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)
where B denotes Baryon number, C the charm, B
∗ the beauty or bottom and T
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 electric 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 (J
p
= 0
− ) and decay weakly predominantly into non-charmed strange
mesons.
Flavor is a generic name to describe different types of quark and lepton.
Generation: The six flavors of quarks and of leptons are grouped into three generations or families. The quarks (d, u), (s, c) and (b, t) are of first, second and third
generations; the corresponding leptons being (e
−
, ν e ), (μ
−
, ν μ ) and (τ
−
, ν τ )
Cabibbo – Kobayashi – Maskowa (CKM) matrix
V i j =
⎛
⎝
V ud V us V ub
V cd V cs V cb
V td V ts V tb
⎞
⎠
The probability for a transition from a quark q to a quark q
is proportional to
V qq
2 , the square of the magnitude of the matrix element. The diagonal elements
of this matrix, V ud , V cs , V tb which correspond to transitions within a family are
short of unity by only a few percent. Hence, transitions u → d, c → s, and t → b
are Cabibbo favoured.
The elements V us , V cd , V cb , and V ts are small but not zero. Hence transitions,
s → u, c → d, b → u and t → s are Cabibbo suppressed.
The elements V ub and V td are nearly zero. Hence transitions b → u and t → 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. Here q
2 is the
square of the four-momentum transfer.
f (q
2 ) ∝
1
q 2 + m 2
(10.9)
541
Gellman’s equation is generalized as:
Q/e = I 3 + 1 / 2 (B + S + C + B
∗
+ T )
(10.8)
where B denotes Baryon number, C the charm, B
∗ the beauty or bottom and T
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 electric 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 (J
p
= 0
− ) and decay weakly predominantly into non-charmed strange
mesons.
Flavor is a generic name to describe different types of quark and lepton.
Generation: The six flavors of quarks and of leptons are grouped into three generations or families. The quarks (d, u), (s, c) and (b, t) are of first, second and third
generations; the corresponding leptons being (e
−
, ν e ), (μ
−
, ν μ ) and (τ
−
, ν τ )
Cabibbo – Kobayashi – Maskowa (CKM) matrix
V i j =
⎛
⎝
V ud V us V ub
V cd V cs V cb
V td V ts V tb
⎞
⎠
The probability for a transition from a quark q to a quark q
is proportional to
V qq
2 , the square of the magnitude of the matrix element. The diagonal elements
of this matrix, V ud , V cs , V tb which correspond to transitions within a family are
short of unity by only a few percent. Hence, transitions u → d, c → s, and t → b
are Cabibbo favoured.
The elements V us , V cd , V cb , and V ts are small but not zero. Hence transitions,
s → u, c → d, b → u and t → s are Cabibbo suppressed.
The elements V ub and V td are nearly zero. Hence transitions b → u and t → 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. Here q
2 is the
square of the four-momentum transfer.
f (q
2 ) ∝
1
q 2 + m 2
(10.9)