# Higgs couplings and fermion masses

As far as I understand, the Higgs mechanism is a crucial component of the standard model, which is responsible for the weak gauge bosons acquiring mass, otherwise forbidden by renormalizability constraints. However, is there any justification for the fermion masses arising from a Yukawa coupling to the Higgs field, or is this just assumed as a handy byproduct with no alternative explanations present? What counters the idea of considering the masses as fundamental parameters, instead of associated with the Yukawa couplings?

• I think LHC observation of Higgs decay also suggests couplings proportional to mass, for heavy fermions like top, bottom, tau. – Mitchell Porter May 31 '18 at 12:39
• see the answer of khzoo here physics.stackexchange.com/q/409189 – anna v May 31 '18 at 16:36

@gj255 's answer is impeccable, of course, but I'll illustrate in detail his point for the mass of the electron, since this is really "the second job of the Higgs", and has nothing to do with the Higgs mechanism--only the SSB! That is, it cares not about the eating of the goldstons by the gauge bosons. However, it makes the gauge theory possible to start with, safeguarding gauge invariance. Weinberg was justifiably proud of this particular technical advance involved in this "second job of the Higgs". There is no other plausible way to generate weakly interacting fermion masses, so, without it, the SM is a non-starter!

First note $e_R$, the right-handed electron is a gauge singlet, but $\begin{pmatrix} \nu_L \\ e_L \end{pmatrix}$ is a gauge SU(2) doublet. So a brute-force mass term $m_e (\bar{e_L} e_R + \bar{e_R}e_L)$ would not be an SU(2) singlet, and the gauge invariance would be broken with nasty, forbidding consequences.

The Higgs doublet, however, saves the day: $$\Phi= \frac{h+v}{\sqrt{2}} e^{2i \vec {\pi}\cdot\vec {\tau} /v}\begin{pmatrix} 0 \\ 1 \end{pmatrix},$$ where h is the neutral Higgs, the 3 $\vec{\pi}$s are the goldstons eaten up by the Ws and the Z and absent in the unitary gauge, of no concern to us here, and v ~ 0.25 TeV is the cornerstone Higgs v.e.v.

Dotting two weak isodoublets together then does yield an SU(2) singlet, preserving gauge invariance: $$-y \overline{ \begin{pmatrix} \nu_{eL} \\ e_L \end{pmatrix} } \cdot \begin{pmatrix} 0 \\ \frac{h+v}{\sqrt 2} \end{pmatrix} ~e_R +\hbox{h.c.},$$ where y is an undetermined dimensionless Yukawa coupling. The gauge theory is thus saved.

We can rewrite this lagrangian term as $$-\frac{y v}{\sqrt 2}(\bar{e_L}e_R +\bar{e_R} e_L) -\frac{y h}{\sqrt 2}(\bar{e_L}e_R +\bar{e_R} e_L).$$ We can then identify $m_e=\frac{y v}{\sqrt 2}$, and thus a different yukawa for every lepton/fermion, really.

But then the trailing term with the Higgs coupling will have its Yukawa coupling strength be $m_e/v$, and correspondingly for other particles: fermion masses cannot avoid proportionality to Yukawas. Weinberg was quite delighted to observe this peculiar feature in his original paper, half a century ago, contrasting the Higgs coupling to the muon versus the electron.

Takeaway: SSB crucial, Higgs mechanism irrelevant, realistic consistent gauge theory impossible without the Higgs. Fermion masses are not an aside at all of the SM; they are the key to it.

It's not possible to give the fermions bare mass terms in a gauge invariant way. In terms of left-handed and right-handed Weyl spinors, the mass term we desire is

$$m( \bar{\psi}_L \psi_R + \bar{\psi}_R \psi_L) \,,$$

but for all fermions in the standard model, $\psi_R$ is a singlet under $\mathrm{SU}(2)$ whilst $\psi_L$ is one component of an $\mathrm{SU}(2)$ doublet.