A Mathematica based Version of the CKMfitter Package

4 Chapter 2. Theory
the bosons of the weak interaction, which have weak charge. The three fundamental
interactions are summarized in Table 2.2.
Interaction Couples to Vector Boson Mass (GeV/c 2 ) J P
strong color 8 gluons 0 1 −
electromagnetic electric charge photon γ 0 1 −
weak weak charge W ± , Z 0 ≈ 10 2 1
Table 2.2: The fundamental interactions
The Standard Model is the combination of Quantum Chromodynamics and the unified
theory of electroweak interactions. It is a renormalizable quantum field theory
and carries the group structure of the gauge group SU(3)C ⊗SU(2)L ⊗U(1)Y , where
C means color, L means left and Y is the electroweak hypercharge.
The QCD, represented by the gauge group SU(3)C, is the theory of strong interactions
between colored quarks and gluons. Its coupling constant αS depends on the
energy scale µ, which leads at high energies to a weak coupling, called Asymptotic
freedom, and a strong coupling at low energies leading to so-called Confinement.
Due to the Confinement, no free quarks or gluons have been observed yet. They
only occur in color-neutral bound states, called hadrons, which can be classified into
baryons and mesons 3 .
The unification of electro-magnetic and weak interaction is described in the model of
S. L. Glashow, S. Weinberg and A. Salam by the gauge group SU(2)L⊗U(1)Y ,
where the fermions are represented by left-handed doublets and right-handed singlets
of the weak isospin. Since gauge invariance requires the absence of explicit mass
terms in the Lagrangian, all particles are initially assumed to be massless. As
a possibility of mass generation without spoiling the gauge invariance, P. Higgs
introduced a complex scalar doublet field Φ = (φ1, φ2), which leads to an additional
term in the SM Lagrangian:
LΦ = � ∂µΦ +� (∂ µ Φ) − V (Φ) (2.1)
with a rotationally symmetric potential V (Φ) = −µ 2 Φ + Φ + λ 2 (Φ + Φ) 2 . The Yukawa
interaction of the quarks with the Higgs field is described by:
LY = −Y d
ij ¯ Q I LiΦd I Rj − Y u
ij ¯ Q I LiεΦ ∗ u I Rj + h.c. , (2.2)
(q = u, d) are complex 3 × 3 matrices, i, j are the labels of the fermion
are the left-handed
represent the right-handed down- and up-type
quark singlets in the basis of weak interaction eigenstates, denoted by I.
where Y q
ij
generation and ε is the 2×2 total antisymmetric tensor. The QI Li
quark doublets, where dI Rj and uI Rj
3 Baryons are composed of three quarks with different color (qiqjqk), mesons are composed of a
quark-antiquark pair (qi ¯q ī), where the indices i, j, k represent the color and ī the anti-color respectively.