Abstract
In order to study the conformal anomaly and related Green functions of the energy-momentum tensor, we need to renormalise the quantum field theory in curved spacetime including sources for all the operators of interest. In this section, which follows [7], we restrict to two dimensions and consider an interaction Lagrangian of the form ℒ = g I O I , where the O I are a complete set of dimension 2 operators. The corresponding sources are simply the position-dependent couplings g I(x). In addition to the conventional renormalisation of the bare parameters, there are then further counterterms depending on the spacetime curvature and on derivatives of the couplings. The latter provide the extra counterterms required to define renormalised Green functions of the composite operators O I , removing the contact-term divergences which arise even in flat spacetime.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The power \(\mu ^{k_{I}\epsilon }\) in g B I fixes the dimension of the bare operator O BI away from n = 2. This allows for composite operators with different combinations of the elementary fields, whose dimension is set by the dimensionality/Weyl invariance of the free action. The renormalised couplings are dimensionless, while with the definition in (2.17) the renormalised operators O I all have dimension n before the physical limit n → 2 is taken. Note that no sum over the index on k I is implied here or in subsequent equations.
- 2.
Notice that there is an arbitrariness in the choice of RG scheme defining the various functions involved here. This can be parametrised by the following changes:
$$\displaystyle{ \delta \beta _{c} = \mathcal{L}_{\beta }b \equiv \beta ^{I}\partial _{ I}b,\qquad \qquad \delta \chi _{IJ} = \mathcal{L}_{\beta }a_{IJ}, }$$which implies
$$\displaystyle{ \delta \tilde{\beta }_{c} = a_{IJ}\beta ^{I}\beta ^{J}\,\qquad \qquad \delta w_{ I} = -\partial _{I}b + a_{IJ}\beta ^{J}. }$$This leaves the form of the consistency relation (2.30) invariant.
- 3.
We further define the two-point functions of the trace T μ μ (x) so that
$$\displaystyle\begin{array}{rcl} i\langle T^{\mu }_{\mu }(x)\ T^{\rho }_{\rho }(y)\rangle & \equiv & g^{\mu \nu }g^{\rho \sigma }i\langle T_{\mu \nu }(x)\ T_{\rho \sigma }(y)\rangle {}\\ & =& \frac{2} {\sqrt{-g}}g^{\mu \nu } \frac{\delta } {\delta g^{\mu \nu }(x)}\left ( \frac{2} {\sqrt{-g}}g^{\rho \sigma } \frac{\delta W} {\delta g^{\rho \sigma }(y)}\right ) - (n + 2)\langle T^{\mu }_{\mu }(x)\rangle \delta (x,y), {}\\ \end{array}$$where the ambiguity in the definition of the renormalised Green function is reflected in the presence of the VEV of T μ μ (x), itself defined by
$$\displaystyle{ \langle T^{\mu }_{\mu }(x)\rangle \equiv g^{\mu \nu }\langle T_{\mu \nu }(x)\rangle. }$$With Minkowski signature, these two-point functions are the usual time-ordered products. Note that throughout this paper δ(x, y) is the delta function density, where \(\delta (x - y) = \sqrt{-g}\delta (x,y)\), and here we have left the dimension n general for future convenience.
Similarly, letting \(\Theta =\beta ^{I}O_{I}\) as the operator part of the trace anomaly, we define
$$\displaystyle\begin{array}{rcl} i\langle \Theta (x)\ \Theta (y)\rangle & \equiv &\beta ^{I}\beta ^{J}i\langle O_{ I}(x)\ O_{J}(y)\rangle {}\\ & =& \frac{1} {\sqrt{-g}}\beta ^{I} \frac{\delta } {\delta g^{I}(x)}\left ( \frac{1} {\sqrt{-g}}\beta ^{J} \frac{\delta W} {\delta g^{J}(y)}\right ) -\beta ^{I}\partial _{ I}\beta ^{J}\langle O_{ J}(x)\rangle \delta (x,y). {}\\ \end{array}$$
References
H. Osborn, Derivation of a four-Dimensional c theorem. Phys. Lett. B 222, 97 (1989)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Shore, G. (2017). Renormalisation and the Conformal Anomaly. In: The c and a-Theorems and the Local Renormalisation Group. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-54000-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-54000-9_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-53999-7
Online ISBN: 978-3-319-54000-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)