Yukawa Interaction · aiwiki.page

A Yukawa interaction is an interaction in quantum field theory that couples a scalar or pseudoscalar field to a pair of fermion fields. Named after Hideki Yukawa, it originated in the description of nuclear forces through meson exchange. In the Standard Model, Yukawa interactions between the Higgs field and fermions generate the masses of quarks and charged leptons after electroweak symmetry breaking. The term Yukawa coupling can denote either this interaction or its coupling coefficient. (damtp.cam.ac.uk)

Mathematical form

For a real scalar field $\phi$ and a Dirac fermion field $\psi$ , the interaction part of the Lagrangian density is conventionally written

\mathcal L_{\mathrm{int}}=-g\phi\,\bar\psi\psi, \qquad \bar\psi=\psi^\dagger\gamma^0,

where $g$ is the coupling coefficient and $\bar\psi$ is the Dirac adjoint. The product $\bar\psi\psi$ is a Lorentz scalar, so the interaction is compatible with special relativity. For a pseudoscalar field $a$ , a parity-preserving coupling instead takes the form

\mathcal L_{\mathrm{int}}=-ig_p a\,\bar\psi\gamma^5\psi.

The factor $i$ makes this term Hermitian when $g_p$ and $a$ are real. The distinction between scalar and pseudoscalar couplings determines their transformation under parity and their spin-dependent effects. (damtp.cam.ac.uk)

In four spacetime dimensions, using natural units $\hbar=c=1$ , the scalar field has mass dimension $1$ and the fermion field dimension $3/2$ . Consequently, $\phi\bar\psi\psi$ has dimension $4$ , and $g$ is dimensionless. Such interactions belong to the class of interactions allowed in perturbatively renormalizable theories. Their renormalized coefficients generally depend on the energy scale through the renormalization group. (damtp.cam.ac.uk)

Particle exchange and the Yukawa potential

A Yukawa vertex connects one spin-zero particle line with two fermion lines in a Feynman diagram. It contributes to processes such as fermion scattering through exchange of a virtual particle, and scalar decay into a fermion–antifermion pair when kinematically permitted. (damtp.cam.ac.uk)

For two slowly moving particles with the same scalar coupling $g$ , exchange of a scalar of mass $m_\phi$ produces, at leading order, the Yukawa potential

V(r)=-\frac{g^2}{4\pi}\frac{e^{-m_\phi r}}{r} \qquad (\hbar=c=1).

Here $r$ is their separation. The exponential suppresses the interaction at distances much greater than its characteristic range,

R=\frac{\hbar}{m_\phi c}.

This is a characteristic decay length, not a sharp cutoff. The potential follows from the three-dimensional Fourier transform of the static propagator $1/(\mathbf q^2+m_\phi^2)$ . Its negative sign indicates attraction in this simple equal-coupling case. In the massless limit, the radial dependence becomes $1/r$ . (damtp.cam.ac.uk)

The interaction term and the potential are distinct concepts: the former specifies a local field coupling, while the latter describes a particular nonrelativistic limit. Pseudoscalar exchange generally introduces spin and momentum dependence rather than only the simple central potential above. (damtp.cam.ac.uk)

Origin in nuclear physics

In 1935, Yukawa proposed that the short-range force binding protons and neutrons could arise through exchange of a massive meson. Relating the mediator’s mass to the force’s range supplied a physical explanation for why nuclear forces act over much shorter distances than electromagnetic forces. Yukawa received the Nobel Prize in Physics in 1949 for predicting mesons on the basis of his work on nuclear forces. (nobelprize.org)

The relevant light mesons are pions, which are pseudoscalars. Accordingly, pion–nucleon interactions require a more elaborate spin-dependent description than elementary scalar exchange. Yukawa-type models describe nuclear forces effectively; they do not replace quantum chromodynamics, the underlying theory of the strong interaction. (damtp.cam.ac.uk)

Role in the Standard Model

Before electroweak symmetry breaking, the gauge-invariant Yukawa terms have the structure

\mathcal L_Y= -\bar Q_LY_d H d_R -\bar Q_LY_u\widetilde H u_R -\bar L_LY_e H e_R +\text{h.c.},

where $H$ is the Higgs doublet, $\widetilde H=i\sigma^2H^*$ , $Q_L$ and $L_L$ are left-handed quark and lepton doublets, and the right-handed fields are electroweak singlets. The coefficients $Y_u,Y_d,Y_e$ are matrices in generation space; “h.c.” denotes the Hermitian conjugate. (pdg.lbl.gov)

Through spontaneous symmetry breaking, the Higgs field acquires a vacuum expectation value $v$ . For a fermion mass eigenstate $f$ ,

m_f=\frac{y_fv}{\sqrt2}, \qquad \mathcal L_{hff}=-\frac{m_f}{v}h\bar f f,

where $h$ is the physical Higgs boson field. Thus, the tree-level Higgs–fermion coupling is proportional to the fermion’s mass. The original Yukawa coefficient $y_f$ differs by a factor of $\sqrt2$ from the coefficient $m_f/v$ of the physical Higgs interaction. (pdg.lbl.gov)

The minimal Standard Model treats Yukawa matrices as input parameters rather than predicting their values. It therefore relates fermion masses to Higgs interactions without explaining the observed mass hierarchy. Its particle content lacks right-handed neutrino fields, so it contains no analogous renormalizable Dirac Yukawa term for neutrinos; adding such fields permits one. (damtp.cam.ac.uk)

References

Hideki Yukawa — Nobel Lecturenobelprize.org
The Nobel Prize in Physics 1949nobelprize.org
Status of Higgs Boson Physics — Particle Data Group, 2016pdg.lbl.gov
Electroweak Model and Constraints on New Physics — Particle Data Grouppdg.lbl.gov