## The Search For The Higgs Boson

We summarise first the relevant formulae for the electroweak interactions of the doublet of the Higgs field (see section 6.2):

φ =

( φ+

φ0

) Y=+1

. (14.1)

We write the Lagrangian for the gauge fields and the scalar doublet:

Ltot = LW + LφW (14.2) where

LW = 1 4 [WµνW

µν +BµνB µν ] (14.3)

while

LφW = (Dµφ)† (Dµφ)− V (φ); V (φ) = µ2φ†φ+ λ(φ†φ)2;

µ2 < 0; (14.4)

with the definition of the covariant derivative:

Dµφ = [∂µ + igWµ · τ 2 + ig′(+

2 )Bµ]φ. (14.5)

We can rewrite the potential of the scalar field as:

V = λ(φ†φ+ µ2

2λ )2 + constant. (14.6)

If we omit the non-essential constant, the minimum of the potential is zero and occurs for:

φ†φ = η2 = −µ 2

2λ > 0 (14.7)

(µ2 < 0). In the unitary gauge we set:

φ(x) =

( 0

) (14.8)

and expand the Lagrangian (14.2) in powers of the fields. The quadratic terms give the mass of the Higgs boson and the masses

of the vector bosons, which are diagonalised as explained in chapter 6. The higher order terms describe interactions of the vector bosons with each other and of the field σ with the vector bosons.