The Standar Model of Particle Physics Lecture I

Transcription

1 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 011

2 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.

3 Particles and forces are a realization of fundamental symmetries of nature Very old story: Noether s theorem in classical mechanics L(q i, q i ) such that L q i = 0 p i = L q i conserved to any symmetry of the Lagrangian is associated a conserved physical quantity: q i = x i p i linear momentum; q i = θ i p i angular momentum. Generalized to the case of a relativistic quantum theory at multiple levels: q i φ j (x) coordinates become fields particles L(φ j (x), µ φ j (x)) can be symmetric under many transformations. To any continuous symmetry of the Lagrangian we can associate a conserved current J µ = L ( µ φ j ) δφ j such that µ J µ = 0

4 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 (φ j rotations ) conservation of isospin -like quantum numbers; local transformations of internal degrees of freedom (φ j (x) rotations ): define the interaction of fermion (s=1/) 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

5 From Global to Local: gauging a symmetry Abelian case A theory of free Fermi fields described by the Lagrangian density L = ψ(x)(i / m)ψ(x) is invariant under a global U(1) transformation (α=constant phase) ψ(x) e iα ψ(x) such that µ ψ(x) e iα µ ψ(x) and the corresponding Noether s current is conserved, J µ = ψ(x)γ µ ψ(x) µ J µ = 0 The same is not true for a local U(1) transformation (α = α(x)) since ψ(x) e iα(x) ψ(x) but µ ψ(x) e iα(x) µ ψ(x)+ige iα(x) µ α(x)ψ(x) Need to introduce a covariant derivative D µ such that D µ ψ(x) e iα(x) D µ ψ(x)

6 Only possibility: introduce a vector field A µ (x) trasforming as A µ (x) A µ (x) 1 g µα(x) and define a covariant derivative D µ according to D µ = µ +iga µ (x) modifying L to accomodate D µ and the gauge field A µ (x) as L = ψ(x)(id/ m)ψ(x) 1 4 Fµν (x)f µν (x) where the last term is the Maxwell Lagrangian for a vector field A µ, i.e. F µν (x) = µ A ν (x) ν A µ (x). 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 µ (x): L int = g ψ(x)γ µ ψ(x)a µ (x) no mass term A µ A µ allowed by the symmetry this is QED.

7 Non-abelian case: Yang-Mills theories Consider the same Lagrangian density L = ψ(x)(i / m)ψ(x) where ψ(x) ψ i (x) (i = 1,...,n) is a n-dimensional representation of a non-abelian compact Lie group (e.g. SU(N)). L is invariant under the global transformation U(α) ψ(x) ψ (x) = U(α)ψ(x), U(α) = e iαa T a = 1+iα a T a +O(α ) where T a ((a = 1,...,d adj )) are the generators of the group infinitesimal transformations with algebra, [T a,t b ] = if abc T c and the corresponding Noether s current are conserved. However, requiring L to be invariant under the corresponding local transformation U(x) U(x) = 1+iα a (x)t a +O(α ) brings us to replace µ by a covariant derivative D µ = µ iga a µ(x)t a

8 in terms of vector fields A a µ(x) that transform as such that A a µ(x) A a µ(x)+ 1 g µα a (x)+f abc A b µ(x)α c (x) D µ U(x)D µ U 1 (x) D µ ψ(x) U(x)D µ U 1 (x)u(x)ψ = U(x)D µ ψ(x) F µν i g [D µ,d ν ] U(x)F µν U 1 (x) The invariant form of L or Yang Mills Lagrangian will then be L YM = L(ψ,D µ ψ) 1 4 TrF µνf µν = ψ(id/ m)ψ 1 4 Fa µνf µν a where F µν = F a µνt a and F a µν = µ A a ν ν A a µ +gf abc A b µa c ν

9 We notice that: as in the abelian case: mass terms of the form A a,µ A a µ are forbidden by symmetry: gauge bosons are massless the form of the interaction between fermions and gauge bosons is fixed by symmetry to be L int = g ψ(x)γ µ T a ψ(x)a a,µ (x) at difference from the abelian case: gauge bosons carry a group charge and therefore... gauge bosons have self-interaction.

10 Feynman rules, Yang-Mills theory: a p b = iδ ab p/ m i µ,c j µ,a k ν,b = igγ µ (T c ) ij = i k [ g µν (1 ξ) k ] µk ν k δ ab α,a q p r γ,c = gf abc( g βγ (q r) α +g γα (r p) β +g αβ (p q) γ) β,b α,a Î Í β,b = ig [ f abe f cde (g αγ g βδ g αδ g βγ )+f ace f bde ( )+f ade f bce ( ) ] Ù Ú Ú γ,c δ,d Ù

11 Spontaneous Breaking of a Gauge Symmetry Abelian Higgs mechanism: one vector field A µ (x) and one complex scalar field φ(x): L = L A +L φ where L A = 1 4 Fµν F µν = 1 4 ( µ A ν ν A µ )( µ A ν ν A µ ) and (D µ = µ +iga µ ) L φ = (D µ φ) D µ φ V(φ) = (D µ φ) D µ φ µ φ φ λ(φ φ) L invariant under local phase transformation, or local U(1) symmetry: φ(x) e iα(x) φ(x) A µ (x) A µ (x)+ 1 g µ α(x) Mass term for A µ breaks the U(1) gauge invariance.

12 Can we build a gauge invariant massive theory? Yes. Consider the potential of the scalar field: V(φ) = µ φ φ+λ(φ φ) where λ>0 (to be bounded from below), and observe that: phi_ 0 0 phi_ µ >0 unique minimum: φ φ = phi_ phi_1 µ <0 degeneracy of minima: φ φ= µ λ

13 µ >0 electrodynamics of a massless photon and a massive scalar field of mass µ (g= e). µ <0 when we choose a minimum, the original U(1) symmetry is spontaneously broken or hidden. φ 0 = ( µ λ ) 1/ = v φ(x) = φ (φ 1 (x)+iφ (x)) L = 1 4 Fµν F µν + 1 g v A µ A µ }{{} massive vector field + 1 ( µ φ 1 ) +µ φ 1 }{{} massive scalar field + 1 ( µ φ ) +gva µ µ φ +... }{{} Goldstone boson Side remark: The φ field actually generates the correct transverse structure for the mass term of the (now massive) A µ field propagator: A µ (k)a ν i ( k) = (g µν k m kµ k ν ) A k +

14 More convenient parameterization (unitary gauge): eiχ(x) v φ(x) = (v +H(x)) U(1) 1 (v +H(x)) The χ(x) degree of freedom (Goldstone boson) is rotated away using gauge invariance, while the original Lagrangian becomes: L = L A + g v Aµ A µ + 1 ( µ H µ H +µ H ) +... which describes now the dynamics of a system made of: a massive vector field A µ with m A =g v ; a real scalar field H of mass m H = µ =λv : the Higgs field. Total number of degrees of freedom is balanced

15 Non-Abelian Higgs mechanism: several vector fields A a µ(x) and several (real) scalar field φ i (x): L = L A +L φ, L φ = 1 (Dµ φ) V(φ), V(φ) = µ φ + λ φ4 (µ <0, λ>0) invariant under a non-abelian symmetry group G: φ i (1+iα a t a ) ij φ j t a =it a (1 α a T a ) ij φ j (s.t. D µ = µ +ga a µt a ). In analogy to the Abelian case: 1 (D µφ) g (T a φ) i (T b φ) i A a µa bµ +... φ min =φ g (T a φ 0 ) i (T b φ 0 ) i A a }{{} µa bµ +... = m ab T a φ 0 0 massive vector boson + (Goldstone boson) T a φ 0 = 0 massless vector boson + massive scalar field

16 Classical Quantum : V(φ) V eff (ϕ cl ) The stable vacuum configurations of the theory are now determined by the extrema of the Effective Potential: V eff (ϕ cl ) = 1 VT Γ eff[φ cl ], φ cl = constant = ϕ cl where Γ eff [φ cl ] = W[J] d 4 yj(y)φ cl (y), φ cl (x) = δw[j] δj(x) = 0 φ(x) 0 J W[J] generating functional of connected correlation functions Γ eff [φ cl ] generating functional of 1PI connected correlation functions V eff (ϕ cl ) can be organized as a loop expansion (expansion in h), s.t.: V eff (ϕ cl ) = V(ϕ cl )+loop effects SSB non trivial vacuum configurations

17 Gauge fixing : the R ξ gauges. Consider the abelian case: L = 1 4 Fµν F µν +(D µ φ) D µ φ V(φ) upon SSB: φ(x) = 1 ((v +φ 1 (x))+iφ (x)) L = 1 4 Fµν F µν + 1 ( µ φ 1 +ga µ φ ) + 1 ( µ φ ga µ (v +φ 1 )) V(φ) Quantizing using the gauge fixing condition: G = 1 ( µ A µ +ξgvφ ) ξ in the generating functional [ Z = C DADφ 1 Dφ exp d 4 x (L 1 )] G det ( ) δg δα (α gauge transformation parameter)

18 L 1 G = 1 A µ ( g µν + ( 1 1 ) ) µ ν (gv) g µν A ν ξ L ghost + = c 1 ( µφ 1 ) 1 m φ 1 φ ( µφ ) ξ (gv) φ + ( [ ξ(gv) 1+ φ )] 1 c v such that: A µ (k)a ν ( k) = φ 1 (k)φ 1 ( k) = i k m A i k m φ 1 (g µν kµ k ν k ) + iξ k ξm A ( k µ k ν k ) φ (k)φ ( k) = c(k) c( k) = i k ξm A Goldtone boson φ, longitudinal gauge bosons

19 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 + ν e new force: small rates/long lifetimes universal: same strength for both hadronic and leptonic processes (n pe ν e, π µ + ν µ, µ e ν e +ν µ,...)

20 violates parity (P) charged currents only affect left-handed particles (right-handed antiparticles) neutral currents not of electromagnetic nature First description: Fermi Theory (1934) L F = G F ( pγ µ (1 γ 5 )n)(ēγ µ (1 γ 5 )ν e ) G F Fermi constant, [G F ] = m (in units of c = h = 1). Easely accomodates a massive intermediate vector boson L IVB = g W + µ J µ +h.c. with (in a proper quark-based notation) provided that, J µ = ūγ µ 1 γ 5 u e d νe q M W d+ ν e γ µ 1 γ 5 u W G F = g 8M W e d νe e

21 Promote it to a gauge theory: natural candidate SU() L, but if T 1, can generate the charged currents (T ± = (T 1 ±it )), T 3 cannot be the electromagnetic charge (Q) (T 3 = σ 3 / s eigenvalues do not match charges in SU() doublets) Need extra U(1) Y, such that Y = T 3 Q! Need massive gauge bosons SSB SU() L U(1) Y SSB U(1) Q L SM = L QCD +L EW where L EW = L ferm EW +L gauge EW +L SSB EW +L Y ukawa EW

22 Strong interactions: Quantum Chromodynamics Exact Yang-Mills theory based on SU(3) C (quark fields only): L QCD = i Q i (id/ m i )Q i 1 4 Fa,µν F a µν with D µ = µ iga a µt a F a µν = µ A a ν ν A a µ +gf abc A b µa c ν Q i (i = 1,...,6 u,d,s,c,b,t) fundamental representation of SU(3) triplets: Q i = Q i Q i Q i A a µ adjoint representation of SU(3) N 1 = 8 massless gluons T a SU(3) generators (Gell-Mann s matrices)

23 Electromagnetic and weak interactions: unified into Glashow-Weinberg-Salam theory Spontaneously broken Yang-Mills theory based on SU() L U(1) Y. SU() L weak isospin group, gauge coupling g: three generators: T i = σ i / (σ i = Pauli matrices, i = 1,,3) three gauge bosons: W µ 1, Wµ, and Wµ 3 ψ L = 1 (1 γ 5)ψ fields are doublets of SU() ψ R = 1 (1+γ 5)ψ fields are singlets of SU() mass terms not allowed by gauge symmetry U(1) Y weak hypercharge group (Q = T 3 +Y), gauge coupling g : one generator each field has a Y charge one gauge boson: B µ Example: first generation ( ) L L = ( ν el e L ) Y = 1/ (ν er ) Y =0 (e R ) Y = 1 Q L = u L d L Y =1/6 (u R ) Y =/3 (d R ) Y = 1/3

24 Three fermionic generations, summary of gauge quantum numbers: SU(3) C SU() L U(1) Y U(1) Q Q i L = ( u L d L ) ( c L s L ) ( t L b L ) 3 u i R = u R c R t R d i R = d R s R b R L i L = ( ν el e L ) ( ν µl µ L ) ( ν τl τ L ) e i R = e R µ R τ R νr i = ν er ν µr ν τr where a minimal extension to include νr i has been allowed (notice however that it has zero charge under the entire SM gauge group!)

25 Lagrangian of fermion fields For each generation (here specialized to the first generation): L ferm EW = L L (id/)l L +ē R (id/)e R + ν er (id/)ν er + Q L (id/)q L +ū R (id/)u R + d R (id/)d R where in each term the covariant derivative is given by D µ = µ igw i µt i ig 1 YB µ and T i = σ i / for L-fields, while T i = 0 for R-fields (i = 1,,3), i.e. ) ( D µ,l ( = µ ig D µ,r = µ +ig 1 YB µ with 0 W + µ W µ 0 i W ± = 1 ( W 1 µ iw ) µ gw 3 µ g YB µ 0 0 gw 3 µ g YB µ )

26 L ferm EW can then be written as L ferm EW = L ferm kin +L CC +L NC where L ferm kin = L L (i /)L L +ē R (i /)e R +... L CC = g W + µ ν el γ µ e L +W µ ē L γ µ ν el +... L NC = g W3 µ [ ν el γ µ ν el ē L γ µ e L ]+ g B µ[y(l)( ν el γ µ ν el +ē L γ µ e L ) + Y(e R ) ν er γ µ ν er +Y(e R )ē R γ µ e R ]+... where W ± = 1 ( W 1 µ iw µ) mediators of Charged Currents W 3 µ and B µ mediators of Neutral Currents. However neither W 3 µ nor B µ can be identified with the photon field A µ, because they couple to neutral fields.

27 Rotate W 3 µ and B µ introducing a weak mixing angle (θ W ) W 3 µ = sinθ W A µ +cosθ W Z µ B µ = cosθ W A µ sinθ W Z µ such that the kinetic terms are still diagonal and the neutral current lagrangian becomes ( L NC = ψγ µ gsinθ W T 3 +g Y cosθ W )ψa µ + ψγ µ ( gcosθ W T 3 g sinθ W Y for ψ T = (ν el,e L,ν er,e R,...). One can then identify (Q e.m. charge) eq = gsinθ W T 3 +g cosθ W Y and, e.g., from the leptonic doublet L L derive that g sinθ W g cosθ W = 0 g sinθ W g cosθ W = e gsinθ W = g cosθ W = e ) ψz µ

28 i Aµ j i Wµ j i Zµ j = ieq f γ µ = ie s w γ µ (1 γ 5 ) = ieγ µ (v f a f γ 5 ) where v f = s w c w Q f + T3 f s W c W a f = T 3 f s W c W

29 Lagrangian of gauge fields where L gauge EW = 1 4 Wa µνw a,µν 1 4 B µνb µν B µν = µ B ν ν B µ W a µν = µ W a ν ν W a µ +gǫ abc W b µw c ν in terms of physical fields: where L gauge EW = L gauge kin +L 3V EW +L 4V EW L gauge kin = 1 ( µw + ν ν W + µ )( µ W ν ν W µ ) 1 4 ( µz ν ν Z µ )( µ Z ν ν Z µ ) 1 4 ( µa ν ν A µ )( µ A ν ν A µ ) L 3V EW = (3-gauge-boson vertices involving ZW + W and AW + W ) L 4V EW = (4-gauge-boson vertices involving ZZW + W, AAW + W, AZW + W, and W + W W + W )

30 µ k ν = i k M V ( g µν k ) µk ν MV W + µ V ρ = iec V [g µν (k + k ) ρ +g νρ (k k V ) µ +g ρµ (k V k + ) ν ] W ν W + µ V ρ Ú Ù = ie C VV (g µν g ρσ g µρ g νσ g µσ g νρ ) Ù Ú Ú Ù Wν V σ where and C γγ = 1, C ZZ = c W s W C γ = 1, C Z = c W s W, C γz = c W s W, C WW = 1 s W

31 The Higgs sector of the Standard Model: SU() L U(1) Y SSB U(1) Q Introduce one complex scalar doublet of SU() L with Y =1/: ( ) φ = φ + φ 0 L SSB EW = (D µ φ) D µ φ µ φ φ λ(φ φ) where D µ φ = ( µ igw a µt a ig Y φ B µ ), (T a =σ a /, a=1,,3). The SM symmetry is spontaneously broken when φ is chosen to be (e.g.): ( ) φ = 1 ( ) 0 µ 1/ with v = (µ < 0, λ > 0) λ The gauge boson mass terms arise from: v (D µ φ) D µ φ (0 v)( gw a µσ a +g B µ )( gw bµ σ b +g B µ)( v 4 [ g (W 1 µ) +g (W µ) +( gw 3 µ +g B µ ) ] + v ) +

32 And correspond to the weak gauge bosons: W ± µ = 1 (W 1 µ iw µ) M W = g v Z µ = 1 g +g (gw3 µ g B µ ) M Z = g +g v while the linear combination orthogonal to Z µ remains massless and corresponds to the photon field: A µ = 1 g +g (g W 3 µ +gb µ ) M A = 0 Notice: using the definition of the weak mixing angle, θ w : cosθ w = g g +g, sinθ w = g g +g the W and Z masses are related by: M W = M Z cosθ w

33 The scalar sector becomes more transparent in the unitary gauge: ( φ(x) = e i v χ(x) τ 0 v +H(x) after which the Lagrangian becomes ) ( SU() φ(x) = 1 L = µ H λvh H4 = 1 M HH 0 v +H(x) ) λ M HH λh4 Three degrees of freedom, the χ a (x) Goldstone bosons, have been reabsorbed into the longitudinal components of the W ± µ and Z µ weak gauge bosons. One real scalar field remains: the Higgs boson, H, with mass M H = µ = λv and self-couplings: H H H= 3i M H v H H H H = 3i M H v

34 From (D µ φ) D µ φ Higgs-Gauge boson couplings: V µ V µ H V ν H= i M V v g µν V ν H = i M V v g µν Notice: The entire Higgs sector depends on only two parameters, e.g. M H and v v measured in µ-decay: v = ( G F ) 1/ = 46 GeV SM Higgs Physics depends on M H

35 Higgs boson couplings to quarks and leptons The gauge symmetry of the SM also forbids fermion mass terms (m Qi Q i L ui R,...), but all fermions are massive. Fermion masses are generated via gauge invariant Yukawa couplings: L Y ukawa EW = Γ ij u Q i Lφ c u j R Γij d Q i Lφd j R Γij e L i Lφl j R +h.c. such that, upon spontaneous symmetry breaking: L Y ukawa EW = Γ ij uū i v +H L u j R Γij d i v +H d L d j R Γij l e i v +H L l j R +h.c. ) = f,i,j f i LM ij f fj R ( 1+ H v +h.c. where M ij f = Γij f v is a non-diagonal mass matrix.

36 Upon diagonalization (by unitary transformation U L and U R ) and defining mass eigenstates: M D = (U f L ) M f U f R f i L = (U f L ) ijf j L and f i R = (U f R ) ijf j R the fermion masses are extracted as L Y ukawa EW = f,i,j f i L [(U f L ) M f U f R ]f j R ( 1+ H ) v ) +h.c. = f,i,jm f ( f L f R + f Rf L ) ( 1+ H v f f H = i m f v = iy t

37 In terms of the new mass eigenstates the quark part of L CC now reads where L CC = g ū i L[(U u L) U d R]γ µ d j L +h.c. V CKM = (U u L) U d R is the Cabibbo-Kobayashi-Maskawa matrix, origin of flavor mixing in the SM. see G. Buchalla s lectures at this school