I am reading Peskin & Schroeder about the renormalisation group flow of the $\phi^4$ theory:

$${\cal L} = \frac{1}{2}(\partial_\mu\phi)^2 +\frac{1}{2}m^2\phi^2 + \frac{\lambda}{4!}\phi^4 $$

P & S writes on the topic (p. 403-404):

"In general, the criterion that the scalar field mass is small compared to the cutoff is equivalent to the statement that $m'^2\sim \Lambda^2$ ($m'^2 =m^2b^{-2n}$ with $b$ as scale transformation parameter and $n$ the number of iterations) only after a large number of iterations of the renormalisation group transformation. This criterion is met whenever the initial conditions for the renormalisation group flow are adjusted so that the trajectory passes very close to a fixed point. In principle, the flow could begin far away, along the direction of an irrelevant operator."

I don't understand the adjustment chosen, i.e. that the trajectory should pass close to the fixed point. The fixed point is characterized by a Lagrangian

$${\cal L}_0 =\frac{1}{2}(\partial_\mu\phi)^2.\tag{12.25}$$

Getting to the fixed means that all parameters and in particular the mass parameter $m'$ would attain small values. However, the scalar field mass operator $\phi^2$ is relevant, during the iteration it would attain values which come closer to the cutoff $\Lambda$ which is large. So this particular flow would rather reach a point like:

$${\cal L} =\frac{1}{2}(\partial_\mu\phi)^2+ \Lambda^2 \phi^2 $$

and not ${\cal L}_0$.

So if somebody could explain it to me I would be very grateful.


1 Answer 1


Peskin is making a much simpler point than you think. Imagine, for example, $$\mathcal{L} = \frac12 (\partial_\mu \phi)^2 + 10^{-10} \Lambda^2 \phi^2 + 10^{-2} \frac{\phi^6}{\Lambda^2}.$$ The RG flow from this Lagrangian gets closer to the fixed point as the irrelevant term decays, then gets further away from the fixed point as the relevant term grows. So the trajectory passes close to the Gaussian fixed point but then goes away.


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