Abstract
With this understanding of the renormalisation group and anomalous Weyl symmetry applied to Green functions of the energy-momentum tensor, we are ready to discuss the derivation of the c-theorem in two dimensions. We present several derivations, each of which brings some fresh insight and suggests possible ways forward in looking for an equivalent theorem in four-dimensional QFTs.
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Notes
- 1.
In n dimensions, the position-space Feynman propagator is
$$\displaystyle\begin{array}{rcl} D_{F}(x; m^{2})& =& \int \frac{d^{n}k} {(2\pi )^{n}}\ e^{-ik.x}\ \frac{i} {k^{2} - m^{2} + i\epsilon } {}\\ & =& \ \theta (x^{2})\frac{(-i)^{n-1}} {(2\pi )^{n/2}} \frac{\pi } {2} \frac{m^{\frac{n} {2} -1}} {(\sqrt{x^{2 } - i\epsilon })^{\frac{n} {2} -1}}H_{\frac{n} {2} -1}^{(2)}(m\sqrt{x^{2 } - i\epsilon }) {}\\ & \ \ +& \theta (-x^{2}) \frac{1} {(2\pi )^{n/2}} \frac{m^{\frac{n} {2} -1}} {(\sqrt{x^{2 } + i\epsilon })^{\frac{n} {2} -1}}K_{\frac{n} {2} -1}(m\sqrt{-x^{2 } + i\epsilon })\ . {}\\ \end{array}$$ - 2.
Note the normalisation of these sampling functions:
$$\displaystyle{ \int _{0}^{\infty }d\lambda ^{2}\,f(\lambda \vert x\vert ) = 2^{8}\,3\,x^{-2}\,\qquad \qquad \int _{ 0}^{\infty }d\lambda ^{2}\,h(\lambda \vert x\vert ) = 2^{7}\,x^{-2}\ . }$$ - 3.
In the QED case, we consider the vacuum polarisation tensor, i.e. the two-point function of the conserved electromagnetic current J μ ,
$$\displaystyle{ \Pi _{\mu \nu } = \left (k^{2}\eta _{\mu \nu } - k_{\mu }k_{\nu }\right )\Pi (\omega ) =\langle J_{\mu }(x)\ J_{\nu }(0)\rangle \big\vert _{\mathrm{ F.T.}}\ . }$$The refractive index n(ω) for photons with frequency ω is identified as
$$\displaystyle{ n(\omega ) - 1 = \Pi (\omega )\, }$$and satisfies the Kramers-Kronig dispersion relation
$$\displaystyle{ n(\infty ) - n(0) = -\frac{2} {\pi } \int _{0}^{\infty }\frac{d\omega } {\omega } \,\mathrm{Im}\,n(\omega ) < 0\, }$$which relates the refractive index in the UV and IR limits. The analogy with (4.45) is even closer if we think of the trace operator T μ μ  ∼ ∂ μ D μ , where D μ is the dilatation current.
References
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Shore, G. (2017). c-Theorem in Two Dimensions. 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_4
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DOI: https://doi.org/10.1007/978-3-319-54000-9_4
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