An even lighter QCD axion
DDESY 21-010IFT-UAM/CSIC-20-143FTUAM-20-21
An even lighter QCD axion
Luca Di Luzio a , Belen Gavela b, c , Pablo Quilez a , Andreas Ringwald a a Deutsches Elektronen-Synchrotron DESY,Notkestraße 85, D-22607 Hamburg, Germany b Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid,Cantoblanco, 28049, Madrid, Spain c Instituto de F´ısica Te´orica, IFT-UAM/CSIC,Cantoblanco, 28049, Madrid, Spain
Abstract
We explore whether the axion which solves the strong CP problem can naturallybe much lighter than the canonical QCD axion. The Z N symmetry proposed byHook, with N mirror and degenerate worlds coexisting in Nature and linked by theaxion field, is considered in terms of generic effective axion couplings. We show thatthe total potential is safely approximated by a single cosine in the large N limit, andwe determine the analytical formula for the exponentially suppressed axion mass.The resulting universal enhancement of all axion interactions relative to those of thecanonical QCD axion has a strong impact on the prospects of axion-like particleexperiments such as ALPS II, IAXO and many others. The finite density axionpotential is also analyzed and we show that the Z N asymmetric background of high-density stellar environments sets already significant model-independent constraints:3 ≤ N (cid:46)
47 for an axion scale f a (cid:46) . × GeV, with tantalizing discoveryprospects for any value of f a and down to N ∼ Z N completions are developed: a composite axion one and aKSVZ-like model with improved Peccei-Quinn quality. E-mail: [email protected] , [email protected] , [email protected] , [email protected] a r X i v : . [ h e p - ph ] J a n ontents Z case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Z N axion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Axion potential in the large N limit . . . . . . . . . . . . . . . . . . . . . 10 Z N axion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Composite Z N axion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Ultra-light QCD axions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 A heavier-than-QCD axion . . . . . . . . . . . . . . . . . . . . . . . . . . 17 f a . . . . . . . . . . . . . . . . . . . . . . . 19 Z N axion potential 24B Fourier series of the Z N axion potential 25C Analytical axion mass dependence from hypergeometric functions 26 Introduction
The axion experimental program is in a blooming phase, with several new experimentsand detection concepts promising the exploration of regions of parameter space thoughtto be unreachable until a decade ago. Many of those experiments are simply prototypes,awaiting the jump to become ‘big-experiments’, or, in the case of more consolidatedtechniques, they are still far from saturating their full physics potential. Nonetheless,they sometimes reach sensitivities which go well-beyond astrophysical limits, albeit oftenstill far from the customary QCD axion window.On the other hand, since axion couplings are inherently ultraviolet (UV) dependent,such early stage experiments already provide valuable probes of the QCD axion parameterspace. Imagine for definiteness that ALPS II would detect a signal in 2021, would it bepossible to interpret that as an axion that solves the strong CP problem? Since thestrong CP problem is one of the strongest motivations for new physics, if an axion-likeparticle (ALP) will be ever discovered, there or elsewhere, it would be compelling toexplore whether it had something to do with the strong CP problem. This work exploreswhether wide regions in the ALP parameter space, well outside the traditional QCDaxion band, may correspond to solutions of the strong CP problem. This is a question ofprofound theoretical and experimental relevance.In axion solutions to the strong CP problem both the axion mass and the couplingsto ordinary matter scale as 1 /f a , where f a is the axion decay constant, denoting thescale of new physics. The precise relation between mass and decay constant depends onthe characteristics of the strong interacting sector of the theory. When QCD is the onlyconfining group to which the axion a couples, in which case we denote the axion mass as m QCD a , they are necessarily linked by the relation [3, 4] m QCD a = √ χ QCD f a (cid:39) m π f π √ m u m d m u + m d f a , (1.1)where χ QCD , m π , f π , m u and m d denote respectively the QCD topological susceptibility,the pion mass, its decay constant, and the up and down quark masses. Equation (1.1) iscompletely model-independent as far as QCD is the only source of the axion mass, and itdefines the “canonical QCD axion”, also often called “invisible axion”. For this axion the aG µν ˜ G µν coupling to the gluon strength G µν is directly responsible for the axion mass,since the only source of explicit breaking of the global axial Peccei-Quinn (PQ) symmetryU(1) PQ is its QCD anomaly. The strength of other axion couplings to Standard Model(SM) fields is instead model-dependent: it varies with the matter content of the UVcomplete axion model.In recent years there have been many attempts to enlarge the canonical QCD axionwindow, by considering UV completions of the axion effective Lagrangian which departedfrom the minimal DFSZ [5, 6] and KSVZ [7, 8] constructions. Most approaches actuallyfocussed on the possibility of modifying the Wilson coefficient of specific axion-SM effec-tive operators [9–15]. That is, the size of the coupling coefficients, at fixed f a , is modified.This has for example allowed to populate new regions of the parameter space by movingvertically the axion band in the axion mass versus coupling plane, see Fig. 1 left. Theresults are then “channel specific”: different couplings c are modified differently. That is, via a global chiral U(1) symmetry, exact although hidden (aka spontaneously broken) atthe classical level and explicitly broken by instanton effects at the quantum level [1, 2].
Different approaches to enlarge the parameter space of axions that solve the strongCP problem. The canonical QCD axion relation is represented by the lower black line inthe { m a , c/f a } parameter space, where c denotes a generic effective axion coupling. Verticaldisplacements, possible within pure QCD axion models (i.e. m a = m QCD a ), are depicted on theleft. Horizontal displacements (via enlarged strong gauge sectors) are illustrated on the rightfor the case of a lighter than usual axion to be explored here. The parameter space of solutions can be alternatively changed by varying the axionmass at fixed f a . This corresponds to horizontal displacements of the canonical axion bandin the parameter space, see right panel in Fig. 1. It always requires that the magnitudeof the relation between the axion mass m a and 1 /f a departs from that in Eq. (1.1):the confining sector of the SM must be enlarged beyond QCD. New instanton sourcesgive then additional contributions to the right-hand side of Eq. (1.1). The practicalconsequence is a universal modification of the parameter space of all axion couplings ata given m a , at variance with the vertical displacement scenarios. This feature could apriori allow for the two mechanisms in Fig. 1 to be distinguished. Examples of horizontal enlargement of the parameter space towards the right of thecanonical QCD axion band are heavy axion models that solve the strong CP problemat low scales (e.g. f a ∼ TeV) [18–34]. The present work explores instead left horizontalshifts: true axions that solve the strong CP problem with m a (cid:28) m QCD a . This avenue ismore challenging, since it requires a new source of PQ breaking aligned with QCD, whosecontribution to the axion mass needs to almost cancel that from QCD without relyingon fine-tunings.A possible mechanism to achieve this lighter-than-usual true axion in a technicallynatural way was recently put forth by Hook [35], in terms of a discrete Z N symmetry. N mirror and degenerate worlds would coexist in Nature, linked by an axion field whichimplements non-linearly the Z N symmetry. One of those worlds is our SM one. All theconfining sectors contribute now to the right-hand side of Eq. (1.1), conspiring by sym-metry to suppress the axion mass without spoiling the solution to the strong CP problem. For instance, via the measurement of the axion coupling to the neutron electric dipole moment(nEDM) operator at CASPER-electric [16, 17], in case the axion would also account for dark matter(DM). The axion-to-nEDM coupling directly follows from the m a – f a relation and so it is unmodified instandard approaches to axion coupling enhancements (left panel in Fig. 1) that still rely on Eq. (1.1). This setup for N = 2 had previously led instead to an enhancement of the axion mass [36], becausethe axion field was assumed to be invariant under the Z transformation. f a , a N -dependent reduced axion mass in spite of allconfining scales being equal to Λ QCD . In other words, for a given value of m a it followsa universal enhancement of all axion interactions relative to those of the canonical QCDaxion. In this paper, we expand on the mathematical properties of the implementation ofthe Z N symmetry and determine the analytic form of the exponential suppression of theaxion mass and its potential in the large N limit. The phenomenological analysis of thenumber of possible mirror worlds N will be next carried out with present and projecteddata.The study will also explore the Z N axion potential at finite density, to confront presentconstraints and prospects from very dense stellar objects and gravitational waves. Ithas been recently pointed out in [37, 38] that a generic reduced-mass axion leads tostrong effects on those systems, raising the effective mass in the dense media. In thescenario considered here, a stellar background made only of SM matter is by nature Z N -asymmetric: we will show analytically how such an asymmetric background breaksthe cancellations which guaranteed an exponentially suppressed axion mass for the Z N symmetric vacuum potential. Limits on the number of possible worlds will be obtainedin turn.The theoretical framework to be used throughout the work described above is that ofeffective axion couplings. Nevertheless, two concrete UV completions of the Z N scenariounder consideration will be developed as well: a model `a la KSVZ [7, 39], and a compositemodel `a la Choi-Kim [40, 41]. The status of the Peccei-Quinn (PQ) quality problem willbe also addressed.An important remark is that we will consider in this paper experiments that can testthe solution to the strong CP problem without further assumptions. Indeed, it is mostrelevant to get a clear panorama on the strong CP problem by itself, given its fundamentalcharacter. In particular, we will not discuss axion or ALP experiments that do rely on theassumption that the DM of the Universe may be constituted by axions. The cosmologicalevolution of the axion field in the Z N scenario under discussion and its contribution tothe DM relic abundance departs drastically from the standard case, and it is discussedin a companion paper [42].The structure of the present paper can be easily inferred from the Table of Contents. In Ref. [35] it was shown how to naturally down-tune the axion mass from its naturalQCD value in Eq. (1.1), exploiting the analyticity structure of the QCD axion potentialin the presence of a Z N symmetry. For pedagogical purposes, before turning to thegeneric Z N case we analyze the (unsuccessful) case of a Z symmetry: the SM plus onedegenerate mirror world linked by an axion which realizes the symmetry non-linearly. Z case Consider the SM plus a complete copy SM (cid:48) , related via a Z symmetry which exchangeseach SM field with its mirror counterpart, while the axion field is shifted by π : Z : SM ←→ SM (cid:48) (2.1) a −→ a + πf a . (2.2)5 − − a/f a . . . . . . . V ( a / f a ) / m π f π V SM ( a/f a ) V SM ( a/f a ) V ( a/f a ) Figure 2: Z axion potential. The mirror contribution to the axion potential V SM (cid:48) ( a/f a )(in green) partially cancels that of the SM, V SM ( a/f a ) (in blue), leading to a total shallowerpotential V ( a/f a ) (in orange). The total potential has a maximum in a/f a = 0 and thus this Z axion does not solve the SM strong CP problem. The Lagrangian, including the anomalous effective couplings of the axion to SM fields,then reads L = L SM + L SM (cid:48) + α s π (cid:16) af a − θ (cid:17) G (cid:101) G + α s π (cid:16) af a − θ + π (cid:17) G (cid:48) (cid:101) G (cid:48) + . . . , (2.3)where θ parametrizes the anomalous QCD coupling, α s is the QCD fine-structure con-stant, the Lorentz indices of the field strength G µν have been obviated, and the dotsstand for possible Z -symmetric portals between the two mirror worlds (see Sect. 2.2.1).Without loss of generality, we can perform a uniform shift in a such that the θ term inEq. (2.3) is set to zero. Therefore, the effective θ -parameter of the SM corresponds to θ eff ≡ (cid:104) a (cid:105) /f a , where (cid:104) a (cid:105) denotes the vacuum expectation value (vev) of the axion field.In the case of an exact Z symmetry, all couplings and masses of the mirror world andthe SM would coincide with the exception of the effective θ -parameter. It is this difference(namely the π shift in the effective θ -parameters of the SM and its mirror) the one respon-sible for displaced contributions to the total axion potential, with destructive interferenceeffects. Were the QCD axion potential to be a simple cosine, the total potential wouldvanish because the two contributions (from QCD and mirror QCD) would have exactlythe same size but opposite sign, i.e. ∝ cos( a/f a ) and ∝ cos( a/f a + π ) = − cos( a/f a )respectively. However, for the true chiral axion potential [43–45] the exact cancellationdisappears and a residual potential –and thus a non-zero axion mass– remains, which atleading chiral order reads (keeping only two flavours) V ( a ) = − m π f π m u + m d (cid:40)(cid:115) m u + m d + 2 m u m d cos (cid:18) af a (cid:19) + (cid:115) m u + m d − m u m d cos (cid:18) af a (cid:19)(cid:41) . (2.4)6his Z -symmetric world would not solve the strong CP problem, though, because a/f a =0 is a maximum of the axion potential, as illustrated in Fig. 2. Indeed, as already pointedout in Ref. [35], a/f a = 0 is a minimum of the potential only for odd values of N , whileit is a maximum for N . Thefore, the simplest viable axion model that solves the strongCP problem with a reduced axion mass incorporates a Z symmetry. Z N axion We consider now N copies of the SM that are interchanged under a Z N symmetry whichis non-linearly realized by the axion field: Z N : SM k −→ SM k +1 (mod N ) (2.5) a −→ a + 2 πk N f a , (2.6)with k = 0 , . . . , N −
1. One of those worlds will be our SM one. The most generalLagrangian implementing this symmetry describes N mirror worlds whose couplings takeexactly the same values as in the SM, with the exception of the effective θ -parameter: foreach copy the effective θ value is shifted by 2 π/ N with respect to that in the neighbour k sector, L = N − (cid:88) k =0 (cid:20) L SM k + α s π (cid:18) θ a + 2 πk N (cid:19) G k (cid:101) G k (cid:21) + . . . (2.7)where L SM k denotes exact copies of the SM total Lagrangian excluding the strong anoma-lous coupling, and the dots stand for Z N -symmetric portal couplings that may connectthose different sectors (to be discussed in Sect. 2.2.1). In this equation θ a ≡ a/f a is theangular axion field defined in the interval [ − π, π ), and a universal (equal for all k sectors)bare theta parameter has been set to zero via an overall shift of the axion field. The SMis identified from now on with the k = 0 sector: to ease the notation, the label k = 0on SM quantities will be often dropped below. Each QCD k sector contributes to the θ a potential, which in the 2-flavour leading order chiral expansion reads V N ( θ a ) = − A N − (cid:88) k =0 (cid:115) z + 2 z cos (cid:18) θ a + 2 πk N (cid:19) , (2.8)where z ≡ m u /m d ≈ . , A ≡ Σ m d ≈ χ (1 + z ) /z , (2.9)and Σ ≡ − (cid:104) uu (cid:105) = − (cid:10) dd (cid:11) = m π f π / ( m u + m d ) (2.10)denotes the chiral condensate [44], while χ ≈ (75 MeV) is the zero temperature QCDtopological susceptibility [45, 46]. Alternatively, the total Z N axion potential can bewritten as V N ( θ a ) = − m π f π N − (cid:88) k =0 (cid:115) − β sin (cid:18) θ a πk N (cid:19) , (2.11)where β ≡ m u m d / ( m u + m d ) = 4 z/ (1 + z ) .7or any N , θ a = 0 is an extrema of the axion potential. Indeed, using the propertysin(2 π ( N − k ) / N ) = − sin (2 πk/ N ) it is straightforward to see that ∂V N ( θ a ) ∂θ a (cid:12)(cid:12)(cid:12)(cid:12) θ a =0 = m π f π f a β N − (cid:88) k =0 sin (cid:0) πk N (cid:1)(cid:113) − β sin (cid:0) πk N (cid:1) = 0 . (2.12)The same holds for any θ a = 2 πn/ N with n ∈ Z , because of the periodicity of thepotential. For N odd the potential V ( θ a ) has N minima located at θ a = {± π(cid:96)/ N } for (cid:96) = 0 , , . . . , N − , (2.13)which includes the origin θ a = 0, while for N even the origin becomes a maximum.This result –valid for any N – can be shown for instance using the exact Fourier seriesexpansion of the potential in Eqs. (2.8)-(2.11) (see final part of Appendix C). It followsthat N odd is required in order to solve the SM strong CP problem (albeit with a 1 / N tuning in the cosmological evolution [35, 42]). The k (cid:54) = 0 worlds have instead non-zeroeffective θ -parameters: θ k ≡ πk/ N for (cid:104) θ a (cid:105) = 0, see Eq. (2.7). A typical shape of theaxion potential for N = 3 is illustrated in Fig. 3. − − − θ a . . . . . V ( θ a ) / m π f π V N =3 ( θ a ) V k =0 ( θ a ) V k =1 ( θ a ) V k =2 ( θ a ) Figure 3: Z axion potential. The contributions from the N = 3 worlds partially canceleach other, leading to an exponentially small total potential V N =3 ( θ a ) (in blue) that exhibits aminimum in θ a = 0. The different effective θ k values translate into slightly different masses for the pionmass in each mirror world, m π ( θ k ). At quadratic order in m π a reduction factor of up to ∼ √ m π ( θ k ) = m π (cid:115) − m u m d ( m u + m d ) sin (cid:18) πk N (cid:19) . (2.14)8nterestingly, nuclear physics would be drastically different in the different mirror copies.In particular, a new scalar pion ( π k ) to nucleon ( N k ) coupling is generated in all worldsbut the SM one (see e.g. Ref. [47]): L χ PT ⊃ c + N − (cid:88) k =0 m u m d sin θ k [ m u + m d + 2 m u m d cos θ k ] / π ak f π N k τ a N k , (2.15)where c + is an O (1) low-energy constant of the baryon chiral Lagrangian. Its impact onthe cosmological histories of the mirror worlds is discussed in Ref. [42] for the Z N scenariounder discussion.Overall, for our world to be that with vanishing effective θ , the ∼
10 orders of magni-tude tuning required by the SM strong CP problem has been traded by a 1 / N adjustment,while N could a priori be as low as N = 3. Renormalizable portals between the SM and its mirror copies (left implicit in Eq. (2.7))are allowed by the Z N symmetry. In the following, we classify for completeness the portaloperators connecting the different k sectors. Higgs portals
The most general Z N symmetric scalar potential for the Higgs doublets H k of the differentmirror worlds includes terms of the form V ( H k ) ⊃ ( N − / (cid:88) i =1 κ i N − (cid:88) k =0 (cid:18) | H k | − v (cid:19) (cid:18) | H k + i | − v (cid:19) (cid:12)(cid:12)(cid:12) (mod N ) , (2.16)where v denotes the Higgs vev and κ i are dimensionless parameters. Note that the Z N -symmetric mixings between different worlds may include next-neighbour, next-to-nextneighbour etc. interactions. All κ i ≥ terms provide renormalizable portals between themirror Higgs copies ( H k (cid:54) =0 ) and the SM Higgs ( H k =0 ). Kinetic mixing
Terms mixing the U(1) kY hypercharge field strengths of mirror worlds are a priori alsoallowed by the Z N symmetry, L ⊃ ( N − / (cid:88) i =1 (cid:15) i N − (cid:88) k =0 F µνk F µν, k + i (cid:12)(cid:12) (mod N ) , (2.17)where F µνk denote here the k -hypercharge field strenghts and (cid:15) i are free dimensionlessparameters.The above renormalizable portals are subject to strong cosmological constraints, asdiscussed in Ref. [42]. This can suggest a ‘naturalness’ issue for the Higgs and the kinetic Although we work in the exact Z N limit, cosmological considerations require the temperature of theSM thermal bath to be higher than that of the other k (cid:54) = 0 sectors [48–50]. Mechanisms to achieve thesedifferent temperatures will be discussed in Ref. [42]. κ i (cid:54) =0 and (cid:15) i (cid:54) =0 → P N symmetry (namely an indepen-dent space-time Poincar´e transformation P in each sector). Those couplings are thenprotected from receiving radiative corrections other than those induced by the explicit P N breaking due to gravitational and axion-mediated interactions, which are presumablysmall. In addition, other terms in the scalar potential which depend on the details of theUV completion of the Z N axion scenario may be present and strongly constrained; anexample is given below in Sect. 3.1. N limit It is non-trivial to sum the series which defines the axion potential, Eq. (2.11). However,the presence of the Z N symmetry allows for the application of powerful mathematicaltools related to its Fourier decomposition and holomorphicity properties, that lead tosimplified expressions in the large N limit. As first noticed in Ref. [35], the fact that the potential in Eq. (2.11) corresponds to aRiemann sum allows one to express it as an integral plus subleading terms, V N ( θ a ) = N − (cid:88) k =0 V (cid:18) θ a + 2 πk N (cid:19) = N π (cid:90) π V ( x ) dx + O ( N ) , (2.18)where the definition of each single-world potential, V (cid:0) θ a + πk N (cid:1) , can be read off Eq. (2.8).Most importantly, the integral does not depend on the field θ a and the amplitude of theaxion potential is thus solely contained in the subleading terms. The latter are nothingbut the error E committed in approximating the Riemann sum by an integral, E N ( V ) = (cid:90) π V ( x ) dx − π N N − (cid:88) k =0 V (cid:18) θ a + 2 πk N (cid:19) . (2.19)Powerful theorems exist that describe the fast convergence of this approximation. It canbe shown, applying complex analysis, that if some conditions are satisfied the convergenceof the rectangular rule is exponential (see e.g. Section 3 in Ref. [53]). More precisely, if V ( θ a ) is a 2 π -periodic function and it can be extended to a holomorphic function V ( w )in a rectangle from 0 to 2 π and from − ib to + ib , then the error of the rectangular rule isconstrained as | E N ( V ) | ≤ πMe N b − , (2.20)where M is an upper limit on V ( w ) in the rectangular region defined above. As aconsequence, the axion mass will be exponentially suppressed for large N . More indetail, let us apply the theorem to the second derivative of the potential, V (cid:48)(cid:48) ( θ a ) = − m π f π z z (cid:0) z (cid:1) cos ( θ a ) + z [3 + cos ( θ a / (cid:2) z + 2 z cos ( θ a ) (cid:3) / , (2.21)10hich can be extended in the complex plane to a holomorphic function until the expressionunder the square root vanishes. Indeed, this function has branch points in w cut = π ± i log z . (2.22)Naively, it is tempting to apply the theorem assuming b = log z in Eq. (2.20). Thisis not possible though, since V (cid:48)(cid:48) ( w ) is not bounded in the rectangular region, due to adivergence in the branch point. As we show in Appendix A, it is possible to optimizethe bound obtained above on the axion mass ( V (cid:48)(cid:48) ( θ a ) /f a ) by allowing a departure fromlog z , b = log z + ∆ b , which leads to ∆ b = 32 1 N , (2.23)where the factor 3 / w cut . Implementing this result in Eq. (2.20), it follows that m a f a ≤ (cid:12)(cid:12) E N ( V (cid:48)(cid:48) ) (cid:12)(cid:12) ≤ πm π f π (cid:114) − z z (cid:18) (cid:19) / N / e − / z −N − . (2.24)In Fig. 4 we compare this analytical bound with the numerical result: our analyticalbound captures the correct dependence on N of the Z N axion mass, m a f a ∝ m π f π (cid:114) − z z N / z N , (2.25)although it misses the overall constant factor. The overall factor will be analyticallydetermined in the following. Nevertheless, the discussion above has the two-fold interestof determining the correct exponential suppression and of being very general, as it onlyrelies on the holomorphicity structure of the potential, and not on the specific form ittakes. As a consequence, the exponential suppression of the axion mass is not spoiledwhen considering the subleading chiral corrections to Eq. (2.11). It is possible to gain further physical insight on the origin of the cancellations in thepotential by constructing its Fourier series expansion. As shown in Appendix B, theFourier series of any scalar potential respecting the Z N shift symmetry only receivescontributions from modes that are multiples of N . Moreover, if the potential can bewritten as a sum of shifted contributions, as it is the case for the Z N axion under discussion–see Eq. (2.18)– then the Fourier series of the total potential V N ( θ a ) can be easily obtainedin terms of the Fourier series of a single V ( θ a ) term, leading to V N ( θ a ) = 2 N ∞ (cid:88) t =1 ˆ V ( t N ) cos( t N θ a ) , (2.26)where ˆ V ( n ) denotes the coefficient of the Fourier series for the single-world potential V ( θ a ), ˆ V ( n ) = − m π f π z (cid:90) π cos( nt ) (cid:113) z + 2 z cos ( t ) dt . (2.27) This result coincides with that in Ref. [35], which defines a = log( c + (cid:112) c − c = ( m u + m d ) m u m d − a can be simplified as a = log( m d /m u ) = − log z .
11t is convenient to express this integral in terms of the Gauss hypergeometric function(see Appendix C and Ref. [54] for conventions and relevant properties),ˆ V ( n ) = ( − n +1 m π f π z z n Γ( n − / − / n ! F (cid:18) − / , n − / n + 1 (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) . (2.28)As shown in Appendix C, in the large N limit this expression further simplifies toˆ V ( n ) (cid:39) ( − n m π f π √ π (cid:114) − z z n − / z n , (2.29)leading to the following expression for the total potential V N ( θ a ) (cid:39) m π f π √ π (cid:114) − z z N − / ∞ (cid:88) t =1 ( − t N t − / z t N cos( t N θ a ) (cid:39) m π f π √ π (cid:114) − z z N − / ( − N z N cos( N θ a ) , (2.30)where in the second line we have kept only the first mode in the expansion, as the highermodes are exponentially suppressed with respect to it. The total potential is thus safelyapproximated by a single cosine . It trivially follows from Eq. (2.30) that θ a = 0 is aminimum of the total potential for N odd, and a maximum for N even. Here and allthrough this work purely constant terms in the potential are obviated, as they have noimpact on the axion mass.Eq. (2.30) can be rewritten as V N ( θ a ) (cid:39) − m a f a N cos( N θ a ) , (2.31)where the Z N axion mass m a in the large N limit is finally given by a compact andanalytical formula, m a f a (cid:39) m π f π √ π (cid:114) − z z N / z N . (2.32)The overall coefficient is thus determined, in addition to exhibiting the z N exponentialsuppression of the potential and the specific N dependence previously argued in Eq. (2.25)from holomorphicity arguments. In summary, in the large N limit the axion mass isreduced with respect to that of the QCD axion by a factor (cid:18) m a m QCD a (cid:19) (cid:39) √ π (cid:112) − z (1 + z ) N / z N − , (2.33)where m QCD a denotes the mass of the canonical QCD axion as given in Eq. (1.1). Thisratio is illustrated in Fig. 4, which compares the numerical behaviour with: a ) the analyt-ical dependence previously proposed in Ref. [35]; b ) that from the holomorphicity boundin Eq. (2.24); c ) the full analytical result in Eq. (2.33). Our analytical results improveon previous ones by Hook on a number of aspects: i ) the explicit determination of theexponential behavior controlled by z N ∼ −N ; ii ) the improved N dependence from thefactor N / ; iii ) the z -dependence of the axion mass in (cid:113) − z z ; iv ) the determination ofthe prefactor 1 / √ π .In practice, the large N results in Eqs. (2.30)-(2.33) turn out to be an excellentapproximation already for N = 3. Note that, although N = 1 denotes the SM world, m QCD a does not correspond to N = 1 in Eq. (2.32),because the latter is only valid in the large N limit.
10 20 30 40 50 60 70 80 90 100
Number of worlds N − − − − − − − − m a m QCD a Numerical resultHook’s fit m a f a ’ m π f π z (1+ z ) (1 / N Holomorphicity boundThis work m a f a ’ m π f π √ π q − z z N / z N Figure 4:
Comparison between different evaluations of the axion mass as a function of N . Ourlarge- N analytical result in Eq. (2.33) (green curve) provides a remarkably good approximationto the numerical evaluation (dots). Up to this point, the analysis has been largely independent from the precise UV comple-tion of the Z N axion scenario. For the sake of illustration, in this section we provide twoUV completions of the axion effective Lagrangian in Eq. (2.7). We also briefly discuss analternative implementation of the Z N symmetry in which the resulting axion is heavierthan usual (rather than lighter). Z N axion Consider N copies of vector-like Dirac fermions Q k ( k = 0 , . . . , N −
1) transforming inthe fundamental representation of QCD k , together with a gauge singlet complex scalar S . The action of the Z N symmetry on these fields is postulated to be Z N : Q k → Q k +1 (mod N ) , (3.1) S → e πi/ N S , (3.2)while the SM Lagrangian and its copies obey Eq. (2.5) under Z N . The most generalLagrangian containing the new degrees of freedom then reads L UV = | ∂ µ S| + N − (cid:88) k =0 (cid:104) Q k i / DQ k + ye πik/ N SQ k P R Q k + h.c. (cid:105) − V ( S , H k ) , (3.3)where P R ≡ (1 + γ ) /
2. It exhibits an accidental U(1) PQ symmetryU(1) PQ : Q k → e − iγ α Q k , (3.4)13 → e iα S , (3.5)that is spontaneously broken by the vev of S , v S , via a proper ‘mexican-hat’ potential V ( S , H k ), whose structure is discussed below. Decomposing the S field in a polar basis, S = 1 √ v S + ρ ) e i av S , (3.6)in terms of canonically normalized radial ( ρ ) and axion modes, the latter can be rotatedaway from the Yukawa term in Eq. (3.3) via an axion-dependent axial transformation Q k → e − iγ (cid:16) a v S + πk N (cid:17) Q k . (3.7)The heavy quarks, with real mass m Q k = y S v S √ , can next be integrated out in orderto obtain the low-energy axion effective field theory. Because the transformation inEq. (3.7) is QCD k anomalous, with anomaly factor 2 N k = 1, the resulting axion effectiveLagrangian is given by δ L UV = N − (cid:88) k =0 α s π (cid:18) av S + 2 πk N (cid:19) G k (cid:101) G k , (3.8)which yields precisely Eq. (2.7), after the identification v S = f a .Furthermore, the presence of the singlet scalar S introduces new scalar portals betweenthe SM and its mirror worlds, in addition to the generic ones in Eq. (2.16). The scalarpotential in the latter equation should thus be enlarged by V ( H k ) −→ V ( S , H k ) = V ( H k ) + δ V , (3.9)with δ V = λ S (cid:18) |S| − f a (cid:19) + κ S (cid:18) |S| − f a (cid:19) N − (cid:88) k =0 (cid:18) | H k | − v (cid:19) . (3.10)Note that, because the Higgs vev v is the same in all k sectors due to the unbroken Z N symmetry, the required hierarchy of scales is obtained with a single fine-tuning between v and f a , as for elementary canonical QCD axions.It is also possible to choose the representations of the Q k fields to transform non-trivially under the electroweak k gauge groups, so that they could e.g. mix with SM k quarks in a Z N invariant way and decay efficiently in the early Universe, thus avoidingpossible issues with colored/charged stable relics in the SM sector [9, 11]. Depending onthe Q k quantum numbers, this would change in turn the value of the electromagnetic-to-QCD anomaly ratio of the PQ current, usually denoted as E/N , which enters theaxion-photon coupling.
The threat posed on traditional QCD axion models by quantum non-perturbative grav-itational corrections [55–63] may also affect the models discussed here, as f a is not veryfar from the Planck scale. These contributions are usually parametrized via effectiveoperators, suppressed by powers of the Planck mass, that could explicitly violate the PQsymmetry and thus spoil the solution to the strong CP problem [55–58]. Note that we crucially removed also the k -dependent phases from the Yukawas, in order to properlyintegrate out the heavy Q k fields. UV sources of PQ breaking can be avoided in some invisible axion constructions within a variety ofextra assumptions or frameworks [64–74], or be arguably negligible under certain conditions [75]. N f a [ G e V ] PQ protected
Figure 5:
Parameter space in the {N , f a } plane that is free from the PQ quality problem, withinthe KSVZ-like UV completion of the reduced-mass Z N axion, for the PQ-breaking parametervalues indicated in the text. In the context of the KSVZ Z N axion model above, the exponentially small axionmass could seem to worsen this threat, increasing the sensitivity to explicit PQ-breakingeffective operators. Interestingly, promoting the in built Z N symmetry to a gauge symme-try leads to an accidental U(1) PQ invariance, that for large N is efficiently protected fromthose extra sources of explicit breaking. Indeed, the lowest-dimensional PQ-violating op-erator in the scalar potential compatible with the Z N symmetry is S N , leading to anexplicitly PQ-breaking contribution to the potential of the form V PQ − break . = k S N M N − + h.c. ⊃ | k | N / − f N a M N − cos ( N θ a + δ ) , (3.11)where M Pl = 1 . × GeV is the Planck mass and δ ≡ Arg k . Considering now V N ( θ a )+ V PQ − break . , expanding for small θ a the axion potential V N ( θ a ) ≈ V N (0)+ m a f a θ a ,and solving the tadpole equation, the induced effective θ parameter in the SM sector reads (cid:104) θ a (cid:105) (cid:39) | k | N f N a M sin δ N / − m a f a M N Pl − | k | N f N a M cos δ (cid:39) √ π | k | sin δ (cid:114) z − z M m π f π √N (cid:18) f a √ zM Pl (cid:19) N , (3.12)where m a from Eq. (2.32) has been used, and in the last step we neglected the secondterm in the denominator in the first line of Eq. (3.12): this is always justified in the (cid:104) θ a (cid:105) (cid:46) − regime.In summary, unlike the customary ad-hoc Z N protection mechanism for the standardKSVZ axion, in the Z N axion scenario under discussion the discrete symmetry is alreadypresent by construction. Note that the scaling with N is slightly different as comparedto the standard KSVZ axion, due to the enhancement factor 1 /z N . But eventually the( f a /M Pl ) N suppression dominates and provides an efficient protection mechanism, eventhough the axion mass is exponentially suppressed. For the sake of an estimate, Fig. 515hows the regions in the {N , f a } plane that saturate the nEDM bound for | k | = 1 andsin δ = 1. Z N axion It is also possible to construct a UV completion of the Z N scenario which correspondsto a dynamical (composite) axion `a la Kim-Choi [40, 41], without extending its exoticfermionic content. In the original version of that model, the SM fields are not chargedunder the PQ symmetry while two exotic massless quarks, ψ and χ , transform underan extra confining “axi-color” group SU( (cid:101) N ) a and one of them, ψ , is also a triplet ofQCD. Upon confinement of the axi-color group at a large scale Λ a ∼ f a (cid:29) Λ QCD , pseudo-Goldstone bosons composed of the exotic quarks emerge. All but one of them are colouredunder QCD and become safely heavy. The light remaining one is the composite axion,whose mass obeys the usual formula for QCD axions Eq. (1.1).We implement the Kim-Choi idea in the framework of our Z N framework withoutincreasing the number of massless exotic fermions representations. The fermion ψ issimply extended to be now a triplet under all QCD k mirror sectors, see Table 1. Theaxion field will thus be unique and will couple to all anomalous terms.SU( (cid:101) N ) a SU(3) c, . . . SU(3) c, k . . .
SU(3) c, N ψ (cid:50) . . . . . . χ (cid:50) . . . . . . (3.13)Table 1: Exotic fermionic sector of the Z N composite axion model. Upon SU( (cid:101) N ) a confinement at the large scale of order f a , the QCD k couplings α ks canbe neglected, and therefore a large global flavor symmetry arises in the exotic fermionicsector: SU(3 N + 1) L × SU(3 N + 1) R × U(1) V . This symmetry is spontaneously brokendown to SU(3 N + 1) L + R × U(1) V by the exotic fermion condensates. Among the resultingGoldstone bosons, the QCD k singlet corresponds to the composite axion. Its associatedPQ current reads (with f PQ ≡ (cid:101) N f a ) j µ PQ = ψγ µ γ ψ − N χγ µ γ χ ≡ f PQ ∂ µ a , (3.14)which corresponds to the only element of the Cartan sub-algebra of SU(3 N + 1) that hasa vanishing anomaly coefficient with respect to SU( (cid:101) N ) a , but a non-vanishing one withrespect to all the QCD k gauge groups.Without further elements the model would be viable, but all mirror worlds wouldhave the same θ -parameter: a heavier than usual axion would result. A simple Z N implementation which leads instead to relatively shifted potentials, and thus to a reducedaxion mass, is to have a relative phase between the argument of the determinant of thequark mass matrix of the mirror worlds,arg (det ( Y u Y d )) k +1 = arg (det ( Y u Y d )) k + 2 π N , (3.15) The U(1) A of the exotic sector is explicitly broken by the SU( (cid:101) N a ) anomaly. Y u and Y d denote the Yukawa matrices for the up and down quark sectors, re-spectively. One of the many possible Z N charge assignments for the quarks that yieldEq. (3.15) is that in which only the right-handed up quarks would transform as Z N : U kR → e i π/ (3 N ) U k +1 R , (3.16)corresponding to a Yukawa quark Lagrangian of the form L Y = − N − (cid:88) k =0 (cid:8) e i πk/ (3 N ) Q L Y u (cid:101) HU R + Q L Y d HD R (cid:9) k + h.c. . (3.17)The resulting low-energy axion effective field theory is then the desired one as in Eq. (2.7).In this Z N composite axion model only the exotic fermions are charged under thePQ symmetry, while the Z N charges are carried solely by SM quarks. This means thatthe Z N and PQ symmetries are not directly linked. As a consequence, gauging the in-built Z N symmetry would not soften the PQ quality problem, contrary to the case ofthe KSVZ Z N axion model discussed earlier above. Our Z N composite axion model isthen subject to the usual PQ quality threat. Standard softening solutions often appliedto composite axion models could be explored, e.g. those based on a chiral gauging of theglobal symmetry of the coset space or on introducing a moose structure [64, 67, 71, 76]. The term ultra-light axions usually refers to the mass range m a ∈ (cid:2) − , − (cid:3) eV (withthe extrema of the interval corresponding respectively to an axion Compton wavelengthof the size of the Hubble horizon and to the Schwarzschild radius of a stellar mass blackhole). As a theoretical motivation for ultra-light axions, the so-called string Axiverse[77] is often invoked, according to which a plenitude of ultra-light axions populatingmass regions down to the Hubble scale 10 − eV is a generic prediction of String Theory,although without a direct reference to the solution of the strong CP problem. On theother hand, according to the usual QCD mass vs. f a relation, Eq. (1.1), axion massesbelow the peV correspond to axion decay constants larger than the Planck mass, andhence they are never entertained within canonical QCD axion models. The Z N axionframework discussed in the present work allows in contrast to populate the sub-peV axionmass region while keeping sub-Planckian axion decay constants, with the advantage ofproviding as well a direct solution to the strong CP problem. As shown in Sec. 4.2, thetantalizing prospects for testing the Z N scenario, through observational data on verydense stellar objects and gravitational waves, can sweep through the discovery region ofthe ultra-light axion range. A remark is in order regarding the Z N charge of the axion in the different sectors. Ifthe implementation of the Z N symmetry would be such that the N world replicas are Note that a factor of 1 / See e.g. Ref. [78] for an ultra-light scalar field whose mass is protected by a discrete Z N symmetrybut does not solve the strong CP problem. Z N : SM k −→ SM k +1 (mod N ) (3.18) a −→ a , (3.19)the potentials of the different mirror worlds would not be relatively shifted but exactlysuperpose. The axion would then be a factor √N heavier than the usual QCD axionin Eq. (1.1). This scenario was proposed in Ref. [36] for a Z symmetry with just onemirror world degenerate with the SM, but its generalization to N copies is trivial. Such a heavier-than-QCD axion solution is viable, and it would transform the ALP arena to the right of the canonical QCD axion band into solutions to the SM strong CP problem. Theaxion Z N charge assignment explored throughout this work, Eq. (2.6), results instead in lighter-than-QCD axions, that is, solutions located to the left of the QCD axion band.Note that this option induces a comparatively much larger impact: a natural exponentialsuppression of the axion mass ∝ z N as the byproduct of the cancellations between themirror potentials, Eq. (2.32), instead of the mild √N enhancement just discussed.All in all, to explore the right-hand side region of the QCD axion band for solutionsto the strong CP problem, other heavy axion scenarios proposed in the literature seemmore efficient and appealing (e.g. those with mirror worlds much heavier than the SM,or scenarios with novel confining scales much larger than Λ QCD , as mentioned in Sect. 1).
The Z N axion with reduced mass can provide a solution to the SM strong CP problem,independently of whether it accounts or not for the DM content of the Universe. It ishence interesting to get a perspective on the experimental panorama that does not rely onthe supplementary assumption that the axion may be the DM particle: all experimentalbounds and prospects below will be independent of that hypothesis. On the other hand,Ref. [42] will focus on experimental probes that do rely on it. From an experimental point of view, a highly relevant axion coupling is that to photons,defined via the Lagrangian term δ L = g aγ aF ˜ F as [45, 99] g aγ = α πf a ( E/N − . , (4.1)where E and N denote model-dependent anomalous electromagnetic and strong contri-butions, respectively. Fig. 6 shows the parameter space of the reference Z N axion model(with E/N = 0) in the coupling vs. mass plane. Predictions for the axion photon cou-pling are obtained by rescaling the Z N axion mass in Eq. (2.32) for different values of N . Present axion limits and projected sensitivities are displayed as filled and transparentareas, respectively.The yellow band depicts the canonical QCD axion solution, which obeys the well-known relation in Eq. (1.1). The oblique lines indicate instead the Z N lighter axionsolutions to the strong CP problem, as a function of the number of mirror worlds N , seeEq. (2.32). Note that the overall effect of a reduced mass axion is simply a shift towards = N = N = K S V Z N = N = N = N = I A X O BabyIAXOBabyIAXO − − − − − − − − − − − − − − m a [eV] HB CAST P V L A S ALPS I
ALPS II
OSQARCROWS SN - γ M87
Hydra A
HESS
Fermi F e r m i S N Chandra α e m . π | g a γ | ∼ f a [ G e V ] − − − − − − − − − | g a γ | [ G e V − ] Figure 6:
Limits on the axion-photon coupling as a function of the axion mass. Laboratory con-strains [79–86] and astrophysical bounds [87–97] are shown in blue and green, respectively. Pro-jected sensitivities appear in translucent colors delimited by dashed lines. The orange obliquelines represent the theoretical prediction for the Z N axion photon couplings assuming E/N = 0for different (odd) number of worlds N . These lines are solid for the regions of the parameterspace in which the KSVZ UV completion of the Z N axion is free from PQ quality problem anddashed otherwise. The secondary vertical axis shows the corresponding axion decay constant f a if E/N = 0 is assumed. Supplementary constraints in case the axion is assumed to accountfor DM can be found in Ref. [42]. Axion limits adapted from Ref. [98]. the left of the parameter space : each of those oblique lines can be considered to be thecenter of a displaced yellow band. It is particularly enticing that experiments set a priori to only hunt for ALPs may in fact be targeting solutions to the strong CP problem. f a This subsection summarizes the model-independent constraints on f a for the Z N sce-nario under discussion. The result of the analysis is illustrated in Fig. 8. Interestingly,apart from the usual constraints stemming from the SN1987A [100] and black hole su-perradiance measurements [101, 102] (depicted in purple), novel bounds apply to theexceptionally light Z N axion due to finite density effects. Indeed, it has been recentlypointed out in Refs. [37, 38] that finite density media may have a strong impact on thephysics of very light axions or ALPs. In those media, the minimum of the total potential19ay be shifted to π . This has a number of phenomenological consequences that spanfrom the modification of the nuclear processes in stellar objects due to θ ∼ O (1), tomodifications in the orbital decay of binary systems (and subsequently in the emittedgravitational waves).For the scenario considered here, the important point is that a background madeonly of ordinary matter breaks the Z N symmetry. This hampers the symmetry-inducedcancellations in the potential which led to a reduced-mass axion in vacuum: the effectiveaxion mass will be larger within a dense medium.We will first elaborate on the Z N axion potential in a nuclear medium. FollowingRefs. [37, 103], one can compute the finite density effects on the axion potential byconsidering the quark condensates in a medium made of non-relativistic neutrons andprotons [104]. Applying the Hellmann-Feynman theorem, the quark condensate at afinite density n N of a given nucleon N can be expressed as (cid:104) qq (cid:105) n N = (cid:104) qq (cid:105) (cid:18) − σ N n N m π f π (cid:19) , (4.2)where (cid:104) qq (cid:105) = (cid:0) (cid:104) uu (cid:105) + (cid:10) dd (cid:11)(cid:1) ≡ − Σ is the quark condensate in vacuum –see Eq. (2.10)–and σ N is defined by σ N ≡ m q ∂M N ∂m q , (4.3)where m q ≡ ( m u + m d ) and M N is the mass of the nucleon N . Because the Z N potential − − − a/f a . . . . . . . V N = ( θ a , n N ) / m π f π . . . . . . . . . . σ N n N f π m π Figure 7:
Example of the in-medium potential dependence as a function of the nuclear densityfor N = 5. For large densities (light green) the total potential develops a minimum in θ a ∼ π . is proportional to the quark condensate, see Eq. (2.8), we can simply obtain the potentialwithin a SM nuclear medium V f.d. N ( θ a , n N ) by weighting the SM (i.e. k = 0) contributionin the vacuum potential by the factor in Eq. (4.2), that is, V f.d. N ( θ a , n N ) (cid:39) (cid:18) − σ N n N m π f π (cid:19) V ( θ a ) + N − (cid:88) k =1 V ( θ a + 2 πk/ N ) (4.4)20 − σ N n N m π f π V ( θ a ) + N − (cid:88) k =0 V ( θ a + 2 πk/ N ) N (cid:29) −−−→ − σ N n N m π f π V ( θ a ) . In the last step of these expressions the large N limit has been taken, which allowedus to neglect the term corresponding to the exponentially reduced axion potential invacuum (see Eq. 2.30). This shows that, if the nucleon density is large enough, the Z N asymmetric background spoils the cancellations among the mirror world contributions tothe potential, in such a way that the total potential in matter is proportional to minus the SM one in vacuum V ( θ a ). Therefore, the minimum of the potential is located at θ a = π . More precisely, V f.d. N ( θ a , n N ) N (cid:29) −−−→ m π f π z (cid:20) σ N n N m π f π (cid:113) z + 2 z cos ( θ a ) − N − / z N √ π (cid:112) − z cos ( N θ a ) (cid:21) , (4.5)which requires σ N n N m π f π (cid:29) z N (4.6)for the minimum to sit at θ a = π . This is illustrated in Fig. 7.A large value of the θ parameter inside dense stellar objects is rich in physical conse-quences, which translates into strong constraints for the Z N scenario. As it was pointedout in Ref. [37], θ ∼ O (1) inside the solar core is excluded due to the increased proton-neutron mass difference (which would prohibit the neutrino line corresponding to theBe -Li mass difference observed by Borexino [105]). Similarly, for θ ∼ π in nearby neu-tron stars (NS), Co would be lighter than Fe [47] and therefore Fe could have beendepleted due to its β -decay to Co . The presence of iron in the surface of neutron starsand its implications in terms of the allowed θ values could be explored through dedicatedX-ray measurements [106]. The corresponding current and projected constraints thatwere derived in Ref. [37] (within the simplifying assumption z = 1) are translated hereto the Z N scenario and further generalized for any z .A conservative criterion consistent with θ = π inside the medium is to impose thatthe axion mass at θ a = 0 becomes tachyonic, i.e. − m T > m T is defined by − m T ≡ d V f.d. N d a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ a =0 = m π f π f a (cid:34) √ π (cid:114) − z z N / z N − σ N n N m π f π z (1 + z ) (cid:35) . (4.7)Requiring this quantity to be positive, it directly follows a limit on the number of worldsallowed by the stellar bounds above: N (cid:46) , (4.8)where the most conservative estimation of σ N has been used. This bound does notapply for the whole range of f a , though, because the argument only makes physical senseas long as the reduced Compton wavelength of the axion is smaller than the stellar object, r core (cid:38) /m f.d.a , where m f.d.a ∼ /f a is the effective axion mass in the medium, (cid:0) m f.d.a (cid:1) = d V f.d. N d a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ a = π = m π f π f a (cid:34) σ N n N m π f π z − z − √ π (cid:114) − z z N / z N (cid:35) . (4.9) While some recent estimates yield σ N = 59(7)MeV [107], in this work we will employ the moreconservative value σ N = 45 ±
15 MeV [103, 104, 108]. Thus, for the solar core, we obtain σ N n N /m π f π =(1 . − . × − , which would translate into N < (48 − r core ∼ .
000 km implies f a (cid:46) . × GeV for the observa-tional constraints to apply. Finally, the area in parameter space excluded is depicted indark blue in Fig. 8. Analogously, the future sensitivity prospects from neutron star dataare depicted in shaded blue. N = N = N = N = N = − − − − − − − − − − m a [eV] Q C D a x i o n SN1987a Superradiance
S u n
PulsarBH-NS inspiral (LIGO/VIRGO and aLIGO)
NS-NS inspiral N e u t r o n S t a r − − − λ [ m ] − − − − − − − − − − − − − − / f a [ G e V − ] Figure 8:
Model-independent constraints on the axion scale f a versus axion mass, from astro-physical data. Regions presently excluded are depicted in solid colors, while the translucentregions circumscribed by dotted lines are projections. The orange oblique lines indicate thetheoretical prediction of the reduced-mass Z N QCD axion scenario, as a function of N : theyare solid where the KSVZ Z N axion is free from the PQ quality problem, and dashed other-wise. Additional constraints which apply if the axion is assumed to account as well for DM arediscussed in Ref. [42]. Even stronger bounds may be established by relaxing the requirement stemming fromEq. (4.7). Indeed, as it can be seen in Fig. 8, long before the mass in θ a = 0 becomestachyonic, the absolute minimum of the potential corresponds to θ ∼ O (1). Thereforeone could constrain larger masses or smaller N values in the Z N scenario than thoseobtained above. This would require, however, a dedicated analysis to ensure that theaxion field would fall into the absolute minimum, so as to overcome the potential barrier;this development lies beyond the scope of the present work.The fact that the position of the minimum of the axion potential depends on thenuclear density of the medium not only modifies the effective θ -parameter inside stellar Our results are analogous to those in Eq. (1.7) of Ref. [37], with their generic parameter (cid:15) identifiedas (cid:15) = m a /m a ( N = 1) (cid:39) π − / (cid:112) − z (1 + z ) N / z N − . Note that the location of the QCD axion line,as well as our projected exclusion regions for neutron stars and gravitational waves, are shifted towardsthe left by a factor of five with respect to those in Refs. [37, 38]. m a ∼ − eV) [113], will bewithin observational reach in the next decades, for a wide range of N values. An axion which solves the strong CP problem may be much lighter than the canonicalQCD axion, down to the range of ultra-light axions, provided Nature has a Z N sym-metry implemented via N degenerate world copies, one of which is our SM. The axionfield realizes the symmetry non-linearly, which leads to exponential cancellations amongthe contributions from each mirror copy to the total axion potential. For large N , wehave shown that the total axion potential is given by a single cosine and we determinedanalytically the –exponentially suppressed– dependence of the axion mass on the num-ber of mirror worlds, using the properties of hypergeometric functions and the Fourierexpansion. In practice, the formula in Eq. (2.32) gives an excellent approximation evendown to N = 3. We have also improved the holomorphicity bounds previously obtained.We compared next the predictions of the theory with present and future data fromexperiments which do not rely on the additional assumption that an axion abundancemay explain the DM content of the Universe. It is particularly enticing that experimentsset a priori to hunt only for ALPs may in fact be targeting solutions to the strong CPproblem. For instance, ALPS II is shown to be able to probe the Z N scenario herediscussed down to N ∼
25 for a large enough axion-photon coupling, while IAXO andBabyIAXO may test the whole N landscape for values of that coupling even smaller, seeFig. 6. In turn, Fermi SN data can only reach N (cid:38)
43 but are sensitive to smaller valuesof the coupling.Highly dense stellar bodies allow one to set even stronger bounds in wide regions of theparameter space. These exciting limits have an added value: they avoid model-dependentassumptions about the axion couplings to SM particles, because they rely exclusivelyon the anomalous axion-gluon interaction needed to solve to the strong CP problem.A dense medium of ordinary matter is a background that breaks the Z N symmetry.This hampers the symmetry-induced cancellations in the total axion potential: the axionbecomes heavier inside dense media and the minimum of the potential is located at θ a = π .From present solar data we obtain the bound N (cid:46)
47 provided f a (cid:46) . × GeV,while larger N values are allowed for smaller f a . Furthermore, we showed that projectedneutron star and pulsar data should allow to test the scenario down to N ∼ f a , see Fig. 8. Furthermore, gravitationalwave data from NS-NS and BH-NS mergers by LIGO/VIRGO and Advanced LIGO willallow to probe all values of N for the remaining f a range, up to the Planck scale andincluding the ultra-light axion mass range.These analytical and phenomenological results have been derived within the model-independent framework of effective couplings. Nevertheless, for the sake of illustration,23e have developed two examples of UV completed models. One is a Z N KSVZ model,which is shown to enjoy an improved PQ quality behaviour: its Z N and PQ symmetriesare linked and in consequence gauging Z N alleviates much the PQ quality problem. Theother UV completion considered in this paper is a Z N version of the composite axion `ala Kim-Choi. While this model is viable, its PQ quality is not improved with respect tothe usual situation, because its Z N and PQ symmetries are independent.This work is intended to be a proof-of-concept that a much-lighter-than usual axionis a viable solution to the strong CP problem, with spectacular prospects of being testedin near future data. It also pinpoints that experiments searching for generic ALPs havein fact discovery potential to solve the strong CP problem.The down-tuned axion considered here could also explain the DM content of theUniverse in certain regions of the parameter space. The impact of such a light axion onthe cosmological history is significant and it will be discussed in a separate paper [42]. Acknowledgments
We thank Gonzalo Alonso- ´Alvarez, Victor Enguita, Mary K. Gaillard, Yann Gouttenoire, Ben-jamin Grinstein, Lam Hui, David B. Kaplan, Philip Sørensen and Neal Weiner for illuminat-ing discussions. M.B.G. and P.Q. are indebted for hospitality to the Theory Department ofColumbia University in New York, where the initial stage of their work took place. The work ofL.D.L. is supported by the Marie Sk(cid:32)lodowska-Curie Individual Fellowship grant AXIONRUSH(GA 840791). L.D.L., P.Q. and A.R. acknowledge support by the Deutsche Forschungsge-meinschaft under Germany’s Excellence Strategy - EXC 2121 Quantum Universe - 390833306.M.B.G. acknowledges support from the “Spanish Agencia Estatal de Investigaci´on” (AEI) andthe EU “Fondo Europeo de Desarrollo Regional” (FEDER) through the projects FPA2016-78645-P and PID2019-108892RB-I00/AEI/10.13039/501100011033. M.B.G. and P. Q. acknowl-edge support from the European Union’s Horizon 2020 research and innovation programmeunder the Marie Sklodowska-Curie grant agreements 690575 (RISE InvisiblesPlus) and 674896(ITN ELUSIVES), as well as from the Spanish Research Agency (Agencia Estatal de Investi-gaci´on) through the grant IFT Centro de Excelencia Severo Ochoa SEV-2016-0597. This projecthas received funding/support from the European Union’s Horizon 2020 research and innovationprogramme under the Marie Sklodowska-Curie grant agreement No 860881-HIDDeN.
A Holomorphicity properties of Z N axion potential In order to determine the parameter b in Eq. (2.20), which controls the exponential suppressionof the axion mass, it is necessary to study the region in the complex plane where the extensionof the potential V ( w ) is holomorphic. As the plots in Fig. 9 illustrate, both the potential and itssecond derivative have branch cuts starting in w cut = π ± i log z . However, the second derivative V (cid:48)(cid:48) ( w ) diverges at the branch point and thus b cannot be extended all the way to log z . In orderto optimize the bound on the axion mass we allow b to depart from log z , b = log z + ∆ b . Takinginto account that V (cid:48)(cid:48) ( w ) for small ∆ b can be approximated by V (cid:48)(cid:48) (cid:0) π + i (log z + ∆ b ) (cid:1) (cid:39) − m π f π (cid:114) − z z (cid:34)
14 1(∆ b ) / + O (cid:0) ∆ b − / (cid:1)(cid:35) , (A.1)the procedure amounts to minimize the function B (∆ b ) that determines the bound | E N ( V ) | ≤ B (∆ b ) (see Eq. (2.20), B (∆ b ) ≡ πM (∆ b ) e N (log z +∆ b ) − πm π f π (cid:114) − z z b ) / e N (log z − ∆ b ) − . (A.2) he requirement dB (∆ b ) d (∆ b ) = 0 shows that the bound is optimized for∆ b = 32 1 N , (A.3)where the factor 3 / Figure 9:
Representation of the complex functions V ( w ) (left) and V (cid:48)(cid:48) ( w ) (right). Colorsrepresent the phase of the corresponding complex function and the brightness represents themodulus. The singularities can be clearly spotted: branch cuts starting from w cut = π ± i log z in both functions and divergences in those same points for V (cid:48)(cid:48) ( w ). B Fourier series of the Z N axion potential We show here that the coefficients of the Fourier series of any Z N symmetric potential, such asthe Z N axion potential in Eq. (2.8), vanish unless the corresponding Fourier mode is a multipleof N . Moreover it will be shown that, when the potential is expressed as V N ( θ a ) = N − (cid:88) k =0 V (cid:18) θ a + 2 πk N (cid:19) , (B.1)the non-vanishing coefficients of the Fourier series can be expressed in terms of the Fouriertransformation of a single term in the sum Eq. (B.1).Let us denote by ˆ V N ( n ) the coefficients of the Fourier series of the total potential, V N ( θ a ) ≡ ∞ (cid:88) n = −∞ e inθ a ˆ V N ( n ) , (B.2)and by ˆ V πk/ N ( n ) the coefficients of the Fourier series of each of the terms in the sum inEq. (B.1), V (cid:18) θ a + 2 πk N (cid:19) ≡ ∞ (cid:88) n = −∞ ˆ V πk/ N ( n ) e inθ a . (B.3)We will stick to the notation that omits the subindex for the first world ( k = 0), ˆ V ( n ) ≡ ˆ V ( n ), V ( θ a ) = ∞ (cid:88) n = −∞ ˆ V ( n ) e inθ a with ˆ V ( n ) = 12 π (cid:90) π V ( x ) e − inx dx . (B.4) ach term in the sum in Eq. (B.1) and Eq. (B.3) corresponds to the function in the first termbut with its argument shifted by πk N . The shift property of the Fourier series relates the Fouriercoefficients of the shifted functions ˆ V πk/ N ( n ) to that of the original function,ˆ V πk/ N ( n ) = e in πk N ˆ V ( n ) . (B.5)Substituting this expression in Eq. (B.3), and inserting the latter in Eq. (B.1), it follows thatthe total potential can be written as V N ( θ a ) = N − (cid:88) k =0 ∞ (cid:88) n = −∞ ˆ V πk/ N ( n ) e inθ a = ∞ (cid:88) n = −∞ ˆ V ( n ) e inθ a N − (cid:88) k =0 e in πk N . (B.6)Comparing this expression with Eq. (B.2), it follows that the coefficients of the Fourier seriesfor the total potential are given byˆ V N ( n ) = ˆ V ( n ) N − (cid:88) k =0 e in πk N . (B.7)Interestingly, these coefficients vanish unless n is a multiple of N If n (mod N ) (cid:54) = 0 = ⇒ N − (cid:88) k =0 e in πk N = 0 = ⇒ ˆ V N ( n ) = 0 , (B.8)If n (mod N ) = 0 = ⇒ N − (cid:88) k =0 e in πk N = N = ⇒ ˆ V N ( n ) = N ˆ V ( n ) . (B.9)To sum up, the Fourier series of the total potential V N ( θ ) can be easily obtained in terms ofthe Fourier series of a single term V ( θ ) and it only receives contributions from the modes thatare multiples of N . In our case of interest the potential is real and even, this translates into V N ( θ a ) = 2 N ∞ (cid:88) t =1 ˆ V ( t N ) cos( t N θ a ) , (B.10)where the factor of two comes from the negative modes and the constant term (i.e. θ a -independent)has been obviated. C Analytical axion mass dependence from hyperge-ometric functions
We show here that the Fourier series coefficients of the single world axion potential in Eq. (2.27),ˆ V ( n ) = − m π f π z (cid:90) π cos( nt ) (cid:113) z + 2 z cos ( t ) dt , (C.1)can be written for large Fourier modes, n (cid:29)
1, as a simple analytical formula that exponentiallydecays with n . Moreover, by applying the result in Appendix B, it will be shown that thepotential for the Z N axion approaches a single cosine and a simple formula for the Z N axionmass follows.Let us start by relating the Fourier series decomposition of the single world potential inEq. (C.1) with the Gauss hypergeometric functions (see for example Eq. (9.112) in Ref. [54]), F (cid:18) p, n + pn + 1 (cid:12)(cid:12)(cid:12)(cid:12) w (cid:19) = w − n π Γ( p ) n !Γ( p + n ) (cid:90) π cos( nt ) dt (cid:16) − w cos t + w (cid:17) p . (C.2) ia the identification w = − z and p = − /
2, ˆ V ( n ) can be written asˆ V ( n ) = ( − n +1 m π f π z z n Γ( n − / − / n ! F (cid:18) − / , n − / n + 1 (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) . (C.3)For convenience, the hypergeometric function can be also expressed as (see Eq. (9.131) fromRef. [54]) F (cid:18) α, βγ (cid:12)(cid:12)(cid:12)(cid:12) w (cid:19) = (1 − w ) − α F (cid:18) α, γ − βγ (cid:12)(cid:12)(cid:12)(cid:12) ww − (cid:19) , (C.4)so that F (cid:18) − / , n − / n + 1 (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) = (cid:16) − z (cid:17) / F (cid:32) − / , / n + 1 (cid:12)(cid:12)(cid:12)(cid:12) z z − (cid:33) . (C.5)The relation in Eq. (C.3) is exact. However, only the modes n which are multiples of N contribute to the potential (see Appendix B), and therefore it is pertinent to focus on thelarge n limit. While the limit of the Gauss hypergeometric function when one or more of itsparameters become large is difficult to compute in general, some asymptotic expansions of thehypergeometric function are known in the literature. In particular, following Ref. [114],lim γ →∞ F (cid:18) α, βγ (cid:12)(cid:12)(cid:12)(cid:12) w (cid:19) = 1 + αβγ w + O (cid:16) ( w/γ ) (cid:17) . (C.6)In turn, the prefactor in Eq. (C.3) simplifies in the large n limit to . lim n →∞ Γ( n − / − / n ! = − √ π n − / . (C.7)Putting all this together, it follows that, in the large n limit, the coefficient of the Fourier seriesfor a single world is given byˆ V ( n ) = ( − n m π f π √ π (cid:114) − z z n − / z n , (C.8)which in turn implies in this limit that the total Z N potential in Eq. (B.10) can be written as V N ( θ a ) = N ∞ (cid:88) t =1 ( − t N m π f π √ π (cid:114) − z z ( t N ) − / z t N cos( t N θ a ) . (C.9)This expression allows us to understand several properties of the total potential. Firstly, it canbe shown now that the total potential approaches a single cosine in the large N , since all theother modes are then exponentially suppressed with respect to the first one,lim N →∞ (cid:12)(cid:12)(cid:12)(cid:12) ˆ V N ( t N )ˆ V N ( N ) (cid:12)(cid:12)(cid:12)(cid:12) = lim N →∞ (cid:12)(cid:12)(cid:12)(cid:12) ˆ V ( t N )ˆ V ( N ) (cid:12)(cid:12)(cid:12)(cid:12) = t − / z ( t − N −→ , (C.10)and thus the potential reads V N ( θ a ) N (cid:29) −−−−→ m π f π √ π (cid:114) − z z N − / . ( − N z N cos( N θ a ) , (C.11)Secondly, we can also obtain an analytical expression for the axion mass that confirms thedependence obtained from the holomorphicity arguments in Section 2.3.1, and completes theexpresion with the correct prefactor, m a f a (cid:39) m π f π √ π (cid:114) − z z N / z N . (C.12) inally, it is now trivial to show that the potential in the large N limit has N minima (maxima)located at θ a = {± π(cid:96)/ N } for (cid:96) = 0 , , . . . , N − , for odd (even) N .The results above assumed the large N limit. However, note that the conclusion about thelocation of the extrema is true for any N . This can be easily seeing after obtaining the exact Fourier expansion of the Z N axion potential in Eq. (2.8), which reads, V N ( θ a ) = − m π f π N ∞ (cid:88) t =1 ( − t N +1 ∞ (cid:88) (cid:96) = t N (2 (cid:96) )!(2 (cid:96) )!2 (cid:96) − (2 (cid:96) − (cid:96) !) ( (cid:96) − t N )!( (cid:96) + t N )! β (cid:96) cos ( t N θ a ) . (C.13)For even N , it trivially follows that θ a = 0 is a maximum, as all factors in this expression arepositive except for the factor (1 − (cid:96) ) <
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