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 Incoming links: Gauge interaction

The free kinetic terms of left- and right-handed Weyl fields are separately invariant under unitary group rotations in flavour space. For

N

massless Dirac fields, the subgroup preserving common fermion number is

U (N)_{L} \times U (N)_{R}

. A common nonzero Dirac mass term couples the two chirality sectors and leaves its diagonal

U (N)

With the index convention

I (N) = 1

for an

S U (N)

b_{0} = \frac{3}{2} I (adj) - \frac{1}{2} \sum_{i} I (R_{i})

. Positive

b_{0}

gives asymptotic freedom, while negative

b_{0}

At energies large compared with fermion masses, the gauge interactions preserve chirality on each fermion line. A photon couples to charged fermions with equal left and right coefficients

e Q_{f}

g_{s} T^{a}

. A Z boson couples to both, with coefficients

(g / cos θ_{W}) (T_{3} - Q_{f} sin^{2} θ_{W})

on left-chiral fields and

(g / cos θ_{W}) (- Q_{f} sin^{2} θ_{W})

on right-chiral fields. A W boson couples only to left-chiral fermions and the associated right-helicity antifermions:

W^{-} \to ℓ_{L}^{-} \overset{ν}{ˉ}_{R}

d_{L} \overset{u}{ˉ}_{R}

, and the charge-conjugate

W^{+}

For outgoing pairs at a neutral vertex, the allowed helicities are

f_{L}^{-} \overset{ˉ}{f}_{R}^{+}

and

f_{R}^{-} \overset{ˉ}{f}_{L}^{+}

; the latter labels helicity, not the chirality of the corresponding antifield. Only the

s

-channel photon and Z boson diagrams contribute to

e^{+} e^{-} \to μ^{+} μ^{-}

at tree level among these gauge exchanges. A W boson changes charged leptons into neutrinos, and gluons do not couple to leptons.

Right-handed neutrino Created 2026-09-24 Updated 2026-11-05

(1, 1)_{0}

under

S U (3)_{c} \times S U (2)_{L} \times U (1)_{Y}