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research30 min studyLesson 611 in Physics

SU(2) left and hypercharge representations

Construct chiral electroweak multiplets with a consistent hypercharge convention, then test electric charges and allowed interactions.

Before you begin
On this page

The question for today

What you’ll understand

  • Translate consistently between two common hypercharge normalizations.
  • Recover electric charges from weak isospin and hypercharge.
  • Check a Yukawa interaction using conjugated representations and chiral fields.

Electroweak gauge symmetry organizes fields before one identifies photons and massive weak bosons. The gauge algebra is that of SU(2) left times U(1) hypercharge, and its representations depend on chirality. We use natural units with Planck's constant and the speed of light set to one, and adopt electric charge Q equal to T-three plus Y. This fixes every hypercharge below. A text using Q equal to T-three plus Y over two assigns twice our numerical hypercharges.

Generators and normalization travel together

On a left-handed doublet the weak generators are half the Pauli matrices; on a weak singlet they vanish. Hypercharge multiplies the identity within an irreducible weak multiplet, so both components of a doublet share Y while their T-three eigenvalues differ. The two gauge couplings are dimensionless in four spacetime dimensions. Rescaling the abelian generator requires inversely rescaling its coupling, leaving their product in the covariant derivative unchanged.

The word left names a Lorentz-chiral representation, not a direction of motion or the lower component of a doublet. A Dirac field splits using chiral projectors, and its two chiral pieces can carry different electroweak representations. Helicity is spin projected onto momentum; for a massive particle it depends on the observer's frame. In the massless particle limit helicity aligns with chirality, with the appropriate reversed particle–antiparticle relation. Treating the two notions as universally interchangeable obscures the gauge structure.

Dμ=∂μ−igTaWμa−ig′YBμ,Ta=σa2 on doublets,Q=T3+YD_\mu=\partial_\mu-igT^aW_\mu^a-ig' YB_\mu,\qquad T^a=\frac{\sigma^a}{2}\ \text{on doublets},\qquad Q=T_3+Y
Our covariant-derivative sign convention yields positive gauge-current terms in the fermion kinetic Lagrangian. g and g′ are dimensionless; gauge fields and derivatives have energy dimension one.↔ Scroll wide equations horizontally.

One generation separates chiral multiplets

For one generation, the lepton doublet contains a left-chiral neutrino and charged lepton with hypercharge minus one half, while the right-chiral charged lepton is a singlet with hypercharge minus one. The quark doublet has hypercharge one sixth, with right-chiral up and down singlets carrying two thirds and minus one third. Color supplies three copies of each quark weak representation but does not change its electric charge.

The minimal renormalizable field inventory has no right-chiral neutrino. Observed neutrino masses therefore require additional structure, such as a singlet field or a higher-dimensional operator. When counting anomalies, it is convenient to rewrite right-chiral particles as left-chiral charge-conjugate fields; the abelian charge then changes sign. Using right-chiral charges directly in a formula written for left-chiral fields would spoil the cancellation. Gauge consistency constrains a generation's charges together, rather than making each neutral-looking term acceptable.

LL:(2,−12),eR:(1,−1),QL:(2,16),uR:(1,23),dR:(1,−13),H:(2,12)L_L:(\mathbf2,-\tfrac12),\quad e_R:(\mathbf1,-1),\quad Q_L:(\mathbf2,\tfrac16),\quad u_R:(\mathbf1,\tfrac23),\quad d_R:(\mathbf1,-\tfrac13),\quad H:(\mathbf2,\tfrac12)
The pair gives (weak representation, our hypercharge); quark color indices are suppressed. H is a complex scalar doublet. These labels are not helicity probabilities.↔ Scroll wide equations horizontally.

Invariant interactions contract every label

An interaction must be a singlet under each gauge factor. The charged-lepton Yukawa operator contracts the conjugate lepton doublet, the Higgs doublet, and the right-chiral charged-lepton singlet. The barred doublet carries opposite hypercharge, making the total zero. For an up-type quark the conjugate Higgs doublet is needed instead; it has hypercharge minus one half. A sum of electric charges alone is insufficient before selecting the Higgs vacuum.

The product-group notation specifies the local algebra and usual representations used here. The global gauge group can involve a quotient linked to the allowed charge spectrum, a separate issue from these elementary vertex checks. A component's electric charge is an eigenvalue of the eventual unbroken generator; it does not specify the whole weak multiplet. Keeping these distinctions visible helps explain why equal electric charges can coexist with different left- and right-chiral weak interactions.

Work it through

Check an electron Yukawa operator

Using our Q=T-three+Y convention, test the hypercharge of L‾LHeR\overline L_LHe_R and derive the two Higgs-component electric charges. Explain why replacing H by its conjugate fails for this charged-lepton operator.

  1. The bar reverses the lepton doublet hypercharge, so the operator's total is +1/2+1/2−1=0+1/2+1/2-1=0. The weak doublet and conjugate-doublet indices can also contract to a singlet.

  2. The Higgs components have T-three eigenvalues +1/2 and −1/2. Adding Y=+1/2 gives electric charges +1 and zero, respectively, in units of the positive elementary charge.

  3. The conjugate Higgs has Y=−1/2, which would give +1/2−1/2−1=−1+1/2-1/2-1=-1 for this operator. It is therefore not hypercharge invariant despite still having two weak indices available for contraction.

The result

The charged-lepton Yukawa operator is invariant with H; its upper and lower Higgs components have charges +1 and 0.

The living notebook

Explore the model

Interactive lab

Interactive lab · vectors

Separate a representation from its components

For each fresh comparison, Reset and set both controls. For each fresh run Reset and set both controls. Compare angle 0 degrees, magnitude 1 with angle 90 degrees, magnitude 1. Predict projections (1,0) and (0,1) with unchanged magnitude. This is a basis-resolution analogy only: the vector is real and two dimensional, without chiral spinors, complex SU(2) transformations, hypercharge, or gauge-anomaly tests.

Separate a representation from its componentsPlayback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Current readings: x component: 1.393; y component: 0.975; Magnitude: 1.7. All parameters are adjustable with the labeled controls below.−7.46−3.7303.737.46−3.5−1.7501.753.5xyvvₓ = 1.39
Vectorx / y projectionsTrace marker
x component
1.393
y component
0.975
Magnitude
1.7
t = 0 s

Playback traces a fixed vector and its projections. Angle and magnitude set the geometry; the moving marker does not change them. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forComponent coordinates can change while a representation’s normalization and dimension remain fixed.

Pause & reason

Check your understanding

A book defines Q=T₃+Ybook/2 while this lesson uses Q=T₃+Y. What is Ybook for our Higgs doublet?

Your turn

Try it yourself

Check the up-type operator Q‾LH~uR\overline Q_L\widetilde H u_R, with H~=iσ2H∗\widetilde H=i\sigma^2H^*. Then infer the electric charges of the two quark-doublet components.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Complex conjugation reverses abelian charge, and the barred quark doublet also reverses its hypercharge.

Compare with the solution

The total hypercharge is −1/6−1/2+2/3=0-1/6-1/2+2/3=0, so the operator can be an electroweak singlet after contracting its weak indices. With Y=1/6, the doublet charges are +1/2+1/6=+2/3+1/2+1/6=+2/3 and −1/2+1/6=−1/3-1/2+1/6=-1/3. In contrast, using H would give total hypercharge one, so it is not the allowed up-type Yukawa structure. Color contracts independently between the barred quark and right-chiral quark.

Keep the idea

Take it with you

  • Hypercharge numbers are meaningful only with the corresponding charge and coupling normalization.
  • Weak multiplets classify chirality, while helicity coincides only in a specified massless limit.
  • A gauge-invariant interaction must contract all weak and color labels and cancel total hypercharge.

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