There is a class of observables in QFT (event shapes, parton density
functions, light-cone distribution amplitudes) whose the renormalization-group
(RG) evolution takes the form of an integro-differential equation:
$$
\mu\partial_{\mu}f\left( x,\mu\right) =\int\mathrm{d}x^{\prime}\gamma\left(
x,x^{\prime},\mu\right) f\left( x^{\prime},\mu\right) .
$$
It is well known for such equations that one should distinguish carefully
between well-posed and ill-posed problems. A classical example of an ill-posed problem is the backward heat equation:
\begin{align*}
\partial_{t}u & =\kappa\partial_{x}^{2}u,\qquad x\in\left[ 0,1\right]
,\qquad t\in\left[ 0,T\right] ,\\
u\left( x,T\right) & =f\left( x\right) ,\qquad u\left( 0,t\right)
=u\left( l,t\right) =0,
\end{align*}
while the forward evolution (i.e., the initial-boundary value problem
$u\left( x,0\right) =f\left( x\right) $) is well-posed. The fact that the
backward evolution is ill-posed (the solution either doesn't exist or doesn't
depend continuously on the initial data) models the time irreversibility in
the sense of the laws of thermodynamics.
Since the renormalization transformation corresponds to integrating out
short-wavelength field modes, the RG transformations are lossy and thus form a
semigroup only. My question is — if there is an explicit example (or a
demonstration) of an ill-posed problem for RG evolution? I mean, RG evolution
equation the solutions (of initial-boundary value problem) of which have some
pathological properties like instability under a small perturbation of initial
data, thus making a numerical solution either not sensible or requiring to
incorporate prior information (like Tikhonov regularization).
Update. Actually, I have two reasons to worry about such ill-posed problems.
The first one: the standard procedure of utilizing the parton density
functions at colliders is to parameterize these function for some soft
normalization scale $\mu\sim\Lambda_{QCD}$ and then use DGLAP equations to
evolve the distributions to the hard scale of the process $Q\gg\mu$. The
direction of such evolution is opposite to «normal» RG procedure (from the small resolution scale $Q^{-1}$ to the large one $\mu^{-1}$). Thus I suspect that such procedure is (strictly speaking) ill-posed.
The second: the observables/distributions mentioned above are matrix elements of some nonlocal operators. Using the operator product expansion (OPE), one
can reduce the corresponding integro-differential equation to a set of
ordinary differential equation for the renormalization constants of local
operators. My intuition says that in this case the RG evolution for the
distribution will be well-posed at least in one RG direction (thus I think the
DGLAP equations are well-posed for the evolution direction $Q\rightarrow\mu$).
Therefore, a complete ill-posed RG evolution appears when the OPE fails.
This post imported from StackExchange Physics at 2016-02-11 14:35 (UTC), posted by SE-user Grisha Kirilin
Q&A (4871)
