Comment about constraints on nanometer-range modifications to gravity from low-energy neutron experiments

A topic of present interest is the application of experimentally observed quantum mechanical levels of ultra-cold neutrons in the earth’s gravitational field for searching short-range modifications to gravity. A constraint on new forces in the nanometer-range published by Nesvizhevsky and Protasov follows from inadequate modelling of the interaction potential of a neutron with a mirror wall. Limits by many orders of magnitude better were already derived long ago from the consistency of experiments on the neutron-electron interaction. PACS numbers: 04.80.Cc, 28.20.Cz, 28.20.-v email: [email protected], [email protected] In a recent experiment, quantum mechanical levels of the neutron in the earth’s gravitational field were observed above a flat neutron optical mirror by measuring the transmission of ultracold neutrons through a narrow horizontal channel formed by the mirror and an absorber, as a function of the channel width [1]. The nuclear Fermi potential of the mirror material is given by UF = 2πħ mn Nb, (1) wheremn is the neutron mass, and b is the coherent neutron scattering length of the mirror nuclei with number density N . The energies of the lowest energy levels of the neutron in the potential well formed by the earth gravitational potential and the mirror are in the peV range whereas a typical value for UF is 100 neV. It was pointed out that additional short-range interactions close to the mirror would modify the transmission pattern. The authors of refs. [2, 3] claim that a competitive limit on non-Newtonian interactions in the nm range follows from the absence of a bound state due to the hypothetic short-range interaction. We show that this claim is not valid. A common parameterisation of short-range modifications of the gravitational interaction employs a Yukawa-type potential between two masses m and M , VG (r) = −G mM r αG exp (−r/λ) . (2) For the extended mass of a flat mirror, integration of eq.(2) provides the effective potential for a neutron situated at distance z outside the mirror with mass density ρ, Veff (z) = −U0 exp (−z/λ) , U0 = 2πGαGmnρλ . (3) The authors of refs. [2, 3] replaced the potential inside the mirror by infinite repulsion and thus consider the potential well U = ∞ for z ≤ 0, and U = −U0 exp (−z/λ) for z > 0, with U0 > 0. The condition for the absence of a bound state in this well, U0mnλ 2 < 0.723ħ, αG > 0, (4)