Trace anomaly and massless scalar degrees of freedom in gravity

The trace anomaly of quantum fields in electromagnetic or gravitational backgrounds implies the existence of massless scalar poles in physical amplitudes involving the stress-energy tensor. Considering first the axial anomaly and using QED as an example, we compute the full one-loop triangle amplitude of the fermionic stress tensor with two current vertices, $⟨T^{\mu \nu }{J}^{\alpha }{J}^{\beta }⟩$, and exhibit the scalar pole in this amplitude associated with the trace anomaly, in the limit of zero electron mass $m\to 0$. To emphasize the infrared aspect of the anomaly, we use a dispersive approach and show that this amplitude and the existence of the massless scalar pole is determined completely by its ultraviolet finite terms, together with the requirements of Poincaré invariance of the vacuum, Bose symmetry under interchange of $J^{\alpha }$ and $J^{\beta }$, and vector current and stress-tensor conservation. We derive a sum rule for the appropriate positive spectral function corresponding to the discontinuity of the triangle amplitude, showing that it becomes proportional to $\delta (k^2)$ and therefore contains a massless scalar intermediate state in the conformal limit of zero electron mass. The effective action corresponding to the trace of the triangle amplitude can be expressed in local form by the introduction of two scalar auxiliary fields which satisfy massless wave equations. These massless scalar degrees of freedom couple to classical sources, contribute to gravitational scattering processes, and can have long range gravitational effects.

Figure 1

The axial anomaly triangle diagram.

Figure 2

Discontinuity or imaginary part of the triangle diagram obtained by cutting two lines.

Figure 3

Tree diagram of the effective action (53) showing the massless propagator $D_{\chi \eta }$ representing the massless $0^{-}$ state in the triangle amplitude (43) to physical photons when the fermion mass $m=0$.

Figure 4

The two kinds of vertices contributing to the stress-tensor amplitude (68).

Figure 5

The two kinds of amplitudes contributing to the stress-tensor amplitude (68).

Figure 6

The spectral function ${\rho }_T$ of (138) in units of $\frac{e^2}{32{\pi }^2}{m}^{-2}$ as a function of $\frac{s}{4m^2}$.

Figure 7

The two-particle intermediate state of a collinear $e^{+}{e}^{-}$ pair responsible for the $\delta \mathrm{\text{−}}\mathrm{fn}$ in (137).

Figure 8

Tree diagram of the effective action (158), which reproduces the trace of the triangle anomaly. The dashed line denotes the propagator $D_{{\psi }^{\prime }\phi }={\square }^{-1}$ of the scalar intermediate state, while the jagged line denotes the gravitational metric field variation $h_{\mu \nu }=\delta g_{\mu \nu }$.

Figure 9

Tree level gravitational scattering amplitude.

Figure 10

Gravitational scattering of photons from the source $T^{\prime \mu \nu }$ via the triangle amplitude.

Figure 11

Gravitational scattering of photons from the trace of a source $T^{\prime \mu }{}_{\mu }$ via massless scalar exchange in the effective theory of (173).