According to the dark dimension DD scenario we might live in a universe with a single compact extra dimension whose mesoscopic size is dictated by the measured value of the cosmological constant. This scenario is based on swampland conjectures that lead to the relation rhorm swampsim mrm KK4 between the vacuum energy rhorm swamp and the size of the extra dimension mrm KK1 mrm KK is the mass scale of a KaluzaKlein tower and on the corresponding result rhorm EFT from the EFT limit. We show that rhorm EFT contains previously missed UVsensitive terms whose presence invalidates the widely spread belief based on existing literature that the calculation gives automatically the finite result rhorm EFTsim mrm KK4 with no need for finetuning. This renders the matching between rhorm swamp and rhorm EFT a nontrivial issue. We then comment on the necessity to find a mechanism that implements the suppression of the aforementioned UVsensitive terms. This should finally allow to frame the DD scenario in a selfconsistent framework also in view of its several phenomenological applications based on EFT calculations.
['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Phenomenology']
Does the Cosmological Constant really indicate the existence of a Dark Dimension
{'No Header': 'International Journal of Geometric Methods in Modern Physics  \n- © World Scientific Publishing Company', 'DOES THE COSMOLOGICAL CONSTANT REALLY INDICATE THE EXISTENCE OF A DARK DIMENSION?': "CARLO BRANCHINA 1 2 † , VINCENZO BRANCHINA 3 ∗ , FILIPPO CONTINO 3 ‡ , 3 /star  \n1  \nARCANGELO PERNACE  \nDepartment of Physics, Chung-Ang University, Seoul 06974, Korea 2 Department of Physics, University of Calabria, and INFN-Cosenza Arcavacata di Rende, I-87036, Cosenza, Italy 3 Department of Physics, University of Catania, and INFN-Catania Via Santa Sofia 64, I-95123 Catania, Italy † carlo.branchina@unical.it ∗ vincenzo.branchina@ct.infn.it ‡ filippo.contino@ct.infn.it /star arcangelo.pernace@ct.infn.it  \nReceived (Day Month Year) Revised (Day Month Year)  \nAccording to the 'dark dimension' (DD) scenario, we might live in a universe with a single compact extra dimension, whose mesoscopic size is dictated by the measured value of the cosmological constant. This scenario is based on swampland conjectures, that lead to the relation ρ swamp ∼ m 4 KK between the vacuum energy ρ swamp and the size of the extra dimension m -1 KK ( m KK is the mass scale of a Kaluza-Klein tower), and on the corresponding result ρ EFT from the EFT limit. We show that ρ EFT contains previously missed UV-sensitive terms, whose presence invalidates the widely spread belief (based on existing literature) that the calculation gives automatically the finite result ρ EFT ∼ m 4 KK (with no need for fine-tuning). This renders the matching between ρ swamp and ρ EFT a non-trivial issue. We then comment on the necessity to find a mechanism that implements the suppression of the aforementioned UV-sensitive terms. This should finally allow to frame the DD scenario in a self-consistent framework, also in view of its several phenomenological applications based on EFT calculations.  \nKeywords : dark energy; Kaluza-Klein theories; dark dimension.", '1. Introduction': 'Theories with large extra dimensions were extensively explored in the nineties in search for a solution to the electroweak naturalness/hierarchy problem [1,2,3]. A recent surge of interest towards the physics of 5D effective field theories (EFTs) with one compact extra dimension of mesoscopic size has followed the dark dimension  \n∗ Corresponding author', '2 C. Branchina, V. Branchina, F. Contino and A. Pernace': "(DD) proposal, which, following arguments based on swampland conjectures [4], and born in a string framework, suggests the existence of a single extra dimension of µ m size. A generic feature of string theories is the existence of towers with an infinite number of states, whose masses are given in terms of a scale µ tow . According to the swampland distance conjecture [5], at large distance in the moduli field space φ one of the tower scales becomes exponentially small, µ tow ∼ e -α | φ | ( α positive O (1) constant), and the DD proposal is related to this asymptotic regime. As stressed in [4], in these regions only two cases seem to arise in string compactifications: a tower of string excitation modes or a tower of Kaluza-Klein (KK) states, i.e. µ tow ∼ M s , m KK (Emergent String Conjecture [6,7]). In general, in d spacetime non-compact dimensions an infinite tower of states contributes to the vacuum energy ρ d an amount ρ d ∼ µ d tow [8,9]. A similar result seems to hold even in the framework of higher dimensional field theories with compact extra dimensions. This is for instance the case for supersymmetric theories with Scherk-Schwarz or brane-localized SUSY breaking [10,11]. Only KK modes are present and the usual calculation gives ρ d ∼ m d KK .  \nGoing back to the string (quantum gravity) framework, when the distance conjecture is implemented in AdS spaces[12]  \nµ tow ∼ | ˜ Λ cc | γ , (1) where ˜ Λ cc is the cosmological constant times the squared Planck mass M 2 P . Even though there is much wider support in AdS, the conjecture is nonetheless extended also to dS spaces, where it forms the basis for the dark dimension proposal[4]. Restricting to the d = 4 case, the one-loop string calculation of ρ 4 gives  \nρ 4 ∼ µ 4 tow . (2)  \nThe authors of [4] note that higher loops might only contribute with higher powers of µ tow , so that (barring cancellation of the µ 4 tow term) the comparison of (2) with (1) gives a  \nγ ≥ 1 4 . (3)  \nThey assume (3) as starting point for their proposal. Moreover, observing that the experimental bounds on possible violations of the 1 /r 2 Newton's law[15] give µ tow ≥ 6 . 6 meV, and that the energy scale associated to the measured value of ˜ Λ cc [16] is of the same order, ˜ Λ 1 / 4 cc ∼ 2 . 31 meV, they infer that (1) is saturated with γ = 1 / 4, and accordingly the 'experimental value' of µ tow is  \nµ exp tow ∼ 2 . 31 meV (4)  \n(order the neutrino scale). Finally they observe that, although it is in principle possible that µ tow = M s , Eq. (4) indicates that this option is 'ruled out by experiments' since we know that physics above the neutrino scale is well described by  \n3  \neffective field theories, and no sign of string excitations is observed at these scales. They then conclude that the only possibility left is an 'EFT decompactification scenario', with a Kaluza-Klein mass m KK ∼ µ exp tow ∼ 2 . 31 meV.  \nThis conclusion takes us from the string theory realm to the EFT terrain, and is crucial to the formulation of the DD proposal. Typically, when physics is described in terms of a string KK tower, the original string theory is replaced by the corresponding higher dimensional EFT with compact extra dimensions. A thorough analysis of this delicate step is one of the goals of the present work.  \nAccording to [17], the strongest bounds for the compactification scale m -1 KK come from the heating of neutron stars due to the surrounding cloud of trapped KK gravitons [17,18], which yields to the upper bounds: m -1 KK < 44 µ mfor n = 1, m -1 KK < 1 . 6 × 10 -4 µ m for n = 2, with more stringent bounds for n > 2. The authors of [4] then conclude that n ≥ 2 is excluded since it is not compatible with m KK ∼ ˜ Λ 1 / 4 cc , and that there should be a single extra dimension, they call it dark dimension , of size ∼ 1 -100 µ m.  \nIn string theory the finite result ρ d ∼ µ d tow arises from modular invariance, that requires to sum over the infinite tower of states. In higher dimensional field theories with compact extra dimensions such an UV-insensitive result for ρ d is obtained performing the calculation in a similar manner, i.e. summing over the infinite number of KK-states (the same is done for the calculation of the 4D Higgs effective potential and Higgs boson mass). Differently from the string theory case, however, in this EFT framework such a way of performing the calculation is less obvious to justify [19]. In fact, this question was at the centre of a heated debate in the early 2000's. Several authors tried to support this way of operating with different arguments [20,21], and even nowadays there are attempts at justifying it from the string theory side [22].  \nThis issue, and more generally the question of developing a well-founded EFT approach to field theories with compact extra dimensions, was recently re-analysed in [23], where the focus was on the problem of the UV-(in)sensitivity of the oneloop Higgs effective potential and Higgs mass. It was shown that within the usual calculations the asymptotics of the loop momenta are mistreated, and that this results in an artificial washout of UV-sensitive terms of topological origin. The latter stem from the boundary conditions that must necessarily be given to define the theory on a multiply connected manifold. Their presence was first pointed out in [23].  \nIn this work we show that, when a proper EFT calculation of the vacuum energy ρ 4 is performed, UV-sensitive terms arise (for an introduction to the cosmological constant problem see for instance [24,25,26]). Moreover we discuss how the EFT logic can and must be consistently applied to theories with compact extra dimensions, although it has been recently argued that no controlled approximation can be obtained cutting a KK tower at a finite value [27].", '2. Vacuum energy': 'For concreteness, in the following we stick to the case (sufficient for our scopes) of a 5D EFT coupled to gravity, where the compact space dimension is in the shape of a circle of radius R . We take the 5D action to be  \nS (4+1) = S (4+1) grav + S (4+1) matter , (5)  \nwhere  \nis the Einstein-Hilbert action in 4 + 1 dimensions and S (4+1) matter the matter action that contains the bosonic and fermionic fields of the theory. We indicate with x the 4D coordinates and with z the coordinate along the compact dimension. After integration over z , the 5D metric (we use the (+ , -, -, -, -) signature)  \nS (4+1) grav = 1 2ˆ κ 2 ∫ d 4 xdz √ ˆ g ( ˆ R2 ˆ Λ cc ) (6)  \nˆ g MN = ( e 2 αφ g µν -e 2 βφ A µ A ν e 2 βφ A µ e 2 βφ A ν -e 2 βφ ) (7)  \nleads to the 4D action [28]  \nS (4) grav = 1 2 κ 2 ∫ d 4 x √ -g [ R2 e 2 αφ ˆ Λ cc +2 α /square φ + ( ∂φ ) 2 2 -e -6 αφ 4 F 2 ] , (8)  \nwhere the 4D constant κ is related to the 5D ˆ κ by  \nκ 2 = ˆ κ 2 2 πR . (9)  \nThe constants α and β satisfy the relation  \n2 α + β = 0 (10)  \nand the canonical radion kinetic term fixes  \nα = 1 √ 12 . (11)  \nFor completeness we recall that the Newton constant κ can be used to write (8) in terms of dimensionful φ and A µ fields through the redefinition  \nφ → φ √ 2 κ , A µ → A µ √ 2 κ . (12)  \nLet us consider the case of a complex 5D scalar field ˆ Φ with action  \nS (4+1) Φ = ∫ d 4 xdz √ ˆ g ( ˆ g MN ∂ M ˆ Φ ∗ ∂ N ˆ Φ -m 2 | ˆ Φ | 2 ) , (13)  \nthat along the compact dimension obeys the non-trivial boundary condition  \nˆ Φ( x, z +2 πR ) = e 2 πiδ ˆ Φ( x, z ) , (14)  \nwhere δ is a generic phase. Defining now ([ δ ] is the integer part of δ )  \nq ≡ δ -[ δ ] , (15)  \nDoes the Cosmological Constant really indicate the existence of a Dark Dimension? 5  \n(14) can be rewritten as  \nˆ Φ( x, z +2 πR ) = e 2 πiq ˆ Φ( x, z ) . (16)  \nIt is then clear that the physical parameter to which we have access is not δ , but rather q . In other words, to implement the non-trivial boundary condition we have to refer to (16) rather than to (14). Consequently, in physical quantities q rather than δ will appear.  \nFor the corresponding 4D action we have  \nS (4) Φ = ∫ d 4 x √ -g ∑ n [ | Dϕ n | 2 -( e 2 αφ m 2 + e 6 αφ ( n + q ) 2 R 2 ) | ϕ n | 2 ] , (17)  \nwhere  \nD µ ≡ ∂ µ -i ( n + q R ) A µ (18)  \nand ϕ n ( x ) are the KK modes of ˆ Φ( x, z ). Taking a constant background for the radion (that for notational simplicity we continue to call φ ) and the trivial background for A µ ,  \nˆ g 0 MN = ( e 2 αφ η µν 0 0 -e 2 βφ ) , (19)  \nfrom (17) we can define the φ -dependent radius R φ ≡ Re -3 αφ ( Re ( β -α ) φ before using (10)) and the φ -dependent mass m 2 φ ≡ m 2 e 2 αφ , so that the KK masses are  \nm 2 n ≡ m 2 φ + ( n + q ) 2 R 2 φ . (20)  \nGoing to Euclidean space and considering the general case, the one-loop contribution to the 4D vacuum energy ρ 4 of a single bosonic or fermionic tower of mass m and boundary charge q is then  \nρ 4 ∼ ( -1) δ if ∑ n ∫ d 4 p (2 π ) 4 log p 2 + ( n + q ) 2 R 2 φ + m 2 φ µ 2 , (21)  \nwhere µ is a subtraction scale, and i = b, f for bosons and fermions respectively.  \nThe right hand side of (21) is calculated according to different strategies. One of them consists in performing the sum over n all the way up to infinity, and the integral in d 4 p with the help b of a cutoff Λ [10,11,29]. Other methods are related to the implementation of the proper time [30], Pauli-Villars [20], thick brane [21], and dimensional regularizations [31]. They all give the same result. For the time being we focus on the first of them; we will comment on the others later. Crucial in getting the UV-insensitive result ρ 4 ∼ R -4 φ = m 4 KK in [10,11,29] is that n is', '6 C. Branchina, V. Branchina, F. Contino and A. Pernace': "sent to infinity while Λ is kept fixed c . As mentioned in the Introduction, however, this way of performing the calculation mistreats the asymptotics of the 5D loop momentum of the original theory. In fact, n/R is the fifth component p 5 of the 5D loop momentum ˆ p ≡ ( p, n/R ). Sending p 5 →∞ while keeping Λ fixed means that in the loop corrections we are (improperly) including first the asymptotics of the fifth component ( n/R ) of the momentum and only later those of the other four components p 1 , p 2 , p 3 and p 4 . As shown in [23], however, a necessary and physical requirement, overlooked in previous literature, is that the asymptotics of all the five components of ˆ p have to be treated on an equal footing. This can be realized considering in (21) a 5D cutoff, ˆ g MN 0 ˆ p M ˆ p N = e -2 αφ p 2 + e -2 βφ n 2 /R 2 ≡ ˜ p 2 + n 2 / ˜ R 2 ≤ Λ 2 , or equivalently through the insertion of a multiplicative smooth cutoff function e -( ˜ p 2 + n 2 / ˜ R 2 ) / Λ 2 (ˆ g MN 0 is the inverse of the Euclidean flat background 5D metric in (19)). Sticking to the first of these two options, defining Λ φ ≡ Λ e αφ , and performing the integration over p , we get (below we write only the contribution of a bosonic tower)  \nρ 4 = 1 64 π 2 [ R φ Λ φ ] ∑ n = -[ R φ Λ φ ] {( Λ 2 φ -n 2 R 2 φ )( m 2 φ + ( n + q R φ ) 2 ) + ( Λ 2 φ -n 2 R 2 φ ) 2 log Λ 2 φ + m 2 φ -n 2 R 2 φ + ( n + q R φ ) 2 µ 2 + ( m 2 φ + ( n + q R φ ) 2 ) 2 log m 2 φ + ( n + q R φ ) 2 Λ 2 φ + m 2 φ -n 2 R 2 φ + ( n + q R φ ) 2 -1 2 ( Λ 2 φ -n 2 R 2 φ ) 2 } ≡ [ R φ Λ φ ] ∑ n = -[ R φ Λ φ ] F ( n ) , (22)  \nwhere the brackets [ ... ] indicate 'integer part' (to simplify the notation, but without loss of generality, we take Λ such that R φ Λ φ is an integer). The sum can be performed using the Euler-McLaurin (EML) formula,  \nρ 4 = ∫ R φ Λ φ -R φ Λ φ dxF ( x ) + F ( R φ Λ φ ) + F ( -R φ Λ φ ) 2 (23) + r ∑ j =1 B 2 j (2 j )! ( F (2 j -1) ( R φ Λ φ ) -F (2 j -1) ( -R φ Λ φ ) ) + R 2 r ,  \nDoes the Cosmological Constant really indicate the existence of a Dark Dimension? 7  \nwhere r is an integer, B n are the Bernoulli numbers, and the rest R 2 r is given by  \nR 2 r = ∞ ∑ k = r +1 B 2 j (2 j )! ( F (2 j -1) ( R φ Λ φ ) -F (2 j -1) ( -R φ Λ φ ) ) = ( -1) 2 r +1 (2 r )! ∫ R φ Λ φ -R φ Λ φ dxF (2 r ) ( x ) B 2 r ( x -[ x ]) , (24)  \nwith B n ( x ) the Bernoulli polynomials. Expanding for m φ / Λ φ , q/ Λ φ /lessmuch 1, we finally get  \nρ 4 = 5 log Λ 2 e 2 αφ µ 2 -2 300 π 2 e 2 αφ R Λ 5 + 5 m 2 +3 q 2 e 4 αφ R 2 180 π 2 e 2 αφ R Λ 3 -35 m 4 +14 m 2 q 2 e 4 αφ R 2 +3 q 4 e 8 αφ R 4 840 π 2 e 2 αφ R Λ+ m 5 60 π e 2 αφ R + 3 log Λ 2 e 2 αφ µ 2 +2 2880 π 2 R 4 e 10 αφ R Λ+ R 4 + O (Λ -1 ) , (25)  \nwhere the rest R 4 is the UV-insensitive term  \nwith  \nR 4 = -x 2 Li 3 ( r b e -x ) + 3 x Li 4 ( r b e -x ) + 3Li 5 ( r b e -x ) 128 π 6 R 4 e 12 αφ + h.c. + 3 ζ (5) 64 π 6 R 4 e 12 αφ + O ( Λ -1 ) (26)  \nr ≡ e 2 πiq , x ≡ 2 πe -2 αφ R √ m 2 . (27)  \nEqs. (25) and (26) are re-written in terms of the original R , Λ and m (rather than R φ , Λ φ and m φ ) to explicitly show the φ -dependence. It is worth to note that the 4D vacuum energy ρ 4 is related to the corresponding 5D one ρ 4+1 through the relation ρ 4 = 2 πRe 2 αφ ρ 4+1 .  \nSeveral comments are in order. First of all we observe that, had we made the calculation in the usual way [10,11,29], all the q -dependent UV-sensitive terms in Eq.(25) (i.e. all the q -dependent terms except those contained in R 4 ) would be absent, while the other UV-sensitive terms are cancelled by SUSY. In fact, while the higher dimensional SUSY imposes m b = m f for the superpartners, q b and q f are necessarily different to have a broken SUSY spectrum at low energies. With the usual calculation we would then get the well-known result ρ 4 ∼ R b 4 -R f 4 ∼ m 4 KK . However, as we explain below, such a result comes from the fact that the UVsensitive terms proportional to powers of q are artificially washed out due to an improper way of treating the asymptotics of the loop momentum.  \nIn this respect, we now show that the interpretation of the 5D theory as a 4D one with an infinite tower of states (if pushed too far) is misleading. Within this interpretational framework, in fact, it is natural to consider that the correct thing to do is to sum the infinitely many ( n → ∞ ) Coleman-Weinberg one-loop contributions brought by each of the towers. Any reference to the original 5D loop", '8 C. Branchina, V. Branchina, F. Contino and A. Pernace': 'momentum ˆ p (and a fortiori to the physical meaning of n ) is lost. If on the contrary we correctly focus on the dynamical origin of the KK states, and recognize them as different momentum eigenstates that appear in the Fourier expansion of the 5D field ˆ Φ( x, z ), it is clear that sending n →∞ while keeping the modulus p of the other four components fixed is unphysical. If our universe has compact extra dimensions, lowenergy 4D physical observables emerge from the piling up of quantum fluctuations above the compactification scale. In implementing such a dressing, it is clear that the components of the loop momenta must be treated in a consistent way, actually on an equal footing. We will further comment on this point later.  \nTo better read the result (25), we stress that it contains four kinds of terms: (i) m - and q -independent UV-sensitive terms; (ii) UV-sensitive terms that depend only on m ; (iii) q -dependent UV-sensitive terms; (iv) UV-insensitive terms. As stressed above, in SUSY theories boson and fermion superpartners have the same mass m , while the boundary charges q are necessarily different to trigger the Scherk-Schwarz mechanism. Therefore:  \n(a) in SUSY theories supersymmetry enforces cancellations between superpartners of all but the q -dependent terms in (25), so that for each supermultiplet the dominant contribution to ρ 4 is controlled by the SUSY breaking parameter q 2 b -q 2 f , and is  \nρ 4 ∼ ( q 2 b -q 2 f ) R 2 e 6 αφ R Λ 3 = ( q 2 b -q 2 f ) m 2 KK R Λ 3 ; (28)  \n(b) in non-supersymmetric theories, each of the higher dimensional fields (each tower in 4D language) gives to the vacuum energy the dominant (uncancelled) contribution  \nρ 4 ∼ e 2 αφ R Λ 5 log Λ 2 e 2 αφ µ 2 = m 2 / 3 KK R 5 / 3 Λ 5 log ( m KK R ) 2 / 3 Λ 2 µ 2 . (29)  \nTherefore, even in the light tower limit m KK → 0 (large negative values of φ ), by no means the UV-insensitive R 4 ∼ m 4 KK term in (25) can overthrow these dominating contributions.  \nThe main point that emerges from our result (25) is that q -dependent UVsensitive terms always arise when non-trivial boundary conditions on multiplyconnected manifolds are realized (necessary for instance to implement the ScherkSchwarz mechanism in SUSY theories), independently of the size of the extra dimensions. On the contrary, the UV-insensitive terms in the rest R 4 originate from the discreteness of the momentum along the circle, generated in the present case by the hierarchy between the size of the 4D box and the radius of the circle.', '3. Vacuum Energy and Dark Dimension': "As already stressed, taking a 5-dimensional supersymmetric EFT with one compact dimension in the shape of a circle of radius R , and performing the calculation in  \nDoes the Cosmological Constant really indicate the existence of a Dark Dimension? 9  \nthe usual manner, for the vacuum energy ρ 4 at the one-loop level we have  \n( ρ 4 ) 1 4 = ( ∑ i R ( i ) 4 ) 1 4 ∼ C m KK , (30)  \nwhere R ( i ) 4 is of the kind (26) (see also (27)). The sum over i includes all the bosonic and fermionic contributions, and C is an O (1) coefficient d . The DD proposal [4] is based on the assumption that in the asymptotic region of the string moduli space where the EFT emerges, the vacuum energy ρ 4 goes as m 4 KK , where m KK is a KaluzaKlein mass (of order the neutrino scale) given by the cosmological constant. The authors argue that the EFT result (30) supports their proposal.  \nHowever, we have shown that (30) results from an improper treatment of the asymptotics of the loop momentum in the original 5D theory, and that the correct result for ρ 4 is given by (25) (with q -independent terms cancelled by SUSY). From this latter equation we see that for a supersymmetric theory with SUSY breaking parameter q 2 b -q 2 f the dominant contribution to ρ 4 is (28), i.e. it goes as m 2 KK R Λ 3 . Far from being UV-finite as m 4 KK , this term is strongly UV-sensitive. We also see that, for a non-supersymmetric theory the term that dominates ρ 4 is (29): it scales with m KK as m 2 / 3 KK log m KK and is UV-sensitive as Λ 5 log Λ. In both cases the expected m 4 KK result is not recovered.  \nTherefore, it seems that the DD proposal can hardly be considered a physical reality. One might think that a possible way to escape from such a conclusion is to admit that what in the literature is usually called EFT limit of string theory, framework in which the DD proposal is formulated, actually gives rise to a new type of Effective Field Theory, far from what is usually intended by the community. At present, however, there is no hint for that, and, as stressed by the authors of [4] themselves, we all know that around and above the neutrino scale physics is well described by the EFT paradigm in the usual and well-known sense.  \nIn this respect we also observe that to calculate the contribution of a KK tower to ρ 4 , in particular to study its UV (in)sensitivity, we could resort to the species scale cutoff Λ sp [32,33], as it is sometimes done within the swampland program (see for instance [34,35,36,37,38]). Referring to our previous examples, we consider the case of a massless 5D field, where the masses of the tower states are given by (20) with m 2 φ = 0. The number N of states with mass below Λ sp is  \nN ≡ n max + | n min | +1 , (31)  \nwith n max and n min solutions of ( n + q ) 2 /R 2 φ = Λ 2 sp , i.e.  \nn max = R φ Λ sp -q ; n min = -R φ Λ sp -q . (32)  \nThe explicit calculation is performed in the Appendix.  \n/negationslash  \nWe stress that when q = 0, cutting the sum over n in (21) with (31) and (32) and the integral over the four-momentum p with Λ sp is not equivalent to the introduction of a cut on the 5D loop momentum ˆ p . As repeatedly underlined in the present work, the latter is the (physically) correct cut to apply. From the calculations in the Appendix we see that, when the cut is imposed on the combination ( n + q ) 2 /R 2 φ rather than e n 2 /R 2 φ , an artificial washout of q -dependent UV-sensitive terms is operated. This again comes from a mistreatment of the 5D loop momentum asymptotics. Similarly to what we have already seen, the application of the Λ sp cut pushes too far the interpretation of the KK modes as massive states of the 4D theory, losing sight of the original physical meaning of n .  \nThese issues were also discussed in [23], where it was shown in full generality that the inclusion of the boundary charge q in the cut (whatever kind of cut) is at the origin of the artificial washout of the q -dependent UV-sensitive terms. In this respect, it is worth to stress that performing the infinite sum while keeping Λ fixed is equivalent to include q in the cut. As explained in the present work, both are physically illegitimate operations. The proper time [30], Pauli-Villars [20], and thick brane [21] regularizations all implement the insertion of q in the cut over n , thus realizing the artificial washout mentioned above. It is worth to point out that the use of dimensional regularization (DR), as done for instance in [27,31], does not help to cope with this kind of issues. By construction, in fact, DR does not detect the full UV-sensitivity of a theory, since in this regularization power 'divergences' are automatically cancelled (see [39] for a careful analysis of DR in comparison with other regularizations). We also note that DR totally masks the presence of UV-sensitive terms in odd dimensions, leading sometimes to the impression that no 'divergences' appear in that case.", '4. Compact dimensions and EFTs': 'The result ρ 4 ∼ m 4 KK (more generally ρ d ∼ m d KK ) is sometimes used to argue for a possible general breakdown of EFT methods. On our side, referring to the original (4 + n ) D theory (with n compact dimensions), we have shown in the previous sections that the EFT approach is perfectly suited to theories with compact extra dimensions, and found for ρ 4 the radically different result (25). An interesting point of view on these issues has been recently given in [27], where the result ρ 4 ∼ m 4 KK is taken for granted but it is argued that it cannot be used as a signal of general departure from the EFT approach.  \nTheir argument goes as follows. Consider a tower of states with mass spectrum m n = f n µ tow . Cutting the sum at n = N means that we include in the theory KK modes up to the N -th one, and exclude the states from the ( N +1)-th up to infinity. In general, integrating out a (finite) set of fields to define a low energy  \nEFT for the lighter ones is a consistent operation only if there is a large mass hierarchy between the fields included and those excluded. When a KK tower is cut, the hierarchy between the heaviest state included and the lightest one excluded is given by f N /f N +1 . Being this ratio O (1) (except for the case N = 0, that defines the 4D EFT), the authors conclude that no EFT estimate with a finite number of KK states can ever be done, and that the (4 + n ) D theory must necessarily contain the infinite tower.  \nThe observation that an infinite tower of massive states cannot be divided into heavy and light fields to define an EFT for the latter ones is certainly true and interesting in its own right. In our opinion, however, this line of reasoning does not apply to higher dimensional theories with compact extra dimensions. In fact, sticking for concreteness to a higher dimensional 5D theory, we should keep in mind that the KK modes are momentum eigenstates of the original 5D fields (and not an infinitely numerable set of massive 4D fields), and that the 5D theory from which the 4D theory derives is an EFT itself. As for any EFT, this means that the 5D momentum ˆ p ≡ ( p, n/R ) in the loops has to be cut at the scale where the theory loses its validity. The fifth component n/R of ˆ p cannot be disentangled from the other four components, p ≡ ( p 1 , p 2 , p 3 , p 4 ), so that the cut in the KK states results from a physically necessary requirement. No large hierarchy between included and excluded momentum modes is ever needed.  \nStarting from the 5D action S (5) Λ , and integrating out the modes in the range [ k, Λ], one obtains the action S (5) k at the lower scale k . Due to the discreteness of p 5 = n/R , the contribution from the related eigenmodes comes in a stepwise fashion. For k < 1 /R no such eigenmodes appear any longer, and the RG evolution becomes effectively of 4D type. It is only in this sense that the 4D theory emerges from the 5D one, and no 4D theory with an infinite tower can ever give an accurate description of the original 5D theory.', '5. Summary and Conclusions': 'In the present work we analysed the recent dark dimension proposal [4], according to which the tiny measured value of the cosmological constant might signal the presence of a single compact extra dimension of mesoscopic size (order µm or so). Moving from swampland arguments, this proposal is based on the idea that the cosmological constant fixes the scale m KK of a KK tower (of order the neutrino scale), and relies on the (widely believed) result ρ EFT ∼ m 4 KK for the vacuum energy of the underlying higher dimensional EFT with one compact extra dimension.  \nAccording to [4], both the relation ρ swamp ∼ m 4 KK (that comes from swampland conjectures formulated in an asymptotic corner of the quantum gravity landscape) and the corresponding result for the vacuum energy coming from the EFT limit are at the basis of the dark dimension scenario. Leaving aside problems that might arise in the string theory framework itself, in the present work we have shown that, due to the presence of previously missed UV-sensitive terms in ρ EFT (proportional  \nto the q charges), the matching between ρ swamp and the corresponding ρ EFT is a delicate issue. In order for such a matching to be realised, a physical mechanism that implements the suppression of the aforementioned UV-sensitive terms is needed.  \nIn fact, contrary to what is typically done in the literature [20,21,29,30,31], we have stressed that in the one-loop calculation of ρ EFT the five components of the 5D loop momentum ˆ p = ( p 1 , p 2 , p 3 , p 4 , n/R ) have to be cut in a coherent manner (for simplicity we stick to the compactification on a circle of radius R , for which p 5 = n/R ). In the usual calculations, where the result ρ EFT ∼ m 4 KK is automatically obtained, the asymptotics of ˆ p are mistreated, and this produces an artificial washout of UV-sensitive terms. Actually, the finite result ρ EFT ∼ m 4 KK comes from an incorrect way of implementing the piling up of the quantum fluctuations. For example, for a 5D supersymmetric model with Scherk-Schwarz SUSY breaking, a UV sensitivity with dominant term m 2 KK R Λ 3 is generated (see (28)), while for a non-SUSY model the dominant UV-sensitive term is m 2 / 3 KK R 5 / 3 Λ 5 log( m 1 / 3 KK R 1 / 3 Λ) (see (29)).  \nWe have also shown that inconsistencies in the way of treating the asymptotics of the loop momenta appear even when the physical cut is implemented resorting to the species scale Λ sp . In fact, when non-vanishing boundary charges q are present (as it is the case with the SUSY breaking Scherk-Schwarz mechanism), the straightforward exclusion of the KK masses (20) above Λ sp does not implement a physically acceptable cut on the fifth component p 5 of the 5D loop momentum ˆ p . Naturally, what we have just said is not in conflict with the existence of the natural physical cutoff Λ sp , that originates from the coupling of N species of fields with gravity, and correctly embodies the scale where gravity itself becomes strong and the EFT language can no longer be used [32,33]. We rather formulate a warning on possible blind uses of Λ sp in KK theories.  \nTo conclude, the results of the present paper indicate that the recently proposed dark dimension scenario, at least the way it has been originally formulated in [4] and further implemented in several applications where the usual (but improper) way of implementing the loop calculation in EFTs are heavily used [40,41,42,43,44, 45,46], requires a great care when the swampland conjectures (string theory side) are confronted with the corresponding EFT limit results. We hope to come back to these issues in the near future.', 'Acknowledgments': "The work of CB has been partly supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2022R1A2C2003567), and partly by the European Union - Next Generation EU through the research grant number P2022Z4P4B 'SOPHYA - Sustainable Optimised PHYsics Algorithms: fundamental physics to build an advanced society' under the program PRIN 2022 PNRR of the Italian Ministero dell'Universit'a e Ricerca (MUR). The work of VB, FC and  \nDoes the Cosmological Constant really indicate the existence of a Dark Dimension? 13  \nAP is carried out within the INFN project QGSKY.", 'Appendix A.': "In this appendix we perform the calculation of the vacuum energy ρ 4 using the species scale cutoff Λ sp . In a 4D theory with N particle states, Λ sp = M p / √ N . In a 5D theory with one compact dimension the identification of Λ sp is done counting the number of KK states that respect the condition m 2 n ≤ Λ 2 sp . The inequality is saturated when  \nm 2 φ + ( n + q R φ ) 2 = Λ 2 sp → n ± = [ -q ± R φ √ Λ 2 sp -m 2 φ ] , (A.1)  \nwhere n ± reduce to n max and n min of (32) in the text when m φ = 0, and the brackets [ ... ] indicate 'integer part' (that in the following we neglect for simplicity). The number of states between n + and n -is  \nN = n + + | n -| +1 = 2 R φ √ Λ sp -m 2 φ +1 (A.2)  \nand the species scale is then obtained as  \nΛ sp = a 3 + X 1 / 3 3 · 2 1 / 3 -2 1 / 3 ( 3 b -a 2 ) 3 · X 1 / 3 (A.3)  \nX = 3 √ 3 √ 4 a 3 c -a 2 b 2 -18 abc +4 b 3 +27 c 2 +2 a 3 -9 ab +27 c (A.4)  \nwith  \nand  \na = m 2 φ + 1 4 R 2 φ ; b = M 2 p 2 R 2 φ ; c = M 4 p 4 R 2 φ . (A.5)  \nExpanding for m φ , R -1 φ /lessmuch M p , we get  \nΛ 2 sp = M 4 / 3 p (2 R φ ) 2 / 3 -M 2 / 3 p 3 (2 R 4 φ ) 1 / 3 + m 2 φ + 1 4 R 2 φ 3 + O ( M -2 / 3 P ) . (A.6)  \nThe first term of this expansion is the one typically referred to in the literature, where only a rough estimate of Λ sp is reported (see for instance [35]).  \nThe contribution of a bosonic (or fermionic, adding an overall minus sign) tower to the vacuum energy is  \nρ 4 ∼ n + ∑ n = n -∫ (Λ sp ) d 4 p (2 π ) 4 log p 2 + ( n + q ) 2 R 2 φ + m 2 φ µ 2 , (A.7)", '14 C. Branchina, V. Branchina, F. Contino and A. Pernace': "where the upper case (Λ sp ) in the integral means that the modulus of the fourdimensional momentum is cut at Λ sp . Performing the integration over p we find  \nρ 4 = 1 64 π 2 n + ∑ n = n -{ -Λ 4 sp +2Λ 2 sp ( m 2 φ + ( n + q ) 2 R 2 φ ) +2 ( m 2 φ + ( n + q ) 2 R 2 φ ) 2 log   m 2 φ + ( n + q ) 2 R 2 φ Λ 2 sp + m 2 φ + ( n + q ) 2 R 2 φ   +2Λ 4 sp log   Λ 2 sp + m 2 φ + ( n + q ) 2 R 2 φ µ 2   } ≡ n + ∑ n = n -G ( n ) . (A.8)  \nAs in the text (see (23), (24) and the text therein, where B i and B i ( x ) are defined), the sum can be calculated by means of the EML formula,  \nρ 4 = ∫ n + n -dxG ( x ) + G ( n + ) + G ( n -) 2 (A.9) + r ∑ j =1 B 2 j (2 j )! ( G (2 j -1) ( n + ) -G (2 j -1) ( n -) ) + R 2 r  \nR 2 r = ∞ ∑ k = r +1 B 2 j (2 j )! ( G (2 j -1) ( n + ) -G (2 j -1) ( n -) ) = ( -1) 2 r +1 (2 r )! ∫ n + n -dxG (2 r ) ( x ) B 2 r ( x -[ x ]) . (A.10)  \nIn the physically meaningful limit m φ , R -1 φ /lessmuch Λ sp , the result for the vacuum energy is  \nρ 4 = 20 log ( 4 M 2 p 5 µ 3 R φ ) +12 π -57 2 -1 / 3 · 3840 π 2 R 2 / 3 φ M 10 / 3 p + -4 log ( 4 M 2 p µ 3 R φ ) -6 π +27 2 -2 / 3 · 2304 π 2 R 4 / 3 φ M 8 / 3 p + 12 π -35 4608 π 2 R 2 φ M 2 p + ( 4 m 2 φ R 2 φ +1 ) log ( M 2 p 2 µ 3 R φ ) -3(5 -4 π ) m 2 φ R 2 φ 1152 π 2 R 2 φ M 2 p + -20 log ( M 2 p µ 3 R φ ) -120 π +309 + 104 log 2 2 -1 / 3 · 124416 π 2 R 8 / 3 φ M 4 / 3 p + 3(19 -8 π ) -4 log ( 4 M 2 p µ 3 R φ ) 2 -1 / 3 · 3456 π 2 R 8 / 3 φ ( m φ R φ ) 2 M 4 / 3 p + 525 π +367log2 -1953 + 35 log ( M 2 p µ 3 R φ ) 2 -2 / 3 1866240 π 2 R 10 / 3 φ M 2 / 3 p  \nwhere  \nDoes the Cosmological Constant really indicate the existence of a Dark Dimension? 15  \n+ 9 log ( M 2 p µ 3 R φ ) +135 π -432 + 99 log 2 2 -2 / 3 46656 π 2 R 10 / 3 φ m 2 φ R 2 φ M 2 / 3 p + 2 log ( M 2 p µ 3 R φ ) +3 π -30 -14 log 2 2 -2 / 3 1728 π 2 R 10 / 3 φ m 4 φ R 4 φ M 2 / 3 p + 61 -18 π +40(17 -6 π ) m 2 φ R 2 φ +80(33 -9 π ) m 4 φ R 4 φ 138240 π 2 R 4 φ (A.11)  \n+ m 5 φ R φ 60 π + R 4 + O ( M -2 / 3 P ) ,  \nwith R 4 given in (26), (27).  \nA few comments are in order. Limiting ourselves to the leading order relation Λ sp ∼ R -1 / 3 φ M 2 / 3 p (see (A.6)), we observe that the powers M 10 / 3 p , M 2 p and M 2 / 3 p correspond to the powers Λ 5 , Λ 3 and Λ respectively in terms of a generic cutoff Λ. However, comparing the coefficients of M 2 p and M 2 / 3 p in (A.11) with the corresponding coefficients of Λ 3 and Λ in (25), we note that they have a different structure. Moreover, powers of M p other than those mentioned above (that do not find any correspondence in (25)) are also present. These differences have a twofold origin: they are due both to the fact that the species scale cut is cylindrical in 5D momentum space (see [23] for a thorough discussion on the difference between the implementation of a cylindrical and a spherical cutoff on the 5D momentum) and to the fact that, as per (A.6), the relation Λ sp ∼ R -1 / 3 φ M 2 / 3 p holds only at the leading (large M p ) order.  \nAn even more important difference between the results (25) and (A.11) is that the latter does not contain any UV-sensitive term proportional to the boundary charge q . Actually (A.11) comes from a physically illegitimate operation. In fact, rather than a cut on p 2 5 = e -2 βφ n 2 /R 2 , Λ sp implements a cut on the 'KK masses' m 2 n = m 2 φ + ( n + q ) 2 /R 2 φ . As discussed in the text, the (unphysical) introduction of the combination n + q in the cutoff is at the origin of the artificial washout of the q -dependent UV-sensitive terms. This makes the result (A.11), and more generally the introduction of the species scale cutoff in higher dimensional theories with compact extra dimensions, unreliable. The species scale cut only arises as a result of a too literal interpretation of the 5D theory in terms of a 4D theory with towers of massive 4D fields. These warnings do not apply to the case of a bona fide 4D theory with a large number N of fields coupled to gravity, where Λ sp truly is the quantum gravity physical cutoff.  \nIt is also worth to point out that (A.11) provides an example of a hard cutoff calculation where no q -dependent UV-sensitive terms are generated. In previous literature the opinion was widely expressed that the use of a hard cutoff was at the origin of UV-sensitive terms, that were considered as spurious [20,21,47]. The above result shows that the presence of these terms is rather due to a correct treatment  \n- 16 C. Branchina, V. Branchina, F. Contino and A. 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