Originally posted by selfAdjoint
Jeff, would it be at all possible to motivate the original Gell-Mann & Low paper that described running in a sort of pre-wilson way? Rather than trying to explain the whole RG approach (which isn't very intuitive even in the advanced textbooks).
The method of what eventually (and misleadingly) came to be known as the renormalization group was originated in work by gell-mann and low to address a failure of perturbation theory in QED at high energies associated with the way naive renormalization prescriptions gave rise to the appearance in amplitudes of factors (familiar to anyone who’s taken an introductory course in QFT) of αln(q²/m²) in which q, m
e and α are respectively internal momentum, electron mass and fine structure constant. The problem is that at high energies such factors will obviously diverge even for small α rendering perturbation theory useless.
The key idea was to introduce coupling constants g
μ defined at a sliding scale μ so that by choosing μ to be of the same order as the energy E typical of the process in question, the factors ln(E/μ) were rendered harmless. Perturbative methods then remain valid as long as g
μ remains small.
In particular, given g
μ, amplitudes could be calculated perturbatively at energy μ+dμ, and then used to calculate g
μ+dμ. By integrating the resulting differential equation the coupling constants at the scale of interest could be related to the coupling constants as conventionally defined.
I’ll sketch the basic idea in terms of an amplitude A(E, g, m) depending on an overall energy scale E, various dimensionless coupling constants and masses represented collectively by g and m with it’s dependence on other dimensionless quantities understood. If A(E,g,m) has dimensionality [mass]
D, then A(E, g, m) = E
DA(1, g, m/E) so that, naively, in the limit E → ∞ we have
A(E, g, m) → E
DA(1, g, m/E).
However A(E, g, m) is usually formally divergent, so to calculate, a cutoff Λ is introduced yielding the regulated quantity A(Λ; E, g, m). Then the conventional renormalization prescription to bury Λ in higher order corrections – and this is the point - yields a renormalized quantity A
R(E, g, m) that doesn’t have this simple power law behaviour but instead the factor E
D is accompanied by powers of ln(E/m).
As discussed above, we introduce a renormalized coupling g(μ) that depends on a sliding energy scale μ which (at least for μ >> m) has no dependence on the scale m of the masses of the theory. Then A(E, g ,m) may be expressed as a function of g
μ & μ, instead of g
E. Dimensional analysis shows such functions may be written as
A(E, g
μ, m/E, μ/E) = E
DA(1, g
μ, m/E, μ/E).
But μ is completely arbitrary so we can take μ = E giving
A(E, g
μ, m, μ) = E
DA(1, g
E, m/E, 1).
Now, since g
E doesn’t depend on m for m << E, there are no large logarithms, so perturbation theory may be used to calculate A(E, g
μ, m, μ) in terms of g
E as long as g
E itself remains sufficiently small. In particular, to any finite order of perturbation theory, as E → ∞, A(E, g
μ, m, μ) has the asymptotic behaviour
A(E, g
μ, m, μ) → E
DA(1, g
E, 0, 1).
Reminding ourselves that the point of the RG as it was originally introduced was to allow calculation at high energies E perturbatively, let’s see how the preceding considerations allow us to calculate g
E. Consider a regulated but unrenormalized amplitude A(Λ; s, g, m) with cut-off Λ and in which s is a collective coordinate for the squares of all external momenta. Working to some given order in g, the conventional renormalized coupling g
R is defined by
g
R ≡ A(Λ; 0, g, m),
that is, g
R ≡ g
μ=0. Then the renormalized amplitude is defined by
A
R ≡ A(s, g
E, m).
Now we define
g
μ ≡ A(Λ; μ, g, m)
which in terms of the renormalized coupling is
g
μ=A(μ, g
E, m).
Then for a real scalar field φ with lagrangian density
L=-\frac{1}{2}\partial_{\nu}\phi\partial_{\nu}\phi-\frac{1}{2}m^2\phi^2<br />
\frac{1}{24}g\phi^4,
this is to second order in the renormalized coupling
(1)[/color] g_\mu=g_R+\frac{3g_R{}^2}{32\pi^2}\int_{0}^{1}dx\ln{[1+\frac{\mu^2x(1-x)}{m^2}]}+O(g_R^2).
The point is that this formula is reliable only if the correction term is smaller than g
R; that is, only if |g
Rln(μ/m)| << 1: If this were the case for μ ≅ E, we wouldn’t need RG methods since ordinary perturbation theory would remain valid.
Thus, rather than working directly with large μ, we must instead proceed in stages: g
μ may be calculated in terms of g
R as long as μ/m isn’t much larger than unity; then g
μ’ may be calculated in terms of g
μ as long as μ’/μ is not much larger than unity, and so on up to g
E.
Instead of discrete stages, this may be done continuously. Dimensional analysis gives the relation between g
μ’ and g
μ as
g
μ’ = G(g
μ, μ’/μ, m/μ).
Differentiating with respect to μ' and then setting μ' = μ yields the differential equation
μdg
μ/dμ=β(g
μ, m/μ)
where
(2)[/color] β(g
μ, m/μ) ≡ [∂G(g
μ, z, m/μ)/∂z]
z=1.
For μ << m we have
μdg
μ/dμ = β(g
μ, 0) = β(g
μ)
which is the gell-mann-low form of the callan-symanzik equation. We are to calculate g
E by integrating this equation with initial value g
M at some scale μ = M chosen large enough that for μ ≥ M, the masses m may be neglected compared with μ, but not so large that ln(M/m) is too big to perturbatively calculate g
M in terms of the conventional renormalized coupling constant g
R. The solution may be formally written
\ln{(E/M)}=\int_{g_M}^{g_E}dg/\beta(g)
as long as β(g) doesn’t vanish between g
M and g
E.
The preceding results don’t rely on perturbation theory, but the functions G and β must usually be calculated perturbatively. As an example, consider (1), but where the bare coupling g is expressed in terms of g
μ instead of g
R. This gives
g_{{\mu}'}=g_\mu-\frac{3g_\mu^2}{32\pi^2}\int_{0}^{1}dx\ln{[\frac{m^2+\mu^2x(1-x)}{ m^2+\mu’^2x(1-x)}]}+O(g_R^3).
Then (2) gives
\beta(g_\mu,m/\mu)= \frac{3g_\mu^2}{16\pi^2}\int_{0}^{1}dx\ln{[\frac{\mu^2x(1-x)}{ m^2+\mu^2x(1-x)}]}+ O(g_R^3 ).
which for μ >> m is
β(g
μ) = 3g
μ²/16pi² + O(g
R³).[/color]
A couple of closing remarks:
Firstly, I pointed out in a previous post in this thread that in the case of QED - studied by gell-man & low - the growth of the coupling with increasing energy can be understood (at least at energies that aren't too high) in terms of shielding of a bodies charge at larger distances by the vacuum polarization it gives rise to.
Secondly, to avoid large logarithms, operator normalizations must also scale (though this scaling behaviour depends generally on that of the couplings as well). But the case that gell-man & low studied was QED which is special in that the scaling of the coupling completely determines that of the operator normalization. In any event, these normalizations satisfy RG equations somewhat analogous to the one's satisfied by couplings.
