Chiral structure of the weak interaction

Every other force in the Standard Model is even-handed: gravity, electromagnetism and the strong interaction treat a particle and its mirror image identically. The weak interaction does not. When a $W^\pm$ boson is emitted or absorbed, it reaches into a fermion, keeps only the left-handed part, and throws the rest away. Written out, the charged-current vertex is $\tfrac{g_W}{\sqrt2}\,\bar u(p')\,\tfrac12\gamma^\mu(1-\gamma^5)\,u(p)$ — a vector-minus-axial-vector ( $V-A$ ) coupling. That innocent-looking factor $\tfrac12(1-\gamma^5)$ is the chiral projection operator $P_L$ , and it is the whole story: a built-in filter that admits left-handed chiral particles and right-handed chiral antiparticles, and nothing else. This is the deepest asymmetry known in the microscopic world. Because a mirror reflection turns a left-handed particle into a right-handed one, and the $W$ refuses the right-handed one, the weak interaction and its mirror image are not the same physics. Parity is not just violated — it is violated maximally, because one chirality couples with weight one and the other with weight exactly zero. This article takes the $V-A$ current as its starting point, shows algebraically why it is a left-handed filter, works out what "left-handed" means for a massive particle (chirality, not helicity), and follows the consequences to the number a laboratory measures: the crushing suppression of $\pi\to e\nu$ relative to $\pi\to\mu\nu$ . The gauge-theory reason nature chose $P_L$ is owned by the electroweak-gauge-group article; here we live inside the current and mine its structure.

Key takeaways

The weak charged-current vertex is $-i\tfrac{g_W}{2\sqrt2}\gamma^\mu(1-\gamma^5)$ — a pure $V-A$ (vector-minus-axial-vector) interaction.
The factor $\tfrac12(1-\gamma^5)=P_L$ is a chiral projector: only left-handed chiral particles and right-handed chiral antiparticles couple to the $W$ ; the "wrong" handedness gives a vanishing current.
Parity is violated maximally because the coupling weights are $1$ and $0$ ; the general strength $\propto 2g_Vg_A/(g_V^2+g_A^2)$ peaks exactly at $|g_V|=|g_A|$ .
Chirality is not helicity for a massive particle — they coincide only as $E\gg m$ ; a left-chiral electron carries a right-helicity admixture of probability $\tfrac12(1-\beta)\approx(m/2E)^2$ .
The filter's most famous fingerprint is helicity suppression: $\Gamma(\pi\to e\nu)/\Gamma(\pi\to\mu\nu)=1.28\times10^{-4}$ , small only because the light electron is forced into its wrong helicity.
Key idea: the weak charged current carries its own left-handed filter — the $\tfrac12(1-\gamma^5)$ factor keeps left-handed chiral particles and right-handed chiral antiparticles and discards the rest, and that structural blindness to the "wrong" handedness is parity violation.

The core insight

The weak charged current carries its own left-handed filter: the $\tfrac12(1-\gamma^5)$ factor projects out left-handed chiral particles and right-handed chiral antiparticles, so the $W$ boson is structurally blind to the "wrong" handedness — and that asymmetry is parity violation itself.

Where this lives today

The $\tfrac12(1-\gamma^5)$ filter is now a load-bearing pillar of the Standard Model. It is why the electroweak theory is built from left-handed doublets and right-handed singlets: the $W$ couples to the doublets only, and the whole structure of generations, mixing (CKM and PMNS) and the Higgs mechanism is grafted onto that chiral scaffold. The same maximal parity violation is now a precision tool: Qweak, MOLLER and P2 scatter longitudinally polarized electrons off nuclei and read a tiny parity-violating asymmetry to extract the weak mixing angle $\sin^2\theta_W$ , and atomic parity violation in cesium does the same on a table top. The neutrino's left-handedness underlies the field's deepest open question: if right-handed neutrinos exist they are sterile, and whether the neutrino is Majorana or Dirac is probed by neutrinoless-double-beta-decay searches like LEGEND and nEXO. And at the LHC the polarization of $W$ bosons in top-quark decays checks that the 1958 $V-A$ vertex still holds at the TeV scale.

The physical picture

Think of the $W$ vertex as a turnstile with a handedness sensor. A fermion arriving is first split into its two chiral halves, $P_R u=u_R$ and $P_L u=u_L$ . The turnstile passes $u_L$ through to the $W$ line and slams shut on $u_R$ : the current built from a right-handed particle spinor is not small, it is exactly zero. For antiparticles the sensor reads the opposite way — the same $\tfrac12(1-\gamma^5)$ keeps the right-handed antiparticle and rejects the left-handed one. The first figure is that turnstile, carrying the relation that makes it work.

Figure

The weak vertex as a handedness filter: the factor ½(1−γ⁵) = P_L passes left-chiral particles and blocks right-chiral ones, so the current from a right-handed particle spinor vanishes, ū_R γ^μ P_L u = 0. Allowed: left-handed particles and right-handed antiparticles

The filter would be a bookkeeping curiosity if chirality were unobservable. It is not, because for a fast particle chirality nearly equals helicity — the projection of spin onto momentum — a directly measurable, geometric quantity. And here the mirror bites: parity reflection flips every momentum ( $\mathbf p\to-\mathbf p$ ) but leaves spin untouched, so it flips helicity, turning an allowed left-handed particle into a forbidden right-handed one. The most vivid consequence is a number. In the decay of a spin-0 charged pion, angular-momentum conservation forces the outgoing charged lepton into the helicity the $V-A$ vertex hates; the light electron resists this far more than the heavy muon, so the electron channel is suppressed by four orders of magnitude. The second figure is that experiment.

Figure

Helicity suppression in π⁻ decay. The spin-0 pion decays back-to-back, so the right-handed antineutrino forces the charged lepton into its wrong (right-handed) helicity; the rate scales as m_ℓ², giving Γ(π→eν)/Γ(π→μν) = 1.28×10⁻⁴

The core equations

The whole subject is the algebra of one vertex. Emission or absorption of a $W$ multiplies the fermion line by a coupling, a gamma matrix for the vector index, and the chiral projector; everything downstream — the filter, parity violation, helicity suppression — is extracted from this.

j^\mu_{CC}=\frac{g_W}{\sqrt2}\,\bar u(p')\,\gamma^\mu\tfrac12(1-\gamma^5)\,u(p),\qquad \text{vertex factor}=-i\frac{g_W}{2\sqrt2}\,\gamma^\mu(1-\gamma^5) \qquad (1)

where: $g_W$ — the universal weak coupling; $u,\bar u$ — Dirac spinors for the fermions; $\gamma^\mu$ — vector matrices; $\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3$ — chirality matrix. The piece $\gamma^\mu\cdot 1$ is the vector part $V$ and $\gamma^\mu\gamma^5$ the axial part $A$ ; the relative sign makes this " $V$ minus $A$ ". The projector $\tfrac12(1-\gamma^5)$ is one of a pair of idempotent operators that split every spinor into two chiral halves; their algebra is what forces the current to be one-sided.

P_L=\tfrac12(1-\gamma^5),\quad P_R=\tfrac12(1+\gamma^5),\qquad P_L+P_R=1,\ \ P_L^2=P_L,\ \ P_R^2=P_R,\ \ P_LP_R=0 \qquad (2)

where: $P_L,P_R$ — left- and right-chiral projectors; $\gamma^5$ satisfies $(\gamma^5)^2=1$ , $\{\gamma^5,\gamma^\mu\}=0$ . Being orthogonal and complete, they split any state uniquely as $u=u_L+u_R$ into pieces the $W$ treats completely differently. The one line that turns "there is a projector at the vertex" into "only one chirality couples" is the anticommutation of $\gamma^5$ with $\gamma^\mu$ , which lets the projector slide through the vector index and collide with itself.

\gamma^\mu P_L=P_R\,\gamma^\mu \ \Longrightarrow\ \bar u_R\,\gamma^\mu P_L\,u=0,\qquad j^\mu_{CC}\propto \bar u'_L\,\gamma^\mu\,u_L \qquad (3)

where: $\gamma^\mu P_L=P_R\gamma^\mu$ follows from $\{\gamma^5,\gamma^\mu\}=0$ , and $\bar u_R=\bar u\,P_L$ because $\gamma^5$ anticommutes with $\gamma^0$ . The upshot: the charged current connects a left-chiral incoming state to a left-chiral outgoing state and nothing else. A general parity-violating vertex need not be maximal. An arbitrary mix of vector and axial couplings exposes how the amount of parity violation depends on their ratio.

\Gamma^\mu=\gamma^\mu\,(g_V-g_A\gamma^5),\qquad \mathcal A_{\text{PV}}\ \propto\ \frac{2\,g_V g_A}{g_V^2+g_A^2} \qquad (4)

where: $g_V,g_A$ — vector and axial coupling strengths; $\mathcal A_{\text{PV}}$ — a parity-violating asymmetry built from the parity-odd $V$ – $A$ interference term. It vanishes if either coupling is zero (pure $V$ or pure $A$ conserves parity) and maximizes at $1$ when $|g_V|=|g_A|$ — the $V-A$ point. Chirality is a Lorentz-invariant label; helicity is frame-dependent. They agree only for massless particles, and the mismatch for a massive one is governed by the velocity $\beta=p/E$ .

P(\text{wrong chirality})=\frac{E-p}{2E}=\tfrac12(1-\beta)\ \xrightarrow{\,E\gg m\,}\ \left(\frac{m}{2E}\right)^2 \qquad (5)

where: $\beta=p/E=v/c$ ; the "wrong chirality" is the right-chiral admixture inside a definite-helicity particle. As $E\to\infty$ it $\to0$ and chirality becomes helicity; at rest ( $\beta=0$ ) it is $\tfrac12$ — a particle at rest has no definite chirality at all. Finally, the observable the filter is best known for. Because the charged lepton is forced into its disfavoured helicity, the amplitude is $\propto m_\ell$ and the rate $\propto m_\ell^2$ , so the two leptonic channels compare cleanly with the coupling cancelling.

\frac{\Gamma(\pi^-\to e^-\bar\nu_e)}{\Gamma(\pi^-\to\mu^-\bar\nu_\mu)}=\frac{m_e^2}{m_\mu^2}\left(\frac{m_\pi^2-m_e^2}{m_\pi^2-m_\mu^2}\right)^2=1.28\times10^{-4} \qquad (6)

where: $m_e=0.511$ , $m_\mu=105.7$ , $m_\pi=139.6$ MeV; the first factor is the helicity penalty $m_\ell^2$ and the bracket is the two-body phase space. The tiny result is a direct, quantitative reading of the $\tfrac12(1-\gamma^5)$ structure.

Derivation AThe weak current as a left-handed filter

The goal is to prove, purely algebraically, that the charged-current vertex $\bar u(p')\gamma^\mu\tfrac12(1-\gamma^5)u(p)$ receives contributions only from the left-chiral components of both spinors, so a current built from a right-handed particle spinor vanishes identically rather than being merely small. The engine is the single relation $\{\gamma^5,\gamma^\mu\}=0$ ; from it and $(\gamma^5)^2=1$ the whole one-sidedness follows in three moves, in any representation of the gamma matrices.

Step 1. Establish the sliding rule. From $\{\gamma^5,\gamma^\mu\}=0$ , $\gamma^\mu\tfrac12(1-\gamma^5)=\tfrac12(1+\gamma^5)\gamma^\mu=P_R\gamma^\mu$ . Thus $\gamma^\mu P_L=P_R\gamma^\mu$ : the projector changes chirality label as it slides past a single gamma matrix. Why: the vector current is the one bilinear that flips chirality, which is why a chiral theory can still have a nonzero current at all.

Step 2. Convert the adjoint. For $u_R=P_Ru$ , the adjoint is $\bar u_R=u^\dagger P_R\gamma^0$ ; since $\gamma^5$ anticommutes with $\gamma^0$ , $P_R\gamma^0=\gamma^0P_L$ , giving $\bar u_R=\bar u\,P_L$ . Why: the bar operation swaps the projector's handedness, so a "right-handed" spinor looks left-handed to its left — bookkeeping where many sign errors hide.

Step 3. Collide the projectors. The right-handed-particle current is $\bar u_R\gamma^\mu\tfrac12(1-\gamma^5)u=\bar u\,P_L\gamma^\mu P_L\,u$ . Slide the first projector using Step 1: $P_L\gamma^\mu=\gamma^\mu P_R$ , so the middle becomes $\gamma^\mu P_R P_L=0$ by orthogonality, and the entire current vanishes. Repeating with the outgoing spinor leaves the surviving current $\bar u'_L\gamma^\mu u_L$ — left-chiral to left-chiral. Why: two projectors of opposite handedness meeting back-to-back annihilate — the mathematical form of "the W is blind to the wrong chirality" — with no suppression factor, exact and representation-independent. *Verified: $\gamma^\mu P_L=P_R\gamma^\mu$ and $P_RP_L=0\Rightarrow\bar u_R\gamma^\mu P_Lu=0$ in the Dirac, Weyl and Majorana representations (verify_aq-pp-chiral-weak-structure.py).*

Go deeper · Derivation B · 1 minHelicity suppression in charged-pion decay

Derivation BHelicity suppression in charged-pion decay

The aim is to derive the ratio $\Gamma(\pi\to e\nu)/\Gamma(\pi\to\mu\nu)$ from the chiral structure alone, showing the smallness of the electron channel is not a phase-space effect but a direct consequence of the $V-A$ vertex fighting angular-momentum conservation. Working in the pion rest frame, we use conservation of total spin projection to fix the required helicities and read off the amplitude's charged-lepton-mass dependence from equation (5) — one of the cleanest quantitative tests of maximal parity violation in nature.

Step 1. Fix the antineutrino helicity. The $\pi^-$ has spin $0$ ; in its rest frame the $\ell^-$ and $\bar\nu_\ell$ emerge back-to-back. The (essentially massless) antineutrino is produced by the $V-A$ current in the right-handed helicity state — the only one that couples. Why: the antineutrino has no mass to hide behind, so the vertex fixes its helicity rigidly, and that single fixed helicity constrains everything else.

Step 2. Force the charged-lepton helicity. Total angular momentum must be zero, so the two spin projections along the decay axis cancel. With the $\bar\nu$ spin along its momentum, the $\ell^-$ spin must point the same way in space — but the $\ell^-$ momentum is opposite, so the $\ell^-$ too comes out right-handed, exactly the helicity $V-A$ disfavours for a particle. Why: angular momentum and the vertex make incompatible demands, so the decay can only proceed through the small wrong-chirality piece of the lepton spinor — the whole origin of the suppression.

Step 3. Extract the mass dependence and evaluate. The overlap with the disfavoured helicity carries an amplitude factor $\propto m_\ell$ (equation 5), so the rate scales as $m_\ell^2$ ; with the phase-space factor $(m_\pi^2-m_\ell^2)^2/m_\pi^2$ the ratio is $\dfrac{\Gamma_e}{\Gamma_\mu}=\dfrac{m_e^2}{m_\mu^2}\left(\dfrac{m_\pi^2-m_e^2}{m_\pi^2-m_\mu^2}\right)^2=2.34\times10^{-5}\times5.49=1.28\times10^{-4}$ . Why: phase space alone would favour the electron; the four-order suppression can only come from the $m_\ell^2$ helicity penalty, an early smoking gun for $V-A$ . *Verified: $m_e^2/m_\mu^2=2.339\times10^{-5}$ , bracket $^2=5.488$ , product $=1.283\times10^{-4}$ ; radiative corrections give the measured $1.230\times10^{-4}$ .*

Go deeper · Derivation C · 1 minWhy V−A produces maximal parity violation

Derivation CWhy V−A produces maximal parity violation

The goal is to show that pure $V-A$ is not just a parity-violating interaction but the most parity-violating one possible, by writing a general vector-plus-axial vertex, forming a parity-odd observable, and showing its magnitude is maximized exactly when the vector and axial strengths are equal. This makes precise the loose phrase "parity is maximally violated" and ties it to the single-chirality nature of the coupling.

Step 1. Classify the currents under parity. Under $\mathsf P$ the vector current transforms as $V^0\to V^0,\ V^i\to-V^i$ , while the axial current transforms oppositely, $A^0\to-A^0,\ A^i\to+A^i$ . A pure $V$ or pure $A$ interaction therefore gives a parity-even squared amplitude. Why: parity acts as a definite sign on each current, so a theory built from one current alone cannot tell a process from its mirror image — you need both to break the symmetry.

Step 2. Build the parity-odd term. For $\Gamma^\mu=\gamma^\mu(g_V-g_A\gamma^5)$ the squared matrix element contains a $V$ – $A$ interference term $\propto g_Vg_A$ . Being the product of one parity-even and one parity-odd current it is odd under $\mathsf P$ , hence the only source of parity violation; a normalized asymmetry reads $\mathcal A_{\text{PV}}=2g_Vg_A/(g_V^2+g_A^2)$ . Why: without the cross term the mirror-image rate is identical, so the whole effect lives in $g_Vg_A$ .

Step 3. Maximize. As a function of $r=g_A/g_V$ , $\mathcal A_{\text{PV}}=2r/(1+r^2)$ peaks at $r=\pm1$ with value $\pm1$ . Hence $|g_V|=|g_A|$ — the $V\mp A$ point — is maximal parity violation, and the weak charged current sits exactly there. Why: the coupling weights to the two chiralities are $g_V-g_A$ and $g_V+g_A$ ; at $|g_V|=|g_A|$ one is zero, so one chirality couples with full strength and the other not at all — the extreme case, which is why $V-A$ is called maximal rather than merely large. *Verified: $\mathcal A_{\text{PV}}=2r/(1+r^2)$ peaks at $r=\pm1$ with $|\mathcal A|=1$ ; pure $V$ ( $g_A=0$ ) or pure $A$ ( $g_V=0$ ) gives $\mathcal A=0$ .*

Limiting cases

Case	Result	Meaning
Massless fermion, $m=0$	chirality $=$ helicity exactly	left-chiral $\equiv$ left-helical; the neutrino is purely left-handed, its antiparticle purely right-handed
Ultra-relativistic, $E\gg m$	wrong-chirality prob. $(m/2E)^2\to0$	helicity $\approx$ chirality; the "wrong" helicity states are strongly suppressed
Rest frame, $\beta=0$	wrong-chirality prob. $\tfrac12$	a particle at rest has no definite chirality — both helicities are equally present in each chiral state
Pure $V$ ( $g_A=0$ ) or pure $A$ ( $g_V=0$ )	$\mathcal A_{\text{PV}}=0$	parity conserved, as in QED and the strong interaction
$\vert g_V\vert=\vert g_A\vert$ (the $V-A$ point)	$\mathcal A_{\text{PV}}=1$	maximal parity violation — one chirality couples, the other not at all: the weak charged current
Nucleon $\beta$ decay	$g_A/g_V=-1.27$ , not $-1$	the fundamental quark coupling is pure $V-A$ ; the strong interaction renormalizes the axial charge of the nucleon

Common trapChirality is the same thing as helicity

It is not — conflating them is the most common error in this subject. Chirality is the eigenvalue of $\gamma^5$ : a Lorentz-invariant label carried by the $\tfrac12(1\pm\gamma^5)$ projectors and the quantity the weak vertex actually reads. Helicity is the projection of spin onto momentum: a frame-dependent, directly measurable quantity. For a massless particle they coincide exactly. For a massive particle they do not: a left-chiral electron is a definite- $\gamma^5$ state, but it contains both helicities, with the "wrong" one weighted by $\tfrac12(1-\beta)\approx(m/2E)^2$ . You can even outrun a massive particle and reverse its helicity by changing frame — but never its chirality. So "the $W$ couples only to left-handed particles" is precise only if "left-handed" means chirality; read as helicity it holds only in the ultra-relativistic limit, and that gap is exactly what produces helicity suppression.

Worked examples

1 · Helicity suppression: π→eν versus π→μν (fully worked)

The charged pion decays almost always to a muon and almost never to an electron, despite the electron channel's larger phase space. Compute $\Gamma(\pi^-\to e^-\bar\nu_e)/\Gamma(\pi^-\to\mu^-\bar\nu_\mu)$ from the $V-A$ structure and compare with experiment. Use $m_e=0.511$ , $m_\mu=105.66$ , $m_\pi=139.57$ MeV.

Step 1. Identify the two competing effects. Phase space favours the electron through the two-body factor $(m_\pi^2-m_\ell^2)^2/m_\pi^2$ , but the $V-A$ helicity penalty $\propto m_\ell^2$ disfavours it. The full ratio is $\dfrac{\Gamma_e}{\Gamma_\mu}=\dfrac{m_e^2}{m_\mu^2}\left(\dfrac{m_\pi^2-m_e^2}{m_\pi^2-m_\mu^2}\right)^2$ .

Step 2. Evaluate the helicity penalty. $\dfrac{m_e^2}{m_\mu^2}=\left(\dfrac{0.511}{105.66}\right)^2=\mathbf{2.34\times10^{-5}}$ — the raw cost of the electron's wrong helicity, five orders of magnitude on its own.

Step 3. Evaluate the phase-space factor. With $m_\pi^2=19480$ , $m_\mu^2=11164$ (MeV $^2$ ) and $m_e^2$ negligible, the bracket $=19480/8316=2.343$ and its square $=\mathbf{5.49}$ — phase space pulls back a factor $5.5$ in the electron's favour, nowhere near enough.

Step 4. Combine and compare. $\dfrac{\Gamma_e}{\Gamma_\mu}=2.34\times10^{-5}\times5.49=\mathbf{1.28\times10^{-4}}$ . The measured value is $1.230\times10^{-4}$ ; the small gap is radiative corrections, absent from the tree-level chiral calculation. The electron channel is suppressed by $\sim$ 8000 — a direct measurement of the $\tfrac12(1-\gamma^5)$ vertex. Had the weak interaction been pure vector, there would be no $m_\ell^2$ penalty and the electron mode would dominate.

2 · The wrong-helicity content and polarization of a β-decay electron

A $V-A$ current emits longitudinally polarized electrons. Take a $\beta$ -decay electron with total energy $E=1.00$ MeV ( $m_e=0.511$ MeV). What fraction of the time is it in the "wrong" (right-handed) chirality, and what is its longitudinal polarization?

Step 1. Momentum and velocity. $p=\sqrt{E^2-m^2}=\sqrt{1.00^2-0.511^2}=0.860$ MeV, so $\beta=p/E=0.860$ — fast but not ultra-relativistic, so chirality and helicity differ meaningfully.

Step 2. Wrong-chirality probability. From equation (5), $P(\text{wrong})=\tfrac12(1-\beta)=\mathbf{0.070}$ . About $\mathbf{7\%}$ of the time the left-chiral electron shows up in the right-handed helicity — a leakage that exists only because the electron is massive.

Step 3. Longitudinal polarization. A purely left-chiral particle has net helicity $P=-\beta$ (electrons; positrons $+\beta$ ). Here $P=-0.860$ : the electrons are $\mathbf{86\%}$ polarized against their motion. This $P=-v/c$ was measured directly in the late 1950s, a clean confirmation that $\beta$ -decay electrons carry the left-handed stamp of the weak current; the residual $7\%$ wrong-helicity content is the massive-particle correction that vanishes as $E\to\infty$ .

Physics of the portrait

The chiral structure of the weak interaction was assembled in barely eighteen months, in a burst that overturned a symmetry everyone had thought sacred. The portrait here is the idea itself — the $V-A$ interaction — built by a relay of theorists and experimentalists, each handing the next a piece.

Origins · 1 minthe V−A interaction and how this was discovered

the V−A interaction

1956–1958, from Lee–Yang to Feynman–Gell-Mann and Sudarshan–Marshak · chirality of the weak charged current

Until 1956 parity conservation was treated as self-evident: surely the laws of physics cannot distinguish left from right. The " $\tau$ – $\theta$ puzzle" — two particles identical in every way but decaying into states of opposite parity — pushed Tsung-Dao Lee and Chen-Ning Yang to ask, in a famous 1956 paper, whether parity had ever actually been tested in the weak interaction. It had not. Within months Chien-Shiung Wu and her collaborators cooled polarized cobalt-60 to near absolute zero and watched its $\beta$ -decay electrons stream out preferentially against the nuclear spin — an unambiguous left-right asymmetry, announced in January 1957. The effect was not small but huge, and Garwin, Lederman and Weinrich confirmed it in muon decay days later. The theoretical synthesis came in 1958, when George Sudarshan and Robert Marshak, and — more visibly — Richard Feynman and Murray Gell-Mann, argued that the weak current is universally $V-A$ , coupling only to the left-handed projection $\tfrac12(1-\gamma^5)$ . The last brick was Goldhaber, Grodzins and Sunyar's ingenious 1958 measurement of the neutrino's helicity, which found it left-handed, exactly as $V-A$ requires. What began as a puzzle about two kaons ended as the discovery that half of every fermion is invisible to the weak force — and that the universe, at its most fundamental level, knows its left hand from its right.

History

1956

— Lee and Yang question whether parity is conserved in the weak interaction

1957

— Wu and collaborators observe parity violation in polarized cobalt-60 β decay

1957

— Garwin, Lederman and Weinrich confirm parity violation in muon decay

1958

— Feynman–Gell-Mann and Sudarshan–Marshak formulate the universal V−A theory

1958

— Goldhaber, Grodzins and Sunyar measure the neutrino to be left-handed

Going further

Texts. Thomson, Modern Particle Physics, §11.3–11.4 and §6.4.1 · Halzen & Martin, Quarks and Leptons, Ch. 12 · Griffiths, Introduction to Elementary Particles, 2nd ed., §9.4–9.7 (helicity suppression) · Peskin & Schroeder, QFT, §3.2 and §20.2 (chirality, massless limit, electroweak couplings) · Schwartz, QFT and the Standard Model, Ch. 29.

Provenance. The vertex $-i\tfrac{g_W}{2\sqrt2}\gamma^\mu(1-\gamma^5)$ and the projector algebra are recalled from aq-pp-weyl-chirality and aq-pp-dirac-adjoint-bilinear-covariants; the helicity–chirality relation $\tfrac12(1-\beta)$ from aq-pp-spin-helicity-states; the coupling and $W$ propagator from aq-pp-weak-charged-current. The gauge-theory reason nature chose $P_L$ is developed in aq-pp-electroweak-gauge-group and used here by name. A verifier should confirm: (1) $\gamma^\mu P_L=P_R\gamma^\mu$ and $\bar u_R\gamma^\mu P_Lu=0$ ; (2) $\mathcal A_{\text{PV}}=2g_Vg_A/(g_V^2+g_A^2)$ peaks at $|g_V|=|g_A|$ ; (3) the pion ratio $1.28\times10^{-4}$ ; (4) $P(\text{wrong})=0.070$ for a 1 MeV electron.

Prerequisites. Weyl Spinors and Chirality · Dirac Adjoint and Bilinear Covariants · Spin and Helicity States · The Weak Charged Current. Leads to. The $V-A$ Interaction · Parity Violation · Helicity Suppression in Pion Decay · Electroweak Gauge Group.

Ask the tutor: Ask me to walk through why $\bar u_R\gamma^\mu P_L u$ vanishes line by line, or to redo the pion helicity-suppression ratio for the tau channel.

What is the difference between chirality and helicity, really?

Chirality is the eigenvalue of the matrix $\gamma^5$ — a Lorentz-invariant label that the weak vertex, through its $\tfrac12(1-\gamma^5)$ factor, directly reads. Helicity is the projection of spin onto the direction of motion, a geometric quantity you can in principle measure. The subtlety is that they are the same thing only for massless particles. For a massive particle a state of definite chirality is a mixture of both helicities: the "wrong" helicity appears with probability $\tfrac12(1-\beta)$ , which shrinks to zero as the particle approaches the speed of light. So a fast electron is almost a helicity eigenstate, but never exactly one. Every apparent paradox in this subject — helicity suppression included — comes from the gap between these two ideas.

Is "only left-handed particles feel the weak force" a law or an accident?

It is a defining structural choice of the Standard Model, encoded by putting left-handed fermions in $SU(2)_L$ doublets and right-handed fermions in singlets. Nothing in our current theory explains why nature picked the left rather than the right, or why it picked a single chirality at all — it is measured, not derived. It could have been otherwise: a mirror-image " $V+A$ " universe is mathematically consistent. This maximal, one-sided coupling is one of the most striking brute facts in physics, and finding a deeper reason for it (left-right symmetric models, grand unification) remains an open program. What is derived, once you accept the $\tfrac12(1-\gamma^5)$ vertex, is everything downstream: parity violation, the neutrino's handedness, helicity suppression.

Does V−A mean right-handed neutrinos simply do not exist?

Not quite — it means that if they exist they are sterile: they do not couple to the $W$ or $Z$ , so no experiment built on the weak interaction can produce or detect them. In the original Standard Model the neutrino was taken to be exactly massless and a right-handed partner was omitted as unnecessary. But the discovery that neutrinos oscillate proves they have mass, and giving a neutrino a Dirac mass requires a right-handed partner. So right-handed neutrinos may well exist; they would just be invisible to the weak force. Whether they do, and whether the neutrino is Dirac or Majorana, is a live experimental question.

Why does the weak current couple left-handed particles but right-handed antiparticles?

Because the same projector $\tfrac12(1-\gamma^5)$ acts oppositely on particle and antiparticle spinors. On a $u$ -spinor it selects the left-chiral component $u_L$ ; on a $v$ -spinor it selects what is conventionally the right-handed antiparticle, because the chirality-to-helicity dictionary flips between particles and antiparticles. Physically this is required by angular-momentum bookkeeping: in a decay like $\pi^-\to\mu^-\bar\nu_\mu$ a left-handed particle emerges alongside a right-handed antiparticle. It is not two rules but one operator seen from both sides.

If the fundamental coupling is pure V−A, why is the neutron's g_A/g_V about −1.27?

At the quark level the coupling is exactly $V-A$ , so $g_A/g_V=-1$ for the elementary $d\to u$ transition. But the neutron and proton are not elementary — they are bound states swarming with gluons and quark-antiquark pairs. The vector current is protected by the conserved-vector-current (CVC) hypothesis, so $g_V$ stays essentially $1$ ; the axial current has no such protection and is renormalized by the strong interaction, pushing $g_A/g_V$ to the measured $-1.2723$ . The deviation from $-1$ is thus a QCD dressing on top of an underlying vertex that remains purely $V-A$ — a symmetry dressed by the strong force without being broken.

How did Wu's cobalt-60 experiment actually show parity violation?

Wu polarized cobalt-60 nuclei by cooling them to about 0.01 K in a magnetic field, aligning their spins, then counted the $\beta$ -decay electrons emitted along versus against the nuclear spin. If parity held, the two directions would be equally populated, because reflecting the setup in a mirror swaps them and the mirror world must look identical. Instead the electrons came out preferentially opposite to the spin — a front-back asymmetry that a mirror would reverse. Since the mirror image is a different rate, the mirror world is not our world, and parity is violated. The size of the asymmetry, consistent with $P=-v/c$ , showed the violation is not small but maximal.