Long Range Forces from Two Neutrino Exchange Revisited

The exchange of two massless neutrinos gives rise to a long range force which couples to weakly charged matter. As has been noted previously in the literature, the potential for this force is Vν(r) ∝ G 2 F /r 5 with monopole-monople, spin-spin and more complicated interactions. Unfortunately, this is far too small to be observed in present day experiments. We calculate Vν(r) explicitly in the electroweak theory, and show that under very general assumptions forces arising from the exchange of two massless fermions can at best yield 1/r5 potentials. Junior Fellow, Harvard Society of Fellows. Email: Hsu@HUHEPL.bitnet, Hsu@HSUNEXT.Harvard. edu The prospect of discovering a new long range force coupling to ordinary matter is exciting from both the theoretical and experimental points of view [1]. Since long range forces require the existence of a massless particle, a logical place to look in the electroweak theory is at the effect due to neutrinos. The exchange of a single neutrino (or in general a single fermion) cannot give rise to a force since the interaction changes the angular momentum of the sources involved. However, the exchange of two neutrinos can leave the quantum numbers of the sources unchanged, and hence can lead to a long range force. One might guess on the basis of dimensional analysis that the potential for this interaction could take the form Vν(r) ∼ G 2 Fm /r, where m is the mass of the source particle. Feynman considered this form when contemplating neutrinos as the mediators of a gravitylike interaction [2]. If this were the interaction, the effects of such a force might be observable in laboratory tests [3]. At r ∼ cm, the resulting force on normal matter would be roughly 10−6 times smaller than that due to gravity if m is the electron mass, and comparable to that of gravity if m is a nucleon mass. The current limit on deviations from a 1/r potential at r ∼ cm are of order 10−4 [3]. If the two neutrino force were measurable, it would provide experimental information on neutrino masses which is complementary to that obtained from standard particle physics experiments long range force experiments are sensitive to extremely small masses. Unfortunately, the form given above for Vν(r) is incorrect. The correct behaviour, which we will derive below, is 1/r. This yields a much smaller effect. The exact form of the interaction is therefore somewhat academic, but seems to us worth computing. The two neutrino force was investigated previously by G. Feinberg and J. Sucher [4] and by A. De Rujula, H. Georgi and S. Glashow (unpublished). In fact, almost all of the results which appear in this paper have been obtained earlier by Feinberg and Sucher. However, our method of computation is different and we feel that it is sufficiently simple to warrant exposition. We became aware of the earlier work only after completing our own calculations. The previous authors come to conclusions similar to ours, but our detailed results disagree slightly with those of Feinberg and Sucher. Our potential Vν(r) is smaller than theirs by a factor of two, and the coefficient of our σ1 · σ2 term is also different. We do not at this time understand the origin of this disagreement. Consider the diagrams shown in figure 1. Since we are interested in a long distance effect, and correspondingly low momentum exchange, it is a good approximation to combine the effects of W and Z exchange into four-fermi operators involving neutrinos and weakly charged source particles. The resulting operator can be Fierz transformed into the following form: O4 = GF √ 2 [ν̄γμ(1 − γ5)ν][ūγ (a− bγ5)u], (1) where a and b depend on the fermion u.