The Standar Model of Particle Physics Lecture I Aspects and properties of the fundamental theory of particle interactions at the electroweak scale Laura Reina Maria Laach School, Bautzen, September 2011 Outline of Lecture I Approaching the study of particles and their interactions: • global and local symmetry principles; −→ consequences of broken or hidden symmetries. −→ Towards the Standard Model of Particle Physics: • main experimental evidence; −→ possible theoretical scenarios. −→ The Standard Model of Particle Physics: • Lagrangian: building blocks and symmetries; −→ strong interactions: Quantum Chromodynamics; −→ electroweak interactions, the Glashow-Weinberg-Salam theory: −→ properties of charged and neutral currents; − breaking the electroweak symmetry: the SM Higgs mechanism. − Particles and forces are a realization of fundamental symmetries of nature Very old story: Noether’s theorem in classical mechanics ∂L ∂L L(qi, q˙i) such that = 0 pi = conserved ∂qi −→ ∂q˙i to any symmetry of the Lagrangian is associated a conserved physical quantity: ⊲ q = x p linear momentum; i i −→ i ⊲ q = θ p angular momentum. i i −→ i Generalized to the case of a relativistic quantum theory at multiple levels: ⊲ q φ (x) coordinates become “fields” “particles” i → j ↔ ⊲ (φ (x), ∂ φ (x)) can be symmetric under many transformations. L j µ j ⊲ To any continuous symmetry of the Lagrangian we can associate a conserved current µ ∂ µ J = L δφj such that ∂µJ = 0 ∂(∂µφj) The symmetries that make the world as we know it . ⊲ translations: conservation of energy and momentum; ⊲ Lorentz transformations (rotations and boosts): conservation of angular momentum (orbital and spin); ⊲ discrete transformations (P,T,C,CP,...): conservation of corresponding quantum numbers; ⊲ global transformations of internal degrees of freedom (φj “rotations”) conservation of “isospin”-like quantum numbers; ⊲ local transformations of internal degrees of freedom (φj(x) “rotations”): define the interaction of fermion (s=1/2) and scalar (s=0) particles in terms of exchanged vector (s=1) massless particles “forces” −→ Requiring different global and local symmetries defines a theory AND Keep in mind that they can be broken From Global to Local: gauging a symmetry Abelian case A theory of free Fermi fields described by the Lagrangian density = ψ¯(x)(i∂/ m)ψ(x) L − is invariant under a global U(1) transformation (α=constant phase) ψ(x) eiαψ(x) such that ∂ ψ(x) eiα∂ ψ(x) → µ → µ and the corresponding Noether’s current is conserved, J µ = ψ¯(x)γµψ(x) ∂ J µ = 0 → µ The same is not true for a local U(1) transformation (α = α(x)) since ψ(x) eiα(x)ψ(x) but ∂ ψ(x) eiα(x)∂ ψ(x)+ igeiα(x)∂ α(x)ψ(x) → µ → µ µ Need to introduce a covariant derivative Dµ such that D ψ(x) eiα(x)D ψ(x) µ → µ Only possibility: introduce a vector field Aµ(x) trasforming as 1 A (x) A (x) ∂ α(x) µ → µ − g µ and define a covariant derivative Dµ according to Dµ = ∂µ + igAµ(x) modifying to accomodate D and the gauge field A (x) as L µ µ 1 = ψ¯(x)(iD/ m)ψ(x) F µν (x)F (x) L − − 4 µν where the last term is the Maxwell Lagrangian for a vector field Aµ, i.e. F (x)= ∂ A (x) ∂ A (x) . µν µ ν − ν µ Requiring invariance under a local U(1) symmetry has: promoted a free theory of fermions to an interacting one; −→ defined univoquely the form of the interaction in terms of a new vector −→ field Aµ(x): = gψ¯(x)γ ψ(x)Aµ(x) Lint − µ no mass term AµA allowed by the symmetry this is QED. −→ µ → Non-abelian case: Yang-Mills theories Consider the same Lagrangian density = ψ¯(x)(i∂/ m)ψ(x) L − where ψ(x) ψ (x) (i = 1,...,n) is a n-dimensional representation of a → i non-abelian compact Lie group (e.g. SU(N)). is invariant under the global transformation U(α) L iαaT a a a 2 ψ(x) ψ′(x)= U(α)ψ(x) , U(α)= e =1+ iα T + O(α ) → a where T ((a = 1,...,dadj)) are the generators of the group infinitesimal transformations with algebra, [T a,T b]= if abcT c and the corresponding Noether’s current are conserved. However, requiring to be invariant under the corresponding local transformation U(x) L U(x)=1+ iαa(x)T a + O(α2) brings us to replace ∂µ by a covariant derivative D = ∂ igAa (x)T a µ µ − µ a in terms of vector fields Aµ(x) that transform as 1 Aa (x) Aa (x)+ ∂ αa(x)+ f abcAb (x)αc(x) µ → µ g µ µ such that 1 D U(x)D U − (x) µ → µ 1 D ψ(x) U(x)D U − (x)U(x)ψ = U(x)D ψ(x) µ → µ µ i 1 F [D ,D ] U(x)F U − (x) µν ≡ g µ ν → µν The invariant form of or Yang Mills Lagrangian will then be L 1 1 = (ψ,D ψ) TrF F µν = ψ¯(iD/ m)ψ F a F µν LY M L µ − 4 µν − − 4 µν a a a where Fµν = Fµν T and F a = ∂ Aa ∂ Aa + gf abcAb Ac µν µ ν − ν µ µ ν We notice that: as in the abelian case: • mass terms of the form Aa,µAa are forbidden by symmetry: gauge −→ µ bosons are massless the form of the interaction between fermions and gauge bosons is −→ fixed by symmetry to be = gψ¯(x)γ T aψ(x)Aa,µ(x) Lint − µ at difference from the abelian case: • gauge bosons carry a group charge and therefore ... −→ gauge bosons have self-interaction. −→ Feynman rules, Yang-Mills theory: ab p iδ F = a b p/ m − i µ c ¤ g = igγ (T )ij µ,c E j k i kµkν ab G = − gµν (1 ξ) δ µ,a ν,b k2 − − k2 α,a p r abc βγ α γα β αβ γ qÎ § = gf g (q r) + g (r p) + g (p q) γ,c − − − Í β,b α,a β,b 2 abe cde αγ βδ αδ βγ ace bde ade bce Ú Ù = ig f f (g g g g )+ f f ( )+ f f ( ) − − ··· ··· Ù Ú γ,c δ,d Spontaneous Breaking of a Gauge Symmetry Abelian Higgs mechanism: one vector field Aµ(x) and one complex scalar field φ(x): = + L LA Lφ where 1 1 = F µν F = (∂µAν ∂ν Aµ)(∂ A ∂ A ) LA −4 µν −4 − µ ν − ν µ and (Dµ =∂µ + igAµ) µ µ 2 2 =(D φ)∗D φ V (φ)=(D φ)∗D φ µ φ∗φ λ(φ∗φ) Lφ µ − µ − − invariant under local phase transformation, or local U(1) symmetry: L φ(x) eiα(x)φ(x) → 1 Aµ(x) Aµ(x)+ ∂µα(x) → g Mass term for Aµ breaks the U(1) gauge invariance. Can we build a gauge invariant massive theory? Yes. Consider the potential of the scalar field: 2 2 V (φ)= µ φ∗φ + λ(φ∗φ) where λ>0 (to be bounded from below), and observe that: 250000 300000 200000 200000 150000 100000 100000 50000 0 0 ±10 ±10 ±15 ±15 ±10 ±10 ±5 ±5 ±5 ±5 0 phi_2 0 phi_1 phi_2 0 0phi_1 5 5 5 5 10 10 10 10 15 15 µ2 >0 unique minimum: µ2 <0 degeneracy of minima: → → µ2 φ∗φ = 0 φ∗φ= − 2λ µ2 >0 electrodynamics of a massless photon and a massive scalar • −→ field of mass µ (g = e). − µ2 <0 when we choose a minimum, the original U(1) symmetry • −→ is spontaneously broken or hidden. µ2 1/2 v 1 φ0 = = φ(x)= φ0 + (φ1(x)+ iφ2(x)) −2λ √2 −→ √2 ⇓ 1 1 1 1 = F µν F + g2v2AµA + (∂µφ )2 + µ2φ2 + (∂µφ )2 + gvA ∂µφ + ... L −4 µν 2 µ 2 1 1 2 2 µ 2 massive vector field massive scalar field Goldstone boson Side remark: The φ2 field actually generates the correct transverse structure for the mass term of the (now massive) Aµ field propagator: i kµkν Aµ(k)Aν ( k) = − gµν + − k2 m2 − k2 ··· − A More convenient parameterization (unitary gauge): i χ(x) e v U(1) 1 φ(x)= (v + H(x)) (v + H(x)) √2 −→ √2 The χ(x) degree of freedom (Goldstone boson) is rotated away using gauge invariance, while the original Lagrangian becomes: g2v2 1 = + AµA + ∂µH∂ H + 2µ2H2 + ... L LA 2 µ 2 µ which describes now the dynamics of a system made of: a massive vector field Aµ with m2 =g2v2; • A a real scalar field H of mass m2 = 2µ2 =2λv2: the Higgs field. • H − ⇓ Total number of degrees of freedom is balanced a Non-Abelian Higgs mechanism: several vector fields Aµ(x) and several (real) scalar field φi(x): 1 λ = + , = (Dµφ)2 V (φ) , V (φ)= µ2φ2 + φ4 L LA Lφ Lφ 2 − 2 (µ2 <0, λ>0) invariant under a non-Abelian symmetry group G: a a φ (1 + iαata) φ t =iT (1 αaT a) φ i −→ ij j −→ − ij j a a (s.t. Dµ =∂µ + gAµT ). In analogy to the Abelian case: 1 1 (D φ)2 ... + g2(T aφ) (T bφ) Aa Abµ + ... 2 µ −→ 2 i i µ φ =φ 1 min 0 ... + g2(T aφ ) (T bφ ) Aa Abµ + ... = −→ 2 0 i 0 i µ 2 mab T aφ = 0 massive vector boson + (Goldstone boson) 0 −→ T aφ = 0 massless vector boson + massive scalar field 0 −→ Classical Quantum : V (φ) V (ϕ ) −→ −→ eff cl The stable vacuum configurations of the theory are now determined by the extrema of the Effective Potential: 1 V (ϕ )= Γ [φ ] , φ = constant = ϕ eff cl −VT eff cl cl cl where δW [J] Γ [φ ]= W [J] d4yJ(y)φ (y) , φ (x)= = 0 φ(x) 0 eff cl − cl cl δJ(x) | | J W [J] generating functional of connected correlation functions −→ Γ [φ ] generating functional of 1PI connected correlation functions eff cl −→ Veff (ϕcl) can be organized as a loop expansion (expansion in ¯h), s.t.: Veff (ϕcl)= V (ϕcl) + loop effects SSB non trivial vacuum configurations −→ Gauge fixing : the Rξ gauges.