Overview of the fields in QFT (FK8017 HT15)

Spinor field Complex scalar field Bispinor field Complex vector field Real scalar multiplet Complex scalar multiplet Real vector multiplet Real scalar field Two-component (Weyl) spinor field Four-component spinor field Real vector field Field: φ(x) ∈ † Field dofs: χ (x) ∈ 2 + hc Field dofs: χ (x), ψ (x) ∈ 2 + hc Field: A (x) ∈ 4 † 4 φ (x) ∈ , i = 1, ..., n † Fields: Ai (x) ∈ 4, i = 1, ..., n R Field dofs: φ(x), φ (x) ∈ C a C a a C µ R Field dofs: Aµ(x),Aµ(x) ∈ C i R φi(x), φi (x) ∈ C, i = 1, ..., n µ R Symmetry: O(1) (no continuous Symmetry: U(1) ' SO(2) = mass Symmetry: O(1) Symmetry: U(1) ' SO(2) = mass Gauge symmetry (if free) Gauge symmetry (if free) Symmetries depend on mass Symmetries depend on mass symmetries) degeneracy of the real doublet degeneracy of the 2-spinor doublet degeneracies (at most O(n)). degenaracies (at most U(n)). To do... (this column is not complete!) Complex field from a real doublet φ1, φ2: Bispinor from a 2-spinor doublet χ1a, χ2a:     φ1(x) φ1(x) 1 1 φ(x) = √ (φ (x) + iφ (x)) χ (x) = √ (χ (x) + iχ (x)) . n . n 2 1 2 a 2 1a 2a  .   .  Φ(x) =  .  ∈ R Φ(x) =  .  ∈ C † 1 1 φ (x) = √ (φ (x) − iφ (x)) ψa(x) = √ (χ1a(x) − iχ2a(x)) 2 1 2 2 φn(x) φn(x)

Symmetrized (hermitian) kinetic term: kinetic term (L = L†) kinetic term (L = L†) kinetic term (alt) (L = L†) kinetic term (alt) (L = L†) kinetic term (L = L) kinetic term (L = L†) kinetic term (L = L†) kinetic term (L = L†) kinetic term (L = L†) 1 µν 1 µ † µ  aa˙  a a˙ aa˙ 1 † µν 1 T µ † µ 1 i i µν L = ∂µφ ∂ φ L = ∂µφ ∂ φ 1 a a˙ L = ψ i∂ ψ + ψ i∂ ψ L = − 4 FµνF L = − FµνF L = ∂µΦ ∂ Φ L = ∂µΦ ∂ Φ L = − GµνG 2 L = 2 χ i∂aa˙ χ + χa˙ i∂ χa aa˙ a˙ a 2 2 4 aa˙ a a˙ where Fµν = 2∂[µAν] where F = 2∂ A i i ijk j k +χ i∂aa˙ χ + χa˙ i∂ χa µν [µ ν] Gµν = 2∂[µA + gf AµAν EoM based kinetic term: ν] kinetic term (alt) kinetic term (alt) (L0 6= L0†) kinetic term (L0 6= L0†) kinetic term (L0 6= L0†) kinetic term (alt) kinetic term (alt) (L0 6= L0†) kinetic term (alt) kinetic term (L0 6= L0†) 0 1 0 † 0 a a˙ a˙ aa˙ 0 1 µ
ν 0 †µ
ν 0 1 T 0 † L = − 2 φ φ L = −φ φ L = χ i∂aa˙ χ 0 a L = 2 A gµν − ∂µ∂ν A L = A g − ∂ ∂ A L = − 2 Φ Φ L = −Φ Φ  L = ψ i∂aa˙ ψ + χa˙ i∂ χa µν µ ν  0 1 µ 0 † µ 0 1 µν 0 1 µ 0 † µ L = L + ∂µ(φ ∂ φ) L = L + ∂ (φ ∂ φ) L = L + ∂µ(Aν F ) 0 † µν L = L + ∂µ(Φ ∂ Φ) L = L + ∂ (Φ ∂ Φ) 2 µ 2 L = L + ∂µ(Aν F ) 2 µ Optional quadratic terms (mass and gauge fixing): mass term mass term mass term mass term mass term mass term quadric term quadratic term mass term

1 2 2 2 † 1 a ∗ a˙
a ∗ a˙ 1 2 µ 2 † µ 1 2 T 2 † 1 2 i i µ − m φ − m φ φ − mχ χa + m χ χ
+ m AµA + m A A + m A A 2 2 a˙ − mψ χa + m χa˙ ψ 2 µ − 2 m Φ Φ − m Φ Φ 2 µ 1 a a˙
∗ − m χ χa + χ χ for m = m a a˙
∗ Free theory = kinetic + opt. mass term 2 a˙ −m ψ χa + χa˙ ψ for m = m gauge fixing term gauge fixing term For the general mass term, For the general mass term, gauge fixing term 1 µ 2 †µ ν − 1 ΦTMΦ, −Φ†MΦ, 1 i µ i ν − 2 ζ (∂µA ) − ζ (∂µA )(∂νA ) 2 − 2 ζ (∂µA )(∂νA ) where M is a positive definite where M is a positive definite Possible self-interaction terms: mixed term mixed terms (+hc) real matrix with k distinct Hermitian matrix with k dis- cubic term cubic term eigenvalues mi each with de- tinct eigenvalues mi each with ghost term for QCD −(A Aµ)(∂ Aν) −(A† Aµ)(∂ Aν) P P − 1 µ φ3 −µ φ†(φ† + φ)φ µ ν µ ν generacy ni (k ≤ n = ni), degeneracy ni (k ≤ n = ni), ∂µη [∂ η˜ + g f ijkη˜jAk ] 3! the internal symmetry group is, the internal symmetry group is, i µ i s µ quartic term quartic term i µ i ijk µ i j k = ∂µη˜ ∂ η +gsf (∂ η˜ )A η quartic term quartic term O(n1) × · · · × O(nk). U(n1) × · · · × U(nk). µ − (A Aµ)2 † µ 2 1 4 † 2 !  T µ − (AµA ) − λ φ − λ (φ φ) ψa := χa χa(x) † χa˙ ghost term for QED 4! Ψ(x) = a˙ , Ψ = , Note: There can be other internal symmetries (in addition to those ψ (x) ψa    T Note: The self-interaction terms for vector fields lead to incon- coming out of mass term degeneracy). ∂ η ∂µη˜ χa(x) † χa˙ µ ΨM(x) = , Ψ = ,  aT sistencies unless their coupling constants are precisely chosen on χa˙ (x) M χ † ψ a Ψ = Ψ β = , the basis of a special type of symmetry, which must involve sev-  T χa˙ † χa eral vector fields. This symmetry underlies the non-Ahelian gauge Ψ = Ψ β =  µ   b M M 0 σ 0 δa χa˙ µ ab˙ theories. γ = µab˙ , β = a˙ σ 0 δ b˙ 0 Free-field examples (with propagators): Massive neutral spin-0 Massive charged spin-0 / KGF Majorana Lagrangian (L= 6 L†) Dirac Lagrangian (L= 6 L†) Massless neutral spin-1 (Maxwell) Massive neutral spin-1 (Proca) 1 2 † 2 µ µ 1   1   L = 2 φ (− − m )φ L = φ (− − m )φ L = Ψ (iγ ∂ − m)Ψ L = Ψ(iγ ∂ − m)Ψ µ ν µ 2 ν Legend: M µ M µ L = 2 A gµν − (1 − ζ)∂µ∂ν A L = 2 A gµν( + m ) − ∂µ∂ν A † µ g : Minkowski metric, diag(1, −1, −1, −1) Meson propagator: Meson propagator: L = i∂µΨ(γ − m)Ψ µν Photon propagator: Massive vector propagator: µ, ν, ... : (World) tensor indices 0 1 µ←→ i i L = Re L = Ψ iγ ∂ µΨ − mΨΨ i∆ (k) = i∆ (k) = 2 µν ζ−1 µ ν
µν 1 µ ν
F 2 2 + F 2 2 + −i g − k k −i g − k k α, β, ... : Four-component (Dirac) spinor indices k − m + iε k − m + iε = L − 1 i∂ (ΨγµΨ) = L0† µν ζ µν m2 ˙ 2 µ iD (k) = iD (k) = a, b, ..., a,˙ b, ... : Two-component (Weyl) spinor indices F 2 + F k2 − m2 + iε+ k + iε µ aa˙ µ aa˙ Dirac fermion propagator: ∂aa˙ := σaa˙ ∂µ, ∂ := σ ∂µ ζ = 1 : Feynman gauge (ξ = ζ−1 = 1) St¨uckelbergLagrangian µ i(γ p + m) −1 iS (p) = µ ζ → ∞ : Landau gauge (ξ = ζ = 0) L = − 1 F F µν + 1 (∂ φ+mA )(∂µφ+mAµ) F p2 − m2 + iε+ 2 µν 2 µ µ Gauge-fixing φ = 0, yields the Proca action.

Possible interaction terms: Prescription for building a consistent Lagrangian: Yukawa coupling (Spinor) QED (Spinor) QED 1. The Lagrangian must be real modulo total derivative (required by CPT invariance). µ µ −g ΨΨ φ Scalar QED −e Ψγ Ψ Aµ −e Ψγ Ψ Aµ 2. All the added terms must be Lorentz-invariant. 4 µ † 3. All the added terms must be of dimension less or equal M . +e φ∂ φ Aµ Yukawa coupling Scalar QED N.b. Any interaction of higher dimension than M4 leads to a nonrenormalizable theory. † µ µ † 4 +e (φ φ)(AµA ) −g ΨΨ φ +e φ∂ φ Aµ Hence, ignoring all the constants, the dimension of any term must be at most M since the 4 † µ dimension of a Lagrangian density is M . +e (φ φ)(AµA ) 0 4 1 3/2 2 [S] = M ,[L] = M ,[∂µ] = [φ] = [Aµ] = M , [Ψ] = [χa] = M ,[Fµν] = M .

Fields in QFT – by M.B.Kocic – Version 1.0 (2016-01-10)