360
11. Loops and Renormalization II: QED
To arrive at a renormalizable theory of the weak interactions it seems to be
necessary to describe them in terms of a gauge theory (recall the ‘universality’
hints mentioned in section 11.6). Yet the mediating gauge field quanta have
mass, which appears to contradict gauge invariance. The remarkable story of
how gauge field quanta can acquire mass while preserving gauge invariance is
reserved for volume 2.
A number of other non-renormalizable interactions are worth mentioning.
Perhaps the most famous of all is gravity, characterized by Newton’s constant
G N , which has the value (1.2 × 10
19 GeV)
−2 . The detection of gravity at energies so far below 10
19 GeV is due, of course, to the fact that the gravitational
fields of all the particles in a macroscopic piece of matter add up coherently.
At the level of the individual particles, its effect is still entirely negligible.
Another example may be provided by baryon and/or lepton violating interactions, mediated by highly suppressed non-renormalizable terms.
2 Such things
are frequently found when the low-energy limit is taken of theories defined
(for example) at energies of order 10
16 GeV or higher.
The stage is now set for the discussion, in volume 2, of the renormalizable
non-Abelian gauge field theories which describe the weak and strong sectors
of the Standard Model.
Problems
11.1 Establish the values of the counter terms given in (11.12).
11.2 Convince yourself of the rule ‘each closed fermion loop carries an additional factor −1’.
11.3 Explain why the trace is taken in (11.14).
11.4 Verify (11.15).
ρ
ρ
11.5 Verify the quoted relation P τ
ρ P
τ = P
ρ where P
ρ = g − q q ν /q
2 (cf
ν
ν
ν
ν
(11.26)).
2
≪
2
11.6 Verify (11.39 ) for q
m .
2
11.7 Verify (11.55 ) for −q
2
≫ m .
11.8 Check the estimate (11.60).
F ˆ μν )
2
11.9 Find the dimensionality of ‘E’ in an interaction of the form E(F ˆ μν
.
Express this interaction in terms of the E ˆ and B ˆ fields. Is such a term finite
or infinite in QED? How might it be measured?
2 The most general renormalizable Lagrangian with the field content, and the gauge
symmetries, of the Standard Model automatically conserves baryon and lepton number
(Weinberg 1996, pp 316-7).
11. Loops and Renormalization II: QED
To arrive at a renormalizable theory of the weak interactions it seems to be
necessary to describe them in terms of a gauge theory (recall the ‘universality’
hints mentioned in section 11.6). Yet the mediating gauge field quanta have
mass, which appears to contradict gauge invariance. The remarkable story of
how gauge field quanta can acquire mass while preserving gauge invariance is
reserved for volume 2.
A number of other non-renormalizable interactions are worth mentioning.
Perhaps the most famous of all is gravity, characterized by Newton’s constant
G N , which has the value (1.2 × 10
19 GeV)
−2 . The detection of gravity at energies so far below 10
19 GeV is due, of course, to the fact that the gravitational
fields of all the particles in a macroscopic piece of matter add up coherently.
At the level of the individual particles, its effect is still entirely negligible.
Another example may be provided by baryon and/or lepton violating interactions, mediated by highly suppressed non-renormalizable terms.
2 Such things
are frequently found when the low-energy limit is taken of theories defined
(for example) at energies of order 10
16 GeV or higher.
The stage is now set for the discussion, in volume 2, of the renormalizable
non-Abelian gauge field theories which describe the weak and strong sectors
of the Standard Model.
Problems
11.1 Establish the values of the counter terms given in (11.12).
11.2 Convince yourself of the rule ‘each closed fermion loop carries an additional factor −1’.
11.3 Explain why the trace is taken in (11.14).
11.4 Verify (11.15).
ρ
ρ
11.5 Verify the quoted relation P τ
ρ P
τ = P
ρ where P
ρ = g − q q ν /q
2 (cf
ν
ν
ν
ν
(11.26)).
2
≪
2
11.6 Verify (11.39 ) for q
m .
2
11.7 Verify (11.55 ) for −q
2
≫ m .
11.8 Check the estimate (11.60).
F ˆ μν )
2
11.9 Find the dimensionality of ‘E’ in an interaction of the form E(F ˆ μν
.
Express this interaction in terms of the E ˆ and B ˆ fields. Is such a term finite
or infinite in QED? How might it be measured?
2 The most general renormalizable Lagrangian with the field content, and the gauge
symmetries, of the Standard Model automatically conserves baryon and lepton number
(Weinberg 1996, pp 316-7).