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Quantum Phenomena in the Realm of Cosmology and Astrophysics

Quantum Phenomena in the Realm of Cosmology and Astrophysics

32 Part II. Dark Energy

32 Part II. Dark Energy from the Vacuum Energy of Quantum Fields That means, the effective action and thus the vacuum energy is proportional to the expression S eff ∝ = ∫ ∞ p c [ p 4 p 2 sin θ dφ dθ dp (2π) 3 4 + m2 p 3 6 − m4 8 p √ 1 + m2 p 2 ln p − m6 48p 2 − ... ] ∞ p c . (4.5) This expression diverges in the ultraviolet limit of the momentum for the first three terms; and is moreover dependent on an infrared cutoff parameter p c . The cutoff p c has been introduced because the expansion (4.4) is only valid for large momenta, and we are not interested in the behaviour for small momenta. The ultraviolet divergences pose a serious problem since there is no more elegant argumentation for the their renormalisation than the assumption of the invalidity of conventional quantum field theories at scales beyond the Planck scale. The usual way to deal with those infinities is thus to introduce another cutoff [50] in phase space, for example at the Planck scale, and disregard particles with momenta higher than that cutoff, leading to the infamous energy density of ρ Λ,th ≃ 10 76 GeV 4 mentioned before. There are alternative renormalisation methods, such as dimensional regularisation [51], which however all serve the purpose of peeling out the singularities explicitly only to then disregard those terms or eliminate them by appropriate counterterms. The concept we would like to put forward in this work to deal with these divergences is to make the divergent contributions from bosons and fermions cancel exactly, so that only the finite parts of the integral remain, which then determines the vacuum energy. The way to achieve this is by considering the one other quantity occurring in the effective action above, i.e. the mass of the particle. If we would like to cancel the divergences between two particles, and taking into account that bosons and fermions give opposite-sign contributions to the effective action, it is clear that by fine-tuning the masses of the particles in the system accordingly, it is possible to get rid of the divergent parts. In order to have the first three terms in the above expression for bosons and fermions cancel each other out there are some relations that have to be fulfilled. The total expression for the vacuum energy for i particle species is S eff ∝ ∑ [ ν i p 4 + m2 i p 3 4 6 i − m4 i 8 ln p − ... ] . (4.6) To cancel the first term in the sums, we demand to have the same number of degrees of freedom on the bosonic and fermionic sector of the system, denoted by b and f, ∑ ν b = ∑ ν f . (4.7) b f

4. Vacuum energy in flat spacetimes 33 To cancel the other two divergent terms in the sum, the masses of the bosons and fermions must obey the relations ∑ m 2 b = ∑ m 2 f , b f ∑ m 4 b = ∑ m 4 f . (4.8) b f Given the assumption that the masses of bosons and fermions can be chosen to fulfil these relations, this is the basic schedule that has to be obeyed in order to achieve the cancellation of divergences. These simple estimations have shown the principle that we employ. In the following sections, we will calculate the vacuum energy exactly, peel out the divergences with slightly different methods, and try to find the exact cancellation conditions that have to be fulfilled. We expect however that they will be in principle the same as what we obtained from these sketchy considerations. 4.1. Exact calculation of the vacuum energy Going back to the integral in question, ∫ d D p (2π) D ln ( −p 2 + m 2) , (4.9) we will apply the technique of dimensional regularisation, which means that the integral will be rewritten in terms of Γ-functions as a function of the number of spacetime dimensions D, which will then be set to a natural number plus an infinitesimal part, i.e. D = 4 − ϵ. The resulting expression will be expanded in terms of the small quantity ϵ, and in the end the limit ϵ → 0 will be taken. From the above expression, we first carry out a Wick rotation, and then calculate the integral as (see e.g. [52]) ∫ d D p (2π) ln ( −p 2 + m 2) = −i Γ(−D/2) 1 . (4.10) D (4π) D/2 (m 2 ) −D/2 This integral is divergent because the Γ-function diverges for negative even values without any possibility of analytic continuation or such tricks, and for D = 4, we have Γ(−2) in the numerator. The expression needs to be processed further using dimensional regularisation, i.e. we substitute D by D = 4 − ϵ, which will make it possible to explicitly isolate the infinities in terms which diverge in the limit ϵ → 0. The result of the integral contains ϵ twice: within the Γ-function, and as power of the mass. It is possible to expand the Γ-function as [53] Γ(−n + ϵ) ≃ (−1)n n! [ 1 ϵ + ψ(n + 1) + ϵ ( )] π 2 2 3 + ψ2 (n + 1) − ψ ′ (n + 1) , (4.11)

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