Rates of Processes with Coherent Production of Different Particles and the GSI Time Anomaly

With the help of an analogy with a double-slit experiment, it is shown that the standard method of calculation of the rate of an interaction process by adding the rates of production of all the allowed final states, regardless of a possible coherence among them, is correct. It is a consequence of causality. The claims that the GSI time anomaly is due to the mixing of neutrinos in the final state of the electroncapture process are refuted. It is shown that the GSI time anomaly may be due to quantum beats due to the existence of two coherent energy levels of the decaying ion with an extremely small energy splitting (about 10 eV) and relative probabilities having a ratio of about 1/99. The standard practice in the calculation of the rates (cross sections and decay rates) of interaction processes is to sum over the rates of production of all the allowed channels with a defined number of particles in the final state, regardless of a possible coherence among them. This practice has been violated in recent papers [1, 2, 3] claiming that the anomalous oscillatory time modulation of the electron-capture decays Pr → Ce + νe , Pm → Nd + νe (1) observed in a GSI experiment [4] is due to neutrino mixing (see Refs. [5, 6, 7, 8, 9, 10, 11, 12, 13]). In the simplest case of two-neutrino mixing, the final νe in the processes in Eq. (1) is a coherent superposition of two massive neutrinos conventionally called ν1 and ν2. Neglecting the neutrino mass effects in the interaction [14, 15, 16, 17, 18, 13], the final electron neutrino state in the processes in Eq. (1) is |νe〉 = cosθ |ν1〉+ sin θ |ν2〉 , (2) where θ is the mixing angle. In Refs. [1, 2, 3] it is claimed that the interference of the massive neutrinos in the final state generates the observed time anomaly. Such claim is in contradiction with standard calculations of decay rates, in which the coherence of the final state is irrelevant: the decay rates are calculated by adding the decay rates with a massive neutrino in the final state. The claim has been criticized in Refs. [19, 20]. Here I want to explain why the standard way of calculation of the rates of interaction processes is correct. For this purpose, it is useful to clarify how interference occurs. 1 As an example, let us consider the well-known double-slit interference experiment with classical or quantum waves. In a double slit experiment an incoming plane wave packet hits a barrier with two tiny holes, generating two outgoing spherical wave packets which propagate on the other side of the barrier. The two outgoing waves are coherent, since they are created with the same initial phase in the two holes. Hence, the intensity after the barrier, which is proportional to the squared modulus of the sum of the two outgoing waves, exhibits interference effects. The interference depends on the different path lengths of the two outgoing spherical waves after the barrier. Here, the important words for our discussion are “after the barrier”. The reason is that we can draw an analogy between the double-slit experiment and an electron-capture decay process of the type in Eq. (1), which can be schematically written as I → F + νe . (3) Taking into account the neutrino mixing in Eq. (2), we have two different decay channels: I → F + ν1 , (4) I → F + ν2 . (5) The initial state in the two decay channels is the same. In our analogy with the double-slit experiment, the initial state I is analogous to the incoming wave packet. The two final states F + ν1 and F + ν2 are analogous to the two outgoing wave packets. The different weights of ν1 and ν2 production due to a possible θ 6= π/4 correspond to different sizes of the two holes in the barrier. In the analogy, the decay rate of I corresponds to the fraction of intensity of the incoming wave which crosses the barrier. I think that it is clear that the fraction of intensity of the incoming wave depends only on the sizes of the holes. It does not depend on the interference effect which occurs after the wave has passed through the barrier. In a similar way, the decay rate of I cannot depend on the interference of ν1 and ν2 which occurs after the decay has happened. Of course, neutrino oscillations caused by the interference of ν1 and ν2 can occur after the decay, in analogy with the occurrence of interference of the outgoing waves in the double-slit experiment, regardless of the fact that the decay rate is the incoherent sum of the rates of production of ν1 and ν2 and the fraction of intensity of the incoming wave which crosses the barrier is the incoherent sum of the fractions of intensity of the incoming wave which pass trough the two holes. The above argument is a simple consequence of causality: the interference of ν1 and ν2 occurring after the decay cannot affect the decay rate. Causality is explicitly violated in Ref. [1], where the decaying ion is described by a wave packet, but it is claimed that there is a selection of the momenta of the ion caused by a final neutrino momentum splitting due to the mass difference of ν1 and ν2. This selection violates causality. In the double-slit analogy, the properties of the outgoing wave packets are determined by the properties of the incoming wave packet, not vice versa. In a correct treatment, all the momentum distribution of the wave packet of the ion contributes to the decay, generating appropriate neutrino wave packets. The authors of Refs.[2, 3] use a different approach: they calculate the decay rate with the final neutrino state |ν〉 = ∑