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Chirality distinguishes the two inequivalent Weyl spinor representations of the proper Lorentz group. For a Dirac spinor, define ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ and the chiral projectorsThey satisfy $P_L{P}_R=0$ and $P_L+P_R=I$. Proper Lorentz transformations preserve each chiral representation, whereas parity exchanges them. Helicity instead measures the projection of spin angular momentum along momentum. For a massive particle it can change under a boost that overtakes the particle, so it must not be identified with chirality. For massless positive-energy particles the left-chiral field annihilates negative-helicity particles and creates positive-helicity antiparticles.

The Standard Model is a chiral gauge theory: left- and right-chiral fields have different gauge group representations. In the convention $Q=T_3+Y$, one generation containsThe minimal field content has no right-handed neutrino. A sterile right-handed field is a possible extension, not a required member of that minimal list. The table displays weak isospin, hypercharge and color charge assignments, not assignments of massive-particle helicity.

The weak charged current contains $\bar{\nu }{\gamma }^{\mu }{P}_{L}e{W}_{\mu }^{+}$ and its Hermitian conjugate, with coupling $g/\sqrt{2}$. Only the left-chiral lepton doublet participates, giving parity violation. Both chiralities can couple to the Z boson, but their coefficients differ:Here $T_3$ is that of the left-chiral component. For the electron, it is $-1/2$. The photon and gluon couplings of a charged massive fermion are vectorlike; this does not make the full electroweak interaction vectorlike.

A bare Dirac mass term $m\bar{\psi }\psi =m({\bar{\psi }}_L{\psi }_R+{\bar{\psi }}_R{\psi }_L)$ joins the two chiral representations. Their different electroweak quantum numbers forbid that term before electroweak symmetry breaking. A Yukawa interaction, such as $-y_e{\bar{L}}_{L}H{e}_R+h.c.$, is gauge invariant; the Higgs doublet expectation value then gives $m_e=y_{e}v/\sqrt{2}$. Up-type quarks use the conjugate doublet. Thus chirality explains the weak charge assignments, their parity violation, and why masses arise through the Higgs mechanism. The consistency of these chiral assignments also requires Standard Model anomaly cancellation.

Gauge boson self-interactions follow from noncommuting generators, rather than a separate assumption that the force carriers should attract one another. With $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }^a{T}^a$ and $[T^a,T^b]=i{f}^{abc}{T}^c$, the Yang-Mills field strength isThe last term is required so that $F_{\mu \nu }$ transforms covariantly under the gauge group. Let $G_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a$. Expanding $\mathcal{L}=-F_{\mu \nu }^a{F}^{a\mu \nu }/4$ gives a quadratic kinetic term and the cubic and quartic Yang-Mills self-interactionsThe structure constants and one gauge coupling fix both types of vertex. For an Abelian group the structure constants vanish: the renormalizable pure electromagnetic field action has no elementary three- or four-photon vertex. Photon scattering through charged-particle loops is a different statement, not a contradiction.

In Quantum chromodynamics, the non-Abelian $SU(3{)}_c$ structure produces three- and four-gluon vertices; the gluons themselves carry color charge. These vertices influence multi-jet production and the renormalization group running of the strong coupling. In the electroweak sector the $SU(2{)}_L$ fields mix with the Abelian field after electroweak symmetry breaking. The resulting charged-vector vertices include $W^{+}{W}^{-}\gamma$ and $W^{+}{W}^{-}Z$, with couplings $e$ and $g\cos {\theta }_W$, together with the related four-boson vertices. Neutral-only triple vertices such as $ZZ\gamma$ are absent at tree level in the Standard Model.

The electroweak triple gauge coupling can be measured using $e^{+}{e}^{-}\to W^{+}{W}^{-}$. The neutrino exchange and neutral-vector exchange diagrams interfere: the gauge-related coefficients cancel the most rapidly growing longitudinal-polarization terms. Their relative strengths and angular dependence therefore test the theory more sharply than merely observing a pair of massive vector bosons. The electron-mass-suppressed Higgs contribution completes the corresponding mass-dependent cancellation. Adding arbitrary independent three- and four-vector couplings would generally destroy these relations.

At the Large Electron–Positron Collider, the L3 experiment fitted charged triple-gauge parameters using hadronic and semileptonic W-pair events, complemented by single-W and single-photon channels. These measurements agreed with the predicted self-couplings. Their interpretation uses reconstructed decay products and differential distributions, rather than direct pictures of a three-boson vertex.^[ref]

For the usual colour SU(3) fixing the fourth coordinate of the SU(4) defining representation,Taking the exterior square givesThe identification ${\Lambda }^{2}3\cong \overline{3}$ follows by contracting an antisymmetric tensor with the invariant three-index Levi-Civita symbol. Combining this with c gives all colour multiplets:A colour triplet has the color charge transformation law of a quark, an antitriplet that of an antiquark, and a colour singlet is colour-neutral, as a lepton is. Thus both parent representations permit quark-like and antiquark-like colour sectors; the vector has one neutral sector and the spinor two. These are colour assignments only. They do not determine particle spin, statistics, electric charge, weak quantum numbers or a full generation of the Standard Model. In a real parent field the conjugate colour sectors may describe one complex colour field and its conjugate, rather than independent particle species.

Global group qualification. The printed chain is correct at the Lie algebra level, $\mathfrak{su}(3)\subset \mathfrak{su}(4)\cong \mathfrak{so}(6)\subset \mathfrak{so}(7)$. Globally its simply connected form is $SU(3)\subset SU(4)\cong \mathrm{Spin}(6)\subset \mathrm{Spin}(7)$. The standard map of SU(4) into $SO(7)$ through $6\oplus 1$ has a nontrivial kernel: the concrete central element $-I_4$ acts as $+I_6$ on ${\Lambda }^{2}4$ and trivially on the singlet representation. Its image is $SO(6)\cong SU(4)/\{\pm I_4\}$, rather than an injective copy of SU(4) under this map.

The global form obstruction for the Spin7 spinor is explicit: the central element $-1$ of the Spin group $\mathrm{Spin}(7)$ maps to the identity in $SO(7)$ but acts as $-I_8$ on the eight-dimensional spin representation. Equivalently $-I_4$ acts as $-I$ on both $4$ and $\overline{4}$ in its restriction. Therefore the spinor does not descend to a genuine SO(7) group representation. A theory containing that spinor needs the Spin group global form, or must understand the source's notation as specifying only its Lie algebra. The displayed colour branchings remain valid with this correction.

Let $\epsilon =i{\sigma }_2=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, so ${\epsilon }^{-1}=-\epsilon$. Direct multiplication of the Pauli matrices gives $\epsilon {\sigma }_i^{\ast }\epsilon ={\sigma }_i$, equivalently $\epsilon {\sigma }_i^{\ast }{\epsilon }^{-1}=-{\sigma }_i$. For $U=\exp (i{\theta }_i{\sigma }_i/2)\in SU(2)$ this impliesTherefore ${\varphi }^c=\epsilon {\varphi }^{\ast }$ transforms as the same weak fundamental representation doublet as $\varphi$, while complex conjugation reverses its hypercharge to $-1/2$. This is the Higgs conjugate doublet. The Yukawa coupling product $\bar{L}{\varphi }^c{u}_R$ is an $SU(2)$ singlet, and its hypercharges add toIt is consequently gauge-invariant and can supply an up-quark mass.

In unitary gauge, ${\varphi }^c=((v+H)/\sqrt{2},0{)}^T$. Writing $y_u$ for the full arbitrary coefficient of the Yukawa coupling term, its physical form isA one-generation field rephasing makes the mass positive, and its Higgs boson coupling is $m_u/v$. The printed Yukawa expression has an additional trailing $m$ after $R^{+}$ which is never defined. If it denotes a constant, absorb it into $y_u=f^{+}m$; in the conventional dimensionless Yukawa notation without that factor, $y_u=f^{+}$. It does not affect the gauge-transformation argument.

For three families, use a singular value decomposition of the two mass matrices: $U_{uL}^{†}{M}_u{U}_{uR}=D_u$ and $U_{dL}^{†}{M}_d{U}_{dR}=D_d$, with positive diagonal entries. The original left-handed weak current then becomes a current between mass eigenstates withA complex $3\times 3$ matrix has eighteen real parameters, and $U^{†}U=I$ imposes nine independent real conditions, leaving nine. A generic unitary matrix can be built from three complex plane rotations and three diagonal phases: each plane rotation supplies one angle and one phase. Thus these are three rotation angles and six phases. Rephase each of the six Dirac quark mass eigenfields, applying the same phase to its two chiralities so the diagonal masses stay real. A common phase changes no entry of $V$, leaving five effective rephasings. Removing five of the six phases leaves three angles and one physical phase. This CKM parameter counting assumes generic nondegenerate masses; degeneracies give additional basis freedom.

Exactly massless neutrinos have an arbitrary unitary flavor rotation that changes no mass term. After diagonalizing the charged-lepton mass matrix, choose the neutrino basis with the same left-handed rotation. The apparent lepton charged-current mixing matrix becomes the identity. Thus mixing of exactly massless neutrinos is removable; it is not an analogue of physical CKM mixing.

For the requested CP symmetry calculation, write $j_{ij}^{\mu }={\bar{u}}_i{\gamma }^{\mu }(1-{\gamma }^5)d_j$. The supplied charge conjugation transformations, with the fermion-reordering sign, giveFor $\Gamma ={\gamma }^{\mu }(1-{\gamma }^5)$, use $C({\gamma }^5{)}^T{C}^{-1}={\gamma }^5$ and the given gamma matrix relation to obtainUnder parity, the current acquires the Lorentz parity matrix and $1+{\gamma }^5$ becomes $1-{\gamma }^5$. Under charge conjugation the charged vector also changes sign and becomes its conjugate. The two minus signs cancel, and the parity matrices cancel in the contracted operator. ConsequentlyWith $k=g/(2\sqrt{2})$, the interaction is $-k\sum (V_{ij}{O}_{ij}+V_{ij}^{\ast }{O}_{ij}^{†})$. Thereforewith the transformed density evaluated at $x_P$. This is the CP transformation of a charged quark current.

Physical noninvariance requires a phase that cannot be removed by quark rephasings. The invariant quartetis the Jarlskog invariant, where $s_{ij}=\sin {\theta }_{ij}$ and $c_{ij}=\cos {\theta }_{ij}$ in the standard three-angle parametrization. If $J\ne 0$, no allowed row/column phases can make the matrix real, and the charged-current interaction violates CP symmetry. For example all three angles $\pi /4$ with $\delta =\pi /2$ give $J=1/(8\sqrt{2})\ne 0$.

The printed noninvariance claim is conditional on a nonzero physical phase. Arbitrary Yukawa matrices do not force it: $V=I$, or any real CKM matrix, gives a CP-invariant charged-current interaction. The correct statement is generic CP violation when the irreducible phase is nontrivial, as explained by quark rephasing and CP conservation.

Finally a right-handed neutrino has gauge representation $(1,1{)}_0$. Its Lorentz-invariant same-chirality bilinear with the field obtained by charge conjugation is therefore a gauge singlet and permits a Majorana mass term, conventionallyThe factor $1/2$ is the usual convention for a field paired with itself; it may be absorbed into the printed coefficient. A Higgs Yukawa interaction can independently give a Dirac mass term. The Majorana term violates lepton number by two units but preserves the gauge symmetries. Corresponding terms for the other charged Standard Model fermions carry nonzero electric charge, and quarks also carry nontrivial color charge. Thus this gauge-invariant right-handed neutrino Majorana mass is allowed while those charged-fermion analogues are forbidden. Merely introducing a right-handed neutrino permits mass; vanishing mass coefficients would still leave it massless.

The derivative convention uses hypercharge Y with $Q=T_3+Y$. For each lepton family $l=e,\mu ,\tau$, the left-handed fields form an SU(2) doublet $L_l=({\nu }_l,l{)}_L^T$, with $Y=-1/2$. Their weak isospin values are $T_3=+1/2,-1/2$ and their electric charges are $0,-1$. The right-handed charged lepton $l_R$ is an SU(2) singlet, with $T_3=0$, $Y=-1$ and charge $-1$. The minimal model used here has no right-handed neutrino. A sterile right-handed neutrino, if added as a singlet with $Y=0$, would have no charged weak current interaction. Antiparticles carry the conjugate representations and opposite electric charges.

Expanding ${\bar{L}}_{l}i{\gamma }^{\mu }{D}_{\mu }{L}_l$ supplies $-g{\bar{L}}_l{\gamma }^{\mu }{A}_{\mu }^a{\sigma }_a{L}_l/2$. The off-diagonal entries of $A^a{\sigma }_a$ are $\sqrt{2}{W}^{+}$ and $\sqrt{2}{W}^{-}$. With the chiral projector $P_L=(1-{\gamma }^5)/2$, one has ${\bar{\nu }}_L{\gamma }^{\mu }{l}_L=\bar{\nu }{\gamma }^{\mu }{P}_{L}l$. Thus the two W boson interactions areFor quarks, separate rotations diagonalize the up-type and down-type mass terms. If their left-handed rotations are $U_{uL}$ and $U_{dL}$, the charged interaction contains the CKM matrix $V=U_{uL}^{†}{U}_{dL}$:There is an implicit sum over color charge. This quark mixing permits transitions between different families, rather than just the paired fields in a single doublet.

Let $J^{\mu }={\sum }_l{\bar{\nu }}_l{\gamma }^{\mu }(1-{\gamma }^5)l+J_{\mathrm{had}}^{\mu }$ and $\kappa =g/(2\sqrt{2})$. At momenta much smaller than $m_W$, neglect derivatives in the heavy W boson equation. Its mass and source terms can be completed to a square:This explicitly performs integrating out a charged vector boson. The resulting Fermi interaction isThe omitted terms contain derivatives and are relatively suppressed by momentum squared divided by $m_W^2$.

For the leptonic pion decay, Lorentz covariance restricts a vacuum-to-spin-zero current matrix element to a constant times $p^{\alpha }$. In the rest frame, rotations eliminate the spatial components. Under parity, the temporal vector current is even, whereas the vacuum is even and the pion is odd, so its temporal matrix element equals its negative and vanishes. The temporal axial current is odd and is allowed. This proves the vacuum-to-pseudoscalar current selection rule, rather than merely assuming an axial current. Define the pion decay constant byThen the vector-minus-axial charged weak current has matrix element $-i{f}_{\pi }{V}_{ud}{p}^{\alpha }$, up to an irrelevant overall phase convention. Consequently the scattering amplitude for the decay can be writtenUse $p=k+q$, the momentum-space Dirac equation ${\bar{u}}_e/k=m_e{\bar{u}}_e$, and $/q{v}_{\nu }=0$ for a massless neutrino. Since $/q(1-{\gamma }^5)=(1+{\gamma }^5)/q$, the neutrino contribution vanishes, leavingThe fermion spin sum is, using the supplied gamma matrix trace identities,Here $(k+q{)}^2=m_{\pi }^2$, $k^2=m_e^2$, and $q^2=0$. The mass term in the trace vanishes; anticommuting ${\gamma }^5$ reduces the remaining trace to twice $\mathrm{tr}(/k/q)$. It follows thatUsing two-body decay phase space, with final momentum magnitude $(m_{\pi }^2-m_e^2)/(2m_{\pi })$, gives the leptonic pseudoscalar decay widthThus the requested $m_e^2(m_{\pi }^2-m_e^2)$ factor is already present in the squared amplitude, and the phase space supplies one more threshold factor.

The physical explanation is helicity suppression. In the pion rest frame the two momenta are opposite. The massless antineutrino has positive helicity; zero total angular momentum would require the electron to have positive helicity too. The charged weak current instead creates a left-chiral electron, whose helicity is negative when its mass vanishes. A nonzero electron mass admits the required opposite helicity with an amplitude proportional to $m_e$. The decay amplitude vanishes for $m_e=0$, and the rate is suppressed by $m_e^2$.

Put $s=q^2>0$ and neglect electron and final-quark masses compared with $\sqrt{s}$. First fix a final color charge; its multiplicity will be restored for the inclusive rate. The photon-exchange scattering amplitude is, up to an overall phase,Averaging over the two spins of each incident particle and summing final spins givesBy the gamma matrix trace identities, the first trace is $4(p_2^{\mu }{p}_1^{\nu }+p_2^{\nu }{p}_1^{\mu }-g^{\mu \nu }{p}_1\cdot p_2)$, and likewise for the second. Contracting the two tensors cancels the terms proportional to $(p_1\cdot p_2)(k_1\cdot k_2)$, givingIn the centre-of-mass frame all four energies are $\sqrt{s}/2$. The first pair of scalar products equals $s(1-\cos \theta )/4$ each, and the second pair equals $s(1+\cos \theta )/4$ each. Hence $\overline{|\mathcal{M}{|}^2}=e^4{Q}_q^2(1+{\cos }^2\theta )$.

For clarity, the printed flux hint has a factor-of-four inconsistency: when $\mathcal{F}=4\sqrt{(p_1\cdot p_2{)}^2-m_1^2{m}_2^2}=2s$, the prefactor multiplying the spin-averaged amplitude is $1/\mathcal{F}$, not $1/(4\mathcal{F})$. Alternatively, $1/(4F_0)$ is correct if $F_0=\sqrt{(p_1\cdot p_2{)}^2-m_1^2{m}_2^2}=s/2$. Use this massless two-body scattering flux normalization, keeping the spin average separate.

To obtain the two-body Lorentz-invariant phase space directly, integrate the momentum-conserving spatial delta function, leaving ${\mathbf{k}}_2=-{\mathbf{k}}_1$. The remaining radial integral is $\int dk\delta (\sqrt{s}-2k)=1/2$, so $d{\Phi }_2/d\Omega =1/(32{\pi }^2)$. The relativistic scattering cross-section is consequentlySince $\int d\Omega (1+{\cos }^2\theta )=2\pi {\int }_{-1}^1(1+x^2)dx=16\pi /3$, the integrated electron-positron annihilation into a quark pair result, for this fixed color, isThe first two printed formulas are thus color-stripped. Distinct final colors are orthogonal states and must be summed, producing $N_c=3$. At leading order each accessible quark flavor contributes independently. Away from thresholds and narrow resonances, quark-hadron duality relates the inclusive parton rate to the observed inclusive electron-positron annihilation into hadrons, givingThis formula concerns the photon channel, assumes $\sqrt{s}$ is large compared with the relevant quark masses and the strong-interaction scale, and does not include Z boson exchange.

Beyond leading order, an exactly exclusive pair of colored quarks is not an observable asymptotic state because of confinement and hadronization. Moreover its virtual corrections have infrared divergences; real soft or collinear gluon emission cannot be excluded if a finite rate is wanted. The inclusive sum includes both the corrected two-parton channel and channels such as $q\bar{q}g$, with their degenerate soft and collinear limits. Real and virtual infrared divergences then cancel for an infrared-safe observable. A specified jet observable can also be meaningful if it defines an infrared-safe observable; a bare two-quark final state is not.

A convenient inclusive calculation uses the optical theorem. For the electromagnetic hadronic current $j_{\mu }={\sum }_f{Q}_f{\bar{q}}_f{\gamma }_{\mu }{q}_f$, defineThe hadronic electromagnetic-current spectral density is obtained from its discontinuity. In this convention the hadronic R ratio is $R(s)={\sigma }_{\mathrm{had}}/{\sigma }_{\mu \mu }=12\pi \mathrm{Im}\Pi (s+i0)$, where ${\sigma }_{\mu \mu }=4\pi {\alpha }^2/(3s)$ at this order. Cutting the forward amplitude sums all allowed hadronic intermediate states, so perturbative Quantum chromodynamics applied to this inclusive quantity implements the required sum of real and virtual contributions.

Finally, write $a={\alpha }_s({\mu }^2)$, $C=1.99-0.11n_f$, $L=\ln (s/{\mu }^2)$, and ${\beta }_0=11-2n_f/3$. The given truncated correction isDifferentiate with respect to $t=\ln {\mu }^2$, using the stated one-loop beta function $da/dt=-{\beta }_0{a}^2/(4\pi )$ and $dL/dt=-1$. The derivative of $a/\pi$ cancels the explicit logarithmic derivative at order $a^2$. Keeping the residual derivative of the truncated expression givesThus the principle of minimal sensitivity, applied to this expression with nonzero coupling and ${\beta }_0\ne 0$, selects the one-loop stationary scale for the hadronic annihilation correctionThis is a prescription based on the residual order-$a^3$ scale dependence of an order-$a^2$ approximation. At strictly order $a^2$, the derivative already vanishes for every renormalization scale; the prescription does not assert exact scale dependence of the observable. It is useful only while the selected running coupling remains perturbative. If $a=0$ or ${\beta }_0=0$, the stated derivative imposes no unique scale.

Use metric signature $(+---)$ and average over the four initial spin states, summing over final spins and the $N_c=3$ quark color charge states. The electrons carry no color charge, so their initial color charge average is trivial. A final color charge average would not give the inclusive relativistic cross-section requested.

Photon exchange gives the tree amplitude, up to an irrelevant overall phase,Massless spin sums give $\sum u\bar{u}=/p$ and $\sum v\bar{v}=/p$. ThereforeThe color charge multiplicity is ${\sum }_{a,b}|{\delta }_{ab}{|}^2=N_c$, not $N_c^2$. Each trace equals $4(a^{\mu }{b}^{\nu }+a^{\nu }{b}^{\mu }-g^{\mu \nu }a\cdot b)$. Contracting them, the terms proportional to $(p_1\cdot p_2)(q_1\cdot q_2)$ cancel, leaving $32[(p_1\cdot q_1)(p_2\cdot q_2)+(p_1\cdot q_2)(p_2\cdot q_1)]$. Thus

In the centre-of-momentum frame every energy is $\sqrt{s}/2$. Let $\theta$ be the angle between $p_1$ and $q_2$. ThenConsequently $\overline{|\mathcal{M}{|}^2}=N_c{e}^4{Q}_q^2(1+{\cos }^2\theta )$.

The supplied two-body Lorentz-invariant phase space formula has a factor-of-two error. Deriving the normalization explicitly,so the massless two-body scattering flux normalization isFor example, integrating the spatial Dirac delta in $d{\Phi }_2$ leaves $[4(2\pi {)}^2{]}^{-1}dkd\Omega \delta (\sqrt{s}-2k)$; its radial integral supplies the factor $1/2$. The printed denominator $64{\pi }^2{p}_1\cdot p_2$ would double the relativistic cross-section and cannot reproduce the requested result with the standard scattering amplitude.

The correctly normalized differential and total results for electron-positron annihilation into a quark pair areand hence

At energies where photon exchange and the parton description apply, sum over kinematically accessible quark flavors. The produced quarks undergo hadronization, so the leading inclusive hadronic rate isThis is the hadronic R ratio. Perturbative QCD adds a leading relative correction ${\alpha }_s(s)/\pi$; thresholds, narrow resonances and, near the Z boson scale, electroweak exchange require additional treatment. The calculation is a photon-exchange, massless result from a tree-level Feynman diagram.

Let $s=q^2>0$, neglect $m_e$, and average over the two spin states of each incoming particle. Only the single-photon channel is being considered. The spectral-density symbol ${\rho }_h$ is not defined in the question, so specify its normalization before calculating. A useful photon-field convention isHere $d{\Pi }_X$ is the product of final-particle measures $d^{3}k/[(2\pi {)}^{3}2E]$, including the appropriate sums and symmetry factors. Lorentz invariance and current conservation give the transverse tensor form. In this convention ${\rho }_h$ is the hadronic photon spectral density, with mass dimension $-2$, rather than a density of a stationary time series.

The electron annihilation amplitude can be written $e\bar{v}(p_{+}){\gamma }_{\mu }u(p_{-})⟨X|A^{\mu }|0⟩$, up to an overall phase. The leptonic tensor after the initial spin average isIt obeys $q^{\mu }{L}_{\mu \nu }=0$ and $g^{\mu \nu }{L}_{\mu \nu }=-s$. In the center-of-momentum frame, the flux denominator in the hint is $2s$, since the relative speed is 2 and each beam energy is $\sqrt{s}/2$. Contracting the tensors gives the inclusive relativistic cross-sectionHere $\alpha$ is the fine-structure constant. The $1/4$ in the initial spin average is essential because the hint's particles were spinless.

For the other common convention, define the hadronic electromagnetic current with the coupling omitted, $J_h^{\mu }={\sum }_f{Q}_f{\bar{q}}_f{\gamma }^{\mu }{q}_f$, and write its inclusive tensor asEquivalently, for $i\int d^{4}x{e}^{iqx}⟨0|T{J}_h^{\mu }(x)J_h^{\nu }(0)|0⟩=(q^{\mu }{q}^{\nu }-q^2{g}^{\mu \nu }){\Pi }_J(q^2)$, one has ${\rho }_J=\mathrm{Im}{\Pi }_J/\pi$. At leading order in the electromagnetic interaction, $⟨X|A^{\mu }|0⟩$ is proportional to $e⟨X|J_h^{\mu }|0⟩/s$. ThusIf the symbol ${\rho }_h$ is instead used for this dimensionless hadronic electromagnetic-current spectral density, the last formula applies with ${\rho }_J$ renamed ${\rho }_h$. The normalization must not be silently switched between these formulas.

At $\sqrt{s}$ well above the strong-coupling scale of Quantum chromodynamics, asymptotic freedom makes production over distances of order $1/\sqrt{s}$ perturbative. The electromagnetic current initially creates a quark-antiquark pair. Subsequent strong interactions produce hadrons, but an inclusive sum over all hadronic final states is much less sensitive to this rearrangement than an exclusive channel. This is the regime in which quark-hadron duality motivates a leading parton model calculation, with radiative and power-suppressed corrections. It is not a pointwise theorem at individual resonances or near thresholds; the comparison is most reliable for sufficiently inclusive or suitably averaged high-energy observables. This argument remains restricted to photon exchange, even where additional electroweak channels could also contribute.

For one active massless quark flavor with charge $Q_{f}e$, the tree-level scattering amplitude isThe two gamma-matrix traces, the initial spin average, and the final color charge sum yieldwhere $t=-s(1-\cos \theta )/2$ and $u=-s(1+\cos \theta )/2$. The massless two-body Lorentz-invariant phase space then givesSumming the distinct final flavors, rather than interfering amplitudes for them, givesAccording to the permitted approximation in the paper, active flavors satisfy $m_f^2<s$ and are treated as massless; the others are omitted. This is the stipulated step approximation, not the exact pair-production threshold $s\ge 4m_f^2$.

The hadronic R ratio is $R=N_c\sum Q_f^2$, relative to the massless muon-pair relativistic cross-section $4\pi {\alpha }^2/(3s)$. If $N_u$ and $N_d$ active flavors have charges $2/3$ and $-1/3$, respectively,For $N_c=3$, the usual sets of three, four, five and six active flavors give $R=2,10/3,11/3,5$. In the two spectral conventions the same leading calculation gives

QCD supplies the underlying strong interaction; Skyrmions provide a mesonic effective description of baryons, and quantized multi-Skyrmions can model nuclei. These are related descriptions at different scales, not three identical theories.

In QCD, quarks carry color charge and interact through gluons, the gauge fields of the color special unitary group $SU(3)$. Its Lagrangian density has the formAsymptotic freedom makes short-distance processes accessible through small-coupling expansions, but nuclear scales involve strongly coupled, confined dynamics. Observable hadrons are color singlets. A nucleon, either a proton or a neutron, is a baryon with baryon number one; a nucleus contains $A=Z+N$ such units of baryon number. Directly extracting all nuclear binding energies, spectra and interactions from QCD is difficult, motivating low-energy effective field theories that preserve its symmetries and relevant degrees of freedom.