---
author:
- 'E. Fischbach'
date:  
title: |
    Many-Body Neutrino-Exchange Interactions and Neutrino Mass:\
    Comment on Phys.  Rev. Lett. [** 120**]{}, 223202 (2018)
---

This Comment corrects an erroneous remark by Stadnik contained in Ref.  [@Stadnik] to the effect that many-body neutrino-mediated forces are suppressed in all types of stars.  As we discuss below, such forces were considered in great detail in Refs.  [@Fischbach; @AoP; @Woodahl] and shown to produce catastrophically large energy densities in neutron stars and white dwarfs, unless neutrinos had a minimum mass $m \gtrsim 0.4$ eV/$c^{2}$. Stadnik attributes his remark to an unpublished preprint [@Smirnov]. 

Many-body effects arising from neutrino-exchange had been considered prior to Ref.  [@Fischbach; @AoP] by other authors including Feynman [@Feynman] and Hartle [@Hartle].  The possibility of catastrophically large effects arising from many-body forces is suggested by the work of Primakoff and Holstein [@Primakoff] dealing with many-body effects in atomic and nuclear systems.  These authors noted that in a nucleus containing $N$ particles the magnitude of the total $k$-body interaction ($k = 2, 3, 4, \ldots$) grows as the binomial coefficient $${N \choose k}= \frac{N!}{k! ( N-k)!}. 
\label{binomial}$$ For a typical neutron star with $N \simeq 10^{57}$, combinatorics can thus lead to catastrophically large effects, as shown in detail in Refs.  [@Fischbach; @AoP] and [@Woodahl]. 

To understand the arguments in Ref.  [@Smirnov] concerning Pauli-suppression of many-body neutrino exchange forces, we consider as an example the Hulse-Taylor binary system PSR 1913+16.  The mass $M_{1}$ of this pulsar is known to be $M_{1} = 1.4411M_{\odot}$, so that the number $N$ of neutrons is then approximately $N = 1.7 \times 10^{57}$, and we can take for its radius $R \simeq 10~{\rm km} \equiv R_{10}$. The relevant parameter determining the many-body neutrino-exchange contributions is then given in terms of the Fermi constant $G_{F}$ by Eq.  (5.34) in Ref.  [@Fischbach; @AoP], $$\eta(N,R_{10}) \equiv \frac{(G_{F}/\hbar c)N}{R_{10}^{2}} = 7.6 \times 10^{12}. 
\label{parameter}$$ It follows from Eq.  (\[parameter\]) that in a neutron star for any spherical volume or sub-volume of radius $R$, the energy grows as $7.6 \times 10^{6}(R/1~{\rm cm})$. As shown in Ref.  [@Fischbach; @AoP] for the Hulse-Taylor pulsar, the ratio of the many-body neutrino-exchange energy $W$ exceeds the total mass $M_{\rm U}$ of the known universe by the ratio $$\frac{W}{M_{\rm U}} \simeq 10^{(2 \times 10^{59} -57 - 99)}. 
\label{M U ratio}$$

It is clear from the preceding discussion that any mechanism that could suppress the many-body neutrino-exchange contribution in any neutron star (or in any sub-volume of a neutron star) to the level required by Eqs.  (\[parameter\]) and (\[M U ratio\]) must be of a universal nature, such as minimum value of the neutrino mass, rather than one depending on the detailed characteristics of individual neutron stars and white dwarfs. 

This conclusion is already evident from Eq.  (14) of Ref.  [@Woodahl] who exhibit the functional form of the simple 2-body exchange in the presence of Pauli-blocking, $$V_{\mu}^{(2)}(r) = \frac{(G_{F}a_{f})^{2}}{4\pi^{3}r^{5}}\left[\cos(2\mu r) + \mu r\sin(2\mu r)\right],
\label{V mu}$$ where $\mu$ is the local chemical potential.  A plot of the oscillatory factor in Eq.  (\[V mu\]) as a function of $x = \mu r$ is shown in Fig.  \[V mu plot\]. 

![ Plot of the oscillatory portion of $V_{\mu}^{(2)}(r)$ given by Eq.  (\[V mu\]).[]{data-label="V mu plot"}](Fig1){height="2.4in"}

The authors of Ref.  [@Smirnov] note that $\cos(2\mu r)$ and $\sin(2\mu r)$ can be rapidly oscillating factors.  However, we see from Fig.  \[V mu plot\] that even if the 2-body oscillatory factor happened to accidentally vanish for a spherical volume of radius $r$ given a particular value of $\mu$, any other volume with radius $r(1 + \epsilon)$, $\epsilon \ll 1$, will again lead to a catastrophically large energy density in light of the results of Eqs.  (\[parameter\]) and (\[M U ratio\]). 

A detailed examination of the 4-body neutrino-exchange contributions illustrates why the requisite cancellations suggested in Refs.  [@Stadnik] and [@Smirnov] are extremely unphysical.  Consider Eq.  (16) of Ref.  [@Woodahl] which gives the 4-body analog of Eq.  (\[V mu\]): $$\tilde{V}_{0}^{(4)} = \frac{(G_{F}a_{n})^{4}}{2\pi^{5}P_{4}S_{4}} \left[\cos(\mu S_{4})
    \left(\frac{3}{S_{4}^{4}} - \frac{3\mu^{2}}{2S_{4}^{2}} + \frac{\mu^{4}}{8}\right)
    + \frac{\mu\sin(\mu S_{4})}{S_{4}}\left(\frac{3}{S_{4}^{2}}- \frac{\mu^{2}}{2}\right)\right],$$

![ Two arbitrary configurations (a) and (b) of four neutrons in a neutron star showing that it is impossible that $S_{4}^{(a)} = S_{4}^{(b)} = r_{13} + r_{32} + r_{24} + r_{41}$ for all mass points.[]{data-label="fig2"}](Fig2a "fig:"){height="2.5in"}![Two arbitrary configurations (a) and (b) of four neutrons in a neutron star showing that it is impossible that $S_{4}^{(a)} = S_{4}^{(b)} = r_{13} + r_{32} + r_{24} + r_{41}$ for all mass points.[]{data-label="fig2"}](Fig2b "fig:"){height="2.5in"}

where $P_{4} = r_{12}r_{23}r_{34}r_{41}$ and $S_{4} = r_{12} + r_{23} + r_{34} + r_{41}$. We see immediately from Fig.  \[fig2\] that, generally, $S_{4}^{(a)} \neq S_{4}^{(b)} = r_{13} + r_{32} + r_{24} + r_{41}$, which represents a separate 4-body contribution to the self-energy.  It follows that even in a simple 4-body diagram one cannot ensure that $\cos(\mu S_{4}^{(a)})$, $\cos(\mu S_{4}^{(b)})$, $\sin(\mu S_{4}^{(a)})$, and $\sin(\mu S_{4}^{(b)})$ will all vanish for all mass points located at any $\vec{r}_{1}$, $\vec{r}_{2}$, $\vec{r}_{3}$, and $\vec{r}_{4}$.

Given the extremely large coefficients arising from Eqs.  (\[binomial\])–(\[M U ratio\]), we conclude that the cancellations suggested in Refs.  [@Stadnik] and [@Smirnov] are extremely unphysical as a mechanism for cancelling the many-body neutrino-exchange contributions to the self-energy of a neutron star or white dwarf.  As shown in Eq.  (5.39) of Ref.  [@Woodahl], the 8-body interaction energy $W^{(8)}$, for example, is according to Eq.  (\[parameter\]) proportional to $\eta(N,R_{10})$ and is given numerically by $$W^{(8)} = 5 \times 10^{77}~{\rm eV} = 3 \times 10^{11} M_{1}c^{2},$$ where $M_{1}$ is the mass of the Hulse-Taylor pulsar.  For the higher order contributions the combinatoric factors make such cancellations unphysical, as they would have to apply to each of an infinite number of possible sub-volumes for $R \gtrsim 10^{-6}$ cm, as well as extremely large numbers of individual neutron stars and white dwarfs. 

It follows that the conclusion of Ref.  [@Fischbach; @AoP] remains correct: For any neutrino species, $m_{\nu}\gtrsim 0.4~{\rm eV}/c^{2}$ is required to ensure that many-body neutrino-exchange forces do not lead to unphysical large many-body neutrino-exchange contributions to the self-energies of neutron stars or white dwarfs. 

Y. V. Stadnik, Phys.  Rev. Lett. [** 120**]{}, 223202 (2018). 

E. Fischbach, Ann.  Phys. ( NY) [**247**]{}, 213 (1996). 

B. Woodahl, M. Parry, S.-J. Tu, and E. Fischbach, in *Beyond the Desert 1997*, ed.  by H. V. Klapdor-Kleingrothaus and H. Päs (Institute of Physics, Bristol, 1998), pp.  108–115. 

A. Yu.  Smirnov and F. Vissani, arXiv:hep-ph/9604443v2 (1996). 

R. P. Feynman, F. B. Morinigo, and W. G. Wagner, *Feynman Lectures on Gravitation*, (Addison-Wesley, Reading, MA, 1995). 

J. B. Hartle, in *Magic Without Magic*, ed.  by J. R. Klauder (Freeman, San Francisco, 1972), p. 259. 

H. Primakoff and T. Holstein, Phys.  Rev. [**55**]{}, 1218 (1939).