Vacuum energy and the cosmological constant problem in $\kappa$-Poincar\'e invariant field theories

We investigate the vacuum energy in $\kappa$-Poincar\'e invariant field theories. It is shown that for the equivariant Dirac operator one obtains an improvement in UV behavior of the vacuum energy and therefore the cosmological constant problem has to be revised.


I. INTRODUCTION
The history of the cosmological constant starts with Albert Einstein, where in his first paper on the application of general relativity (GR) to cosmology [1], he aimed to construct a static universe with a finite average density of matter. In doing so he concluded that one needs to introduce an additional term to the gravitational equation of motion, namely the cosmological constant λ.
One can visualize λ as the curvature of the empty space. However, in GR curvature is connected with energy, momentum and pressure of matter. If we transfer the λ-term to the right hand side of the Einstein equation one sees that the empty space produces the same gravitational field as when the space contains matter (or more precisely an ideal fluid) with density ρ λ = λ 8πG and pressure p λ = −ρ λ . In this sense we can speak of an energy density and pressure of the vacuum. The quantities p λ and ρ λ are the same in all coordinate systems (Lorentz-transformed moving relative to one another), so they will never appear in experiments with elementary particles, neither in atomic or molecular physics, since the cosmological constant term will always cancel out in the energy conservation law [2]. Cosmological constant term only reveals himself in the gravitational phenomena on a large scale, i.e. cosmology.
The first problem with vacuum energy producing a gravitational effect was identified by W. Nernst [3], and later it developed into what is nowadays known as the cosmological constant problem. More precisely, the cosmological constant problem is the disagreement between the observed value of the cosmological constant (vacuum energy density) and the theoretically large value of zero-point energy obtained within the quantum field theory (QFT). Namely, depending on the Planck energy cutoff and other factors, the discrepancy can go up to 120 orders of magnitude, being "the worst prediction in the history of physics". For more details on the history and development on the attempts of solving the cosmological constant problem, the reader is advised to look at some of the following papers [4][5][6][7].
The nature of the vacuum energy in QFT highly depends on the UV structure of the theory under question [8]. It is determined that the classical (smooth) notion of spacetime is no longer adequate at Planck scale in order to reconcile GR with quantum mechanical axioms [9][10][11]. Properties of spacetime at the Planck scale are very different from what we expect from GR. Namely, various models including string theory [12], loop quantum gravity [13], and noncommutative (NC) geometry [14], suggest that the spacetime might have a certain discrete structure at the quantum gravity scale. Combining together GR and quantum uncertainty principle one can predict a very general class of NC spaces [9][10][11]. Once the NC structure of spacetime is assumed, NC field theories arises very naturally [15][16][17]. These NC field theories can have features like non-locality, UV/IR mixing and completely different UV behavior from their commutative counterparts [18][19][20][21][22][23][24][25][26].
Motivated by recent papers [27,28] we want to further investigate the nature of vacuum energy in NC field theories. In [27] the hypothesis that in models with an explicit breaking of Lorentz symmetry one can cure or improve the cosmological constant problem was tested. The result found there is that for certain Lorentz violating models one indeed obtains an improvement, while for the NC field theories examined there the result was negative. Furthermore, in the followup paper [28] it was found that by examining appropriate measure on Hilbert space one can indeed obtain improvement of UV behavior of the vacuum energy in NC field theories. In the present work we will be interested in a very specific, but widely known Lie algebraic type of NC space, the so called κ-Minkowski space [29][30][31] and the field theories on it.
NC field theories on κ-Minkowski space were widely investigated [32][33][34][35][36][37][38][39][40][41][42][43][44][45], but only recently the full investigation of the quantum properties of the 2-point and 4-point functions was explored in [46,47]. In [46] a natural ⋆-product for κ-Minkowski space was used to investigate various classes of κ-Poincaré invariant scalar field theories with quartic interactions. κ-Poincaré symmetry induces a twisted trace which defines a KMS weight for the NC C * -algebra that models the κ-Minkowski space. It was found that in all examined NC field theories, the twist generates different one-loop contributions to the 2-point functions which are at most UV linearly divergent. Some of the theories are free of UV/IR mixing. The one-loop contribution to the 4-point function is even found UV finite [47] for some models whose kinetic operators are related to the square of the Dirac operator involved in the construction of an equivariant spectral triple [49] aiming to encode the geometry of κ-Minkowski space. This is partly due to the large spatial momentum damping in the propagator which decays as 1/p 4 . This strong decay of the propagator at large (spatial) momenta could ensure the perturbative renormalizability to all orders.
In this paper we will analyze the vacuum energy in κ-Poincaré invariant field theories for three different choices of kinetic operators: the Casimir operator [31], the square of modular Dirac operator [48], and square of equivariant Dirac operator [49]. Exploiting the κ-Poincaré symmetry and properties of the ⋆-product for κ-Minkowski space we formulate action functional and show that in the case of the square of equivariant Dirac operator one obtains a significant improvement in the UV behavior of the vacuum energy. Therefore one needs to revise the cosmological constant problem accordingly.
The paper is organized as follows. In section II we outline the standard computation of vacuum energy using partition function and heat kernel. We present the construction of the κ-Poincaré invariant field theories in section III. In section IV the vacuum energy for three different choices of kinetic operators is calculated. We conclude with some final remarks in section V. Technical details on construction of the ⋆-product and appropriate Hilbert product are given in the Appendices.

II. VACUUM ENERGY AND COSMOLOGICAL CONSTANT
Consider a Euclidean bosonic field theory defined by the partition function where is the action functional and F (x, y) = d 4 p e ip(x−y)F (p) is the kernel of a generalized kinetic operator F (and its Fourier transform). Then the partition function Z is simply given by Z = (DetF ) −1/2 and the effective action W = ln Z can be written in terms of the heat kernel [8,50] where s is a real parameter, H(s) = Tr e −sF is the trace of the heat kernel. The cutoff 1/M 2 is introduced because in the standard field theory the effective action W is usually divergent. Recall that the heat kernel H(s; x, x ′ ) = (x|e −sF |x ′ ) is defined as a solution of the heat equation It is usually more convenient to perform calculations in the momentum space, so that the trace of the heat kernel becomes where V is the volume of the spacetime. The vacuum energy density ρ vac is defined as and it is often identified with the cosmological constant λ, but the relation between them is given by λ = 8πGρ vac .
For the standard massless bosonic quantum field theory in 4-d space we have F = ∂ µ ∂ µ , so that the heat kernel becomes and the vacuum energy is given by If we consider the cutoff M to be proportional to the Planck mass M P l ≈ 10 18 GeV, then we have a huge discrepancy of almost 120 orders of magnitude between the vacuum energy of the quantum field theory and the observed cosmological constant ρ ob vac ≈ 10 −47 GeV 4 , leading to the "worst prediction of theoretical physics".
However, if there is a change in the UV behavior of the kinetic operator this could drastically affect the structure of the vacuum energy [8,27,28]. Therefore we will investigate κ-Poincaré invariant field theories because they are characterized by kinetic operators which exhibit significant modifications in the UV regime.

III. κ-POINCARÉ INVARIANT FIELD THEORIES
A. ⋆-product for κ-Minkowski space κ-Minkowski space can be viewed as the universal enveloping algebra of the Lie algebra g defined by where κ > 0 is a deformation parameter of mass dimension. Furthermore, one can define a Lie group G by exponentiation from which follows the group law A convenient presentation of κ-Minkowski space is provided by a mere adaptation of the Wigner-Weyl quantization scheme giving rise to the celebrated Moyal product by replacing the Heisenberg group with G. This latter approach, whose main steps are recalled in the Appendix (see also [46,51]), leads to the following expressions for the ⋆-product and the adjoint for any functions f, g ∈ F (S c ). Here S c is the set of Schwartz functions on R 4 with compact support in the first variable and F is the Fourier transform. For all physical purposes it will be sufficient to identify the algebra modeling κ-Minkowski M κ with the algebra F (S c ).

B. κ-Poincaré algebra and invariant action
In the present study, we require the action functional S κ to be κ-Poincaré invariant which is a reasonable assumption regarding the important role played by the (classical) Poincaré symmetries in standard QFT together with the fact that κ-Poincaré algebra can be viewed as describing the quantum symmetries of the κ-Minkowski space-time. These symmetries are encoded algebraically by the Hopf algebra of κ-Poincaré algebra P κ . One way to present P κ is by using the 11 elements: P i are the momenta, N i are boosts, M i are rotations, and E is the shift operator related to time translations E = e − P 0 κ . They all satisfy the following commutation relations κ-Poincaré algebra can be endowed by a Hopf algebra structure. Let us define the coproduct ∆ : P κ → P κ ⊗ P κ , counit ε : P κ → C and antipode S : P κ → P κ satisfying and The algebra M κ is a left-module over the Hopf algebra P κ which can be viewed as the algebra of quantum symmetries of the corresponding noncommutative space. κ-Minkowski space 1 is a dual of the Hopf subalgebra generated by P i and E. The structure of M κ as a left module over the Hopf algebra P κ can be expressed for any f ∈ M κ as Notice that the generators of spatial translations are not derivations of the algebra M κ due to the coproduct (15).
To define a well behaved field theory we need to find some reasonable action functional. It is natural to impose the following three conditions on the physical action functional S κ [φ]: [33,46,51] • The action S κ [φ] is positive and real.
• S κ [φ] reduces to the standard scalar field theory in the commutative limit κ → ∞.
In order to find an action S κ [φ] invariant under P κ we consider functional of the following form where φ, L[φ] ∈ M κ . Now, by using (17) This way we have proven that the action of the form (19) fulfills the first condition and is invariant under the action of P κ .
Since the following properties hold it is easy to see which defines a positive map d 4 x : M κ + → R + where M κ + denotes the set of positive elements of M κ . It is important to note that the Lebesgue integral does not define a cyclic trace. Namely The operator σ is an algebra automorphism often called twist. It gives rise to a twisted trace Tr(a ⋆ b) = Tr((σ ⊲ b) ⋆ a) which signals the occurrence of a KMS condition. The operator σ t = e i 3tP 0 κ defines a group of automorphisms called the modular group of the KMS weight (for more details see [46]).
In order to construct real action functional we use a natural Hilbert product 2 given by Notice that the Hilbert product (24) is R-valued for all f, g ∈ M κ that satisfy f, g = g, f . Therefore, the reality condition for the action functional will be automatically fulfilled if we use where K is some self-adjoint kinetic operator satisfying φ, Kφ = Kφ, φ . It is straightforward to see that P µ and E are self-adjoint with respect to the Hilbert product (24). In constructing the kinetic operator we assume it to be a pseudo-differential operator given by for any f in the domain of K dense in the Hilbert space H ∼ = L 2 (R 4 ), andK(p) is the Fourier transform of the kernel of K. The self-adjointness of K impliesK(p) =K(p).
According to the discussion made so far and using (25) and (26) we can finally define the action functional that satisfies all three conditions by 3 If we compare this expression with (3) we get that the relevant operator for calculating the vacuum energy is given by So, in order to calculate vacuum energy in various κ-Poincaré invariant theories we just need to extract the functioñ F (p).
We have a few natural choices for the explicit form of the kinetic operator in (27). In this paper we will investigate three such choices: the Casimir operator, the square of modular Dirac operator and the equivariant Dirac operator.

IV. VACUUM ENERGY IN Pκ-INVARIANT FIELD THEORIES
We have set up everything for calculating the vacuum energy. All we need to do now is to evaluate (6) and (4) for different kinetic operators, that is (28).

A. The Casimir operator
The Casimir operator C κ commutes with all the generators of the κ-Poincaré algebra and as such we can use it as the kinetic operator. This choice fulfills all the conditions discussed in the previous section. Also this is in the complete analogy with the commutative case, where the Casimir operator of the Poincaré algebra serves as the kinetic operator. The Casimir operator C κ in the Majid-Ruegg basis [31] is given by so that the function (28) that we need in order to calculate the vacuum energy is given bỹ The relevant integral in calculating the heat kernel (6) where we used the integral ∞ 0 dx x 2 e −ax 2 = √ π 4a 3/2 in the first line and the substitution y = e − p 0 κ in the second line. The integral in (31) is UV divergent, that is, it is singular for y → 0. To illustrate this let us put a cutoff on the variable p 0 such that ±∞ → ±Λ, then the integral we need to solve is given by Now, one has to be careful with the regularization and interpreting the cutoff in p 0 , that is Λ as the UV limit. Namely, one has to take in account the κ-Poincaré symmetry. Motivated by the Hopf algebra structure of the P κ , especially the deformed translation algebra generated by P i and E (see (15)), it is more natural to interpret y = e − p 0 κ as related to the physical quantity replacing p 0 in the NCFT. More precisely, the Casimir operator (29) can be written as so that the relevant quantity for the P κ -covariant quantum field theories, which reduces to p 0 in the commutative limit is given by P 0 . Putting a cutoff |P 0 | ≤ M enables us to derive the appropriate cutoff on the variable p 0 Taking the relation between the cutoffs (34) and (32), the integral in (31) and the corresponding effective action W Cκ reads (35) so that the vacuum energy becomes We see that the vacuum energy behaves as ρ vac ∝ M 6 κ 2 in the UV limit, which is more divergent when compared with the commutative case. In case that the deformation parameter is equal to the UV cutoff κ = M we recover the commutative behavior ρ vac ∝ M 4 . The result obtained here is in agreement with the investigation performed in [27,28] where, even though there they used a different integration measure, the same behavior for the vacuum energy was discovered.

B. Modular operator
Another choice for the kinetic operator comes from studying modular spectral triples [48]. It is related to the Casimir operator by where we used the expansion for the incomplete Gamma function Γ − 7 6 , ax 3 ≈ 6 7(ax) 7/6 + O(x −1/2 ). The corresponding vacuum energy is ρ vac ∝ M 13 κ 5 which is even more divergent that in the case of Casimir operator. In the limit κ = M we recover the commutative behavior.

C. Equivariant Dirac operator
In this subsection we investigate the square of the equivariant Dirac operator. This operator comes from the construction of the equivariant spectral triple for the κ-Minkowski space [49] and is related to the Casimir operator by The relevant function is given byF K (p) = e − p 0 κ K κ and for the sake of future convince we write it in the following formF where A = κ 2 4 (y + 2y 2 − 2y 3 + 2y 4 + y 5 ), B = 1 2 (y + y 3 ), C = y 2 The integral we need to obtain in order to calculate the heat kernel is given by We will first deal with the integral over the radial variable p and for that we need the following integral where M (a, b, z) is the confluent hypergeometric function. Since we will be interested in the the most divergent parts we look at the limit s −→ 0 where we have 4 ∞ 0 p 2 dp e −s(Bp 2 +Cp 4 ) ≈ 1 4(sC) 3/4 + O(s −1/4 ) so that (43) becomes The integral in (43) is once again divergent and therefore we will proceed with the same strategy as in the former cases and use the cutoff regularization procedure. In doing so, we get Now, after taking into account (4), (6) and (46) we get for the vacuum energy The vacuum energy for the square of the equivariant Dirac operator has a quite better behavior then the commutative theory since ρ vac ∝ κM 3 . This improvement in the UV behavior of the quantum field theory was already demonstrated in [46] where the properties of the 2-point function was investigated, and also in [47] where it was shown that the β-function for the various versions of NC φ 4 -theory vanishes. In the limit κ = M we recover the usual commutative behavior.

V. FINAL REMARKS
The quantum fluctuations of the vacuum contribute to the expectation value of the energy-momentum tensor in a way that mimics the cosmological constant. In QFT the vacuum energy is highly nontrivial, as in the case of a simple quantum harmonic oscillator in the ground state, each mode of every field has its own contribution to the zero-point energy. This energy arises from virtual particle-antiparticle pairs, i.e. loops. The corresponding energy-momentum tensor can be written as T µν = −ρ vac g µν . Even though it would appear on the right hand side of the Einstein's equation, vacuum energy has the form of a cosmological constant, therefore we can absorb it in order to redefine λ −→ λ ren = λ + 8πGρ vac . Equivalently, we may absorb the bare cosmological constant λ appearing in the Einstein's equation into the energy density of the vacuum [52]. This way one turns the cosmological constant problem into a fine tunning one.
In this work we have presented a construction of κ-Poincaré invariant field theory and investigated the vacuum energy for several choices of kinetic operator. It is shown that in the case of the square of equivariant Dirac operator one obtains an improved UV behavior of the zero-point energy. Therefore one needs to revise the cosmological constant problem. Even though in the case of the square of the equivariant Dirac we obtained a significant improvement in the UV behavior it is still not sufficient to resolve the cosmological constant problem. Namely, if we would assume that the UV cutoff is proportional to the Planck mass M = M P l ≈ 10 18 GeV, and compared the theoretical estimate obtained in (47) with the observed one ρ obs vac ≈ 10 −47 GeV 4 , then we could get a bound on the deformation parameter κ that would be far too low, indicating that we should have already observed NC effects in various experiments. However, one can speculate that the new energy scale, governed by the deformation parameter κ, is in the interval κ ∈ [1TeV, M P l ]. This assumption is supported by the fact that more or less above the energy of 1TeV we did not explore particle interactions experimentally. In doing so, we would get a discrepancy with the observations ranging between 10 and 120 orders of magnitude. Therefore one is forced to understand the expression (47) as a starting point of the renormalization procedure, i.e. as part that will be reabsorbed by the counter-term. In doing so one transforms the cosmological constant problem into a fine tuning one, but with a sufficiently better starting point for tuning since it was shown that the quantum corrections in the NC version of the φ 4 theory are finite, namely the β-function vanishes perturbatively [47].
When introducing curvature in consideration then things get much more complicated and interesting [64]. One encounters the so called running cosmological constant [65]. In order to provide a small value of the observed cosmological constant one can also introduce the vacuum term which cancels the induced one at some point in the very far infrared cosmic scale [66]. Namely, one can even show that the renormalized vacuum energy in curved background is ρ vac ≈ m 2 H 2 , where m is the mass of the scalar field and H the Hubble scale [67]. Therefore one is tempted to extend the aforementioned ideas to the NC setting. We leave this line of research for future work.
There have been other investigations concerning NC effects on the cosmological constant problem. The implications to cosmology steaming from the spectral triple approach to NC geometry was reported in [68]. In [69,70] the study of the cosmological constant problem as an eigenvalue problem of a certain Sturm-Liouville problem was performed. The authors of [69,70] employed the NC effects into the Weeler-De Witt equation through minimal length. It would be very interesting and challenging to accommodate the two aforementioned approaches for the case of κ-Poincaré symmetry and ⋆-product for κ-Minkowski space.
One has to be aware that the calculations performed in this paper are within the framework of NC field theory, so we are really only able to comment on the NC effect on the vacuum or zero-point energy and not on the whole cosmological constant problem. Namely, noncommutativity also affects the geometric or "pure" gravity side of the Einstein's equation. In numerous approaches [53][54][55][56][57][58][59][60] we have witnessed the NC corrections to Einstein's equation. Moreover, in [61][62][63] it has been indicated that the nature of the cosmological constant could be entirely noncommutative. Therefore one needs to find the proper NC correction to the geometric side of the Einstein's equation and then together with the result obtained in this paper investigate further all the consequences.
One of the most obvious manifestation of vacuum energy (besides the cosmological constant) is the so called Casimir effect [71]. It would be very interesting to calculate the modification of the scalar Casimir force between two parallel plates due to noncommutativity of the spacetime and compare it with the existing results in the literature [72][73][74][75][76].
Acknowledgment T.J. is very grateful to Andjelo Samsarov for numerous and fruitful discussions and to Ivica Smolić for comments. This work was partially supported by Croatian Science Foundation under the project (IP-2018-01-4872) "Black holes, nonlinear fields and quantum spaces" and it was partially supported by the H2020 CSA Twinning project No. 692194, RBI-T-WINNING.
Appendix A: κ-Minkowski space and the ⋆-product κ-Minkowski space can be viewed as the universal enveloping algebra of the Lie algebra g defined by where κ is the deformation parameter usually related to Planck mass or some other quantum gravity scale. The coordinates x 0 and x i are self-adjoint operators acting on a suitable Hilbert space. Since the derived algebra [g, g] is nilpotent it follows that the Lie algebra g is solvable and therefore the associated Lie group G is also solvable. In order to investigate further the κ-Minkowski space it is useful to use the framework of group algebras and C ⋆dynamical systems [51,77]. Notice that the group G is not unimodular and therefore we have a distinct left and right invariant Haar measure, which we denote by dµ and dν respectively. These measures are related by a modular function dν(s) = ∆ G (s)dµ(s). For the sake of simplicity let us consider d = 1 case. In doing so, we see that the Lie group G is completely characterized by defining where p 0 , p 1 ∈ R are coordinates on the group manifold and can be interpreted as momenta. Using (A1) and (A2) one obtains the group law for the multiplication in the group G W (p 0 , p 1 )W (q 0 , q 1 ) = W (p 0 + q 0 , p 1 + e − p 0 κ q 1 ) (A3) It is easy to see that the unit element I G and the inverse W −1 are given by Notice that the ⋆-product of 2-dim κ-Minkowski algebra can be obtained using Weyl quantization [78,79]. To illustrate this we have to start from L 1 (G), that is the convolution algebra of G and a representation π u : G −→ B(H) that is a (strongly continuous) unitary representation of G, where H is a suitable Hilbert space and B(H) is the C *algebra of bounded operators on H. We define the convolution algebra with respect to the right invariant measure aŝ G = (L 1 (G), •) where the algebra multiplication is the convolution defined by The unitary representation of the convolution algebraĜ is given by π : L 1 (G) −→ B(H) and so that it is a non-degenerate * -representation. One can easily check that π(f ) † = π(f * ), π(f • g) = π(f )π(g), ∀f, g ∈Ĝ (A7) where the † denotes the adjoint operation acting on the operators with respect to the Hilbert product (24) u In order to obtain the ⋆-product we will use the Weyl quantization map [78]. This way one makes the identification between the functions on G with functions on R 2 . First, let and similarly one finds Finally, by using the fact that the right invariant measure is dν(p 0 , p 1 ) = dp 0 dp 1 and the expression for right convolution product (A5), one obtains (f ⋆ g)(x 0 , x 1 ) = dp 0 dy 0 e −iy0p0 f (x 0 + y 0 , x 1 )g(x 0 , e − p 0 κ x 1 ) (A14) and f † (x 0 , x 1 ) = dp 0 dy 0 e −iy0p0 f (x 0 + y 0 , e − p 0 κ x 1 ) (A15) for any f and g ∈ F [S c ], where S c denotes the space of Schwartz functions on R 2 with compact support.