12
1. The Particles and Forces of the Standard Model
when the constituents are no longer close, the energy of the system is greater
than the sum of the short distance (free) quark masses. In potential models
(see section 1.3.6), the effect is least pronounced for the ‘heavy’ quarks (m q
greater than about 1 GeV). For example, the ground state of the Υ(b ¯
b) lies at
about 9.46 GeV, which is close to the average value of 2m b as given in Table
1.2. For ψ(c¯ c) the ground state is at about 3 GeV, somewhat greater than
2m c . For the three lightest quarks, and especially for the u and d quarks, the
position is quite different: for example, the proton (uud) with a mass of 938
MeV is far more massive than 2m u + m d . Here the ‘spring’ is responsible for
about 300 MeV per quark.
While this picture is qualitatively useful, it is clearly model dependent,
as would be even a more sophisticated quark model. To do the job properly,
we have to go to the actual QCD Lagrangian, and use it to calculate the
hadron masses with the Lagrangian masses as input. This can be done through
a lattice simulation of the field theory, as will be described in chapter 16
of volume 2. Independently, another handle on the Lagrangian masses is
provided by the fact that the QCD Lagrangian has an extra symmetry (‘chiral
symmetry’) which is exact when the quark masses are zero. This is, in fact,
an excellent approximation for the u and d quarks, and a fair one for the
s quark. The symmetry is, however, dynamically (‘spontaneously’) broken
by QCD, in such a way as to generate (in the case m u = m d = 0) the
nucleon mass entirely dynamically, along with a massless pion. The small
Lagrangian masses can then be treated perturbatively in a procedure called
‘chiral perturbation theory’. These essential features of QCD will be treated
in chapter 18 of volume 2. For the moment, we accept the values in Table 1.2;
Nakamura et al. (2010) contains a review of quark masses.
1.3 Particle interactions in the Standard Model
1.3.1 Classical and quantum fields
In the world of the classical physicist, matter and force were clearly separated.
The nature of matter was intuitive, based on everyday macroscopic experience;
force, however, was more problematical. Contact forces between bodies were
easy to understand, but forces which seemed capable of acting at a distance
caused difficulties.
That gravity should be innate, inherent and essential to matter, so
that one body can act upon another at a distance, through a vacuum,
without the mediation of anything else, by and through which action
and force may be conveyed from one to the other, is to me so great
an absurdity, that I believe no man who has in philosophical matters
1. The Particles and Forces of the Standard Model
when the constituents are no longer close, the energy of the system is greater
than the sum of the short distance (free) quark masses. In potential models
(see section 1.3.6), the effect is least pronounced for the ‘heavy’ quarks (m q
greater than about 1 GeV). For example, the ground state of the Υ(b ¯
b) lies at
about 9.46 GeV, which is close to the average value of 2m b as given in Table
1.2. For ψ(c¯ c) the ground state is at about 3 GeV, somewhat greater than
2m c . For the three lightest quarks, and especially for the u and d quarks, the
position is quite different: for example, the proton (uud) with a mass of 938
MeV is far more massive than 2m u + m d . Here the ‘spring’ is responsible for
about 300 MeV per quark.
While this picture is qualitatively useful, it is clearly model dependent,
as would be even a more sophisticated quark model. To do the job properly,
we have to go to the actual QCD Lagrangian, and use it to calculate the
hadron masses with the Lagrangian masses as input. This can be done through
a lattice simulation of the field theory, as will be described in chapter 16
of volume 2. Independently, another handle on the Lagrangian masses is
provided by the fact that the QCD Lagrangian has an extra symmetry (‘chiral
symmetry’) which is exact when the quark masses are zero. This is, in fact,
an excellent approximation for the u and d quarks, and a fair one for the
s quark. The symmetry is, however, dynamically (‘spontaneously’) broken
by QCD, in such a way as to generate (in the case m u = m d = 0) the
nucleon mass entirely dynamically, along with a massless pion. The small
Lagrangian masses can then be treated perturbatively in a procedure called
‘chiral perturbation theory’. These essential features of QCD will be treated
in chapter 18 of volume 2. For the moment, we accept the values in Table 1.2;
Nakamura et al. (2010) contains a review of quark masses.
1.3 Particle interactions in the Standard Model
1.3.1 Classical and quantum fields
In the world of the classical physicist, matter and force were clearly separated.
The nature of matter was intuitive, based on everyday macroscopic experience;
force, however, was more problematical. Contact forces between bodies were
easy to understand, but forces which seemed capable of acting at a distance
caused difficulties.
That gravity should be innate, inherent and essential to matter, so
that one body can act upon another at a distance, through a vacuum,
without the mediation of anything else, by and through which action
and force may be conveyed from one to the other, is to me so great
an absurdity, that I believe no man who has in philosophical matters