Extrinsic CPT Violation in Neutrino Oscillations

In this talk, we investigate extrinsic CPT violation in neutrino oscillations in matter with three flavors. Note that extrinsic CPT violation is different from intrinsic CPT violation. Extrinsic CPT violation is one way of quantifying matter effects, whereas intrinsic CPT violation would mean that the CPT invariance theorem is not valid. We present analytical formulas for the extrinsic CPT probability differences and discuss their implications for long-baseline experiments and neutrino factory setups. Introduction. Recently, there have been several studies on CPT violation in order to incorporate the results of the LSND experiment [1], which require a third mass squared difference. However, this is not compatible with three neutrino flavors. Therefore, in most of the phenomenological studies on CPT violation, different mass squared differences and mixing parameters for neutrinos and antineutrinos are introduced by hand leading to four mass squared differences and eight mixing parameters. Thus, it is possible to include the results of the LSND experiment. Note that the results of the LSND experiment will be tested by the MiniBooNE experiment (September 2002 → ∼ 2005) [2]. Furthermore, note that another possible description of the results of the LSND experiment are sterile neutrinos. However, sterile neutrinos have, in principle, been excluded by the SNO experiment [3]. Moreover, the first KamLAND data are consistent with the LMA solution [4], which means that there is no need for fundamental CPT violation. Eccentric or extrinsic CPT violation? Let us denote the neutrino oscillation transition probabilities by Pαβ ≡ P(να → νβ ). Then, the CP, T, and CPT probability differences (pds) are defined as ∆PCP αβ ≡ Pαβ −Pᾱβ̄ , ∆P T αβ ≡ Pαβ −Pβα , and ∆P CPT αβ ≡ Pαβ −Pβ̄ ᾱ . Now, intrinsic (eccentric) CPT violation (or fundamental or genuine CPT violation) is due to violation of the CPT invariance theorem, whereas extrinsic CPT violation (or matter-induced or fake CPT violation) is due to presence of ordinary matter. Here, we will assume that the CPT invariance theorem is valid. This implies for the CP and T pds that the intrinsic and extrinsic effects are mixed, whereas for the CPT pds there are extrinsic effects only. Therefore, non-zero (extrinsic) CPT pds show matter effects, and thus, they are one way of quantifying such effects. From conservation of probability, we obtain ∑α=e,μ,τ,... ∆PCPT αβ = ∑β=e,μ,τ,... ∆P CPT αβ = 0. Note that not all of these equations are linearly independent. For three neutrino flavors, 1 In collaboration with: Magnus Jacobson. Talk presented at the 5th International Workshop on Neutrino Factories & Superbeams (NuFact’03), Columbia University, New York, USA, June 5-11, 2003. we have nine CPT pds for neutrinos. However, only four are linearly independent. Choosing, e.g., ∆PCPT ee , ∆PCPT eμ , ∆PCPT μe , and ∆PCPT μμ as the known ones, the other five can be expressed in terms of these. Furthermore, we have ∆PCPT αβ = −∆P CPT β̄ ᾱ , i.e., the CPT pds for antineutrinos do not give any further information. The CPT probability differences. In vacuum, the CPT pds are ∆PCPT αβ = 0, α,β = e,μ,τ , whereas, in matter, they are given by ∆PCPT αβ = |[S f (t, t0)]βα | 2 −|[S̄ f (t, t0)]αβ | 2, where S f ≡ S f (t, t0) and S̄ f ≡ S̄ f (t, t0) are the evolution operators for neutrinos and antineutrinos, respectively. We have calculated S f and S̄ f explicitly using first order perturbation theory in the small leptonic mixing angle θ13. These explicit expressions for S f and S̄ f can be found in Ref. [5]. Two of the CPT pds (with an arbitrary matter density profile) are: ∆PCPT ee ≃ |β̄ |2−|β |2 and ∆PCPT eμ ≃ c23 ( |β |2 −|β̄ |2 ) − 2c23s23I ( β fC− β̄ f̄ ∗Ā∗ ) , where β and β̄ describe a part of the two flavor neutrino evolution in the (1,2)-subsector, f and f̄ are some functions, and Ā and C are complicated functions that can be found in Ref. [5]. In matter of constant density in the low-energy region (V . δ ≪ ∆), the CPT pds ∆PCPT ee and ∆PCPT μe are calculated to be [5] ∆P ee ≃ 8s 2 12c 2 12 cos2θ12 ( δLcos δL 2 −2sin δL 2 ) sin δL 2 V δ +O ( (V/δ ) ) , (1) ∆P μe ≃ −8s 2 12c 2 12c 2 23 cos2θ12 ( δLcos δL 2 −2sin δL 2 )