Yukawa Couplings and Fermion Masses

Gauge invariance forbids bare masses for the chiral electroweak fermions, but the Higgs doublet permits Yukawa matrices. Once its neutral component has $H^0=(v+h)/\sqrt2$, each matrix produces both a mass matrix and a Higgs interaction with the same flavor structure:

$$M_f=\frac{v}{\sqrt2}Y_f, \qquad \mathcal L_{h\bar ff}=-\frac{h}{v}\,\bar fM_ff .$$

Diagonalizing the up- and down-type matrices leaves their relative left-handed rotation in the charged current; it does not create tree-level flavor-changing neutral Higgs couplings in the minimal model.

Required background. The Higgs doublet and electroweak symmetry breaking supplies $H\sim(\mathbf1,\mathbf2)_{1/2}$ and its vacuum direction. Weyl fields and chirality supplies the relation between chiral fields and Dirac masses.

Helpful background. Multiplets, invariants, and selection rules supplies a systematic way to enumerate the gauge-singlet contractions.

Gauge-invariant Yukawa operators

For $Q=T_3+Y$, use

$$\begin{aligned} Q_L&\sim(\mathbf3,\mathbf2)_{1/6},& u_R&\sim(\mathbf3,\mathbf1)_{2/3},& d_R&\sim(\mathbf3,\mathbf1)_{-1/3},\\ L_L&\sim(\mathbf1,\mathbf2)_{-1/2},& e_R&\sim(\mathbf1,\mathbf1)_{-1},& H&\sim(\mathbf1,\mathbf2)_{1/2}. \end{aligned}$$

Generation indices are suppressed in

$$\mathcal L_Y= -\bar Q_LY_dHd_R -\bar Q_LY_u\widetilde H u_R -\bar L_LY_eHe_R+\mathrm{h.c.}, \qquad \widetilde H=i\tau^2H^* .$$

The conjugate doublet has $Y=-1/2$, which is why it is required for the up-type operator. Every term is a color and weak singlet and has total hypercharge zero. A bare term $\bar f_Lf_R$ fails the weak-representation check because $f_L$ is part of a doublet while $f_R$ is a singlet.

In a renormalizable gauge,

$$H= \begin{pmatrix} \phi^+\\ (v+h+i\chi)/\sqrt2 \end{pmatrix}, \qquad \widetilde H= \begin{pmatrix} (v+h-i\chi)/\sqrt2\\ -\phi^- \end{pmatrix}.$$

Keeping the neutral vacuum and radial field gives

$$\mathcal L_Y\supset -\bar d_LM_dd_R\left(1+\frac{h}{v}\right) -\bar u_LM_uu_R\left(1+\frac{h}{v}\right) -\bar e_LM_ee_R\left(1+\frac{h}{v}\right) +\mathrm{h.c.},$$

with $M_f=vY_f/\sqrt2$. The full component expansion also fixes the charged- and neutral-Goldstone couplings required by gauge identities. The construction and its electroweak charge checks are developed in Schwartz 2014, §29.3.2, pp. 595–599.

The minimal field table contains no $\nu_R$, so it gives no renormalizable neutrino Yukawa operator and leaves the neutrinos massless. Dirac or Majorana neutrino masses require additional fields or effective operators and are not silently included in $Y_e$.

Mass diagonalization and flavor misalignment

For each charged species choose unitary matrices satisfying

$$U_{fL}^\dagger M_fU_{fR}=\widehat M_f =\operatorname{diag}(m_{f_1},\ldots,m_{f_N}), \qquad m_{f_i}\ge0.$$

Writing weak-basis fields as $f_L=U_{fL}f_L^{\,m}$ and similarly on the right makes the mass and radial-Higgs interactions diagonal:

$$\mathcal L_{\rm mass+h} =-\sum_{f,i}m_{f_i}\bar f_i^{\,m}f_i^{\,m} -\frac{h}{v}\sum_{f,i}m_{f_i}\bar f_i^{\,m}f_i^{\,m}.$$

Thus the tree-level one-Higgs coupling is $m_f/v$. This proportionality is a consequence of one doublet generating the mass and the interaction; additional scalar multiplets or higher-dimensional operators can break it.

The quark charged current transforms instead as

$$\bar u_L\gamma^\mu d_L =\bar u_L^{\,m}\gamma^\mu \underbrace{U_{uL}^\dagger U_{dL}}_{V_{\rm CKM}} d_L^{\,m}.$$

The neutral gauge currents are unchanged because the same unitary matrix appears on both sides of a generation-universal generator. The neutral Higgs interaction is diagonal for the same reason that $M_f$ is diagonal. Flavor physics begins with the relative matrix $V_{\rm CKM}$, not with an individual $U_{fL}$, whose entries depend on the weak basis.

What is physically invariant

Individual entries and phases of $Y_u$ or $Y_d$ are not observables. Useful invariant data include

• the singular values of each Yukawa matrix, equivalently the eigenvalues of $Y_f^\dagger Y_f$;
• traces and determinants built from $Y_fY_f^\dagger$;
• relative invariants involving both sectors, including commutators of $Y_uY_u^\dagger$ and $Y_dY_d^\dagger$.

For $N$ quark generations, the two complex Yukawa matrices contain $4N^2$ real parameters. The $U(N)_{Q_L}\times U(N)_{u_R}\times U(N)_{d_R}$ field redefinitions remove $3N^2-1$, because common baryon number remains unbroken. The $N^2+1$ physical parameters can be organized as

$$2N\ \text{masses} +\frac{N(N-1)}2\ \text{angles} +\frac{(N-1)(N-2)}2\ \text{phases}.$$

For one generation there is no mixing angle or physical quark-sector phase. Degenerate or vanishing masses enlarge the allowed field-redefinition freedom, so some parameters in a generic mixing parametrization then cease to be identifiable.

Checks and failure modes

Hypercharge check. Include the minus sign on $\bar Q_L$ and use $Y(\widetilde H)=-1/2$. The sums must vanish before any vacuum value is inserted.

Dimension check. In four spacetime dimensions $Y_f$ is dimensionless, $v$ has mass dimension one, and $M_f$ has dimension one.

Alignment check. The same biunitary transformation must diagonalize the mass term and the one-Higgs Yukawa term. A tree-level off-diagonal $h\bar f_if_j$ term indicates either inconsistent rotations or physics beyond the one-doublet renormalizable model.

Charged-current check. $V_{\rm CKM}$ is unitary if the diagonalization acts on the complete set of canonical quark fields. Apparent nonunitarity can arise after integrating out additional fermions, but not from the minimal rotation above.

Common pitfalls

Treating a Yukawa entry as a measured coupling. Matrix entries change under generation-basis rotations. Quote masses, mixing invariants, or a completely specified basis.

Using $H$ for the up-type mass. With the displayed hypercharges the up operator needs $\widetilde H$. Replacing it by $H$ violates $U(1)_Y$.

Diagonalizing the left-handed sectors together. The two independent mass matrices generally require different left rotations. Their mismatch is precisely the physical charged-current mixing matrix.

Handoff

Flavor and Higgs calculations receive

$$\left\{\widehat M_u,\widehat M_d,\widehat M_e;\ V_{\rm CKM}=U_{uL}^\dagger U_{dL};\ g_{h f_if_j}=\frac{m_{f_i}}v\delta_{ij};\ \text{basis and phase convention}\right\}.$$

The weak vertices are completed on charged and neutral currents. Rephasing invariants and measurable mixing belong to Quark Flavor and CP; neutrino masses and mixing belong to Neutrino and Lepton Physics.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §29.3.2, pp. 595–599. DOI.