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 j ⊲ To any continuous symmetry of the Lagrangian we can associate a conserved current
∂ J = L δφj such that ∂ J = 0 ∂(∂ φj) 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) → → and the corresponding Noether’s current is conserved,
J = ψ¯(x)γ ψ(x) ∂ J = 0 → 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) → →
Need to introduce a covariant derivative D such that
D ψ(x) eiα(x)D ψ(x) → Only possibility: introduce a vector field A (x) trasforming as 1 A (x) A (x) ∂ α(x) → − g and define a covariant derivative D according to
D = ∂ + igA (x) modifying to accomodate D and the gauge field A (x) as L 1 = ψ¯(x)(iD/ m)ψ(x) F ν (x)F (x) L − − 4 ν where the last term is the Maxwell Lagrangian for a vector field A , i.e.
F (x)= ∂ A (x) ∂ A (x) . ν ν − ν 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 A (x): = gψ¯(x)γ ψ(x)A (x) Lint − no mass term A A allowed by the symmetry this is QED. −→ → 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 ∂ by a covariant derivative
D = ∂ igAa (x)T a − a in terms of vector fields A (x) that transform as 1 Aa (x) Aa (x)+ ∂ αa(x)+ f abcAb (x)αc(x) → g such that
1 D U(x)D U − (x) → 1 D ψ(x) U(x)D U − (x)U(x)ψ = U(x)D ψ(x) → i 1 F [D ,D ] U(x)F U − (x) ν ≡ g ν → ν The invariant form of or Yang Mills Lagrangian will then be L 1 1 = (ψ,D ψ) TrF F ν = ψ¯(iD/ m)ψ F a F ν LY M L − 4 ν − − 4 ν a a a where F ν = F ν T and
F a = ∂ Aa ∂ Aa + gf abcAb Ac ν ν − ν ν We notice that:
as in the abelian case: • mass terms of the form Aa, Aa are forbidden by symmetry: gauge −→ 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, (x) Lint −
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
c ¤ g = igγ (T )ij ,c
E j
k i k kν ab G = − g ν (1 ξ) δ ,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 A (x) and one complex scalar field φ(x): = + L LA Lφ where 1 1 = F ν F = (∂ Aν ∂ν A )(∂ A ∂ A ) LA −4 ν −4 − ν − ν and (D =∂ + igA )
2 2 =(D φ)∗D φ V (φ)=(D φ)∗D φ φ∗φ λ(φ∗φ) Lφ − − − invariant under local phase transformation, or local U(1) symmetry: L φ(x) eiα(x)φ(x) → 1 A (x) A (x)+ ∂ α(x) → g Mass term for A 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)= 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 2 >0 unique minimum: 2 <0 degeneracy of minima: → → 2 φ∗φ = 0 φ∗φ= − 2λ 2 >0 electrodynamics of a massless photon and a massive scalar • −→ field of mass (g = e). − 2 <0 when we choose a minimum, the original U(1) symmetry • −→ is spontaneously broken or hidden.
2 1/2 v 1 φ0 = = φ(x)= φ0 + (φ1(x)+ iφ2(x)) −2λ √2 −→ √2
⇓ 1 1 1 1 = F ν F + g2v2A A + (∂ φ )2 + 2φ2 + (∂ φ )2 + gvA ∂ φ + ... L −4 ν 2 2 1 1 2 2 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) A field propagator: i k kν A (k)Aν ( k) = − g ν + − 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 = + A A + ∂ H∂ H + 2 2H2 + ... L LA 2 2 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 = 2 2 =2λv2: the Higgs field. • H −
⇓ Total number of degrees of freedom is balanced a Non-Abelian Higgs mechanism: several vector fields A (x) and several (real) scalar field φi(x): 1 λ = + , = (D φ)2 V (φ) , V (φ)= 2φ2 + φ4 L LA Lφ Lφ 2 − 2 ( 2 <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. D =∂ + gA T ). In analogy to the Abelian case: 1 1 (D φ)2 ... + g2(T aφ) (T bφ) Aa Ab + ... 2 −→ 2 i i φ =φ 1 min 0 ... + g2(T aφ ) (T bφ ) Aa Ab + ... = −→ 2 0 i 0 i 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 inh ¯), s.t.:
Veff (ϕcl)= V (ϕcl) + loop effects
SSB non trivial vacuum configurations −→ Gauge fixing : the Rξ gauges. Consider the abelian case:
1 ν = F F +(D φ)∗D φ V (φ) L −4 ν − upon SSB: 1 φ(x)= ((v + φ1(x)) + iφ2(x)) √2
⇓ 1 1 1 = F ν F + (∂ φ + gA φ )2 + (∂ φ gA (v + φ ))2 V (φ) L −4 ν 2 1 2 2 2 − 1 − Quantizing using the gauge fixing condition:
1 G = (∂ A + ξgvφ2) √ξ in the generating functional 1 δG Z = C A φ φ exp d4x G2 det D D 1D 2 L− 2 δα (α gauge transformation parameter) −→ 1 1 1 G2 = A g ν ∂2 + 1 ∂ ∂ν (gv)2g ν A L− 2 −2 − − ξ − ν 1 1 1 ξ (∂ φ )2 m2 φ2 + (∂ φ )2 (gv)2φ2 + 2 1 − 2 φ1 1 2 2 − 2 2 + φ =c ¯ ∂2 ξ(gv)2 1+ 1 c Lghost − − v such that: i k kν iξ k kν A (k)Aν ( k) = − g ν + − − k2 m2 − k2 k2 ξm2 k2 − A − A i φ (k)φ ( k) = − 1 1 − k2 m2 − φ1 i φ (k)φ ( k) = c(k)¯c( k) = − 2 2 − − k2 ξm2 − A Goldtone boson φ2, longitudinal gauge bosons ⇐⇒ Towards the Standard Model of particle physics
Translating experimental evidence into the right gauge symmetry group.
Electromagnetic interactions QED • → ⊲ well established example of abelian gauge theory ⊲ extremely succesful quantum implementation of field theories ⊲ useful but very simple template Strong interactions QCD • → ⊲ evidence for strong force in hadronic interactions ⊲ Gell-Mann-Nishijima quark model interprets hadron spectroscopy ⊲ need for extra three-fold quantum number (color) − (ex.: hadronic spectroscopy, e+e hadrons, . . .) → ⊲ natural to introduce the gauge group SU(3)C → ⊲ DIS experiments: confirm parton model based on SU(3)C ⊲ . . . and much more! Weak interactions most puzzling ... • → − ⊲ discovered in neutron β-decay: n p + e +¯νe → ⊲ new force: small rates/long lifetimes ⊲ universal: same strength for both hadronic and leptonic processes − − − − − (n pe νe , π +¯νµ, e ν¯e + νµ, . . .) → → → ⊲ violates parity (P) ⊲ charged currents only affect left-handed particles (right-handed antiparticles) ⊲ neutral currents not of electromagnetic nature ⊲ First description: Fermi Theory (1934)
GF µ F = (¯pγµ(1 γ5)n)(¯eγ (1 γ5)νe) L √2 − −
−2 GF Fermi constant, [GF ]= m (in units of c =¯h = 1). → ⊲ Easely accomodates a massive intermediate vector boson
g + − IVB = Wµ Jµ + h.c. L √2 with (in a proper quark-based notation)
− 1 γ5 µ 1 γ5 J =uγ ¯ µ − d +ν ¯eγ − e µ 2 2 u e u e W D E −→ D g E
¥ D ¥ D provided that, d ν¯e d ν¯e 2 2 2 GF g q MW = 2 ≪ −→ √2 8MW ⊲ Promote it to a gauge theory: natural candidate SU(2)L, but if 1,2 1 2 3 T can generate the charged currents (T ± =(T iT )), T ± cannot be the electromagnetic charge (Q) (T 3 = σ3/2’s eigenvalues do not match charges in SU(2) doublets) ⊲ Need extra U(1) , such that Y = T 3 Q! Y − ⊲ Need massive gauge bosons SSB → ⇓
SSB SU(2) U(1) U(1) L × Y −→ Q
= + LSM LQCD LEW where = ferm + gauge + SSB + Y ukawa LEW LEW LEW LEW LEW Strong interactions: Quantum Chromodynamics
Exact Yang-Mills theory based on SU(3)C (quark fields only): 1 = Q¯ (iD/ m ) Q F a, ν F a LQCD i − i i − 4 ν i with D = ∂ igAa T a − F a = ∂ Aa ∂ Aa + gf abcAb Ac ν ν − ν ν
Q (i = 1,..., 6 u,d,s,c,b,t) fundamental representation of • i → → SU(3) triplets: → Qi
Qi = Qi
Qi Aa adjoint representation of SU(3) N 2 1 = 8 massless gluons • → → − T a SU(3) generators (Gell-Mann’s matrices) → Electromagnetic and weak interactions: unified into Glashow-Weinberg-Salam theory
Spontaneously broken Yang-Mills theory based on SU(2) U(1) . L × Y
SU(2)L weak isospin group, gauge coupling g: • → ⊲ three generators: T i = σi/2 (σi = Pauli matrices, i =1, 2, 3) µ µ µ ⊲ three gauge bosons: W1 , W2 , and W3 1 ⊲ ψL = 2 (1 γ5)ψ fields are doublets of SU(2) 1 − ⊲ ψR = 2 (1 + γ5)ψ fields are singlets of SU(2) ⊲ mass terms not allowed by gauge symmetry
′ U(1)Y weak hypercharge group (Q = T3 + Y ), gauge coupling g : • → ⊲ one generator each field has a Y charge ⊲ one gauge boson:→ Bµ Example: first generation
νeL LL = (νeR)Y =0 (eR)Y =−1 eL Y =−1/2
uL QL = (uR)Y =2/3 (dR)Y =−1/3 dL Y =1/6 Three fermionic generations, summary of gauge quantum numbers:
SU(3)C SU(2)L U(1)Y U(1)Q
2 uL cL tL Qi = 3 2 1 3 L 6 1 dL sL bL − 3 i 2 2 uR = uR cR tR 3 1 3 3 i 1 1 d = dR sR bR 3 1 R − 3 − 3
νeL νµL ντL 0 Li = 1 2 1 L − 2 eL L τL 1 − i e = eR R τR 1 1 1 1 R − − i νR = νeR νµR ντR 1 1 0 0
i where a minimal extension to include νR has been allowed (notice however that it has zero charge under the entire SM gauge group!) Lagrangian of fermion fields
For each generation (here specialized to the first generation):
ferm = L¯ (iD/)L +¯e (iD/)e +¯ν (iD/)ν +Q¯ (iD/)Q +¯u (iD/)u +d¯ (iD/)d LEW L L R R eR eR L L R R R R where in each term the covariant derivative is given by
i i 1 D = ∂ igW T ig′ YB − − 2 and T i = σi/2 for L-fields, while T i = 0 for R-fields (i = 1, 2, 3), i.e.
+ 3 ′ ig 0 Wµ i gWµ g Y Bµ 0 Dµ,L = ∂µ − − √ − − 2 3 ′ 2 Wµ 0 0 gWµ g Y Bµ − − ′ 1 D = ∂ + ig YB µ,R µ 2 µ with 1 1 2 W ± = W iW √2 ∓ ferm can then be written as LEW ferm = ferm + + LEW Lkin LCC LNC where
ferm = L¯ (i∂/)L +e ¯ (i∂/)e + ... Lkin L L R R g + CC = W ν¯eLγ eL + W −e¯Lγ νeL + ... L √2 g g = W 3 [¯ν γ ν e¯ γ e ]+ ′ B [Y (L)(¯ν γ ν +e ¯ γ e ) LNC 2 eL eL − L L 2 eL eL L L + Y (eR)¯νeRγ νeR + Y (eR)¯eRγ eR]+ ... where
1 1 2 W ± = W iW mediators of Charged Currents √2 ∓ → W 3 and B mediators of Neutral Currents. →
⇓ 3 However neither W nor B can be identified with the photon field A , because they couple to neutral fields. 3 Rotate W and B introducing a weak mixing angle (θW )
3 W = sin θW A + cos θW Z B = cos θ A sin θ Z W − W such that the kinetic terms are still diagonal and the neutral current lagrangian becomes
µ 3 ′ Y µ 3 ′ Y NC = ψγ¯ g sin θW T + g cos θW ψAµ+ψγ¯ g cos θW T g sin θW ψZµ L 2 − 2 for ψT =(ν , e ,ν , e ,...). One can then identify (Q e.m. charge) eL L eR R → 3 Y eQ = g sin θ T + g′ cos θ W W 2 and, e.g., from the leptonic doublet LL derive that
′ g g 2 sin θW 2 cos θW = 0 − ′ g sin θW = g′ cos θW = e g sin θ g cos θ = e −→ − 2 W − 2 W − i
µ A ¤ g = ieQ γ − f
E j i
µ W ie ¤ g = γ (1 γ5) 2√2sw −
E j i
µ Z ¤ g = ieγ (v a γ ) f − f 5
E j where 3 sw Tf vf = Qf + − cw 2sW cW 3 Tf af = 2sW cW Lagrangian of gauge fields
1 1 gauge = W a W a, ν B B ν LEW −4 ν − 4 ν where
B = ∂ B ∂ B ν ν − ν W a = ∂ W a ∂ W a + gǫabcW bW c ν ν − ν ν in terms of physical fields:
gauge = gauge + 3V + 4V LEW Lkin LEW LEW where
gauge 1 + + µ −ν ν −µ = (∂µW ∂ν W )(∂ W ∂ W ) Lkin − 2 ν − µ − 1 µ ν ν µ 1 µ ν ν µ (∂µZν ∂ν Zµ)(∂ Z ∂ Z ) (∂µAν ∂ν Aµ)(∂ A ∂ A ) − 4 − − − 4 − − − − 3V = (3-gauge-boson vertices involving ZW +W and AW +W ) LEW − − 4V = (4-gauge-boson vertices involving ZZW +W , AAW +W , LEW − − − AZW +W , and W +W W +W ) k i k kν
G = − g ν ν k2 M 2 − M 2 − V V + Wµ
Ú g = ieCV [g ν (k+ k )ρ + gνρ(k kV ) + gρ (kV k+)ν ] Vρ − − − − −
Ù − Wν + Wµ Vρ
2 ′ Ú Ù = ie C (2g g g g g g ) V V ν ρσ − ρ νσ − σ νρ
Ù Ú − ′ Wν Vσ where cW Cγ = 1 , CZ = −sW and 2 cW cW 1 Cγγ = 1 , CZZ = 2 , CγZ = , CW W = 2 − −sW sW sW SSB The Higgs sector of the Standard Model: SU(2)L U(1)Y U(1)Q × −→
Introduce one complex scalar doublet of SU(2)L with Y =1/2:
+ φ SSB 2 2 φ = EW =(D φ)†D φ φ†φ λ(φ†φ) φ0 ←→ L − −
a a a a where D φ =(∂ igW T ig′Y B ), (T =σ /2, a=1, 2, 3). − − φ The SM symmetry is spontaneously broken when φ is chosen to be (e.g.): 1/2 1 0 2 φ = with v = − ( 2 < 0,λ> 0) √2 λ v The gauge boson mass terms arise from:
1 a a b b 0 (D φ)†D φ + (0 v) gW σ + g′B gW σ + g′B + −→ 8 v 2 1 v 2 1 2 2 2 2 3 2 + g (W ) + g (W ) +( gW + g′B ) + −→ 2 4 − And correspond to the weak gauge bosons:
1 1 2 v W ± = (W iW ) MW = g √2 ∓ −→ 2
1 3 2 2 v Z = (gW g′B ) MZ = g + g′ 2 2 2 g + g′ − −→ while the linear combination orthogonal to Z remains massless and corresponds to the photon field:
1 3 A = (g′W + gB ) MA = 0 2 2 g + g′ −→
Notice: using the definition of the weak mixing angle, θw:
g g′ cos θw = , sin θw = 2 2 2 2 g + g′ g + g′ the W and Z masses are related by: MW = MZ cos θw The scalar sector becomes more transparent in the unitary gauge:
i χ(x) τ e v 0 SU(2) 1 0 φ(x)= φ(x)= √2 v + H(x) −→ √2 v + H(x) after which the Lagrangian becomes
2 2 3 1 4 1 2 2 λ 3 1 4 = H λvH H = MH H MH H λH L − − 4 −2 − 2 − 4 Three degrees of freedom, the χa(x) Goldstone bosons, have been reabsorbed into the longitudinal components of the W ± and Z weak gauge bosons. One real scalar field remains: 2 2 2 the Higgs boson, H, with mass M = 2µ = 2λv H − and self-couplings:
H H H M 2 M 2 H = 3i H = 3i H − v − v2 H H H From (D φ)†D φ Higgs-Gauge boson couplings: −→
µ µ V V H M 2 M 2 H V µν V µν = 2i v g = 2i v2 g ν ν V V H Notice: The entire Higgs sector depends on only two parameters, e.g.
MH and v v measured in µ-decay: 1/2 SM Higgs Physics depends on MH v =(√2GF )− = 246 GeV −→ Higgs boson couplings to quarks and leptons
The gauge symmetry of the SM also forbids fermion mass terms i i (mQi QLuR,...), but all fermions are massive.
⇓ Fermion masses are generated via gauge invariant Yukawa couplings:
Y ukawa = ΓijQ¯i φcuj ΓijQ¯i φdj ΓijL¯i φlj +h.c. LEW − u L R − d L R − e L R such that, upon spontaneous symmetry breaking: v + H v + H v + H Y ukawa = Γiju¯i uj Γijd¯i dj Γij¯li lj +h.c. LEW − u L √2 R − d L √2 R − e L √2 R H = f¯i M ijf j 1+ +h.c. − L f R v f,i,j where v M ij =Γij f f √2 is a non-diagonal mass matrix. Upon diagonalization (by unitary transformation UL and UR)
f f MD =(UL)†Mf UR and defining mass eigenstates:
i f j i f j fL′ =(UL)ijfL and fR′ =(UR)ijfR the fermion masses are extracted as
Y ukawa i f f j H = f¯′ [(U )†M U ]f ′ 1+ +h.c. LEW L L f R R v f,i,j H = m f¯′ f ′ + f¯′ f ′ 1+ f L R R L v f,i,j
f m H = i f = iy − v − t f In terms of the new mass eigenstates the quark part of now reads LCC g i u d j CC = u¯′ [(U )†U ]γ d +h.c. L √2 L L R L where u d VCKM =(UL)†UR is the Cabibbo-Kobayashi-Maskawa matrix, origin of flavor mixing in the SM.
⇓ see G. Buchalla’s lectures at this school