β-decay

Lorentz invariance has been tested rather poorly in the weak interaction in comparison to the electromagnetic interaction. This work discusses which tests on the weak interaction may be relevant. In particular, it considers exploiting the spin degrees of freedom in β-decay for testing rotational invariance. The relation between the various phenomenological tests of Lorentz invariance is shown using a new theoretical framework.

Lorentz invariance means that physical laws are invariant under boosts and rotations. There are many experimental tests of Lorentz invariance. Arguably, most and the most precise tests were done on the electromagnetic interaction. In contrast, very few tests have been made on the weak interaction, even though the StandardModel originated from –and has been shaped by– the details of the weak interaction, i.e. the violation of Parity (P) and Charge conjugation (C) on one hand and the violation of the combined CP symmetry on the other. Of course, the importance of Lorentz invariance demands testing it for all interactions. Manifestations of Lorentz Invariance Violation (LIV) in the weak interaction can be searched for in low-energy experiments, such as in β-decay.

Presently, one of the main efforts in fundamental physics is the unification of the Standard Model with General Relativity, in what is mostly referred to as quantum gravity models. Certain models of quantumgravity contain terms which violate Lorentz invariance and CPT symmetry. Requiring a theory that identifies the appropriate observables, Kostelecký and coworkers have developed a theoretical framework named “Standard-Model Extension” (SME) that contains all the properties of the StandardModel and General Relativity, but additionally contains all possible terms violating Lorentz and CPT symmetry resulting from spontaneous breaking of Lorentz invariance. It also follows from this phenomenological approach that observables for the different interactions are a priori independent. Therefore, it is insufficient to test only the electromagnetic interaction.

We have started an experimental and theoretical program on LIV considering charged currents in the weak interaction, focusing on β-decay. A theoretical framework has been formulated that gives guidance to possible experiments [1]. It also shows to what extent experiments can be related. In this theory Lorentz symmetry breaking is implemented by modifying the propagation of the W boson. The theoretical motivation can be found in reference [1]. Here we will discuss the relevant results for β-decay experiments. In our experimental work we focus on the spin degree of freedom which was not at all considered before.

The β-decay rate, ignoring Coulomb and induced recoil effects, is given in the StandardModel by

where β is the velocity of the β particle in units of the light velocity, <J>/J describes the direction and degree of nuclear polarization of the parent nucleus, σ is the spin vector of the β particle, and h bar / Gamma is the lifetime of the nucleus. A and G are the well-known parity-violating parameters: A is the β asymmetry or “Wu” parameter and G the longitudinal polarization of the outgoing β particle (G = 1 for β^- and G = -1 for β⁺).

If there is a preferred direction in space (i.e. Lorentz symmetry breaking), eq. (1) will be modified. In that case we expect it to be of the form

The ξ_i are the magnitudes of the Lorentz symmetry breaking terms. Here n_i are the preferred directions in space given in the laboratory frame. The preferred direction is expected to be independent of the Earth's rotation. Assuming that the preferred direction is fixed in the Sun-centered inertial reference frame (N), one can express the parameters in terms of the lab-centered frame (n) by using a rotation matrix R, which depends on the colatitude of the experiment (θ_l) and the Earth's sidereal rotation frequency (ω = 2π/[23h 56m] ).

The components n are defined such that the z-axis is perpendicular to the Earth's surface, the x-axis points in the north-south direction and the y-axis completes the right-handed coordinate system by pointing from west to east.

To calculate these parameters yourself, please go to our transformation-calculator.

[1] J.P. Noordmans, H.W.Wilschut, R.G.E. Timmermans - Lorentz violation in neutron and allowed nuclear beta decay, Phys Rev C 87 (2013)055502 (arXiv1302.2730)

[2] H.W. Wilschut et al. - A new approach to test Lorentz invariance (arXiv:1303.6419v2)