Sensitivity of CPT Tests with Neutral Mesons

Neutral-meson interferometry is a powerful tool for investigating the discrete symmetry CPT. This product of charge conjugation C, parity reflection P, and time reversal T is known to be an invariance of local relativistic quantum field theories of point particles in flat spacetime [1]. Among the various tests of CPT [2], the sharpest published bounds are obtained with the neutralkaon system. As an example, the CPT figure of merit rK ≡ |mK −mK |/mK has recently been constrained to rK < 1.3 × 10−18 at the 90% confidence level by the experiment E773 at Fermilab [3,4]. In neutral-meson interferometry, bounds on CPT violation are extracted using a phenomenological description of the meson time evolution. Denote by P 0 any of the possible neutral mesons K, D, B d, B 0 s produced using the strong interaction, and combine the Schrödinger wave functions of P 0 and its opposite-flavor antiparticle P 0 into a two-component object Ψ. Then, the time evolution of Ψ is governed by a 2×2 effective hamiltonian Λ through the equation i∂tΨ = ΛΨ. Off-diagonal components of Λ drive flavor oscillations between P 0 and P 0. Two possible kinds of CP violation can be studied within this formalism. The one usually considered involves T violation with CPT invariance and is controlled by a parameter P . In the kaon system, for example, a nonzero value of K is well established [2]. The other involves CPT violation with T invariance. It is controlled by a complex parameter δP ≈ ∆Λ/∆λ, where ∆Λ ≡ (Λ11 − Λ22)/2 is half the diagonal-element difference in Λ and ∆λ is the eigenvalue difference. In the kaon system, a bound on rK constrains δK . The parameter δP can be bounded experimentally whether or not a nonzero value has any theoretical basis. However, a framework for CPT violation based on conventional quantum field theory does exist. The idea is that apparent low-energy CPT and Lorentz breaking might arise spontaneously within a more fundamental theory that is otherwise CPT and Lorentz invariant [5]. Any apparent breaking at the level of the standard model would then merely reflect a feature of the vacuum rather than a fundamental property of the theory [6]. Potentially observable effects within this general framework have been studied in neutral-meson systems [7–10], in QED [11], and in baryogenesis [12]. Apparent Lorentz and CPT violation of this type can be incorporated in a general extension of the minimal SU(3) × SU(2) × U(1) standard model that preserves gauge invariance and renormalizability [13]. In the underlying theory, the spontaneous breaking generates constant background expectation values as usual, but the fields involved are Lorentz tensors instead of Higgs scalars. In the standard-model extension, nonzero expectation values appear as coupling constants with Lorentz indices. For example, an expectation value aμ would allow a CPTand Lorentz-violating term −aμψγψ for a fermion ψ. The present work investigates the sensitivity of neutral-meson experiments to indirect CPT-violating effects produced in the standard-model extension [14]. Most of the theoretical considerations apply to any of the four neutral-meson systems. For definiteness, in the discussion of CPT tests some emphasis is placed on the E773 experiment mentioned above. The results are of immediate interest for CPT tests because at present the framework of the standard-model extension seems to be the only available consistent theoretical basis for a nonzero δP within conventional quantum field theory [15]. The first step is to obtain an explicit expression for δP within the standard-model extension. A key point is that the parameter δP must be C violating but P and T preserving. This is because the strong-interaction states P , P 0 are eigenvectors of parity with the same eigenvalue, so the linear combinations forming the physical eigenstates PS , PL of Λ are parity eigenstates too. Parity is therefore preserved during the time evolution of a neutral-meson state, so any CP violation appearing in Λ is really C violation with P invariance. In the lagrangian L for the standard-model extension, the parameters controlling the Lorentz and CPT violation are assumed suppressed by the (small) dimensionless ratio of the relevant light energy scale to the Planck scale [13]. Thus, only contributions linear in these parameters could produce observable CPT violation in experiments with neutral mesons. Also, since ∆Λ is flavor-diagonal, any term in L with both CPT breaking and flavor changing would affect δP at most as the square of a small parameter and hence can be disregarded. Remarkably, an inspection shows that only one type of term in the standard-model extension is flavor diagonal while violating C but preserving P and T. For each quark field q it has the form −aq0qγ q, where aq0 is the zeroth component of a background expectation value aμ that