Divergences in quantum gravity

2.2 Divergences in quantum gravity

In general, the loop momentum integrals in a quantum field theory will diverge in the ultraviolet where the momenta in the loops become arbitrarily large. Unless these divergences are of the right form they indicate that a theory cannot be interpreted as fundamental, but is instead valid only at low energies. Gauge theories such as quantum chromodynamics are renormalizable: Divergences from high energy scales can be absorbed into redefinitions of the original parameters appearing in the theory. In quantum gravity, on the other hand, it is not possible to re-absorb divergences in the original Lagrangian for a very simple reason: The gravity coupling $√ ------- κ = 32πGN$ , where $GN$ is Newton’s constant, carries dimensions of length (in units where $¯h = c = 1$ ). By dimensional analysis, any divergence must be proportional to terms with extra derivatives compared to the original Lagrangian and are thus of a different form. This may be contrasted to the gauge theory situation where the coupling constant is dimensionless, allowing for the theory to be renormalizable.

The problem of non-renormalizability of quantum gravity does not mean that quantum mechanics is incompatible with gravity, only that quantum gravity should be treated as an effective field theory [140 , 64 , 51 , 85 , 101 ] for energies well below the Planck scale of 10¹⁹ GeV (which is, of course, many orders of magnitude beyond the reach of any conceivable experiment). In an effective field theory, as one computes higher loop orders, new and usually unknown couplings need to be introduced to absorb the divergences. Generally, these new couplings are suppressed at low energies by ratios of energy to the fundamental high energy scale, but at sufficiently high energies the theory loses its predictive power. In quantum gravity this happens at the Planck scale.

Quantum gravity based on the Feynman diagram expansion allows for a direct investigation of the non-renormalizability issue. For a theory of pure gravity with no matter, amazingly, the one-loop divergences cancel, as demonstrated by ’t Hooft and Veltman¹ [132 ]. Unfortunately, this result is “accidental”, since it does not hold generically when matter is added to the theory or when the number of loops is increased. Explicit calculations have shown that non-supersymmetric theories of gravity with matter generically diverge at one loop [132 , 43 , 42 ], and pure gravity diverges at two loops [66 , 136 ]. The two-loop calculations were performed using various improvements to the Feynman rules such as the background field method [130 , 46 , 1 ].

Supersymmetric theories of gravity are known to have less severe divergences. In particular, in any four-dimensional supergravity theory, supersymmetry Ward identities [73 , 72 ] forbid all possible one-loop [75 ] and two-loop [71 , 134 ] divergences. There is a candidate divergence at three loops for all supergravities including the maximally extended $N = 8$ version [39 , 82 , 84 , 81 ]. However, no explicit three-loop (super)gravity calculations have been performed to confirm the divergence. In principle it is possible that the coefficient of a potential divergence obtained by power counting can vanish, especially if the full symmetry of the theory is taken into account. As described in Section 7, this is precisely what does appear to happen [19 , 129 ] in the case of maximally supersymmetric supergravity.

The reason no direct calculation of the three-loop supergravity divergences has been performed is the overwhelming technical difficulties associated with multi-loop gravity Feynman diagrams. In multi-loop calculations the number of algebraic terms proliferates rapidly beyond the point where computations are practical. As a particularly striking example, consider the five-loop diagram in Figure 4 , which, as noted in Section 7, is of interest for ultraviolet divergences in maximal $N = 8$ supergravity in $D = 4$ . In the standard de Donder gauge this diagram contains twelve vertices, each of the order of a hundred terms, and sixteen graviton propagators, each with three terms, for a total of roughly 10³⁰ terms, even before having evaluated any integrals. This is obviously well beyond what can be implemented on any computer. The standard methods for simplifying diagrams, such as background-field gauges and superspace, are unfortunately insufficient to reduce the problem to anything close to manageable levels. The alternative of using string-based methods that have proven to be useful at one-loop and in certain two-loop calculations [27 , 25 , 119 , 55 , 56 , 57 , 120 ] also does not as yet provide a practical means for performing multi-loop scattering amplitude calculations [112 , 113 , 47 , 114 , 63 ], especially in gravity theories.

Figure 4: An example of a five-loop diagram.