Derivations of Atomic Ionization Effects Induced by Neutrino Magnetic Moments

A recent paper [M.B. Voloshin, Phys. Rev. Lett. 105, 201801 (2010)] pointed out that our earlier derivations of atomic ionization cross-section due to neutrino magnetic moments (arXiv:1001.2074v2) involved unjustified assumptions. We confirm and elaborate on this comment with these notes. We caution that the results of the sum-rule approach in this paper contradict the expected behaviour in atomic transitions.

Advances in low energy detectors make it relevant to evaluate atomic effects induced by possible neutrino electromagnetic interactions A recent paper (Ref. [1]) observed that there are unjustified assumptions implicit in our previous derivations of the atomic ionization (AI) cross-section induced by neutrino magnetic moment (µ ν ) (Ref. [2]). This comment is correct.
We use the pre-defined notations in Ref. [2] and work with positive q 2 for clarity. The dσ/dT formula in Eq.10 is due to integration of Eq.8 over dΩ which, implicitly, is integration over q 2 . However, the q 2 → 0 limits have been taken in the assignments of the form factors F a (q 2 , T ) and q 2 F b (q 2 , T ). There is an unjustified assumption in Eq.10 that the form factors are constant within the integration range of q 2 from 0 to ∼ 4E 2 ν (where E ν ∼few MeV for reactor neutrinos). Consequently, Eq.10 as well as the results that follow are invalid.
The q 2 -dependent components of Eq.8 in the laboratory frame can be written as: There are no experimental constraints on F a (q 2 , T ) and F b (q 2 , T ). It is natural to expect the electron mass scale (m e ) plays an important role. In the case where the form factors are exclusively defined by m e , such that they are suppressed at q 2 > 2m e T (∼(0.1 MeV) 2 ), the AI effects would be small compared to the freeelectron cross-section. If, however, a higher mass scale like that of the atomic mass may have even a minor role to play in the process, the form factors can be finite up to q 2 ∼ E 2 ν . Large AI contributions are possible in this scenario and the discussions of Ref. [2] would still hold.
It is instructive to note how the equivalent photon approximation approach of Ref. [2] does produce valid results in two similar, but non-identical, problems.
1. µ ν -induced deuteron disintegration with solar neutrinos [3] − the form factors are defined by the nucleon mass scale (∼GeV) and so can be taken as constant within the range of integration up to q 2 ∼ (10 MeV) 2 , so that Eq.10 remains valid.

Charge-induced AI processes with relativistic minimum ionizing particles [4]
− the kinematics involves an additional (1/q 2 ) weight factor. The integral is dominated by contributions at q 2 → 0 and insensitive to the behaviour of the form factors at large q 2 . It is adequate to describe them by the physical photoelectric cross-section at q 2 = 0. Ref.
[1] adopted a sum-rule approach to arrive at an inclusive cross-section of µ ν -induced scattering with atomic electrons. This is given in Eq.13 as: In atomic transitions where binding energies (∆ b ) are involved, the cross-sections are expected to have a sharp increase across the transition edge from T < ∆ b to T > ∆ b . The sum-rule results, however, represent a continuous cross-section, smoothed to < 10 −3 . In addition, the contribution of photoelectric cross-section σ γ to the inclusive process is negative, which implies the total cross-section actually decreases across the transition edge. Both features contradict the expected behaviour. The results should therefore be taken with caution. We note that an alternative derivation using Hartree-Fock techniques results in a cross-section resembling that for free electrons scattering modified by step-functions [5].
We are grateful to Prof. Voloshin for pointing out our error in Ref. [2], and for the subsequent stimulating and in-depth discussions.