Flavour Physics and Neutrinos

Derivation. Derives Phenomenon 103.2. The natural time scale for a decay driven by the interaction that also binds the hadron is the time light takes to cross it, \(R/c=10^{-15}\,\mathrm{m}/c=3.3\times 10^{-24}\,\mathrm{s}\), and \(\tau_{\Delta}\) is within a factor of two of it: the \(\Delta\) decays as fast as it can. The ratio \(\tau_{\Lambda}/\tau_{\Delta}=4.7\times 10^{13}\) therefore cannot be absorbed into phase space, which varies as a power of the energy release and can account for two or three orders of magnitude at most, nor into a centrifugal barrier, which for the observed spins is absent. The only available conclusion is that the interaction responsible for the decay is not the interaction responsible for the production. Comparing \(\tau_{\Lambda}\) with the muon lifetime \(2.197\times 10^{-6}\,\mathrm{s}\) and allowing for the fifth-power dependence of Equation (101.53) on the energy release identifies the decay interaction as the weak one.

The inference is then sharp. Something is conserved by the interaction that makes these particles and violated by the interaction that destroys them. That is the whole content of the next subsection, and it is the template for every conserved flavour quantum number in this chapter: exact for two of the three forces, broken by the third.

Derives Proposition 103.4. Additivity and conservation give \(\sum_{\text{final}}S=0\), so a final state containing one particle with \(S=-1\) must contain another with \(S=+1\); a final state \(\Lambda^{0}\pi^{0}\) with \(S=-1\) is forbidden however much phase space favours it, which is what is seen. For the second statement, let \(h\) be the lightest hadron with \(S=-1\). A strong decay must conserve \(S\), so its products together carry \(S=-1\); but every hadron with \(S=-1\) is at least as heavy as \(h\), and any accompanying non-strange hadron has positive mass, so no such final state is open. The decay must therefore proceed through an interaction that violates \(S\), and by Phenomenon 103.2 that interaction is weak. Applying the argument to \(\Lambda^{0}\) (\(S=-1\)) and to \(K^{+}\) (\(S=+1\)) accounts for the long lifetimes of both.

Derivation. Derives Phenomenon 103.7. Cabibbo's hypothesis is that the down-type state entering the charged current is a rotation of the mass eigenstates, \(d'=d\cos\theta_{C}+s\sin\theta_{C}\), carrying the same overall coupling \(G_{F}\) as the leptonic current [Cabibbo:1963]. A strangeness-conserving amplitude then acquires a factor \(\cos\theta_{C}\) and a strangeness-changing one a factor \(\sin\theta_{C}\), so their rates stand in the ratio \(\tan^{2}\theta_{C}\) and the two effects are not independent: the suppression of the second is precisely the deficit of the first against the leptonic strength, which is what restores universality. Reading \(\sin\theta_{C}=\abs{V_{us}}\) from Equation (103.5) gives \(\theta_{C}=12.96^\circ\) and hence \(\cos\theta_{C}=0.9745\), which is \(\abs{V_{ud}}\) to within one part in a thousand — one angle, measured twice over in unrelated processes. The global fit quoted later as \(\theta_{12}=13.00^\circ\) in Equation (103.22) is the same angle: the \(0.04^\circ\) between the two is the difference between the direct kaon determination of \(\abs{V_{us}}\) used here and the fitted \(\lambda=0.22501\) used there, and nothing else. The rate ratio it predicts, \(\tan^{2}\theta_{C}=0.0530\), is the observed factor of about twenty. That the squares sum to unity is the two-generation case of the unitarity discussed in Section 103.4.3; that they sum to unity only to a part in \(10^{3}\) is a measurement, not an approximation, and it is recorded here rather than smoothed away.

Derives Proposition 103.8. \(\abs{V_{ud}}^{2}=0.948033\), \(\abs{V_{us}}^{2}=0.050310\) and \(\abs{V_{ub}}^{2}=1.46\times 10^{-5}\), summing to \(0.998358\). The uncertainty propagates as \(\delta\Delta=2\left[\left(\abs{V_{ud}}\delta\abs{V_{ud}}\right)^{2} +\left(\abs{V_{us}}\delta\abs{V_{us}}\right)^{2}\right]^{1/2}\), the \(V_{ub}\) term being negligible, which gives \(2\left[(3.12\times 10^{-4})^{2}+(1.79\times 10^{-4})^{2}\right]^{1/2} =7.2\times 10^{-4}\). Hence \(1-\Delta_{\mathrm{CKM}} =1.64\times 10^{-3}\pm7.2\times 10^{-4}\), a deficit of \(2.3\sigma\).

Derivation. Derives Phenomenon 103.10. The tree-level statement is Theorem 101.59, proved there for two generations and extended to three in Theorem 104.38(ii): the neutral-current coefficient depends only on \(T^{3}\) and \(Q\), hence is the same for every generation, hence is proportional to the identity in generation space, and \(U^{\dagger}\identity U=\identity\) for any unitary \(U\). Nothing survives the rotation to the mass basis. This is the Glashow–Iliopoulos–Maiani mechanism [Glashow:1970], and it is a consequence of unitarity together with the requirement that every fermion of a given charge and chirality sit in a complete weak multiplet. The two hypotheses are what a search for flavour-changing neutral currents actually tests.

At tree level the cancellation is exact and the observed rate would be zero, which is also wrong. The measured \(6.84\times 10^{-9}\) is small but not absent, and it is produced at one loop, where the cancellation is only partial. That calculation is Theorem 103.11 below, and it is where the mechanism stops being an algebraic identity and starts predicting a number.

Derives Theorem 103.11. The sum rule \(\sum_{i}\lambda_{i} =\sum_{i}V_{id}^{\phantom{*}}V_{is}^{*}=\delta_{ds}=0\) is the \((d,s)\) column relation of Theorem 101.62. Write \(A(m_{i})=A(0)+A'(0)x_{i}+O(x_{i}^{2})\), where the expansion is legitimate for \(m_{i}c^{2}\ll M_{W}c^{2}\) and where \(A(0)\) is by construction independent of \(i\). Then

which is Equation (103.8). Two remarks fix its meaning. First, the cancellation is not a fine tuning: it is the statement that the loop, evaluated with degenerate internal masses, is a flavour-diagonal operator, and the mixing matrix cannot rotate a multiple of the identity into anything else. Second, the surviving term is a difference, since \(\sum_{i}\lambda_{i}=0\) allows \(m_{i}^{2}\) to be shifted by any constant; writing the sum over the first two generations alone gives \(\lambda_{c}\left(m_{c}^{2}-m_{u}^{2}\right)c^{4}\), and the mass of the light quark drops out of everything.

Derivation. Derives Phenomenon 103.14. The three resonances differ in one respect: how their valence quarks can reach the final state. The \(\rho\) is \(u\bar{u}\) and \(d\bar{d}\), and its decay \(\rho\to\pi\pi\) can proceed with both valence quarks carried into the final-state hadrons, one \(q\bar{q}\) pair being created from the vacuum. The \(\phi\) is \(s\bar{s}\); its dominant decay \(\phi\to K\bar{K}\) likewise carries both valence quarks through, but only just, the phase space being \(32\,\mathrm{MeV}\), and its decay to three pions — which requires the \(s\bar{s}\) pair to annihilate — is suppressed despite the far larger phase space. The \(J/\psi\) is \(c\bar{c}\) below the threshold for a pair of charmed mesons, so every decay to light hadrons requires the \(c\) and \(\bar{c}\) to annihilate.

That is the rule of Okubo [Okubo:1963], Zweig [Zweig:1964] and Iizuka [Iizuka:1966]: a decay whose quark-line diagram falls apart into two disconnected pieces is suppressed. Stated as an empirical rule it explains the \(\phi\); stated dynamically it explains the \(J/\psi\) and the size of the suppression. The disconnected diagram requires at least three gluons, because one gluon carries colour and cannot make a colour singlet, and two gluons cannot make a state of odd charge conjugation (Theorem 105.9 is the same argument for photons). The rate is therefore of order \(\alpha_{s}^{6}\) evaluated at a scale set by the charm mass, and by Phenomenon 102.41 that coupling is already small: \(\alpha_{s}\left(m_{c}c^{2}\right)\approx0.35\), so \(\alpha_{s}^{6}\approx1.8\times 10^{-3}\) against a light-quark process running at \(\alpha_{s}\approx1\). Asymptotic freedom, discovered the year before, is what turns an empirical rule into a number of the right order.

Derives Proposition 103.15. Under \(CP\) the charged-current term of Equation (101.67) maps to itself with \(V\mapsto V^{*}\) (Equation (105.16) gives the transformation of the bilinear). If \(V\) can be made real by a rephasing that leaves the rest of the Lagrangian invariant, then \(V=V^{*}\) in that basis and the interaction is \(CP\) symmetric. Equation (101.68) with \(N=2\) gives \((N-1)(N-2)/2=0\) surviving phases, so the Cabibbo matrix is real and \(CP\) is conserved however \(\theta_{C}\) is chosen; with \(N=3\) one phase survives and cannot be removed, so \(V\neq V^{*}\) in any basis unless the phase is a multiple of \(\pi\).

Derivation. Derives Phenomenon 103.17. The universality claim is testable to a fraction of a per cent by one number, and the test is the cleanest in the whole of flavour physics because it involves no hadrons at all. Equation (101.53) gives the rate of a purely leptonic charged-current decay,

whose fifth power of the mass is the dominant factor. Since the \(\tau\) has hadronic decay channels as well, the prediction is for the branching fraction

Inserting \(\tau_{\tau}=2.9034\times 10^{-13}\,\mathrm{s}\), \(\tau_{\mu}=2.1969811\times 10^{-6}\,\mathrm{s}\), \(m_{\tau}c^{2}=1776.93\,\mathrm{MeV}\) and \(m_{\mu}c^{2}=105.6583755\,\mathrm{MeV}\), so that \(\left(m_{\tau}/m_{\mu}\right)^{5}=1.3453\times 10^{6}\) and the phase-space ratio is \(1.00019\), gives \(\mathcal{B}=0.1778\). The measured value is \(\mathcal{B}\left(\tau\to e\nu\bar{\nu}\right)=0.1782(4)\) [Navas:2024]. Agreement to two parts in a thousand, over a factor \(1.3\times 10^{6}\) in rate, is what “the same coupling” means quantitatively; expressed as a coupling ratio, \(g_{\tau}/g_{\mu}\) differs from unity by less than \(2\times 10^{-3}\).

Derivation. Derives Phenomenon 103.19. The mechanism is Proposition 104.43: a doublet whose two members have different masses breaks the custodial symmetry of Theorem 104.40, and the resulting shift in the ratio of \(W\) and \(Z\) self-energies grows as the squared mass splitting, \(\Delta\rho\propto\left(m_{t}^{2}-m_{b}^{2}\right)c^{4}\). Because the dependence is quadratic rather than logarithmic, a measurement of the \(Z\) lineshape and asymmetries at the per-mille level of Section 104.6.3 pins \(m_{t}\) to about ten per cent. That is a real prediction and it was confirmed: the indirect and direct values agree within their uncertainties, which is a test of the loop structure of the theory and not merely of its tree-level content. The same fit subsequently bounded the Higgs mass, whose contribution is only logarithmic and therefore far weaker — and that bound too was confirmed, in Phenomenon 104.59.

Derives Proposition 103.20. \(\Lambda_{\mathrm{QCD}}\approx0.2\,\mathrm{GeV}\) by Equation (102.45), and the hadronization time is the inverse of that energy in the sense of Equation (103.1), \(\hbar/0.2\times 10^{9}\,\mathrm{eV}=3.3\times 10^{-24}\,\mathrm{s}\): seven times Equation (103.14). The width itself scales as \(m_{t}^{3}\), since \(t\to bW\) is a two-body decay into a real \(W\) with the top mass setting both the coupling's phase space and the longitudinal polarization enhancement, so the inequality gets stronger with mass and is a statement about the top specifically. The consequences are experimental and considerable: the top's spin information survives into its decay products, unrandomized by hadronization, and the object reconstructed in a detector is a bare colour triplet in flight — never, by Phenomenon 102.65, an asymptotic state.

Derivation. Derives Phenomenon 103.22. The invisible width is the total width minus the sum of the widths into observed final states, and the species count is that quantity divided by the width into one neutrino pair. The latter is not measured; it is computed, and the computation is short.

By Equation (101.75) the \(Z\) couples to a fermion \(f\) through vector and axial couplings

and the partial width of a vector boson of mass \(M_{Z}\) into a pair of fermions light compared with it is

\(N_{c}^{f}\) being \(3\) for quarks and \(1\) for leptons. The dimensional check is worth doing once: \(G_{F}/(\hbar c)^{3}\) carries \(/\mathrm{J}^{2}\), multiplying it by an energy cubed leaves an energy, and the remaining \(\hbar^{-1}\) converts that energy into a rate — so Equation (103.17) is \(/\mathrm{s}\), and \(\hbar\Gamma\) is the width in energy units usually quoted.

For a neutrino, \(T^{3}=+\tfrac{1}{2}\) and \(Q=0\), so \(g_{V}=g_{A}=\tfrac{1}{2}\) and the bracket is \(\tfrac{1}{2}\):

the full one-loop value being \(167.2\,\mathrm{MeV}\). For a charged lepton, \(T^{3}=-\tfrac{1}{2}\) and \(Q=-1\), so with \(\sin^{2}\theta_{W}=0.2312\) — the \(\overline{\mathrm{MS}}\) value at the scale \(M_{Z}c^{2}\) [Navas:2024], the fourth digit being meaningless without that scheme and that scale, which is why Equation (104.62) quotes the same quantity to three figures — the bracket is \(\left(-0.0376\right)^{2}+\left(-\tfrac{1}{2}\right)^{2}=0.2514\) and \(\hbar\Gamma_{\ell\ell}=83.4\,\mathrm{MeV}\), against the measured \(83.985(86)\,\mathrm{MeV}\) [Schael:2006] — the \(0.7\%\) shortfall being exactly the radiative correction the tree-level formula omits. The count is then

the tree-level version of which is \(0.5/0.2514=1.9888\), differing from the full value by one part in a thousand. With the measured \(\Gamma_{\mathrm{inv}}=499.0(15)\,\mathrm{MeV}\) and \(\hbar\Gamma_{\ell\ell}=83.985(86)\,\mathrm{MeV}\) [Schael:2006], the ratio is \(499.0/83.985=5.942\), and dividing by \(1.9912\) gives \(N_{\nu}=2.984\), which is Equation (103.15).

The residual \(2\sigma\) deviation from three deserves a word, because it is not a hint of a fourth neutrino. It is comparable with the radiative corrections just displayed, and the largest of those enters not the widths but the luminosity: the \(t\)-channel photon exchange in small-angle Bhabha scattering, which is what LEP counted to normalize every cross section. A re-analysis of that effect published in 2019 — not in this book's bibliography, and therefore named without a citation rather than cited to a key that does not exist — raised the central value to \(2.996\pm0.007\). The measurement counts three, and the apparent tension was an artefact of the normalization, not of the neutrino sector.

Derivation of the form. Derives Definition 103.25. Write \(V=R_{23}(\theta_{23})\,\Phi(\delta)\,R_{13}(\theta_{13})\, \Phi(-\delta)\,R_{12}(\theta_{12})\), where \(R_{ij}\) is a real rotation in the \(ij\) plane and \(\Phi(\delta)=\diag(1,1,\ee^{\ii\delta})\). Any unitary \(3\times3\) matrix can be brought to this form because such a matrix has \(9\) real parameters — the CKM matrix is unitary with no determinant condition imposed, so the count is that of \(\U(3)\) and not of \(\SU(3)\), which has \(8\) — of which \(3\) are the angles of the \(\SO(3)\) subgroup and \(6\) are phases, and \(5\) of the phases are removed by rephasing the quark fields, per Proposition 101.63. Multiplying out the five factors gives Equation (103.21). The placement of \(\delta\) is a convention: any rephasing moves it around the matrix without changing a single observable, since every observable is a rephasing invariant of the type of Equation (105.77). That the phase can be attached to \(V_{ub}\), the smallest element, is what makes the standard parametrization convenient and what makes \(CP\) violation small even though \(\delta\) is of order unity.

Derivation. Derives Definition 103.26. Expand Equation (103.21) with \(s_{12}=\lambda\), \(s_{23}=A\lambda^{2}\), \(s_{13}\ee^{-\ii\delta}=A\lambda^{3}(\rho-\ii\eta)\) and \(c_{ij}=1-\tfrac{1}{2}s_{ij}^{2}+\ldots\), keeping terms to \(O(\lambda^{3})\) in each entry. The \((1,1)\) entry is \(c_{12}c_{13}=1-\tfrac{1}{2}\lambda^{2}+O(\lambda^{4})\), the \((1,2)\) entry is \(s_{12}c_{13}=\lambda+O(\lambda^{5})\), and the \((2,1)\) entry is \(-s_{12}c_{23}-c_{12}s_{23}s_{13}\ee^{\ii\delta} =-\lambda+O(\lambda^{6})\), so the upper-left \(2\times2\) block is a rotation through the Cabibbo angle up to \(O(\lambda^{4})\). The \((3,1)\) entry is \(s_{12}s_{23}-c_{12}c_{23}s_{13}\ee^{\ii\delta} =A\lambda^{3}-A\lambda^{3}(\rho+\ii\eta) =A\lambda^{3}\left(1-\rho-\ii\eta\right)\) to this order, which is the entry carrying the interesting phase. Note that the expansion is not a mathematical approximation imposed for convenience: \(\lambda^{2}\) is \(0.0506\) and \(\lambda^{4}\) is \(2.6\times 10^{-3}\), so the hierarchy is a property of the measured matrix.

Derives Proposition 103.30. For three complex numbers \(z_{1}+z_{2}+z_{3}=0\), the triangle they close has area \(\tfrac{1}{2}\abs{\mathrm{Im}\left(z_{1}z_{2}^{*}\right)}\), since the cross product of the two vectors \(z_{1}\) and \(-z_{3}=z_{1}+z_{2}\) is \(\mathrm{Im}\left(z_{1}\overline{z_{1}+z_{2}}\right) =\mathrm{Im}\left(z_{1}z_{2}^{*}\right)\). With \(z_{1}=V_{ud}V_{ub}^{*}\) and \(z_{2}=V_{cd}V_{cb}^{*}\) this is \(\mathrm{Im}\left(V_{ud}V_{ub}^{*}V_{cd}^{*}V_{cb}\right)\), which is a Jarlskog quartet and therefore equals \(\pm J\) by Equation (105.77). Since Equation (105.77) makes every quartet equal up to sign, and each of the six relations produces one, all six triangles have area \(J/2\). A degenerate triangle — three collinear complex numbers — has \(J=0\) and by Proposition 105.27 conserves \(CP\). That the area is a rephasing invariant while the individual angles are not is the reason \(J\), and not \(\delta\), is the physical measure of \(CP\) violation.

Derivation. Derives Phenomenon 103.31. Two checks, one of the angles and one relating angles to sides.

First the angles. Equation (103.32) sums to \(173^\circ\pm6^\circ\), consistent with \(180^\circ\) at \(1.2\) standard deviations; and the constraint \(\alpha+\beta+\gamma=\pi\) is an identity of Definition 103.29, not an input to any of the three measurements. That the three numbers, extracted from \(B\to\pi\pi\) and \(\rho\rho\), from \(B\to J/\psi K_{S}\), and from \(B\to DK\) respectively, sum correctly is a non-trivial test.

Now the apex. Definition 103.29 places it at \(\left(\bar{\rho},\bar{\eta}\right)\), so

Inserting Equation (103.24) gives \(\beta=22.7^\circ\), hence \(\sin2\beta=0.713\), and \(\gamma=65.7^\circ\). The directly measured values are \(\sin2\beta=0.70\pm0.02\), from the time-dependent asymmetry in \(B^{0}\to J/\psi K_{S}\) [Aubert:2001] [Abe:2001], and \(\gamma=66^\circ\pm3^\circ\), from interference between \(b\to c\) and \(b\to u\) amplitudes in \(B\to DK\). Both agree. The first of them is the sharper statement, because \(\sin2\beta\) is measured to three per cent in a tree-dominated decay whose theoretical uncertainty is below one per cent — and it agrees with a value obtained from \(\Delta m_{d}\), \(\Delta m_{s}\), \(\varepsilon_{K}\) and \(\abs{V_{ub}/V_{cb}}\), that is from three loop processes and one tree process, none of them a \(CP\) asymmetry.

What is being tested is the following. There are four parameters. The observables listed number well over a dozen. Any one of them could have landed outside the region the others define, and that would have falsified the hypothesis that a single phase in a unitary matrix accounts for all \(CP\) violation in the quark sector. None has. The global fits [Charles:2005] quantify the agreement, and the residual freedom in the apex is now a few per cent in each coordinate.

Derives Proposition 103.34. Write \(\ket{K^{0}}=\left(\ket{K_{L}}+\ket{K_{S}}\right)/\sqrt{2}\) and \(\ket{\bar{K}^{0}}=\left(\ket{K_{L}}-\ket{K_{S}}\right)/\sqrt{2}\), which inverts Equation (103.35) up to the \(CP\)-violating admixture of order \(2\times 10^{-3}\) that is neglected here and restored in Section 103.5.2. Each mass eigenstate evolves with its own phase and its own damping, \(\ket{K_{i}(t)}=\ee^{-\ii m_{i}c^{2}t/\hbar}\ee^{-\Gamma_{i}t/2} \ket{K_{i}}\), so

whose squared modulus is Equation (103.37). The strangeness of the beam at time \(t\) is measured directly, by the sign of the charged lepton in a semileptonic decay: the rule \(\Delta S=\Delta Q\) (Proposition 101.56) says a \(K^{0}\) gives \(\ell^{+}\) and a \(\bar{K}^{0}\) gives \(\ell^{-}\). Counting the two charges as a function of decay position along a beam line therefore traces Equation (103.37), and the cosine's period measures \(\Delta m_{K}\). With \(x=\Delta m_{K}c^{2}/\hbar\bar{\Gamma}=0.946\) from Table 103.2, roughly one oscillation occurs per lifetime, which is the optimum: much less and there is no interference, much more and it averages away.

Derivation. Derives Phenomenon 103.36. The pion has intrinsic parity \(-1\), so a state of two pions with relative orbital angular momentum \(L\) has parity \((-1)^{2}(-1)^{L}=(-1)^{L}\); the kaon has spin zero, so \(L=0\) and the parity is \(+1\). Charge conjugation exchanges \(\pi^{+}\) with \(\pi^{-}\), which for \(L=0\) is the identity, so the state also has \(C=+1\). The two-pion state is therefore an eigenstate of \(CP\) with eigenvalue \(+1\). The three-pion state available to the same system has, by the same counting together with the small energy release that suppresses non-zero orbital angular momenta, \(CP=-1\). If \(CP\) were a symmetry of the decay interaction the two final states would be reached from two orthogonal kaon states, and whichever state decays to three pions could never decay to two. That one and the same long-lived state does both is thus a direct proof that \(CP\) is not conserved [Christenson:1964]. What the argument does not settle is where the violation sits — in the mixing that defines the long-lived state, or in the decay amplitude itself. Separating them required the measurement of \(\epsilon'/\epsilon\) a generation later, and only that measurement excluded a superweak interaction acting solely on the mixing.

Derives Proposition 103.38. From Equation (103.37) with \(\Gamma_{L}=\Gamma_{S}=\Gamma\), the mixed and unmixed probabilities are \(\tfrac{1}{2}\ee^{-\Gamma t} \left[1\mp\cos\left(\Delta m\,c^{2}t/\hbar\right)\right]\). Using \(\int_{0}^{\infty}\ee^{-\Gamma t}\dd t=\Gamma^{-1}\) and \(\int_{0}^{\infty}\ee^{-\Gamma t}\cos\omega t\,\dd t =\Gamma/\left(\Gamma^{2}+\omega^{2}\right)\) with \(\omega=\Delta m\,c^{2}/\hbar=x\Gamma\), and normalizing by the total \(\Gamma^{-1}\),

Solving \(r=\chi/(1-\chi)=0.21\) for \(x\) gives \(x^{2}=2r/(1-r)=0.532\), hence \(x=0.73\).

Derives Proposition 103.40. Divide Equation (103.40) by its \(B_{s}\) counterpart; every factor independent of the light spectator quark cancels. Inverting with the measured oscillation frequencies \(\Delta m_{d}c^{2}/\hbar=5.069\times 10^{11}\,/\mathrm{s}\) and \(\Delta m_{s}c^{2}/\hbar=1.7765\times 10^{13}\,/\mathrm{s}\) [Navas:2024] [Aaij:2013], that is \(\Delta m_{d}c^{2}=3.336\times 10^{-4}\,\mathrm{eV}\) and \(\Delta m_{s}c^{2}=1.169\times 10^{-2}\,\mathrm{eV}\),

against \(8.6\times 10^{-3}/4.15\times 10^{-2}=0.207\) from the individual entries of Table 103.1. Note the units. The literature quotes these as angular frequencies \(\Delta mc^{2}/\hbar\) in \(/\mathrm{ps}\) — \(0.5069\,/\mathrm{ps}\) and \(17.765\,/\mathrm{ps}\) — while this book quotes the energy splitting \(\Delta mc^{2}\) in \(\mathrm{eV}\). The conversion is \(\hbar=6.5821196\times 10^{-16}\,\mathrm{eV}\,\mathrm{s}\), so \(1\,/\mathrm{ps}\) corresponds to \(6.582\times 10^{-4}\,\mathrm{eV}\), and the ratio the derivation uses is of course the same either way.

Derivation. Derives Phenomenon 103.41. Proposition 105.28 gives the asymmetry as \(S_{f}\sin\left(\Delta m_{d}c^{2}t/\hbar\right) -C_{f}\cos\left(\Delta m_{d}c^{2}t/\hbar\right)\) with \(S_{f}=2\,\mathrm{Im}\,\lambda/\left(1+\abs{\lambda}^{2}\right)\) and \(\lambda=\left(q/p\right) \left(\bar{\mathcal{A}}_{f}/\mathcal{A}_{f}\right)\). For \(f=J/\psi K_{S}\) three things conspire to make this clean. The decay \(b\to c\bar{c}s\) is dominated by a single tree amplitude, so \(\abs{\bar{\mathcal{A}}/\mathcal{A}}=1\) to within a per cent and \(C_{f}\approx0\); the mixing phase \(q/p\) is, from the box of Equation (103.40), twice the phase of \(V_{tb}^{*}V_{td}\); and the \(K^{0}\)–\(\bar{K}^{0}\) mixing in the final state contributes twice the phase of \(V_{cs}^{*}V_{cd}\). Collecting,

which is exactly the angle of Equation (103.31); hence \(S=\sin2\beta\) with no hadronic quantity anywhere in the relation. That is the whole reason the measurement is decisive: the hadronic matrix elements cancel between the two interfering paths because both end in the same final state.

Why is the effect a thousand times larger than in the kaon? Because \(CP\) violation requires interference between two amplitudes of comparable size and different weak phase. In the kaon, the \(CP\)-violating amplitude is a small admixture in a state dominated by \(CP\)-conserving physics, so the asymmetry carries the small factor \(\varepsilon\). In \(B^{0}\to J/\psi K_{S}\) the two interfering paths — decay without mixing, and decay after mixing — have equal magnitude by construction, since \(\abs{q/p}=1\) and \(\abs{\lambda}=1\), so the asymmetry is the full \(\sin\) of the phase difference. Nothing is small except the phase, and the phase is not small.

Derivation. Derives Phenomenon 103.42. Taking the difference of two asymmetries rather than either alone is what makes the measurement possible: production asymmetries between \(D^{0}\) and \(\bar{D}^{0}\) in \(pp\) collisions, and detection asymmetries between positive and negative tracks, are common to the two final states and cancel in the difference, while the \(CP\)-odd parts do not, since \(U\)-spin relates the two channels with opposite sign. The residual is then a genuine asymmetry between a decay and its conjugate. Its magnitude, \(1.5\times 10^{-3}\), is of the order the Kobayashi–Maskawa phase predicts once the small mixing factor and the interference with the dominant Cabibbo-favoured amplitude are accounted for — but the hadronic matrix elements of charm decays are not calculable to better than a factor of a few, so the correct statement is that the observation is consistent with the Standard Model and does not test it sharply.

Derivation. Derives Phenomenon 103.44. The three Sakharov conditions are derived as Theorem 105.41 and the value of \(\eta\) from \(\Omega_{b}h^{2}\) as Proposition 105.42; neither is repeated. What is needed here is the size of the effect the CKM phase can produce, and the argument is dimensional.

By Proposition 105.27, all \(CP\) violation in the quark sector is proportional to the invariant \(\det\comm{M_{u}M_{u}^{\dagger}c^{4}}{M_{d}M_{d}^{\dagger}c^{4}}\), which carries \(J\) multiplied by the six squared-mass differences of Equation (105.79) and has SI dimension \(\mathrm{J}^{12}\). Any asymmetry generated at a temperature \(T\) is a pure number, so it can depend on that invariant only through the dimensionless ratio Equation (105.106), and evaluating it with the running quark masses at \(k_{B}T\approx100\,\mathrm{GeV}\), where the baryon-number-violating processes are active, gives Equation (105.107),

to be compared with \(\eta=6.1\times 10^{-10}\): the ratio \(\eta/d_{CP}=3\times 10^{11}\), a shortfall of eleven orders of magnitude, and detailed calculations [Gavela:1994] do not improve it. The reason is structural. \(CP\) violation in this theory requires all six quark masses to be distinct, and at \(100\,\mathrm{GeV}\) five of the six are negligible against the temperature, so the effect is suppressed by twelve powers of small mass ratios. The phase is of order unity; the masses kill it.

The third Sakharov condition fails as well, and independently. A departure from equilibrium requires the electroweak transition to be first order, and for the measured Higgs mass of Equation (104.66) it is not: it is a smooth crossover, as Remark 104.65 records. Two of the three conditions are therefore not met by the Standard Model with its measured parameters. The origin of Equation (103.45) is unknown, and this treatise says so; it is carried to Section 133.5.

Derives Proposition 103.47. Every oscillation amplitude has the structure \(\sum_{i}U_{\beta i}^{\phantom{*}}\ee^{-\ii\varphi_{i}} U_{\alpha i}^{*}\), in which each index \(i\) appears once on \(U\) and once on \(U^{*}\). Under Equation (103.48), \(U_{\alpha i}\mapsto U_{\alpha i}\ee^{\ii\alpha_{i1}/2}\) for every \(\alpha\), so the product \(U_{\beta i}U_{\alpha i}^{*}\) acquires \(\ee^{\ii\alpha_{i1}/2}\ee^{-\ii\alpha_{i1}/2}=1\). The same cancellation occurs in any amplitude with one neutrino line entering and one leaving, that is, in any lepton-number-conserving process. It fails only where two neutrino lines are contracted with each other, which requires a Majorana propagator and violates lepton number by two units — the process of Section 103.7.2, and the reason that is the only practical test.

The counting is worth recording. For \(N\) Dirac neutrinos the phase count is \((N-1)(N-2)/2\), as for quarks, because the neutrino fields may be rephased. For \(N\) Majorana neutrinos a rephasing \(\nu_{i}\to\ee^{\ii\phi}\nu_{i}\) is forbidden — it would multiply the Majorana mass term \(\overline{\nu^{c}}\nu\) by \(\ee^{2\ii\phi}\) — so only the charged-lepton fields may be rephased, \(N-1\) of those phases being usable, and the surviving count is \(N(N-1)/2\): three phases for \(N=3\), one Dirac and two Majorana.

Derivation. Derives Phenomenon 103.48. Equation (103.49) follows by inserting the measured \(\sin^{2}\theta_{12}=0.307\), \(\sin^{2}\theta_{23}=0.546\) and \(\sin^{2}\theta_{13}=0.0220\) of Table 103.3 into Equation (103.21), with \(\delta_{CP}\) set to zero, which shifts the entries of the lower two rows by at most a few units in the second decimal. The corresponding angles are

to be compared with Equation (103.22), where the largest is \(13.0^\circ\) and the smallest \(0.21^\circ\). The comparison is not between two numbers but between two patterns: the quark matrix is a small perturbation of the identity, with a hierarchy \(1:\lambda:\lambda^{3}\); the lepton matrix has no hierarchy at all except that one of its nine entries is small.

Derivation. Derives Phenomenon 103.50. Take two flavour states related to two states of definite mass \(m_{1}\neq m_{2}\) by a single angle,

The mass states, not the flavour states, are the ones that propagate with a definite phase. For a neutrino of energy \(E\) far above its rest energy, \(E_{i}=\sqrt{p^{2}c^{2}+m_{i}^{2}c^{4}}\approx E+m_{i}^{2}c^{4}/2E\), so after travelling a distance \(L\) in a time \(L/c\) the component \(\ket{\nu_{i}}\) has acquired the phase

The first term is common to both components and unobservable. The amplitude to find the other flavour is the projection of the propagated state on \(\ket{\nu_{\beta}}\),

and with \(\abs{\ee^{-\ii\varphi_{2}}-\ee^{-\ii\varphi_{1}}}^{2} =2\left[1-\cos(\varphi_{1}-\varphi_{2})\right] =4\sin^{2}\tfrac{1}{2}(\varphi_{1}-\varphi_{2})\) its squared modulus is

Three consequences follow, and all three are what is seen. The effect is an interference between two mass components, so it measures a difference of squared masses and never a mass: the oscillation experiments cannot say how heavy a neutrino is, only that two of them differ. It vanishes if the masses are equal or if \(\theta=0\), so its observation proves both that the mass states are non-degenerate and that the flavour and mass bases are misaligned. And it depends on \(L\) and \(E\) only through their ratio, which is why the atmospheric, reactor and accelerator data — taken at baselines from tens of metres to thousands of kilometres — collapse onto one curve when plotted against \(L/E\).

Derives Proposition 103.52. Multiply numerator and denominator of the phase in Equation (103.52) by \(c\): the numerator becomes \(\Delta m^{2}c^{4}\), an energy squared, and the denominator \(4\hbar cE\), also an energy squared multiplied by a length. Hence

which is manifestly dimensionless when \(L\) is a length. Now insert \(\hbar c=1.9732698\times 10^{-7}\,\mathrm{eV}\,\mathrm{m}\) from Equation (103.1), and convert the units of the three inputs: \(L=\left(L/\mathrm{km}\right)\times10^{3}\,\mathrm{m}\) and \(E=\left(E/\mathrm{GeV}\right)\times10^{9}\,\mathrm{eV}\). Then

and the prefactor is the pure number

This is the constant that appears as “\(1.27\)” in every experimental paper on the subject, where it is quoted rather than derived; it is \(\left(4\hbar c\right)^{-1}\) in disguise, and nothing else. Setting \(\varphi=\pi\) and solving for \(L\) gives \(\pi/1.26693=2.4797\), which is Equation (103.55).

Derives Theorem 103.54. The amplitude is \(A_{\alpha\to\beta}=\sum_{i}U_{\beta i}^{\phantom{*}} \ee^{-\ii m_{i}^{2}c^{3}L/2\hbar E}U_{\alpha i}^{*}\), the common phase of Equation (103.51) having been dropped. Then

Split the double sum into \(i=j\), which gives \(\sum_{i}\abs{U_{\alpha i}}^{2}\abs{U_{\beta i}}^{2}\), and the off-diagonal terms, which pair \((i,j)\) with \((j,i)\). Writing \(W_{ij}=U_{\alpha i}^{*}U_{\beta i}U_{\alpha j}U_{\beta j}^{*}\), the coefficient standing in front of \(\ee^{-\ii\varphi_{ij}}\) in the display above is \(U_{\beta i}U_{\alpha i}^{*}U_{\beta j}^{*}U_{\alpha j} =\left(U_{\alpha i}^{*}U_{\beta i}\right) \left(U_{\alpha j}U_{\beta j}^{*}\right)=W_{ij}\) itself, and the \((j,i)\) term is its complex conjugate because \(W_{ji}=W_{ij}^{*}\) and \(\varphi_{ji}=-\varphi_{ij}\). The paired contribution is therefore \(2\,\mathrm{Re}\left(W_{ij}\ee^{-\ii\varphi_{ij}}\right) =2\,\mathrm{Re}\,W_{ij}\cos\varphi_{ij} +2\,\mathrm{Im}\,W_{ij}\sin\varphi_{ij}\) with \(\varphi_{ij}=\Delta m^{2}_{ij}c^{3}L/2\hbar E\) — the sign of the second term being the sign of the \(CP\)-violating term in Equation (103.56), so it is worth stating why it is a plus: \(\ee^{-\ii\varphi}\) contributes \(-\ii\sin\varphi\), and \(\mathrm{Re}(-\ii W)=+\,\mathrm{Im}\,W\). Using \(\cos\varphi=1-2\sin^{2}(\varphi/2)\) and the unitarity relation \(\sum_{i}U_{\alpha i}^{*}U_{\beta i}=\delta_{\alpha\beta}\), whose squared modulus supplies \(\sum_{i}\abs{U_{\alpha i}}^{2}\abs{U_{\beta i}}^{2} +2\sum_{i>j}\mathrm{Re}\,W_{ij}=\delta_{\alpha\beta}\), gives Equation (103.56). Replacing \(U\) by \(U^{*}\), which is what \(CP\) conjugation does, flips the sign of the imaginary part and leaves the real part alone, which is the last statement.

Derives Proposition 103.56. The relevant term of the effective Lagrangian of Equation (101.6), Fierz-rearranged so that the neutrino fields are paired with each other, is

of dimension \(\mathrm{J}/\mathrm{m}^{3}\), since each bilinear carries \(/\mathrm{m}^{3}\) by Definition 100.2 and \(G_{F}\) carries \(\mathrm{J}\,\mathrm{m}^{3}\). Average the electron bilinear over the medium. For an unpolarized, isotropic electron gas at rest, the spatial components of the vector current average to zero and the axial current averages to the net spin density, also zero; the surviving component is \(\avg{\bar{\psi}_{e}\gamma^{0}\psi_{e}}=n_{e}\), the electron number density. The neutrino bilinear with \(\gamma^{0}\) counts the neutrino number density, so the average Lagrangian is a potential energy per neutrino of \(V_{e}=\sqrt{2}G_{F}n_{e}\): one factor of \(2\) from the \(\left(\identity-\gamma^{5}\right)\) acting on a left-chiral neutrino, against the \(1/\sqrt{2}\) of the Fermi normalization. The neutral current contributes as well, but identically for all three flavours, so it adds a multiple of the identity to the Hamiltonian and cannot affect oscillation — the reason only the charged-current term appears in Equation (103.57). For an antineutrino, \(C\) conjugation reverses the sign of the vector current, giving \(-V_{e}\); this sign is the whole of the mass-ordering sensitivity discussed below. The numerical evaluation is Equation (103.58), using \(G_{F}\) in SI from Equation (101.8) — not \(G_{F}/(\hbar c)^{3}\), the confusion Definition 101.8 exists to prevent.

Derives Theorem 103.57. The evolution of the flavour amplitudes with time is \(\ii\hbar\,\pp_{t}\Psi=\Ham\Psi\), and for an ultrarelativistic neutrino of fixed momentum the free part of \(\Ham\) is \(\sqrt{p^{2}c^{2}+m_{i}^{2}c^{4}}\approx pc+m_{i}^{2}c^{4}/2E\) in the mass basis. Dropping the term proportional to the identity and rotating to the flavour basis with Equation (103.50),

the matter term being diagonal in the flavour basis by Proposition 103.56. Subtracting \(\tfrac{1}{2}V_{e}\identity\), which shifts both levels equally and changes no probability,

A real symmetric traceless \(2\times2\) matrix of this form is diagonalized by a rotation through \(\theta_{m}\) with \(\tan2\theta_{m}=B/A\), which after multiplying numerator and denominator by \(2E\) is Equation (103.59); its eigenvalue difference is \(\sqrt{A^{2}+B^{2}}\), which multiplied by \(2E\) is Equation (103.60). The angle is \(\pi/4\) exactly when \(A=0\), which is Equation (103.61); and since \(V_{e}>0\) for matter, \(A\) can vanish only if \(\Delta m^{2}\cos2\theta>0\). For an antineutrino \(V_{e}\to-V_{e}\) and the condition becomes \(\Delta m^{2}\cos2\theta<0\). The resonance therefore occurs for neutrinos or for antineutrinos according to the sign of the splitting, and that is the only place in this chapter where a sign, rather than a magnitude, is observable.

Derives Proposition 103.59. The adiabatic theorem states that a system in a non-degenerate instantaneous eigenstate of a slowly varying Hamiltonian stays in it, with a transition probability that vanishes as the variation is slowed; the general statement and its proof belong to Section 82.6.1, which reserves them. Its condition is that the rate of change of the Hamiltonian's eigenbasis be small compared with the level splitting divided by \(\hbar\). Here the eigenbasis is fixed by \(\theta_{m}\), and the fastest variation is at resonance. Differentiating \(\tan2\theta_{m}=B/A\) from the proof of Theorem 103.57, in which only \(A\) depends on \(r\),

the last step using \(V_{e}=\Delta m^{2}c^{4}\cos2\theta/2E\) at resonance; hence \(\dd\theta_{m}/\dd r=\cot2\theta/\left(2L_{\rho}\right)\) in magnitude. The two eigenvalues of the traceless Hamiltonian there are \(\pm\Delta m^{2}_{m}c^{4}/4E\) with \(\Delta m^{2}_{m}c^{4}=\Delta m^{2}c^{4}\sin2\theta\) by Equation (103.60), so the ratio of the phase rate to the rotation rate is

Numerically, with \(\Delta m^{2}c^{4}\sin^{2}2\theta =7.53\times 10^{-5}\,\mathrm{eV}^{2}\times0.851 =6.41\times 10^{-5}\,\mathrm{eV}^{2}\), \(2E\cos2\theta=2\times1.91\times 10^{6}\,\mathrm{eV}\times0.386 =1.47\times 10^{6}\,\mathrm{eV}\) and \(L_{\rho}/\hbar c=6.6\times 10^{7}\,\mathrm{m}/1.9733\times 10^{-7}\,\mathrm{eV}\,\mathrm{m} =3.34\times 10^{14}\,/\mathrm{eV}\), the product is \(1.5\times 10^{4}\). The solar density profile is exponential over the relevant region, which is what makes \(L_{\rho}\) a constant, and the margin is four orders of magnitude: the conversion is adiabatic and the survival probabilities of Example 103.58 are exact in that limit.

Derivation. Derives Phenomenon 103.62. By Equation (101.11), the electron spectrum of an allowed transition is

\(T_{e}\) being the electron kinetic energy. For \(m_{\beta}=0\) this is \(\left(Q-T_{e}\right)^{2}\) near the endpoint, so the Kurie function of Definition 101.19 is linear and reaches zero at \(T_{e}=Q\); for \(m_{\beta}\neq0\) the spectrum terminates at \(T_{e}=Q-m_{\beta}c^{2}\) and approaches it with infinite slope. Two consequences fix the design.

First, the sensitivity is to \(m_{\beta}^{2}\), not to \(m_{\beta}\): the leading distortion of the rate integrated over the last interval \(\Delta\) below the endpoint is a relative shift of order \(m_{\beta}^{2}c^{4}/\Delta^{2}\). Halving the mass sensitivity therefore costs a factor of four in statistics, which is why progress in this field is slow and why an experiment reports a limit on \(m_{\beta}^{2}\) that can and does fluctuate negative.

Second, the region carrying the information is minute. Since \(\dd\Gamma/\dd T_{e}\propto\left(Q-T_{e}\right)^{2}\), the fraction of all decays occurring within \(\Delta\) of the endpoint is

which for \(\Delta=1\,\mathrm{eV}\) and \(Q=18.574\,\mathrm{keV}\) is \(1.6\times 10^{-13}\). An experiment sensitive to \(0.2\,\mathrm{eV}/c^{2}\) must therefore combine a source of enormous activity with a spectrometer of electronvolt resolution at \(18.6\,\mathrm{keV}\) — a relative resolution of \(5\times 10^{-5}\) — and reject every background event in the same window. That is what KATRIN's \(10\,\mathrm{m}\)-diameter magnetic adiabatic collimation spectrometer exists to do.

Tritium is chosen because \(Q\) is the smallest of any convenient superallowed emitter, maximizing Equation (103.68), and because its nuclear matrix element is simple. The residual model dependence is molecular, not nuclear: the daughter \(^{3}\mathrm{HeT}^{+}\) ion is left in a distribution of rotational, vibrational and electronic excitations spread over some \(1\,\mathrm{eV}\), and that distribution is calculated, not measured. It is the leading systematic of Equation (103.67).

Derivation. Derives Phenomenon 103.64. Furry computed the transition \((A,Z)\to(A,Z+2)+2e^{-}\) in 1939 [Furry:1939]: the antineutrino emitted at one nucleon vertex is absorbed as a neutrino at the other. Two conditions must hold for the amplitude not to vanish, and each supplies a factor.

The neutrino must be its own antiparticle, since the particle emitted at the first vertex is what must be absorbed at the second. This is the Majorana condition, and it is what makes the process a test of Section 103.7.4 rather than of anything else.

The neutrino must have mass. The charged current emits a right-helicity antineutrino and absorbs a left-helicity neutrino (Proposition 101.32), so the two vertices demand opposite helicities of the same particle. The amplitude therefore requires the “wrong-helicity” component of the propagating state. Equation (101.32) gives the chirality weight of that component as \(\tfrac{1}{2}(1-\beta)\simeq m^{2}c^{4}/4E^{2}\), which is a probability, so the amplitude carries its square root, \(m_{i}c^{2}/2E\). The internal neutrino carries a virtual momentum of order the inverse internucleon spacing, \(\hbar c/r\approx100\,\mathrm{MeV}\), so the suppression factor is of order \(m_{i}c^{2}/100\,\mathrm{MeV}\), one factor per exchanged mass eigenstate; the factor of two, and the shape of the neutrino propagator between the two nucleons, are conventionally absorbed into the nuclear matrix element defined below rather than carried separately. Summing coherently over the three mass states with the mixing factors \(U_{ei}\) from each vertex, the amplitude is proportional to \(\sum_{i}U_{ei}^{2}m_{i}\) — squared, not modulus-squared, because both vertices are of the same type — which is \(m_{\beta\beta}\) of Equation (103.65). Hence

with \(G^{0\nu}\) a phase-space factor, carrying \(/\mathrm{yr}\), calculable exactly from the \(Q\) value and the Coulomb field, and \(M^{0\nu}\) the dimensionless nuclear matrix element.

Two structural remarks. First, Equation (103.70) is a rate proportional to a squared mass, so a half-life limit improved by a factor \(100\) improves the mass limit only by a factor \(10\) — the same quadratic wall that, in the derivation of Phenomenon 103.62, makes every halving of the kinematic mass sensitivity cost a factor of four in statistics. Second, the electron sum-energy spectrum of the neutrinoless mode is a line at \(Q\), sitting on the continuum of the two-neutrino mode, and the two-neutrino mode is therefore an irreducible background whose tail is controlled only by energy resolution. This is why the two leading techniques are a high-purity germanium diode, whose resolution at \(Q=2039\,\mathrm{keV}\) is about \(3\,\mathrm{keV}\), and a very large mass of xenon-loaded scintillator, which trades resolution for target mass.

Derives Proposition 103.66. Write \(m_{\beta\beta}=\abs{c_{12}^{2}c_{13}^{2}m_{1} +s_{12}^{2}c_{13}^{2}m_{2}\ee^{\ii\alpha_{21}} +s_{13}^{2}m_{3}\ee^{\ii\alpha_{31}}}\). In the inverted ordering \(m_{3}\approx0\) and \(m_{1}c^{2}\approx m_{2}c^{2}\approx \sqrt{\abs{\Delta m^{2}_{32}}c^{4}}=50\,\mathrm{meV}\), so the third term is negligible and the first two, being nearly equal in mass, can cancel only to the extent that their coefficients differ: the minimum over \(\alpha_{21}\) is

using \(\cos2\theta_{12}=0.386\) from Table 103.3. The maximum, at \(\alpha_{21}=0\), is \(49\,\mathrm{meV}/c^{2}\). In the normal ordering with \(m_{1}\to0\) the three terms are \(0\), \(s_{12}^{2}c_{13}^{2}\sqrt{\Delta m^{2}_{21}c^{4}}/c^{2} =2.6\,\mathrm{meV}/c^{2}\) and \(s_{13}^{2}\sqrt{\Delta m^{2}_{31}c^{4}}/c^{2} =1.1\,\mathrm{meV}/c^{2}\), which are comparable and can cancel exactly for a suitable phase. There is therefore no floor.

The consequence for the programme is sharp and should be stated without optimism. Equation (103.71) is at the level of the inverted-ordering band, and an experiment covering that band completely would, if it saw nothing, exclude the inverted ordering provided the neutrino is a Majorana particle. It would not exclude the Majorana hypothesis, because the normal ordering permits \(m_{\beta\beta}=0\) exactly. The asymmetry is inherent: the experiment can discover, and can exclude one ordering, but can never refute the Majorana nature.

Derives Proposition 103.67. The relic number density per species follows from decoupling in thermal equilibrium, the general treatment of which belongs to Section 48.3: a fermion species in equilibrium at temperature \(T_{\nu}\) has \(n_{\nu}=\tfrac{3}{4}\times\tfrac{2\zeta(3)}{\pi^{2}} \left(k_{B}T_{\nu}/\hbar c\right)^{3}\) per degree of freedom, and \(T_{\nu}=\left(4/11\right)^{1/3}T_{\gamma}\) because the electron–positron annihilation that follows decoupling heats the photons and not the neutrinos. With \(T_{\gamma,0}=2.7255\,\mathrm{K}\) this gives \(n_{\nu}=1.12\times 10^{8}\,/\mathrm{m}^{3}\) per species. Dividing the mass density \(\Sigma\,n_{\nu}\) by the critical density \(\rho_{c}=1.87834\times 10^{-26}\,\mathrm{kg}/\mathrm{m}^{3}\,h^{2}\) gives Equation (103.73). The naive arithmetic, \(\rho_{c}c^{2}/n_{\nu}\) per unit \(h^{2}\), comes to \(94.1\,\mathrm{eV}\); the standard constant \(93.14\,\mathrm{eV}\) includes the one-per-cent correction from the fact that decoupling is not instantaneous, which leaves the neutrinos slightly hotter than \(\left(4/11\right)^{1/3}T_{\gamma}\).

For the free streaming: a relic neutrino of mass \(m\) has, at redshift \(z\), a typical momentum \(p\approx3.15k_{B}T_{\nu}(1+z)/c\) and hence a speed \(v/c\approx3.15k_{B}T_{\nu}(1+z)/mc^{2}\). Today \(T_{\nu}=\left(4/11\right)^{1/3}\times2.7255\,\mathrm{K} =1.9454\,\mathrm{K}\), so \(k_{B}T_{\nu}=1.676\times 10^{-4}\,\mathrm{eV}\) and \(3.15k_{B}T_{\nu}=5.28\times 10^{-4}\,\mathrm{eV}\); for \(mc^{2}=0.05\,\mathrm{eV}\) this gives \(v/c=1.06\times 10^{-2}\), that is \(v=3.2\times 10^{3}\,\mathrm{km}/\mathrm{s}\). Over a Hubble time it therefore travels a comoving distance of order \(v/H\), and perturbations of smaller wavelength are erased because the neutrinos simply leave them. The suppression coefficient \(8\) follows from solving the equation for the growth of density perturbations, which belongs to Section 48.5.1, with a component that does not cluster; the essential point for this chapter is that the effect is proportional to \(\Omega_{\nu}\), hence to \(\Sigma\), and is therefore a measurement of the sum of the masses and of no other combination.

Derivation. Derives Phenomenon 103.68. The microwave background alone constrains \(\Sigma\) only weakly, because a neutrino of mass below about \(0.6\,\mathrm{eV}/c^{2}\) is still relativistic at recombination and its effect on the acoustic peaks is degenerate with other parameters; the constraint comes from combining it with a low-redshift measurement of the expansion history and of the clustering amplitude, which is what the baryon acoustic scale of Phenomenon 48.15 supplies. Extracting Equation (103.74) from those data is a global fit, not a closed-form derivation, and it is quoted here for comparison with Equation (103.67) rather than reproduced.

The comparison is the point. Equation (103.74) is a factor \(20\) stronger than the kinematic bound — and rests on the assumptions that the dark energy is a cosmological constant, that the primordial spectrum is a power law, and that no other light relic contributes. Each has been examined and none is established independently. Relaxing them typically weakens the bound by a factor of two to three, which is the honest uncertainty to attach to it.