Yukawa interaction

A Yukawa interaction couples left- and right-handed fermions through a scalar field. Replacing the Higgs field by its vacuum expectation value turns a gauge-invariant Standard Model Yukawa interaction into a fermion mass term.

The combined transformation $\psi \mapsto e^{-i\alpha {\gamma }_5/2}\psi$ and $({\varphi }_1+i{\varphi }_2)\mapsto e^{i\alpha }({\varphi }_1+i{\varphi }_2)$ preserves this interaction. The Dirac adjoint transforms with the same axial exponential as $\psi$. For potential $-{\mu }^2{r}^2/2+\lambda r^4/4$, $v=\mu /\sqrt{\lambda }$ gives a radial scalar of squared mass $2\lambda v^2$, a massless Goldstone boson and fermion physical mass $|g|v$. Cubic scalar vertices appear after the shift: $hhh$ has factor $-6i\lambda v$ and $h\pi \pi$ has factor $-2i\lambda v$.

A nonderivative interaction of two real scalar legs with a fermion pair in $2+1$ dimensions. The engineering dimensions of relativistic fields are $[\varphi ]=1/2$ and $[\psi ]=1$, so $g$ is a marginal coupling by classical power counting. The internal symmetry $\varphi \mapsto -\varphi$ forbids odd-scalar interactions, but allows a sextic scalar interaction. The two-component Dirac field representation must be distinguished from four-component gamma matrices in four spacetime dimensions.

Four quartic Yukawa vertices on a fermion loop with one internal scalar chord give six external scalar legs, two loops and overall logarithmic degree. The coefficient is nonzero: in a constant scalar background $v$, the induced cubic Yukawa coupling is $y=gv$ and the fermion mass is $\mu =m+g{v}^2/2$. The two-loop sunset diagram in the quantum effective potential has a logarithmic pole proportional to $y^2(4{\mu }^2-{\mathcal{B}}^2)$, where ${\mathcal{B}}^2$ is the background scalar squared mass. Its $v^6$ term is $g^4{v}^6$ when the tree sextic coupling vanishes. Thus zero renormalized sextic coupling at one scale does not eliminate the sextic counterterm or make that zero stable under renormalization.

With zero scalar sextic coupling, the only potentially logarithmically divergent one-loop six-scalar vertex has three quartic Yukawa vertices on a fermion loop. Its leading numerator is $(/k{)}^3=k^2/k$, whose trace vanishes. A mass insertion or external momentum lowers the degree and makes the integral convergent. This is not vanishing of every odd gamma-matrix trace: for ${\gamma }^0={\sigma }_3$, ${\gamma }^1=i{\sigma }_1$, ${\gamma }^2=i{\sigma }_2$, one has $\mathrm{tr}({\gamma }^0{\gamma }^1{\gamma }^2)=-2i$. No one-loop sextic counterterm is needed, although a finite six-point vertex may remain.

The physical Higgs boson interaction $\mathcal{L}\supset -(m_e/v)h\bar{e}e$ after electroweak symmetry breaking. It is distinguished from the unbroken-phase electron Yukawa coupling $y_e=\sqrt{2}{m}_e/v$. This small coefficient explains the electron Yukawa suppression of Higgs production in direct electron–positron annihilation.

For one four-component Dirac field with scalar interaction $-y\bar{\psi }\psi \varphi$ and the mostly-plus propagator convention, the fermion loop numerator is $4[M^2-k\cdot (k-p)]$. Reduce it to a bubble and tadpole: $-4y^2[(2M^2+p^2/2)J-I]$. Dimensional regularization gives pole parts $J=2/(\epsilon 16{\pi }^2)$ and $I=-2M^2/(\epsilon 16{\pi }^2)$. The local pole then requires both wavefunction renormalization and a scalar mass counterterm.

For a real scalar coupled through $-g\bar{\psi }\varphi \psi$, the scalar-fermion Feynman vertex is $-ig$. Two-fermion scattering has scalar exchange in the $t$ and $u$ channels with a relative minus from exchanging identical external fermions. Fermion-antifermion scattering has $t$ exchange and $s$ annihilation, again with a relative fermionic minus for a fixed state ordering. Fermion-scalar scattering has the two orderings along one open fermion line, using internal momenta $p_1+p_2$ and $p_1-p_4$; these have the same sign. The Mandelstam variables specify the propagator denominators.

The pseudoscalar Yukawa interaction with coefficient $-g$ has the displayed Feynman vertex. Its two-fermion tree-level Feynman diagrams are the $t$ and $u$ exchanges; antisymmetry of the final identical fermions requires their relative minus. Fermion-antifermion scattering has $t$ exchange and $s$ annihilation, again with a relative minus. With initial and final states ordered as $b_1^{†}{d}_2^{†}|0⟩$ and $b_3^{†}{d}_4^{†}|0⟩$, the normal-ordered current contributes $-d^{†}d\bar{v}{\gamma }^{5}v$ in the exchange channel but $db$ and $b^{†}{d}^{†}$ in the annihilation channel. These fermionic signs fix the overall sign as well as their interference; changing an external-state phase changes both terms together.

The Standard Model lepton doublet $L=({\nu }_L,e_L{)}^T$ has hypercharge $-1/2$, the electron singlet $e_R$ has $-1$, and the Higgs doublet has $1/2$. Therefore $-y_e\bar{L}\varphi e_R+h.c.$ is gauge-invariant even though a bare ${\bar{e}}_L{e}_R$ mass is not. The vacuum expectation value generates a Dirac mass $|y_e|v/\sqrt{2}$. In unitary gauge, the Higgs boson coupling is $-(m_e/v)h\bar{e}e$ after rephasing the electron.

A pseudoscalar Yukawa interaction couples a Dirac field bilinear containing the chirality matrix to a spin-zero field. In the standard mostly-minus gamma matrix convention with ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ and the usual Dirac adjoint, $ig\bar{\psi }{\gamma }^5\psi \varphi$ is Hermitian for real $g$ and real $\varphi$. If $\varphi$ is a pseudoscalar, the interaction also preserves parity symmetry in quantum field theory; if $\varphi$ is parity even, it breaks that symmetry. Moving the factor $i$ into the definition of ${\gamma }^5$ changes the displayed vertex coefficient but not the interaction. Each vertex has one bosonic leg and two oppositely oriented fermion legs, and preserves Dirac fermion number conservation.

For an unbroken parity symmetry in quantum field theory with $\varphi$ a pseudoscalar, pure-scalar local counterterms of mass dimension at most four are even in $\varphi$. A four-vertex fermion loop requires a quartic counterterm, even when the initial Lagrangian density shows only a Yukawa interaction. The one-point and three-point pure-boson vertices vanish at all orders: their available external four-momenta cannot form a parity-odd Lorentz scalar.

With the usual Dirac adjoint, Hermitian ${\gamma }^5$ and $\{{\gamma }^0,{\gamma }^5\}=0$, taking the adjoint gives ${\psi }^{†}{\gamma }^5{\gamma }^0\psi =-{\psi }^{†}{\gamma }^0{\gamma }^5\psi$. Consequently a real pseudoscalar with coupling $-g\varphi \bar{\psi }{\gamma }^5\psi$ has a Hermitian interaction only if $g^{\ast }=-g$. Writing $g=i{g}_P$ with real $g_P$ gives the familiar interaction $-i{g}_P\varphi \bar{\psi }{\gamma }^5\psi$. Formal Euler-Lagrange field equations obtained with independent spinor and adjoint variables can still be computed for the original symbolic coupling; it must not silently be identified with a real physical coupling.

After a scalar vacuum expectation value, a Yukawa matrix produces a bilinear $-{\bar{f}}_L{M}_f{f}_R+h.c.$. Independent unitary left/right rotations diagonalize $M_f$ with nonnegative masses. Extra scalar-dependent operators can make the physical scalar coupling matrix fail to diagonalize with the mass matrix.

A Yukawa matrix contains the flavor-space couplings between left-handed fermions, right-handed fermions, and a scalar field. Its singular values determine fermion masses after symmetry breaking, while mismatches between its diagonalizing transformations produce flavor mixing.

Let the left and right fermions transform in $D\oplus \epsilon$, where $D$ is the two-dimensional irreducible representation of the symmetric group $S_3$ and $\epsilon$ is the sign representation. Let the scalar field be a trivial representation. In matching orthonormal bases, invariance of ${\bar{\psi }}_L\lambda {\psi }_R$ says that $\lambda$ commutes with $D\oplus \epsilon$. The Schur lemma forces each diagonal irreducible block to be scalar and both mixed blocks to vanish, giving the displayed form. Thus after a vacuum expectation value $h$, the physical fermion mass matrix has singular values $|ha|,|ha|,|hb|$: unbroken $S_3$ enforces a degenerate doublet. In a common nonorthonormal basis with invariant Gram matrix $K$, the doublet coefficient is $aK$; canonical normalization recovers $a{I}_2$.

A singular value decomposition of the charged-lepton Yukawa matrix uses independent unitary rotations of left- and right-handed leptons. Their gauge-covariant kinetic terms are unchanged because each rotation mixes identical Standard Model representations. After electroweak symmetry breaking, the nonnegative singular values give Dirac masses $m_i=y_{i}v/\sqrt{2}$ and diagonal Higgs boson couplings $m_i/v$. When the neutrinos are massless, rotating their left-handed fields with the charged partners preserves the diagonal weak charged current.