Interaction Between Massive and Massless Gravitons by Perturbing Topological Field Theory

We test the Wu gauge theory of gravity with massive gravitons in the perturbing topological field theory framework. We show that the computation of the correlation function between massive and massless gravitons is reported up to 4-loop and appears to be unaffected by radiative correction. This result ensures the stability of the linking number between massive and massless gravitons with respect to the local perturbation, a result with potential wider applications in cosmology.


Introduction
Various authors have attempted to derive General Relativity from a gauge-like principle, involving invariance of physics under transformations of the locally (i.e., in the tangent space at each point) acting Lorentz or Poincaré group. [1−4] Gauge theory of gravity is based on the principle of local gauge invariance. Since the model requires strict local gravitational gauge symmetry, [4] gauge theory is a pertubatively renormalizable quantum model. In the original model, all gauge gravitons are massless. [5] The massive gravitons were described effectively by Fierz and Pauli. [6] According to their theory, the existence of massive gravitons would violate the local gauge symmetry of the Lagranian. [7] Building on this earlier work, Wu proposed a mechanism, which introduces massive gravitons without violating the local gauge symmetry of the Lagranian. [4,8−10] The third-order gravitational gauge field C a 3µ is massive if mass is very large. Such a field has no contribution to the long-range gravitational force. Long-range gravitational force results exclusively from the contribution of the fouth-order gravitational gauge field C a 4µ and obeys inverse square law.
If the mass term of gravitational gauge field is extremely small, however, the third-order gravitational gauge field C a 3µ will contribute to the middle range gravitational force with approximate range L ≈ hc/m (where h is the Plank constant and c is the speed of light). For graviton mass 2 × 10 −7 eV the gravitational force range will be about one metre.
Recent results from cosmological observation, especially from Cosmic Microwave Background (CMB) tem-perature anisotropy, suggest that our Universe is essentially flat and that it consists of mainly dark matter and dark energy. [11] A natural origin for dark matter and dark energy is to regard it as consisting of massive gravitons. There are indications that those massive gravitons with mass 2 × 10 −7 eV can produce today's acceleration of the Universe. [12−13] In the traditional gauge treatment of gravity the Lorentz group is localized. The gravitational field is, thus, not represented by gauge potential, but by the metric field g µν .
Here we propose an alternative understanding of gravity resulting from the extension of Wu's gauge theory of gravity with massive gravitons into the framework of perturbing topological field theory. Based on this, we calculate the Feynman diagrams for the interaction between massive and massless gravitons in the topological field theory. [14−22] These calculations provide insights to interactions between gravity and dark energy. They could, therefore, have important repercussions for current cosmological problems.

Interaction Between Massive and Massless Gravitons in Topological Field Theory
Taking Wu's gauge model as our starting point, [4,8−10,23−25] we introduce two gravitational gauge fields (C a µ , C a 2µ ) simultaneously. Since (C a µ , C a 2µ ) are vectors in Lie algebra, [8−9] they can be expanded as These correspond with two gauge covariant derivatives Vol. 57 and two different field strengths, given by The Lagrangian of the system is given by where m is the constant mass parameter. The action of the system is given by Equation (4) gives the mass term in gravitational gauge fields. To obtain the Eigenstates of mass matrix the following rotation is needed where the angle θ is given by After transformation (6), the Lagrangian of the system is given by where F a 30µν , F a 40µν are given by: From the above follows that the gauge field C a 3µ is massive, with mass m, whereas the gravitational gauge field C a 4µ , is massless.
To investigate the interaction between the massive gravitational gauge field C a 3µ and the massless gravitational gauge field C a 4µ we use topological field theory.
Our computation of the correlation function between the massive and massless gravitons is analogous with that given in the literature. [22] γ2 dy ν C a 4ν (10) Fig. 1 Linking between γ1, γ2.
For the case of the two gravitational gauge fields C a 3µ and C a 4µ lying on two smooth, closed, non-intersecting curves γ 1 and γ 2 (Fig. 1), the computation of the correlation function is reported up to 4-loop. This computation shows that the correlation function is unaffected by radiative correction. This result ensures the stability of the linking number with respect to the local perturbation. [22] The local perturbation can be added to the Chern-Simons action given by withF 4µ α = 1 2 ε µνρ F 4 aνρ and τ being an arbitrary parameter with negative mass dimension, reflecting the power-counting nonrenormalizability of the perturbation. The Feynman diagrams for the interaction between massive and massless gravitons in the topological field theory are similar to those described at [22]. To calculate the correlator function (10) we use the action The Lagrange multiple b is introduced to this function to implement the Landau gauge.
To evaluate the Feynman diagrams we use the following equations: From Eqs. (13) we obtain where∂ ab µν = g ab ε µνρ ∂ ρ , the transverse derivative operator.
The Feynman diagrams that contribute to the correlation function (10) are of two-loop order (see Fig. 2). Therefore, they correspond to the following integral: Let us analyze the first term of the above expression. Making use of the propagators (14), we obtain The terms containing the derivatives ∂ µ and ∂ ν do not contribute, as they correspond to total derivatives on closed curves. The term described by expression (17) then becomes This can be also obtained by regularizing the delta functions with coinciding arguments through the point-splitting procedure already used by Polyakov: [26] More precisely, whenever a product of n delta functions with coinciding arguments occurs, it is understood as where the limit ε → 0 is meant to be taken at the end of all calculations. Expression (18) becomes, Whatever the order of integration, we obtain, before taking the limit, an expression containing δ 3 (x − y), which leads to a null result.  Analysis of the second term of (16) gives similar results. The two-loop diagram of Fig. 2, therefore, does not contribute to the correlator (10). Concerning the higher-order contributions in the perturbation theory, the results are of a similar nature. The topologically distinct diagrams contributing to the 3-and 4-loop are given in Figs. 3, 4, and 5.
For instance, a typical contraction from Fig. 3 is proportional to while the diagram of Fig. 4 gives All terms in all possible diagrams may be, then, shown to be proportional to δ 3 (x − y) (or its derivatives). One may easily convince oneself that this mechanism also applies to any order in perturbation theory. As it is always x = y, all these diagrams amount to a null correction to the basic diagram, so that the correlation function (10) for two closed smooth nonintersecting curves γ 1 and γ 2 gives their linking number to all orders:

Conclusion
Equation (18) links massive graviton, C a 3µ representing dark energy, with the massless graviton, C a 4µ representing the gravity. This expression of the interaction between gravity and dark energy may have potentially very important implications to cosmology. We find that the inter-actions between dark energy and gravitons are independent of metric. These interactions are, therefore, within the framework of perturbing topological field theory. The topological properties of these interactions are represented by knots and links. The size, exact shape, location etc. of these knots and links are not of immediate concern for the problem at hand (interaction between dark mater/energy and gravitons).
Equation (18), that links massive and massless gravitons, is independent of the exact location, size and shape of the two knots. Equation (18) depends only on the topological relationship of the knots with each other. This invariant may have a physical interpretation: it may represent the work done to move a massive graviton (dark energy) around one knot in three dimensional space while a massless graviton runs around the other knot.