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Quantum Phenomena in the Realm of Cosmology and Astrophysics

Quantum Phenomena in the Realm of Cosmology and Astrophysics

34 Part II. Dark Energy

34 Part II. Dark Energy from the Vacuum Energy of Quantum Fields where ψ(n) is the Digamma function, and ψ ′ (n) the Trigamma function. The expansion of the Γ-function then reads ( Γ −2 + ϵ ) ≃ 1 2 ϵ + 1 − γ [ ] π 2 2 + ϵ γ2 + 2(1 − γ) + ) + O(ϵ 2 ) . (4.12) 12 2 The mass term can be expanded for small ϵ as well, m −ϵ = µ −ϵ [ µ m ) ϵ ≃ µ −ϵ ( 1 + ϵ ln µ ] m + O(ϵ2 ) , (4.13) where µ is an auxiliary parameter with the dimension of a mass, introduced to make the argument of the logarithm dimensionless. Truncating at linear order in ϵ, we end up with an expression for the effective action as S eff ≃ − iµ−ϵ (4π) 2 m4 [ 1 ϵ + ln µ m + 1 − γ 2 + O(ϵ) ] . (4.14) Thus, we see that in order to cancel the divergent terms proportional to 1/ϵ, we need to require that the sum over all quartic powers of the masses of the system vanish, ∑ m 4 b = ∑ f b m 4 f , (4.15) which will then ensure the effective energy to be finite. This will also cancel the constant term 1 − γ/2 in the effective action. Thus for the convergent part, there is only the term S eff,conv = − iµ−ϵ (4π) 2 m4 ln µ m (4.16) left, which should be tuned to result in the observed cosmological constant by requiring the condition to be fulfilled. ∑ m 4 b ln µ − ∑ m b f b m 4 f ln µ m f = ρ Λ (4.17) In this argumentation, we ended up with one condition to cancel the divergences, and one to tune the convergent remainder to a specific value ρ Λ . However, we failed to recover also the condition ∑ ν b = ∑ ν f . (4.18) b f to balance the degrees of freedom of the particles - a condition that was predicted by the naive estimations of the vacuum energy previously. The reason for this is that within the formalism of dimensional regularisation as applied here the quadratically divergent contributions to the integral are lost due to the application of Veltman’s rule [54, 55].

4. Vacuum energy in flat spacetimes 35 Concretely, an integral like the one in Eq. (4.10) can always be reformulated as an integral over a fraction as ∫ Consider the identity ∫ d D p (2π) ln ( −p 2 + m 2) ∫ = D d D p (2π) D m 2 p 2 (p 2 + m 2 ) = ∫ d D ∫ p (2π) D dm 2 1 p 2 + m 2 . (4.19) [ ] d D p 1 (2π) D p − 1 . (4.20) 2 p 2 + m 2 By using the formula for Schwinger’s proper time integral [56], ∫ i d D k (4πis) D/2 e−im2s +m = 2) (2π) D e−is(−k2 , (4.21) and taking into account the integral representation of the Γ-function, we obtain ∫ d D p m 2 ∫ (2π) D p 2 (p 2 + m 2 ) ≡ d D p 1 Γ(1 − D/2) 1 = , (4.22) (2π) D p 2 + m2 (4π) D/2 (m 2 ) −D/2+1 which consequently implies ∫ d D p 1 (2π) D p = 0 , (4.23) 2 known as Veltman’s formula [52,54,55]. That shows that in using the technique of dimensional regularisation, the existence of an additional quadratic pole in the integrand is lost. On the contrary, using the formalism of a cutoff regularisation, the quadratic divergence is obtained correctly with an ultraviolet cutoff at p Λ : ∫ p Λ 0 d D p 1 (2π) D p 2 + m = − i ( ) p 2 2 (4π) 2 Λ − m 2 ln p2 Λ . (4.24) m 2 In the formalism of a dimensional regularisation, lost quadratic poles as in Eq. (4.23) can be recovered by considering the following reparameterization of the masses: m 2 i → m 2 0 + m 2 i , (4.25) where the square of the masses are rewritten as the sum of a constant part m 2 0 and an individually different part m 2 i . m 2 0 is the same for each particle species, and the m 2 i characterise the differences between the particles. The effective action integral then reads ∫ d D p (2π) ln ( ) −p 2 + m 2 D 0 + m 2 Γ(−D/2) i = −i (4π) D/2 (m2 0 + m 2 i ) D . (4.26) Doing the same expansions as before, expanding the Gamma function and the mass term, and neglecting terms of linear order in ϵ, we obtain a similar expression as before, [ ] S eff ≃ − iµ−ϵ (m 4 i + 2m 2 0 m 2 i + m 4 0) 1 (4π) 2−ϵ/2 ϵ + 1 − γ 2 + ln µ √ + O(ϵ) . (4.27) m 2 i + m 2 0

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