Higgs Physics (in the SM and in the MSSM) Abdelhak DJOUADI (LPT CNRS & U. Paris-Sud) • The Higgs in the Standard Model • Higgs decays • Higgs production at hadron colliders • Implications of the discovery for the SM • The Higgs beyond the Standard Model • The MSSM Higgs sector • Implications of the discovery for the MSSM • What next? GGI Firenze, 1–2/10/2014 Higgs Physics – Abdelhak Djouadi – p.1/74
1. The Higgs in the Standard Model SM is based on the gauge symmetry G SM ≡ SU ( 3 ) C × SU ( 2 ) L × U ( 1 ) Y • SU ( 2 ) L × U ( 1 ) Y describes the electromagnetic+weak=EW interaction: – between the three families of quarks and leptons: f L / R = 1 2 ( 1 ∓ γ 5 ) f I 3L , 3R � ν e L , R = e − = ± 1 � R , Q = ( u 2 , 0 ⇒ L = d ) L , u R , d R e − f f ⇒ Y L = − 1 , Y R = − 2 , Y Q = 1 3 , Y u R = 4 3 , Y d R = − 2 Y f = 2Q f − 2I 3 3 Same holds for the two other generations: ( µ, ν µ , c , s ) and ( τ, ν τ , t , b ) . There is no ν R field (and neutrinos are thus exactly and stay massless). – mediated by the W i µ (isospin) and B µ (hypercharge) gauge bosons corresping to the 3 generators (Pauli matrices) of SU(2) and are massless T a = 1 2 τ a ; [ T a , T b ] = i ǫ abc T c and [ Y , Y ] = 0 . Lagrangian simple: with fields strengths and covariant derivatives as QED W a µν = ∂ µ W a ν − ∂ ν W a µ + g 2 ǫ abc W b µ W c ν , B µν = ∂ µ B ν − ∂ ν B µ ψ , T a = 1 � ∂ µ − igT a W a µ − ig ′ Y � 2 τ a D µ ψ = 2 B µ 4 B µν B µν + ¯ F Li iD µ γ µ F Li + ¯ f Ri iD µ γ µ f R i L EW = − 1 µν W µν a − 1 4 W a GGI Firenze, 1–2/10/2014 Higgs Physics – Abdelhak Djouadi – p.2/74
1. The Higgs in the Standard Model But if gauge boson and fermion masses are put by hand in L EW V V µ V µ and/or m f ¯ 1 2 M 2 ff terms: breaking of gauge symmetry. This statement can be visualized by taking the example of QED where the photon is massless because of the local U ( 1 ) Q local symmetry: Ψ ( x ) → Ψ ′ ( x )= e ie α ( x ) Ψ ( x ) , A µ ( x ) → A ′ µ ( x )= A µ ( x ) − 1 e ∂ µ α ( x ) • For the photon (or B field) mass for instance we would have: A A µ A µ → 1 1 A ( A µ − 1 e ∂ µ α )( A µ − 1 e ∂ µ α ) � = 1 A A µ A µ 2 M 2 2 M 2 2 M 2 and thus, gauge invariance is violated with a photon mass. • For the fermion masses, we would have e.g. for the electron: � � 1 2 ( 1 − γ 5 ) + 1 m e ¯ ee = m e ¯ e 2 ( 1 + γ 5 ) e = m e ( ¯ e R e L + ¯ e L e R ) manifestly non–invariant under SU(2) isospin symmetry transformations as e L is in an SU(2) doublet while e R is in an SU(2) singlet. We need a less “brutal” way to generate particle masses in the SM: ⇒ The Brout-Englert-Higgs mechanism ⇒ the Higgs particle H. GGI Firenze, 1–2/10/2014 Higgs Physics – Abdelhak Djouadi – p.3/74
1. The Higgs in the Standard Model Brout-Englert-Higgs: spontaneous electroweak symmetry breaking ⇒ � � φ + introduce a new doublet of complex scalar fields: Φ = , Y Φ =+ 1 φ 0 with a Lagrangian density that is invariant under SU ( 2 ) L × U ( 1 ) Y L S = ( D µ Φ ) † ( D µ Φ ) − µ 2 Φ † Φ − λ ( Φ † Φ ) 2 µ 2 > 0 : 4 scalar particles.. µ 2 < 0 : Φ develops a vev: V( � ) V( � ) � 0 | Φ | 0 � = ( 0 2 ) √ v / 1 with ≡ v = ( − µ 2 /λ ) � � 2 > > 0 0 2 2 + v � > 0 � < 0 = 246 GeV – symmetric minimum: unstable – true vacuum: degenerate ⇒ to obtain the physical states, write L S with the true vacuum (diagonalised fields/interactions). GGI Firenze, 1–2/10/2014 Higgs Physics – Abdelhak Djouadi – p.4/74
1. The Higgs in the Standard Model • Write Φ in terms of four fields θ 1 , 2 , 3 ( x ) and H(x) at 1st order: Φ ( x ) = e i θ a ( x ) τ a ( x ) / v 2 ( θ 2 + i θ 1 1 2 ( 0 1 v + H ( x ) ) ≃ v + H − i θ 3 ) √ √ • Make a gauge transformation on Φ to go to the unitary gauge: Φ ( x ) → e − i θ a ( x ) τ a ( x ) Φ ( x ) = 1 2 ( 0 v + H ( x ) ) √ • Then fully develop the term | D µ Φ ) | 2 of the Lagrangian L S : | D µ Φ ) | 2 = � 2 µ − i g 2 � τ a � �� 2 W a � ∂ µ − ig 1 2 B µ Φ 2 � �� � � � 0 − ig2 ∂ µ − i 2 ( g 2 W 3 2 ( W 1 µ − iW 2 µ + g 1 B µ ) µ ) � � = 1 � � v + H − ig2 ∂ µ + i 2 2 ( g 2 W 3 µ − g 1 B µ ) 2 ( W 1 µ + iW 2 µ ) � � = 1 2 ( ∂ µ H ) 2 + 1 µ | 2 + 1 8 g 2 2 ( v + H ) 2 | W 1 µ + iW 2 8 ( v + H ) 2 | g 2 W 3 µ − g 1 B µ | 2 • Define the new fields W ± µ and Z µ [ A µ is the orthogonal of Z µ ]: g 2 W 3 g 2 W 3 W ± = µ − g 1 B µ µ + g 1 B µ 1 √ √ 2 ( W 1 µ ∓ W 2 µ ) , Z µ = , A µ = √ g 2 2 + g 2 g 2 2 + g 2 1 1 with sin 2 θ W ≡ g 2 / � g 2 2 + g 2 1 = e / g 2 GGI Firenze, 1–2/10/2014 Higgs Physics – Abdelhak Djouadi – p.5/74
1. The Higgs in the Standard Model • And pick up the terms which are bilinear in the fields W ± , Z , A : µ W − µ + 1 Z Z µ Z µ + 1 W W + A A µ A µ M 2 2 M 2 2 M 2 ⇒ 3 degrees of freedom for W + L , W − L , Z L and thus M W ± , M Z : � M W = 1 2 vg 2 , M Z = 1 g 2 2 + g 2 2 v 1 , M A = 0 , √ 2G F ) 1 / 2 ∼ 246 GeV . with the value of the vev given by: v = 1 / ( ⇒ the photon stays massless and U ( 1 ) QED is preserved as it should. • For fermion masses, use same doublet field Φ and its conjugate field ˜ Φ = i τ 2 Φ ∗ and introduce L Yuk which is invariant under SU(2)xU(1): u , ¯ u , ¯ d ) L ˜ L Yuk = − f e ( ¯ e , ¯ ν ) L Φe R − f d ( ¯ d ) L Φd R − f u ( ¯ Φu R + · · · = − 1 v + H ) e R · · · = − 1 e L )( 0 2 f e ( ¯ ν e , ¯ 2 ( v + H ) ¯ e L e R · · · √ √ 2 , m d = f d v ⇒ m e = f e v 2 , m u = f u v √ √ √ 2 With same Φ , we have generated gauge boson and fermion masses, while preserving SU(2)xU(1) gauge symmetry (which is now hidden)! What about the residual degree of freedom? GGI Firenze, 1–2/10/2014 Higgs Physics – Abdelhak Djouadi – p.6/74
1. The Higgs in the Standard Model It will correspond to the physical spin–zero scalar Higgs particle, H. 2 ( ∂ µ H ) 2 , comes from | D µ Φ ) | 2 term. The kinetic part of H field, 1 Mass and self-interaction part from V ( Φ ) = µ 2 Φ † Φ + λ ( Φ † Φ ) 2 : V = µ 2 v + H ) + λ 2 ( 0 , v + H )( 0 2 | ( 0 , v + H )( 0 v + H ) | 2 Doing the exercise you find that the Lagrangian containing H is, 2 ( ∂ µ H ) 2 − λ v 2 H 2 − λ v H 3 − λ 2 ( ∂ µ H )( ∂ µ H ) − V = 1 L H = 1 4 H 4 H = 2 λ v 2 = − 2 µ 2 . The Higgs boson mass is given by: M 2 The Higgs triple and quartic self–interaction vertices are: g H 3 = 3i M 2 H / v , g H 4 = 3iM 2 H / v 2 What about the Higgs boson couplings to gauge bosons and fermions? They were almost derived previously, when we calculated the masses: V ( 1 + H / v ) 2 , L m f ∼ − m f ( 1 + H / v ) L M V ∼ M 2 ⇒ g Hff = im f / v , g HVV = − 2iM 2 V / v , g HHVV = − 2iM 2 V / v 2 Since v is known, the only free parameter in the SM is M H or λ . GGI Firenze, 1–2/10/2014 Higgs Physics – Abdelhak Djouadi – p.7/74
1. The Higgs in the Standard Model Constraints on M H from pre–LHC experiments: LEP, Tevatron... Indirect Higgs boson searches: Direct searches at colliders: H looked for in e + e − → ZH H contributes to RC to W/Z masses: e + Z H W/Z W/Z Z ∗ e − H Fit the EW precision measurements: M H > 114 . 4 GeV @95% CL we obtain M H = 92 + 34 − 26 GeV, or 1 CL s 6 LEP Theory uncertainty -1 10 ∆α had = ∆α (5) 5 0.02761 ± 0.00036 -2 0.02747 ± 0.00012 10 incl. low Q 2 data 4 -3 Observed 10 ∆χ 2 3 Expected for background -4 10 2 -5 114.4 115.3 10 1 -6 Excluded Preliminary 10 0 100 102 104 106 108 110 112 114 116 118 120 20 100 400 M H (GeV) m H [ GeV ] M H < ∼ 160 GeV at 95% CL Tevatron M H � = 160 − 175 GeV GGI Firenze, 1–2/10/2014 Higgs Physics – Abdelhak Djouadi – p.8/74
1. The Higgs in the Standard Model Scattering of massive gauge bosons V L V L → V L V L at high-energy W + W + H H W − W − Because w interactions increase with energy ( q µ terms in V propagator), W ⇒ σ ( w + w − → w + w − ) ∝ s : ⇒ unitarity violation possible! s ≫ M 2 Decomposition into partial waves and choose J=0 for s ≫ M 2 W : � � �� M 2 M 2 M 2 s a 0 = − 1 + H + s log 1 + H H H s − M 2 M 2 8 π v 2 H For unitarity to be fullfiled, we need the condition | Re( a 0 ) | < 1 / 2 . s ≫ M 2 M 2 • At high energies, s ≫ M 2 H , M 2 H W , we have: a 0 − → − H 8 π v 2 unitarity ⇒ M H < ∼ 870 GeV ( M H < ∼ 710 GeV) s ≪ M 2 s H • For a very heavy or no Higgs boson, we have: a 0 − → − 32 π v 2 unitarity ⇒ √ s < ∼ 1 . 7 TeV ( √ s < ∼ 1 . 2 TeV) Otherwise (strong?) New Physics should appear to restore unitarity. GGI Firenze, 1–2/10/2014 Higgs Physics – Abdelhak Djouadi – p.9/74
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