10.1 Basic Concepts and Formulae 541
Gellman’s equation is generalized as:
Q/e=I 3 +^1 / 2 (B+S+C+B∗+T) (10.8)whereBdenotes Baryon number,Cthe charm,B∗the beauty or bottom andT
the top.
Quarks are not observed as free particles as they are confined in hadrons.
In order to save pauli’s principle, a new quantum number called “colour” is
assigned to quarks. This has nothing to do with ordinary colour. The quarks appear
in three colours, red, blue, and green. The antiquarks have anticolour. The observed
hadrons are colourless. Color plays a role in strong interactions similar to charge in
electromagnetic interaction.
The strong color field between quarks is mediated by massless gluons analogous
to electro-magnetic field mediated by photons. While a photon does not carry elec-
tric charge, gluon itself carries color charge. There are eight types of gluons.
Charmonium(cc) is the state formed from the charmed anti-charmed quark pair.
D mesons(D^0 ,D±) contain a charmed quark or antiquark. They are pseudoscalar
like pions (Jp = 0 −) and decay weakly predominantly into non-charmed strange
mesons.
Flavoris a generic name to describe different types of quark and lepton.
Generation: The six flavors of quarks and of leptons are grouped into three gener-
ations or families. The quarks (d, u), (s, c) and (b, t) are of first, second and third
generations; the corresponding leptons being (e−,νe),(μ−,νμ) and (τ−,ντ)
Cabibbo – Kobayashi – Maskowa (CKM) matrix
Vij=⎛
⎝
VudVusVub
VcdVcsVcb
VtdVtsVtb⎞
⎠
The probability for a transition from a quarkqto a quarkq′is proportional to
∣
∣Vqq′
∣
∣^2 , the square of the magnitude of the matrix element. The diagonal elementsof this matrix,Vud,Vcs,Vtbwhich correspond to transitions within a family are
short of unity by only a few percent. Hence, transitionsu→d,c→s, andt→b
are Cabibbo favoured.
The elementsVus,Vcd,Vcb, andVtsare small but not zero. Hence transitions,
s→u,c→d,b→uandt→sare Cabibbo suppressed.
The elementsVubandVtdare nearly zero. Hence transitionsb→uandt→d
are Cabibbo forbidden.
The boson propagator: The rate of a particular reaction mediated by boson exchange
is proportional to the square of the amplitude f(q^2 ) multiplied by a phase factor
and determines the cross-section or the decay of an unstable particle. Hereq^2 is the
square of the four-momentum transfer.
f(q^2 )∝1
q^2 +m^2