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The neutral Sigma baryon and Lambda baryon both have valence content $uds$, but different spin-isospin couplings. An electromagnetic transition changes those internal couplings without changing any quark flavour. In the simple ground-state quark model, the nonzero magnetic transition is associated with the up- and down-quark electromagnetic currents; the strange-quark spin operator alone cannot change the $ud$ pair from spin one to spin zero. Four-momentum conservation gives photon energy $E_{\gamma }=(M_{\Sigma }^2-M_{\Lambda }^2)/(2M_{\Sigma })$ in natural units.

Rapidity is a dimensionless parameter for a signed collinear velocity $u$ with $|u|<c$, defined using the inverse hyperbolic functions byHere $c$ is the speed of light, and $\tanh$ is the hyperbolic tangent. Zero velocity has zero rapidity; negative velocity has negative rapidity. Every finite real rapidity gives $|u|<c$, while $u\to \pm c$ requires $\theta \to \pm \infty$. Thus a light ray has no finite rapidity in this parametrization. The header's formula uses natural units $c=1$: its $v$ is the numerical velocity $u/c$.

For the inertial frames in the velocity-addition formula, $S^{\prime }$ moves at signed velocity $V$ along the $x$-axis of $S$. Set $\varphi =\mathrm{artanh}(V/c)$. Re-expressing the same particle in the moving inertial frame givesThe subtraction is a passive change of reference frame; the inverse adds the particle's rapidity in $S^{\prime }$ to the frame's rapidity relative to $S$. For example, $V=0.6c$ and $u^{\prime }=0.6c$ give $\varphi ={\theta }^{\prime }=\ln 2$, so $\theta =\ln 4$ and $u=c\tanh (\ln 4)=15c/17$, exactly as in the velocity-addition formula.

For a massive particle of rest mass $m>0$ moving in one spatial dimension, its energy and signed momentum areThe hyperbolic cosine is the particle's Lorentz factor, and the hyperbolic sine is $\gamma u/c$. The hyperbolic Pythagorean identity immediately gives the relativistic energy-momentum relation $E^2-c^2{p}^2=m^2{c}^4$. In the older $c=1$ notation, writing the rest mass as $M$ recovers $v=\tanh \theta$, $E=M\cosh \theta$, and $p=M\sinh \theta$. These expressions use natural units in which the factors of $c$ are numerically one; the preceding formulas restore those factors explicitly. ^[ref].

The useful simplification follows from the hyperbolic-function identity $\tanh (a+b)=(\tanh a+\tanh b)/(1+\tanh a\tanh b)$: the nonlinear velocity-addition formula becomes ordinary addition of rapidities. See iterated collinear boosts with equal rapidity for a plotted example. A common collinear Lorentz boost shifts every rapidity by the same constant, so a rapidity difference is invariant. This makes rapidity useful for relativistic factorized scattering in one spatial dimension, where two-body S-matrices depend on the difference of the incoming rapidities, with particle species fixed.

The reheating temperature characterizes the thermal radiation bath after a nonthermal component transfers its energy to other particles and those particles reach thermal equilibrium. Use natural units and the reduced Planck mass $M=(8\pi G{)}^{-1/2}$. In the sudden decay of a dominant nonrelativistic relic, decay rate $\Gamma$ and the Friedmann equation give ${\rho }_R\simeq 3M^2{\Gamma }^2$ at $H\simeq \Gamma$. Equating this with equilibrium radiation energy density givesHere $g_{\ast ,\rho }$ counts the effective relativistic energy degrees of freedom, so the formula can be implicit when the particle content changes with temperature. For example, increasing $\Gamma$ by a factor four doubles this estimate at fixed $g_{\ast ,\rho }$.

The residual bath just before decay can be much colder, while gradual reheating can temporarily produce a maximum bath temperature above the eventual radiation-domination value. Prompt energy transfer is also distinct from prompt thermalization. A scalar scalar potential alone therefore does not fix $T_R$ without interactions and a transfer history.^[ref]

In a low-energy allowed weak three-body decay, negligible final-state masses and a nearly constant effective matrix element give a fifth-power energy-release dependence. For muon decay in natural units, $\Gamma \simeq G_F^2{m}_{\mu }^5/(192{\pi }^3)$. This four-fermion approximation requires momentum transfers small compared with the W boson mass. Extrapolation to a heavy particle able to emit a real W changes the decay kinematics and the mass scaling, so it is not a reliable heavy-particle lifetime prediction.

For a canonical real scalar field with metric signature $(-,+,+,+)$ and scalar potential $V$, variation of its minimally coupled action gives the displayed equation. A homogeneous field in an FLRW metric therefore obeys $\ddot{\phi }+3H\dot{\phi }+V^{\prime }(\phi )=0$ in natural units. Its energy density is ${\dot{\phi }}^2/2+V$ and its pressure is ${\dot{\phi }}^2/2-V$, which satisfy the cosmological continuity equation.

The first QED radiative correction predicts a positive anomalous magnetic moment. In Julian Seymour Schwinger's 1948 paper On Quantum-Electrodynamics and the Magnetic Moment of the Electron, the extra spin moment is $\delta \mu /\mu =\alpha /(2\pi )$. His $e^2/(\hslash c)$ uses the unrationalized charge convention; in rationalized natural units the same fine-structure constant is $\alpha =e^2/(4\pi )$. He explicitly connected the result to mass and charge renormalization.^[ref]

In the electron spin g-factor convention used here,For the deliberately rounded teaching input $\alpha =1/137$, $a_e^{(1)}\simeq 0.0011617149$ and $g_s^{(1)}\simeq 2.0023234298$. The correction is about $0.116\%$ of the Dirac equation spin moment. These digits describe the stipulated calculation, not the precision of a measured fine-structure constant or a current experimental comparison.

If a beta spectrum has nonconstant shape factor $C(T)$, divide by it as well as by the Coulomb and phase-space factors before taking a square root. A leading unique first-forbidden shape contains $p_e^2+p_{\nu }^2$ in natural units, with Coulomb and nuclear corrections. Dividing only by a Coulomb Fermi function does not remove that shape. A straight uncorrected line must therefore be treated as an approximation, not an exact consequence of forbidden-transition selection rules.

Soft-collinear effective theory is an effective field theory of quantum chromodynamics for processes containing energetic particles moving close to lightlike directions together with lower-momentum radiation. It retains the modes needed for that hierarchy and organizes interactions by power counting. Fleming's overview describes energetic collinear particles interacting with soft quanta and applies the framework to meson-decay observables.^[ref]

The geometry is clearer in light-cone coordinates. Choose the Minkowski metric with signature $(+,-,-,-)$ and defineA particle moving nearly along the positive $z$ direction has one large component $p^{-}$, a smaller transverse momentum and a still smaller $p^{+}$. A common light-cone power counting assignment isIndividual terms in $p^2$ then scale as $Q^2{\lambda }^2$; they can cancel for a particle on its mass shell. Virtuality and energy are therefore different measures of scale.

For an original exact example in natural units, take $p^{-}=100\mathrm{GeV}$, $p^{+}=1\mathrm{GeV}$ and $|{\mathbf{p}}_{\perp }|=10\mathrm{GeV}$. Then $E=50.5\mathrm{GeV}$, $p_z=49.5\mathrm{GeV}$ andThe momentum is energetic and exactly lightlike in this example. Removing all energetic fields from a low-virtuality theory would discard this mode. The numerical components illustrate the stated convention and are not data from a collider.

The original2001 construction by Christian W. Bauer, Sean Fleming, Dan Pirjol and Iain W. Stewart identifies separated scales and determines effective-theory currents and coefficient evolution.^[ref] Comparing calculations with the underlying theory fixes Wilson coefficients; renormalization group evolution then organizes scale dependence. The resulting effective Lagrangian is an expansion with a specified domain of validity, rather than a replacement for QCD in every kinematic regime. Precise assignments for lower-energy modes depend on which hierarchy the application requires.

Starobinsky inflation can be formulated as a metric theory of gravity with a squared Ricci scalar in the action. In a convention with positive de Sitter Ricci scalar, use natural units, reduced Planck mass $M$ and mass scale $m>0$:The higher-curvature term carries an extra scalar degree of freedom, the scalaron. Set $F=1+R/(3m^2)>0$ and perform a conformal transformation to $g_{\mu \nu }^E=F{g}_{\mu \nu }$, with canonical field $\varphi =\sqrt{3/2}M\log F$. Its Einstein frame scalar potential isThis scalar representation explains the flat large-field plateau without assuming an unrelated matter potential. The pure $R+R^2$ contribution has this scalar representation.^[ref].

Near $\varphi =0$, $V\simeq m^2{\varphi }^2/2$, allowing an oscillatory post-inflationary phase. On the plateau, the leading slow-roll approximation gives ${ϵ}_V\simeq 3/(4N^2)$ and ${\eta }_V\simeq -1/N$ for $N$ remaining number of e-folds. Hence $n_s\simeq 1-2/N$ and $r\simeq 12/N^2$. These are model predictions, not an observational measurement; the mapping from $N$ to a measured scale also depends on reheating.

The sudden-decay approximation replaces a dominant nonrelativistic species $X$ by radiation near $H=\Gamma$, neglecting expansion during the conversion. Use natural units, reduced Planck mass $M=(8\pi G{)}^{-1/2}$ and relic mass $m$. Define the pre-decay yield $Y=n_X/s_{-}$, where $n_X$ is its number density and $s_{-}=(2{\pi }^2/45)g_{\ast ,s}{T}_{-}^3$ is the residual bath's entropy density. Relic domination givesAfter conversion and prompt thermalization, conservation of energy giveswhich defines the reheating temperature. The entropy count $g_{\ast ,s}$ and energy count $g_{\ast ,\rho }$ have different definitions and need not be identical. The yield in the first equation is defined before the newly produced entropy dilutes it.

If the relic energy greatly exceeds the residual radiation energy, the post-decay temperature can greatly exceed $T_{-}$. The related entropy production by decay of a dominant relic follows by comparing the two equilibrium baths at the same scale factor. The assumptions are relic domination, nonrelativistic relics and rapid conversion/thermalization; a gradual decay or incomplete equilibrium requires coupled evolution instead.

Use natural units $c=1$ and assume open phase space, $M>m_1+m_2+m_3$. In the parent's rest frame, define $s_{23}=(p_2+p_3{)}^2$. Four-momentum conservation givesThe upper recoil-mass limit follows from $E_1\ge m_1$. For positive $m_2,m_3$, the lower recoil-mass limit follows in their pair rest frame and is attained when their relative momentum vanishes. If both recoil particles are massless, $s_{23}=2E_2{E}_3(1-\cos {\theta }_{23})$ reaches zero with nonzero parallel momenta. If exactly one is massless, $s_{23}$ approaches the nonzero daughter's squared mass only when the massless particle becomes soft. Thus that upper $E_1$ bound is a supremum for a genuine three-body event with all massless particles of positive energy. If $m_1=0$, the lower endpoint is likewise a soft limit. Multiplying the mass expressions for $E_1$ by $c^2$ restores energy units.

Combine the two massless neutrinos in ${\tau }^{-}\to {\ell }^{-}{\bar{\nu }}_{\ell }{\nu }_{\tau }$ into $q$. In the parent rest frame, four-momentum conservation givesThe lower bound follows from the nonnegative Minkowski norm of the sum of two future-directed null four-vectors; the upper bound follows from $E_{\ell }^{\ast }\ge m_{\ell }$. Substituting gives the displayed energy bounds. They are kinematic: they do not specify the weak interaction decay amplitude or the probability density function inside the interval.

If the charged daughter is also massless, put $x=2E_{\ell }^{\ast }/m_{\tau }\in [0,1]$. A parent with Lorentz factor $\gamma$ and speed $\beta$ in natural units producesHere ${\theta }^{\ast }$ is the daughter angle relative to the boost direction in the parent rest frame. Two opposite parent momenta therefore need not produce two opposite visible daughter momenta: the invisible system can recoil differently in each particle decay.

For one daughter of mass $m$ and two massless daughters, set $r=m^2/M^2$, where $M$ is the parent mass. In natural units, integrating over all directions and all kinematically allowed final momenta givesThis follows by splitting the final state into a massless spectator and a two-particle cluster of invariant squared mass $s$, and using $d{\Phi }_3=d{\Phi }_2(M;\sqrt{s},0)d{\Phi }_2(\sqrt{s};m,0)ds/(2\pi )$. The expression counts phase space alone; a physical decay rate must weight it with the squared decay amplitude. Near threshold the dimensionless factor vanishes as $(1-r{)}^3/3$.

In a Friedmann equation written $H^2=8\pi G\rho /3+\Lambda /3-k{c}^2/a^2$, the symbol $\Lambda$ has units of inverse time squared. If ${\Lambda }_{\mathrm{geom}}$ is the geometrical constant measured in inverse length squared, $\Lambda =c^2{\Lambda }_{\mathrm{geom}}$. Natural units hide this distinction.

For a two-body W boson decay into a charged lepton $\ell$ and a neutrino, neglect both daughter masses and use natural units $c=1$. Let $\Delta \varphi$ be the angle between the daughter transverse momenta. Identifying the missing transverse momentum with the neutrino's givesThis observable uses no $p_{z\nu }$. The 1983 UA1 experiment paper already used transverse mass as a lower bound on parent mass, before making additional decay-model assumptions for a mass estimate.^[ref]

The bound follows directly from the energy–momentum relation. For each massless daughter write its four-momentum, using rapidity $y$ and azimuth $\varphi$, asUsing the Minkowski inner product with positive time sign, $p^2=E^2-|\mathbf{p}{|}^2$, the two daughter four-momenta give the true parent invariant mass $M$:Since $\cosh u\ge 1$, $m_T\le M$, with equality at equal daughter rapidities when both transverse momenta are nonzero. Thus the unmeasured longitudinal information can enlarge the true invariant mass, but cannot lower it below this ideal transverse mass.

For an original example, stipulate $p_{T\ell }=p_{T\nu }=40\mathrm{GeV}$ and $\Delta \varphi =\pi$. Then $m_T=80\mathrm{GeV}$. Equal rapidities give $M=80\mathrm{GeV}$; choosing $y_{\ell }-y_{\nu }=1$ with exactly the same transverse readout gives $M=80\cosh (1/2)\simeq 90.210\mathrm{GeV}$. One measured transverse configuration therefore permits different true parent masses.

Virtuality characterizes the invariant momentum scale of an internal or otherwise off-mass shell momentum. One common convention uses $p^2$ itself; for a massive particle, departure from its physical mass shell is measured by $p^2-m^2$ in natural units. The convention must be stated: an on-shell massive particle has $p^2=m^2$, not generally zero.

With the Minkowski metric signature $(+,-,-,-)$ and $c=1$,It is invariant under Lorentz transformations, unlike the energy component by itself. A massless on-shell particle has $p^2=0$ even when its energy is large. Positive and negative values describe timelike and spacelike four-momenta respectively; a spacelike transfer is not the four-momentum of an isolated real massive particle.

An original scattering-kinematics example uses idealized massless incoming and outgoing lepton momenta, in GeV:Both satisfy $k^2=k^{\prime 2}=0$. Their transferred momentum is $q=k-k^{\prime }=(1,-2.4,0,1.8)$, givingIn deep inelastic scattering, the conventional positive scale is $Q^2=-q^2$, so this example has $Q^2=8\mathrm{Ge}{\mathrm{V}}^2$. The PDG review defines the transfer and this sign convention when introducing scattering invariants.^[ref] Measuring the scattered lepton's energy and direction supplies the kinematic information needed for that inference.

The example is not an event measurement or a neglect of four-momentum conservation. An internal momentum can obey the vertex conservation equations while failing a real particle's on-shell mass relation. In soft-collinear effective theory, separating large energy from smaller virtuality is a reason to retain energetic nearly lightlike degrees of freedom.^[ref]

Changing the metric signature changes signs in the formula, and different authors may reserve the word virtuality for $p^2$, $|p^2|$ or the displacement from a massive mass shell. Reporting the defining equation prevents these notational conventions from being mistaken for physical disagreement.

In natural units with unreduced Planck mass $M_P=G^{-1/2}$, the radiation-era Hubble parameter scales as $\sqrt{g_{\ast }}{T}^2/M_P$, while a low-energy weak interaction rate scales as $G_F^2{T}^5$. Their equality gives a decoupling scale of order one MeV. Distinct weak processes have distinct rate coefficients: neutrino decoupling and cosmological weak freeze-out of neutron-proton conversion are nearby events, not an identical precise temperature inferred from the power laws alone.