Is it accidental or deeply meaningful that the Higgs and the left-handed lepton have the same quantum numbers?

Precisely, under $SU(3) \times SU(2) \times U(1)$:

  1. the Higgs has the quantum numbers: singlet under $SU(3)$, doublet under $SU(2)$, charge 1/2 under $U(1).$ Denote as $$(1,2)_{1/2}.$$

  2. the left-handed lepton has the quantum numbers: singlet under $SU(3)$, doublet under $SU(2)$, charge -1/2 under $U(1).$ Denote as $$(1,2)_{-1/2}.$$


1 Answer 1


Well, do they?

In your ("modern", half-scale) conventions, $Q=T_3+Y_W$, you observe the lepton isodoublets with $Y_W=-1/2$, $$ l_L = \begin{pmatrix} \nu\\ e^-_L\end{pmatrix} $$ the isosinglets with $Y_W=-1$ $$ e^-_R, $$ (and, arguably, $\nu_R$ with $Y_W=0$). They all have lepton number 1, which is thus lacking from the Higgs doublet, with $Y_W=1/2$, $$ H=\begin{pmatrix} H^+\\ H_0\end{pmatrix}, $$ and the conjugate with $Y_W=-1/2$, $$ \tilde H=\begin{pmatrix} H_0^*\\ -H^-\end{pmatrix}. $$

The situation is somewhat analogous to the quark sector, whose color in the Yukawa coupling term is matched by the antiquark's; whereas in the Yukawas of the lepton sector it is just the lepton number that's matched by the antilepton's.

You then see that , e.g., $$ \frac{m_e}{v} ~ \overline{l_L}\cdot H e ^ -_R $$ has the hypercharges (and hence charges) cancelling to 0=1/2+1/2-1, as required of a Lagrangian term.

Likewise for a Dirac neutrino mass term, $$ \frac{m_\nu}{v} ~ \overline{l_L}\cdot \tilde H \nu_R, $$ hence $Y_W= 1/2 -1/2 +0$. So, even outside the quark Yukawa, the SM has covered the entire waterfront.

You might, if you insisted, declare this fact not a coincidence, but a necessity, starting from the Y=0 right handed neutrino, so a Dirac mass term would force the hypercharges of the lepton doublet and the conjugate Higgs to be the same. But, to me, this appears like a shoulder-shrugging tautology, not a deep conceptual cornerstone...

  • $\begingroup$ Thanks +1, ("modern") conventions, you mean "modern" in the sense of? $\endgroup$ Mar 21, 2021 at 1:46
  • $\begingroup$ Explained in Wikipedia. Older , original, usage uses twice these eigenvalues. $\endgroup$ Mar 21, 2021 at 2:12
  • $\begingroup$ thanks I will accept it after I fully digest! +1 $\endgroup$ Mar 22, 2021 at 18:54

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