# Local and Covariant Flow Relations for OPE Coefficients in Lorentzian Spacetimes

September 18, 2022

For Euclidean-signature quantum field theories with renormalizable
self-interactions, Holland and Hollands have shown the operator product
expansion (OPE) coefficients satisfy "flow equations'': For a (renormalized)
self-coupling parameter $\lambda$, the partial derivative of any
OPE coefficient with respect to $\lambda$ is given by an integral
over Euclidean space of a sum of products of other OPE coefficients
evaluated at $\lambda$. These Euclidean flow equations were proven
to hold order-by-order in perturbation theory, but they are well defined
non-perturbatively and thus provide a possible route towards giving
a non-perturbative construction of the interacting field theory. The
purpose of this paper is to generalize the Holland and Hollands results
for flat Euclidean space to curved Lorentzian spacetimes in the context
of the solvable "toy model'' of massive Klein-Gordon scalar field
theory on globally-hyperbolic curved spacetimes, with the squared
mass, $m^{2}$, viewed as the "self-interaction parameter''. There
are a number of difficulties that must be overcome to carry out this
program. Even in Minkowski spacetime, a serious difficulty arises
from the fact that all integrals must be done over a compact region
of spacetime to ensure convergence. However, there does not exist
any Lorentz-invariant function of compact support, so any flow relations
that involve only integration over a compact region cannot be Lorentz
covariant. We show how covariant flow relations can be obtained by
the addition of "counterterms'' that cancel the non-covariant dependence
on the cutoff function in a manner similar to that used in the Epstein-Glaser
renormalization scheme. The necessity of integration over a finite
region also effectively introduces an "infrared cutoff scale''
$L$ into the flow relations, which gives rise to undesirable behavior
of the OPE coefficients under scaling of the metric and $m^{2}$.
(This behavior also occurs in the Euclidean case.) We show how to
modify the flow relations so that the dependence on $L$ is systematically
removed, thereby yielding flow relations compatible with almost homogeneous
scaling of the fields. A potentially even more serious difficulty
arises in curved spacetime simply due to the fact that the flow relations
involve integration over a spacetime region. Such an integration will
cause the OPE coefficients to depend non-locally on the spacetime
metric, in violation of the requirement that the quantum fields should
depend locally and covariantly on the spacetime metric. We show how
this difficulty can be overcome by replacing the metric with a local
polynomial approximation carried to suitable order about the OPE expansion
point. We thereby obtain local and covariant flow relations for the
OPE coefficients of Klein-Gordon theory in an arbitrary curved Lorentzian
spacetime. As a byproduct of our analysis, we prove that the field
redefinition freedom in the Wick fields (i.e. monomials of the scalar
field and its covariant derivatives) can be characterized by the freedom
to add a smooth, covariant, and symmetric function $F_{n}(x_{1},\dots,x_{n};z)$
to the identity OPE coefficients, $C_{\phi\cdots\phi}^{I}(x_{1},\dots,x_{n};z)$,
for the elementary $n$-point products. We thereby obtain an explicit
construction of any renormalization prescription for the nonlinear
Wick fields in terms of the OPE coefficients $C_{\phi\cdots\phi}^{I}$.
The ambiguities inherent in our procedure for modifying the flow relations
are shown to be in precise correspondence with the field redefinition
freedom of the Klein-Gordon OPE coefficients. In an appendix, we develop
an algorithm for constructing local and covariant flow relations in
Lorentzian spacetimes beyond our "toy model'' based upon the associativity
properties of the OPE coefficients. We illustrate our method by applying
it to the flow relations of $\lambda\phi^{4}$-theory.

Keywords:

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