The charged and neutral pion masses revisited

Abstract Results from neutrino mass and oscillation experiments now set the mass of the muon neutrino to less than 2 eV/c2. This fact, together with our former measurement of the muon momentum in pion decay at rest, p μ + = ( 29.792 00 ± 0.000 11 ) MeV/c, allows us to directly determine the charged pion mass with 1 ppm precision which constitutes the most precise value of the charged pion mass to date, m π + = ( 139.570 21 ± 0.000 14 ) MeV / c 2 . This value is within 1.44 σ of the Particle Data Group's compilation of the charged pion mass value, m π ± = ( 139.570 61 ± 0.000 24 ) MeV/c2. From p μ + we derive the kinetic energy of the muon, T μ + = ( 4.119 84 ± 0.000 03 ) MeV and the mass difference, m π + − m μ + = ( 33.911 84 ± 0.000 14 ) MeV/c2. From our new m π + value, assuming CPT invariance ( m π − = m π + ) and our measured mass difference D π = m π − − m π 0 = ( 4.593 64 ± 0.000 48 ) MeV/c2 we obtain a new value for the neutral pion mass, m π 0 = ( 134.976 57 ± 0.000 50 ) MeV/c2. One also obtains a new quantitative measure of CPT invariance in the pion sector: ( m π + − m π − ) / m π ( av ) = ( − 2.9 ± 2.0 ) ⋅ 10 − 6 , an improvement by two orders of magnitude.

For the special case of a pion at rest, p π + = 0, follows Hence Eq. (2) now reads m π + = m 2 μ + + p 2 μ + + m 2 ν μ + p 2 μ + . (5) For the measurement of the muon momentum, we used a single focusing semicircular magnetic spectrometer [8,9] with a homogeneous field. Details of the apparatus are described in Refs. [3,7]. The results from the five different experimental periods (Mark I to V) are listed in Table 1 and displayed in Fig. 1.

Table 1
Our results for the muon momentum from pion decay at rest.

Mark
Year These measurements were originally intended to determine the mass of the muon neutrino m ν μ , or its upper limit, respectively. This can be obtained from Eq. (5) as For the numerical evaluation we used the validity of the CPT theorem, m π + = m π − and we obtained an upper limit, m ν μ ≤ 170 keV/c 2 with 95% confidence [7]. Now, with the establishment of neutrino oscillations [10,11], the scenario has changed significantly. It was B. Pontecorvo's prediction that neutrinos might oscillate [12,13] that led to the formulation of the Pontecorvo-Maki-Nakagawa-Sakawa (PMNS) mixing matrix to explain the phenomenon of neutrino oscillations [14].  This matrix parameterizes the transformation of mass eigenstates The number of light neutrinos has been restricted to three [15,16] and so far no hints for heavy sterile neutrinos were found in pion decay [17][18][19][20][21][22][23]. Thus, the PMNS matrix reads with the numerical values of the matrix elements [ The upper limit of the electron neutrino mass has been measured at the level of m ν e ≤ 2 eV/c 2 [16,25,26]. This mass value stands for the "effective" electron neutrino mass which is the weighted sum of the mass eigenstates, In all these experiments, the energy or momentum resolution cannot resolve the mass splitting of the mass eigenstates: the mass differences m 21 and m 32 are experimentally found to be in the range of millielectronvolt [16,[27][28][29][30].
As can be seen from Eq. (8), m ν e is predominantly m ν 1 . That is, also the masses of m ν 2 and m ν 3 must be equal or less than ∼2 eV/c 2 and consequently the muon and tau neutrinos are equal or less than ∼2 eV/c 2 . Thus, our measurements of the muon momentum from pion decay at rest should be re-interpreted as a precise direct determination of the mass of the positively charged pion, m π + .
According to Eq. (5), the uncertainty m π + is limited by the uncertainties of p μ + , m μ + , and m ν μ . Differentiation of Eq. (5) leads to the following sensitivities: results for m π − from pionic atoms. Red symbols and lines: results for m π + from muon momentum in pion decay at rest. The π − measurements of Ref. [35] were re-analyzed after the π + results of Ref. [6] were published in view of the large discrepancy. The re-analysis resulted in two solutions in Ref. [ For the mass of the muon, we use m μ = (105.658 374 5 ± 0.000 002 4) MeV/c 2 [16,31,32] and for the mass of the neutrino we use (conservatively) m ν μ = (2.0 ± 2.0) · 10 −6 MeV/c 2 [16]. The momentum of the muon from pion decay at rest is p μ + = (29.792 00 ± 0.000 11) MeV/c, cf. Table 1. As can be seen, the contributions from the uncertainties of the masses of the muon and the neutrino to the total uncertainty of m π + are negligible compared to the uncertainty of p μ + . The result for the mass of the positively charged pion is thus This is the most precise value for the charged pion mass with a precision of 1 ppm. The re-analysis of our experimental results, in view of the more stringent limit from the neutrino sector yields practically the same value as published earlier [7]. The new main aspect here is that we no longer, as in our earlier analysis, rely on a value for the neutrino mass based on cosmological considerations [33,34] but rather a value from experimental data from the neutrino sector [16,[27][28][29][30].
Our new result is now effectively independent of the neutrino mass in the range up to ∼2 eV/c 2 and thus constitutes a direct precise measurement of the pion mass and no longer a lower limit.
The measured values of m π − from pionic atoms and m π + from our measurements are shown in Fig. 2. Our result is more precise than and within 1.44σ of the recent compilation of the Particle Data Group for m π ± [16]: m π ± = (139.570 61 ± 0.000 24) MeV/c 2 (14) which uses the three most recent pionic atom experiments [36][37][38]. 2 Our mass value agrees also with those of the single measurements, m π − = (139.569 95 ± 0.000 35) MeV/c 2 [36] solution B (15) m π − = (139.570 71 ± 0.000 53) MeV/c 2 [37] ( 16) In particular, the agreement with the most precise single measurement of m π − , m π − = (139.570 77 ± 0.000 18) MeV/c 2 [38] is only fair (2.43σ ): To summarize, with this negligibly small neutrino mass, we can rewrite Eq. (5) as: yielding T μ + the kinetic energy of the muon from the decay of the pion at rest as: From Eq. (20) we also obtain the difference of the masses of the pion and muon: m π + −m μ + = T μ + + p μ + = (33.911 84 ±0.000 14) MeV/c 2 . (22) Furthermore, by considering the charged states of the pion mass separately and comparing the PDG value, Eq. (14), which is based solely on π − measurements, with our π + -value one has a quantitative measure of the CPT invariance in the pion sector. Using the PDG nomenclature one obtains: This is two orders of magnitude more precise than the best value so far, (2 ± 5) · 10 −4 [39]. Our result is consistent within 1.45 σ with the CPT theorem, which predicts that a particle and its antiparticle have equal masses.