Emergent/Composite axions

Hidden theories coupled to the SM may provide emergent axions, that are composites/bound-states of the hidden fields. This is motivated by paradigms emerging from the AdS/CFT correspondence but it is a more general phenomenon. We explore the general setup and find that UV-sourced interactions of instanton densities give rise to emergent axions in the IR. We study the general properties of such axions and argue that they are generically different from both fundamental and composite axions that have been studied so far.

6. Higher Interactions 25 6.1 One-loop correction to the propagator due to quartic interactions 27 6.2 Implications for the emergent axion propagator 28 6.3 One loop corrections due to a cubic vertex 29 6.4 Implications for the emergent axion propagator 30 7. The holographic axion 31

Introduction and results
The notion of an axion goes back to the seminal work of Peccei and Quinn, [1] who introduced it in order to provide a natural solution to the strong CP-problem. The original theory was not renormalizable because of the axion coupling to the QCDinstanton density [2,3]. Quickly afterwards, renormalizable theories of axions were constructed [4,5,6], that also made the axion weakly interacting so as to avoid direct experimental constraints. Such axions came under the name of "invisible" axions and their couplings to matter have been determined using anomalies and the chiral Lagrangian [7]- [12]. They have become over the years the objects of both theoretical and experimental scrutiny, especially as they are prime candidates for the dark matter of the universe, but also candidates for the inflaton, [13]- [17]. Axions as scalar fields in an effective field theory are special. They always have a perturbative shift symmetry. It is this perturbative symmetry and the fact that they couple to instanton densities that provides a definition of what is an axion. The issue of the symmetry is however subtle: in all the cases we know and which make sense as QFTs, such a symmetry is broken to a discrete symmetry (at best) due to non-perturbative effects. In weakly-coupled theories, such effects are associated to instantons.
In the canonical case of the QCD axion, such instanton-related effects are responsible for giving a mass and a potential to the axion (see [11] for a recent exposition).
Axions are ubiquitous in string theory (see [18] for a review). They appear in two forms. Either as pristine massless scalars as the Ramond-Ramond (RR) axion of type IIB string theory. Or as internal components of antisymmetric form gauge fields upon compactification, as well as off-diagonal components of the metric. Both types unite in that they are generalized gauge fields of string theory and therefore have accompanying gauge symmetries [19,20]. It is these symmetries that provide the perturbative Peccei-Quinn symmetries in String theory [21].
It is also the case that continuous shift symmetries, that appear as would-be global symmetries in string theory, are broken by non-perturbative effects, at best to discrete ones, in agreement with the postulate and evidence that there are no global continuous symmetries in string theory. The argument is quite general 1 : RR axions couple to the world volume of D-branes as shown in general in [22]. The same D-branes, wrapped around an appropriate Euclidean internal cycle, provide instanton effects in string theory [23]- [26]. The nature of these effects depends on the amount of supersymmetry. In the case of maximal supersymmetry they do not generate a potential, but affect higher derivative terms like the R 4 corrections, [27,28], that are reproduced by the AdS/CFT correspondence [29,30,31]. With less or no supersymmetry, they generate a (super)potential for the axions [32]- [36] as it happens for QCD. In both cases, the end result is the same: the would-be global symmetry is broken to a discrete subgroup. Moreover, even this remaining discrete symmetry is gauged in string theory, because the original shift symmetry was a remnant of a gauge symmetry. There are subtleties in the above that arise when the axionic symmetries are coupled to anomalous U(1)'s [32]- [36], but do not change the final result.
In all the above descriptions, the axion fields are fundamental fields in the theory, be it a string theory or a QFT. In this paper, we would like to discuss a novel type of axion, the composite or emergent axion. The composite axion idea is not new, [37]- [43]. In all its realizations, it involved a strongly-coupled and confining theory and the role of the axion was played by one of the axi-pions that remains massless in perturbation theory. The characteristic energy scale, f , for the composite axion was the strong coupling scale of the axi-gauge theory, f ∼ Λ a , that is responsible for the interaction generating the bound state, since the mass of the axion is generated by QCD effects. Therefore, to generate a reasonable composite axion that is "invisible" one needs Λ a Λ QCD . It was also realized that there are phenomenological differences in such realizations compared to fundamental axions. As an example, in simple models, the axion does not couple to the lepton sector.
What we consider here is different in several respects and for this we will name the composite axions we discuss "emergent" axions. The first is that the strongly coupled gauge theory that is responsible for the composite axion (we shall henceforth call it the hidden sector) is a large N, strongly-coupled (and therefore holographic) gauge theory 2 . The second is that in such a context, the axion is an avatar of the generation of emergent gravity at the same time, [44]. The third is that the hidden-sector theory and the SM are coupled via interactions that are irrelevant in the IR. This fact ensures that one can have an invisible emergent axion even if the strong coupling scale of the hidden gauge theory Λ h Λ QCD . Of course, the paradigm for such a type of composite/emergent axion is already in the AdS/CFT correspondence. In the holographic context, the (gauge singlet) bulk fields of the dual string theory are thought of as composites made out of the generalized gluons of the (hidden) quantum field theory. In the standard examples, this theory is a CFT, and the notion of a bound state as a "particle" is somewhat imprecise. One can make it however more particle-like by breaking conformal invariance and producing a non-conformal strongly coupled theory with a non-trivial characteristic scale.
We start by briefly describing some of the ideas in [44]. Their purpose was to describe the SM of particle physics coupled to (semiclassical) gravity (and maybe other interactions) as emerging from four-dimensional QFT's, one of which is the SM. For simplicity, we assume that beyond the SM there is a single hidden QFT that is strongly coupled and at large N.
The hidden theory will be, among other things, the source of gravity for the SM fields. The reason that we require large N and strong coupling from the hidden theory is that it should generate semiclassical gravity according to the dictums of the holographic correspondence.
The key point is the coupling of the hidden theory to the SM. From some general considerations that are detailed in [44], we would like to couple the two theories in the UV. Modulo subtleties, the only UV complete way of doing this is via a messenger sector that is composed of bifundamental fields, charged under both the hidden gauge group and the SM gauge group. The messenger fields have large masses, M , much above all the mass scales of the SM. Integrating them out, we obtain the effective action for the low-energy theory that involves the SM and the hidden theory coupled by double (or multiple) trace interactions, that are all irrelevant 3 . This is the reason that the novel interactions that are triggered between the two theories are weak at low energies.
Generically, upon integrating out the "invisible" hidden theory, the inter-theory high-energy interactions can be resolved by effective, low-energy dynamical fields coupled to the standard model. One of the them, associated to the hidden stress tensor, appears as a dynamical metric coupled to the SM [44,45]. In principle, all single-trace operators of the hidden theory give rise to emergent low-energy fields, coupled to the SM model. For scalar operators, this is detailed in sections 4 and 5. However, most of them acquire effective masses, that are of the order of the messenger mass scale M . Therefore, they are irrelevant for low-energy physics. This is analyzed in detail in section 5. There are only three classes of operators that are protected by symmetries, and therefore have generically low masses.
• Instanton densities. They are protected by their topological invariance. They give rise to emergent axions.
There are however, less common symmetries that have not been considered in [44]. An example is higher form symmetries. The only continuous example we know of in D = 4 is an antisymmetric tensor conserved current, which can always be mapped to a free U(1) gauge field. Such a current, if generically coupled, will be broken and therefore the associated particle will acquire a mass. All other examples we know of, couple to non-local discrete symmetries.
In the rest of the paper we pursue the topic of emergent axions. We assume a setup where there is a four-dimensional hidden theory which is coupled to the visible theory (the SM) via a set of bifundamental messenger fields with typical masses of order M , and we assume M to be much bigger than any SM scale, or hidden field theory scale. Our main goal is a hidden theory with a large N gauge group, and this is the reason why we sometimes call it QFT N . However, we shall assume the large N limit only when necessary.
We would like to compute the direct couplings of the operator A ≡ T r[F ∧ F ] in the "hidden" large-N QFT to the associated operators of the SM.
We expect that • It will couple to all analogous operators of the SM z I = T r[F I ∧ F I ] where I = 1, 2, 3 labels one of the simple factors of the SM gauge group. It might also couple to other CP-odd scalar operators (likeψψH − cc).
• It will not couple to other CP-even operators 4 .
Our first goal will be to understand how these couplings are generated.
To do this we must write the couplings of the messenger fields to the SM as well as to the large-N QFT that we denote by QFT N .
We denote the messenger fields by φ ai , where the index a = 1, 2, · · · N is the color index of the hidden QFT N while i is a gauge index of a gauge group of the SM. φ ai can be a vector boson, a fermion or a scalar. Note that in order to be able to write UV-complete messenger couplings to the SM, all the SM fields should be bi-fundamentals 5 , χ ij , with respect to the gauge groups of the SM. Again, here χ ij can be a scalar, a fermion or a vector boson. Note that there are many ways to write the SM fields as bifundamentals, but all these ways have been carefully classified in [48], and they involve the inclusion of at least one more and typically several extra U(1)s. These U(1) are generically anomalous. Finally, we label the QFT N fields as A ab as they are in the adjoint (or bifundamentals) of the large-N gauge group. There is a set of anomaly cancellation conditions that the messenger sector should satisfy that are presented in appendix A.
The couplings of messengers to the SM fields and to the QFT N fields can be schematically written as where we have suppressed possible derivatives and other space-time indices. The notation is sketchy and the spin of the various fields is not indicated. The relevant coupling constants are the appropriate gauge couplings. For now we collectively denote them by λ. There may be also quartic couplings in (1.1) but they do not affect our analysis.
Our results can be summarized as follows.
• Generic couplings in the messenger sector generate interactions of hidden and observable instanton densities. This will be shown by an explicit one-loop calculation in a typical example with messengers being fermions. It will be clear that this result is generic.
• There will be many other double (or higher) trace couplings generated by the messenger sector. With the exception of conserved operators discussed earlier, they will lead to interactions that are highly suppressed in the IR (E M ). The reason is that the corresponding emergent fields will acquire masses of order M .
• We analyse the induced couplings in the observable sector of general scalarscalar interactions between the hidden and the observable theory in section 4 by assuming a semiclassical (quadratic) approximation. This is valid if both theories belong to one of the following two options: a large N theory or a perturbative theory. The effect of generic higher interactions in this picture will be analyzed in section 6 and will be shown that it does not change the leading picture.
• The inter-theory couplings will be reformulated in section 5 in terms of scalar fields that represent the associated composite operators of the hidden theory.
Their quadratic action will be determined in terms of hidden and observable two-point functions.
• It will be shown that, for generic scalar operators, the associated composite particles have O(M ) masses and are therefore irrelevant in the low-energy effective theory.
• It will also be shown that for scalar operators that are instanton densities, the effective action for their composites has a different parametric dependence. The hidden instanton density generates an emergent axion coupled to the SM. The characteristic decay constant of the emergent axion is where m hidden is the characteristic scale of the hidden theory. The corrections to this scale from SM quantum effects are suppressed.
• The mass µ of the emergent axion has two contributions. The first is due to SM quantum effects and is ∼ Λ 2 QCD /f as with standard axions. However, unlike fundamental axions, the emergent axion mass has also a contribution from the hidden theory that is dominant and of order m hidden . The reason is that in the theoretical setup we analyze, the associated PQ symmetry is broken by non-perturbative effects both in the hidden and in the visible theory.
• In the case where the hidden theory is a CFT, the emergent axion mass and decay constant arises entirely from SM corrections, and their parametric dependence is different Therefore the mass is comparable to that of hadrons and moreover it is no longer "invisible". For energies Λ QCD E f it is strongly coupled and it is interesting to derive the resulting interaction it mediates 6 .
• When the hidden theory is a holographic theory then the description that we developed needs a further amendment, so that the Lagrangian description is closer to the dynamics. If the instanton density in the hidden theory has a gapped and discrete spectrum as in QCD 7 then a better "resolution" of the interaction is to introduce an axion field for each pole of the two-point correlator of the instanton density. This amounts to the presence of an infinite number of four-dimensional axions and as usual the dynamics organizes them into a single five dimensional axion field in the emergent fifth holographic dimension.
The appropriate picture then for the hidden theory is as a bulk five-dimensional holographic theory that interacts with the SM represented as a brane immersed in the five dimensional space-time at an appropriate radial direction corresponding to the messenger mass cutoff M .
This picture is developed in section 7. It resembles the DGP setup, [49], but now the field at stake is an axion. Moreover the hierarchy between bulk and brane kinetic terms that marred the DGP setup now is naturally explained from the dual QFT picture.
An analysis of the effective interaction mediated by the axion on the brane can be done along the same lines as for the DGP case, [49,50,51]. We find that the interaction at short and long distances is the one of a four-dimensional massive scalar. Depending on some parameters, at intermediate distances there may be a phase where the axion interaction is five-dimensional. In such a regime all emergent axion resonances contribute equitably to the relevant interactions.
There are many directions that remain open and are related to the ideas presented in this paper. The phenomenology and potential model building of emergent axions is unexplored. In reference [44] it was argued that one or more massive anomalous U(1)'s may form part of the SM as in the case of orientifolds, [53,48,54,55,56]. Their low energy effective action was analyzed in [57]. The interplay of the emergent axions with the anomalous U(1) gauge bosons is interesting to explore and the string theory knowledge is a good guide in this direction [58]- [60].
The cosmology is also novel as in the holographic case the description is quite different from the standard one. The role of the emergent axion as an inflaton may be investigated along the lines of [61]. The role of such an axion as dark matter is also interesting to analyze.
The emergent axion may also play the role of a relaxion, [62]. Indeed the bulk axion can combine with other relevant perturbations of the bulk theory, along the lines of [51] to provide a different mechanism for relating the hierarchy problem associated with the EW scale and the self-tuning of the cosmological constant [63].

The effective cross-couplings
We start by considering the messenger interactions between the hidden QFT N and the visible theory (which for us will be the SM). We perform a simple calculation to assess the emergent double trace couplings that will be our focus in this paper. We consider in particular the effective terms in the action which are generated by messengers going around a loop.
If we only have QFT N external states, such corrections will affect the action of QFT N . At large N, this correction is suppressed by 1/N compared to the leading corrections due to the hidden theory itself, as the messengers transform in the fundamental of SU(N). If we only include SM external fields, then we obtain corrections to the SM couplings that are O(N ). 8 We are interested in corrections that involve external fields from both sides as these will generate interactions in the IR between QFT N and the SM. By gauge invariance, the minimal numbers of fields we need on each side is two, so the leading correction is generated by box diagrams.
Our setup is perturbative, but also indicative of the general case, that is not necessarily perturbative.

The exceptional couplings
Before we discuss the generic case we start from the exceptions. They involve U(1) gauge fields that can be made gauge invariant with a single field strength , namely F µν . There are good reasons to believe that QFT N does not have U (1) gauge fields as they decouple in the holographic limit. However, the SM has the hypercharge, as well as possible anomalous U(1)'s. Couplings of F Y µν to appropriate QF T N gauge invariant operators are special as T r[Q Y ] = 0. They will be discussed in a companion paper [46].
There can be couplings also to the SM anomalous U(1)'s. We shall not consider them here.

The generic couplings
We consider the generic couplings induced between QFT N and SM. They involve single-trace, gauge invariant operators of QFT N of any spin coupled to similar gauge invariant operators of the SM.
We focus on operators involving two non-abelian gauge fields in QFT N and two gauge fields on the SM side. The reason is that such operators include the very special case of the instanton densities. They are special as they are protected by perturbative shift symmetries and they are expected to induce effective emergent axion couplings on the SM side [44]. This class of calculations is in the "light by light scattering" class and various aspects have been addressed in the literature, [64]- [67].
We insert therefore two gauge fields from QFT N and two from the SM. The terms we are interested in will come from fermion messengers going around the loop (fig 1). Similar terms are expected to appear also from bosonic messengers. We shall not study bosonic messenger contributions here, as they are expected to be similar.  Figure 1: The 1-loop diagram with two gauge fields from QFT N and two from the SM, in a "double line" notation. The a, b and i, j indexes denote the colours of the QFT N and the SM respectively. The lines denote the index contractions of the hidden theory and the SM.  Figure 2: The three box diagrams. Apart from the first "box-1", we have to add the contribution from two more diagrams with p 3 , ρ ↔ p 4 , σ ("box-2") and p 1 , µ ↔ p 4 , σ ("box-3") exchanged.
There are six box diagrams, three of them are represented in fig 2, the other three diagrams are similar to the previous ones but with the internal fermion line going anti-clockwise instead of clockwise. The full amplitude can be written as We can hold T SM 1 in the first position using cyclicity of the trace. However all the traces yield the same result because they split in two independent traces for instance Up to traces, there is no difference between the same diagrams computed with two pairs of different non-abelian bosons or four identical abelian bosons. The result holds even for the non-abelian case 9 , since the gauge groups live on different sectors, and split to separate traces like in (2.2). At the low energy limit where all external momenta p 1 , p 2 , p 3 , p 4 M the result is (more details of the computation are presented in the appendix B) The A (0) and A (2) terms vanish due to gauge invariance. The A (4) term, which is the interesting part for us, has a very long and complicated form and cannot be presented here. Instead we use the spinor helicity formalism and manipulate separately the different helicity configurations. A short description of the method can de found in appendix F, for a more detailed discussion [68].

The box amplitude in different helicity configurations
In this section we evaluate the amplitude in different helicity configurations. In particular, we use the spinor-helicity formalism [68,69,70], briefly reviewed in Appendix F. It helps to expose a great number of cancelations that remove superfluous terms thanks to smart kinematic/gauge choices. Our results assume very compact and neat forms which can be used to evaluate various couplings in the effective action.
In principle, one has to deal with 16 different helicity configurations (±±±±). However, those with the same number of +'s and −'s are connected by permutations. In addition, configurations which are related via parity + ↔ − yield similar results. Therefore, we end up with three different classes of configurations, with four/zero, three/one and two/two +'s and −'s.
we can extract three independent combinations After all these manipulations the amplitude reads By parity/conjugation [p i p j ] → p i p j , the all-minus configuration produces a similar result Helicity configurations (−+++) and (+−−−) In principle one has 8 configurations, but not all are independent. In fact they are connected by cyclic transformations. The only two independent configurations are (−+++) and (+−−−).
In any of these two cases we choose q 1 =p 3 and q 2 =q 3 =q 4 =p 1 as auxiliary momenta, which leads to Evaluating the amplitudes we obtain as expected.

Helicity configuration (−−++)
The four helicity configurations with two consecutive pluses and minuses can be obtained by cyclic permutations of (−−++). In this case we choose q 1 =q 2 =p 3 and q 3 =q 4 =p 2 as auxiliary momenta so that all the products between polarizations are zero except With this choices we obtain The scalar products in the last formula are not independent, indeed We finally get and similarly for (++−−)

Effective action
In this section we study the effective action after integrating out the heavy fermions (messengers). Our goal is to extract the couplings in the action by comparing them with the box amplitudes evaluated in section 2.2.
Integrating out the heavy fermions, the effective Lagrangian takes the form where L G denotes the corrections to QFT N , L F denotes the corrections to the SM and L F,G denotes mixed terms that couple the two theories. Focusing on the mixed part, the most general form is where (F G) = F µν G µν , F µν = 1 2 µνρσ F ρσ and the coefficients a i , b i and c i are real. In the above F µν are the (abelianized) SM field strengths, while G µν are the (abelianized) field strengths of QFT N .

Fixing the coefficients with helicity configurations
In order to fix the coefficients we compare the amplitude computed in section 2.2 with the corresponding amplitude encoded in the effective theory in all the independent helicity configurations. As we already mentioned, we use the spinor helicity formalism in order to maximally simplify our results by choosing wisely the auxiliary momenta associated to the polarizations.

Effective action with fixed helicities
Each field strength can be split into self-dual and anti-self-dual components F = When the field strength is associated to massless particles we can identify helicity with chirality using Weyl spinors. Using as a basis the self-dual and anti-self-dual Lorentz generators σ µν andσ µν in the spinorial representations we can write F ± as Since (σ µν ) αβ is symmetric in the spinor indices, f + can be thought of as the symmetric tensor product of two left handed commuting Weyl spinors. Indeed where u α (k) is a massless commuting Weyl spinor of definite (positive) helicity h = + 1 2 and light-like momentum k µ = u α (k)σ µ ααūα (k), so that (f + ) αβ has helicity h = +1. Due to Lorentz invariance, tensors with positive and negative helicities (or more correctly left and right chirality) are orthogonal to each other therefore, the terms that appear in the effective action (3.2) read Expanding the effective Lagrangian in (3.2) we obtain As already mentioned, in principle one has 16 different helicity configurations, however terms with an odd number of ± helicities identically vanish. We also note that in a non-chiral Lagrangian the coefficients b i must vanish due to parity symmetry. In other words, exchanging F + ↔ F − and G + ↔ G − should leave the Lagrangian invariant.
Terms with M 0 or M −2 don't appear in the effective action thus we expect that the contributions A (0) and A (2) must also vanish in any helicity configuration.

Comparison with the effective action
In this section we compare our results in section 2.2 with the corresponding amplitudes obtained from the effective action. We have noticed that the configurations (++++) and (−−−−) on the one end and (−−++) and (++−−) on the other end yield similar results with the same coefficients. This suggests that amplitudes are invariant under the swap of plus and minus helicities, so that the effective action must be non-chiral. Therefore, all coefficients b i in (3.10) vanish Now let us compute the amplitude from the effective action. The configuration (++++) yields The configuration (−+++) gives The configuration (−−++) gives while (−+−+) gives Comparing the amplitudes we obtain the following coefficients for the effective action and the quartic interaction part of the effective Lagrangian becomes where appropriate traces are understood in the hidden and SM part. That agrees with similar results found by S. Marchesani under the supervision of Ya. Stanev [71]. For comparison with the QED case (Euler-Heisenberg action), where only one gauge boson is present, we set G = F and the new parameters are The factor of 6 is due to the permutation symmetries of the interaction terms, in fact the 4-point vertex F F F F has an extra factor of 6=3!=4!/2!2! compared to F F GG.

The effective interactions
The effective Lagrangian we obtained, (3.17) describes double trace interactions between the hidden theory QFT N and the SM. The relevant operators that participate are scalars as well as four index tensors in the hidden theory and the corresponding ones in the SM, The final one-loop interaction Lagrangian can be written as The large N estimate of this calculation agrees with the generic large N estimates made in appendix A of [44] provided there is a redefinition of the normalization of the operators so that the two normalization agree. At large N, and strong coupling, the tensor fields are expected to acquire large anomalous dimensions 11 and are not therefore expected to be important at low energy. The scalar coupling, as we show in the next section will also not be very relevant at low energy. It is only the coupling of the instanton densities that will turn out to be important and give rise to an emergent axion.

The general scalar cross interaction and its emergent resolution
We would now like to reinterpret the interaction between two scalar operators in the two theories and in particular the instanton densities, in an effective description.
To do this, we would like to address the more general problem of a scalar-scalar interaction between two theories and its IR "resolution". We consider two theories T 1 and T 2 coupled via an interaction of the form where O 1 is an operator of dimension ∆ 1 belonging to T 1 and O 2 is an operator of dimension ∆ 2 belonging to T 2 . This may be an UV coupling, defining an interacting pair of theories in the ultimate UV. It may be also the definition of the coupled theory at a finite energy cutoff. For instance this is the case at the scale of the messenger mass M . In that case with λ 0 dimensionless. We assume that the hidden theory T 1 is a theory at large N. We shall explicitly show the dependence on N in the sequel. In particular, the coupling λ 0 ∼ O(1) when O 1 is a standard normalized single trace operator so that all its correlators are of order O(N 2 ). We assume that the two theories are such that higher than three-point (connected) functions are suppressed compared to two-point functions. Examples of such theories are near-free theories (interactions suppressed by small couplings) and large N-theories (interactions of gauge-invariant operators suppressed by 1/N ).
Consider now the generating functional for the correlators of By performing a Hubbard-Stratonovich transformation we can write that allows us to express the complete generating functional as follows where Z 1 , Z 2 are the Schwinger functionals of the respective uncoupled theories We henceforth work at the quadratic order 12 in which 1 is the translationally invariant, unperturbed twopoint correlation function of O 1 in theory T 1 and with a similar expression defining Z 2 . Using (4.7) and performing the integral over ζ 1 , ζ 2 in (4.5) explicitly, we obtain the quadratic order generating functional (expressed in momentum space) as Equivalently one can use the saddle point equations (4.5) (since they are exact for Gaussian integrals) and substitute them back to obtain (4.8). More explicitly, one can rewrite the matrix appearing in (4.8) as The scaling dimension of G 11 (p) is 2(∆ 1 − 2) and the one of G 22 (p) is 2(∆ 2 − 2). We notice that the interaction between the two theories modifies the non-interacting correlators and create cross-correlations between the two sectors. As an example, the new correlator for O 2 in momentum space is In particular, in the regime where G 11 → 0, it is given by the initial correlator G 22 while in the regime where G 11 → ∞, it is given by −1/λ 2 G 11 . We now consider the IR expansion of the correlators. It depends crucially on the detailed physics of each theory. In a theory with a single scale, which is also its mass gap m the IR expansion in p m reads If ∆ 2 is an integer, then starting with the term p 2(∆ 2 −2) logs of momentum appear in the expansion.
If the theory has a UV scale Λ but also other smaller IR scales like m Λ then for generic scalar operators the larger scale dominates the coefficients in the expansion, There is however a scalar operator in the gauge theory that is special and for which this scaling is not valid. This is the (CP-odd) instanton density operator that is the focus of the present paper. It is well known from studies in QFT, [72] and holography, [73] that the correlators of the instanton density are UV insensitive. The reason is that the θ angle in QCD is not renormalized, as shown rigorously on the lattice (see [73,74,75] for a detailed discussion). This is also true in holography, whereby the bulk axion field dual to the instanton density does not have a potential and the procedure of holographic renormalisation allows to derive its correlation functions [76,77]. However, in holographic QCD there is a non-trivial (and non-perturbative) β-function for θ driven by the vev of the instanton density on the (non-trivial) YM vacuum (encoded in the topological susceptibility), [73]. Notwithstanding this, all θ-dependent contributions to the vacuum energy are cutoff independent. Therefore, even though there is a non-trivial UV structure in the gauge theory, the two-point function scales as in (4.12) where m is the characteristic IR scale of the gauge theory, and UV scales do not appear in the correlator. This is an important feature that distinguishes this operator from all other scalar operators and has important IR consequences as we shall see further on.
We also similarly parametrize in the IR i O 1 (p)O 1 (−p) 1 N 2 (a 0 + a 2 p 2 + a 4 p 4 + · · · ) (4.14) If there is a non-trivial CFT in the IR, then there are also non-analytic contributions. For example, for an operator with IR dimension ∆ IR we have instead 15) The dependence of the coefficients a n on the various scales of the theory follow our discussion above for O 2 (p)O 2 (−p) 1 .
In the case that one of the two theories is a CFT, one finds that there is no mass term in the IR and that the expansion starts with fractional powers of momentum.
We will comment here on the N -dependence of the various functions and parameters in the case the hidden theory T 1 is a large N theory. As was shown in [44], the interaction

Integrating in a new (pseudo)-scalar
We would like now to interpret the presence of the interaction (4.1) from the point of view of theory T 2 as due to a novel dynamical scalar, coupled linearly to the operator O 2 . In a sense, we consider T 1 as the "hidden" theory and the new coupling is a channel of communication between the hidden and the visible theory induced by the interaction (4.1) in the UV. Therefore we imagine that we probe theory T 2 and we can perform experiments involving only the operator O 2 of T 2 . 13 We wish to ask to which extent we can represent the effects of T 1 and its interaction to T 2 as coming from an "emergent" dynamical field coupled linearly to O 2 . We henceforth call the theory T 1 as the "hidden sector" while the theory T 2 will represent for us the visible sector.
We consider a new scalar field χ coupled to the operator O 2 as follows where K is an operator that we want to determine using consistency with the results of section 4. We have inserted a dimensionfull coupling g in the interaction of χ with O 2 so that the scalar χ is dimensionless. Because of this, the mass dimension of g is 4 − ∆ 2 and the operator K has scaling dimension 4 in mass.
In order to determine the correct form of K, we now compute the O 2 correlator by integrating out the scalar field. To do this properly we introduce again a source for O 2 From the definitions we have 3) that allows us to compute at quadratic order Differentiating twice with respect to θ 2 we obtain the corrected two-point function If we wish to reproduce the two-point function obtained in the previous section, in (4.11), we must match it to (5.6). We find This expresses the consistency relation between the two descriptions. To study the physical properties of the field we integrated in, we now consider a generic expansion for the IR behaviour of two-point functions. As mentioned both theories have a UV scale M that is the messenger scale. Therefore, for generic operators the IR behavior of correlators below the mass gaps of the respective theories is as follows Using (4.2) we obtain so that the field χ has mass dimension zero. Parametrizing then For a generic scalar operator O 1 as argued before we have a n ∼ M 2(∆ 1 −2)−n and we obtain It is clear that the induced interaction is weak but the mass scale of the scalar is a very high scale, the messenger scale.
On the other hand if the operators O 1,2 are the instanton densities, then as we have mentioned earlier their two-point function is not UV sensitive and in this case a n =ā n m 2(∆ 1 −2)−n 1 , b n =b n m 2(∆ 1 −2)−n 2 (5.14) where m 1,2 are the IR mass scales of the hidden and visible theories T 1,2 andā n ,b n are dimensionless and typically O(1) coefficients. If the hidden theory were YM then m 1 is Λ Y M . In this special case we obtain instead If m 1 M then this is an emergent weakly-coupled axion-like field that as we have shown earlier is coupled to the SM instanton densities (which are represented here by O 2 ).
As mentioned above, the case of interest here is when O 1 is the "hidden" instanton density and the O 2 is one of the observable (SM) instanton densities. In this case ∆ 1 = ∆ 2 = 4 to a high accuracy. We have assumed that m 1 M . If m 1 is also comparable or smaller than SM scales then this resembles a standard PQ axion.
Moreover (5.15) is the dominant contribution to its mass as m 1,2 M . This is definitely a different situation compared to a fundamental axion field. Its origin here is not in a continuous global symmetry of a QFT at a higher scale, but an "emergent PQ symmetry" arising from a hidden instanton density.
For the instanton densities, as ∆ is an integer, the low energy structure of the two-point functions is of the form presented in (4.15). Therefore, a non-analytic term appears in the emergent axion induced terms a fourth order in derivatives.
There is an interesting limit to discuss, corresponding to the hidden theory being a CFT, with m 1 → 0. In that case, the only non-trivial scale is the messenger scale that breaks scale invariance in the UV and according to our earlier discussion, in such a case the O 1 O 1 correlator 14 starts in the IR as p 4 log p 2 . The emergent semiclassical axion is "massless" and non-local as the inverse propagator starts at O(p 4 log p 2 ). This is reminiscent of the Witten-Weinberg theorem, [78], although this theorem applies to emergent massless gravitons and photons 15 . Our results suggests that there must be an analogue of this for massless axions.
The SM quantum effects associated to the SM instanton densities provide corrections to the axion action. In perturbation theory they provide a renormalization of the kinetic terms but no mass renormalization. The associated one loop diagram is calculated in appendix D. Non-perturbative QCD effects will provide also a mass correction of order Λ 2 QCD /f that will be added to the axion mass originating in the hidden theory. The EM instanton density is not expected to contribute to the axion mass.
We can discuss such corrections in general by computing the effective mass of χ which is affected by the mixing with the visible theory operator O 2 . Indeed, a 14 We assume that there are no spurious contact terms in the correlator and it is defined in agreement with conformal invariance. 15 In the case of emergent gravitons and photons it can be shown that when they are massless their effective theory is non-local in agreement with the Witten-Weinberg theorem, [45] calculation of the two-point function of χ gives In the generic case where one or both of the operators O 1 are generic, the effective mass of χ is the messenger scale, M . Therefore both f 2 and µ 2 remain of order O(M 2 ). One can in principle imagine a fine-tuned situation where µ M but this is not justified at this point.
If both O 1 and O 2 are UV-protected operators (instanton densities), we can expand Then, expanding the propagator as we obtain the renormalization of f 2 due to visible theory effects to be Note that for the case that interests us, ∆ 1 = ∆ 2 = 4, this correction is small, as m 1,2 M . The renormalized mass is 16 where we have neglected the small corrections to f 2 . In particular the visible theory contribution to the mass is m 4 2 which should identified with Λ 4 QCD for the SM 17 . However, here the axion has also a mass contribution that originates in the hidden sector and is proportional to the hidden topological susceptibility, m 2 1 [2,3]. In the case where the hidden theory is a CFT, m 1 = 0 and the whole twoderivative effective action of the emergent axion originates in the SM quantum corrections namely the g 2 G 22 part of the propagator in (5.16) We obtain in this case 16 It may be that the effective mass turns out to be negative. This usually happens when the intertheory interaction can become strong. This is not the case here as the interaction is irrelevant. 17 In the general case it is m 4 2 m2 M 2∆2−8 .
In the case of the instanton density (∆ 2 = 4)they give, This is an emergent axion that now is local at low energies, with a mass of order Λ QCD but which is relatively strongly coupled, as f Λ QCD . This is excluded by experiments.
We conclude that in the general case where we have multiscalar couplings between various scalar operators of the two theories, generically these lead to emergent interactions via scalars that are very heavy (and therefore not very relevant for low energy physics) as their masses are at the messenger scale M . However, the instanton density 18 of the hidden theory gives rise to an emergent axion-like field that couples (weakly) to the SM model instanton densities 19 .

Higher Interactions
As we discussed in section 4, we assumed a perturbative structure for the two theories T 1,2 and treated both of them in the quadratic approximation. This was implemented in (4.7)-(4.11). The only interaction present was the O 1 O 2 deformation coupling the two sectors.
One can further extend the analysis to the non-linear regime by considering interactions both inherently present in each of the uncoupled sectors as well as further cross-interactions between them. The functional (4.5) contains in principle all the possible self-interactions of each sector so in that sense it is exact. To accommodate higher point cross-interactions one can deform it with terms such as In the rest of this section we will reserve the capital Latin indices I, J, K, L as indicators of the two theories, hence they can take the two values 1, 2. As an example a generic four-operator interaction will take the form One would then have to expand the total functional in powers of the external sources to obtain the appropriate correlator. This can be performed in a perturbative expansion in powers of the various couplings. The expansion is around the Gaussianapproximation resulting in (4.8). Instead of listing all the possible diagrams, it is convenient to rewrite the functional (4.8) as and add the interactions by deforming the functional S q (φ I ) with the following terms This ansatz encapsulates both the interactions present in each uncoupled theory itself as well as cross interactions. The advantage of this rewriting is that one can easily take into account quantum corrections at any loop order, the couplings V IJK at tree level correspond to the bare couplings λ IJK , while quantum corrections renormalise them 20 . One such effect of interactions is to renormalise the two-point function via loop corrections. This is the main effect we will be interested in, since it affects the propagator of the emergent (pseudo) scalar (5.1) as shown in (6.6) below. In particular loop corrections cause a shift (or renormalisation) in the quadratic matrix part of S G of the form where with Σ IJ (Λ, p) we denote the matrix elements of the loop corrections, that generically depend on the momentum p, but also on any cutoff scale which we denote by Λ. In particular for us such a cutoff will be set by the messenger scale and therefore Λ = M . In Appendix E we describe this computation and provide explicit expressions for the matrix elements Σ IJ (p, Λ) in terms of integrals, considering quartic and cubic interactions. Their physical properties will be analysed below in different regimes.
To obtain the dressed correlators, one can invert the renormalised matrix (we assume that Σ IJ is symmetric) to obtain 20 We also note that Γ[φ I ] = S G [φ I ] + S int [φ I ] can be thought of as the effective action for the coupled theories computed at a given loop order, for more details see [79].
One can now match the G ren 22 element of the correlator that corresponds to O 2 O 2 ren with the expression for the corrected correlator (5.6) that arises from the presence of a dynamical "emergent" field with kinetic operator K(p), see (5.1). This identifies the latter as follows (up to the messenger cutoff scale M ) where in the last line we assumed that the renormalisation effects are perturbative in nature and therefore the matrix elements Σ IJ are parametrically small and we can keep the leading correction. With this identification we can now study the effects that interactions have in interpreting the effects of the "hidden" theory T 1 as coming from some dynamical "emergent" field coupled linearly to the operator O 2 of T 2 beyond the Gaussian regime of the previous section. As an example, we will now discuss in detail the physical implications of cubic and quartic corrections in several regimes of interest.

One-loop correction to the propagator due to quartic interactions
We now consider quartic interactions and find their correction to the matrix correlator. The procedure involving the computation of one loop diagrams is discussed in Appendix E.2. We use the two-point function (4.11) to compute the one loop diagrams (E.3)with a cutoff method. We will use Latin indices I, J, K, L = 1, 2 to label the fields φ 1 , φ 2 . The simplest case for the four-point vertex is the isotropic one for which V (4) . From the relevant Feynman graphs, thanks to momentum conservation at the four-vertex, it is easy to see that there cannot be any external momentum dependence and thus Σ To perform the computation one has to use the matrix propagator (4.10) with As described in the Appendix E.2, all the matrix elements of Σ where the precise constant numerical coefficients A IJ can be found using (E.10) together with (E.4). One notices that upon expressing the couplings in terms of dimensionless parameters the matrix elements scale as expected from dimensional analysis in (6.4). In particular the leading correction is In the case that one or the two theories develops a mass gap m in the IR, the leading cutoff dependence still keeps the same form, up to corrections in the form of powers of the dimensionless ratio m/M .
We note that even though the correction matrix Σ (4) IJ is momentum independent, the momentum dependence will come from inverting the matrix in order to compute the propagator as shown in passing from (6.4) to (6.5). In addition it is easy to see that for integer ∆ 1 , ∆ 2 one can get logarithmic divergences. The structure of the integrals then is that given in (E.9). We will describe such cases with more detail in section 6.2.

Implications for the emergent axion propagator
We now work out the implications of quartic interactions to the emergent axion propagator. The matrix K(p) given by (6.6) is going to be identified as the inverse propagator of an emergent field that we integrate in, see (5.1). This emergent field captures the dominant physical effects of the theory T 1 which we refer to as the hidden sector. Since we work in perturbation theory we can only trust small corrections around the non-perturbative quadratic result given in (5.7). We will again re-express the coupling constants in terms of dimensionless parameters The coupling g for the emergent field χ in case that this field is identified with an emergent axion can be conveniently expressed as g = N M 4−∆ 2 to make the axions dimensionless. All the constants in this section will depend on these dimensionless parameters. We also notice that Σ 12 ∼ M 4−∆ 1 −∆ 2 , Σ 11 ∼ M 4−2∆ 1 , Σ 22 ∼ M 4−2∆ 2 which are the expected canonical dimensions of these elements that shift the original inverse propagator elements appropriately. We also recall that ∆ 1 is the dimension of the hidden sector operator while ∆ 2 refers to the dimension of the SM operator.
Using (6.6) together with (6.10) we find where a 0 , b 0 , c 0 are numerical constants and V 0 4 1 so that the perturbative approximation is trustworthy. The first term depends only on the cutoff and is either to be subtracted or vanishes in the limit of large cutoff M (i.e. with g = N M 4−∆ 2 it is found to vanish as M −4 ). The b 0 term results in a constant wavefunction renormalisation. The momentum dependent term shifts the quadratic solution (5.7) by a small amount and therefore provides the most interesting effect. Let us now list the following cases depending on the conformal dimensions of the operators: • The relevant operator case ∆ 1 < 2.
In this case the c 0 term deformation is relevant and shifts perturbatively the pole of the propagator from p = 0.
• At ∆ 1 = 2 one finds a logarithmic scaling for the correction in terms of the cutoff M • Beyond that (∆ 1 > 2), one finds that the deformation is irrelevant and does not affect the non-perturbative quadratic solution in the IR. Therefore operators with ∆ 1 = 4 do lead to perturbative corrections to the quadratic result (5.7) except from an overall wave-function renormalisation.
• A special case is given by a standard model operator of dimension ∆ 2 = 4. In such a case it is easy to see from (6.10) that the matrix elements Σ 22 , Σ 12 → 0 which leads to a 0 = b 0 = 0 i.e. no wavefunction renormalisation. If furthermore ∆ 1 > 2 (as in our case for a "hidden" instanton density ∆ 1 = 4), the effects of such a combination of operators are exactly captured by the quadratic result.
We therefore conclude that operators with ∆ = 4 are special in that they are protected by UV effects to one loop order (assuming quartic interactions) that could effect both an overall wavefunction renormalisation and/or a shift in the poles of the propagator of the non-perturbative Gaussian treatment. We will now proceed to the study of cubic interactions.

One loop corrections due to a cubic vertex
In this section we will repeat the analysis performed for the quartic case 6.1, now for the cubic vertex. The relevant computation is presented with more detail in appendix E.3. One notices two important differences with the quartic case. The one loop correction scales as (V 3 ) 2 since one needs two cubic vertices to form a one loop graph, and the matrix Σ (1-loop) (3) IJ (p, M ) now depends both on the messenger scale cutoff M and the momentum p. In addition for dimensional reasons it is found than in a p/M expansion, the leading term is momentum independent and scales accordingly to the dimension of the related matrix element of the inverse correlator (4.10) similarly to what happens in the quartic case (6.8.) The corrections can then be organised in an expansion n a n ( p M ) 2n where the momentum can appear only in even powers due to rotational invariance of the integrals. We can directly proceed analysing the physical regimes and properties of the cubic corrections.

Implications for the emergent axion propagator
We will now discuss in more detail the physical regimes of the cubic corrections in a similar spirit to 6.2 We will again use the coupling constants scaling in terms of the bare constants as m All the arbitrary constants in this section will depend on these dimensionless parameters.
Using the results of the appendix E.3, we find that the matrix elements Σ   (6.13) with B IJ , C IJ numerical coefficients. These matrix elements take the expected scaling and are presented in a perturbative fashion for small momenta. Notice again that ∆ 1 is the hidden operator dimension while ∆ 2 refers to the SM operator.
From these elements, we obtain the cubic corrected propagator of the emergent field χ up to first order in p/M with a, b, c's numerical constants. The results of the previous section 6.2 apply with the following extra modifications: • There are now further higher derivative terms compared to the quartic case parametrised by a 1 , b 1 , c 1 . All these terms are irrelevant in the IR compared to the leading terms and supressed by powers of the cutoff.
• There is the possibility of new poles (an infinite number of them) arising from all these higher derivative terms. Since to fully clarify such a possibility would require computations up to the cutoff our perturbative approach does not provide a systematic and consistent method to analyse such physical effects. A better approach to study such an infinite number of resonances based on holography is described in section 7.
By studying the leading terms, the conclusions of the previous section remain unaltered. In particular for ∆ 2 = 4 we get that the a and b-terms vanish completely. The c-terms that are left are found to be irrelevant for ∆ 1 > 2 or in other words the effects of "hidden" fields vanish for such operators. These conclusions are in line with the previous sections. Operators of conformal dimension ∆ = 4 do not get any corrections with cutoff dependence even after including perturbative interactions.

The holographic axion
We now investigate the special case where the hidden theory T 1 is a large N holographic theory.
The general action can be written as where the interaction term has been defined in (4.1), S 1 is the action of the holographic theory, and S 2 the action of the SM. Applying the holographic correspondence, we can write 21 where on the left, the expectation value is taken in the holographic theory T 1 . S bulk [a] is the bulk gravity action, z is the holographic coordinate, a is the bulk field dual to the operator O 1 of dimension ∆ = 4 and the gravitational path integral has boundary conditions for a to asymptote to the operator O 2 near the AdS boundary. We have also neglected the other bulk fields. By inserting a functional δ-function we may rewrite (7.2) as If we now integrate φ(x) first in the path integral transform, we obtain the Legendre transform of the Schwinger functional of the bulk axion which becomes the bulk effective action. This corresponds in holography to switching boundary conditions at the AdS boundary from Dirichlet to Neumann, and where k(x) is the expectation value of the operator O 1 . We finally obtain We may imagine the SM action as coupled at the radial scale z 0 ∼ 1/M to the bulk action. Following holographic renormalisation [76,77], we may then rewrite the full bulk+brane action of the emergent axion as S total = S bulk + S brane (7.5) whereâ(x) ≡ a(z 0 , x) is the induced axion on the brane. As we will be interested at energies E M we can ignore higher axion terms like a 2 a on the brane. In the bulk action, (7.5) we have neglected the graviton and other scalar fields dual to other scalar operators of the "hidden" holographic theory T 1 . The factor Z in (7.5) in general depends on the various other scalars fields. For the case of holographic YM this action has been studied in detail in [73,80]. The graviton also couples to the SM action and provides emergent gravity, [45]. Importantly, there is a bulk potential for the axion but it is due to instantons and therefore it is exponentially suppressed at large N. We have therefore neglected it. To all orders in 1/N, the bulk axion has only derivative interactions. Finally the boundary conditions for the bulk action are Neumann. It should be noted that what we have here is a close analogue of the DGP mechanism, [49], with two differences: here we have an axion and also the bulk data are non-trivial.
In the boundary action (7.7) γ is the induced four-dimensional metric. The first term in (7.7) is the coupling of the axion to the SM Instanton densities descending from (4.1) . The kinetic and mass terms of the axion in the brane action come from the quantum effects of the SM fields, as explained at the end of section 5. The ellipsis stands for the rest of the SM action as well as higher derivative corrections to the brane axion field action.
The main difference in the physics of an emergent axion originating in a holographic theory is that due to the strong coupling effects there is an infinity of axionlike resonances coupled to the SM instanton densities. They correspond to the poles of the two-point function of the instanton density of the "hidden" holographic theory. If the holographic theory is gapless, then there is a continuum of modes and as mentioned earlier in such a case the induced axionic interaction is non-local. If the theory has a gap as large-N YM then there is a tower of nearly stable states at large N that are essentially the 0 +− glueball trajectory and act as the KK modes of the bulk axion that couple with variable strengths to the SM instanton densities.
To investigate these interactions we analyze the propagator of the axion on the SM brane. To do this we introduce a δ-function source for the axion on the brane and we solve the bulk+brane equations in the linearized approximation, assuming a trivial profile for the bulk axion 22 while the metric and other scalars have the holographic RG flow profile of a Lorentz-invariant QFT, namely The calculation follows similar ones in [81,82,83,84] which we reproduce here, where we work in Euclidean 4d space and primes stand for derivatives with respect to z. We Fourier transform along the 4 space-time dimensions to obtain (7.10) where p 2 = p i p i is the (Euclidean) momentum squared. Later on we will also use p = p 2 .
To solve this, we must first solve this equation for z > z 0 and for z < z 0 obtaining two branches of the bulk propagator, G IR (p, z) and G U V (p, z) respectively. The IR part, G IR (p, z) depends on a single multiplicative integration constant as the regularity constraints in the interior of the bulk holographic geometry fix the extra integration constant. G U V (p, z) is defined with Neumann boundary conditions at the AdS boundary and depends on two integration constants. In the absence of sources and fluctuations on the SM brane, the propagator is continuous with a discontinuous z-derivative at the SM brane 23 In that case there is a single multiplicative integration constant left and the standard AdS/CFT procedure extracts from this solution the two-point function of the bulk instanton-density. We denote this bulk axion propagator in the absence of the brane as G 0 (p, z; z 0 ) and satisfies In our case the presence of an induced action on the SM brane changes the matching conditions to 13) The general solution can be written in terms of the bulk propagator G 0 with Neumann boundary conditions at the boundary as follows 24 [51] G(p, z; z 0 ) = G 0 (p, z; z 0 ) 1 + (M 2 p 2 + Λ 4 )G 0 (p, z 0 ; z 0 ) (7.14) The propagator on the brane is obtained by setting z = z 0 and becomes The general structure of the bulk axion propagator G 0 is known, [51] and is as follows.
The position of the brane z 0 determines a bulk curvature energy scale, R 0 . In the case of simple bulk RG flows 25 we obtain The IR expansion above is valid for all holographic RG flows. It starts having nonanalytic terms starting at p 4 log p 2 as is the case with the bulk axion field, [51]. The expansion coefficients can be determined either analytically or numerically from the bulk holographic RG flow solution. The UV expansion in (7.16) is given, expectantly, by the flat space result. Using (7.16) we now investigate the axion interaction on the SM brane from (7.15). It is known that G 0 (p, z 0 ; z 0 ) is monotonic as a function of p, vanishes at large p and attains its maximum at p = 0 compatible with (7.16). Therefore at short enough distances, p → ∞, the axion propagator becomes which is the propagator of a massless four-dimensional scalar. For sufficiently small momenta, p m, we obtain (7.18) In a simple holographic theory and for the instanton density we have, [51]  where m is the characteristic scale of the dual QFT, is the IR AdS length andd n are dimensionless numbers of order O(1). The expansion in (7.16) is valid for p m. We may rewrite (7.18) using (7.19) as We may then recast (7.18) as the propagator of a massive four-dimensional scalar with effective mass and coupling strengths where as usual (M P ) 3 ∼ N 2 . Moreover, the coefficient of the p 4 term is dimensionless and of order O(N 2 ).
For M p m we have instead Depending on the hierarchy of the various scales of the problem at intermediate distances, it may be that (M 2 p 2 + Λ 4 )G 0 (p, z 0 ; z 0 ) 1 and the axion propagator may behave as a 5-dimensional massless scalar In such a regime, all axion resonances contribute equitably and the resumed result is as above.

A. Cancellation of gauge anomalies in the messenger sector
In this appendix we discuss the detailed anomaly cancelation requirements for the messenger sector and its coupling to the SM. We assume that QFT N and the SM are separately anomaly free. We must impose that the messenger fermions do not introduce gauge anomalies. For the QFT N gauge group (that we assume to be SU(N)) the condition is that there is an equal number of N andN . Similarly, anomaly freedom for the SM implies that there must be an equal number of 3 and3, 2 and2 (for the cancellation of the Witten SU(2) anomaly) and hypercharge Q andQ.
All spinors are assumed to be left-handed Weyl spinors. In order to allow masses for all the messengers we must have n 3 =n3, n3 =n 3 , n 1 =n 1 , Q 1 = −Q 1 .
In terms of the above table of charges the conditions for absence of anomalies become • SU (N ) 3 3n 3 +3n3+2n 2 +2n2+n 1 = 3n 3 +3n3+2n 2 +2n2+n 1 → n 2 +n2 =n 2 +n2 (A.1) • Witten SU(2) n 2 +n 2 + n2 +n2 = even (automatic) (A.3) • U (1) 3 n 1 Q 3 1 +n 1Q Taking into account anomaly cancelation, we obtain for the following charge assignments If we include a "realistic" SM sector that is fully bifundamental, then we know from previous work on orientifold embeddings of the SM that it must include at least a couple of extra "anomalous" U(1)s, [53,55,48]. These give rise to more anomalies to be considered. We shall not consider them further in this paper.
Ignoring the SM interactions (that are weak) the messengers have a fermionic chiral symmetry that may be broken by mass terms to a vectorial subgroup.
The most important point here is that the overall U (1) A is anomalous. We have This analysis must be amended by replacing the SM with one of the quivers found in orientifolds, [48]. There might be new constraints on the extra anomalous U(1)'s that appear in that case.

B. The calculation of the box diagrams
In this appendix we present some intermediate steps of the amplitude (2.1). Focusing on the first color-ordered diagram A (1234) (box-1 in figure 2) we have Notice here that there is no logarithmic divergence in the full amplitude (sum of all three box diagrams) since, due to gauge invariance, 4 external momentum factors are produced in the numerator by integration. The amplitude is a sum of several terms with different number of γ 5 in the trace. After some computations we get two different classes of terms with no (scalar) or one (proportional to µνρσ ) γ 5 . In the non-chiral case, the second part vanishes.
Following the usual procedure we perform the Feynman trick by shifting the loop momentum p → P in order to have perfect squares of P in the denominator, and we drop all odd parts of P which vanish upon integration. The form of the amplitude becomes A µνρσ g µν g ρσ + g µρ g νσ + g µσ g νρ 4 1 + log The A µνρσ , B µν and C are functions of the external momenta, the external polarizations and the Feynman parameters. For dimensional reasons A does not contain M while B and C have at most M 2 and M 4 terms respectively. The integral over momenta P µ P ν P ρ P σ is divergent. We use dimensional regularization to integrate over loop momentum and get (B.4). The divergence cancels when we sum all box diagrams in figure 2.
In order to simplify our computations, we assume M 2 p 2 's and we expand our expressions by bringing ∆'s in the nominator.
Therefore, all terms in (B.4) become polynomials of the Feynman parameters x 1 , x 2 , x 3 , x 4 which can be easily integrated out giving the final result for the amplitude Adding all box diagrams we get the final result which has the same form as (B.5).

C. The multiscalar case
In this appendix we generalize the analysis of section 4 to the case of multiple scalar interactions linking the theories T 1 and T 2 . We start again from the two decoupled theories T 1 and T 2 and a set of scalar operators, O i (x), i = 1, · · · , M belongin to T 1 and O i (x), a = 1, · · · , M belonging to T 2 .
We define the partition function in the presence of sources and the Schwinger functional which has as an expansion (in momentum space)

Consider now the generating functional for the correlators of
If we ignore three-and higher-point functions we have and In momentum space and for translationally invariant theories In momentum space the action in (C.4) can be written as and the final Schwinger functional reads Therefore the new correlators for O i in momentum space are This formula generalizes the one in (4.11).

D. Renormalization of axion coupling
In this appendix we indicate how the quantum effects of SM (gauge) fields do not generate a mass for the axion in perturbation theory. We use EM as an example and do the calculation at one-loop.
Consider the theory defined by the (bare) action 1) involving a scalar field a and a U(1) gauge field A µ . µνρσ is the standard completely antisymmetric Levi-Civita tensor, with 0123 = 1.
We vary the action 2) and obtain the equations of motion by integrating by parts We add the standard gauge fixing term 1 2g 2 ξ (∂ µ A µ ) 2 to the Lagrangian and derive the Feynman rules in momentum space. Fourier transforming the action and scaling the gauge fields so that the kinetic term is simply normalized we obtain from which we read the photon propagator the scalar propagator as well as the scalar-photon-photon vertex derivative terms. In particular note that there is no cutoff-dependent mass generated by the one-loop corrections. The general expression for this renormalization to all loop orders involves where G(p) is the two-point function of the instanton density, in momentum space. It is well known, that in perturbation theory this correlation function has no constant piece. For EM this is the whole story. However in a non-abelian theory strong YM dynamics generate a constant piece that is related to the topological susceptibility, [2,3,12,72,74,75]. This will contribute a renormalization of the axion mass term as argued in the main text.

E. One loop corrections through interactions E.1 Path integrals with indices
In order to set up the notations for the one-loop computation we will now consider path integrals where one has several fields combined into a vector. The latin indices I, K = (1, 2) take two values and denote the φ 1 , φ 2 fields. We work in Euclidean space, but is easy to extend these formulae into Lorentzian time by Wick rotating τ = −it. Let us define the path integral where A IK is a matrix operator and thus the determinant is generically both a functional and a matrix one. To go to the non linear level one deforms this Gaussian theory with a generic potential V (φ I ) so that From this formal expression, upon taking derivatives one can construct all the Feynman diagrams of the theory and compute correlation functions once the field content of the theory is specified. Note also that we do not write explicitly the overall normalization since there are functional determinants involved that need regularization and a more thorough study. For the correlation functions that we are interested in, such subtleties do not matter.
We will now study the one loop corrections to (6.2), (4.10), (4.11) due to interaction terms (6.3). To read the one-loop corrected action (in a cutoff regulated fashion), we simply use the abstract expressions above to match any Feynman diagram with the corresponding integral. The momenta running through the loops are defined up to a cutoff Λ which in the main text is the messenger scale M . To fully renormalise the interacting theory, one can further follow a Wilsonian procedure using the path integral above. Here one makes a foliation in momentum space with a cutoff Λ and integrates out the "fast" modes fluctuating in a thin slice of momenta (Λ − δΛ, Λ). These are the modes that run in the loop, but a general diagram has also dependence on the external momenta. This procedure for a general graph leads to the one-loop renormalisation of the corresponding term in the effective action.

E.2 Quartic vertex
The first example is a potential with a quartic vertex V (4) IJKL (p). In this case one has the one loop diagram with two external lines (tadpole graph) given in fig. 3 that evaluates to k J I Figure 3: The one-loop "tadpole" graph This expression is readily derived by expanding (E.2) and keeping the quartic vertex term. In our case the form of A −1 IK (p) = G IK (p) is given in (4.11). We will now compute it for the simplest tensor structure for the quartic vertex, for which V (4) with N = V 4 /16π 2 . Expanding in powers of the cutoff Λ using (E.11) one finds the matrix (6.8) given in the main text. Other cases (such as a propagator with a mass gap) follow in a similar fashion.

E.3 Cubic vertex
For the cubic interaction the one-loop correction is represented in fig. 4 and evaluates to Figure 4: The one-loop graph from cubic vertices.
We will be interested again in the case of a common vertex of strength V 3 so that one finds 3! = 6 possible terms that need to be summed over for each element. All the integrals can be re-expressed in terms of the following general form in terms of a normalization prefactor and having indices a, b, c that correspond to various momentum exponents. Such integrals can be computed in a cutoff expansion as before using (E.7), (E.10). To compute them, we use (k + p) 2 = k 2 (1 + 2p · k/k 2 + p 2 /k 2 ), expand for large k and keep only the integrals with even powers of k, since integrals with odd powers of k vanish (they are not rotationally invariant). If we just do power counting we immediately find the result in a similar fashion to the quartic case as an expansion in inverse powers of the cutoff. Due to consistency the first term will start again as in the case of the quartic vertex dictated by dimensional consistency (6.4) but now each power of the cutoff Λ can be traded for a power of external p. There can be no odd powers of p since these integrals are not rotationally invariant and vanish. This of course is a perturbative treatment for small p/Λ.
When p ∼ Λ one needs to perform a more thorough study, since our low-energy approximation breaks down. To simplify the discussion in the main text we have picked a concrete interaction term of the form ∼ V 3 φ 2 1 φ 2 . This is natural in case the "hidden sector" is more strongly coupled than the visible SM and therefore graphs with more hidden sector fields appearing in internal propagators are contributing more.

E.4 Cutoff regulated integrals
We will perform cutoff-regularization to the one loop integrals in order to keep power law divergences. The integrals we need are (for d = 4, a One again finds similarly logarithmically divergent terms in cases where a+1−nb = m with m, n integers.
Every massless vector can be written in terms of Weyl spinors using the spinor completeness relation (for massless Dirac spinors it is −k / = u(k)ū(k) + v(k)v(k)) k αα = k µ σ µ αα = −|k] α k|α kα α = k µσαα µ = −|k α [k| α (F.8) The signs are due to the Minkowski metric that here is chosen to be η = diag(−, +, +, +). The scalar product between two massless momenta can be written in terms of spinors' products where q is a massless (largely arbitrary) momentum different to k, it is an auxiliary variable that accounts for gauge invariance or, equivalently, the incomplete determination of the polarization of a massless gauge vector. In order to illustrate this point, let us consider the change of q into another momentum q. Writing the polarizations in terms of q one obtains µ + (k; q)= where we have used the Schouten identity. The extra term proportional to k µ is the usual term in a gauge transformations, so we can thought of a specific q as a choice of a gauge. Indeed for each polarization we can chose a different auxiliary momentum q so much so that not only k· ± (k; q) = 0 but also q· ± (k; q) = 0. The polarizations µ ± (k; q) so defined have helicity h = ±1, it's easy to see that using the transformations defined above (F.11).
In the computations of diverse amplitudes we encounter all sorts of scalar products between polarizations and momenta with all the possible helicities. To this end we write them here in terms of spinors using the above formulae and obtain: [qk] kp [qp] (F.13) + (p 1 , q 1 )· + (p 2 , q 2 ) = q 1 q 2 [p 1 p 2 ] q 1 p 1 q 2 p 2 − (p 1 , q 1 )· − (p 2 , q 2 ) = [q 1 q 2 ] p 1 p 2 [q 1 p 1 ][q 2 p 2 ] (F.14) + (p 1 , q 1 )· − (p 2 , q 2 ) = q 1 p 2 [p 1 q 2 ] q 1 p 1 [q 2 p 2 ] (F. 15) We can point out the advantages that this formalism yields. Thanks to gauge invariance, the choice of auxiliary momenta is largely free, so we can choose them in order to set as many scalar products as possible to zero. The usual choices are the following: • The product between two polarizations 1 and 2 with equal helicities can be set to zero by choosing q 1 = q 2 as auxiliary momenta; • The product between two polarizations with opposite helicities can be set to zero by choosing either q 1 = p 2 or q 2 = p 1 as auxiliary momenta; • The products between a polarization (k; q) and a momentum p can be set to zero by choosing q = p as auxiliary momentum; We also deal with terms involving the 4-index Levi-Civita tensor. Using momentum conservation 26 they can always be written in terms of at least one polarization and one momentum of the same external state. When the helicity is fixed, the field strength has fixed chirality and is either self-dual or anti-self-dual 16) where f µν = k µ ν (k) − k ν µ (k). This simplification allows us to express the contractions with Levi-Civita tensors in terms of spinor products that can be further manipulated as explained above.