The matrix element for muon capture by a proton is calculated to O(p 3 ) within heavy baryon chiral perturbation theory using the new O(p 3 ) Lagrangian of Ecker and Mojžiš. External nucleon fields are renormalized using the appropriate definition of the wave function renormalization factor Z N . Our expression for Z N differs somewhat from that found in existing literature, but is the one which is consistent with the Lagrangian we use and the one which ensures, within our approach, the nonrenormalization of the vector coupling as required by the conserved vector current. Expressions for the standard muon capture form factors are derived and compared to experimental data and we determine three of the coefficients of the Ecker -Mojžiš Lagrangian, namely, b 7 , b 19 , and b 23 .Typeset using REVT E X 1

This comment is in response to Nieves and Pal [1] who dispute our claim [2] that a classical gravitational field may possibly distinguish between Dirac and Majorana neutrinos, described in terms of Gaussian wave packets propagating in a Lense-Thirring background, where the distinction is manifested in spin-gravity corrections to the neutrino oscillation length. They contend that our model for Majorana neutrinos is incorrect and that any conclusions relating to this neutrino type are not reliable. Furthermore, they suggest that any distinction between the two neutrino types will be suppressed by factors of m/E where m is the neutrino mass and E is its mean energy of propagation, and claim that this distinction is unobservable when m/E ≪ 1. We beg to disagree.As noted above, our model for the Dirac and Majorana neutrinos is motivated by a wave packet approach in quantum mechanics, as opposed to a plane-wave expansion in quantum field theory. This is an essential detail which Nieves and Pal fail to acknowledge. In addition, the gravitational field [2] is incorporated in terms of a gravitational phase Φ G , giving rise to an interaction Hamiltonian H ΦG with spin-dependent features, to be evaluated in terms of time-independent perturbation theory. These two details are important for framing the context underpinning our reply. Regarding the technical concerns, we agree that the Majorana condition they note in their eq. (1) is certainly true for a fermion field operator. However, we again emphasize that our perspective is quantum mechanical, so our treatment of the Majorana condition must be described in terms of wave functions. Adopting their notation, our approach is to identify [3,4] Unlike what Nieves and Pal claim, it indeed follows [3] that W 1,2 is a solution of the free particle equation / k W 1,2 = ± m W 1,2 [5] for wave functions. However, a more precise implementation of (1) with our notation leads to a Majorana wave packet model with the formwhereand ξ(k) is the Gaussian function [2]. Clearly, it follows from (2) that |ψ c 1(2) Maj. = ± |ψ 1(2) Maj. . This leads to a modification of our eq. (13) for the Majorana matrix element [2], which isOur plots [2], as applied to the SN 1987A data, are completely unaffected by the adjustments because the corrections only alter the contributions coupled to M ΩR 2 /r 2 , which are all exponentially damped compared to the M/r contributions. As for the m/E suppression issue raised by Nieves and Pal, this is of no relevance because all such terms are automatically excluded within the construction of |W 1,2 (k) Maj. , so everything presented in [2] is of leading order. As shown in (5), and also present in our eq. (13)

It is shown that the Pauli-Lubanski spin vector defined in terms of curvilinear co-ordinates does not satisfy Lorentz invariance for spin-1/2 particles in noninertial motion along a curved trajectory. The possibility of detecting this violation in muon decay experiments is explored, where the noninertial contribution to the decay rate becomes large for muon beams with large momenta and trajectories with radius of curvature approaching the muon's Compton wavelength scale. A new spacelike spin vector is derived from the Pauli-Lubanski vector that satisfies Lorentz invariance for both inertial and noninertial motion. In addition, this spin vector suggests a generalization for the classification of spin-1/2 particles, and has interesting properties that are applicable for both massive and massless particles.

We study the relative motion of nearby free test particles in cosmological spacetimes, such as the FLRW and LTB models. In particular, the influence of spatial inhomogeneities on local tidal accelerations is investigated by means of simple spherically symmetric models. The implications of our results for the dynamics of the solar system are briefly discussed. That is, on the basis of the models studied in this paper, we estimate the tidal influence of the cosmic gravitational field on the orbit of the Earth around the Sun and show that the corresponding temporal rate of variation of the astronomical unit is negligibly small.

The effects of isospin violation on the neutral weak magnetic form factor of the proton are studied using two-flavor chiral perturbation theory. The first nonzero contributions appear at O(p 4 ) in the small-momentum expansion, and the O(p 5 ) corrections are also calculated. The leading contributions from an explicit ⌬͑1232͒ isomultiplet are included as well. At such a high order in the chiral expansion, one might have expected a large number of unknown parameters to contribute. However, it is found that no unknown parameters can appear within loop diagrams, and a single tree-level counterterm at O(p 4 ) is sufficient to absorb all divergences. The momentum dependence of the neutral weak magnetic form factor is independent of the counterterm, and is consistent with previous quark model estimates. Removal of higher-energy physics from the counterterm leaves a pion-cloud contribution to the form factor which is approximately 0.02 nuclear magnetons at q 2 ϭ0.