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For $N_p$ protons, exposure $T$ and efficiency $ϵ$, the expected count in this channel is $\mu =ϵN_p{\Gamma }_{\mu {\pi }^0}T$. A partial decay rate at the quoted upper bound gives one event per year whenMore generally, $M_p\simeq 11\mathrm{kilotonnes}\mu /[ϵ(T/\mathrm{yr})]$ at the limiting rate. This is an estimate of the required proton inventory, not necessarily the whole detector mass. For water, each molecule contains ten protons and has mass approximately $18$ atomic mass units; the corresponding ideal inventory is about $19.7\mathrm{kilotonnes}$ of water. Finite efficiency and a useful sample larger than one event increase the required exposure.

An upper limit is not a measured nonzero rate. The true rate could be much smaller, including zero, so no finite target guarantees a measurable signal. The estimate describes sensitivity to a rate near the existing limit.

A large, optically clean water Cherenkov detector provides many protons, a transparent radiator and a relatively inexpensive homogeneous target. A surrounding array of inward-facing photomultipliers records hit times and charge. Reconstructing the vertex and cone intersections identifies track directions and visible energies; an outer active veto rejects entering particles. Water has refractive index near $1.33$, and the charged decay products are above its relevant thresholds.

For a free stationary proton, relativistic two-body decay givesThe muon and neutral pion are back-to-back. The pion decays essentially at the vertex into two gamma rays: $c{\tau }_{{\pi }^0}\simeq 2.4\times 10^{-8}\mathrm{m}$ before its modest Lorentz factor is included. Each gamma ray converts and initiates an electromagnetic shower, producing a diffuse ring. The muon gives a relatively sharp ring and, if it stops, a delayed Michel Positron on the $2.2\mathrm{microsecond}$ scale. The ideal topology therefore has one muon-like and two electron-like rings, although overlaps or a weak photon may merge or hide a ring.

Require the two-photon invariant mass to match $m_{{\pi }^0}$, the total reconstructed mass to match $m_p$, and small total momentum. Bound oxygen protons have Fermi motion and their hadronic products can rescatter, so those cuts need realistic nuclear corrections; the water target also contains free hydrogen protons. Install the detector deep underground to suppress cosmic-ray muons, with shielding and an active veto. Important irreducible backgrounds are atmospheric-neutrino interactions producing a muon and neutral pion, potentially with an undetected recoil nucleon.

In the supplied Dirac representation, ${\gamma }^5$ exchanges the upper and lower blocks. ThereforeDefine $\eta =[I-\mathbf{\sigma }\cdot \mathbf{p}/(E+m)]\chi$. Acting with this projector givesFor completeness, its two independent components areThe equal-and-opposite block structure verifies ${\gamma }^5{\psi }_L=-{\psi }_L$. Projection isolates a chiral component of the massive Dirac solution; that component alone generally does not satisfy an independent massive Dirac equation, since the mass couples both chiralities.

The charged weak current contains $\bar{\psi }{\gamma }^{\mu }{P}_L\psi$. For an ultrarelativistic external fermion, it selects negative helicity; its antifermion partner has positive helicity. Wrong-helicity charged-current amplitudes vanish in the massless limit and are suppressed by powers of $m/E$ at finite mass. The active neutrino is consequently left-helicity and its antineutrino right-helicity in this limit.

This is a charged-current statement, not a prohibition on every right-helicity Standard Model fermion. Electromagnetic and strong vector interactions act on both chiralities and conserve helicity along a massless fermion line. A neutral-current $Z$ vertex also couples to both charged-fermion chiralities with unequal strengths. For example, a right-helicity electron can scatter electromagnetically: it is not excluded from the Standard Model. In vector-current pair annihilation the massless fermion and antifermion have opposite helicities; a purely charged-current vertex keeps only the left-fermion/right-antifermion choice. Yukawa interactions connect left and right fields and must be distinguished from these vector-current selection rules.

Use the explicit charged channels $D^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$ and $K^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$, labeling the two positive pions as 1,2 and the negative one as 3. With $m_{\pi }=0.139\mathrm{GeV}$, the opposite-charge pair mass squares have the rangesTheir exact correlated boundaries follow by taking the pair rest-frame angle to its endpoints. For general daughter masses, define $\Lambda (a,b,c)=a^2+b^2+c^2-2ab-2ac-2bc$. At fixed $s=s_{13}$,The charged-pion thresholds give the following numerical comparison:

The area is about $3150$ times smaller for the kaon. Including the $M_0^{-2}$ phase-space prefactor, its integrated labelled three-body phase space is about $220$ times smaller; neither comparison predicts a ratio of weak decay widths without the matrix elements.

A ${\rho }^0\to {\pi }^{+}{\pi }^{-}$ intermediate state produces horizontal or vertical bands near $s_{13}$ or $s_{23}=m_{\rho }^2\simeq (0.775\mathrm{GeV}{)}^2$ in the charm decay. Scalar pion-pair amplitudes, including broad low-mass $S$-wave contributions and accessible $f_0(980)$, can also populate the plot. The kaon pair mass cannot exceed $0.355\mathrm{GeV}$, so it cannot contain an on-shell ${\rho }^0$ or $f_0(980)$ band; smooth nonresonant and rescattering effects remain possible. Channels containing neutral pions use their supplied $0.135\mathrm{GeV}$ mass and shift these boundaries slightly without changing the conclusion. The figure shows kinematic domains and representative accessible bands, not an assumed measured intensity.

The charged two-pion and three-pion channels can be assigned to the same spin-zero kaon by their parent mass and lifetime. For two spinless pions, total angular momentum zero forces their relative orbital angular momentum to zero, so the product of their intrinsic parities givesFor three spinless pions, use Jacobi orbital angular momenta $l,L$. Coupling them to total zero requires $l=L$, and henceThus both channels cannot conserve parity in decays of one parent with a definite parity. The weak decay interaction permits the observed coexistence.

Dalitz plots identify the three-body events, determine their internal kinematics, and allow amplitude and angular-momentum analyses through their populations and resonance structures. Together with the spin and parent-identity evidence, they support the parity assignment. They are not a standalone parity-odd observable: reversing all three momenta leaves every $s_{ij}$, and therefore every point on the plot, unchanged. A flat distribution by itself cannot establish parity violation without the channel and spin information.

The lepton mixing matrix relates charged-current neutrino flavour fields to mass eigenfields: ${\nu }_{\alpha L}={\sum }_i{U}_{\alpha i}{\nu }_{iL}$. Nondegenerate masses then give relative propagation phases and flavour oscillations. This is the three-massive-neutrino extension of the minimal massless-neutrino Standard Model.

A unitary three-by-three matrix has nine real parameters: three rotation angles and six phases. For Dirac leptons, the independent field rephasings remove five phases, because one common phase acts trivially. Three angles and one physical Dirac phase remain. For Majorana neutrinos two additional phases cannot be removed, but they cancel from ordinary oscillation probabilities.

The phases printed in the lower two rows do not give a unitary matrix for general $\delta$. A consistent convention is $U=R_{23}{U}_{13}(\delta )R_{12}$, where the real rotations use the indicated mixing angles andThe lower-row phase factors in the resulting full matrix are $e^{+i\delta }$, opposite to the $e^{-i\delta }$ in $U_{e3}$. For the source's all-minus assignment, the scalar product of the electron row with the conjugated muon row is $-2i{s}_{23}{s}_{13}{c}_{13}\sin \delta$, which is nonzero at $\delta =\pi /2$ and nonzero mixing. This supplies an explicit counterexample. The electron-row moduli used below are unaffected by this correction.^[ref]

Restore the unit conversion in the phase,Equivalently use mass units $\mathrm{eV}/c^2$ and energies in $\mathrm{eV}$, with $\hslash c=1.97327\times 10^{-7}\mathrm{eV}\mathrm{m}$. The numerical conversion factor isThusThe familiar coefficient $1.267$ uses $L$ in metres and $E$ in MeV, or kilometres and GeV; it would be a factor of one thousand too small with the units requested here.^[ref] The quoted differences with units ${\mathrm{eV}}^2$ are interpreted as squared-mass differences, despite the source's abbreviated $\Delta m_{ij}$ notation.

With the supplied angles, $c_{13}^4{\sin }^{2}2{\theta }_{12}=0.70545$ and ${\sin }^{2}2{\theta }_{13}=0.11698$. At $E=3\mathrm{MeV}$ and with the two atmospheric-scale splittings approximated by a common $3\times 10^{-3}{\mathrm{eV}}^2$, the requested sketch is represented byThe fast period is $2.480\mathrm{km}$ and its first disappearance maximum is at $1.240\mathrm{km}$. The slow period is $92.99\mathrm{km}$, giving slow minima near $46.49$ and $139.48\mathrm{km}$. At each baseline the fast oscillations lie between $1-0.70545{\sin }^2(0.0337849L)$ and that value minus $0.11698$. Averaging over unresolved fast oscillations subtracts half that amplitude, rather than erasing the slow oscillation.

The equal atmospheric splittings are an approximation: exact squared masses require $\Delta m_{13}^2=\Delta m_{12}^2+\Delta m_{23}^2$. For example, choosing magnitudes $\Delta m_{31}^2=3.00\times 10^{-3}$ and $\Delta m_{21}^2=0.08\times 10^{-3}{\mathrm{eV}}^2$ gives $\Delta m_{32}^2=2.92\times 10^{-3}{\mathrm{eV}}^2$. This changes the fine fast pattern while retaining the two physical scales. The figure deliberately displays the source's common-frequency approximation and labels the fast-averaged curve separately from the monoenergetic result.

Place a far detector near the first atmospheric disappearance maximum, approximately $1.2\mathrm{km}$ for the source's monochromatic energy and splitting, and a nearly identical near detector a few hundred metres from the reactor complex. Their flux-corrected comparison largely cancels reactor power, emitted-spectrum and absolute-efficiency uncertainties. For a broad reactor spectrum one fits energy-dependent disappearance instead of treating that single baseline as a universal exact optimum.

Detect ${\bar{\nu }}_{e}p\to e^{+}n$ in a hydrogen-rich target, for example liquid scintillator with neutron-capture enhancement. The prompt Positron signal estimates neutrino energy; the delayed neutron-capture signal supplies a time-and-position coincidence that strongly reduces random backgrounds. Use sufficient target mass and exposure, low radioactivity, overburden and a muon veto to reduce cosmogenic backgrounds, calibrated relative energy response and efficiency, and monitored reactor powers and baselines. Account for multiple reactor cores, spent-fuel backgrounds and accidental coincidences. At kilometre baselines the slow solar phase is small, so the disappearance depth directly probes ${\sin }^2(2{\theta }_{13})$ with little sensitivity to the CP phase or mass ordering.

Assume first an ultrarelativistic charged-current interaction with negligible electron mass in the centre-of-mass spin analysis. Choose the incident antineutrino along $+z$. Its right helicity gives spin $+z$; the interacting left-helicity electron moves along $-z$ and also has spin $+z$. The total initial projection is $+1$. The final antineutrino and selected electron likewise have total spin $+1$ along the outgoing antineutrino direction. The spin-one rotation therefore givesAn unpolarized electron target changes the normalization through the spin average, not this selected-current shape. For electron antineutrinos the charged-current diagram is the $s$-channel $W^{-}$ graph. A contact approximation further requires $m_e^2\ll s\ll m_W^2,m_Z^2$.

The full Standard Model also has neutral-current $Z$ exchange. In the contact limit it supplies a right-electron contribution as well as interference in the left-electron channel. With $g_L=1/2+{\sin }^2{\theta }_W$, $g_R={\sin }^2{\theta }_W$, the full high-energy shape is proportional to $g_L^2(1+\cos {\theta }^{\star }{)}^2/4+g_R^2$. Thus the pure squared angular form is an explicitly stated charged-current approximation, not an exact prediction for all Standard Model contributions. At momentum transfers comparable to the weak-boson masses, propagators must also be retained.^[ref]

Neglecting the electron mass in the centre-of-mass kinematics, take $k=(E^{\ast },0,0,E^{\ast })$, $p=(E^{\ast },0,0,-E^{\ast })$, and $k^{\prime }=(E^{\ast },E^{\ast }\sin {\theta }^{\star },0,E^{\ast }\cos {\theta }^{\star })$. Then $p\cdot k=2E^{\ast 2}$ and $p\cdot k^{\prime }=E^{\ast 2}(1+\cos {\theta }^{\star })$. HenceThe constant Jacobian $|d\cos {\theta }^{\star }/dy|=2$ givesIncluding both Standard Model chiral channels in the contact approximation instead gives $d\sigma /dy\propto g_L^2(1-y{)}^2+g_R^2$. The right-electron term is flat in $y$.

In the electron-rest laboratory frame $p=(m_e,0)$, the invariant definition immediately givesThis last equality is exact. With electron mass retained in the centre-of-mass relation, the angular formula becomes $y=(s-m_e^2)(1-\cos {\theta }^{\star })/(2s)$, approaching the boxed massless relation at high energy.

Treat a nucleon as incoherent collinear spin-half quarks and antiquarks carrying fractions $x$ of its momentum. Neglect their masses and transverse motion, use a charged-current four-fermion interaction, and take $Q^2\ll m_W^2$. The subprocess energy is $\hat{s}\simeq xs$. Antineutrino scattering on a quark has the same selected-current helicity factor $(1-y{)}^2$ as in part a; crossing to an antiquark changes the allowed spin pairing and gives a flat distribution. Multiplying the elementary strength by the respective parton densities givesHere $q,\bar{q}$ are the effective weak flavour combinations with the isoscalar proton-neutron average and CKM weights absorbed into their definitions. With only light flavours and charge symmetry, the relevant up and down combinations coincide on an isoscalar target. Rescaling those density definitions requires rescaling the prefactor consistently.

The equation is a leading-order scaling result: it neglects logarithmic QCD evolution, heavy-quark thresholds, higher-twist and target-mass corrections, nuclear modifications, and weak-propagator suppression. The current produces ${\mu }^{+}$ for an incident muon antineutrino. At higher momentum transfer the weak propagator supplies a factor approximately $[m_W^2/(m_W^2+Q^2){]}^2$, and at low energy resonance and quasielastic scattering invalidate a pure DIS description.

For a nucleon at rest and beam energy well above its mass scales, $s\simeq 2m_{N}E$. If the leading parton distributions are approximately independent of beam energy, integration over $x,y$ therefore makes both total cross sections approximately proportional to $E$. The plotted plateaus are evidence for scattering on approximately pointlike constituents in this range, with the departures expected from QCD evolution and threshold effects.

Define the momentum-weighted contents $Q={\int }_0^{1}xq(x)dx$ and $\bar{Q}={\int }_0^{1}x\bar{q}(x)dx$. Since ${\int }_0^1(1-y{)}^{2}dy=1/3$, the two cross sections obeyRead the original high-energy plateaus as approximately $0.67$ and $0.33$ in the plotted $10^{-38}\mathrm{c}{\mathrm{m}}^2/GeV$ units. With $R={\sigma }_{\bar{\nu }}/{\sigma }_{\nu }\simeq 1/2$ and $r=\bar{Q}/Q$,Using the literal readings $0.33/0.67$ gives $r=0.1905$, so twenty percent is an appropriate graphical estimate, not a precision fit. A purely valence target would have $R=1/3$, while a fully symmetric sea would give $R=1$.

The neutrino-antineutrino DIS sea momentum ratio is a ratio of momentum fractions in the effective flavour combinations, not an unweighted count of antiquarks. The cross-section integral contains $xq(x)$: it cannot determine $\int \bar{q}/\int q$ without additional assumptions about their shapes. A distribution growing at small $x$ can have a large number integral while its momentum integral remains finite.

The dominant semileptonic decay is $t\to b{W}^{+}\to b{\ell }^{+}{\nu }_{\ell }$. Replacing $b$ by $s$ or $d$ weights the amplitude by $V_{ts}$ or $V_{td}$ rather than $V_{tb}$. Neglecting the down-type masses,with the source's rounded matrix entries. For the $W$, three leptonic widths and two open up-type quark rows with three colours give nine common width units. Thus, ignoring QCD and final-mass corrections,Summing all three flavours gives approximately one third; summing only electron and muon modes gives approximately $22.2\%$. These are per-top branching ratios, not the separate branching ratio of a specified two-top event topology.

An isolated high-transverse-momentum charged lepton, missing transverse momentum from the neutrino, and a tagged bottom jet form a much cleaner signature than an entirely hadronic final state amid QCD jets. Electrons and muons are the easiest charged-lepton flavours to identify cleanly. Tau decays add neutrinos and often hadrons, making identification and reconstruction more difficult. The figure contains the quark/lepton-level charged-current diagram as well as the rare-loop examples used below.

The weak charged current couples an up-type quark to the rotated down field $d_{iL}^{\prime }={\sum }_j{V}_{ij}{d}_{jL}$. A neutral current is flavour universal within equal-charge quarks. Rotating it into the mass basis therefore givesThe off-diagonal terms vanish by CKM unitarity, so no tree-level flavour-changing neutral current is generated. In the two-generation description, the orthogonal strange-charm partner completes the rotation whose single down-strange combination alone would have produced an off-diagonal neutral coupling. The complete multiplet supplies the cancellation.

At loop level, a flavour-changing amplitude has the structureIf all internal masses were equal, $F$ would be common and ${\sum }_i{V}_{ai}{V}_{bi}^{\ast }=0$. Subtracting an arbitrary common value makes the suppression explicit:This is the Glashow-Iliopoulos-Maiani mechanism. Unequal masses leave small loop-level flavour-changing neutral currents, so the statement that they are never observed would be too strong: the mechanism suppresses them and forbids them at tree level. The cancellation requires complex CKM products; the rounded table of absolute values alone must not be summed as if every phase were positive.

The literal $t\to qZ\gamma$ final state has a three-vertex loop using the Standard Model quartic vertex $W^{+}{W}^{-}Z\gamma$. The two other vertices are $t\to d_i{W}^{+}$ and $d_i{W}^{+}\to q$, with $q=u,c$. The internal down-type quark joins those two vertices, while two charged-vector propagators join them through the quartic vertex emitting both neutral bosons. The bottom panel of the figure shows this topology, satisfying the three-vertex hint without dropping the photon. Its additional electromagnetic coupling is contained in the four-leg vertex; counting vertices alone does not determine coupling order.

The companion single-$Z$ penguin diagram illustrates $t\to qZ$: it instead attaches the neutral boson to the internal quark line and has three cubic vertices. These are different final states. A complete gauge-invariant amplitude requires all companion attachments; for $qZ\gamma$ these include photon radiation from charged lines as well as the quartic graph.

Both flavour sums have the formwhere the loop function also depends on the final-state kinematics. After cancellation of a mass-independent part by GIM mechanism, the bottom-quark mass gives the largest surviving contribution: $m_b^2\gg m_s^2,m_d^2$, while $|V_{tb}{V}_{cb}|\simeq 0.042$ is not unusually small compared with the strange combination. The larger $|V_{cb}|$ than $|V_{ub}|$ then favours final charm. Schematically,The same flavour preference applies to the separate two-body modes $t\to cZ$ and $t\to c\gamma$. This estimates a relative flavour preference, not an absolute branching ratio. Loop factors and mass-difference suppression remain, so these channels are fully compatible with GIM. A resolved extra photon also introduces another electromagnetic coupling and three-body phase space; an experimental $Z\gamma$ definition needs photon-energy and isolation cuts to distinguish hard radiation from the soft limit.

For a $t\to cZ$ search, the cleanest neutral-boson reconstruction is $Z\to e^{+}{e}^{-}$ or ${\mu }^{+}{\mu }^{-}$: require an isolated opposite-sign same-flavour pair near the $Z$ mass and a jet forming the top mass with that pair. In top-pair events, tagging the other top through its ordinary $bW$ decay provides a bottom jet and, if its $W$ is leptonic, another isolated lepton and missing transverse momentum. Kinematic reconstruction and heavy-flavour information help reject vector-boson-plus-jets and other backgrounds. Hadronic or invisible $Z$ decays have higher rate but less distinctive backgrounds or incomplete mass reconstruction.

For the literal $t\to cZ\gamma$ channel, additionally require a prompt isolated photon and reconstruct the top with $Z+\gamma +$jet. For the alternative $t\to c\gamma$, use an isolated energetic photon and jet near the top mass, again exploiting the other top as a tag. These are search signatures; no present-day observation claim is needed to interpret the historical question.

Work in the W boson rest frame, taking the spin-quantization axis along its laboratory motion. Neglect charged-lepton and neutrino masses. The weak charged current produces a negative-helicity ${\ell }^{-}$ and a positive-helicity antineutrino. Their momenta are opposite, so both spin Vectors point opposite to the charged lepton's direction $\hat{\mathbf{n}}$; the final total spin projection along $\hat{\mathbf{n}}$ is $-1$.

Let ${\lambda }_W=+1,0,-1$ denote the parent spin projection along $z$. In the given probability amplitude, use $j=1$ and the rotation element $d_{-1,{\lambda }_W}^1(\theta )$. The supplied spin-one rotation elements giveThe transverse states have spin projections $\pm 1$; the longitudinal state has projection zero, not total spin zero. The original diagram below distinguishes momenta from spins and shows the corresponding normalized angular profiles.

With relativistically normalized one-particle states, Fermi's golden rule becomesIn the parent rest frame, the three-momentum delta function fixes ${\mathbf{p}}_2=-{\mathbf{p}}_1$. The energy delta function fixes the common magnitude $p^{\ast }$; its radial Jacobian converts the remaining phase-space factor intoOnly one momentum magnitude remains because the two final momenta are constrained, not independent. Explicitly,For leptonic W boson decay both final masses are small compared with $m_W$, including the tau mass at the precision requested, so $p^{\ast }\simeq m_W/2$.