section_path: Theoretical aspects/Construction of the Standard Model Lagrangian/Yukawa sector
section_name: Yukawa sector
paragraph_text: Yukawa sector
  
  The Yukawa interaction terms are:
  
  L Yukawa = (Y u) m n (Q ¯ L) m φ ~ (u R) n + (Y d) m n (Q ¯ L) m φ (d R) n + (Y e) m n (ℓ ¯ L) m φ (e R) n + h. c. {\displaystyle {\mathcal {L}}_{\text{Yukawa}}=(Y_{\text{u}})_{mn}({\bar {Q}}_{\text{L}})_{m}{\tilde {\varphi }}(u_{\text{R}})_{n}+(Y_{\text{d}})_{mn}({\bar {Q}}_{\text{L}})_{m}\varphi (d_{\text{R}})_{n}+(Y_{\text{e}})_{mn}({\bar {\ell }}_{\text{L}})_{m}{\varphi }(e_{\text{R}})_{n}+\mathrm {h.c.} }
  
  where Y u {\displaystyle Y_{\text{u}}}, Y d {\displaystyle Y_{\text{d}}}, and Y e {\displaystyle Y_{\text{e}}} are 3 × 3 matrices of Yukawa couplings, with the mn term giving the coupling of the generations m and n, and h.c. means Hermitian conjugate of preceding terms. The fields Q L {\displaystyle Q_{\text{L}}} and ℓ L {\displaystyle \ell _{\text{L}}} are left-handed quark and lepton doublets. Likewise, u R, d R {\displaystyle u_{\text{R}},d_{\text{R}}} and e R {\displaystyle e_{\text{R}}} are right-handed up-type quark, down-type quark, and lepton singlets. Finally φ {\displaystyle \varphi } is the Higgs doublet and φ ~ = i τ 2 φ ∗ {\displaystyle {\tilde {\varphi }}=i\tau _{2}\varphi ^{*}} is its charge conjugate state.
  
  The Yukawa terms are invariant under the SU (2) L × U (1) Y {\displaystyle \operatorname {SU} (2)_{\text{L}}\times \operatorname {U} (1)_{\text{Y}}} gauge symmetry of the Standard Model and generate masses for all fermions after spontaneous symmetry breaking.