Multi-loop zeta function regularization and spectral cutoff in curved spacetime

Abstract

We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard zeta function regularization at one loop and, on the other hand, a natural generalization of this method to higher loops. In particular, to any Feynman diagram is associated a generalized meromorphic zeta function. For the one-loop vacuum diagram, it is directly related to the usual spectral zeta function. To any loop order, the renormalized amplitudes can be read off from the pole structure of the generalized zeta functions. We focus on scalar field theories and illustrate the general formalism by explicit calculations at one-loop and two-loop orders, including a two-loop evaluation of the conformal anomaly.

2D gravitational Mabuchi action on Riemann surfaces with boundaries

“…[21]), the asymptotic behaviour of the eigenvalues for large n is λ n ∼ n A and, hence the spectral ζ-functions are defined by converging sums for Re s > 1, and by analytic continuations for all other values. In particular, they are well-defined…”

“…[21]). A straightforward formal manipulation shows that ζ ′ (0) ≡ d ds ζ(s)| s=0 provides a formal definition of − n≥0 log λ n , i.e.…”

“…[21]. While the field X is dimensionless, the ϕ n scale as A −1/2 ∼ µ where µ is some arbitrary mass scale (even if m = 0), and the c n as µ −1 .…”

“…The regularization-renormalization of determinants in terms of the ζ-function may appear as rather ad hoc, but it can be rigorously justified by introducing the spectral regularization [21]. The regularized logarithm of the determinant then equals ζ ′ (0) + ζ(0) log µ 2 plus a diverging piece ∼ AΛ 2 (log Λ 2 µ 2 + const), where Λ is some cutoff.…”

“…Next, the variation of ℓ 2 (x, y) was given e.g. in [6,7,21]. In the limit x → y one simply has ℓ 2 (x, y) ≃ g ab dx a dx b = e 2σ(y) g (0) ab dx a dx b which shows that one has δℓ 2 (x, y) ≃ 2δσ(y) ℓ 2 (x, y) and, hence, lim x→y δ log ℓ 2 (x, y)µ 2 = 2 δσ(y) .…”