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

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 $$(1,2)_{-1/2}.$$

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...

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