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Assume the charmonium state annihilates through a virtual photon, with lepton universality and negligible higher-order corrections. For a vector meson of mass $M$, the partial decay width into a charged lepton pair contains both the relativistic two-body decay factor ${\beta }_{\ell }$ and a vector-current spin factor:^[ref] For example, contracting the vector polarization sum with the Dirac trace for the two final fermions gives a factor proportional to $M^2+2m_{\ell }^2$, explaining why retaining only ${\beta }_{\ell }$ is insufficient. Using $m_{\tau }=1.777\mathrm{GeV}$, $m_{\mu }=0.1057\mathrm{GeV}$ and $M=3.686\mathrm{GeV}$ givesThe supplied total decay width also gives ${\Gamma }_{\mu \mu }=2.352\mathrm{keV}$ and ${\Gamma }_{\tau \tau }\simeq 0.91\mathrm{keV}$. It cancels from the branching fraction ratio.

The decisive change at $3.770\mathrm{GeV}$ is crossing the open-charm D meson-pair threshold, approximately $3.73\mathrm{GeV}$. The strong interaction can now create a light quark-antiquark pair and rearrange $c\bar{c}$ into $D\bar{D}$, without annihilating the heavy pair. This increases the total decay width from hundreds of keV to tens of MeV. The leptonic decay partial decay width is additionally smaller because $\psi (3770)$ is predominantly a $1{}^3{D}_1$ state, whereas $\psi (3686)$ is predominantly $2{}^3{S}_1$; the short-distance annihilation overlap is suppressed for nonzero orbital angular momentum. A roughly hundredfold increase in total decay width together with an order-ten reduction in leptonic partial decay width explains the order-thousand reduction in branching fraction. Attributing all three orders of magnitude to the threshold alone would miss the second effect.^[ref]

Matched near and far particle detectors are a good design. A near particle detector, close enough that the fast oscillation phase is small, measures the unoscillated reactor spectrum and normalization. A far particle detector near the first fast maximum measures the disappearance. Their appropriately geometry-corrected ratio cancels much of the uncertain reactor flux and common cross section and detection efficiency. Relative calibration, different backgrounds, multiple reactor distances and any residual near-site oscillation still need treatment. The near site should retain enough shielding to suppress cosmic backgrounds.^[ref]

The radiation length $X_0$ is the characteristic length for high-energy electron energy loss by bremsstrahlung; smaller $X_0$ means more radiative loss for the same traversed thickness. To limit that loss, minimize $x/X_0$, choosing less material or a larger $X_0$, rather than minimizing $X_0$ itself. At reactor Positron energies of a few MeV, ionization and subsequent annihilation matter more than high-energy bremsstrahlung. A sufficiently large scintillation counter measures the deposited energy, so containment of the Positron and annihilation photons, light yield and calibration are the relevant design issues.^[ref]

The initial total momentum is zero, so the outgoing neutron recoils with momentum magnitude $E_{\gamma }/c$. Put $W=(m_p+m_{\pi })c^2$. Conservation of relativistic energy givesFor example the rest energies $938.2721$, $139.5704$ and $939.5654\mathrm{MeV}$ for the proton, charged pion and neutron give this value.^[ref] The result is smaller than the initial-to-final rest-energy difference of about $138.3\mathrm{MeV}$, because some released energy becomes neutron recoil kinetic energy. Atomic binding of a captured pion is neglected in the stated stationary-particle model.

With outgoing physical photon polarization vectors, the Feynman rule givesThe overall phase does not affect the decay width. Replacing either polarization vector by its own four-momentum gives zero by antisymmetry. Thus the Ward identity permits replacement of each physical photon polarization sum by $-{\eta }^{\mu \nu }$ inside this amplitude. The printed sum is not an equality of free-standing physical two-polarization tensors: for a photon moving along $z$, its physical $00$ component is zero whereas $-{\eta }_{00}=-1$. Gauge-dependent longitudinal terms cancel only after contraction with the transverse amplitude.

The contraction of the two Levi-Civita symbols givessince $q_1^2=q_2^2=0$ and $2q_1\cdot q_2=m_a^2$. No initial spin average is needed for the spin-zero axion. The result is independent of direction in its rest frame. Including the identical-particle factor explained in solution therefore givesOmitting $1/2!$ would overestimate the physical rate by two. The normalization agrees with the standard axion-photon convention.^[ref]

The characteristic thickness over which a high-energy electron loses all but a fraction $1/e$ of its energy through bremsstrahlung. At fixed thickness $x$, smaller $X_0$ increases the radiative loss; the relevant dimensionless measure is $x/X_0$. Low-energy charged particles can instead be dominated by collisional stopping power.^[ref]

A particle track projected onto the plane perpendicular to a uniform magnetic field $B$ is a circular arc with radius $R=p_{\perp }/(|q|B)$. Here $p_{\perp }$ is the momentum component perpendicular to the magnetic field; it is the collider transverse momentum when the field follows the beam axis. For chord length $L$ and sagitta $s$, circle geometry givesFor a short arc with $s\ll L$, this yieldsThe measurement uncertainty follows by differentiating the inverse relation, treating $B$ and $L$ as known. A fixed spatial measurement uncertainty therefore makes fractional momentum uncertainty grow with momentum, whereas increasing the lever arm improves it quadratically in this idealization. Multiple Coulomb scattering, field uncertainties and alignment add other contributions; a helical particle track also requires its pitch to recover the full momentum.^[ref]

Here $x=m_W^2/m_t^2$, and the tree-level decay width neglects the bottom quark mass. The open two-body W channel gives a top quark lifetime of a few $10^{-25}\mathrm{s}$, shorter than the hadronization time. A fifth-power Sargent's rule extrapolation misses this physical regime.^[ref]