4
$\begingroup$

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

$\endgroup$

1 Answer 1

5
$\begingroup$

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...

$\endgroup$
3
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.