AND H IGGS - BOSON PHYSICS AT THE LHC Carlo Oleari Universit` a di - PowerPoint PPT Presentation

I NTRODUCTION TO E LECTROWEAK THEORY AND H IGGS - BOSON PHYSICS AT THE LHC Carlo Oleari Universit` a di Milano-Bicocca, Milan GGI, Firenze, September 2007 • Theoretical introduction • • • Constraints on the Higgs boson • Higgs boson signals at the LHC •

The Standard Model (SM) • A quick introduction to non-Abelian gauge theories: many formulae but • they will look familiar! – QED – Yang-Mills theories – Electroweak interactions • Spontaneous symmetry breaking and mass generation: the Higgs boson • • Theoretical bounds on the mass of the Higgs boson • • Experimental bounds on the mass of the Higgs boson • Exercise: Please, do the exercises! You will be given all the elements to solve them. Carlo Oleari Introduction to EW theory and Higgs boson physics at the LHC 1

Abelian gauge theory: QED We start with a Lagrangian (density) ψ ( x ) ( i ∂ L 0 = ¯ / − m ) ψ ( x ) invariant under a GLOBAL U(1) symmetry ( θ is constant) e iq θ ψ ( x ) ψ ( x ) → ∂ µ ψ ( x ) e iq θ ∂ µ ψ ( x ) → From Noether’s theorem, there is a conserved current: ∂ µ J µ ( x ) = 0 J µ ( x ) = q ¯ ψ ( x ) γ µ ψ ( x ) = ⇒ To gauge this theory, we promote the GLOBAL U(1) symmetry to local symmetry: e iq θ ( x ) ψ ( x ) ψ ( x ) → ∂ µ ψ ( x ) e iq θ ( x ) ∂ µ ψ ( x ) + iqe iq θ ( x ) ψ ( x ) ∂ µ θ ( x ) → Carlo Oleari Introduction to EW theory and Higgs boson physics at the LHC 2

Covariant derivative Invent a new derivative D µ such that e iq θ ( x ) ψ ( x ) = U ( x ) ψ ( x ) ψ ( x ) → e iq θ ( x ) D µ ψ ( x ) = U ( x ) D µ ψ ( x ) D µ ψ ( x ) → i.e. both ψ ( x ) and D µ ψ ( x ) transform the same way under the U(1) local symmetry D µ ≡ ∂ µ + iqA µ where A µ transforms under the local gauge symmetry as A µ → A µ − ∂ µ θ ( x ) The commutator of the covariant derivatives gives the electric and the magnetic fields, i.e. the field strength tensor µν ( x ) = 1 iq [ D µ , D ν ] = 1 iq [ ∂ µ + iqA µ , ∂ ν + iqA ν ] = ∂ µ A ν ( x ) − ∂ ν A µ ( x ) F F µν is invariant under a gauge transformation. Carlo Oleari Introduction to EW theory and Higgs boson physics at the LHC 3

From global to local symmetry From ψ ( x ) ( i ∂ L 0 = ¯ / − m ) ψ ( x ) invariant under GLOBAL U(1), to ¯ L 1 = ψ ( x ) ( iD / − m ) ψ ( x ) ψ ( x ) ( i ∂ ¯ / − m ) ψ ( x ) − q ¯ ψ ( x ) γ µ ψ ( x ) A µ ( x ) = invariant under LOCAL U(1) and interpret A µ ( x ) as the photon field and J µ = q ¯ ψγ µ ψ as the electromagnetic current. The only missing ingredient is the kinetic term for the photon field L 2 = L 1 − 1 µν ( x ) F µν ( x ) 4 F L 2 cannot contain a term proportional to A µ A µ (a mass term for the photon field) since this term is not gauge invariant under the local U(1) A µ → A µ − ∂ µ θ ( x ) Carlo Oleari Introduction to EW theory and Higgs boson physics at the LHC 4

Non-Abelian (Yang-Mills) gauge theories The starting point is a Lagrangian of free or self-interacting fields, that is symmetric under a GLOBAL symmetry L ψ ( ψ , ∂ µ ψ )   ψ 1 where   .  .  ψ =  = multiplet of a compact Lie group G .    ψ n The Lagrangian is symmetric under the transformation ψ → ψ ′ = U ( θ ) ψ UU † = U † U = 1 U ( θ ) = exp ( igT a θ a ) unitary matrix If U is unitary, the T a are hermitian, and are called group generators (they “generate” infinitesimal transformation around the unity � θ 2 � U ( θ ) = 1 + igT a θ a + O If U ∈ SU( N ) matrix (unitary and det U = 1), then there are N 2 − 1 traceless, hermitian generators T a = λ a / 2. Show this. Exercise:

Gauging the symmetry The generators satisfy the relation � T a , T b � = i f abc T c and the f abc are called the structure functions of the group G . The starting hypothesis is that L is invariant under G ψ ′ = U ( θ ) ψ L ψ ( ψ , ∂ µ ψ ) = L ψ ( ψ ′ , ∂ µ ψ ′ ) Gauging the symmetry means to allow the parameters θ a to be function of the space-time coordinates θ a → θ a ( x ) so that = ⇒ U → U ( x ) � θ 2 � U ( x ) = 1 + igT a θ a ( x ) + O Carlo Oleari Introduction to EW theory and Higgs boson physics at the LHC 6

From ∂ µ → D µ We obtain a LOCAL invariant Lagrangian if we make the substitution L ψ ( ψ , ∂ µ ψ ) → L ψ ( ψ , D µ ψ ) D µ = ∂ µ − igA a µ ( x ) T a ≡ ∂ µ − igA µ ( x ) with the transformation properties � � 1 + ig θ a T a + O ( θ 2 ) ψ ( x ) → U ( x ) ψ ( x ) = ψ ( x ) U ( x ) D µ ψ ( x ) = U ( x ) D µ U − 1 ( x ) U ( x ) ψ ( x ) D µ → i.e. the covariant derivative must transform as µ θ c + O ( θ 2 ) D µ → U ( x ) D µ U − 1 ( x ) A a µ → A a µ + ∂ µ θ a ( x ) + g f abc A b implying We can build a kinetic term for the A a µ fields from µν T a = i µν = F a F a µν = ∂ µ A a ν − ∂ ν A a µ + g f abc A b µ A c F g [ D µ , D ν ] with ν which transforms homogeneously under a local gauge transformation µν F µν → Tr UF µν U − 1 UF µν U − 1 = Tr F µν U − 1 µν F µν F a µν F µν µν → UF = ⇒ ≡ Tr F F a µν F µν where F a is gauge invariant ( F a µν in not singularly gauge-invariant). a

The Lagrangian for gauge and matter field Gauge invariant Yang-Mills (YM) Lagrangian for gauge and matter fields L YM = − 1 µν F µν 4 F a + L ψ ( ψ , D µ ψ ) a where ∂ µ − igA a = D µ µ T a ∂ µ A a ν − ∂ ν A a F a µ + g f abc A b µ A c = µν ν � T a , T b � i f abc T c = Carlo Oleari Introduction to EW theory and Higgs boson physics at the LHC 8

Remarks on Yang-Mills theories µ A µ • Mass terms A a • a for the gauge bosons are NOT gauge invariant! No mass term is allowed in the Lagrangian. Gauge bosons of (unbroken) YM theories are massless. � �� � µν F µν ∂ µ A a ν − ∂ ν A a ∂ µ A ν a − ∂ ν A µ a + g f abc A µ • From the F a µ + g f abc A b µ A c b A ν • = part of the a c ν Lagrangian, we have cubic and quartic gauge boson self interactions • • gauge invariance, Lorentz structure and renormalizability (absence of higher powers of fields and covariant derivatives in L ) determines gauge-boson/matter couplings and gauge-boson self interaction • if G = SU c ( N = 3) and the fermion are in triplets, •     ψ red ψ 1     ψ =  =  ψ blue   ψ 2     ψ green ψ 3 we have the QCD Lagrangian and N 2 − 1 = 8 gauge bosons = gluons. Derive the form of the three- and four-gluon vertex starting from gauge invariance, Exercise: Lorentz structure and renormalizability of the Lagrangian.

Electroweak sector From experimental facts (charged currents couple only with left-handed fermions, the existence of a massless photon and a neutral Z . . . ), the electroweak group is chosen to be SU(2) L × U(1) Y . ψ L ≡ 1 ψ R ≡ 1 2 ( 1 − γ 5 ) ψ 2 ( 1 + γ 5 ) ψ ψ = ψ L + ψ R      ν e  ν eL L L ≡ 1 ν eR ≡ 1 e R ≡ 1  = 2 ( 1 − γ 5 ) 2 ( 1 + γ 5 ) ν e 2 ( 1 + γ 5 ) e  e e L ⇒ three gauge bosons: W 1 , W 2 and W 3 . • SU(2) L : weak isospin group. Three generators = • The generators for doublets are T a = σ a / 2, where σ a are the 3 Pauli matrices (when acting on � σ a , σ b � the gauge singlet e R and ν R , T a ≡ 0), and they satisfy = i ǫ abc σ c . The gauge coupling will be indicated with g . • U(1) Y : weak hypercharge Y . One gauge boson B with gauge coupling g ′ . • One generator (charge) Y ( ψ ) , whose value depends on the corresponding field.

Gauging the symmetry: fermionic Lagrangian Following the gauging recipe (for one generation of leptons. Quarks work the same way) L ψ = i ¯ / L L + i ¯ / ν eR + i ¯ L L D ν eR D e R D / e R where T i = σ i i T i − ig ′ Y ( ψ ) D µ = ∂ µ − igW µ T i = 0 B µ i = 1, 2, 3 or 2 2 L ψ ≡ L kin + L CC + L NC L L ∂ ν eR ∂ e R ∂ i ¯ L kin = / L L + i ¯ / ν eR + i ¯ / e R L L γ µ σ 1 L L γ µ σ 2 g L L γ µ σ + L L + g L L γ µ σ − L L g W 1 2 L L + g W 2 W + W − µ ¯ µ ¯ µ ¯ µ ¯ √ √ L CC = 2 L L = 2 2 g ν L γ µ e L + g e L γ µ ν L W + W − √ √ = µ ¯ µ ¯ 2 2 e L γ µ e L ] + g ′ g � ν eL γ µ ν eL − ¯ ν eL γ µ ν eL + ¯ e L γ µ e L ) 2 W 3 L NC = µ [ ¯ Y ( L ) ( ¯ 2 B µ � e R γ µ e R ν eR γ µ ν eR + Y ( e R ) ¯ + Y ( ν eR ) ¯ with 1 � � σ ± = 1 � σ 1 ± i σ 2 � W ± W 1 µ ∓ iW 2 µ = √ µ 2 2

Electroweak unification e L γ µ e L ] + g ′ � g e L γ µ e L ) ν eL γ µ ν eL − ¯ ν eL γ µ ν eL + ¯ 2 W 3 L NC = µ [ ¯ 2 B µ Y ( L ) ( ¯ � ν eR γ µ ν eR + Y ( e R ) ¯ e R γ µ e R + Y ( ν eR ) ¯ Neither W 3 µ nor B µ can be interpreted as the photon field A µ , since they couple to neutral fields.       1 / 2 Y ( L ) ν eL 0       − 1 / 2 Y ( L )  e L   0          Ψ ≡ T 3 ≡ Y ≡       0 Y ( ν eR ) ν eR             Y ( e R ) e R 0 Ψ γ µ Y Ψ γ µ T 3 Ψ W 3 µ + g ′ ¯ L NC = g ¯ 2 Ψ B µ Carlo Oleari Introduction to EW theory and Higgs boson physics at the LHC 12