Laura Reina
TASI 2004, Boulder Outline of Part I
Understanding the Electroweak Symmetry Breaking as a first step • towards a more fundamental theory of particle physics.
The Higgs 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 Higgs field. −→ First step: calculate the SM Higgs boson decay branching ratios. • 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 Higgs boson(s): • . The Higgs Hunter Guide, J. Gunion, H.E. Haber, G. Kane, and S. Dawson . Introduction to the physics of Higgs bosons, S. Dawson, TASI Lectures 1994, hep-ph/9411325 . Introduction to electroweak symmetry breaking, S. Dawson, hep-ph/9901280 . Higgs Boson Theory and Phenomenology, M. Carena and H.E. Haber, hep-ph/0208209 Breaking the Electroweak Symmetry: Why and How?
The gauge symmetry of the Standard Model (SM) forbids • gauge boson mass terms, but:
M ± = 80.426 0.034 GeV and M = 91.1875 0.0021 GeV W ± Z ±
⇓ 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): ' = Higgs Mechanism in the SM or its extensions (MSSM, etc.) ⇒ Little Higgs models −→ . Strongly coupled dynamics at the TeV scale: Technicolor in its multiple realizations. −→ . Extra dimensions beyond the 3+1 space-time dimensions 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. The story begins in 1964 . . . with Englert and Brout; Higgs; Hagen, Guralnik and Kibble 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 phi_2 0 0phi_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 µ ¶
⇓ 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{z φ2 field actually} | generates{z the} correct| transverse{z } structure for the mass term of the (now massive) Aµ field propagator: i kµkν Aµ(k)Aν ( k) = − gµν + h − i k2 m2 − k2 ··· − A µ ¶ 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 | {z } T aφ =0 massive vector boson + (Goldstone boson) 0 6 −→ 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) h | | iJ Z 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 µν µ = F F +(D φ)∗D φ V (φ) L −4 µν µ − upon SSB: 1 φ(x)= ((v + φ1(x)) + iφ2(x)) √2
⇓ 1 1 1 = F µν F + (∂µφ + gAµφ )2 + (∂µφ gAµ(v + φ ))2 V (φ) L −4 µν 2 1 2 2 2 − 1 − Quantizing using the gauge fixing condition:
1 µ G = (∂ A + ξgvφ2) √ξ µ in the generating functional 1 δG Z = C A φ φ exp d4x G2 det D D 1D 2 L− 2 δα Z ·Z µ ¶¸ µ ¶ (α gauge transformation parameter) −→ 1 1 1 G2 = A gµν ∂2 + 1 ∂µ∂ν (gv)2gµν A L− 2 −2 µ − − ξ − ν µ µ ¶ ¶ 1 1 1 ξ (∂ φ )2 m2 φ2 + (∂ φ )2 (gv)2φ2 + 2 µ 1 − 2 φ1 1 2 µ 2 − 2 2 ··· + φ =c ¯ ∂2 ξ(gv)2 1+ 1 c Lghost − − v · µ ¶¸ such that: i kµkν iξ kµkν Aµ(k)Aν ( k) = − gµν + − h − i k2 m2 − k2 k2 ξm2 k2 − A µ ¶ − A µ ¶ i φ (k)φ ( k) = − h 1 1 − i k2 m2 − φ1 i φ (k)φ ( k) = c(k)¯c( k) = − h 2 2 − i h − i k2 ξm2 − A Goldtone boson φ2, longitudinal gauge bosons ⇐⇒ The Higgs sector of the Standard Model : SSB SU(2)L U(1)Y U(1)Q × −→ Introduce one complex scalar doublet of SU(2)L with Y =1/2:
+ φ µ 2 2 φ = =(D φ)†Dµφ µ φ†φ λ(φ†φ) Ã φ0 ! ←→ L − −
a a a a where D φ =(∂ igA τ ig0Y B ), (τ =σ /2, a=1, 2, 3). µ µ − µ − φ µ The SM symmetry is spontaneously broken when φ is chosen to be (e.g.): h i 1/2 1 0 µ2 φ = with v = − (µ2 < 0,λ> 0) h i √2 λ Ã v ! µ ¶ The gauge boson mass terms arise from:
µ 1 a a bµ b µ 0 (D φ)†Dµφ + (0 v) gAµσ + g0Bµ gA σ + g0B + −→ ··· 8 Ã v ! ··· 2 ¡ ¢ ¡ ¢ 1 v 2 1 2 2 2 2 3 2 + g (A ) + g (A ) +( gA + g0B ) + −→ ··· 2 4 µ µ − µ µ ··· £ ¤ And correspond to the weak gauge bosons:
1 1 2 v W ± = (A A ) MW = g µ √2 µ ± µ −→ 2
0 1 3 2 2 v Z = (gA g0Bµ) MZ = g + g0 µ 2 2 µ 2 g + g0 − −→ p p 0 while the linear combination orthogonal to Zµ remains massless and corresponds to the photon field:
1 3 Aµ (g0A + gBµ) MA =0 2 2 µ g + g0 −→
Notice: using thep definition of the weak mixing angle, θw:
g g0 cos θw = , sin θw = 2 2 2 2 g + g0 g + g0 p p 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 Ã v + H(x) ! −→ √2Ã v + H(x) ! after which the Lagrangian becomes
2 2 3 1 4 1 2 2 λ 3 1 4 = µ H λvH H = MH H MH H λH L − − 4 −2 − r 2 − 4 Three degrees of freedom, the χa(x) Goldstone bosons, have been 0 reabsorbed into the longitudinal components of the Wµ± and Zµ 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 µ From (D φ)†D φ Higgs-Gauge boson couplings: µ −→
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 −→ Also: remember Higgs-gauge boson loop-induced couplings:
γ,Z γ,Z
H H
γ γ
g
H
g
They will be discussed in the context of Higgs boson decays. Finally: 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:
ij i c j ij i j ij i j Yukawa = Γ Q¯ φ u Γ Q¯ φd Γ L¯ φl + h.c. L − u L R − d L R − e L R such that, upon spontaneous symmetry breaking:
1 0 v φ(x)= mf =Γf √ √2Ã v + H(x) ! −→ 2 and f m H = i f = iy − v − t f SM Higgs boson decay branching ratios We can now calculate branching ratios and total width of the SM HIggs boson: ¢ ¢
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Observe difference between light and heavy Higgs These curves include: tree level + QCD and EW loop corrections . Tree level decays: H ff¯ and H V V → → At lowest order:
GF MH 2 3 Γ(H ff¯) = Ncf m β → 4√2π f f 3 GF MH 3 2 Γ(H V V ) = δV 1 τV + τ βV → 16√2π − 4 V µ ¶ 2 2 (βi = √1 τi, τi =4m /M , δW,Z =2, 1, (Nc)l,q =1, 3) − i H Ex.1: Higher order corrections to H qq¯ → QCD corrections dominant:
3GF MH 2 3 Γ(H qq¯)QCD = m¯ q(MH )βq [∆QCD +∆t] → 4√2π
2 αs(MH ) αs(MH ) ∆QCD = 1+5.67 + (35.94 1.36NF ) π − π µ ¶ 2 2 2 αs(MH ) 2 MH 1 2 m¯ q(MH ) ∆t = 1.57 ln + ln π − 3 m2 9 M 2 µ ¶ · t H ¸ Consist of both virtual and real corrections:
Q Q Q
H H g H g ¯ ¯ ¯ Q Q Q
Q Q Q g t H g H g H g ¯ g ¯ Q ¯ Q Q
Q Q Q g g H g H H g g ¯ ¯ g Q Q ¯ Q Large Logs absorbed into MS quark mass • 2b0 γ αs(µ) 0 Leading Order :m ¯ Q(µ)=m ¯ Q(mQ) αs(mQ) µ ¶ f (αs(µ)/π) Higher order :m ¯ Q(µ)=m ¯ Q(mQ) f (αs(mQ)/π) where (from renormalization group equation)
12 25 25 f(x) = x [1+1.014x + ...] for mc <µ
H
g
2 3 GF αsMH H Γ(H gg)= Aq (τq) → 36√2π3 ¯ ¯ ¯ q ¯ ¯X ¯ 2 2 ¯ ¯ where τq =4mq/MH and ¯ ¯ 3 AH (τ)= τ [1+(1 τ)f(τ)] q 2 −
2 1 arcsin √τ τ 1 f(τ)= 2 ≥ 1 1+√1 τ − 4 ln 1 √1 τ iπ τ < 1 − − − − Main contribution from toph quark optimali situation to use −→ Low Energy Theorems to add higher order corrections. QCD corrections dominant:
q g g t,b t,b q¯ Hg t,b H g H
g g g
Difficult task since decay is already a loop effect. However, full massive calculation of Γ(H gg(q),qqg¯ ) agrees with → m M result at 10% t À H (NL) (NL) (NL) αs Γ(H gg(q),qqg¯ ) = ΓLO(α (MH )) 1+ E → s π · ¸ M2 ¿4m2 (NL) H q 95 7 E NL −→ 4 − 6
Dominant soft/collinear radiation do not resolve the Higgs boson coupling to gluons QCD corrections are just a (big) rescaling factor −→ NLO QCD corrections almost 60 70% of LO result in the low mass region: −
0.8 δ Γ Γ δ (H→gg) = LO(1+ ) 0.75 µ = MH M = 175 GeV 0.7 t
0.65
0.6
0.55
0.5
0.45
0.4 50 100 200 500 1000
MH [GeV] solid line full massive NLO calculation −→ dashed line heavy top limit (M 2 4m2) −→ H ¿ t
NNLO corrections calculated in the heavy top limit: add 20% perturbative stabilization. −→ Low-energy theorems, in a nutshell.
Observing that: • In the p 0 limit: the interactions of a Higgs boson with the SM H → particles arise by substituting H M M 1+ (i = f, W, Z) i −→ i v µ ¶ In practice: Higgs taken on shell (p2 = M 2 ), and limit p 0 is limit of H H H → small Higgs masses (e.g.: M 2 4m2). H ¿ t Then • 1 ∂ lim (X Y + H)= Mi (X Y ) pH 0 A → v ∂M A → → i i X very convenient!
Equivalent to an Effective Theory described by: • α H = s Gaµν Ga (1 + O(α )) Leff 12π µν v s including higher order QCD corrections. For completeness:
2 2 3 GF α MH 2 H H Γ(H γγ)= Ncf e A (τf )+ A (τW ) → 128√2π3 ¯ f f W ¯ ¯ f ¯ ¯X ¯ ¯ ¯ where (f(τ) as in H gg): ¯ ¯ → ¯ ¯ AH = 2τ [1+(1 τ)f(τ)] f − AH (τ) = [2+3τ +3τ(2 τ)f(τ)] W − −
2 G2 M 2 αM 3 M 2 3 Γ(H Zγ)= F W H 1 Z AH (τ , λ )+ AH (τ , λ ) → 64π4 − M 2 ¯ f f f W W W ¯ µ H ¶ ¯ f ¯ ¯X ¯ ¯ ¯ H H ¯ ¯ where the form factors Af (τ, λ) and AW (τ,¯ λ) can be found in the ¯ literature (see, e.g., M. Spira, hep-ph/9705337).
For both decays, both QCD and EW corrections are very small ( 1 3%). ' − Present theoretical accuracy on SM Higgs branching ratios
1 bb
-1 10 + - τ τ gg cc SM Higgs Branching Ratio SM Higgs Branching
-2 + - 10 W W
γγ -3 10 100 110 120 130 140 150 160 (GeV) MH
Example: MH =120 GeV Decay mode: b¯b W W ∗ τ +τ − cc¯ gg γγ Theory 1.4% 2.3% 2.3% 23% 5.7% 2.3%
Mainly due to: pole masses mc and mb, and αs(µ).