V-A structure of the weak interaction

In QED the electron talks to the photon through a single, blameless object: the vector current $j^\mu=\bar u(p')\gamma^\mu u(p)$ . It transforms as a genuine four-vector, it conserves parity, and for thirty years everyone assumed the weak interaction would look the same. It does not. In 1957 Wu and her collaborators cooled polarised $^{60}$ Co in a magnetic field and watched its $\beta$ -decay electrons stream out preferentially opposite to the nuclear spin. Since spin is an axial vector (unmoved by a mirror) and momentum is a polar vector (reversed by one), a parity-conserving theory demands equal counts along and against the spin. The counts were not equal. Parity — the symmetry between a process and its mirror image — is broken, and broken as violently as it can be. The repair is dictated by Lorentz invariance. Any current built from two Dirac spinors is a combination of just five bilinear covariants — scalar, pseudoscalar, vector, axial-vector, and tensor — and no others. A spin-1 (vector-boson) exchange singles out the vector and axial-vector pieces, so the most general weak vertex is $j^\mu\propto\bar u(p')(g_V\gamma^\mu+g_A\gamma^\mu\gamma^5)u(p)$ . The vector and axial currents transform oppositely under parity, so squaring the sum produces a cross term $g_Vg_A\,j_V\!\cdot\!j_A$ that flips sign in the mirror — a built-in parity-violating handle. Experiment fixes the mixture with brutal economy: $g_V=-g_A$ . The vertex collapses to $\gamma^\mu(1-\gamma^5)$ , the " $V\!-\!A$ " form, which is nothing but twice the left-chiral projector. The weak charged current therefore couples to left-handed particles and right-handed antiparticles only — neutrinos come out purely left-handed, electrons come out polarised by $-v/c$ , and the entire chiral architecture of the Standard Model follows from this one sign.

Key takeaways

QED and QCD use a pure vector current $\bar u\gamma^\mu u$ , which conserves parity; the weak interaction does not.
Lorentz invariance allows exactly five bilinear covariants — scalar, pseudoscalar, vector, axial-vector, tensor (component counts 1, 1, 4, 4, 6) — and spin-1 exchange picks out V and A.
Under parity the vector and axial currents transform oppositely, so the interference term $j_V\!\cdot\!j_A$ flips sign: this is the mechanism of parity violation.
Experiment fixes $g_V=-g_A$ , giving the $V\!-\!A$ current $\bar u\gamma^\mu\tfrac12(1-\gamma^5)u=\bar u_L\gamma^\mu u_L$ — the left-chiral projector.
Consequences are measured and sharp: neutrinos are left-handed ( $h=-1$ ), $\beta$ -decay electrons carry longitudinal polarisation $P=-v/c$ , and the muon Michel parameter is $\rho=3/4$ .
Key idea: mixing a parity-even vector current with a parity-odd axial-vector current in equal and opposite measure ( $g_V=-g_A$ ) produces a $V\!-\!A$ vertex whose interference term flips sign under parity — maximal parity violation, and coupling to left-chiral fermions alone.

The core insight

Mixing a parity-even vector current with a parity-odd axial-vector current in equal and opposite strength ( $g_V=-g_A$ ) builds the $V\!-\!A$ weak vertex $\gamma^\mu(1-\gamma^5)$ , whose vector–axial interference flips sign under parity — the maximal parity violation Wu observed, and the statement that only left-chiral fermions feel the charged weak force.

Where this lives today

$V\!-\!A$ is no longer a phenomenological guess; it is the low-energy shadow of a gauge principle. The Standard Model builds it in from the start by placing left-handed fermions in $SU(2)_L$ weak doublets and right-handed fermions in singlets that the $W$ cannot reach, so the $\gamma^\mu(1-\gamma^5)$ vertex is derived rather than fitted, and the Fermi constant is $G_F/\sqrt2=g^2/8M_W^2$ . Precision now runs in two directions. Downward, experiments hunt for any deviation from pure $V\!-\!A$ : the TWIST measurement of the muon Michel spectrum ( $\rho=0.74977\pm0.00026$ ), the Nab and other $\beta$ -decay correlation experiments measuring the electron–neutrino angular coefficient, and searches for scalar and tensor couplings that new heavy bosons would induce — all currently consistent with $V\!-\!A$ to the sub-percent level, turning a $1958$ hypothesis into a stringent test of physics beyond the Standard Model. Sideways, the axial coupling of the nucleon, $g_A/g_V=-1.2756$ , is where $V\!-\!A$ meets the strong interaction: the vector current is protected at $g_V=1$ by the conserved-vector-current hypothesis, while the axial current is renormalised by roughly $27\%$ (PCAC), and computing that number from first principles is now a benchmark target for lattice QCD, tied directly to the neutron-lifetime puzzle. The left-handed neutrino of $V\!-\!A$ also frames the deepest open question in the sector — whether the right-handed states that the charged current ignores exist at all, and if so whether neutrinos are their own antiparticles.

The physical picture

Wu's experiment is the whole story compressed into one correlation. A nucleus is prepared with its spin pointing up; its decay electrons come out preferentially downward, against the spin. Now reflect the apparatus in a mirror. The nuclear spin, an axial vector, is unchanged by the reflection, but every momentum reverses — so the mirror-world nucleus emits its electrons upward, along the spin. Nature and its mirror image disagree. The pseudoscalar correlation $\langle\mathbf J\rangle\!\cdot\!\mathbf p_e$ is nonzero, and a nonzero pseudoscalar is the signature of parity violation. The first figure sets the real world beside its forbidden reflection.

Figure

Wu's cobalt-60 experiment as a parity mirror. The nuclear spin J (an axial vector) is unchanged by reflection, but electron momenta reverse, so the intensity I(θ) ∝ 1 − (v/c)cosθ becomes 1 + (v/c)cosθ in the mirror. Nature emits electrons opposite the spin, the mirror world along it — the two differ, so parity is violated

Why does the vertex have to be $V\!-\!A$ ? Because Lorentz invariance leaves almost no freedom. A current sewn from two spinors is one of only five kinds — scalar, pseudoscalar, vector, axial-vector, tensor — distinguished by how many components they carry and what spin of boson they can exchange. A spin-1 $W$ selects the vector and axial-vector, and their equal-and-opposite mixture is what nature chose. The second figure is the sharpest quantitative fingerprint of that choice: $V\!-\!A$ predicts that every $\beta^-$ electron leaves with its spin pointing backward along its motion, a longitudinal polarisation $P=-v/c$ that climbs to $-1$ as the electron turns relativistic.

Figure

The V−A prediction for β⁻-decay electron longitudinal polarisation P = −v/c = −β, plotted against kinetic energy T (electron mass 0.511 MeV). A 0.5 MeV electron is 86% left-handed, a 1 MeV electron 94%, and P → −1 as T grows — the direct, measured signature that the weak current couples to left-chiral fermions

The core equations

The starting point is a counting fact about the Dirac algebra: the $4\times4$ complex matrices sandwiched between two spinors form a 16-dimensional space, and Lorentz covariance sorts them into exactly five families. These are the bilinear covariants — the only currents a relativistic theory is allowed to write.

\bar\psi\,\Gamma\,\phi,\qquad \Gamma\in\{\,\mathbb{1},\ \gamma^5,\ \gamma^\mu,\ \gamma^\mu\gamma^5,\ \sigma^{\mu\nu}\,\},\qquad \sigma^{\mu\nu}=\tfrac{i}{2}[\gamma^\mu,\gamma^\nu] \qquad (1)

where: $\mathbb{1}$ gives the scalar $\bar\psi\phi$ (1 component); $\gamma^5$ the pseudoscalar (1); $\gamma^\mu$ the vector (4); $\gamma^\mu\gamma^5$ the axial-vector (4); $\sigma^{\mu\nu}$ the antisymmetric tensor (6). The counts sum to $1+1+4+4+6=16$ , the dimension of the algebra. Read it as: there are no other Lorentz-covariant fermion currents — any weak vertex is a combination of these five and nothing else. The number of components fixes the spin of the boson that can be exchanged: scalar and pseudoscalar go with a spin-0 mediator, vector and axial-vector with spin-1, tensor with spin-2. A spin-1 mediator (the $W$ ) therefore restricts the weak vertex to the vector and axial-vector pieces, giving the most general Lorentz-invariant charged-current form.

j^\mu=\bar u(p')\big(g_V\gamma^\mu+g_A\gamma^\mu\gamma^5\big)u(p)=g_V\,j^\mu_V+g_A\,j^\mu_A \qquad (2)

where: $g_V,g_A$ — real vector and axial couplings; $j^\mu_V=\bar u\gamma^\mu u$ ; $j^\mu_A=\bar u\gamma^\mu\gamma^5 u$ . QED is the special case $g_A=0$ (pure vector, parity-conserving). The reading: two numbers $g_V,g_A$ parametrise every possible spin-1 weak vertex, and experiment must pin their ratio. The two currents behave oppositely in a mirror, and this is the crux. Under parity the vector current transforms as a proper (polar) four-vector while the axial current picks up an extra sign, so their scalar product — the interference term — is odd.

j^\mu_V\xrightarrow{\,P\,}(+j^0_V,\,-\vec\jmath_V),\quad j^\mu_A\xrightarrow{\,P\,}(-j^0_A,\,+\vec\jmath_A),\quad j_V\!\cdot\!j_A\xrightarrow{\,P\,}-\,j_V\!\cdot\!j_A \qquad (3)

where: the time components carry the labelling sign, the space components the opposite; $j_V\!\cdot\!j_V$ and $j_A\!\cdot\!j_A$ are each parity-invariant, but the cross term $j_V\!\cdot\!j_A$ flips sign. Read it as: any observable built from the interference of vector and axial currents is a pseudoscalar, and a nonzero pseudoscalar means parity is violated. Experiment does not leave the mixture free — it demands the maximal choice $g_V=-g_A$ . The vertex then collapses onto the left-chiral projector, and the vector and axial pieces enter with equal magnitude.

j^\mu_{\mathrm{CC}}=\bar u(p')\,\gamma^\mu\tfrac12(1-\gamma^5)\,u(p)=\bar u_L(p')\,\gamma^\mu u_L(p),\qquad g_V=-g_A \qquad (4)

where: $\tfrac12(1-\gamma^5)=P_L$ is the left-handed chiral projector; $u_L=P_Lu$ . Because $\gamma^\mu(1-\gamma^5)=2\gamma^\mu P_L$ , the current keeps only the left-chiral parts of both fields. The reading: the charged weak current couples to left-handed particles and right-handed antiparticles, and to nothing else. At energies far below the $W$ mass the $W$ propagator shrinks to a point and the interaction becomes Fermi's effective current–current contact term — the form in which $V\!-\!A$ was originally written and is still used for low-energy decays.

\mathcal{L}_{\mathrm{CC}}=-\frac{G_F}{\sqrt2}\big[\bar\psi_1\gamma^\mu(1-\gamma^5)\psi_2\big]\big[\bar\psi_3\gamma_\mu(1-\gamma^5)\psi_4\big],\qquad \frac{G_F}{(\hbar c)^3}=1.1664\times10^{-5}\,\mathrm{GeV}^{-2} \qquad (5)

where: $G_F$ — the Fermi constant, setting the strength of every low-energy weak process; each bracket is a $V\!-\!A$ current. This is $g_V=-g_A$ enforced at both vertices. Read it as: muon decay, $\beta$ -decay, and pion decay are all one interaction with one coupling, written in the $V\!-\!A$ form.

Derivation AThe five bilinear covariants and how parity sorts them

The goal is to show that a relativistic theory can build only five kinds of current from a pair of Dirac spinors, to count the components of each, and to work out how each transforms under a mirror reflection — because it is precisely those parity signatures that tell us which currents can conspire to violate parity. We start from the dimension of the Dirac matrix algebra, group its basis into Lorentz-covariant families, and read off the spin of the boson each can exchange, ending with the parity table that the rest of the article depends on.

Step 1. Count the algebra. The gamma matrices are $4\times4$ , so the space of matrices $\Gamma$ that can sit inside $\bar\psi\Gamma\phi$ is 16-dimensional. A complete, Lorentz-covariant basis is $\{\mathbb{1};\ \gamma^5;\ \gamma^\mu;\ \gamma^\mu\gamma^5;\ \sigma^{\mu\nu}\}$ with dimensions $1+1+4+4+6=16$ . Why: any $\Gamma$ whatsoever is a linear combination of these sixteen, so there are no currents beyond the five families — the classification is exhaustive, not illustrative.

Step 2. Assign Lorentz type. Under a Lorentz transformation $\bar\psi\phi$ is invariant (scalar, S); $\bar\psi\gamma^5\phi$ is invariant up to the sign of a parity flip (pseudoscalar, P); $\bar\psi\gamma^\mu\phi$ carries one vector index (vector, V); $\bar\psi\gamma^\mu\gamma^5\phi$ also carries one index but with a relative parity sign (axial-vector, A); $\bar\psi\sigma^{\mu\nu}\phi$ carries two antisymmetric indices (tensor, T). Why: the index structure is what a mediating field must match, tying each covariant to a definite boson spin.

Step 3. Read off the mediator spin. A spin-0 field couples to S or P (no free index), a spin-1 field to V or A (one index), a spin-2 field to T (two indices). The observed charged-current mediator is the spin-1 $W$ , so only V and A survive as candidates. Why: this is the single physical input that reduces five families to two — everything downstream is about the V–A mixture alone.

Step 4. Tabulate parity. Using $\psi\to\gamma^0\psi$ under $P$ : $\bar\psi\phi\to+\bar\psi\phi$ ; $\bar\psi\gamma^5\phi\to-\bar\psi\gamma^5\phi$ ; the vector current transforms as $(+,-)$ in its time and space components, the axial as $(-,+)$ — opposite. Why: this opposite behaviour is the whole engine of parity violation; a product of a V and an A current inherits both signs and comes out odd, which the next derivation exploits.

Go deeper · Derivation B · 1 minWhy the V−A interference is what violates parity

Derivation BWhy the V−A interference is what violates parity

The aim is to prove that a vertex mixing vector and axial currents violates parity only through their interference, that either current used alone conserves it, and that the violation is largest exactly when the two couplings are equal in magnitude — the $|g_V|=|g_A|$ that nature chose. We take the general current of equation (2), form the parity-relevant contraction that enters any rate, expand it, and apply the parity signatures of the previous derivation term by term.

Step 1. Form the contraction. A transition rate is built from a current dotted into another (or its conjugate); schematically the parity-sensitive scalar is $j\!\cdot\!j=(g_Vj_V+g_Aj_A)\!\cdot\!(g_Vj_V+g_Aj_A)=g_V^2\,j_V\!\cdot\!j_V+g_A^2\,j_A\!\cdot\!j_A+2g_Vg_A\,j_V\!\cdot\!j_A$ . Why: expanding the square exposes three pieces whose parity we can test independently — two "diagonal" and one "cross".

Step 2. Apply parity to the diagonal terms. From equation (3), $j_V\!\cdot\!j_V\to+j_V\!\cdot\!j_V$ and $j_A\!\cdot\!j_A\to+j_A\!\cdot\!j_A$ : both are invariant. Why: a vector dotted into itself, or an axial into itself, has its two sign flips cancel — so a pure-V theory (QED) or a pure-A theory conserves parity outright.

Step 3. Apply parity to the cross term. The mixed product picks up a single net sign, $j_V\!\cdot\!j_A\to-\,j_V\!\cdot\!j_A$ . Hence the whole contraction is not parity-invariant precisely because of the $2g_Vg_A\,j_V\!\cdot\!j_A$ piece. Why: this is the term that generates pseudoscalar observables like $\langle\mathbf J\rangle\!\cdot\!\mathbf p_e$ — a mirror-asymmetric emission, exactly Wu's result.

Step 4. Maximise. The parity-violating cross term scales as $g_Vg_A$ relative to the diagonal $g_V^2+g_A^2$ ; the asymmetry $2g_Vg_A/(g_V^2+g_A^2)$ reaches its extreme value $\mp1$ when $|g_V|=|g_A|$ . The weak interaction sits at that extreme with $g_V=-g_A$ . Why: "maximal parity violation" is not rhetoric — it is the statement that the V and A couplings are equal in size, so the interference is as large as algebra permits.

Go deeper · Derivation C · 1 minV−A is the left-chiral projector

Derivation CV−A is the left-chiral projector

The target is to show that the choice $g_V=-g_A$ turns the weak vertex into a projector onto left-handed chirality, so that the charged weak current couples to left-chiral particles and right-chiral antiparticles alone — and to extract the two headline consequences, that the massless neutrino is purely left-handed and that a massive electron couples with its "wrong" helicity suppressed by $(1-\beta)/2$ . We substitute the maximal couplings, rewrite with chiral projectors, and then track a definite-helicity spinor.

Step 1. Substitute $g_V=1,\ g_A=-1$ . The bracket becomes $\gamma^\mu-\gamma^\mu\gamma^5=\gamma^\mu(1-\gamma^5)=2\gamma^\mu P_L$ with $P_L=\tfrac12(1-\gamma^5)$ . Why: the equal-and-opposite mixture is not one current minus another in some loose sense — it is algebraically twice the left-handed projector, the cleanest possible object.

Step 2. Move the projector onto both fields. Using $P_R\gamma^\mu=\gamma^\mu P_L$ and the barred identity $\overline{P_L\psi}=\bar\psi P_R$ , one finds $\bar u\,\gamma^\mu P_L\,u=\bar u\,P_R\gamma^\mu P_L\,u=\overline{(P_Lu)}\,\gamma^\mu\,(P_Lu)=\bar u_L\gamma^\mu u_L$ . Why: the single $\gamma^\mu$ shuffles a right-projector into a left-projector, so the current is left-chiral on entry and on exit — the weak boson never touches a right-chiral field.

Step 3. Read the massless consequence. For a massless fermion chirality equals helicity, so a left-chiral neutrino is a left-helicity ( $h=-1$ ) neutrino; the right-handed neutrino, carrying no left-chiral part, is sterile to the charged current. Antiparticles flip: the antineutrino is right-handed. Why: this is why only left-handed neutrinos and right-handed antineutrinos have ever been seen in weak decays — the vertex forbids the others.

Step 4. Restore mass. A definite-helicity massive electron is a mix of chiralities; its coupling to the left-chiral current carries the "wrong-helicity" weight $(1-\beta)/2$ for the right-helicity piece, vanishing as $\beta\to1$ . The observable shadow is the longitudinal polarisation $P=-\beta=-v/c$ of $\beta^-$ electrons. Why: chirality, not helicity, is what the vertex selects, so a slow electron is not perfectly polarised — the polarisation only reaches $-1$ in the relativistic limit, which is exactly the curve in figure 2.

Limiting cases

Case	Result	Meaning
Pure vector ( $g_A=0$ )	parity conserved	the QED/QCD case — no V·A interference, mirror-symmetric
Pure axial ( $g_V=0$ )	parity conserved	an axial-only theory is also mirror-symmetric on its own
Maximal mix ( $g_V=-g_A$ )	maximal parity violation	equal V and A; vertex $=\gamma^\mu(1-\gamma^5)$ , couples left-chiral only
Massless fermion	chirality $=$ helicity	neutrino purely left-handed, $h=-1$ ; antineutrino right-handed
Massive, $\beta\to1$	wrong-helicity weight $(1-\beta)/2\to0$	$\beta$ -electron polarisation $P=-\beta\to-1$
Nucleon axial current	$g_A/g_V\simeq-1.27$ , not $-1$	strong-interaction renormalisation (PCAC); vector part protected at $g_V=1$ by CVC
Energy $\ll M_W$	Fermi contact $G_F/\sqrt2$	$V\!-\!A$ four-fermion interaction, the low-energy face of $W$ exchange

Common trapReading "left-handed" as helicity instead of chirality

The slogan "the weak force couples only to left-handed particles" is right, but the word "left-handed" means chirality — the eigenvalue of $\gamma^5$ — not helicity, the spin projected on the momentum. They coincide only for a massless particle. A massive right-helicity electron is not invisible to the weak interaction: it still carries a left-chiral component, and it couples with amplitude-squared weight $(1-\beta)/2$ — fully $7\%$ for a $1$ MeV electron. So the charged current does act on "wrong-helicity" fermions, just with a mass-suppressed strength that vanishes as $\beta\to1$ . A second, related slip is to call the violation "small": it is not. Because $|g_V|=|g_A|$ , parity violation is maximal — the interference term is as large as the algebra of equation (3) allows, and there is no parity-conserving remainder hiding underneath. Say "chirality" when you mean the vertex projector $P_L$ , and reserve "helicity" for the spin a polarimeter actually measures; conflating them is the single most common error in reading $V\!-\!A$ .

Worked examples

1 · Longitudinal polarisation of a β-decay electron (fully worked)

The $V\!-\!A$ interaction predicts that $\beta^-$ electrons emerge longitudinally polarised with $P=-v/c$ . Find $P$ for an electron of kinetic energy $T=1.00$ MeV ( $m_e=0.511$ MeV), and state what fraction of it couples with the "wrong" (right-) chirality.

Step 1. Total energy and momentum. $E=T+m_e=1.00+0.511=1.511$ MeV, and $p=\sqrt{E^2-m_e^2}=\sqrt{1.511^2-0.511^2}=\sqrt{2.283-0.261}=\sqrt{2.022}=\mathbf{1.422\ MeV}$ .

Step 2. Speed and polarisation. $\beta=p/E=1.422/1.511=0.941$ , so the $V\!-\!A$ prediction is $P=-\beta=\mathbf{-0.941}$ . The electron is $94\%$ left-handed: its spin points, on average, backward along its flight.

Step 3. The wrong-chirality fraction. A right-helicity electron couples through its left-chiral admixture with weight $(1-\beta)/2=(1-0.941)/2=\mathbf{0.029}$ — under $3\%$ . So even a modest $1$ MeV electron is close to a pure left-chiral state, and the approximation "electron $=$ left-handed" is already good to a few percent. A 1 MeV β-electron has $P=-0.941$ , with only $2.9\%$ wrong-chirality coupling; the polarisation rises toward $-1$ as $T$ grows (figure 2).

2 · The Michel parameter as a test of pure V−A

Muon decay $\mu^-\to e^-\bar\nu_e\nu_\mu$ is described by a general four-fermion interaction. How many independent couplings does the most general Lorentz-invariant contact interaction have, what does pure $V\!-\!A$ predict for the electron energy spectrum, and how well is it confirmed?

Step 1. Count the couplings. The most general four-fermion interaction is a sum over the five bilinear covariants $S,P,V,A,T$ at each current, giving up to ten complex coupling constants. Pure $V\!-\!A$ sets all of them to zero except a single vector–axial combination with $g_V=-g_A$ . This is a strong, falsifiable claim: it forbids any scalar or tensor admixture.

Step 2. Predict the spectrum shape. The shape of the outgoing electron energy spectrum is summarised by the dimensionless Michel parameter $\rho$ . Pure $V\!-\!A$ predicts the sharp value $\rho=\tfrac34=\mathbf{0.750}$ ; a scalar or tensor contamination would pull $\rho$ away from $3/4$ . The number $3/4$ falls straight out of the $(1-\gamma^5)$ structure with no free parameters.

Step 3. Compare with experiment. The TWIST experiment measured $\rho=0.74977\pm0.00026$ , agreeing with $3/4$ to the fourth decimal and bounding any non- $V\!-\!A$ coupling to the sub-percent level. The measured $\rho=0.74977$ versus the $V\!-\!A$ prediction $0.750$ is one of the cleanest confirmations that the weak charged current is purely $V\!-\!A$ .

Physics of the portrait

The $V\!-\!A$ current was written down in a rush of 1957–58, in the immediate aftershock of parity's fall, by two pairs of physicists who arrived at the same left-handed vertex from different directions.

Origins · 1 minGeorge Sudarshan and how this was discovered

George Sudarshan

1931–2018 · the V−A theory of the weak interaction

Ennackal Chandy George Sudarshan was an Indian-American theorist whose fingerprints are on half a dozen corners of twentieth-century physics — the diagonal representation of quantum coherence that bears his name, the quantum Zeno effect, tachyon kinematics — but his first great result, arrived at as a graduate student with Robert Marshak at Rochester, was the universal $V\!-\!A$ form of the weak interaction. In the confusion after Wu's 1957 experiment the data seemed to favour scalar and tensor couplings, a reading that Sudarshan and Marshak argued was contaminated by mismeasured recoil experiments; against the grain of the community they proposed in 1957 that the charged weak current is purely $\gamma^\mu(1-\gamma^5)$ , left-chiral and universal across leptons and hadrons. They presented it at the Padua–Venice conference that autumn, but their full paper appeared only in 1958, by which time Feynman and Gell-Mann had published the same $V\!-\!A$ interaction in a more celebrated paper — the two independent routes converged on an identical vertex, and the priority remained a sore point for decades. The physical content was revolutionary: it unified $\beta$ -decay, muon decay, and pion decay under one coupling $G_F$ , it made the neutrino a two-component left-handed Weyl fermion, and it wrote maximal parity violation into the fabric of the weak force. Everything the modern electroweak theory does with $SU(2)_L$ doublets is the gauge-theoretic completion of the left-chiral current Sudarshan and Marshak first insisted on. Sudarshan was nominated for the Nobel Prize several times and never received one — a fact many colleagues, including the $V\!-\!A$ story, regard as among the more conspicuous omissions in the prize's history.

History

1956

— Lee and Yang argue parity conservation is untested in weak decays

1957

— Wu, Ambler and collaborators observe maximal parity violation in ⁶⁰Co

1957

— Sudarshan and Marshak propose the universal V−A current at Padua–Venice

1958

— Feynman and Gell-Mann publish the V−A theory of the weak interaction

1958

— Goldhaber, Grodzins and Sunyar measure the neutrino helicity as −1

Going further

Texts. Thomson, Modern Particle Physics, §11.2–§11.3 and Table 11.2 (parity violation and the $V\!-\!A$ structure; the source this article grounds) · Halzen & Martin, Quarks and Leptons, §12.1–§12.4 (bilinear covariants and the charged current) · Griffiths, Introduction to Elementary Particles, 2nd ed., §10.4–§10.7 · Peskin & Schroeder, An Introduction to QFT, §3.4 (bilinears) and §20.2 · Schwartz, QFT and the Standard Model, Ch. 29 (weak interactions). The key checkpoints of this article are: the component count $1+1+4+4+6=16$ ; the opposite parity of the vector and axial currents and the sign flip of $j_V\!\cdot\!j_A$ ; the identities $\gamma^\mu(1-\gamma^5)=2\gamma^\mu P_L$ and $\bar u\gamma^\mu P_Lu=\bar u_L\gamma^\mu u_L$ ; the electron polarisation $P=-\beta=-0.941$ at $T=1$ MeV with wrong-chirality weight $(1-\beta)/2=0.029$ ; and the $V\!-\!A$ Michel prediction $\rho=3/4$ against the measured value $0.74977$ .

Prerequisites. The Dirac Adjoint and Bilinear Covariants · Parity Violation · Gamma Matrices and the Clifford Algebra. Leads to. The Chiral Structure of the Weak Interaction · The Weak Charged Current · Helicity in Pion Decay.

Ask the tutor: Ask me to expand $(g_V\gamma^\mu+g_A\gamma^\mu\gamma^5)$ into its parity pieces line by line, or to compute the $\beta$ -electron polarisation $-v/c$ and wrong-chirality fraction for any decay energy you name.

Why are there exactly five bilinear covariants and no more?

Because the object $\Gamma$ between two Dirac spinors is a $4\times4$ complex matrix, and that space is only sixteen-dimensional. A complete Lorentz-covariant basis of those sixteen degrees of freedom is $\mathbb{1}$ (scalar, 1), $\gamma^5$ (pseudoscalar, 1), $\gamma^\mu$ (vector, 4), $\gamma^\mu\gamma^5$ (axial-vector, 4), and $\sigma^{\mu\nu}$ (tensor, 6), summing to sixteen. Any other product of gamma matrices — say $\gamma^\mu\gamma^\nu\gamma^\rho$ — reduces, via the Clifford algebra $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}$ , to a combination already on the list. So "five" is not a convention or a truncation; it is the exhaustive classification of relativistic currents, and it is why the weak vertex has nowhere to hide beyond S, P, V, A, T.

Is it "V minus A" or "V and A" — what does the minus sign mean?

Both descriptions are correct and they refer to the same vertex. "V and A" means the current contains both a vector piece $g_V\gamma^\mu$ and an axial piece $g_A\gamma^\mu\gamma^5$ ; "V minus A" specifies the particular ratio nature chose, $g_V=-g_A$ , so that the two enter with equal magnitude and opposite sign. Written out, $g_V\gamma^\mu+g_A\gamma^\mu\gamma^5=\gamma^\mu-\gamma^\mu\gamma^5=\gamma^\mu(1-\gamma^5)$ — literally "vector minus axial". The minus sign is physically loaded: it is what makes the combination equal to twice the left-handed projector $P_L$ , and hence what makes the weak force couple to left-chiral fermions rather than right. Flip the sign to $\gamma^\mu(1+\gamma^5)$ and you would have a right-handed "V+A" world, which is not the one we live in.

Does V−A mean the weak force never acts on right-handed electrons?

Not quite — the precise statement is about chirality, not helicity. The vertex projects onto left chirality, and for a massless particle that is the same as left helicity, so a right-helicity massless fermion would indeed be untouched. But a real electron has mass, and a right-helicity massive electron still contains a left-chiral component whose weight is $(1-\beta)/2$ . So the weak interaction does couple to it, only with a strength suppressed by how relativistic it is — $7\%$ at $1$ MeV, falling toward zero at high energy. The right-handed neutrino is the genuine exception: being (nearly) massless and carrying essentially no left-chiral part, it is sterile to the charged current, which is why it has never been observed in a weak decay.

Why is the neutron's g_A/g_V about −1.27 rather than exactly −1?

At the level of fundamental fields the leptonic weak vertex is exactly $\gamma^\mu(1-\gamma^5)$ , so $g_A/g_V=-1$ precisely. The neutron, however, is not a fundamental field — it is a bound state of quarks swimming in gluons, and the strong interaction dresses its currents. The vector current is protected: the conserved-vector-current hypothesis (CVC) ties it to the conserved electromagnetic current, so $g_V=1$ survives the strong dressing intact. The axial current is only partially conserved (PCAC), and the strong interaction renormalises it upward by about $27\%$ , giving the measured $g_A/g_V=-1.2756$ . That single number encodes a great deal of low-energy QCD, which is why computing it from the lattice is a serious ongoing effort and why it feeds directly into the neutron lifetime.

Is V−A exact, or just a low-energy approximation?

The four-fermion contact form of equation (5) is an approximation — it is what $W$ exchange looks like when the momentum transfer is far below $M_W\approx80$ GeV, so the $W$ propagator can be shrunk to a point and its strength packaged into $G_F$ . At energies approaching $M_W$ this contact picture breaks down and you must use the full propagator. But the chiral structure — that the current is $\gamma^\mu(1-\gamma^5)$ , coupling to left-chiral fermions — is not an approximation at all; it is exact within the Standard Model, built in by the $SU(2)_L$ gauge symmetry. So " $V\!-\!A$ " as a statement about handedness is fundamental, while " $V\!-\!A$ " as a pointlike four-fermion interaction is the tree-level low-energy limit.

How did experiment decide on V−A rather than scalar or tensor couplings?

This was genuinely contested in 1957. The early electron–neutrino angular-correlation data in $\beta$ -decay actually favoured scalar and tensor couplings, and it took Sudarshan and Marshak's insistence that those experiments were flawed — later vindicated when the measurements were redone — to make the case for $V\!-\!A$ . Three clean observations settled it: the electron longitudinal polarisation came out at $-v/c$ (a V/A signature, not S/T, which predict $+v/c$ ); the Goldhaber measurement pinned the neutrino helicity at $-1$ ; and the muon Michel spectrum matched $\rho=3/4$ . Each of these is what $V\!-\!A$ predicts and what a scalar or tensor interaction does not, and together they excluded the alternatives. Modern precision decays continue this program, now as a search for tiny non- $V\!-\!A$ pieces that new physics might add.