass term by making bi-unitary transformations,we get oscillation in the neutrino sector as well as in the charge lepton sector[7, 8].In the Pontecorvo framework of neutrino oscillation, the flavor eigenstates (which are producedin the weak interaction) can be written as a linear superposition of mass eigenstates, | ν (cid:96) (cid:105) = (cid:88) (cid:96) U ∗ (cid:96)j | ν j (cid:105) (2)where | ν j (cid:105) is the mass eigenstate of neutrino. The probability of transition from ν (cid:96) → ν (cid:96) (cid:48) is givenby P ( ν (cid:96) → ν (cid:96) (cid:48) ) = (cid:12)(cid:12)(cid:12) (cid:88) j (cid:54) = k U (cid:96) (cid:48) j e − ∆ m kj L E U ∗ (cid:96)j (cid:12)(cid:12)(cid:12) (3)where U (cid:96) (cid:48) j is the probability amplitude to find the state | ν j (cid:105) in flavor state | ν (cid:96) (cid:48) (cid:105) , E is the energyof the neutrino beam and L is the distance between source and detector. The kinematically phase2n the neutrino oscillation is given by φ kj = ∆ m kj L E = 1 . (cid:16) ∆ m kj eV (cid:17) (cid:16) Lkm (cid:17) (cid:16)
GeVE (cid:17) (4)In this letter, we report an additive VSR induced phase to the QFT induced phase[9] due tothe magnetic moment of neutrino. The modified VSR invariant Dirac equation can be written as (cid:16) /p − m ν /n n.p (cid:17) ν L = 0 (5)Applying the modified Dirac operator from the left-hand side of Eq.(5), we get p = m ν . Since thethe dispersion relation remain unchanged, the free particle propagation does not change in VSR.The modified Dirac spinor for positive energy solution can be written as [7] u (cid:48) ( p ) ≈ (cid:16) − m ν /nn.p (cid:17) u ( p ) (6)where u ( p ) is the positive energy Dirac spinor. The matrix element for neutrino interacting with theelectromagnetic field can be expressed as : i ¯ u (cid:48) ( p (cid:48) , s (cid:48) ) Γ µs, s (cid:48) ( p, p (cid:48) ) u (cid:48) ( p, s ) A µ . In the n.A = 0 gauge,the effective interaction of the neutrino with electromagnetic field due to its magnetic moment canbe written as [7] i ¯ u (cid:48) ( p (cid:48) , s (cid:48) ) Γ µs, s (cid:48) ( p, p (cid:48) ) u (cid:48) ( p, s ) A µ ≈ iµ ν m ν u ( p (cid:48) , s (cid:48) ) n ν n β ˜ F µν σ βν ( n.p ) ( n.p (cid:48) ) u ( p, s ) (7)In the non-relativistic limit, the interaction term can be expanded as iµ ν m ν u ( p (cid:48) , s (cid:48) ) n ν n β ˜ F µν σ βν ( n.p ) ( n.p (cid:48) ) u ( p, s ) ≈ iµ ν χ † [( n . σ )( n . B ) − σ . B + ( n × E ) . σ ] χ (8)The corresponding interaction Hamiltonian can be written as [7, 10] H V SR = µ ν [ σ . B − ( n . σ )( n . B ) − ( n × E ) . σ ] (9)Neutrino magnetic moment has been studied in the beyond standard model physics with nonzeroneutrino mass[11–19]. The neutrino magnetic moment due to quantum correction is proportionalto its mass eigenstate [11, 12] µ ν i = 3 eG f m ν i √ π (10)We can drop the third term in Eq.(11), because the Earth and compact objects like neutron starsdo not have its own electric field. The additive phase due to µ ν σ . B in the flavor oscillation has3een reported in Ref.[9]. The new additive phase to the kinematic phase ( φ kj ) in the neutrinoflavor oscillation due to the neutrino magnetic moment in the VSR framework can be written as φ new = φ kj φ V SR = − φ kj µ ν ( n . σ )( n . B ) (11)where φ V SR = − µ ν ( n . σ )( n . B ). The new phase φ V SR depends upon the angle between the VSRpreferred direction with the spin of the neutrino as well as the external magnetic field. Furthermore,Earth rotation changes the angle between the VSR preferred axis and neutrino beam, hence wemay observe a change in the probability amplitude of conversion from one flavor to another in theneutrino oscillation data. This phase is also linearly proportional to the neutrino masse. In thenext section, we will discuss about time variation in the neutrino oscillation in one sidereal daydue to this VSR phase.
2. DAILY VARIATION IN THE NEUTRINO OSCILLATION DATA DUE TO VSR PHASE
Let us assume the observer is at latitude λ in local laboratory coordinate system XY Z andthe astronomical equatorial system is denoted by xyz . The Earth rotation axis is parallel to the z axis of the equatorial system. Let the unit vector along the x, y, z axis be ˆ x, ˆ y, ˆ z respectivelyand unit vector along X, Y, Z axis be ˆ X, ˆ Y ˆ Z respectively. We take ˆ x towards vernal equinox,ˆ Z is vertically upward in the local frame, ˆ X and ˆ Y vectors are tangential to the surface pointingtowards north and west respectively. Let β is the right ascension of ˆ j at initial time t=0. Thesetwo coordinates are related byˆ x = cos( θ + β ) ˆ Y − sin( θ + β )(cos λ ˆ Z − sin λ ˆ X )ˆ y = cos( θ + β ) ˆ Y + cos( θ + β )(cos λ ˆ Z − sin λ ˆ X )ˆ z = cos λ ˆ X + sin λ ˆ Z (12)The preferred direction n makes an angle ( θ e , φ e ) in the equatorial coordinate system, where θ e and φ e are polar angle and right ascension. The vector n in the local laboratory system is givenby ˆ n = [sin λ sin θ e sin( θ + β − φ e ) + cos λ cos θ e ] ˆ X + [sin θ e cos( θ + β − φ e )] ˆ Y + [cos θ e sin λ − sin θ e cos λ sin( θ + β − φ e )] ˆ Z (13)where θ = πT t , T is the one sidereal day. Let us take the magnetic field in the ˆ Z direction,i.e. B = B ˆ Z and the spin orientation of neutrino in the laboratory frame is given by σ =4 ( h ) ϕ VS R FIG. 1: The time variation of the VSR phase ( φ V SR ) in an arbitrary unit for one sidereal day. Here wechoose θ e = π/ , β − φ e = π/ λ = π/ σ ˆ X + σ ˆ Y + σ ˆ Z . The quantum mechanical operator acts on the spin state | p, σ (cid:105) as follows: n . σ | p, σ (cid:105) = s j | p, σ (cid:105) , where s j is the eigenvalue which is equals to ±
1. Then, the new phase due toVSR is given by φ V SR = − µ ν ( n . σ )( n . B )= ± µ ν B [cos θ e sin λ − sin θ e cos λ sin( θ + β − φ e )]= ± . × − B Gauss [cos θ e sin λ − sin θ e cos λ sin( θ + β − φ e )] (14)Due to the rotation of the Earth, the angle between the preferred axis and magnetic field changeswith time (t), so the neutrino oscillation phase show time variation. Hence, the conversion proba-bility of neutrino from one flavor to another also show time variation in the laboratory coordinatesystem. The time variation of φ V SR is shown in Fig.(1) and time period of this phase is one siderealday.
3. SUMMARY AND CONCLUSIONS
In the VSR, an additional kinematic phase is induced in the neutrino flavor oscillation due tothe neutrino magnetic moment. This phase show time dependence in the neutrino flavor oscillationwith a time period of one sidereal day. Since Earth magnetic field is ∼ Gauss[20] and 10 Gauss[21] respectively, hence this VSR induced oscillation iseighteen and eleven orders of magnitude smaller than standard flavor oscillation in these compactobjects. Large neutrino magnetic moment in the order of 10 − µ B ( ∼ − eV /Gauss ) has been5roposed[22–24]. Such large neutrino magnetic moment enhances the φ V SR effect by eight ordersof magnitude in all the above scenario and that could bring the effect to possible observability. Ifneutrino mass has VSR origin, then this novel feature of time variation in the neutrino oscillationis expected.
Acknowledgements:
I would like to thank Prof. Dharam Vir Ahluwalia for discussions andacknowledges the hospitality provided by IUCAA, Pune, India, during my visit where the workhas been initiated. I also thank my friends and colleagues Tripurari Srivastava and Arpan Das forreading this draft and giving useful suggestions. [1] J. Collins, A. Perez, D. Sudarsky, L. Urrutia, and H. Vucetich, Phys. Rev. Lett. , 191301 (2004),gr-qc/0403053.[2] P. Jain and J. P. Ralston, Phys. Lett. B621 , 213 (2005), hep-ph/0502106.[3] S. Groot Nibbelink and M. Pospelov, Phys. Rev. Lett. , 081601 (2005), hep-ph/0404271.[4] J. Polchinski, Class. Quant. Grav. , 088001 (2012), 1106.6346.[5] A. G. Cohen and S. L. Glashow, Phys. Rev. Lett. , 021601 (2006), hep-ph/0601236.[6] A. G. Cohen and S. L. Glashow (2006), hep-ph/0605036.[7] A. Dunn and T. Mehen (2006), hep-ph/0610202.[8] A. C. Nayak and P. Jain, Phys. Rev. D96 , 075020 (2017), 1610.01826.[9] D. V. Ahluwalia and C.-Y. Lee, EPL , 61001 (2017), 1705.09066.[10] J. Fan, W. D. Goldberger, and W. Skiba, Phys. Lett.
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