Quantum Field Theory: Standard Model and Electroweak Symmetry Breaking

José Ignacio Illana

Taller de Altas Energías Benasque, September 17, 2013 1 Outline

1. Quantum Field Theory: Gauge Theories B The symmetry principle B Quantization of gauge theories B Spontaneous Symmetry Breaking 2. The Standard Model B Gauge group and particle representations B The SM with one family: electroweak interactions B Electroweak SSB: Higgs sector, gauge boson and fermion masses B Additional generations: fermion mixings B Complete Lagrangian and Feynman rules 3. Phenomenology of the Electroweak Standard Model B Input parameters, experiments, observables, precise predictions B Global fits 2 1. Gauge Theories

3 The symmetry principle free Lagrangian

Lagrangian of a free fermion field ψ(x): • (Dirac) = ψ(i/∂ m)ψ ∂/ γµ∂ , ψ = ψ†γ0 L0 − ≡ µ

Invariant under global U(1) phase transformations: ⇒ iqθ ψ(x) ψ0(x) = e− ψ(x) , q, θ (constants) R 7→ ∈

By Noether’s theorem there is a conserved current: ⇒ µ µ µ j = q ψγ ψ , ∂µ j = 0

and a Noether charge: Z 3 0 Q = d x j , ∂tQ = 0

1. Gauge Theories 4 The symmetry principle free Lagrangian

A quantized free fermion field: • Z 3 d p  (s) ipx † (s) ipx ψ(x) = ap u (p)e + b v (p)e 3p ∑ ,s − p,s (2π) 2Ep s=1,2

– is a solution of the Dirac equation (Euler-Lagrange):

(i/∂ m)ψ(x) = 0 , (/p m)u(p) = 0 , (/p + m)v(p) = 0 , − −

– is an operator from the canonical quantization rules (anticommutation):

† † 3 3 ap , a = bp , b = (2π) δ (p k)δ , ap , ak = = 0 , { ,r k,s} { ,r k,s} − rs { ,r ,s} ···

that annihilates/creates particles/antiparticles on the Fock space of fermions

1. Gauge Theories 5 The symmetry principle free Lagrangian

For a quantized free fermion field: •

Normal ordering for fermionic operators (H spectrum bounded from below): ⇒ † † † † : ap a : a ap ,: bp b : b bp ,r q,s ≡ − q,s ,r ,r q,s ≡ − q,s ,r

The Noether charge is an operator: ⇒ Z Z 3 3 0 d p  † †  : Q : = q d x : ψγ ψ : = q 3 ∑ ap,sap,s bp,sbp,s (2π) s=1,2 −

Q a† 0 = +q a† 0 (particle), Q b† 0 = q b† 0 (antiparticle) k,s | i k,s | i k,s | i − k,s | i

1. Gauge Theories 6 The symmetry principle gauge symmetry dictates interactions

To make invariant under local gauge transformations of U(1): • L0 ≡ iqθ(x) ψ(x) ψ0(x) = e− ψ(x) , θ = θ(x) R 7→ ∈ perform the minimal substitution:

∂ D = ∂ + ieqA (covariant derivative) µ → µ µ µ

where a gauge field Aµ(x) is introduced transforming as:

1 iqθ(x) A (x) A0 (x) = A (x) + ∂ θ(x) D ψ e D ψ ψD/ψ inv. µ 7→ µ µ e µ ⇐ µ 7→ − µ The new Lagrangian contains interactions between ψ and A : ⇒ µ ( µ coupling e = eq ψγ ψAµ ∝ Lint − charge q

(= e jµ A ) − µ 1. Gauge Theories 7 The symmetry principle gauge invariance dictates interactions

Dynamics for the gauge field add gauge invariant kinetic term: • ⇒ 1 (Maxwell) = F Fµν F = ∂ A ∂ A F L1 −4 µν ⇐ µν µ ν − ν µ 7→ µν

The full U(1) gauge invariant Lagrangian for a fermion field ψ(x) reads: • 1 = ψ(iD / m)ψ F Fµν (= + + ) Lsym − − 4 µν L0 Lint L1 The same applies to a complex scalar field φ(x): • 1 = (D φ)†Dµφ m2φ†φ λ(φ†φ)2 F Fµν Lsym µ − − − 4 µν

1. Gauge Theories 8 The symmetry principle non-Abelian gauge theories

A general gauge symmetry group G is an N-dimensional compact Lie group • iT θa g G , g(θ) = e− a , a = 1, . . . , N ∈ θa = θa(x) R , T = Hermitian generators , [T , T ] = i f T (Lie algebra) ∈ a a b abc c Tr T T 1 δ , structure constants: f = 0 Abelian { a b} ≡ 2 ab abc f = 0 non-Abelian abc 6 Finite-dimensional irreducible representations are unitary: ⇒   ψ1    .  d-multiplet : Ψ(x) Ψ0(x) = U(θ)Ψ(x) , Ψ =  .  7→   ψd d d matrices : U(θ) [given by T algebra representation] × { a}

1. Gauge Theories 9 The symmetry principle non-Abelian gauge theories

Examples: G N Abelian • U(1) 1 Yes SU(n) n2 1 No (n n matrices with det = 1) − × – U(1): 1 generator (q), one-dimensional irreps only

– SU(2): 3 generators

fabc = eabc (Levi-Civita symbol) 1 Fundamental irrep (d = 2): Ta = 2 σa (3 Pauli matrices) ∗ adj Adjoint irrep (d = N = 3): (T ) = i f ∗ a bc − abc – SU(3): 8 generators 123 458 678 √3 147 156 246 247 345 367 1 f = 1, f = f = 2 , f = f = f = f = f = f = 2 1 − Fundamental irrep (d = 3): Ta = 2 λa (8 Gell-Mann matrices) ∗ adj Adjoint irrep (d = N = 8): (T ) = i f ∗ a bc − abc (for SU(n): fabc totally antisymmetric)

1. Gauge Theories 10 The symmetry principle non-Abelian gauge theories

To make invariant under local gauge transformations of G: • L0 ≡

Ψ(x) Ψ0(x) = U(θ)Ψ(x) , θ = θ(x) R 7→ ∈ substitute the covariant derivative:

a ∂µ Dµ = ∂µ igWe µ , We µ TaW → − ≡ µ a where a gauge field Aµ(x) per generator is introduced, transforming as:

† i † We µ(x) We 0 (x) = UWe µ(x)U (∂µU)U DµΨ UDµΨ ΨD/Ψ inv. 7→ µ − g ⇐ 7→

The new Lagrangian contains interactions between Ψ and Wa: ⇒ µ ( µ a coupling g int = g Ψγ TaΨWµ ∝ L charge Ta

µ a (= g ja Wµ)

1. Gauge Theories 11 The symmetry principle non-Abelian gauge theories

Dynamics for the gauge fields add gauge invariant kinetic terms: • ⇒

1 n µνo 1 a a,µν † (Yang-Mills) = Tr We µνWe = W W We µν UWe µνU LYM −2 −4 µν ⇐ 7→

We µν DµWe ν DνWe µ = ∂µWe ν ∂νWe µ ig[We µ, We ν] ≡ − − − Wa = ∂ Wa ∂ Wa + gf WbWc ⇒ µν µ ν − ν µ abc µ ν

contains cubic and quartic self-interactions of the gauge fields Wa: ⇒ LYM µ 1 = (∂ Wa ∂ Wa)(∂µWa,ν ∂νWa,µ) Lkin −4 µ ν − ν µ − 1 = gf (∂ Wa ∂ Wa)Wb,µWc,ν Lcubic −2 abc µ ν − ν µ 1 = g2 f f WaWbWc,µWd,ν Lquartic −4 abe cde µ ν

1. Gauge Theories 12 Quantization of gauge theories propagators

The (Feynman) propagator of a scalar field: • Z 4 † d p i ip (x y) D(x y) = 0 T φ(x)φ (y) 0 = e− · − − h | { } | i (2π)4 p2 m2 + ie − is a Green’s function of the Klein-Gordon operator:

2 4 i (x + m )D(x y) = iδ (x y) De(p) = − − − ⇔ p2 m2 + ie − The propagator of a fermion field: • Z 4 d p i ip (x y) S(x y) = 0 T ψ(x)ψ(y) 0 = (i/∂x + m) e− · − − h | { } | i (2π)4 p2 m2 + ie − is a Green’s function of the Dirac operator:

4 i ( /i∂x m)S(x y) = iδ (x y) Se(p) = − − − ⇔ /p m + ie −

1. Gauge Theories 13 Quantization of gauge theories propagators

BUT the propagator of a gauge field cannot be defined unless is modified: • L 1 1 (e.g. modified Maxwell) = F Fµν (∂µ A )2 L −4 µν −2ξ µ

    ∂ ∂ µν 1 µ ν Euler-Lagrange: L ∂µ L = 0 g  1 ∂ ∂ Aµ = 0 ∂Aν − ∂(∂µ Aν) ⇒ − − ξ

– In momentum space the propagator is the inverse of:     2 µν 1 µ ν i kµkν k g + 1 k k Deµν(k) = gµν + (1 ξ) − − ξ ⇒ k2 + ie − − k2

Note that ( k2gµν + kµkν) is singular! ⇒ − One may argue that above will not lead to Maxwell equations . . . ⇒ L unless we fix a (Lorentz) gauge where:

µ µ µ ∂ A = 0 A A0 = A + ∂ Λ with ∂ ∂ Λ ∂ A µ ⇐ µ 7→ µ µ µ µ ≡ − µ

1. Gauge Theories 14 Quantization of gauge theories gauge fixing (Abelian case)

The extra term is called Gauge Fixing: • 1 = (∂µ A )2 LGF −2ξ µ

modified equivalent to Maxwell Lagrangian just in the gauge ∂µ A = 0 ⇒ L µ the ξ-dependence always cancels out in physical amplitudes ⇒

Several choices for the gauge fixing term (simplify calculations): R gauges • ξ

igµν (’t Hooft-Feynman gauge) ξ = 1 : Deµν(k) = −k2 + ie   i kµkν (Landau gauge) ξ = 0 : Deµν(k) = gµν + k2 + ie − k2

1. Gauge Theories 15 Quantization of gauge theories gauge fixing (non-Abelian case)

For a non-Abelian gauge theory, the gauge fixing terms: •

1 µ a 2 GF = ∑(∂ Wµ) L −2ξ a

allow to define the propagators:   ab iδab kµkν De (k) = gµν + (1 ξ) µν k2 + ie − − k2

BUT, unlike the Abelian case, this is not the end of the story . . .

1. Gauge Theories 16 Quantization of gauge theories Faddeev-Popov ghosts

Add Faddeev-Popov ghost fields c (x) in the adjoint irrep: • a adj adj adj = (∂µc )(D ) c = (∂µc )(∂ c gf c Wc ) D = ∂ igT Wc LFP a µ ab b a µ a − abc b µ ⇐ µ µ − c µ Computational trick: anticommuting scalar fields, just in loops as virtual particles

iδab De (k) = [( 1) sign for closed loops! (like fermions)] ab k2 + ie − Faddeev-Popov ghosts needed to preserve gauge symmetry: ⇒

= (g k2 k k )Π(k2) µν − µ ν

1. Gauge Theories 17 Quantization of gauge theories complete Lagrangian

Then the complete quantum Lagrangian is •

+ + Lsym LGF LFP

Note that in the case of a massive vector field ⇒ 1 1 (Proca) = F Fµν + M2 A Aµ L −4 µν 2 µ it is not gauge invariant – The propagator is:

i  kµkν  Deµν(k) = gµν + k2 M2 + ie − M2 −

1. Gauge Theories 18 Spontaneous Symmetry Breaking discrete symmetry

Consider a real scalar field φ(x) with Lagrangian: • 1 1 λ = (∂ φ)(∂µφ) µ2φ2 φ4 invariant under φ φ L 2 µ − 2 − 4 7→ −

1 = (φ˙ 2 + ( φ)2) + V(φ) ⇒ H 2 ∇ 1 1 V = µ2φ2 + λφ4 2 4

µ2, λ R (Real/Hermitian Hamiltonian) and λ > 0 (existence of a ground state) ∈ 2 (a) µ > 0: min of V(φ) at φcl = 0 r µ2 (b) µ2 < 0: min of V(φ) at φ = v − , in QFT 0 φ 0 = v = 0 (VEV) cl ≡ ± λ h | | i 6 –A quantum field must have v = 0 φ(x) v + η(x) , 0 η 0 = 0 a 0 = 0 ⇒ ≡ h | | i | i

1. Gauge Theories 19 Spontaneous Symmetry Breaking discrete symmetry

At the quantum level, the same system is described by η(x) with Lagrangian: • 1 λ = (∂ η)(∂µη) λv2η2 λvη3 η4 not invariant under η η L 2 µ − − − 4 7→ − (mη = √2λ v)

Lesson: ⇒ (φ) had the symmetry but the parameters can be such that the ground state of L the Hamiltonian is not symmetric (Spontaneous Symmetry Breaking)

Note: ⇒ One may argue that (η) exhibits an explicit breaking of the symmetry. However L this is not the case since the coefficients of terms η2, η3 and η4 are determined by just two parameters, λ and v (remnant of the original symmetry)

1. Gauge Theories 20 Spontaneous Symmetry Breaking continuous symmetry

Consider a complex scalar field φ(x) with Lagrangian: • † µ 2 † † 2 iqθ = (∂ φ )(∂ φ) µ φ φ λ(φ φ) invariant under U(1): φ e− φ L µ − − 7→ r v µ2 λ > 0, µ2 < 0 : 0 φ 0 , v = − h | | i ≡ √2 | | λ Take v R+. In terms of quantum fields: ∈ 1 φ(x) [v + η(x) + iχ(x)], 0 η 0 = 0 χ 0 = 0 ≡ √2 h | | i h | | i 1 1 λ 1 = (∂ η)(∂µη) + (∂ χ)(∂µχ) λv2η2 λvη(η2 + χ2) (η2 + χ2)2 + λv4 L 2 µ 2 µ − − − 4 4 Note: if veiα (complex) replace η by (η cos α χ sin α) and χ by (η sin α + χ cos α) −

The actual quantum Lagrangian (η, χ) is not invariant under U(1) ⇒ L U(1) broken one scalar field remains massless: m = √2λ v, m = 0 ⇒ η χ 1. Gauge Theories 21 Spontaneous Symmetry Breaking continuous symmetry

Another example: consider a real scalar SU(2) triplet Φ(x) • 1 T µ 1 2 T λ T 2 iT θa = (∂ Φ )(∂ Φ) µ Φ Φ (Φ Φ) inv. under SU(2): Φ e− a Φ L 2 µ − 2 − 4 7→ that for λ > 0, µ2 < 0 acquires a VEV 0 ΦTΦ 0 = v2 (µ2 = λv2)   h | | i − ϕ1(x)   1 Assume Φ(x) =  ϕ2(x)  and define ϕ (ϕ1 + iϕ2)   ≡ √2 v + ϕ3(x) 1 λ 1 = (∂ ϕ†)(∂µ ϕ) + (∂ ϕ )(∂µ ϕ ) λv2 ϕ2 λv(2ϕ† ϕ + ϕ2)ϕ (2ϕ† ϕ + ϕ2)2+ λv4 L µ 2 µ 3 3 − 3 − 3 3 − 4 3 4 Not symmetric under SU(2) but invariant under U(1): ⇒ iqθ ϕ e− ϕ (q = arbitrary) ϕ ϕ (q = 0) 7→ 3 7→ 3 SU(2) broken to U(1) 3 1 = 2 broken generators ⇒ − 2 (real) scalar fields (= 1 complex) remain massless: m = 0, m = √2λ v ⇒ ϕ ϕ3 1. Gauge Theories 22 Spontaneous Symmetry Breaking continuous symmetry

Goldstone’s theorem: [Nambu ’60; Goldstone ’61] ⇒ The number of massless particles (Nambu-Goldstone bosons) is equal to the number of spontaneously broken generators of the symmetry

Hamiltonian symmetric under group G [T , H] = 0 , a = 1, . . . , N ⇒ a By definition: H 0 = 0 H(Ta 0 ) = Ta H 0 = 0 | i ⇒ | i | i

– If 0 is such that Ta 0 = 0 for all generators | i | i non-degenerate minimum: the vacuum ⇒ – If 0 is such that T 0 = 0 for some (broken) generators a | i a0 | i 6 0 a0 degenerate minimum: chose one (true vacuum) and e iTa θ 0 = 0 ⇒ − 0 | i 6 | i a0 excitations (particles) from 0 to e iTa θ 0 cost no energy: massless! ⇒ | i − 0 | i

1. Gauge Theories 23 Spontaneous Symmetry Breaking gauge symmetry

Consider a U(1) gauge invariant Lagrangian for a complex scalar field φ(x): • 1 = F Fµν + (D φ)†(Dµφ) µ2φ†φ λ(φ†φ)2 , D = ∂ + ieqA L −4 µν µ − − µ µ µ 1 inv. under φ(x) φ (x) = e iqθ(x)φ(x) , A (x) A (x) = A (x) + ∂ θ(x) 7→ 0 − µ 7→ µ0 µ e µ If λ > 0, µ2 < 0, the in terms of quantum fields η and χ with null VEVs: L 1 φ(x) [v + η(x) + iχ(x)] , µ2 = λv2 Comments: ≡ √2 − 1 1 1 √ = F Fµν + (∂ η)(∂µη) + (∂ χ)(∂µχ) (i) mη = 2λ v L −4 µν 2 µ 2 µ mχ = 0 2 2 2 2 λ 2 2 2 1 4 λv η λvη(η + χ ) (η + χ ) + λv (ii) M = eqv (!) − − − 4 4 A | | + eqvA ∂µχ + eqA (η∂µχ χ∂µη) (iii) Term A ∂µχ (?) µ µ − µ 1 1 (iv) Add GF + (eqv)2 A Aµ + (eq)2 A Aµ(η2 + 2vη + χ2) L 2 µ 2 µ

1. Gauge Theories 24 Spontaneous Symmetry Breaking gauge symmetry

Removing the cross term and the (new) gauge fixing Lagrangian: • 1 = (∂ Aµ ξM χ)2 LGF −2ξ µ − A

total deriv. 1 1 1 z }| { + = F Fµν + M2 A Aµ (∂ Aµ)2 + M ∂ (Aµχ) ⇒ L LGF − 4 µν 2 A µ − 2ξ µ A µ 1 1 + (∂ χ)(∂µχ) ξM2 χ2 + ... 2 µ − 2 A

and the propagators of Aµ and χ are: " # i kµkν Deµν(k) = gµν + (1 ξ) k2 M2 + ie − − k2 ξM2 − A − A i De(k) = k2 ξM2 − A χ has a gauge-dependent mass: actually it is not a physical field! ⇒ 1. Gauge Theories 25 Spontaneous Symmetry Breaking gauge symmetry

A more transparent parameterization of the quantum field φ is • 1 φ(x) eiqζ(x)/v [v + η(x)] , 0 η 0 = 0 ζ 0 = 0 ≡ √2 h | | i h | | i

iqζ(x)/v 1 φ(x) e− φ(x) = [v + η(x)] ζ gauged away! 7→ √2 ⇒

1 1 Comments: = F Fµν + (∂ η)(∂µη) L −4 µν 2 µ (i) m = √2λ v λ 1 η λv2η2 λvη3 η4 + λv4 − − − 4 4 (ii) MA = eqv | | 1 2 µ 1 2 µ 2 + (eqv) A A + (eq) A A (2vη + η ) (iii) No need for GF 2 µ 2 µ L

This is the unitary gauge (ξ ∞): just physical fields ⇒ →

1. Gauge Theories 26 Spontaneous Symmetry Breaking gauge symmetry

Higgs mechanism: [Anderson ’62; Higgs ’64; Englert, Brout ’64; Guralnik, Hagen, Kibble ’64] ⇒ The gauge bosons associated with the spontaneously broken generators become massive, the corresponding would-be Goldstone bosons are unphysical and can be absorbed, the remaining massive scalars (Higgs bosons) are physical (the smoking gun!)

– The would-be Goldstone bosons are ‘eaten up’ by the gauge bosons (‘get fat’) and disappear (gauge away) in the unitary gauge (ξ ∞) → Degrees of freedom are preserved ⇒ Before SSB: 2 (massless gauge boson) + 1 (Goldstone boson) After SSB: 3 (massive gauge boson) + 0 (absorbed would-be Goldstone)

– For loops calculations, ’t Hooft-Feynman gauge (ξ = 1) is more convenient: Gauge boson propagators are simpler, but ⇒ Goldstone bosons must be included in internal lines ⇒

1. Gauge Theories 27 Spontaneous Symmetry Breaking gauge symmetry

Comments: •

– After SSB the FP ghost fields (unphysical) acquire a gauge-dependent mass, due to interactions with the scalar field(s):

iδab Deab(k) = k2 ξM2 + ie − A

– Gauge theories with SSB are renormalizable [’t Hooft, Veltman ’72]

UV divergences appearing at loop level can be removed by renormalization of parameters and fields of the classical Lagrangian predictive! ⇒

1. Gauge Theories 28 2. The Standard Model

29 Gauge group and particle representations [Glashow ’61; Weinberg ’67; Salam ’68] [Gell-Mann ’64; Zweig ’64]

The Standard Model is a gauge theory based on the local symmetry group: •

SU(3)c SU(2)L U(1)Y SU(3)c U(1)Q | {z } ⊗ | {z⊗ } → ⊗ | {z } strong electroweak em

with the electroweak symmetry spontaneously broken to the electromagnetic

U(1)Q symmetry by the Higgs mechanism

The particle (field) content: (ingredients: 12 flavors + 12 gauge bosons + H) • Fermions I II III Q Bosons

1 2 spin 2 Quarks f uuu ccc ttt 3 spin 1 8 gluons strong interaction f ddd sss bbb 1 W ,Z weak interaction 0 − 3 ± Leptons f νe νµ ντ 0 γ em interaction f e µ τ 1 spin 0 Higgs origin of mass 0 −

Q f = Q f 0 + 1

2. The Standard Model 30 Gauge group and particle representations

The fields lay in the following representations (color, weak isospin, hypercharge): • Multiplets SU(3) SU(2) U(1) I II III Q = T + Y c ⊗ L ⊗ Y 3       2 1 1 1 uL cL tL 3 = 2 + 6 Quarks (3, 2, 6 )       d s b 1 = 1 + 1 L L L − 3 − 2 6 (3, 1, 2 ) u c t 2 2 3 R R R 3 = 0 + 3 1 1 1 (3, 1, ) dR sR bR = 0 − 3 − 3 − 3       1 1 1 νeL νµL ντL 0 = 2 2 Leptons (1, 2, 2 )       − − e µ τ 1 = 1 1 L L L − − 2 − 2

(1, 1, 1) eR µR τR 1 = 0 1 − − −

(1, 1, 0) νeR νµR ντR 0 = 0 + 0 1 Higgs (1, 2, 2 ) (3 families of quarks & leptons) From now on just the electroweak part (EWSM): SU(2) U(1) ⇒ L⊗ Y 2. The Standard Model 31 The EWSM with one family (of quarks or leptons)

Consider two massless fermion fields f (x) and f (x) with electric charges • 0 Q = Q + 1 in three irreps of SU(2) U(1) : f f 0 L⊗ Y

0 1 1 = i f ∂/f + i f 0∂/f 0 f = (1 γ ) f , f 0 = (1 γ ) f 0 LF R,L 2 ± 5 R,L 2 ± 5   fL = iΨ1∂/Ψ1 + iψ2∂ψ/ 2 + iψ3∂ψ/ 3 ; Ψ1 =   , ψ2 = fR , ψ3 = fR0 fL0 |{z} |{z} | {z } (1, y2) (1, y3) (2, y1)

To get a Langrangian invariant under gauge transformations: • iy β(x) iT αi(x) σi Ψ (x) U (x)e− 1 Ψ (x), U (x) = e− i , T = (weak isospin gen.) 1 7→ L 1 L i 2 iy β(x) ψ (x) e− 2 ψ (x) 2 7→ 2 iy β(x) ψ (x) e− 3 ψ (x) 3 7→ 3

2. The Standard Model 32 The EWSM with one family covariant derivatives

i Introduce gauge fields W (x) (i = 1, 2, 3) and B (x) through covariant derivatives: ⇒ µ µ  σi i DµΨ1 = (∂µ igWe µ + ig0y1Bµ)Ψ1 , We µ Wµ  − ≡ 2  Dµψ2 = (∂µ + ig0y2Bµ)ψ2 F  ⇒ L  Dµψ3 = (∂µ + ig0y3Bµ)ψ3 

where two couplings g and g0 have been introduced and

† i † We µ(x) U (x)We µ(x)U (x) (∂µU (x))U (x) 7→ L L − g L L 1 Bµ(x) Bµ(x) + ∂µβ(x) 7→ g0 Add gauge invariant kinetic terms for the gauge fields ⇒ 1 1 j = Wi Wi,µν B Bµν , Wi = ∂ Wi ∂ Wi + ge W Wk LYM −4 µν − 4 µν µν µ ν − ν µ ijk µ ν (include self-interactions of the SU(2) gauge fields) and B = ∂ B ∂ B µν µ ν − ν µ 2. The Standard Model 33 The EWSM with one family mass terms forbidden

Note that mass terms are not invariant under SU(2) U(1) , since LH and RH ⇒ L⊗ Y components do not transform the same:

m f f = m( fL fR + fR fL)

Mass terms for the gauge bosons are not allowed either ⇒

Next the different types of interactions are analyzed ⇒

2. The Standard Model 34 The EWSM with one family charged current interactions   3 √ † µ 1 Wµ 2Wµ F gΨ1γ We µΨ1 , We µ =   • L ⊃ 2 √2W W3 µ − µ charged current interactions of LH fermions with complex vector boson field W : ⇒ µ g µ † 1 1 2 CC = f γ (1 γ5) f 0Wµ + h.c. , Wµ (Wµ + iWµ) L 2√2 − ≡ √2

ℓ d W W ν u

ν u

W W

ℓ d

2. The Standard Model 35 The EWSM with one family neutral current interactions

The diagonal part of • µ µ µ µ gΨ γ We µΨ g0Bµ(y Ψ γ Ψ + y ψ γ ψ + y ψ γ ψ ) LF ⊃ 1 1 − 1 1 1 2 2 2 3 3 3 neutral current interactions with neutral vector boson fields W3 and B ⇒ µ µ We would like to identify Bµ with the photon field Aµ but that requires:

y = y = y and g0y = eQ impossible! 1 2 3 j j ⇒ Since they are both neutral, try a combination: ⇒       3 Wµ cW sW Zµ sW sin θW , cW cos θW    −    ≡ ≡ Bµ ≡ sW cW Aµ θW = weak mixing angle

3 µ      NC = ∑ ψjγ gT3sW + g0yjcW Aµ + gT3cW g0yjsW Zµ ψj L j=1 − − σ with T = 3 (0) the third weak isospin component of the doublet (singlet) 3 2 2. The Standard Model 36 The EWSM with one family neutral current interactions

To make A the photon field: • µ

e = gsW = g0cW Q = T3 + Y   Q f 0 where the electric charge operator is: Q1 =   , Q2 = Q f , Q3 = Q f Q 0 0 f 0

Electroweak unification: g of SU(2) and g of U(1) are related ⇒ 0 The hyperchages are fixed in terms of electric charges and weak isospin: ⇒ 1 1 y = Q = Q + , y = Q , y = Q 1 f − 2 f 0 2 2 f 3 f 0

µ = e Q f γ f A + ( f f 0) LQED − f µ →

RH neutrinos are sterile: y = Q = 0 ⇒ 2 f 2. The Standard Model 37 The EWSM with one family neutral current interactions

The Z is the neutral weak boson field: • µ Z µ = e f γ (v a γ ) f Z + ( f f 0) LNC f − f 5 µ → with fL 2 fL T3 2Q f sW T3 v f = − , a f = 2sW cW 2sW cW

The complete neutral current Lagrangian reads: • = + Z LNC LQED LNC

f f

γ Z f = u,d,ℓ f = u,d,ν,ℓ

2. The Standard Model 38 The EWSM with one family gauge boson self-interactions

Cubic: • iecW n µν † † µ ν † µνo YM 3 = W Wµ Zν WµνW Z WµWνZ L ⊃ L − sW − − n o + ie WµνW† A W† Wµ Aν W†W Fµν µ ν − µν − µ ν with

F = ∂ A ∂ A Z = ∂ Z ∂ Z W = ∂ W ∂ W µν µ ν − ν µ µν µ ν − ν µ µν µ ν − ν µ

W W

γ Z

W W

2. The Standard Model 39 The EWSM with one family gauge boson self-interactions

Quartic: • 2  2  e  † µ † µ† ν YM 4 = 2 WµW WµW WνW L ⊃ L −2sW − 2 2 e cW n † µ ν † µ νo 2 WµW ZνZ Wµ Z WνZ − sW − 2 e cW n † µ ν † µ ν † µ νo + 2WµW Zν A Wµ Z Wν A Wµ A WνZ sW − − n o e2 W†Wµ A Aν W† AµW Aν − µ ν − µ ν

W W W γ W γ W Z

W W W γ W Z W Z

Note: even number of W and no vertex with just γ or Z

2. The Standard Model 40 Electroweak symmetry breaking setup

Out of the 4 gauge bosons of SU(2) U(1) with generators T , T , T , Y we • L⊗ Y 1 2 3 need all to be broken except the combination Q = T3 + Y so that Aµ remains massless and the other three gauge bosons get massive after SSB Introduce a complex SU(2) Higgs doublet ⇒     φ+ 1 0 Φ =   , 0 Φ 0 =   φ0 h | | i √2 v

with gauge invariant Lagrangian (µ2 = λv2): − † µ 2 † † 2 = (DµΦ) D Φ µ Φ Φ λ(Φ Φ) , DµΦ = (∂µ igWe µ + ig0y Bµ)Φ LΦ − − − Φ   1 0 take yΦ = (T3 + Y) 0 = Q   = 0 2 ⇒ | i v T , T , T Y 0 = 0 { 1 2 3 − } | i 6

2. The Standard Model 41 Electroweak symmetry breaking gauge boson masses

Quantum fields in the unitary gauge: •   n σi o 1 0 Φ(x) exp i θi(x)   ≡ 2v √2 v + H(x)

  1 physical Higgs field n σi o 1 0 H(x) Φ(x) exp i θi(x) Φ(x) =   7→ − 2v √2 v + H(x) ⇒ 3 would-be Goldstones θi(x) gauged away

– The 3 dof apparently lost become the longitudinal polarizations of W± and Z that get massive after SSB:

2 2 2 2 g v † µ g v µ 1 Φ M = W W + ZµZ MW = MZcW = gv L ⊃ L 4 µ 8c2 ⇒ 2 |{z} W 2 |{z} MW 1 2 2 MZ

2. The Standard Model 42 Electroweak symmetry breaking Higgs sector

1 4 In the unitary gauge (just physical fields): = + + 2 + λv ⇒ LΦ LH LM LHV 4

1 1 M2 M2 q = ∂ H∂µ H M2 H2 H H3 H H4 , M = 2µ2 = √2λ v LH 2 µ − 2 H − 2v − 8v2 H − H H H H

H H H

 2   2  2 † µ 2 H 1 2 µ 2 H + 2 = M W W 1 + H + + M Z Z 1 + H + LM LHV W µ v v2 2 Z µ v v2

W W H Z Z H H H

W W H Z Z H 2. The Standard Model 43 Electroweak symmetry breaking Higgs sector

Quantum fields in the R gauges: • ξ   + φ (x) + Φ(x) =   , φ−(x) = [φ (x)]∗ 1 [v + H(x) + iχ(x)] √2

1 4 = + + 2 + λv LΦ LH LM LHV 4

+ µ 1 µ + (∂ φ )(∂ φ−) + (∂ χ)(∂ χ) µ 2 µ µ + † µ µ + iM (W ∂ φ W ∂ φ−) + M Z ∂ χ W µ − µ Z µ + trilinear interactions [SSS, SSV, SVV]

+ quadrilinear interactions [SSSS, SSVV]

2. The Standard Model 44 Electroweak symmetry breaking gauge fixing

To remove the cross terms W ∂µφ+, W†∂µφ , Z ∂µχ and define propagators add: • µ µ − µ

1 µ 2 1 µ 2 1 µ 2 GF = (∂µ A ) (∂µZ ξZ MZχ) ∂µW + iξW MW φ− L −2ξγ − 2ξZ − − ξW | |

Massive propagators for gauge and (unphysical) would-be Goldstone fields: ⇒   γ i kµkν Deµν(k) = gµν + (1 ξγ) k2 + ie − − k2   Z i kµkν χ i De (k) = gµν + (1 ξZ) ; De (k) = µν k2 M2 + ie − − k2 ξ M2 k2 ξ M2 + ie − Z − Z Z − Z Z   W i kµkν φ i De (k) = gµν + (1 ξW ) ; De (k) = µν k2 M2 + ie − − k2 ξ M2 k2 ξ M2 + ie − W − W W − W W

(’t Hooft-Feynman gauge: ξγ = ξZ = ξW = 1)

2. The Standard Model 45 Electroweak symmetry breaking Faddeev-Popov ghosts

The SM is a non-Abelian theory add Faddeev-Popov ghosts c (x) (i = 1, 2, 3) • ⇒ i 1 i c1 (u+ + u ) , c2 (u+ u ) , c3 cW uZ sW uγ ≡ √2 − ≡ √2 − − ≡ −

µ k FP = (∂ ci)(∂µci igeijkcjWµ) + interactions with Φ L | {z− } | {z } U kinetic + [UUV] U masses + [SUU]

Massive propagators for (unphysical) FP ghost fields: ⇒

uγ i uZ i u i De (k) = , De (k) = , De ± (k) = k2 + ie k2 ξ M2 + ie k2 ξ M2 + ie − Z Z − W W

(’t Hooft-Feynman gauge: ξZ = ξW = 1)

2. The Standard Model 46 Electroweak symmetry breaking Faddeev-Popov ghosts

µ µ µ µ FP = (∂µuγ)(∂ uγ) + (∂µuZ)(∂ uZ) + (∂µu+)(∂ u+) + (∂µu )(∂ u ) L − −  µ µ iecW µ µ  + ie[(∂ u+)u+ (∂ u )u ]Aµ [(∂ u+)u+ (∂ u )u ]Zµ  − − − − sW − − −   µ µ † iecW µ µ † [UUV] ie[(∂ u+)uγ (∂ uγ)u ]Wµ + [(∂ u+)uZ (∂ uZ)u ]Wµ  − − − sW − −   µ µ iecW µ µ  + ie[(∂ u )uγ (∂ uγ)u+]Wµ [(∂ u )uZ (∂ uZ)u+]Wµ − − − sW − − 2 2 2 ξZ MZ uZuZ ξW MW u+u+ ξW MW u u − − − − −    1 1 +
 eξZ MZ uZ HuZ φ u + φ−u+ − 2s c − 2s −  W W W  " 2 2 !#  1 + cW sW [SUU] eξ M u+ (H + iχ)u+ φ u − u − W W 2s − γ − 2s c Z  W W W  " !#  2 2  1 cW sW  eξW MW u (H iχ)u φ− uγ − uZ − − 2sW − − − − 2sW cW

2. The Standard Model 47 Electroweak symmetry breaking fermion masses

We need masses for quarks and leptons without breaking gauge symmetry • Introduce Yukawa interactions: ⇒     + 0   φ   φ ∗ Y = λd uL dL   dR λu uL dL   uR L − φ0 − φ − −     + 0   φ   φ ∗ λ` νL `L   `R λν νL `L   νR + h.c. − φ0 − φ − −     0 + c φ ∗ φ where Φ iσ2Φ∗ =   transforms under SU(2) like Φ =   ≡ φ φ0 − − After EW SSB, fermions acquire masses: ⇒ 1 n o v Y (v + H) λd dd + λu uu + λ` `` + λν νν m f = λ f L ⊃ −√2 ⇒ √2

2. The Standard Model 48 Additional generations Yukawa matrices

There are 3 generations of quarks and leptons in Nature. They are identical copies • with the same properties under SU(2) U(1) differing only in their masses L ⊗ Y

Take a general case of n generations and let uI, dI, νI, `I be the members of ⇒ G j j j j family j (j = 1, . . . , nG). Superindex I (interaction basis) was omitted so far

General gauge invariant Yukawa Lagrangian: ⇒         φ+ φ0 I I (d) I ∗ (u) I Y = ∑ ujL djL   λjk dkR +   λjk ukR L −  φ0 φ jk − −        φ+ φ0  I I (`) I ∗ (ν) I + νjL `jL   λjk `kR +   λjk νkR + h.c. φ0 φ  − − (d) (u) (`) (ν) where λjk , λjk , λjk , λjk are arbitrary Yukawa matrices

2. The Standard Model 49 Additional generations mass matrices

After EW SSB, in n -dimensional matrix form: • G  H  n o 1 + d I M d I + u I M u I + lI M lI + ν I M ν I + h.c. LY ⊃ − v L d R L u R L ` R L ν R with mass matrices

(d) v (u) v (`) v (ν) v (Md)ij λ (Mu)ij λ (M`)ij λ (Mν)ij λ ≡ ij √2 ≡ ij √2 ≡ ij √2 ≡ ij √2

Diagonalization determines mass eigenstates dj, uj, `j, νj ⇒ I I I I in terms of interaction states dj , uj , `j , νj , respectively Each M can be written as ⇒ f M = H = S† S M M† = H2 = S† 2 S f f U f f M f f U f ⇐⇒ f f f f M f f q with H M M† a Hermitian positive definite matrix and unitary f ≡ f f U f – Every H f can be diagonalized by a unitary matrix S f – The resulting is diagonal and positive definite M f 2. The Standard Model 50 Additional generations fermion masses and mixings

In terms of diagonal mass matrices (mass eigenstate basis): • = diag(m , m , m ,...) , = diag(m , m , m ,...) Md d s b Mu u c t = diag(m , m , m ,...) , = diag(m , m , m ,...) M` e µ τ Mν νe νµ ντ

 H  n o 1 + d d + u u + l l + ν ν LY ⊃ − v Md Mu M` Mν where fermion couplings to Higgs are proportional to masses and

d S dI u S uI l S lI ν S ν I L ≡ d L L ≡ u L L ≡ ` L L ≡ ν L d S dI u S uI l S lI ν S ν I R ≡ dUd R R ≡ uUu R R ≡ `U` R R ≡ νUν R  Neutral Currents preserve chirality  NC does not change flavor I I I I ⇒ L f L f L = fL fL and fR fR = fR fR 

GIM mechanism [Glashow, Iliopoulos, Maiani ’70] ⇒ 2. The Standard Model 51 Additional generations quark sector

However, in Charged Currents (also chirality preserving and only LH): • I I † u L d L = uL Su Sd dL = u LVd L

† with V S S the (unitary) CKM mixing matrix [Cabibbo ’63; Kobayashi, Maskawa ’73] ≡ u d g µ † CC = ∑ uiγ (1 γ5) Vij dj Wµ + h.c. ⇒ L 2√2 ij −

dj ui

W Vij W Vij∗

ui dj

If u or d had degenerate masses one could choose S = S (field redefinition) ⇒ i j u d and flavor would be conserved in the quark sector. But they are not degenerate S and S are not observable. Just masses and CKM mixings are observable ⇒ u d

2. The Standard Model 52 Additional generations quark sector

How many physical parameters in this sector? •

– Quark masses and CKM mixings determined by mass (or Yukawa) matrices

– A general n n unitary matrix, like the CKM, is given by G × G n2 real parameters = n (n 1)/2 moduli + n (n + 1)/2 phases G G G − G G Some phases are unphysical since they can be absorbed by field redefinitions:

iφ iθ i(θ φ ) u e i u , d e j d V V e j− i i → i j → j ⇒ ij → ij Therefore 2n 1 unphysical phases and the physical parameters are: G − (n 1)2 = n (n 1)/2 moduli + (n 1)(n 2)/2 phases G − G G − G − G −

2. The Standard Model 53 Additional generations quark sector

Case of n = 2 generations: 1 parameter, the Cabibbo angle θ : ⇒ G C   cos θC sin θC V =   sin θ cos θ − C C Case of n = 3 generations: 3 angles + 1 phase. In the standard parameterization: ⇒ G

     iδ    Vud Vus Vub 1 0 0 c13 0 s13e− 13 c12 s12 0         V = V V V  = 0 c s   0 1 0   s c 0  cd cs cb   23 23   − 12 12  V V V 0 s c s eiδ13 0 c 0 0 1 td ts tb − 23 23 − 13 13  iδ  c12c13 s12c13 s13e− 13   δ13 only source =  s c c s s eiδ13 c c s s s eiδ13 s c  of CP violation − 12 23 − 12 23 13 12 23 − 12 23 13 23 13  ⇒   in the SM ! s s c c s eiδ13 c s s c s eiδ13 c c 12 23 − 12 23 13 − 12 23 − 12 23 13 23 13 with c cos θ 0, s sin θ 0 (i < j = 1, 2, 3) and 0 δ 2π ij ≡ ij ≥ ij ≡ ij ≥ ≤ 13 ≤ 2. The Standard Model 54 Additional generations lepton sector

If neutrinos were massless we could redefine the (LH) fields no lepton mixing • ⇒ But they have (tiny) masses because there are neutrino oscillations!

Neutrinos are special: • they may be their own antiparticle (Majorana) since they are neutral

If they are Majorana: • – Mass terms are different to Dirac case (neutrino and antineutrino may mix) – Intergenerational mixings are richer (more CP phases)

2. The Standard Model 55 ? lepton sector

About Majorana fermions •

– A Dirac fermion field is a spinor with4 independent components: 2 LH+2 RH (left/right-handed particles and antiparticles)

ψ = P ψ , ψ = P ψ , ψc (ψ )c = P ψc , ψc (ψ )c = P ψc L L R R L ≡ L R R ≡ R L T where ψc Cψ = iγ2ψ (charge conjugate) with C = iγ2γ0, P = 1 (1 γ ) ≡ ∗ R,L 2 ± 5

– A Majorana fermion field has just2 independent components since ψc η ψ: ≡ ∗ c c ψL = ηψR , ψR = ηψL

where η = iη (CP parity) with η 2 = 1. Only possible if neutral − CP | |

2. The Standard Model 56 ? lepton sector

About mass terms • c c c c ψRψL = ψ ψ , ψLψR = ψ ψ (∆F = 0) L R  R L c c ψ ψL , ψLψ  L L ( ∆F = 2) c c | | ψRψR , ψRψR  1 1 = m ψ ψ + m ψc ψ + m ψ ψc + h.c. ⇒ −Lm D R L 2 L L L 2 R R R | {z } | {z } Dirac term Majorana terms

– A Dirac fermion can only have Dirac mass term – A Majorana fermion can have both Dirac and Majorana mass terms

In the SM: m from Yukawa coupling after EW SSB (m = λ v/√2) ⇒ ∗ D D ν m forbidden by gauge symmetry ∗ L m compatible with gauge symmetry! ∗ R 2. The Standard Model 57 ? lepton sector

About mass terms (a more transparent parameterization) • Rewrite previous mass terms introducing a doublet of Majorana fermions:   0 0 0c 0 0c c 0 0c 0 0c χ1 χ1 = χ1 = χ1L + χ1L ψL + ψL χ = χ = χL + χL   , ≡ ≡ χ0 χ0 = χ0c = χ0 + χ0c ψc + ψ 2 2 2 2L 2L ≡ R R   1 0c 0 mL mD m = χL M χL + h.c. with M =   ⇒ −L 2 mD mR

M is a square symmetric matrix diagonalizable by a unitary matrix e: ⇒ U T 0 0c c e M e = = diag(m0 , m0 ) , χ = eχ (χ = e∗χ ) U U M 1 2 L U L L U L To get real and positive eigenvalues m = η m (physical masses) take χ0 = ξ : i i i0 L U L c ξ1 = χ1L + η1χ1L = ediag(√η1, √η2) , (physical fields) ηi = CP parities U U c ξ2 = χ2L + η2χ2L

2. The Standard Model 58 ? lepton sector

About mass terms (a more transparent parameterization) • – Case of only Dirac term (mL = mR = 0)     0 mD 1 1 1 M =   e =   , m10 = mD , m20 = mD m 0 ⇒ U √2 1 1 − D −

Eigenstates Physical states ⇒ 1 0 0 1 c c χ = (χ χ ) = (ψ ψ ) ξ1 = χ1L + η1χ [η1 = 1] 1L √ 1L 2L √ L R 1L − 2 − 2 − c ξ2 = χ2L + η2χ2L [η2 = +1] 1 0 0 1 c χ2L = (χ1L + χ2L) = (ψL + ψR) √2 √2 with masses m1 = m2 = mD 1 1 = m ( χ χ + χ χ ) = m (ξ ξ + ξ ξ ) = m (ψ ψ + ψ ψ ) ⇒ −Lm 2 D − 1 1 2 2 2 D 1 1 2 2 D L R R L

One Dirac fermion = two Majorana of equal mass and opposite CP parities

2. The Standard Model 59 ? lepton sector

About mass terms (a more transparent parameterization) • – Case of seesaw (type I) [Yanagida ’79; Gell-Mann, Ramond, Slansky ’79; Mohapatra, Senjanovic ’80] (m = 0, m m ) L D  R     r 0 mD cos θ sin θ mD mν M =   e =   , θ (negligible) m m ⇒ U sin θ cos θ ' mR ' mN D R − 2 mD m1 mν m2 mN mR ≡ ' mR  ≡ '

c ξ1 ν = ψL + η1ψL [η1 = 1] 1 c 1 c ≡ − m = mν νLνL + mN NR NR + h.c. ξ N = ψc + η ψ [η = +1] ⇒ −L 2 2 2 ≡ R 2 R 2

Perhaps the observed neutrino νL is the LH component of a light Majorana ν (then ν = RH) and light because of a very heavy Majorana neutrino N

e.g. m v 246 GeV , m m 1015 GeV m 0.1 eV D ∼ ' R ∼ N ∼ ⇒ ν ∼ X 2. The Standard Model 60 Additional generations lepton sector

Lepton mixings • – From Z lineshape: there are nG = 3 generations of νL [νi (i = 1, . . . , nG)] (but we do not know (yet) if neutrinos are Dirac or Majorana fermions) – From neutrino oscillations: neutrinos are light, non degenerate and mix

να = ∑ Uαi νi νi = ∑ Uα∗i να | i i | i ⇐⇒ | i α | i

mass eigenstates νi (i = 1, 2, 3) / interaction states να (α = e, µ, τ)

U matrix is unitary (negligible mixing with heavy neutrinos) and analogous ⇒ to Su, Sd, S` defined for quarks and charged leptons except for: – ν fields have both chiralities – If neutrinos are Majorana, U may contain two additional physical (Majorana) phases (irrelevant and therefore not measurable in oscillation experiments) c that cannot be absorbed since then field phases are fixed by νi = ηiνi

2. The Standard Model 61 Additional generations lepton sector

Lepton mixings • The so called PMNS matrix U [Pontecorvo ’57; Maki, Nakagawa, Sakata ’62; Pontecorvo ’68] – does not change Neutral Currents (unitarity), but – introduces intergenerational mixings in Charged Currents:

g µ CC = ∑ `α γ (1 γ5) Uαi νi Wµ + h.c. L 2√2 αi − (basis where charged leptons are diagonal)

νi ℓj

W Uji W Uji∗

ℓj νi

2. The Standard Model 62 Additional generations lepton sector

Standard parameterization of the PMNS matrix: ⇒

  Ue1 Ue2 Ue3   U = U U U   µ1 µ2 µ3 Uτ1 Uτ2 Uτ3

  iδ13  iα  c12 c13 s12 c13 s13 e− e 1 0 0      iδ13 iδ13  iα2 =  s12 c23 c12 s23 s13 e c12 c23 s12 s23 s13 e s23 c13   0 e 0 − − −    s s c c s eiδ13 c s s c s eiδ13 c c 0 0 1 12 23 − 12 23 13 − 12 23 − 12 23 13 23 13 (Majorana (different values than in CKM) phases)

[θ13 θ , θ23 θatm and θ13 (not yet δ13) measured in oscillations] ≡ ≡

2. The Standard Model 63 Complete SM Lagrangian fields and interactions

= + + + + + L LF LYM LΦ LY LGF LFP

Fields: [F] fermions [S] scalars • [V] vector bosons [U] unphysical ghosts

Interactions: [FFV] [FFS] [SSV] [SVV] [SSVV] • [VVV] [VVVV] [SSS] [SSSS] [SUU] [UUVV]

2. The Standard Model 64 Complete SM Lagrangian Feynman rules

Feynman rules for generic couplings normalized to e (all momenta incoming): • (i ) [FFV ] ieγµ(g g γ ) = ieγµ(g P + g P ) L µ V − A 5 L L R R [FFS] ie(g g γ ) = ie(c P + c P ) S − P 5 L L R R [SVµVν] ieKgµν [S(p )S(p )V ] ieG(p p ) 1 2 µ 1 − 2 µ [V (k )V (k )V (k )] ieJ g (k k ) + g (k k ) + g (k k )  µ 1 ν 2 ρ 3 µν 2 − 1 ρ νρ 3 − 2 µ µρ 1 − 3 ν [V (k )V (k )V (k )V (k )] ie2C 2g g g g g g  µ 1 ν 2 ρ 3 σ 4 µν ρσ − µρ νσ − µσ νρ 2 [SSVµVν] ie C2gµν also [UUVV]

[SSS] ieC3 also [SUU] 2 [SSSS] ie C4

Note: g = g g Attention to symmetry factors! L,R V ± A c = g g L,R S ± P 2. The Standard Model 65 Complete SM Lagrangian Feynman rules (’t Hooft-Feynman gauge)

+ + FFV f i fjγ f i fjZ uidjW djuiW− νi`jW `jνiW− f 1 1 1 1 gL Q f δij g+δij Vij Vij∗ U∗ji Uji − √2sW √2sW √2sW √2sW f gR Q f δij g δij 0 0 0 0 − −

fL 2 fL f T3 2Q f sW T3 g v f a f v f = − a f = ± ≡ ± 2sW cW 2sW cW

2. The Standard Model 66 Complete SM Lagrangian Feynman rules (’t Hooft-Feynman gauge)

+ FFS f i fj H f i fjχ uidjφ djuiφ− m m m 1 fi i fL fi 1 mui 1 dj cL δij 2T3 δij + Vij Vij∗ −2sW MW −2sW MW √2sW MW −√2sW MW m m 1 m fi i fL m fi 1 dj 1 uj cR δij + 2T3 δij Vij + Vij∗ −2sW MW 2sW MW −√2sW MW √2sW MW

+ FFS νi`jφ `jνiφ− m 1 mνi 1 `j cL + U∗ji Uji √2s MW −√2s MW W m W 1 `j 1 mνi cR U∗ji + Uji −√2sW MW √2sW MW

2. The Standard Model 67 Complete SM Lagrangian Feynman rules (’t Hooft-Feynman gauge)

+ SVV HZZ HW W− φ±W∓γ φ±W∓Z K M /s c2 M /s M M s /c W W W W W − W − W W W

SSV χHZ φ±φ∓γ φ±φ∓Z φ∓HW± φ∓χW± i c2 s2 1 i G 1 W − W −2sW cW ∓ ± 2sW cW ∓2sW −2sW

+ + VVV γW W− ZW W− J 1 c /s − W W

2. The Standard Model 68 Complete SM Lagrangian Feynman rules (’t Hooft-Feynman gauge)

+ + + + + VVVV W W W−W− W W−ZZ W W−γZ W W−γγ 2 1 cW cW C 2 2 1 sW − sW sW −

+ SSVV HHW−W HHZZ SSS HHH SSSS HHHH 2 2 1 1 3M 3MH C2 C H C s2 s2 c2 3 4 2 2 2 W 2 W W −2MW sW −4MW sW

– Would-be Goldstone bosons in [SSVV], [SSS] and [SSSS] omitted – Faddeev-Popov ghosts in [UUVV] and [SUU] omitted – All Feynman rules from FeynArts (same conventions):

http://www.ugr.es/local/jillana/SM/FeynmanRulesSM.pdf

2. The Standard Model 69 3. Phenomenology

70 Input parameters

Parameters: • 17 +9 = 1 1 1 1 9 +346

formal: g g0 v λ λ f VCKM UPMNS practical: α MW MZ MH m f

where e = gsW = g0cW and

2 e 1 MW v α = MW = gv MZ = MH = √2λ v m f = λ f 4π 2 cW √2

Many (more) experiments ⇒ After Higgs discovery, for the first time all parameters measured! ⇒

3. Phenomenology 71 Input parameters

Experimental values [Particle Data Group ’13] • – Fine structure constant: 1 α− = 137.035 999 074 (44) from Harvard cyclotron (ge)

– The SM predicts MW < MZ in agreement with measurements: M = (91.1876 0.021) GeV from LEP1/SLD Z ± M = (80.385 0.015) GeV from LEP2/Tevatron/LHC W ± – Top quark mass: m = (173.2 0.9) GeV from Tevatron/LHC t ± – Higgs boson mass: M = (125.9 0.4) GeV from LHC H ± – ...

3. Phenomenology 72 Observables and experiments

Low energy observables • – ν-nucleon (NuTeV) and νe (CERN) scattering: asymmetries CC/NC and ν/ν¯ s2 ⇒ W – atomic parity violation (Ce, Tl, Pb): asymmetries e N eX due to Z-exchange between e and nucleus s2 R,L → ⇒ W – muon decay (PSI): lifetime G2 m5 1 F µ 2 2 = Γµ = 3 f (me /mµ) τµ 192π G ⇒ F f (x) 1 8x + 8x3 x4 12x2 ln x ≡ − − −

Fermi theory ( q2 M2 ) −  W  2 z }| { ie ρ igρδ δ 4GF ρ GF πα i = eγ νL − νLγ µ i (eγ νL)(νLγρµ) ; = M √2s q2 M2 ≡ √2 √2 2s2 M2 W − W W W 3. Phenomenology 73 Observables and experiments

Low energy observables • Fermi constant provides the Higgs VEV (electroweak scale): ⇒   1/2 v = √2G − 246 GeV F ≈ Consistency checks: e.g. ⇒ From muon lifetime:

5 2 G = 1.166 378 7(6) 10− GeV− F × If one compares with (tree level result) G πα πα F = = √2 2s2 M2 2(1 M2 /M2 )M2 W W − W Z W using measurements of MW, MZ and α there is a discrepancy that disappears when quantum corrections are included

3. Phenomenology 74 Observables and experiments

e+e f¯ f • − → 2 dσ f α n h i = N β 1 + cos2 θ + (1 β2 ) sin2 θ G (s) dΩ c 4s f − f 1 o +2(β2 1) G (s) + 2β cos θ G (s) f − 2 f 3

G (s) = Q2Q2 + 2Q Q v v Reχ (s) +( v2 + a2)(v2 + a2 ) χ (s) 2 1 e f e f e f Z e e f f | Z | G (s) =( v2 + a2)a2 χ (s) 2 2 e e f | Z | G (s) = 2Q Q a a Reχ (s) + 4v v a a χ (s) 2 3 e f e f Z e f e f | Z | s f with χZ(s) , Nc = 1 (3) for f = lepton (quark), β f = velocity ≡ s M2 + iM Γ − Z Z Z 2 f 2πα h i q σ(s) = N β (3 β2 )G (s) 3(1 β2 )G (s) , β = 1 4m2 /s c 3s f − f 1 − − f 2 f − f

3. Phenomenology 75 Observables and experiments

Z production (LEP1/SLD) • M , Γ , σ , A , A , R , R , R M , s2 Z Z had FB LR b c ` ⇒ Z W from e+e f¯ f at the Z pole (γ Z interference vanishes). Neglecting m : − → − f

+ Γ(e e−)Γ(had) σhad = 12π 2 2 MZΓZ Γ(bb¯) Γ(cc¯) Γ(had) Rb = Rc = R` = + Γ(had) Γ(had) Γ(` `−)   f αM Γ(Z f f¯) Γ( f f¯) = N Z (v2 + a2 ) → ≡ c 3 f f

Forward-Backward and (if polarized e−) Left-Right asymmetries due to Z:

σF σB 3 Ae + Pe σL σR 2v f a f AFB = − = A f ALR = − = AePe with A f 2 2 σF + σB 4 1 + Pe Ae σL + σR ≡ v f + a f

3. Phenomenology 76 Observables and experiments

W-pair production (LEP2) W production (Tevatron/LHC) • • e+e WW 4 f (+γ) pp/pp¯ W `ν (+γ) − → → → → `

MW ⇒ M ⇒ W Top-quark production (Tevatron/LHC) • pp/pp¯ t¯t 6 f → →

m ⇒ t

3. Phenomenology 77 Observables and experiments

Higgs production (LHC) • pp H + X and H decays to different channels M → ⇒ H [ggF] [VBF] [VH] [ttH]

gg fusion WW, ZZ fusion Higgs-strahlung t¯t fusion

1 WW bb

-1 gg ZZ [ggF] 10 oo LHC HIGGS XS WG 2013

cc [VBF] 10-2

aa Za [VH] [%] Uncert Total + BR Higgs 10-3

[ttH] µµ 10-4 80 100 120 140 160 180 200 MH [GeV] 3. Phenomenology 78 Precise determination of parameters

Experimental precision requires accurate predictions quantum corrections • ⇒ (complication: loop calculations involve renormalization)

– Correction to GF from muon lifetime:

GF GF πα = [1 + ∆r(mt, MH)] √2 → √2 2(1 M2 /M2 )M2 − W Z W when loop corrections are included:

+ + + + 2 loops ···

Since muon lifetime is measured more precisely than MW, it is traded for GF: s ! M2 4πα M2 (α, G , m , M ) = Z 1 + 1 [1 + ∆r(m , M )] W F t H 2 √ 2 t H ⇒ − 2GF MZ

(correlation between MW, mt and MH, given α and GF)

3. Phenomenology 79 Precise determination of parameters

Indirect constraints from LEP1/SLD Direct measurements from LEP2/Tevatron

MH(MW, mt) Allowed regions for MH LHC excluded

[LEPEWWG 2013] 80.5 LHC excluded LEP2 and Tevatron LEP1 and SLD 68% CL ]

80.4 GeV [

W m

80.3 6_ mH [GeV] 114 300 600 1000 155 175 195

mt [GeV]

3. Phenomenology 80 Precise determination of parameters

– Corrections to vector and axial couplings from Z pole observables:

f f f f v g = v + ∆g a g = a + ∆g f → V f V f → A f A 2 sW f 1 h f f i z }| { f sin2 θ = 1 Re(g /g ) (1 M2 /M2 ) κ ⇒ eff 4 Q − V A ≡ − W Z Z | f | (Two) loop calculations are crucial and point to a light Higgs:

[Awramik, Czakon, Freitas] hep-ph/0608069 2 loops s2 = 0.22290 0.00029 (tree) W ± sin2 θlept = 0.23148 0.000017 (exp) −•| eff ± + QCD −

1 loop

3. Phenomenology 81 Precise determination of parameters

In addition, experiments and observables testing the flavor structure of the SM: • flavor conserving: dipole moments, . . . flavor changing: b sγ,... → very sensitive to new physics through loop corrections ⇒ Extremely precise measurements are: – muon anomalous magnetic moment: a = (g 2)/2 µ µ − aexp = 116 592 089 (63) 10 11 [Brookhaven ’06] µ × − aQED = 116 584 718 10 11 [QED: 5 loops] µ − exp SM 11 × aµ a = 287 (80) 10− EW 11 µ aµ = 154 10− [W, Z, H: 2 loops] − × × 3.6σ ! ahad = 6 930 (48) 10 11 [e+e had] µ × − − → aSM = 116 591 802 (49) 10 11 µ × − 2 2 – electron magnetic moment (new physics suppressed by a factor of me /mµ):  exp: ge/2 = 1.001 159 652 180 76 (27)  1 α− = 137.035 999 074 (44) theo: QED (8 loops!)  ⇒

3. Phenomenology 82 Global fits

Fit input data from a list of observables (EWPO): • b,c,` 2 lept MH, MW, ΓW, MZ, ΓZ, σhad, AFB , Ab,c,`, Rb,c,`, sin θeff ,...

2 finding the χmin for ndof = 13 (14) when MH is included (excluded):

2 2 αs(MZ), ∆αhad(MZ), GF, MZ, 9 fermion masses, MH | {z } | {z } 1 (QCD) 17 4=13 (CKM irrelevant) −

[Gfitter 2013] http://gfitter.desy.de

2 ndof χmin p-value 14 20.7 0.11 SM describes data to 1.6σ (about 90% CL) ⇒ 13 19.3 0.11

3. Phenomenology 83 Global fits

Compare direct measurements of these observables with fit values: • (0.2) (0.0) (0.2) (0.4) (1.0) (2.7) (0.0) (0.6) (0.0) (0.0) (0.0) (0.0) (0.0) (-1.7) (-1.0) (-0.7) (-1.7) (-0.8) (-2.1) (-1.4) (-0.3) 0.1 0.2 0.0 0.2 0.9 2.5 0.0 0.6 0.0 0.0 0.0 0.3 0.0 -1.1 -2.1 -0.1 -1.6 -1.9 -1.2 -0.8 -0.7

May 13 SM fitter meas G m ) / meas measurement measurement H H - O fit (O with M w/o M -3 -2 -1 0 1 2 3 Plot inspired by Eberhardt et al. [arXiv:1209.1101] t ) ) Z Z c b 0 c 0 b c b H W W 2 Z 0,l FB 0,c FB 0 lep 0,b FB K R R 0 had FB A A m M M K m m M A R A A m (M (Q (LEP) (SLD) l l (5) had lept eff A A _ O 6 2 sin some tensions (none above 3σ): A (SLD), Ab (LEP), R ,... ⇒ ` FB b 3. Phenomenology 84 Global fits

Fits prefer a somewhat lighter Higgs: ⇒

2 5 May 13 r

6 G fitter SM 4.5 SM fit SM fit w/o M measurement 4 H 2m ATLAS measurement [arXiv:1207.7214] 3.5 CMS measurement [arXiv:1207.7235] 3

2.5

2

1.5

1 1m

0.5

0 60 70 80 90 100 110 120 130 140

MH [GeV]

3. Phenomenology 85 Global fits

In general, impressive consistency of the SM, e.g.: ⇒

80.5 kin 68% and 95% CL fit contours mt Tevatron average ± 1m

w/o MW and mt measurements [GeV]

W 80.45 68% and 95% CL fit contours M w/o M , m and M measurements W t H M world average ± 1m 80.4 W

80.35

80.3

80.25 May 13 =50 GeV =125.7 =300 GeV =600 GeV SM M H M H M H M H G fitter

140 150 160 170 180 190 200

mt [GeV]

3. Phenomenology 86 Summary

The SM is a gauge theory with spontaneous symmetry breaking (renormalizable) • Confirmed by many low and high energy experiments with remarkable accuracy, • at the level of quantum corrections, with (almost) no significant deviations

In spite of its tremendous success, it leaves fundamental questions unanswered: • why 3 generations? why the observed pattern of fermion masses and mixings? and there are several hints for physics beyond: – phenomenological: – conceptual: (g 2) gravity is not included ∗ µ − ∗ neutrino masses hierarchy problem ∗ ∗ dark matter ∗ baryogenesis The SM is an Effective Theory ∗ ⇒ cosmological constant valid up to electroweak scale? ∗ 87 88 89 Kinematics

90 Cross-section

m m p1, 1 p3, 3 . . . p , m m 2 2 pn+2, n+2

n+2 3 1 d pj dσ(i f ) = 2(2π)4δ4(p p ) 2 2 2 1/2 i f ∏ 3 → 4 ( p p ) m m |M| − (2π) 2Ej { 1 2 − 1 2} j=3 B Sum over initial polarizations and/or average over final polarizations if the initial state is unpolarized and/or the final state polarization is not measured

B Divide the total cross-section by a symmetry factor S = ∏ ki! if there are ki identical particles of species i in the final state i

Kinematics 91 Cross-section case 2 2 in CM frame →

m m p1, 1 p3, 3

q = p = p p = p = p 1 − 2 3 − 4 m m p2, 2 p4, 4

Z Z 3 3 Z 4 4 d p3 d p4 p dΩ dΦ2 (2π) δ (p1 + p2 p3 p4) 3 3 = | |2 ⇒ ≡ − − (2π) 2E3 (2π) 2E4 16π ECM

and if m = m 4 ( p p )2 m2m2 1/2 = 4E q 1 2 ⇒ { 1 2 − 1 2} CM| |

dσ 1 p (1, 2 3, 4) = | | 2 dΩ → 64π2E2 q |M| CM | |

Kinematics 92 Decay width

n 3 1 2 4 4 d pj dΓ(i f ) = (2π) δ (P p f ) ∏ 3 → 2M |M| − j=1 (2π) 2Ej

m p1, 1 dΓ 1 p case 1 2 P, M (i 1, 2) = | | 2 → dΩ → 32π2 M2 |M| m p2, 2

B Note that masses M, m1 and m2 fix final energies and momenta: M2 m2 + m2 M2 m2 + m2 E = − 2 1 E = − 1 2 1 2M 2 2M 1/2 [M2 (m + m )2][M2 (m m )2] p = p = p = − 1 2 − 1 − 2 | | | 1| | 2| 2M

Kinematics 93 Loop calculations

94 Structure of one-loop amplitudes

Consider the following generic one-loop diagram with N external legs: • p1 p2

q + k1

m1

q m0

mN 1 −

q + kN 1 − pN pN 1 − N 1 − k1 = p1, k2 = p1 + p2,... kN 1 = ∑ pi − i=1 It contains general integrals of the kind: • D i Z d q qµ qµ TN µ4 D 1 ··· P 2 µ1...µP − D 2 2 2 2 2 2 16π ≡ (2π) [q m0][(q + k1) m1] [(q + kN 1) mN 1] − − ··· − − − Loop calculations 95 Structure of one-loop amplitudes

B D dimensional integration in dimensional regularization B Integrals are symmetric under permutations of Lorentz indices B Scale µ introduced to keep the proper mass dimensions B P is the number of q’s in the numerator and determines the tensor structure of the integral (scalar if P = 0, vector if P = 1, etc.). Note that P N ≤ 1 2 B Notation: A for T , B for T , etc. For example, the scalar integrals A0, B0, etc. B The tensor integrals can be decomposed as a linear combination of the Lorentz covariant tensors that can be built with gµν and a set of linearly independent momenta [Passarino, Veltman ’79]

B The choice of basis is not unique

Here we use the basis formed by gµν and the momenta ki, where the the tensor coefficients are totally symmetric in their indices [Denner ’93]

This the basis used by the computer package LoopTools [www.feynarts.de/looptools]

Loop calculations 96 Structure of one-loop amplitudes

We focus here on: •

Bµ = k1µB1

Bµν = gµνB00 + k1µk1νB11

Cµ = k1µC1 + k2µC2 2 Cµν = gµνC00 + ∑ kiµkjνCij i,j=1

Cµνρ = ...

We will see that the scalar integrals A and B and the tensor integral coefficients • 0 0 B1, B00, B11 and C00 are divergent in D = 4 dimensions (ultraviolet divergence, equivalent to take cutoff Λ ∞ in q) → It is possible to express every tensor coefficient in terms of scalar integrals • (scalar reduction) [Denner ’93]

Loop calculations 97 Explicit calculation

Basic ingredients: • – Euler Gamma function:

Γ(x + 1) = xΓ(x)

Taylor expansion around poles at x = 0, 1, 2, . . . : − − 1 x = 0 : Γ(x) = γ + (x) x − O ( 1)n 1 x = n : Γ(x) = − γ + 1 + + + (x + n) − n!(x + n) − ··· n O where γ 0.5772 . . . is Euler-Mascheroni constant ≈ – Feynman parameters: ! 1 Z 1 n (n 1)! = dx1 dxn δ ∑ xi 1 − n a a a 0 ··· − [x a + x a + x a ] 1 2 ··· n i=1 1 1 2 2 ··· n n

Loop calculations 98 Explicit calculation

– The following integrals, with e 0+, will be needed: → D n  n D/2 Z d q 1 ( 1) i Γ(n D/2) 1 − = − − (2π)D (q2 ∆ + ie)n (4π)D/2 Γ(n) ∆ − D 2 n 1  n D/2 1 Z d q q ( 1) i D Γ(n D/2 1) 1 − − = − − − − ⇒ (2π)D (q2 ∆ + ie)n (4π)D/2 2 Γ(n) ∆ − Let’s solve the first integral in Euclidean space: q0 = iq0 , q = q , q2 = q2 , B E E − E Z dDq 1 Z dDq 1 = i( 1)n E (2π)D (q2 ∆ + ie)n − (2π)D (q2 + ∆)n − E (equivalent to a Wick rotation of 90◦). The second integral follows from this Im q0 δ = q2 +∆ ǫ √δ +i 90 − 2√δ ◦ Re q0 ǫ +√δ i − 2√δ

Loop calculations 99 Explicit calculation

In D-dimensional spherical coordinates: Z dDq 1 Z Z ∞ 1 E = d dq qD 1 D 2 n ΩD E E − 2 n A B (2π) (qE + ∆) 0 (qE + ∆) ≡ I × I Z 2πD/2 where = dΩ = IA D Γ(D/2)

Z ∞ D Z Z Z ∞ D x2 D ∑D x2 D 1 x2 since (√π) = dx e− = d x e− i=1 i = dΩD dx x − e− ∞ 0 − Z  Z ∞ Z  1 D/2 1 t 1 = dΩD dt t − e− = dΩD Γ(D/2) 2 0 2 2 and, changing variables: t = qE, z = ∆/(t + ∆), we have

 n D/2 Z 1  n D/2 1 1 − n D/2 1 D/2 1 1 1 − Γ(n D/2)Γ(D/2) B = ∂z z − − (1 z) − = − I 2 ∆ 0 − 2 ∆ Γ(n) Z 1 α 1 β 1 Γ(α)Γ(β) where Euler Beta function was used: B(α, β) = dz z − (1 z) − = 0 − Γ(α + β) Loop calculations 100 Explicit calculation Two-point functions

q + k1 p p m1

m0 q

Z D µ µ ν i µ µν 4 D d q 1, q , q q B0, B , B (args) = µ − { } 16π2 { } (2π)D q2 m2
(q + p)2 m2 − 0 − 1 B k1 = p 2 B The integrals depend on the masses m0, m1 and the invariant p :

2 2 2 (args) = (p ; m0, m1)

Loop calculations 101 Explicit calculation Two-point functions

Using Feynman parameters, • 1 Z 1 1 = dx 2 a1a2 0 [a x + a (1 x)] 1 2 −

Z 1 Z D µ µ ν µ ν i µ µν 4 D d q 1, A , q q + A A 2 B0, B , B = µ − dx D { − 2 2 } ⇒ 16π { } 0 (2π) (q ∆ ) − 2 with

∆ = x2 p2 + x(m2 m2 p2) + m2 2 1 − 0 − 0

a = (q + p)2 m2 1 − 1 a = q2 m2 2 − 0 and a loop momentum shift to obtain a perfect square in the denominator:

qµ qµ Aµ, Aµ = xpµ → − Loop calculations 102 Explicit calculation Two-point functions

Then, the scalar function is: • Z 1 Z D i 4 D d q 1 2 B0 = µ − dx D 2 2 16π 0 (2π) (q ∆ ) − 2 Z 1 ∆2 B0 = ∆e dx ln 2 + (e)[D = 4 e] ⇒ − 0 µ O − 2 where ∆ γ + ln 4π and the Euler Gamma function was expanded around e ≡ e − x = 0 for D = 4 e, using xe = exp e ln x = 1 + e ln x + (e2): − { } O  2 D/2   iΓ(2 D/2) 1 − i ∆ 4 D = 2 + ( ) µ − − D/2 2 ∆e ln 2 e (4π) ∆2 16π − µ O Comparing with the definitions of the tensor coefficientes we have: • Z 1 Z D µ i µ 4 D d q A 2 B = µ − dx D 2 2 16π − 0 (2π) (q ∆ ) − 2 Z 1 1 ∆2 B1 = ∆e + dx x ln 2 + (e)[D = 4 e] ⇒ −2 0 µ O − Loop calculations 103 Explicit calculation Two-point functions

and Z 1 Z D 2 µν µ ν i µν 4 D d q (q /D)g + A A 2 B = µ − dx D 2 2 16π 0 (2π) (q ∆ ) − 2 1 B = (p2 3m2 3m2)(∆ + 2γ 1) + (e)[D = 4 e] ⇒ 00 −12 − 0 − 1 e − O − Z 1 1 2 ∆2 B11 = ∆e dx x ln 2 + (e)[D = 4 e] 3 − 0 µ O −

where qµqν have been replaced by (q2/D)gµν in the integrand and the Euler Gamma function was expanded around x = 1 for D = 4 e: − −  1 D/2 iΓ(1 D/2) 1 − i 1 4 D = ( + ) + ( ) µ − D−/2 2 ∆2 ∆e 2γ 1 e − (4π) 2Γ(2) ∆2 16π 2 − O

Loop calculations 104 Explicit calculation Three-point functions

p1

q + k1

m1 q m0 p2 p1 m2 −

q + k2

p − 2 Z D µ µ ν i µ µν 4 D d q 1, q , q q C0, C , C (args) = µ − { } 16π2 { } (2π)D q2 m2
(q + p )2 m2 (q + p )2 m2 − 0 1 − 1 2 − 2 B It is convenient to choose the external momenta so that:

k1 = p1, k2 = p2.

B The integrals depend on the masses m0, m1, m2 and the invariants: (args) = (p2, Q2, p2; m2, m2, m2), Q2 (p p )2. 1 2 0 1 2 ≡ 2 − 1 Loop calculations 105 Explicit calculation Three-point functions

Using Feynman parameters, • Z 1 Z 1 x 1 − 1 = 2 dx dy 3 a1a2a3 0 0 [a x + a y + a (1 x y)] 1 2 3 − − Z 1 Z 1 x Z D µ µ ν µ ν i µ µν 4 D − d q 1, A , q q + A A 2 C0, C , C = 2µ − dx dy D { − 2 3 } ⇒ 16π { } 0 0 (2π) (q ∆ ) − 3 with

∆ = x2 p2 + y2 p2 + xy(p2 + p2 Q2) + x(m2 m2 p2) + y(m2 m2 p2) + m2 3 1 2 1 2 − 1 − 0 − 1 2 − 0 − 2 0 a = (q + p )2 m2 1 1 − 1 a = (q + p )2 m2 2 2 − 2 a = q2 m2 3 − 0 and a loop momentum shift to obtain a perfect square in the denominator: µ µ qµ qµ Aµ, Aµ = xp + yp → − 1 2 Loop calculations 106 Explicit calculation Three-point functions

Then the scalar function is: • Z 1 Z 1 x Z D i 4 D − d q 1 2 C0 = 2µ − dx dy D 2 3 16π 0 0 (2π) (q ∆ ) − 3 Z 1 Z 1 x − 1 C0 = dx dy [D = 4] ⇒ − 0 0 ∆3

Comparing with the definitions of the tensor coefficientes we have: • Z 1 Z 1 x Z D µ i µ 4 D − d q A 2 C = 2µ − dx dy D 2 3 16π − 0 0 (2π) (q ∆ ) − 3 Z 1 Z 1 x − x C1 = dx dy [D = 4] ⇒ 0 0 ∆3 Z 1 Z 1 x − y C2 = dx dy [D = 4] 0 0 ∆3

Loop calculations 107 Explicit calculation Three-point functions

Z 1 Z 1 x Z D 2 µν µ ν i µν 4 D − d q (q /D)g + A A 2 C = 2µ − dx dy D 2 3 16π 0 0 (2π) (q ∆ ) − 3 Z 1 Z 1 x 2 − x C11 = dx dy [D = 4] ⇒ − 0 0 ∆3 Z 1 Z 1 x 2 − y C22 = dx dy [D = 4] − 0 0 ∆3 Z 1 Z 1 x − xy C12 = dx dy [D = 4] − 0 0 ∆3 Z 1 Z 1 x 1 1 − ∆3 C00 = ∆e dx dy ln 2 + (e)[D = 4 e] 4 − 2 0 0 µ O − 2 where ∆ γ + ln 4π and qµqν was replaced by (q2/D)gµν in the integrand e ≡ e − In C the Euler Gamma function was expanded around x = 0 for D = 4 e: 00 −  2 D/2   iΓ(2 D/2) 1 − i 1 ∆ 4 D = 3 + ( ) µ − −D/2 2 ∆e ln 2 e (4π) Γ(3) ∆3 16π 2 − µ O Loop calculations 108 Note about Diracology in D dimensions

Attention should be paid to the traces of Dirac matrices when working in D • dimensions (dimensional regularization) since

µ ν ν µ µν µν µν γ γ + γ γ = 2g 14 4, g gµν = Tr g = D × { } Thus, the following identities involving contractions of Lorentz indices can be proven:

µ γ γµ = D γµγνγ = (D 2)γν µ − − γµγνγργ = 4gνρ (4 D)γνγρ µ − − γµγνγργσγ = 2γσγργν + (4 D)γνγργσ µ − −

Loop calculations 109