In 1 we pointed out that in the Dark Dimension scenario 2 theoretical issues arise when the prediction for the vacuum energy rho that is obtained from swampland conjectures in string theory is confronted with the corresponding result for rho in the effective field theory EFT limit. One of the problems concerns the widely spread belief that in higher dimensional EFTs with compact dimensions the vacuum energy is automatically finite. On the contrary our analysis shows that rho contains previously missed UVsensitive terms. Our work was challenged in 3. Here we show why in our opinion the claims in 3 are flawed and provide further support to our findings. We conclude presenting ideas on the physical mechanism that should dispose of the large UV contributions to rho.
['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Phenomenology']
Dark Dimension and the Effective Field Theory limit
{'No Header': '- International Journal of Geometric Methods in Modern Physics © World Scientific Publishing Company', 'DARK DIMENSION AND THE EFFECTIVE FIELD THEORY LIMIT': 'CARLO BRANCHINA 1 2 † , VINCENZO BRANCHINA 3 ∗ , FILIPPO CONTINO 3 ‡ , ARCANGELO PERNACE 3 ⋆  \n1  \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 ⋆ arcangelo.pernace@ct.infn.it  \nReceived (Day Month Year) Revised (Day Month Year)  \nIn [1] we pointed out that in the Dark Dimension scenario [2] theoretical issues arise when the prediction for the vacuum energy ρ , that is obtained from swampland conjectures in string theory, is confronted with the corresponding result for ρ in the effective field theory (EFT) limit. One of the problems concerns the widely spread belief that in higher dimensional EFTs with compact dimensions the vacuum energy is automatically finite. On the contrary, our analysis shows that ρ contains (previously missed) UV-sensitive terms. Our work was challenged in [3]. Here we show why in our opinion the claims in [3] are flawed, and provide further support to our findings. We conclude presenting ideas on the physical mechanism that should dispose of the large UV contributions to ρ .  \nKeywords : dark energy; Kaluza-Klein theories; dark dimension.', '1. Introduction': 'In search for a solution to the naturalness/hierarchy problem, in the nineties the possibility of having large extra dimensions (millimetre size) was considered [4,5]. Recently, these ideas were applied to the so-called dark dimension (DD) scenario [2], according to which we might well live in a (4 + 1)D universe with one compact dimension of micrometre size. In a nutshell a , the DD proposal [2] combines swampland conjectures with observational data and phenomenological bounds, and is based on', '2 C. Branchina, V. Branchina, F. Contino and A. Pernace': "the assumption that our universe lies in a unique (asymptotic) corner of the quantum gravity landscape, where only two kinds of light tower states are expected: towers of string excitations or towers of KK states. Indicating with µ tow the tower scale, and with Λ cc the measured vacuum energy (cosmological constant times M 2 P , with M P the Planck mass), the application of the distance conjecture to (A)dS vacua relates the two as  \nµ tow ∼ ∣ ∣ ∣ ∣ Λ cc M 4 P ∣ ∣ ∣ ∣ α M P . (1)  \nFurthermore, combining indications from one-loop string calculations with the Higuchi bound [6], the proponents of the DD scenario restrict the parameter α to the range  \n1 4 ≤ α ≤ 1 2 . (2)  \nThen, considering bounds set by the non-observation of deviations from Newton's force law, they infer that µ tow is close to the energy scale associated to the cosmological constant (that coincides with the neutrino scale)  \nµ tow ∼ Λ 1 / 4 cc ∼ meV . (3)  \nThey also observe that, for (1) to be consistent with (3), it is necessary to take α at the lower bound of (2), α = 1 / 4.  \nThe authors then make the following crucial observation. Let us quote from [2]: 'since we can describe physics above the neutrino scale with effective field theory' (EFT), the string scenario is 'ruled out experimentally'. They then conclude that the light tower must be a KK tower, µ tow = m KK , so that (1) becomes  \nΛ cc ∼ m 4 KK . (4)  \nFinally, to determine the number of extra compact dimensions, they compare bounds on m KK that come from calculations of the heating of neutron stars due to the surrounding cloud of trapped KK gravitons [7], [8], [9] with the value of m KK in (4). This allows them to conclude that there is only one extra dimension of size ∼ µ m.  \nIn a recent paper [1], we analysed aspects of this scenario, and pointed out possible theoretical issues. Below we briefly recall the scope and the main results of [1]. As already said, according to the proponents of the DD scenario, above the m KK scale our universe is described by a (4+1)D EFT with one compact dimension of micrometer size. This proposal has triggered a large amount of work, where (4 + 1)D EFT models that could realize the DD scenario are proposed, and their phenomenological consequences studied (see for instance [10,11,12,13,14,15,16,17, 18,19,20]). In [1], our goal was to investigate on the matching between the string theory framework in which the basic relation (4) of the DD proposal is obtained and its low energy EFT limit.  \nTo this end, we considered a supersymmetric (4+1)D theory with ScherkSchwarz SUSY breaking (and also a non-supersymmetric theory) with compact dimension in the shape of a circle b of radius R and calculated the one-loop vacuum energy ρ 1 l . According to existing literature (see for instance [21,22,23,24], and for the analogous calculation of the effective potential [25,26]), in (4 + n )D theories with n compact dimensions ρ 1 l should be finite and proportional to m 4 KK ,  \nρ 1 l ∼ m 4 KK , (5)  \nwith no need for fine-tuning. In [1] we showed that the right hand side of (5) also contains UV-sensitive terms (see the first line in the right hand side of Eq. (6) below), that were missed in previous literature c . Therefore, the matching between the prediction (4) for the vacuum energy Λ cc , that comes from swampland conjectures in string theory, and the corresponding result in the low energy field theory limit is not as straightforward as it was thought on the basis of the result (5). In this respect, it is worth to stress again (see the paragraph below Eq. (3)) that such a matching is crucial for the DD scenario.  \nIn [1] we show why it is commonly thought that ρ 1 l is finite and proportional to m 4 KK : this result comes from a mistreatment of the loop momentum asymptotics. Performing the calculation properly, we found that (in addition to the finite term ∼ m 4 KK and to divergences that cancel out in SUSY theories) the one-loop contribution to the vacuum energy ρ 1 l contains 'UV-sensitive' terms proportional to powers of the boundary charges. These divergences do not disappear, not even in a SUSY theory. This is due to the fact that, to trigger the Scherk-Schwarz mechanism, the boundary charges q b and q f of the boson and fermion superpartners need to be different.  \nOur work generated a debate that finally resulted in [3], where our results were challenged. Below we present a deep analysis of the arguments given in [3], and show why in our opinion these authors (in their attempt to confute our results) bring confusion on the whole subject using flawed arguments that lead them to incorrect conclusions. In view of the great wealth of theoretical and phenomenological work triggered by the DD scenario, we believe it worth trying to bring clarity on the subject and continuing its investigation. In the course of this analysis, we will bring new arguments that further support the conclusions of our previous work.  \nThe rest of the paper is organised as follows. In section 2, we deeply investigate the arguments brought by the authors of [3] against our work, showing that they misuse our results and draw incorrect conclusions. Moreover, we present the global", '4 C. Branchina, V. Branchina, F. Contino and A. Pernace': 'framework in which the DD scenario is framed, that allows to better understand the flaws contained in the claims made in [3]. In section 3, we show that arguments typically used in the literature (including [3]) to support the widely spread belief that in higher dimensional theories with compact dimensions the vacuum energy is automatically finite are misleading. In section 4, we present our conclusions, together with ideas on the physical mechanism that in our opinion should dispose of the large UV contributions to ρ 1 l .', '2. Global framework of the DD scenario and misuse of our results': "As anticipated above, our results were challenged by the authors of [3]. Below we show that they misuse and distort our findings, and how this leads them to incorrect conclusions. Moreover, we present the global framework in which the DD scenario is framed (see Fig. 1), which further helps in bringing clarity on the whole subject. Before doing that, however, it is worth mentioning that [3] starts with the claim that we question the swampland relation (4), while what we did in [1] is totally different, and this was far from being our goal (see the Introduction above).  \nOur results for ρ 1 l are in Eqs. (25) and (26) of [1]. For ease of presentation (and with no loss of generality), from now on we concentrate on the SUSY theory case only (unless expressly indicated otherwise), and write the contribution to ρ 1 l from a single couple of superpartners for the massless case (i.e. taking m = 0 in (25) and (26) of [1])  \nρ 1 l = ( q 2 b -q 2 f ) Rm 2 KK 60 π 2 Λ 3 -( q 4 b -q 4 f ) R 1 3 m 10 3 KK 280 π 2 Λ + 3 [ Li 5 ( e 2 πiq b ) -Li 5 ( e 2 πiq f ) + h.c. ] 128 π 6 m 4 KK , (6)  \nwhere q b and q f are the boson and fermion boundary charges respectively, and Li 5 ( x ) is the polylogarithm function Li n ( x ) for n = 5. Λ is the physical UV cutoff of the (4+1)D theory defined by the action S (4+1) = S (4+1) grav + S (4+1) matter , with (4+1)D metric ˆ g MN parametrized in terms of the graviphoton A µ and the radion field ϕ (see Appendix B, and equations (6)-(12) in [1]). The KK scale m KK is given in terms of the radion field vev (here indicated with ϕ )  \nm KK = e √ 3 2 ϕ M P R -1 ≡ R -1 ϕ . (7)  \nTo be phenomenologically viable, the (4 + 1)D model that implements the DD scenario has to include the SM on a 3-brane. An important consequence is that the physical cutoff Λ of the (4 + 1)D theory is an upper bound for the SM cutoff Λ SM (the derivation of Eq. (8) below is in Appendix B)  \nΛ SM ≤ Λ . (8)  \nAs we will see, this result is important for our analysis. Let us also note that m KK is the cutoff of the 'reduced 4D theory', which besides brane fields contains the zero  \n5  \nmodes of bulk fields. Moreover, it is worth to observe that, although both the Standard Model and this 'reduced theory' are 4-dimensional EFTs, they are certainly not the same theory, and their cutoffs (Λ SM and m KK respectively) should not be confused. In Fig. 1 we provide a comprehensive picture of the global framework in which the DD scenario is framed, described in detail in the caption below.  \nIn Eq. (6), the divergences (i.e. UV-sensitive terms) are in the first line, the finite term in the second one. Limiting ourselves to consider only the most divergent term of ρ 1 l (and doing the same for the non-SUSY theory, whose result for ρ 1 l is not reported above but can be found in our work [1]), we have  \nρ 1 l ∼ m 2 KK R Λ 3 and ρ 1 l ∼ m 2 / 3 KK R 5 / 3 Λ 5 , (9)  \nfor the SUSY and the non-SUSY case respectively. These latter equations are Eqs. (28) and (29) of [1]. As stressed above, one of the main goals of our work [1] is to point out that ρ 1 l contains previously unnoticed divergences (that do not disappear even in a SUSY theory) and that, before comparing ρ 1 l with the measured value of the vacuum energy Λ cc , a fine-tuning is needed.  \nFor the purposes of the present analysis, it is of fundamental importance to stress that the authors of [3] write our Eqs. (9) as (see Eqs. (4) of [3])  \nΛ cc ∼ m 2 KK R Λ 3 and Λ cc ∼ m 2 / 3 KK R 5 / 3 Λ 5 , (10)  \nthus operating in (9) the replacement  \nρ 1 l → Λ cc . (11)  \nSo doing, they take the result ρ 1 l of our one-loop calculation in [1] to directly coincide (without any fine-tuning) with the measured value of the vacuum energy.  \nAlthough this is opposite to what we did (we repeat ourselves, in [1] our goal was to point out the presence of unexpected UV-sensitive terms), the replacement (11) is not in itself a problem. It is a theoretically legitimate possibility that the cutoff Λ is a physical scale that appears in physical quantities, with no need for fine-tuning. However, the consequences of such a choice have to be fully taken into account. Anticipating on the results, we will show that the replacement ρ 1 l → Λ cc has crucial physical consequences that the authors of [3] failed to recognize. It is precisely this replacement that leads them to incorrect conclusions. Let us note that, for reasons that will be soon apparent, the right hand side of (10) 1 should contain all the terms that appear in the right hand side of (6), not only the first one. However, since this has no impact on the present discussion, we can equivalently refer to (10) 1 or more completely to (6).  \nIt is now crucial to realise that, once ρ 1 l is chosen to coincide with the measured vacuum energy Λ cc , Eqs. (10) completely determine Λ. As we will see, the authors of [3] base their analysis on Eqs. (10) and, failing to realise that they do not have this freedom, freely play with the cutoff Λ. Below we determine the value of Λ imposed by their choice (11), and investigate on the possibility that the resulting Λ can really play the role of the theory UV cutoff. For this to be true, Λ cannot  \nFig. 1. Pictorial representation of the energy scales and theories involved in the Dark Dimension scenario. The grey region indicates a possible UV-completion of the (4 + 1)D theory between its (rescaled, see text) physical cutoff Λ ϕ and the string scale M s . A particular case could be that Λ ϕ directly coincides with M s . Below Λ ϕ (that coincides with the Standard Model physical cutoff Λ SM , see text) the (4 + 1)D theory of the DD scenario (blue region), that contains bulk fields and SM (with possibly other) brane fields, takes place. Below m KK , the theory is well described by a 4 D EFT (green region) containing the 0-modes of the bulk fields together with brane fields. The components p and p 5 of the (4 + 1)D momentum ˆ p ≡ ( p, p 5 ), with p ≡ ( p 1 , p 2 , p 3 , p 4 ) and p 5 the fifth component, are reported. The fully colored blue and green rectangles indicate that p takes continuous values (non-compact dimensions), while the blue lines indicate that p 5 takes discrete values starting at m KK (compact dimension).  \n<!-- image -->  \nviolate well-tested phenomenology, in particular it should not enter in conflict with the Standard Model.  \nLet us determine now the value of Λ imposed by (11). First of all, we recall that from (4) we have m KK ∼ Λ 1 / 4 cc . A simple inspection of (10) 1 (or (6) with the replacement (11)) shows that this implies d  \nR Λ 3 ∼ m 2 KK ∼ (meV) 2 . (12)  \nThe above relation is the unavoidable consequence of (11). Starting from (12), it is now easy to show that the whole construction put forward in [3] is flawed. In fact, using the relation ( ˆ M P is the five-dimensional Planck scale)  \nˆ M 3 P = (2 πR ) -1 M 2 P , (13)  \nd The same is true for (10) 2 .  \nEq. (12) is straightforwardly rewritten as  \nΛ 3 ∼ ( m 2 KK M 2 P ) ˆ M 3 P , (14)  \nfrom which we obtain (remember that m KK ∼ meV)  \nΛ ∼ 10 -21 ˆ M P ≤ 10 -2 GeV , (15)  \nwhere the last inequality stems from ˆ M P ≤ M P . Eq.(15) seems to set a too low upper bound on the cutoff Λ of the (4 + 1)D theory. We will see in a moment that this is really the case.  \nThe problem comes from the relation between Λ and the 4D cutoff Λ SM of the Standard Model, that, as already said, lives on a 3-brane. In Appendix B (to which we refer for the calculation) we show that Λ SM is given by  \nΛ SM = Λ ϕ ≡ e 1 √ 6 ϕ M P Λ , (16)  \nwhere ϕ is the vev of the radion field (see comments above Eq. (7) and Appendix B) and Λ ϕ is the (4 + 1)D cutoff for the momenta rescaled by the factor e 1 √ 6 ϕ M P , that are nothing but the momenta from the 4D observer perspective (see Eq. (B.9) in Appendix B). Since the asymptotic scenario of the DD proposal is realized for negative values of ϕ (see Appendix A), Eqs. (15) and (16) imply  \nΛ SM < 10 -2 GeV , (17)  \nan unacceptably low upper bound on Λ SM .  \nWe can go further and determine the value that Λ SM should have if the replacement ρ 1 l → Λ cc operated by the authors of [3] could be accepted. In fact, observing that the species scale Λ sp [29,30] is given by  \nΛ sp ∼ m 1 / 3 KK M 2 / 3 P ∼ e 1 √ 6 ϕ M P ˆ M P (18)  \n(see for instance [31] and the Appendix of [1]), comparing (16) with (18) and using (15) we immediately have  \nΛ SM ∼ Λ ˆ M P Λ sp ∼ 10 -21 Λ sp . (19)  \nFinally, since from (18) we have Λ sp ∼ 10 9 GeV,  \nΛ ϕ = Λ SM ∼ 10 -12 GeV = 10 -3 eV , (20)  \na too low value, as already known from (17).  \nEq. (20) definitely shows that what the authors of [3] do when they present our Eqs. (9) as Eqs. (10) (that are nothing but (9) with the replacement ρ 1 l → Λ cc ), is physically not allowed. They operate an inconsistent replacement, and Eqs. (10) are deprived of physical meaning. The point is that the arguments they use in their attempt to confute our work completely depend on this replacement, actually cannot even be formulated without it. Their arguments are thus unfounded and inconsistent.", '8 C. Branchina, V. Branchina, F. Contino and A. Pernace': "There is a simpler way to obtain (20) for Λ SM . From (7) and (16) it is easily seen that R Λ 3 = R ϕ Λ 3 ϕ , so that Eqs. (10) can be written as  \nΛ cc ∼ m 2 KK R ϕ Λ 3 ϕ and Λ cc ∼ m 2 / 3 KK R 5 / 3 ϕ Λ 5 ϕ . (21)  \nTherefore, inserting R ϕ = m -1 KK in (21), we immediately find that Λ cc ∼ m 4 KK is obtained only if  \nΛ ϕ = Λ SM ∼ m KK ∼ 10 -3 eV , (22)  \nthat is nothing but (20). Eq. (22) (and equivalently Eq. (20)) shows a further drawback of the (illegitimate) replacement ρ 1 l → Λ cc . From this equation in fact we see that the latter forces the (4 + 1)D cutoff Λ ϕ to be squeezed upon m KK . This leaves no space to the (4 + 1)D EFT needed for the DD scenario, and one is only left with a 4D theory valid below m KK ( ∼ meV), above which the theory directly merges to its UV completion (see Fig. 1, where the choice Λ ϕ = m KK makes the blue region disappear). Therefore, if the replacement ρ 1 l → Λ cc operated in [3] could be accepted, this would close the door to the DD scenario. Certainly, this was not the original intention of the authors in [3].  \nThe above analysis shows that Eqs. (10) are physically inconsistent, and obviously no use can be made of them. In particular, they cannot be used to confute our results. Unfortunately, this is precisely what the authors of [3] do, thus developing untenable arguments and claims.  \nThe present section could end here. Nevertheless, in the hope that this can bring further clarity and help the reader gain more insight in the whole subject, we now provide a detailed analysis of their arguments, that (as already said) are based on the inconsistent Eqs. (10). To follow their presentation closely, below we consider both the SUSY and the non-SUSY cases, i.e. both their Eqs. (10) 1 and(10) 2 .  \nOne of the authors' first points is to compare the (A)dS distance conjecture (1), that for the reader's convenience we rewrite here as ( µ tow in (1) is clearly m KK )  \nm KK ∼ M 1 -4 α P Λ α cc , (23)  \nwith Eqs. (10), that again for convenience we write as  \nm KK ∼ ( R Λ 3 ) -1 / 2 Λ 1 / 2 cc ; m KK ∼ ( R Λ 3 ) -5 / 2 Λ 3 / 2 cc . (24)  \nThis comparison leads them to affirm that for the parameter α we claim the values α = 1 / 2 and α = 3 / 2, for the SUSY and non-SUSY case respectively.  \nNothing could be further from us. Such, allegedly ours, claim actually results from (i) the replacement (11) that the authors of [3] operate in our equations (9), thus obtaining Eqs. (10), and (ii) the comparison of these latter equations with (23). Since we know that Eqs. (10) cannot be written (simply do not exist), we never could have made such a claim. Having clarified this point, we further observe that even if for a moment we follow the authors' line of reasoning, it is immediate to see from (23) and (24) that the values α = 1 / 2 and α = 3 / 2 (for the SUSY and  \nnon-SUSY case, respectively) can be derived only if it is also verified that  \nR Λ 3 ∼ M 2 P . (25)  \nClosing for a moment our eyes on the phenomenological impossibility of having (20), what we have just found would mean that, for (1) and (10) to give α = 1 / 2 and α = 3 / 2 (for the SUSY and non-SUSY case, respectively), Eqs. (12) and (25) should both be fulfilled at the same time, which is obviously an absurd requirement.  \nLet us move now to another of the arguments developed in [3]. The authors consider the Higuchi bound e , that in the present case reads  \nΛ cc ≤ ( 3 2 M 2 P ) m 2 KK , (27)  \nand comparing (27) with (10) they get  \nR Λ 3 ≲ M 2 P and R Λ 3 ≲ 10 -48 M 2 P (28)  \nfor the SUSY and non-SUSY cases, respectively f . As we know, Eqs.(28) would not even exist without the inconsistent replacement ρ 1 l → Λ cc in our equations (9). Nevertheless, closing again an eye on this point, we now follow their reasoning. Concerning (28) 1 , they observe that this relation does not constrain R Λ 3 too strongly, while for (28) 2 they say it imposes a too strong constraint on Λ. This latter claim deserves a comment. Using the relation (13), Eq. (28) 2 can be written as  \nΛ ≲ 10 -16 ˆ M P . (29)  \nAt the same time, using again R Λ 3 = R ϕ Λ 3 ϕ (see comments above Eq.(21)), Eq. (28) 2 can also be written in terms of Λ ϕ as  \nΛ SM = Λ ϕ ≲ 10 5 m KK ∼ 10 2 eV . (30)  \nA simple look at (29) and (30) shows that the trouble with (28) 2 is not that it implies a too low value for the cutoff Λ (as stated in [3]), but rather that it implies a too low value for the SM cutoff Λ SM , as we see from (30). Yet another contradiction is found if we observe that, combining (16) with (30), at threshold (i.e. when in (30) Λ ϕ ∼ 10 2 eV) we have  \nΛ ≳ 10 2 eV . (31)  \nIn fact, few lines after their claim that (28) 2 implies a too low value for Λ, the authors of [3] make another claim, namely that the 'correct choice' for Λ is Λ = m KK (see below). As we have just seen, however, at threshold (28) 2 implies (31),  \ne We remind here that according to the Higuchi bound the physical mass M of a spin-2 particle in dS space and the physical value of the cosmological constant satisfy [6],  \nM 2 ≥ 2 3 Λ cc M 2 p . (26)  \nf To obtain (28) 2 , Λ cc ∼ 10 -120 M 4 P is used.  \nwhile Λ = m KK means Λ ∼ meV, that is five orders of magnitude below the lower bound in(31). The patent contradiction between these two claims of theirs can hardly be missed.  \nSuccessively, in [3] the authors make yet another use of their Eqs. (10). They observe that these equations still contain the radius R , and claim that the latter should be replaced by m -1 KK , so that additional powers of m KK appear in these equations. Before proceeding with our analysis, we must observe that, so doing, the authors miss the crucial radion dependence in the relation between R and m KK (see (7)).  \nAnyway, let us go on and follow their arguments. Invoking the UV-IR mixing occurring in string theory and/or quantum gravity, the authors of [3] say that the UV cutoff Λ of an EFT coupled to quantum gravity should depend on the IR scale, that in this case is m KK . They then observe that the largest possible cutoff is the species scale Λ sp ∼ m 1 / 3 KK M 2 / 3 P (see (18)), which in fact depends on m KK . Now, replacing in Eqs. (10) R with m -1 KK and Λ with Λ sp , they finally get g  \nΛ cc ∼ m 2 KK M 2 P and Λ cc ∼ m 2 / 3 KK M 10 / 3 P . (32)  \nFrom these results, they conclude that this first of their 'UV-IR mixing motivated' attempts is not satisfactory, as (32) 2 violates the Higuchi bound.  \nThen, still invoking the 'UV-IR mixing', they proceed by making another trial for Λ, that they eventually dub as the 'correct choice'. In this second attempt they insert in Eqs. (10) Λ = m KK (and R = m -1 KK , as for the previous case). A first thing worth to note in connection with this new 'trial' is that a simple look to Eqs. (10) shows that, since they are fixing all the scales to m KK , the right hand side of these equations is trivially proportional to m 4 KK . The authors of [3] seem to be satisfied by this result, and claim that with this 'correct choice' of Λ they are able to remove what they call the 'ambiguity' in the EFT calculation and fix what they consider as the unique cutoff required by UV-IR mixing. In our opinion, this is (at least) too cheap a way to get the result 'vacuum energy ∼ m 4 KK ', and to implement the string UV-IR connection in the field theory limit. But this is not the worst. What certainly kills the possibility of taking Λ = m KK is what we have already seen. This choice is not allowed as it implies a ridiculously small cutoff for the Standard Model h .  \nh It is probably worth to note that if instead of the authors' replacement R = m -1 KK one makes the correct replacement R ϕ = m -1 KK (see (7)), from (32) we have that Λ cc ∼ m 4 KK is obtained for Λ ϕ = m KK rather than Λ = m KK . In any case, both Λ ϕ = m KK and Λ = m KK are unacceptably", '3. Casimir energy and finite temperature field theory': "We finally comment on two additional arguments that the authors of [3] use to lend further support to their views i . Referring to the finiteness of string theory calculations, their goal is to give examples where finite results are obtained also in field theory. To this end, they consider (i) the calculation of the one-loop free energy F 1 l in finite temperature QFT, where the T -dependent contribution goes like F 1 l T ∼ T 4 with no need for any 'string regularization' (to use their words), and (ii) the calculation of the Casimir energy.  \nLet us begin with the finite temperature field theory case. As stressed in [32], despite their apparent similarity, there is a profound difference between the sum over the integer n that appears in F 1 l and the analogous sum in the calculation of the vacuum energy ρ 1 l in KK theories. In fact, both for ρ 1 l and F 1 l we have an expression of the kind (below d = 4 for the (4 + 1)D KK theory and d = 3 for finite temperature field theory)  \nρ 1 l , F 1 l ∼ 1 2 ∑ n ∫ d d p log( p 2 + m 2 + f n ) . (33)  \nFor the case of KK theories ( ρ 1 l ) the integer n is related to the fifth component of the (4 + 1)D loop momentum ˆ p , that is cut at Λ. Therefore n and p are intertwined between one another and are to be cut accordingly [1,32]. This in particular implies an upper bound on the integer n . As a result, our Eqs. (25) and (26) of [1], here reported for the massless SUSY case in Eq. (6), are obtained. For the finite temperature field theory case ( F 1 l ), on the contrary, the integer n and the 3 D momentum p are not intertwined. The d 3 p integral provides the trace over quantum fluctuations (and is cut at a certain momentum cutoff), while the ∑ n gives the statistical average (mixed states), and as such has to be performed all the way up to infinity to implement ergodicity. This infinite sum is at the origin of the finiteness j of the T-dependent contribution F 1 l T to F 1 l . In fact,  \nF 1 l T = T 2 ∞ ∑ n = -∞ ∫ d 3 p log( p 2 + m 2 + f n ) -1 2 ∫ d 3 p √ p 2 + m 2 ∼ T 4 = finite . (34)  \nAs mentioned above and discussed at length in [1] and [32], performing in the KK case too the sum over n up to infinity (while keeping p finite) mistreats the loop momentum. This is what causes the (artificial) disappearance of the q -dependent UV-sensitive terms in (6), giving the impression that there is a cancellation similar to what happens in the finite temperature case.  \nlow.  \nMoving now to the example of the Casimir energy E C , we observe that the latter is nothing but the difference between the vacuum energy ρ R calculated in the KK theory with one compact dimension (that for our illustrative purposes we took as a circle of radius R ) and the corresponding ρ ∞ calculated in the decompactification limit (i.e. in the theory with all non-compact dimensions): E C = ρ R -ρ ∞ . Now, if we perform the one-loop calculation of ρ 1 l according to previous literature, meaning that in (33) we perform the sum over n up to infinity, E 1 l C takes a form in all similar to (34), with T 4 replaced by m 4 KK . However, we have shown in [1] that (contrary to what is found in previous literature) the subtraction ρ R -ρ ∞ is not in general sufficient to ensure the disappearance of all the UV-sensitive terms. This is for instance the case of KK theories with Scherk-Schwarz SUSY breaking considered in the present work.", '4. Summary, conclusions and open questions': 'In summary, we have shown that the criticisms raised by the authors of [3] against our work [1] are unfounded. Their objections rely on Eqs. (10), that in turn arise from the replacement (11), namely ρ 1 l → Λ cc , operated by the authors of [3] in our original results for ρ 1 l (here reported in Eqs. (9)). We have shown that (11) is inconsistent and leads to arguments deprived of physical meaning. The most striking consequence of the replacement (11) is that for the physical cutoff Λ SM of the Standard Model it necessarily implies Λ SM ∼ m KK ∼ 10 -3 eV (see Eq. (20)), which is obviously absurd. This crucial consequence of their replacement, that makes their arguments inconsistent, was overlooked by the authors of [3]. Moreover, we have shown that, even closing an eye on this severe shortcoming, the arguments presented by the authors of [3] have further flaws and lack of internal coherence.  \nBefore ending this work, we would like to make a couple of additional comments on some open questions connected with the subject discussed in the present work. Concerning the UV-IR mixing invoked in [3], there are several works where the impact that modular invariance in string theory can have on its low energy EFT limit is studied (see for instance [33,34,35,36]), and it is certainly of the greatest importance to continue the investigation in these directions. However, we think that such a delicate issue cannot be trivially shortcut with the choice Λ = m KK considered in [3] and discussed above.  \nFinally, we would like to spend one word on the general framework where, in our opinion, the DD scenario should be framed. In [1] and in the present work we considered the contribution to the vacuum energy that comes from the bulk. The reason is that in the paper that contains the DD proposal [2], as well as in the paper [3] that attempts to confute the results of our work [1], this is the only part of the vacuum energy taken into account. Naturally, there is also a contribution from the Standard Model, that, as already said, lives on a 3-brane. Since the SM physical cutoff Λ SM is at least of the TeV order, this contribution cannot be ignored. In this respect, it is important to stress that it has been already objected in the  \npast (well before the formulation of the DD proposal) that the SM contribution to the vacuum energy seems to invalidate any attempt to get a small cosmological constant in theories with compact extra dimensions [37,38].  \nThe viability of the DD scenario is certainly subject to the existence of a physical mechanism that could dispose of this too large contribution. We expect that, for both bulk and brane contributions, such a mechanism (if any) should be provided by the piling up of quantum fluctuations in the UV → IR renormalization group (RG) flow that takes into account the whole (4 + 1)D theory, including the SM in the 3-brane. In such a framework, the UV-complete theory (string theory/quantum gravity) should provide the boundary (i.e. UV) value of the vacuum energy (more generally the boundary values of all the physical parameters), so that RG equations should dictate how the measured vacuum energy is reached in the IR (see [39] for examples where this mechanism is realized in d = 3 and d = 4 scalar theories). Work is in progress in this direction [40].', 'Acknowledgments': "We thank S. Abel, G. Dvali, H. P. Nilles, A. Pilaftsis and M. Scalisi for useful discussions. 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 AP is carried out within the INFN project QGSKY.", 'Appendix A. Dark Dimension scenario': "For the reader's convenience, in this Appendix we provide an expanded version of the introduction to the DD scenario given in the text. The proposal is formulated combining observational data and phenomenological bounds with swampland conjectures. It is based on the assumption that our universe lies in a unique corner of the quantum gravity landscape. This region is determined by the distance conjecture [41], according to which at large distances in the moduli field space ϕ the scale µ tow of one of the towers present in these asymptotic regions becomes exponentially small, µ tow ∼ e -α | ϕ | , with α positive O (1) constant. When this conjecture is applied to (A)dS vacua k , the relation  \nµ tow ∼ ∣ ∣ ∣ ∣ Λ cc M 4 P ∣ ∣ ∣ ∣ α M P (A.1)  \nk The AdS version of the conjecture is supported by a larger amount of arguments in string theory. The relation (A.1) is nevertheless assumed to be valid also in dS space.  \nis established [42].  \nCombining indications from one-loop string calculations with the Higuchi bound[6], the authors of [2] restrict α to the range  \n1 4 ≤ α ≤ 1 2 . (A.2)  \nMoreover, according to the emergent string conjecture [43,44], they observe that in these asymptotic regions only two possible kinds of towers are expected: either a tower of string excitations, or one of KK states.  \nSince their goal is to apply these conjectures/results to our universe, they begin by observing that a light tower of states causes significant deviations from the gravitational Newton's force law at the energy scale l µ tow , and that torsion balance experiments provide the strongest bounds to deviations of the 1 /r 2 law. Noticing that this law was verified down to scales around 30 µ m[45], which implies that µ tow has to satisfy the bound  \nµ tow ≥ 6 . 6 meV , (A.3)  \nand that this value of µ tow is close to the energy scale associated to the cosmological constant, Λ 1 / 4 cc ∼ 2 . 31 meV, they conclude that µ tow must satisfy the relation  \nµ tow ≥ Λ 1 / 4 cc , (A.4)  \notherwise we would have already detected deviations from Newton's law. They then observe that the only way to make the 'experimental bound' (A.4) consistent with the 'theoretical swampland bound' (A.1) is to take α = 1 / 4 (the quotation marks are to indicate that we are using their words). This leads them to conclude that: (i) 'there is a tower of states starting at the neutrino scale in our universe'; (ii) 'this tower is either a light string tower or a light KK tower'. Concerning this latter point, they observe that 'since we can describe physics above the neutrino scale with Effective Field Theory ' (our italics), the string scenario is 'ruled out experimentally', as no experiment has ever detected a string spectrum at these scales.  \nThe authors conclude that experiments tell us that the light tower must be a KK tower, µ tow = m KK , so that (in their words) 'the only possibility left' is that of a 'scenario of decompactification', where (A.1) becomes Λ cc ∼ m 4 KK , that is nothing but (4). The last step in the construction of their proposal is the determination of the number of extra compact dimensions. To this end, they compare bounds on m KK that come from calculations of the heating of neutron stars due to the surrounding cloud of trapped KK gravitons [8,9] with the value of m KK given in (4), that leads to the conclusion that there can be only one extra dimension of size ∼ µ m.  \nTherefore, assuming that our universe lies in a unique corner of the quantum gravity landscape, and combining swampland conjectures with phenomenological  \nl For a KK tower this is due to gravity propagating on the extra dimensions, while for the string case local physics breaks down at the string scale.  \nand observational data, the Dark Dimension scenario predicts that at low energies physics is described by a (4 + 1)D EFT with a compact dimension of micrometer size.", 'Appendix B. Standard Model cutoff': 'In this Appendix we derive the relation between the cutoff Λ of a (4 + 1)D theory and the 4D cutoff Λ SM of the Standard Model localized on a 3-brane. Let us consider the (4+1)D theory (with compact space dimension in the shape of a circle of radius R ) defined by  \nS = S grav + S mat , (B.1)  \nwhere  \nS grav = 1 2ˆ κ 2 ∫ d 4 xdz √ ˆ g ( ˆ R2 ˆ Λ cc ) (B.2)  \nis the (4 + 1)D Einstein-Hilbert action and as an example for the matter action we take  \nS mat = ∫ d 4 xdz √ ˆ g ( ˆ g MN ∂ M ˆ Φ ∗ ∂ N ˆ Φ -m 2 | ˆ Φ | 2 ) , (B.3)  \nwith ˆ Φ a (4 + 1)D scalar field that obeys the boundary condition ˆ Φ( x, z +2 πR ) = ˆ Φ( x, z ). We indicate with x the 4D coordinates and with z the coordinate along the compact dimension. Using the signature (+ , -, -, -, -), the (4 + 1)D metric is parametrized as  \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 , ) (B.4)  \nwhere A µ is the so called graviphoton and ϕ the radion field. Considering only zero modes for ˆ g MN , i.e. g µν ( x ), A µ ( x ) and ϕ ( x ) only depend on x , and integrating over z , for the 4D gravitational action S (4) grav we get [46]  \nS (4) grav = 1 2 κ 2 ∫ d 4 x √ -g [ R2 e 2 αϕ ˆ Λ cc +2 α □ ϕ + ( ∂ϕ ) 2 2 -e -6 αϕ 4 F 2 ] , (B.5)  \nwhere the 4D constant κ = M 2 P is related to the (4 + 1)D ˆ κ = ˆ M 3 P through the relation κ 2 = ˆ κ 2 / (2 πR ). The fields ϕ and A µ in the above equation are dimensionless (dimensionful fields are obtained through the redefinition ϕ → ϕ/ ( √ 2 κ ) , A µ → A µ / ( √ 2 κ )), and we used 2 α + β = 0. The canonical kinetic term in (B.5) for the radion field is obtained taking α = 1 / √ 12.  \nConsidering the Fourier decomposition of ˆ Φ( x, z ), for the 4 D matter action (B.3) we have  \nS (4) mat = ∫ d 4 x √ -g ∑ n [ | Dφ n | 2 -( e √ 2 3 ϕ M P m 2 + e √ 6 ϕ M P n 2 R 2 ) | φ n | 2 ] , (B.6)  \nwhere D µ ≡ ∂ µ -i ( n/R ) A µ , and φ n ( x ) are the KK modes of ˆ Φ( x, z ).', '16 C. Branchina, V. Branchina, F. Contino and A. Pernace': 'Taking a constant background for the radion (that for notational simplicity we continue to indicate with ϕ ) and the trivial background for A µ , the metric (B.4) becomes  \nˆ g 0 MN = ( e √ 2 3 ϕ M P η µν 0 0 -e -2 √ 2 3 ϕ M P ) . (B.7)  \n√  \nFrom (B.6) we define the ϕ -dependent radius R ϕ ≡ Re -3 2 ϕ M P . With such a definition, we immediately see that, when computing radiative corrections, the (4 + 1)D momentum ˆ p ≡ ( p, n/R ) is cut as  \nˆ p 2 = e -√ 2 3 ϕ M P ( p 2 + n 2 R 2 ϕ ) ≤ Λ 2 . (B.8)  \nThis latter equation is conveniently rewritten as  \np 2 + n 2 R 2 ϕ ≤ Λ 2 ϕ , (B.9)  \nwhere we defined Λ ϕ ≡ Λ e 1 √ 6 ϕ M P . In terms of the dimensionless ϕ of (B.4) and (B.5), and before using α = 1 / √ 12, it is Λ ϕ = e αϕ Λ = m 1 / 3 KK R 1 / 3 Λ.  \nAs p 2 in (B.9) is nothing but the modulus of the four-momentum on the brane, this equation tells us that Λ ϕ is the cutoff Λ SM of the SM (or more generally of the BSM model that lives on the 3-brane, where fields have n = 0). Therefore:  \nΛ SM = Λ ϕ = Λ e 1 √ 6 ϕ M P . (B.10)  \nFinally, as the DD scenario is realized for negative values of ϕ , from (B.10) we see that Λ SM ≤ Λ, i.e. the SM cutoff is lower than the cutoff of the (4 + 1)-dimensional EFT that implements the DD scenario.  \nBefore closing this Appendix, let us note that here we considered a spherical cutoff. 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