Higgs Boson Phenomenology Lecture I

Transcription

1 iggs Boson Phenomenology Lecture I Laura Reina TASI 2011, CU-Boulder, June 2011

2 Outline of Lecture I Understanding the Electroweak Symmetry Breaking as a first step towards a more fundamental theory of particle physics. The iggs mechanism and the breaking of the Electroweak Symmetry in the Standard Model. Toy model: breaking of an abelian gauge symmetry. Quantum effects in spontaneously broken gauge theories. The Standard Model: breaking of the SU(2) L U(1) Y symmetry. Fermion masses through Yukawa-like couplings to the iggs field. First step: calculate the SM iggs boson decay branching ratios.

3 Some References for Part I Spontaneous Symmetry Breaking of global and local symmetries: An Introduction to Quantum Field Theory, M.E. Peskin and D.V. Schroeder The Quantum Theory of Fields, V. II, S. Weinberg Theory and Phenomenology of the iggs boson(s): The anatomy of the electro-weak symmetry breaking I: the iggs boson in the standard model, A. Djouadi, Phys. Rep. 457 (2008) 1, hep-ph/ The anatomy of the electro-weak symmetry breaking II: the iggs bosons in the minimal supersymmetric model A. Djouadi, Phys. Rep. 457 (2008) 1, hep-ph/ iggs boson physics, L. Reina, lectures given at TASI 2004, hep-ph/

4 Breaking the Electroweak Symmetry: Why and ow? The gauge symmetry of the Standard Model (SM) forbids gauge boson mass terms, but: M W ± = ±0.023GeV and M Z = ±0.0021GeV Electroweak Symmetry Breaking (EWSB) Broad spectrum of ideas proposed to explain the EWSB: Weakly coupled dynamics embedded into some more fundamental theory at a scale Λ (probably TeV): = iggs Mechanism in the SM or its extensions (MSSM, etc.) Little iggs models Strongly coupled dynamics at the TeV scale: Technicolor in its multiple realizations. Extra dimensions beyond the 3+1 space-time dimensions

5 Different but related... Explicit fermion mass terms also violate the gauge symmetry of the SM: introduced through new gauge invariant interactions, as dictated by the mechanism of EWSB intimately related to flavor mixing and the origin of CP-violation: new experimental evidence on this side will give further insight.

6 The story begins in with Englert and Brout; iggs; agen, Guralnik and Kibble

7 Spontaneous Breaking of a Gauge Symmetry Abelian iggs 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 2 φ φ λ(φ φ) 2 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.

8 Can we build a gauge invariant massive theory? Yes. Consider the potential of the scalar field: V(φ) = µ 2 φ φ+λ(φ φ) 2 where λ>0 (to be bounded from below), and observe that: phi_2 0 0 phi_ µ 2 >0 unique minimum: φ φ = phi_ phi_1 µ 2 <0 degeneracy of minima: φ φ= µ2 2λ

9 µ 2 >0 electrodynamics of a massless photon and a massive scalar field of mass µ (g= e). µ 2 <0 when we choose a minimum, the original U(1) symmetry is spontaneously broken or hidden. φ 0 = ( µ2 2λ ) 1/2 = v 2 φ(x) = φ (φ 1 (x)+iφ 2 (x)) L = 1 4 Fµν F µν g2 v 2 A µ A µ }{{} massive vector field ( µ φ 1 ) 2 +µ 2 φ 2 1 }{{} massive scalar field ( µ φ 2 ) 2 +gva µ µ φ }{{} Goldstone boson Side remark: The φ 2 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 2 m 2 kµ k ν ) A k 2 +

10 More convenient parameterization (unitary gauge): eiχ(x) v φ(x) = (v +(x)) 2 U(1) 1 2 (v +(x)) The χ(x) degree of freedom (Goldstone boson) is rotated away using gauge invariance, while the original Lagrangian becomes: L = L A + g2 v 2 2 Aµ A µ ( µ µ +2µ 2 2) +... which describes now the dynamics of a system made of: a massive vector field A µ with m 2 A =g2 v 2 ; a real scalar field of mass m 2 = 2µ2 =2λv 2 : the iggs field. Total number of degrees of freedom is balanced

11 Non-Abelian iggs mechanism: several vector fields A a µ(x) and several (real) scalar field φ i (x): L = L A +L φ, L φ = 1 2 (Dµ φ) 2 V(φ), V(φ) = µ 2 φ 2 + λ 2 φ4 (µ 2 <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 2 (D µφ) g2 (T a φ) i (T b φ) i A a µa bµ +... φ min =φ g2 (T a φ 0 ) i (T b φ 0 ) i A a }{{} µa bµ +... = m 2 ab T a φ 0 0 massive vector boson + (Goldstone boson) T a φ 0 = 0 massless vector boson + massive scalar field

12 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

13 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 2 ((v +φ 1 (x))+iφ 2 (x)) L = 1 4 Fµν F µν ( µ φ 1 +ga µ φ 2 ) ( µ φ 2 ga µ (v +φ 1 )) 2 V(φ) Quantizing using the gauge fixing condition: G = 1 ( µ A µ +ξgvφ 2 ) ξ in the generating functional [ Z = C DADφ 1 Dφ 2 exp d 4 x (L 12 )] G2 det ( ) δg δα (α gauge transformation parameter)

14 L 1 2 G2 = 1 2 A µ ( g µν 2 + ( 1 1 ) ) µ ν (gv) 2 g µν A ν ξ L ghost + = c 1 2 ( µφ 1 ) m2 φ 1 φ ( µφ 2 ) 2 ξ 2 (gv)2 φ ( [ 2 ξ(gv) 2 1+ φ )] 1 c v such that: A µ (k)a ν ( k) = φ 1 (k)φ 1 ( k) = i k 2 m 2 A i k 2 m 2 φ 1 (g µν kµ k ν k 2 ) + iξ k 2 ξm 2 A ( k µ k ν k 2 ) φ 2 (k)φ 2 ( k) = c(k) c( k) = i k 2 ξm 2 A Goldtone boson φ 2, longitudinal gauge bosons

15 The iggs sector of the Standard Model : SU(2) L U(1) Y SSB U(1) Q Introduce one complex scalar doublet of SU(2) L with Y =1/2: ( ) φ = φ + φ 0 L = (D µ φ) D µ φ µ 2 φ φ λ(φ φ) 2 where D µ φ = ( µ iga a µτ a ig Y φ B µ ), (τ a =σ a /2, a=1,2,3). The SM symmetry is spontaneously broken when φ is chosen to be (e.g.): ( ) φ = 1 ( ) 0 µ 2 1/2 with v = (µ 2 < 0, λ > 0) 2 λ v The gauge boson mass terms arise from: (D µ φ) D µ φ (0 v)( ga a µσ a +g B µ )( ga bµ σ b +g B µ)( v 2 4 [ g 2 (A 1 µ) 2 +g 2 (A 2 µ) 2 +( ga 3 µ +g B µ ) 2] + v ) +

16 And correspond to the weak gauge bosons: W ± µ = 1 2 (A 1 µ ±A 2 µ) M W = g v 2 Z 0 µ = 1 g2 +g 2(gA3 µ g B µ ) M Z = g 2 +g 2 v 2 while the linear combination orthogonal to Z 0 µ remains massless and corresponds to the photon field: A µ = 1 g2 +g 2(g A 3 µ +gb µ ) M A = 0 Notice: using the definition of the weak mixing angle, θ w : cosθ w = g g2 +g 2, sinθ w = g g2 +g 2 the W and Z masses are related by: M W = M Z cosθ w

17 The scalar sector becomes more transparent in the unitary gauge: ( φ(x) = e i v χ(x) τ 2 0 v +(x) after which the Lagrangian becomes ) ( SU(2) φ(x) = 1 2 L = µ 2 2 λv = 1 2 M2 2 0 v +(x) ) λ 2 M λ4 Three degrees of freedom, the χ a (x) Goldstone bosons, have been reabsorbed into the longitudinal components of the W ± µ and Z 0 µ weak gauge bosons. One real scalar field remains: the iggs boson,, with mass M 2 = 2µ 2 = 2λv 2 and self-couplings: = 3i M2 v = 3i M2 v 2

18 From (D µ φ) D µ φ iggs-gauge boson couplings: V µ V µ V ν = 2i M2 V v g µν V ν = 2i M2 V v 2 g µν Notice: The entire iggs sector depends on only two parameters, e.g. M and v v measured in µ-decay: v = ( 2G F ) 1/2 = 246 GeV SM iggs Physics depends on M

19 Also: remember iggs-gauge boson loop-induced couplings: γ,z γ,z γ γ g g They will be discussed in the context of iggs boson decays.

20 Finally: iggs 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 Yukawa = 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: ( ) φ(x) = 1 0 m f = Γ f 2 v 2 v +(x) and f f = i m f v = iy t

21 SM iggs boson decay branching ratios and width Branching ratios ττ bb gg WW ZZ tt LC IGGS XS WG 2010 Γ [GeV] LC IGGS XS WG cc γγ Zγ M [GeV] M [GeV] Observe difference between light and heavy iggs These curves include: tree level + QCD and EW loop corrections. Exercise: Calculate first order predictions and compare to DECAY (M. Spira).