Limits given below are for the square root of $\mathit m{}^{{\mathrm {2(eff)}}}_{{{\mathit \nu}_{{\mu}}} }{}\equiv\sum_{i}\vert U_{{{\mathit \mu}} i}\vert ^2$ ${{\mathit m}^{2}}_{{{\mathit \nu}_{{i}}}}$.

In some of the COSM papers listed below, the authors did not distinguish between weak and mass eigenstates.

OUR EVALUATION is based on OUR AVERAGE for the ${{\mathit \pi}^{\pm}}$ mass and the ASSAMAGAN 1996 value for the muon momentum for the ${{\mathit \pi}^{+}}$ decay at rest. The limit is calculated using the unified classical analysis of FELDMAN 1998 for a Gaussian distribution near a physical boundary. WARNING: since $\mathit m{}^{{\mathrm {2(eff)}}}_{{{\mathit \nu}_{{\mu}}} }$ is calculated from the differences of large numbers, it and the corresponding limits are extraordinarily sensitive to small changes in the pion mass, the decay muon momentum, and their errors. For example, the limits obtained using JECKELMANN 1994 , LENZ 1998 , and the weighted averages are 0.15, 0.29, and 0.19 MeV, respectively.

VALUE (MeV) CL% DOCUMENT ID TECN  COMMENT $\bf{ <0.19}$ OUR EVALUATION $<0.17$ 90 1 ASSAMAGAN 1996 SPEC ${{\mathit m}^{2}}_{{{\mathit \nu}}}$ = $-0.016$ $\pm0.023$ • • We do not use the following data for averages, fits, limits, etc. • • $<0.15$ 2 DOLGOV 1995 COSM Nucleosynthesis $<0.48$ 3 ENQVIST 1993 COSM Nucleosynthesis $<0.3$ 4 FULLER 1991 COSM Nucleosynthesis $<0.42$ 4 LAM 1991 COSM Nucleosynthesis $<0.50$ 90 5 ANDERHUB 1982 SPEC ${{\mathit m}^{2}}_{{{\mathit \nu}}}$= $-0.14$ $\pm0.20$ $<0.65$ 90 CLARK 1974 ASPK ${{\mathit K}_{{\mu3}}}$ decay 1  ASSAMAGAN 1996 measurement of ${{\mathit p}_{{\mu}}}$ from ${{\mathit \pi}^{+}}$ $\rightarrow$ ${{\mathit \mu}^{+}}{{\mathit \nu}}$ at rest combined with JECKELMANN 1994 Solution$~$B pion mass yields ${{\mathit m}^{2}}_{{{\mathit \nu}}}$ = $-0.016$ $\pm0.023$ with corresponding Bayesian limit listed above. If Solution$~$A is used, ${{\mathit m}^{2}}_{{{\mathit \nu}}}$ = $-0.143$ $\pm0.024$ MeV${}^{2}$. Replaces ASSAMAGAN 1994 . 2  DOLGOV 1995 removes earlier assumptions (DOLGOV 1993 ) about thermal equilibrium below $\mathit T_{{\mathrm {QCD}}}$ for wrong-helicity Dirac neutrinos (ENQVIST 1993 , FULLER 1991 ) to set more stringent limits. 3  ENQVIST 1993 bases limit on the fact that thermalized wrong-helicity Dirac neutrinos would speed up expansion of early universe, thus reducing the primordial abundance. FULLER 1991 exploits the same mechanism but in the older calculation obtains a larger production rate for these states, and hence a lower limit. Neutrino lifetime assumed to exceed nucleosynthesis time, $\sim{}1~$s. 4  Assumes neutrino lifetime $>1~$s. For Dirac neutrinos only. See also ENQVIST 1993 . 5  ANDERHUB 1982 kinematics is insensitive to the pion mass.

References:

ASSAMAGAN 1996 PR D53 6065 Upper Limit of the ${{\mathit \nu}_{{\mu}}}$ Mass and Charged Pion Mass from Momentum Analysis of a Surface Muon Beam DOLGOV 1995 PR D51 4129 Bounds on Dirac Neutrino Masses from Nucleosynthesis ENQVIST 1993 PL B301 376 Cosmological Neutrino Mass Limit Revisited FULLER 1991 PR D43 3136 New Cosmological Limit on Neutrino Mass LAM 1991 PR D44 3345 Cosmological Bound on Dirac Neutrino Mass via ${{\mathit \gamma}}$ ${{\mathit \gamma}}$ $\rightarrow$ ${{\mathit \pi}^{0}}$ $\rightarrow$ ${{\mathit \nu}}{{\overline{\mathit \nu}}}$ ANDERHUB 1982 PL 114B 76 Determination of an Upper Limit of the Mass of the Muonic Neutrino from the Pion Decay in Flight CLARK 1974 PR D9 533 Neutrino Mass Limits from the ${{\mathit K}_L^0}$ $\rightarrow$ ${{\mathit \pi}^{\pm}}{{\mathit \ell}^{\mp}}{{\mathit \nu}}$ Decay Spectra