14
G. Altarelli and S. Forte
Leptons are colourless and thus do not interact strongly (they are not hadrons) but
have electroweak charges, in particular electric charges −1 for charged leptons (e − ,
μ − and τ − ) while it is 0 for neutrinos (ν e , ν μ and ν τ ). Quarks and leptons are
grouped in 3 “families” or “generations” with equal quantum numbers but different
masses. At present we do not have an explanation for this triple repetition of fermion
families:
u u u ν e
d d d e
,
c c c ν μ
s s s μ
,
t t t ν τ
b b b τ
.
(2.1)
The QCD sector of the SM has a simple structure but a very rich dynamical
content, including the observed complex spectroscopy with a large number of
hadrons. The most prominent properties of QCD are asymptotic freedom and
confinement. In field theory the effective coupling of a given interaction vertex is
modified by the interaction. As a result, the measured intensity of the force depends
on the transferred (four)momentum squared, Q 2 , among the participants. In QCD
the relevant coupling parameter that appears in physical processes is α s = e 2
s /4π
where e s is the coupling constant of the basic interaction vertices of quark and
gluons: qqg or ggg (see Eq. (2.30)). Asymptotic freedom means that the effective
coupling becomes a function of Q 2 : α s (Q 2 ) decreases for increasing Q 2 and
vanishes asymptotically. Thus, the QCD interaction becomes very weak in processes
with large Q 2 , called hard processes or deep inelastic processes (i.e. with a final state
distribution of momenta and a particle content very different than those in the initial
state). One can prove that in 4 space-time dimensions all pure-gauge theories based
on a non commuting group of symmetry are asymptotically free and conversely.
The effective coupling decreases very slowly at large momenta with the inverse
logarithm of Q 2 : α s (Q 2 ) = 1/b log Q 2 // 2 where b is a known constant and is
an energy of order a few hundred MeV. Since in quantum mechanics large momenta
imply short wavelengths, the result is that at short distances the potential between
two colour charges is similar to the Coulomb potential, i.e. proportional to α s (r)/r,
with an effective colour charge which is small at short distances. On the contrary the
interaction strength becomes large at large distances or small transferred momenta,
of order Q . In fact all observed hadrons are tightly bound composite states
of quarks (baryons are made of qqq and mesons of q ¯
q), with compensating colour
charges so that they are overall neutral in colour. In fact, the property of confinement
is the impossibility of separating colour charges, like individual quarks and gluons
or any other coloured state. This is because in QCD the interaction potential between
colour charges increases at long distances linearly in r. When we try to separate the
quark and the antiquark that form a colour neutral meson the interaction energy
grows until pairs of quarks and antiquarks are created from the vacuum and new
neutral mesons are coalesced and observed in the final state instead of free quarks.
For example, consider the process e + e − → q ¯
q at large center of mass energies.
The final state quark and antiquark have large energies, so they separate in opposite
directions very fast. But the colour confinement forces create new pairs in between
them. What is observed is two back-to-back jets of colourless hadrons with a number
G. Altarelli and S. Forte
Leptons are colourless and thus do not interact strongly (they are not hadrons) but
have electroweak charges, in particular electric charges −1 for charged leptons (e − ,
μ − and τ − ) while it is 0 for neutrinos (ν e , ν μ and ν τ ). Quarks and leptons are
grouped in 3 “families” or “generations” with equal quantum numbers but different
masses. At present we do not have an explanation for this triple repetition of fermion
families:
u u u ν e
d d d e
,
c c c ν μ
s s s μ
,
t t t ν τ
b b b τ
.
(2.1)
The QCD sector of the SM has a simple structure but a very rich dynamical
content, including the observed complex spectroscopy with a large number of
hadrons. The most prominent properties of QCD are asymptotic freedom and
confinement. In field theory the effective coupling of a given interaction vertex is
modified by the interaction. As a result, the measured intensity of the force depends
on the transferred (four)momentum squared, Q 2 , among the participants. In QCD
the relevant coupling parameter that appears in physical processes is α s = e 2
s /4π
where e s is the coupling constant of the basic interaction vertices of quark and
gluons: qqg or ggg (see Eq. (2.30)). Asymptotic freedom means that the effective
coupling becomes a function of Q 2 : α s (Q 2 ) decreases for increasing Q 2 and
vanishes asymptotically. Thus, the QCD interaction becomes very weak in processes
with large Q 2 , called hard processes or deep inelastic processes (i.e. with a final state
distribution of momenta and a particle content very different than those in the initial
state). One can prove that in 4 space-time dimensions all pure-gauge theories based
on a non commuting group of symmetry are asymptotically free and conversely.
The effective coupling decreases very slowly at large momenta with the inverse
logarithm of Q 2 : α s (Q 2 ) = 1/b log Q 2 // 2 where b is a known constant and is
an energy of order a few hundred MeV. Since in quantum mechanics large momenta
imply short wavelengths, the result is that at short distances the potential between
two colour charges is similar to the Coulomb potential, i.e. proportional to α s (r)/r,
with an effective colour charge which is small at short distances. On the contrary the
interaction strength becomes large at large distances or small transferred momenta,
of order Q . In fact all observed hadrons are tightly bound composite states
of quarks (baryons are made of qqq and mesons of q ¯
q), with compensating colour
charges so that they are overall neutral in colour. In fact, the property of confinement
is the impossibility of separating colour charges, like individual quarks and gluons
or any other coloured state. This is because in QCD the interaction potential between
colour charges increases at long distances linearly in r. When we try to separate the
quark and the antiquark that form a colour neutral meson the interaction energy
grows until pairs of quarks and antiquarks are created from the vacuum and new
neutral mesons are coalesced and observed in the final state instead of free quarks.
For example, consider the process e + e − → q ¯
q at large center of mass energies.
The final state quark and antiquark have large energies, so they separate in opposite
directions very fast. But the colour confinement forces create new pairs in between
them. What is observed is two back-to-back jets of colourless hadrons with a number