Helicity and Chirality; The Majorana Fermion · Elementary Particle Physics

Spin admits three labels and they are not interchangeable. The interaction that matters in the rest of the book couples to the one an experiment cannot measure directly.

🎯 Why this matters

Every measured weak asymmetry is therefore diluted, and by a factor that depends on the particle’s energy. The disagreement between what the interaction selects and what a detector reports vanishes only in the massless limit.

The Dirac bispinor of §2.5 has four components arranged as two two-component spinors, φ and χ. This section asks what those two entries of each mean — and the answer is that there are three different useful answers, two of which are constantly confused with each other, including in the professional literature. The book says so and then refuses to do it. So will this page: left and right will always mean chirality, never helicity.

Three ways to label a spin

The three labellings, and what each requires

Label↕	Eigenstate of	What it needs	When you use it
polarization	$\sigma_z$	an externally defined axis — typically a magnetic or electric field	Whenever an apparatus picks out a direction. This is the everyday laboratory notion.
helicity	$\tfrac{1}{2}\,\boldsymbol\sigma\cdot\hat{\mathbf p}$	the particle to be moving — the momentum supplies the axis	When there is no field but there is a direction of travel, which is to say almost always in this book. It is what a detector actually measures.
chirality	$\gamma^5$	nothing at all — it is a property of the spinor, not of the motion	When you write the interaction. The weak interaction couples to chirality, so the Standard Model is written in this basis.

The middle and bottom rows are the ones that get conflated, and the confusion is understandable: at high energy they agree. The rest of this page is about where they do <strong>not</strong>.

\htmlClass{t-h}{h} = \frac{1}{2}\,\frac{\htmlClass{t-s}{\boldsymbol\sigma}\cdot\htmlClass{t-p}{\mathbf p}}{|\htmlClass{t-p}{\mathbf p}|}

(2.55)

Helicity is a scalar product of a polar and an axial vector, and every one of its properties follows from that: invariant under rotations, odd under parity, and — because a boost can reverse p but not σ — not invariant under a boost.

Every symbol, one at a time

Hover or tap a symbol above — it lights up in the equation and its meaning, units and type appear here.

Why helicity is frame-dependent, in one picture. The middle panel is what the observer-velocity slider below produces once β_obs exceeds the particle’s own β.

💡 What this really says — two properties, exactly opposite

Everything on this page follows from one table:

	Lorentz invariant?	conserved for a free particle?
helicity	no (unless massless)	yes
chirality	yes	no (the mass term mixes L and R)

Why helicity is not invariant. A massive particle travels slower than light, so you can always board a faster frame and overtake it. In that frame its momentum has reversed while its spin has not, so the projection of spin on momentum has changed sign. Only for a rigorously massless fermion is the speed the same in every frame, and only then is helicity invariant.

Why helicity is conserved. The helicity operator commutes with the Dirac Hamiltonian. A free fermion that starts with one helicity keeps it.

Why chirality is invariant. It is the eigenvalue of γ⁵, a matrix built from the γ’s, and it does not refer to the motion at all.

Why chirality is not conserved. Split the Dirac Lagrangian and the kinetic term separates cleanly into left and right pieces via the chiral projectors — but the mass term does not:

m_f\,\bar\psi_f\psi_f = m_f\left(\bar\psi_L\psi_R + \bar\psi_R\psi_L\right).

Every term couples one chirality to the other. A mass is precisely a coupling between left and right, which is why a massless fermion has separately conserved L and R counts and a massive one does not. Hold on to that sentence — it is the whole of the Higgs mechanism in Ch. 9 in embryo.

⚙️ Engineer’s bridge — an invariant that drifts, and a constant that transforms

You are used to quantities that are both invariant and conserved, so a pair that splits those properties is worth pausing on.

Helicity is like a quantity measured in a particular frame. Its value does not change with time, but it is not the same number for two observers — like the Doppler-shifted frequency of a tone. Constant for you, constant for me, different values.
Chirality is like a label attached to the signal itself. Every observer agrees on it, but it can change as the system evolves — like a modulation index that all observers measure identically while the modulator alters it.

And the coupling between them is the mass. In the chiral basis (switch the widget below to it) the Dirac equation reads

i\gamma^\mu\partial_\mu \psi_L = m\,\psi_R, \qquad i\gamma^\mu\partial_\mu \psi_R = m\,\psi_L,

which is a pair of coupled first-order equations with the mass as the coupling constant — formally identical to two resonators exchanging energy, or to a two-state system with an off-diagonal term. Set $m = 0$ and they decouple completely: two independent, non-interacting systems, and their occupation numbers are separately conserved. Turn the mass on and they beat against each other.

The same structure returns in Ch. 8 for K⁰–K̄⁰ mixing and in Ch. 10 for neutrino oscillation, both times as an off-diagonal mass term coupling two states. It is worth recognising the shape now.

Where it breaks: the two-state analogy is exact for a Hermitian 2 × 2 system, and the systems it points at are not that. K⁰–K̄⁰ and the neutrinos both decay, so the effective Hamiltonian is non-Hermitian, its eigenvalues are complex, and the eigenvectors are not orthogonal — which is the whole reason §8.5 can define an impurity parameter ε at all. Recognising the shape is the right instinct; expecting the textbook two-level results to carry over unmodified is not.

The helicity content of a chiral state

Chirality is the useful label for writing interactions, but it is not what a detector measures — a detector measures helicity. So the practical question is: given a left-chiral fermion, what helicity will you find?

🪜 From φ and χ to ⟨h⟩ = −β

Step 1 of 6 — The two spinors are not independent(2.59)

\begin{pmatrix} E - m & -\boldsymbol\sigma\cdot\mathbf p \\ \boldsymbol\sigma\cdot\mathbf p & -(E+m)\end{pmatrix}\begin{pmatrix}\varphi \\ \chi\end{pmatrix} = 0

Why you may do this: Writing the Dirac equation for a plane wave gives a 2 × 2 block equation. It is not two separate conditions on φ and χ — it ties them together.

So you cannot choose φ and χ freely. Fixing one fixes the other, and the link is through σ·p.

Bettini Eqs. (2.59)–(2.66). The result is exact at every energy, not just in the ultra-relativistic limit.

🌀 Helicity is not chirality

energy in units of the mass, E/mE/m = 5.012 · β = 0.979892

A left-chiral fermion — what helicity will you measure?

negative helicity (the “right” one)0.989946

positive helicity (the “wrong” one)0.010054

amplitudes  a∓ = 1 ∓ p/(E+m) = 0.1831, 1.8169
wrong-helicity probability = 0.010054
the (m/2E)² estimate       = 0.009953
⟨h⟩ = (a₊² − a₋²)/(a₊² + a₋²) = -0.979892
−β                            = -0.979892

⟨h⟩ = −β exactly, at every energy — Eq. (2.66). A left-chiral fermion is a definite-helicity particle only in the limit β → 1; at rest the two helicities are equally likely and chirality says nothing about spin direction at all.

Now run alongside it — what survives the boost?

observer's velocity β_obs0.0000

particle in the lab       β  = 0.979892
seen by the observer      β' = 0.979892
   the observer is still behind it

helicity	negative, unchanged
chirality	left, unchanged — always

Because the particle is massive you can always find a frame moving faster than it — set β_obs above 0.9799 and its momentum reverses while its spin does not, so the helicity changes sign. Helicity is not Lorentz invariant. Chirality, an eigenvalue of γ⁵, is.

exact wrong-helicity probability
the (m/2E)² limit

The wrong-helicity admixture falls as (m/2E)², so it is a 1 % effect at E = 5m and a part in ten thousand at E = 50m. This is why 'left-handed' and 'left-chiral' can be used interchangeably at collider energies and must not be at low ones — and why neutrino mass, however small, means the weak interaction is not quite chirally pure (Ch. 10).

	polarization	helicity	chirality
eigenstate of	σ_z, along an axis you chose	σ·p̂, along the motion	γ⁵
needs	an external field to define the axis	the particle to be moving	nothing
Lorentz invariant?	no — the axis is frame-dependent	NO, unless massless	YES, always
conserved for a free particle?	no	YES	NO, the mass term mixes L and R
directly measurable?	yes	yes	no — you measure helicity and infer it
what the weak interaction couples to	—	—	THIS (Ch. 7)

Read the two middle rows of the invariance/conservation pair: helicity and chirality have exactly opposite properties, which is why neither alone is the whole story and why the literature confuses them. The book's convention, followed here: left and right always mean chirality, never helicity.

🔢 Worked example — how “wrong” is the wrong helicity?

Take an electron from a weak decay, which the weak interaction produces left-chiral. What fraction is measured with the wrong (positive) helicity?

P_{\text{wrong}} \simeq \left(\frac{m}{2E}\right)^2 .

particle	$m$	$E$	$P_{\text{wrong}}$
electron from β decay	0.511 MeV	1 MeV	$6.5\times10^{-2}$
electron from β decay	0.511 MeV	10 MeV	$6.5\times10^{-4}$
muon at the LHC	106 MeV	50 GeV	$1.1\times10^{-6}$
a 0.1 eV neutrino	$10^{-7}$ MeV	1 MeV	$2.5\times10^{-15}$

The last row is the interesting one. Neutrino masses are so small that at any energy anyone has ever detected one, the wrong-helicity component is astronomically suppressed — which is why for fifty years the weak interaction appeared to couple to helicity rather than chirality, and why the distinction could be ignored. It stopped being ignorable when neutrinos turned out to have mass at all (Ch. 10).

The Majorana fermion

📐 Physics you need first — what “its own antiparticle” would require

Some particles are their own antiparticles: the photon, the π⁰, the Z⁰. All of them have every charge equal to zero — electric charge, colour, weak charge, lepton number, baryon number — because an antiparticle differs from its particle by the sign of all of those, and a particle can only equal its own antiparticle if there is nothing to flip.

Every one of those examples has integer spin. Majorana asked in 1937 whether a spin-½ particle could do it too — a Majorana fermion .

The bookkeeping is immediate. A Dirac field is complex, and its four complex components are two spin states for the particle plus two for the antiparticle. If the antiparticle is the particle, you need only two — so the field must have half as many degrees of freedom, which means it must be real.

The requirement is therefore: find a set of gamma matrices for which $i\gamma^\mu\partial_\mu$ is a real operator, so that a real field can satisfy the equation. That needs all four γ to be purely imaginary — and they exist.

\gamma^0 = \sigma_2 \otimes \sigma_1, \quad \gamma^1 = i\sigma_1 \otimes \mathbf 1, \quad \gamma^2 = i\sigma_3 \otimes \mathbf 1, \quad \gamma^3 = i\sigma_2 \otimes \sigma_2 \tag{2.67}

Switch the widget below to the Majorana representation and check: every entry is imaginary, and the anticommutation relation $\{\gamma^\mu,\gamma^\nu\} = 2g^{\mu\nu}$ still holds, index pair by index pair. It is the same algebra in a different basis — nothing physical has been assumed, only a convenient set of coordinates chosen.

🔢 The Dirac algebra, as matrices

Dirac: The book’s choice, Eq. (2.36). γ⁰ is diagonal, which makes the non-relativistic limit transparent: the upper two components are the particle and the lower two the antiparticle, and at low energy they decouple.

γ⁰

1	0	0	0
0	1	0	0
0	0	-1	0
0	0	0	-1

γ¹

0	0	0	1
0	0	1	0
0	-1	0	0
-1	0	0	0

γ²

0	0	0	−i
0	0	i	0
0	i	0	0
−i	0	0	0

γ³

0	0	1	0
0	0	0	-1
-1	0	0	0
0	1	0	0

γ⁵ = iγ⁰γ¹γ²γ³

0	0	1	0
0	0	0	1
1	0	0	0
0	1	0	0

Check the defining relation yourself

{γ^μ, γ^ν} =μ =ν =

γ⁰γ¹ + γ¹γ⁰

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

2 g⁰¹ · 𝟙 = 2 × 0 × 𝟙

✓ they agree, in every representation

Try all sixteen pairs, then switch representation and try again. The matrices change completely; this relation does not. That is what it means to say the choice of representation carries no physics.

Why exactly five bilinear covariants — a dimension count

Covariant	Transforms as	Independent components
ψ̄ψ	scalar	1
ψ̄γ⁵ψ	pseudoscalar	1
ψ̄γ^μψ	vector	4
ψ̄γ^μγ⁵ψ	axial vector	4
ψ̄σ^μνψ	tensor	6
total		16

Sixteen is 4² — the dimension of the space of all 4×4 matrices. So the five covariants are not a list somebody collected: they are a complete basis, sorted into the pieces that transform into themselves under Lorentz transformations. Any 4×4 matrix you could put between ψ̄ and ψ is a combination of these, which is why the list stops at five. Nature then uses only two of them, V and A — and that is a physical fact, not a mathematical one (§2.8, Ch. 7).

💡 What this really says — why a Majorana fermion cannot be massless

This is the striking result of §2.9, and the argument is two lines.

A massless fermion must be an eigenstate of γ⁵ — that is the content of the massless limit above, where the two chiralities decouple and each propagates independently. So we would need

\gamma^5\psi = \pm\psi .

But in the Majorana representation $\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3$ is purely imaginary, and ψ is real. A purely imaginary matrix acting on a real vector gives a purely imaginary vector, which cannot equal $\pm$ (a real vector) unless both are zero.

So the two requirements are incompatible: real field and chirality eigenstate cannot hold at once. A Majorana fermion must have a mass.

The consequence is a live experimental question. If neutrinos are Majorana particles, then lepton number is not conserved, and neutrinoless double-β decay should occur (§10.5–10.6). Nobody has seen it. That is currently the sharpest handle on whether the neutrino is a Dirac or a Majorana fermion — and, as §2.4 noted, the extension of Dirac theory to neutrinos has never had experimental support in the first place.

⚠️ On the word “left”

The book issues a warning worth repeating, because the site follows it everywhere:

Unfortunately, one often encounters in the literature the use of the ‘left’ and ‘right’ terms also for negative and positive helicity. We shall avoid this confusing language: with left and right we shall always mean negative and positive chirality.

Where papers say “left-handed neutrino” they usually mean negative helicity, because that is what is measured. At neutrino energies the two coincide to a part in $10^{15}$ , so nothing goes wrong numerically — but the two words describe different operators with opposite invariance and conservation properties, and the distinction becomes load-bearing the moment a mass matters.

Reproduce it

import numpy as np

print("left-chiral fermion: amplitudes a± = 1 ∓ p/(E+m); <h> should be exactly -beta")
for Em in (1.001, 1.5, 5.0, 50.0, 1000.0):
    beta = np.sqrt(1 - 1/Em**2)
    r = np.sqrt((Em - 1)/(Em + 1))                 # = p/(E+m)
    ap, am = 1 - r, 1 + r
    P = ap**2/(ap**2 + am**2)
    h = (ap**2 - am**2)/(ap**2 + am**2)
    print(f"   E/m = {Em:8.3f}: beta = {beta:.6f}, P(wrong) = {P:.2e}, "
          f"(m/2E)^2 = {(1/(2*Em))**2:.2e}, <h> = {h:.6f}, -beta = {-beta:.6f}")

print("wrong-helicity fraction for real cases:")
for name, m, E in (("e from beta decay at   1 MeV", 0.511, 1.0),
                   ("e from beta decay at  10 MeV", 0.511, 10.0),
                   ("muon at 50 GeV              ", 105.66, 50e3),
                   ("0.1 eV neutrino at 1 MeV    ", 1e-7, 1.0)):
    print(f"   {name} : {(m/(2*E))**2:.1e}")

s = [np.array([[0,1],[1,0]], dtype=complex), np.array([[0,-1j],[1j,0]], dtype=complex),
     np.array([[1,0],[0,-1]], dtype=complex)]
I2, Z = np.eye(2, dtype=complex), np.zeros((2,2), dtype=complex)
blk = lambda a,b,c,d: np.block([[a,b],[c,d]])
beta_m = blk(I2, Z, Z, -I2)
gD = [beta_m] + [beta_m @ blk(Z, x, x, Z) for x in s]
g5 = 1j*gD[0]@gD[1]@gD[2]@gD[3]
L, R = 0.5*(np.eye(4) - g5), 0.5*(np.eye(4) + g5)
print("chiral projectors L = (1-g5)/2, R = (1+g5)/2 (Question 2.1):")
print(f"   L^2 = L {np.allclose(L@L, L)} ; R^2 = R {np.allclose(R@R, R)} ; "
      f"L R = 0 {np.allclose(L@R, 0)} ; L + R = 1 {np.allclose(L+R, np.eye(4))}")

kr = np.kron
gM = [kr(s[1],s[0]), 1j*kr(s[0],I2), 1j*kr(s[2],I2), 1j*kr(s[1],s[1])]      # Eq. (2.67)
g5M = 1j*gM[0]@gM[1]@gM[2]@gM[3]
print(f"Majorana representation (2.67): every gamma purely imaginary "
      f"{all(np.allclose(x.real, 0) for x in gM)}")
print(f"   and gamma^5 purely imaginary too: {np.allclose(g5M.real, 0)}")
print("   so gamma^5 psi is imaginary for real psi -> no chirality eigenstate"
      " -> no massless Majorana fermion")

prints

left-chiral fermion: amplitudes a± = 1 ∓ p/(E+m); <h> should be exactly -beta
 E/m =    1.001: beta = 0.044688, P(wrong) = 4.78e-01, (m/2E)^2 = 2.50e-01, <h> = -0.044688, -beta = -0.044688
 E/m =    1.500: beta = 0.745356, P(wrong) = 1.27e-01, (m/2E)^2 = 1.11e-01, <h> = -0.745356, -beta = -0.745356
 E/m =    5.000: beta = 0.979796, P(wrong) = 1.01e-02, (m/2E)^2 = 1.00e-02, <h> = -0.979796, -beta = -0.979796
 E/m =   50.000: beta = 0.999800, P(wrong) = 1.00e-04, (m/2E)^2 = 1.00e-04, <h> = -0.999800, -beta = -0.999800
 E/m = 1000.000: beta = 0.999999, P(wrong) = 2.50e-07, (m/2E)^2 = 2.50e-07, <h> = -0.999999, -beta = -0.999999
wrong-helicity fraction for real cases:
 e from beta decay at   1 MeV : 6.5e-02
 e from beta decay at  10 MeV : 6.5e-04
 muon at 50 GeV               : 1.1e-06
 0.1 eV neutrino at 1 MeV     : 2.5e-15
chiral projectors L = (1-g5)/2, R = (1+g5)/2 (Question 2.1):
 L^2 = L True ; R^2 = R True ; L R = 0 True ; L + R = 1 True
Majorana representation (2.67): every gamma purely imaginary True
 and gamma^5 purely imaginary too: True
 so gamma^5 psi is imaginary for real psi -> no chirality eigenstate -> no massless Majorana fermion

🔑 If you remember only three things

Chirality is what the theory uses; helicity is what you see. They coincide for a massless particle and drift apart at the rate m/E, which is why low-energy tests are the hard ones.
A label can be conserved without being invariant, or invariant without being conserved. Losing track of which one you hold is the standard error on this page.
Mass is what forbids a Majorana neutrino from being simple. The moment a fermion has mass it can no longer be an eigenstate of γ⁵, and the two-line argument on this page is the whole obstruction.

Where this goes next

§3.3–3.4 is parity, the operation helicity is odd under — and §3.6–3.7 is the discovery that nature does not respect it.
Ch. 7 is where “the weak interaction couples to chirality” becomes the V−A structure, using exactly two of the five bilinear covariants of §2.5.
§9.10–9.11 is the Higgs mechanism, which exists because a mass term couples left to right and the weak interaction treats them differently — so masses cannot simply be written down.
§10.5–10.6 is the search for neutrinoless double-β decay, the experiment that would prove the neutrino is a Majorana fermion.

✅ Check yourself — the two words that are not the same word

0/6 answered · 0 correct

1.Helicity and chirality have exactly opposite properties. Which pair is right?
2.Why is helicity not invariant under a boost, while chirality is?
3.In the widget, slide E/m down towards 1. The wrong-helicity probability rises towards ½. What does that mean physically?
4.For a 0.1 eV neutrino at 1 MeV the wrong-helicity fraction is (m/2E)² ≈ 2.5 × 10⁻¹⁵. Why does that matter historically?
5.In the chiral basis the Dirac equation becomes iγ^μ∂_μψ_L = mψ_R and iγ^μ∂_μψ_R = mψ_L. What does that structure say?
6.Majorana's argument that a self-conjugate fermion cannot be massless runs in two lines. What are they?