What Decay Trees Reveal About Particle Energy

A particle decay is a measured rearrangement of energy. Following its products through further decays gives a tree whose energy must balance at every split. For IDWT, this offers a practical way to test how an oscillating field pattern becomes other patterns, their motion and radiation.

The first calculation covers 19 selected channels involving muons, taus, W and Z bosons, the Higgs, and pion and kaon controls. It gives absolute kinematic energies and ratios. It does not yet measure the underlying field amplitudes or derive sector stiffness.

A Split With a Complete Energy Ledger

Write masses as rest energies in MeV, setting $c=1$. For a parent at rest with rest energy $M$, decaying into two products of rest energies $m_1,m_2$,

$$Q=M-m_1-m_2,\qquad E_1=\frac{M^2+m_1^2-m_2^2}{2M},\qquad E_2=M-E_1.$$

The products' kinetic energies are $T_i=E_i-m_i$, so $T_1+T_2=Q$. These are energy–momentum constraints; they do not prescribe a mechanism for the interaction. See PDG Kinematics.

The table uses the dated PDG 2025 rest energies: pion 139.57039 MeV, kaon 493.677 MeV, muon 105.6583755 MeV and tau 1776.93 MeV. Neutrino masses and radiative corrections are neglected for these nominal two-body values. Pions and kaons are composite controls, not elementary sector assignments. Sources: pion, kaon, muon, tau.

There is also a direct recoil measurement: Assamagan and colleagues measured pion-decay muon momentum as $29.79200\pm0.00011$ MeV/$c$. Converting that momentum gives muon kinetic energy $4.119835\pm0.000030$ MeV. Fitted pion masses include related recoil information, so this comparison is not an independent significance test. Original experiment.

Follow One Branch to Its End

For the positively charged sequence, with the corresponding antiparticles and neutrinos,

Tau
  ├─ Pion + neutrino
  │    └─ Muon + neutrino
  │         └─ Positron + two neutrinos
  ├─ Muon + two neutrinos → positron + two more neutrinos
  └─ Positron + two neutrinos

Alternative branches are alternatives, not simultaneous products. Along the pion branch, the intermediate rest masses cancel:

$$Q_{\rm tree} =(m_\tau-m_\pi)+(m_\pi-m_\mu)+(m_\mu-m_e) =m_\tau-m_e=1776.41900105\ {\rm MeV}.$$

That is the final total kinetic energy in the original tau rest frame, under the stated neutrino and radiation approximations. It is not obtained by adding daughter energies calculated in different rest frames. A moving intermediate transfers inherited motion to its products. The required boost is

$$E_{\rm original}=\gamma(E_{\rm intermediate}+\beta p\cos\theta).$$

For example, the muon from the tau's pion branch has energy between 514.650 and 891.606 MeV in the tau rest frame. The decay angle determines where within that interval an event falls.

Three Products Require a Spectrum

Muon decay produces a charged lepton and two neutrinos. Their combined energy is fixed, but each event divides it differently. A measured spectrum therefore supplies information that a mass table cannot.

In the angle-integrated leading Michel spectrum, neglecting electron mass and radiation, the charged-lepton mean energy obeys

$$\frac{\langle E_e\rangle}{m_\mu}=\frac3{10}+\frac\rho{15}.$$

The measured parameter $\rho=0.74979\pm0.00026$ gives approximately $0.349986\pm0.0000173$ within that approximation. At $\rho=3/4$ the fraction is exactly $7/20$, leaving $13/20$ for the two neutrinos together. These are mean fractions, not fixed shares in each event. Spectrum.

The Bridge to Stiffness and Amplitude

A specified harmonic field coordinate has

$$E_{\rm osc}=\frac12KA^2,\qquad \omega^2=\frac K{\mathcal I},$$

where $A$ is its amplitude, $K$ its restoring stiffness and $\mathcal I$ its modal inertia. Once $K$ and the internal oscillation energy are supplied by a physical mode model, the amplitude follows. Relative amplitudes can be calculated without an absolute field scale when that scale cancels.

The energy tree helps establish which energy is available for that inference. A muon emerging from pion decay and one emerging from kaon decay have very different kinetic energies but the same intrinsic rest-frequency. Recoil energy cannot simply be counted as additional intrinsic muon oscillation.

The next IDWT calculation must supply the spatial shapes and their kinetic and restoring integrals, then reproduce the tree's energy distributions with one interaction law. The tree supplies a concrete empirical ledger for that calculation. Its conservation identities are standard physics; the proposed same-field explanation and stiffness inference remain the research questions.

Continue with the introduction to IDWT or the sector-structure argument.