MODERN COSMOLOGY

The SM of particle physics 189

Ta b l e 5. 1 .The fermionic spectrum of the SM.

Generations

Fermions I II III SU( 2 )L⊗U( 1 )Y

EbL≡

(

νb

e−b

)

L

(

νe

e−

)

L

(

νμ

μ−

)

L

(

ντ

τ−

)

L

( 2 ,− 1 )

ebR e−R μ−R τ−R ( 1 ,− 2 )

QbL≡

(

ub

db

)

L

(

u

d

)

L

(

c

s

)

L

(

t

b

)

L

( 2 , 1 / 3 )

ubR uR cR tR ( 1 , 4 / 3 )

dbR dR sR bR ( 1 ,− 2 / 3 )

leptonic number and the baryonic one: the fermions belonging to the fieldsEbL
andebRare called leptons and trasform under the leptonic symmetryU( 1 )Lwhile
the ones belonging toQbL,ubRanddbRbaryons and trasform underU( 1 )B.
The Lagrangian for the gauge fields reads:

LK gauge=−^14 (∂μWνa−∂νWμa+abcWμbWνc)

×(∂μWνa−∂νWμa+ab

′c′

Wb

′

νW

c′

μ)

−^14 (∂μBν−∂νBμ)(∂μBν−∂νBμ). (5.2)

5.2.1 The Higgs mechanism and vector boson masses

The gauge symmetry protects the gauge bosons from having mass. Unfortunately
the weak interactions require massive gauge bosons in order to explain the
experimental behaviour. However, adding a direct mass term for gauge bosons
breaks explicitly the gauge symmetry and spoils renormalizability. To preserve
such nice feature of gauge theories, it is necessary to break spontaneously the
symmetry. This is achieved through the Higgs mechanism. We introduce in the
spectrum a scalar fieldH, which transforms as a doublet underSU( 2 )L, carries
hypercharge while it is colourless. The Higgs doublet has the following potential
VHiggs, kinetic termsLKHand Yukawa couplings with the fermionsLHf:

VHiggs=−μ^2 H†H+λ(H†H)^2

LKH=−

(

∂μH+igWμaTaH+i

g′

2

BμH

)†(

∂μH+igWμaTaH+i

g′

2

BμH

)