Light Sterile Neutrinos and a High-Quality Axion from a Holographic Peccei-Quinn Mechanism

UMN-TH-4021/21

Peter Cox,1, ∗ Tony Gherghetta,2, † and Minh D. Nguyen2, ‡ 1School of Physics, The University of Melbourne, Victoria 3010, Australia 2School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455 USA We present a 5D axion-neutrino model that explains the Standard Model fermion mass hierarchy and flavor structure, while simultaneously generating a high-quality axion. The axion and right- handed neutrinos transform under a 5D Peccei-Quinn gauge symmetry, and have highly suppressed profiles on the UV brane where the symmetry is explicitly broken. This setup allows neutrinos to be either Dirac, or Majorana with hierarchically small sterile neutrino masses. The axion decay constant originates from the IR scale, which in the holographically dual 4D description corresponds to the confinement scale of some new strong dynamics with a high-quality global Peccei-Quinn symmetry that produces a composite axion and light, composite sterile neutrinos. The sterile neutrinos could be observed in astrophysical or laboratory experiments, and the model predicts specific axion–neutrino couplings.

INTRODUCTION work.

Two unsettled issues in the Standard Model are neu- The PQ symmetry forbids an explicit or spontaneously trino masses and the strong CP problem. A natural so- generated bulk Majorana mass for the right-handed neu- lution to the origin of the neutrino masses is the Type-I trinos, leading to an accidental lepton number symmetry. seesaw mechanism [1–5] with Majorana masses at an in- However, explicit PQ (and L) violating terms are allowed 10 termediate scale & 10 GeV. On the other hand, the on the UV brane, and the fundamental Majorana mass most popular solution to the strong CP problem is the scale is tied to explicit Planck-scale PQ (and L) viola- Peccei-Quinn (PQ) mechanism [6], where the axion is a tion. Despite this connection, the sterile neutrino mass pseudo-Nambu-Goldstone boson [7, 8] that results from eigenstates can be naturally light. This is because the the spontaneous breaking of a global U(1)PQ symmetry. right-handed neutrino profiles can be localized towards These two solutions appear to be unrelated; however, the the IR brane, away from the explicit symmetry viola- similarity of the PQ-breaking and Majorana mass scales tion. In fact, the sterile neutrino masses can range from suggests there could be an underlying mechanism respon- the intermediate scale down to the eV scale (in the see- sible for both neutrino masses and the axion. saw mechanism limit), or even lower to the theoretical Any axion solution to the strong CP problem must Dirac limit. The model also predicts axion couplings to address the axion quality problem, which requires extra- both the active and sterile neutrinos; however, these are neous, explicit violations of the global PQ symmetry to well below the current experimental limits [15]. be sufficiently suppressed compared to that arising from non-perturbative QCD. Recently, a possible solution was A connection between neutrino masses and axions was given in Ref. [9], where the axion propagates in a slice of first discussed in the context of the grand unified group AdS5. The PQ symmetry is gauged in the bulk, and the 0 SO(10) × U(1) , where the U(1)PQ symmetry is realised axion profile is suppressed near the sources of explicit PQ 0 as a linear combination of U(1)B−L and U(1) [16]. Other symmetry violation on the UV brane. The warped geom- models based on the DFSZ axion with a connection to arXiv:2107.14018v1 [hep-ph] 29 Jul 2021 etry [10] can also naturally explain fermion mass hierar- neutrino masses include [17–19]. In contrast, our 5D chies [11], and a holographic DFSZ-type [12, 13] axion model automatically addresses the axion quality prob- model that incorporates Standard Model flavor was pre- lem while simultaneously explaining the hierarchies of sented in Ref. [14], giving predictions for flavor-violating the Standard Model fermion masses and flavor structure axion–fermion couplings. The 5D framework, therefore, in the quark and lepton sectors. Furthermore, by the provides a natural setting to seek a connection between AdS/CFT correspondence [20], the 5D model is dual to a neutrino masses and the axion. strongly-coupled 4D theory where the intermediate scale In this Letter, we extend the model of Refs. [9, 14] to is dynamically generated by dimensional transmutation. include neutrino masses. Right-handed neutrinos are in- The axion is identified as a composite pseudo-Nambu- troduced into the bulk and are charged under the U(1)PQ Goldstone boson, and the right-handed neutrinos are also symmetry. The model can explain neutrinos as either composite states. While right-handed neutrinos propa- Dirac or Majorana states, with hierarchies in the effective gating in a slice of AdS5 were previously considered in 4D neutrino Yukawa couplings and/or right-handed neu- Refs [21–28], our setup is the first model to amalgamate trino masses naturally generated within the 5D frame- neutrinos with axion physics. 2

AXION–NEUTRINO MODEL Li Ei Ni Hu Hd Φ 2 2 2 2 U(1)PQ 2 sin β 4 sin β 2 −2 cos β −2 sin β 1 Consider a 5D U(1) gauge theory with a complex 1 1 1 PQ U(1)Y − −1 0 − 0 scalar field Φ propagating in a slice of AdS bounded by 2 2 2 5 U(1)L 1 1 1 0 0 0 UV and IR branes located at zUV and zIR. The metric U(1)Φ 0 0 0 0 0 1 in 5D coordinates xM = (xµ, z) is given by 1 TABLE I: U(1) charges of the bulk fields. ds2 = dx2 + dz2 ≡ g dxM dxN , (1) (kz)2 MN where the AdS curvature scale k M , with M = (5) (5) . P P ye,ν are 3 × 3 complex matrices and yN , bN are complex 2.435 × 1018 GeV the reduced Planck mass. The Yang- symmetric matrices. Mills–scalar action is given in [9], where it is also shown The model contains four U(1) symmetries in the bulk: that the usual global PQ symmetry that acts on the 4D the hypercharge and PQ gauge symmetries, and acciden- axion corresponds to a particular bulk U(1)PQ gauge tal global lepton number and U(1)Φ symmetries. The transformation. charges of the fields are given in table I. The PQ charges The complex, PQ-charged scalar field Φ obtains a VEV of the Higgs fields have been chosen so that there is no mixing between the axion and the longitudinal compo- η(z) = k3/2 λ(kz)4−∆ + σ(kz)∆ , (2) nent of the Z boson, where tan β = vu/vd is the ratio of where ∆ is related to the bulk scalar mass-squared, the Higgs VEVs. 2 2 mΦ = ∆(∆ − 4)k . In the dual 4D interpretation [29] The U(1)PQ and U(1)Φ symmetries are spontaneously of our setup, σ is proportional to the PQ-breaking con- broken by the VEV of Φ and explicitly broken by the densate in the CFT, which has dimension ∆. The coeffi- scalar potential on the UV boundary. This results in a cient λ is associated with explicit breaking of the U(1)PQ single pseudo-Nambu-Goldstone boson, identified as the symmetry on the UV brane. Note that the boundary con- axion (see Refs. [9, 14] for details). Furthermore, the lep- ditions are such that the 5D gauge symmetry reduces to ton number and PQ symmetries are explicitly broken by (5) a global symmetry on the UV brane (guaranteeing there yN and bN . Note that the U(1)PQ gauge symmetry for- is no massless 4D U(1)PQ gauge boson), and therefore bids corresponding terms in the bulk. This has important the symmetry can be explicitly broken there (see Ref. [9] phenomenological consequences, since such terms would for details). lead to sterile neutrino zero-mode masses of order the In Ref. [14] this model was extended to include bulk PQ-breaking scale, as might be expected for a high-scale fermion and Higgs fields, creating a 5D version of the seesaw. The UV boundary terms, on the other hand, DFSZ axion model [12, 13] that could simultaneously ad- can naturally give rise to hierarchically smaller sterile dress both the axion quality and fermion mass hierarchy neutrino masses, as will be shown. These may then be problems. Here, we further extend the model to incorpo- accessible to experiments. rate the neutrino sector. Neutrino masses are obtained by including bulk right-handed neutrinos Ni (i = 1, 2, 3) with 5D Yukawa couplings and UV boundary localized Zero-mode profiles Majorana masses and Φ coupling terms. The relevant 1 part of the action is given by The equations of motion for the 5D fields can be solved

Z zIR via the usual expansion in Kaluza-Klein (KK) modes. 5 √ SN = −2 d x −g The massless 4D zero-modes are then identiﬁed with the zUV SM fermions and the axion. 1 (5) (5) The scalar ﬁelds are parameterized as × √ y LiNjHu + y LiEjHd + h.c. k ν,ij e,ij (5) vu i a (xµ,z) 1 y vu u 1 c N,ij c Hu = √ e , + bN,ijN Nj + ΦN Nj + h.c. δ(z − zUV ) , 2 0 2 i k3/2 i v i µ 0 (3) d v ad(x ,z) Hd = √ e d , 2 1 νi where Li = ei are the SU(2) lepton doublets and Ei ia(xµ,z) are the SU(2) singlet leptons. The 5D Yukawa couplings Φ = η(z) e , (4)

where vu (vd) are the up (down)-type Higgs VEVs sat- 2 2 2 isfying (vu + vd)/k = v , with v ' 246 GeV. In general, 1 Note that the delta function is deﬁned such that 2 R zIR dz δ(z − zUV vu,d are z-dependent, but for simplicity we take them to zUV )f(z) = f(zUV ). be constant, which requires a tuning between the Higgs 3

(5) mass terms in the bulk and on the IR boundary. Further- order-one parameters ci, ye,ν,N , and bN . These 5D pa- more, the Higgs hierarchy problem is not addressed in the rameters can then be constrained by fitting to the two current model. The 5D fields au,d(x, z) and a(x, z) are neutrino mass-squared differences and the Pontecorvo- the neutral Nambu-Goldstone bosons propagating in the Maki-Nakagawa-Sakata (PMNS) matrix. bulk. Note that we have ignored the radial components Substituting eqs. (4) and (8) into (3) gives rise to the and the electromagnetically-charged Nambu-Goldstone neutrino (zero-mode) 6 × 6 mass matrix M defined as bosons since they play no role in the discussion. ! The equations of motion for a (x, z) and a(x, z) are 1 Z 0 m ~νc u,d − d4x ~ c D L + h.c., (9) coupled and the 5D fields are expanded in terms of the ~νL NR T ~ 2 mD MM NR same set of 4D modes, c c c c µ 0 0 µ with (~ν , N~R) ≡ (ν , ν , ν ,N1R,N2R,N3R). The a(x , z) = fa (z)a (x ) + ..., L 1L 2L 3L µ 0 0 µ Dirac mass matrix is given by au,d(x , z) = f (z)a (x ) + .... (5) au,d √ Z zIR (5) 2vu dz The (approximately) massless zero-mode a0(xµ) is iden- mij = y √ f 0 (z)f 0 (z) , (10) D ν,ij k (kz)5 LiL NjR tified with the axion. The profile f 0(z) was calculated zUV a p v (−1 + 2c )(1 + 2c ) 1 in Ref. [9], where it was shown that explicit breaking of u Li Nj − −min(cL ,cN ) ' √ (kzIR) 2 i j , the PQ symmetry on the UV brane causes the profile to 2k(cL − cN ) ∆ i j become suppressed by (z/zIR) as z → zUV . Away from (11) the UV boundary the profile is approximately constant and given by and, taking kzUV = 1, the Majorana mass matrix is √ 0 ∆ − 1 ij (5) 0 2 f (z) ≈ z , (6) MM = yN,ij(λ + σ) + bN,ij (fN (zUV )) , (12) a σ IR iR 0 1 −1−2c ∆ ' k yˆ c + (kz ) Ni , (13) where σ0 = σ(kzIR) . 1. The profile also determines N,ij Ni IR 0 −1 2 the value of the axion decay constant Fa = fa (zIR) , which is of order the IR scale, z−1. (5) IR withy ˆN,ij ≡ y (λ + σ) + bN,ij. Eqs. (11) and (13) The remaining scalar zero-mode profiles were obtained N,ij also assume zUV zIR, cL > 1/2, and cN > −1/2. in Ref. [14] and are approximately given by i i Notice that for cN > 0, the effective Majorana masses are suppressed relative to the PQ-breaking or IR scale. For g2kσ f 0 (z) ≈ v X √5 0 c < 0, on the other hand, the right-handed neutrinos au,d u,d Hu,d N 4∆ ∆ − 1 have masses of order the IR scale, and can no longer be " ! # z2 z 2(∆−1) ∆z2 treated as approximately massless modes with profiles × − ∆ + UV , (7) zIR zIR zIR given by eq. (8). We therefore restrict our discussion to cN > 0. This corresponds to right-handed neutrinos that where XHu,d are the PQ charges of the Higgs fields, and are localized towards the IR brane and therefore in the g5 is the 5D U(1)PQ gauge coupling. dual 4D description are mostly composite. For cL > 0, After imposing appropriate boundary conditions, the the left-handed neutrinos are mostly elementary. The KK expansions of the 5D fermions Li,(Ei and Ni) con- Majorana mass matrix (12) can be made diagonal and µ tain left- (right-) handed 4D chiral zero-modes LiL(x ), non-negative via a unitary rotation of the Ni, and we µ µ (EiR(x ) and NiR(x )) respectively. These have pro- work in this basis wherey ˆN is diagonal. files [11] The neutrino mass matrix in (9) is diagonalized by a T 0 2−c unitary matrix U to give U MU = diag(mνi ), where f (z) = N (kz) Li , LiL Li mνi are the six neutrino mass eigenvalues. The mass 0 2+c f (z) = N (kz) Ei , EiR Ei eigenstates νi are 0 2+cN f (z) = NN (kz) i , (8) ! ! NiR i ~νc ~ν ~ν = U † L + U T L . (14) where c , c , c are order-one constants that ~ ~ c Li Ei Ni NR NR parametrize the bulk lepton masses (= ci k) and NLi , NE , NN are normalization factors. 2 −1 T i i In the seesaw limit , kMM mDk 1, the six eigenstates split into two distinct sets. One set contains the Neutrino flavor structure

2 P 21/2 The overlap of the bulk fermion proﬁles completely de- The Frobenius norm kAk = (Aij ) , for a matrix A. termines the neutrino ﬂavor structure in terms of the i,j 4

“active” light neutrinos that are mostly SU(2) doublets where ω is an arbitrary real parameter. The 5D fermion and the other set contains the heavier “sterile” neutri- kinetic and boundary Ni terms are not invariant under nos that are mostly Standard Model gauge singlets. The this transformation, giving rise to mixing matrix is then approximately [30] Z zIR 5 d x au M M i ∂M N¯iγ Ni − ν¯iγ νi 1 † † 4 1 − Θ Θ U Θ U z (kz) vu U' 2 ν N + O(kΘk3) , UV −ΘU 1 − 1 ΘΘ† U ν 2 N − 2ω N¯ γM N +ν ¯ γM ν (15) i i i i −1 T where Θ = MM mD, and Uν , UN are the 3 × 3 ma- (5) yN,ij au trices that diagonalize the active and sterile neutrinos c i(2ω−1) v − bN,ij+ Φ Ni Nje u +h.c. δ(z−zUV ) . respectively. To leading order, the active masses are k3/2 T −1 T (18) diag(mνi ) ' −Uν (mDMM mD)Uν , while the sterile masses are diag(mNi ) ' MM . The PMNS matrix is The terms proportional to ω do not contribute to the e† e† VPMNS = AL Uν , where AL is the unitary matrix that S-matrix. This is seen by identifying the lepton num- rotates the left-handed charged leptons to the mass ba- ber current in the second line of (18). The (anomalous) sis. After using the phase freedom of the charged leptons Ward-Takahashi identity for lepton number then guar- to remove three redundant phases, it can be expressed antees that contributions to the S-matrix from the ω- using the standard parameterization. dependent terms in the second and third lines cancel.

Interestingly, when cLi > cNj ∀ i, j, the active neu- Restricting our focus to the zero-modes, the axion– trino masses do not depend on the IR scale. This neutrino couplings are then is due to the correlation between the effective Dirac Z zIR d5x f 0 (z) and Majorana masses in eqs. (11) and (13), such that i (∂ a0) au N¯ (f 0 (z))2γµN 2 4 µ iR NiR iR mν ∝ v /k. The 4D dual description provides an alterna- zUV (kz) vu tive viewpoint on the neutrino mass mechanism in which − ν¯ (f 0 (z))2γµν . (19) there are elementary, Planck-scale Majorana fermions iL LiL iL that mix with the composite right-handed neutrinos. In- cluding the (mostly) elementary left-handed neutrinos, We have neglected additional UV boundary terms, which the active neutrino masses then arise from a seesaw mech- lead to axion–sterile neutrino couplings, since they are anism. highly suppressed by the fa or fau profile at z = zUV . We also consider the possibility that neutrinos are Integrating over the profiles and rotating to the fermion mass basis we obtain the 4D effective action Dirac fermions (MM = 0). In this case it is convenient to instead define the mass eigenstate Dirac fermions by Z ∂ a0 S ⊃ i d4x µ ν¯ γµ (cV ) − (cA) γ5 ν , (20) 4D 2F i ν ij ν ij j ν† ν† a ν = A νL + A NR , (16) L R where γ5 = diag(1, −1). The vector and axial-vector V A ν† e† couplings are given by c = i Im(ξ ) and c = Re(ξ ), where A m Aν = diag(m ), and V = A Aν . ν ν ν ν L D R νi PMNS L R where Note that the Dirac limit does not necessarily require Z zIR 0 that lepton number is preserved on the UV brane. In- (ξν)ij dz f (z) = au U † F kk U , (21) stead, the Dirac limit can actually be obtained while as- 4 ik kj Fa zUV (kz) vu suming Planck-scale lepton number violation on the UV with F = diag((f 0 )2, (f 0 )2, (f 0 )2, (f 0 )2, (f 0 )2, brane [24]. By formally taking c 1 in (13), the ef- L1L L2L L3L N1R N2R N (f 0 )2). Note that the vector (axial-vector) couplings fective 4D Majorana mass M → 0. For simplicity, we N3R M in eq. (20) are symmetric (anti-symmetric). will not consider this pseudo-Dirac limit and instead just In the Dirac neutrino case, the axion–neutrino cou- analyse the Majorana (seesaw-mechanism) limit and the plings are similar to those obtained for the charged pure Dirac limit. fermions in [14], and can be written as

V,A Z zIR 0 (cν )ij dz f (z) Axion–neutrino couplings au = 4 Fa zUV (kz) vu × (Aν †) (f 0 )2(Aν ) ∓ (Aν†) (f 0 )2(Aν) . The axion–neutrino couplings are obtained by ﬁrst re- R ik NkR R kj L ik LkL L kj moving the au-dependence in eq. (3) via a 5D ﬁeld redef- (22) inition of the form Finally, note that for on-shell fermions the correspond- 1 1 i(ω+ ) au(x,z) V,A νi(x, z) → e 2 vu νi(x, z) , ing matrix elements are proportional to (c )ij(mi ∓ 1 1 mj), so that the axion–active neutrino couplings can be i(ω− ) au(x,z) Ni(x, z) → e 2 vu Ni(x, z) , (17) neglected in most cases. 5

PHENOMENOLOGY Dirac neutrinos

The phenomenological predictions of the 5D model are In the Dirac neutrino case,y ˆN = 0. Given our assump- obtained by taking the 5D parameters to be ∼ O(1). tion of O(1) 5D Yukawa couplings, the neutrino mass Hierarchies in 4D parameters are then generated as a scale is solely determined by the cL and cN parameters. consequence of localisation in the extra dimension. The This case is similar to the quark sector studied in [14]. relevant 5D parameters in the lepton sector are Figure 1 (left) shows the range of ci parameters that can produce the measured values of the lepton masses c , c , c , y(5) , y(5) , yˆ . (23) Li Ei Ni e,ij ν,ij N,ij and PMNS angles/phase. Note that for the right-handed charged leptons |c | 0.5, while for the right-handed These parameters are then constrained by the nine ob- Ei . neutrinos c 1. This simply reflects the fact that servables Ni & the neutrino masses are much smaller than the charged 2 mei , ∆mν,ij, θij, δ , (24) lepton masses. Furthermore, the structures seen in figure 1 for both the charged leptons and neutrinos can be where m are the charged lepton masses, ∆m2 are ei ν,ij understood from the fact that the effective Dirac mass the neutrino mass-squared differences, θ are the PMNS ij term depends only on min(c , c ) (see eq. (11)). The mixing angles and δ is the PMNS Dirac phase. L N/E (5) hierarchies in the charged leptons can also clearly be seen, We perform a numerical scan of the ci and ye,ν ,y ˆN pa- whereas the neutrinos need not be hierarchical. rameter space. The ci are restricted to satisfy |cL,E,N | . Since the Higgs VEV profile is flat, the overlap between 4, which ensures that the bulk fermion masses remain the left and right-handed fermions is responsible for gen- below the 5D cutoff. For the Majorana case, the c are Ni erating the hierarchies in the 4D effective Yukawa cou- also restricted to be in the range [0, 1.4] to ensure that plings. The c parameter ranges in figure 1 correspond to the effective Majorana masses remain below the IR scale the left (right)-handed fermions being localized towards but sufficiently greater than the Dirac masses (at least 1 the UV (IR) branes. Finally, for c < 0, the overlap inte- eV). The 5D Yukawa couplings are restricted to satisfy L (5) gral in (10) is suppressed relative to the electroweak VEV 0.01 ≤ |y |, yˆ ≤ 3. To increase the efficiency, the scan −n e,ν N by a factor of (kzIR) , with n > 1/2; hence, there are (5) is done in two stages. First, ye,ν are fixed to random val- no solutions in this region as the charged lepton masses ues, and the ci andy ˆN fit to the experimental values of would be too small. the lepton masses by performing a χ2 minimization us- The axial-vector axion–neutrino couplings are shown −6 ing Minuit [31]. In the second stage, all parameters are in Figure 3 (top panel), for mν1 > 10 eV. The flavor- A −5 floated to fit both the masses and PMNS matrix elements diagonal couplings are approximately cν ' 2 × 10 . A using the values from the first stage as initial seeds for Only the coupling (cν )33 is shown in the figure, but A A the minimization routine. Finally, we discard points for (cν )11 and (cν )22 are of the same order of magnitude. 2 A which χ /nd.o.f > 1, where nd.o.f = 9. In addition, there are flavor non-diagonal cν couplings −6 The charged lepton MS masses are run up to the PQ- which are much smaller (. 10 ). Almost identical val- breaking scale, 1010 GeV, in order to compare with the ues are obtained for the off-diagonal vector couplings model predictions.3 The PMNS matrix elements do not (note that the diagonal vector couplings are unphysical, run significantly and we use the low-scale values. The up to electroweak anomalies). neutrino mass differences and PMNS angles and phase These axion-neutrino couplings can, in principle, be are taken from the fit in Ref. [32], assuming a normal constrained by astrophysical neutrinos scattering off relic hierarchy for the neutrinos. Qualitatively similar results axions [15], particle emission in double-β decay experi- are obtained for an inverted mass hierarchy spectrum. ments [34], or Planck satellite measurements [35]. To In the following two sections we present the results for compare with the experimental bounds, we first con- the Dirac and Majorana neutrino cases. As a benchmark vert the axial-vector couplings to axial couplings, gaνν ∼ A point, we assume the following parameter values: σ0 = cν mν /Fa with L ⊃ gaνν aνγ¯ 5ν. However, the mν /Fa 7 0.1, λ = 0.1, ∆ = 10, tan β = 3, kzIR = 10 , and k = MP , factor suppresses the predicted gaνν couplings to be 9 which leads to an axion decay constant Fa ' 8.12 × 10 well below the current most stringent experimental limit −4 −7 GeV (corresponding to an axion mass, ma ' 7 × 10 gaνν . 10 [15]. eV [33]). The value of ∆ is chosen to sufficiently suppress the axion profile on the UV brane and therefore solve the axion quality problem [9]. Majorana neutrinos

In the Majorana neutrino case, we consider the pa- −1 T 3 To improve the numerical stability of the ﬁt, we use an enlarged rameter space where kMM mDk 1 and the neutrino uncertainty of 0.1% for the charged lepton masses instead of the mass hierarchy is partially generated by the seesaw mech- experimental uncertainty. anism. Nevertheless, the cN parameters have an impor- 6

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-0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

FIG. 1: The distribution of 5D fermion mass parameters, ci, for Dirac (left) and Majorana (right) neutrinos. The

closed shapes (line symbols) denote cEi (cNi ) values, and the colors blue, red and green correspond to the largest, next-to-largest and smallest cN,E parameters, respectively.

tant role, since they determine the scale of MM and mD via eqs. (10) and (12). We utilise the mixing matrix in + + +++++++++⨯+⨯++++ eq. (15), which gives an excellent approximation in the +++++⨯⨯⨯+*+⨯+++++ +++⨯++*⨯+⨯+⨯++*⨯⨯++++⨯++⨯+⨯ parameter space we consider. +++⨯+⨯⨯+⨯⨯+*+*⨯+++*+⨯*+++⨯+*+⨯+⨯⨯++⨯+⨯ +⨯+⨯*⨯++⨯⨯+⨯*+⨯++⨯*⨯+*⨯+⨯+⨯+++⨯⨯+⨯+⨯+*⨯⨯ The results of the scan are shown in Figure 1 (right). ⨯*++⨯⨯+*⨯+⨯*⨯+*⨯+*+⨯*+⨯*+⨯*+⨯++⨯+⨯+⨯*+⨯+⨯⨯+ ⨯+⨯*⨯⨯*⨯⨯+*⨯+*⨯*+⨯*+⨯+⨯*⨯+⨯++⨯++⨯ Notice that the distribution of the cE parameters is simi- **⨯+*⨯⨯*⨯**+⨯+⨯+⨯*+⨯+* **⨯⨯***⨯*⨯*⨯⨯*⨯⨯⨯*+⨯*⨯⨯⨯*+⨯++⨯+ lar to the Dirac neutrino case. On the other hand, the cN ***⨯⨯**⨯⨯⨯*⨯*+⨯*⨯ *+⨯**⨯⨯**⨯*⨯***⨯+⨯**⨯** values are smaller compared to the Dirac case, since part ***⨯*+***⨯*⨯*⨯**⨯ ⨯ *****⨯*⨯**⨯* of the neutrino hierarchy is now obtained from the seesaw ⨯**⨯*⨯********* ******⨯** * mechanism. The range of c values in Figure 1 again cor- ******* responds to left (right)-handed fermions localized on the ****** * UV (IR) brane, and to composite right-handed fermions in the dual 4D theory. Interestingly, and as discussed below eq. (13), sterile neutrino masses that are hierarchically smaller than the FIG. 2: The range of sterile neutrino masses arising IR scale are naturally obtained for c > 0. This can be N from the 5D mass parameters c . The largest, clearly seen in Figure 2. The assumption that the 5D pa- Ni next-to-largest and smallest c parameters are rameters are O(1) with flat priors results in a preference N represented by + (blue), × (red) and ∗ (green), for light sterile neutrinos. respectively. Only sterile neutrino masses below the Finally, we discuss the axion–neutrino couplings. The benchmark IR scale = 2.4 × 1011 GeV are shown. axial-vector couplings are shown in Figure 3. The vector couplings are similar in magnitude and not shown. The flavor-diagonal axial-vector couplings are much smaller than in the Dirac neutrino case (the diagonal vector couplings are identically zero). This is because, neglect- sterile masses, as seen in figure 3 (bottom). A similar A A ing active-sterile mixing, they depend only on the left- range is found for (cν )1j and (cν )2j (not shown). These handed profiles, f 0 , which are UV localized and have axial-vector couplings can be converted to axial couplings LiL A a small overlap with the IR-localized fau . The active– gaνN ∼ cν mN /Fa, with L ⊃ gaνN aνγ¯ 5N. Again, the sterile axial-vector couplings are generated through the suppression factor mN /Fa means the predicted couplings active–sterile mixing and hence are suppressed for large are well below current experimental limits [15]. 7

plicit (non-QCD) violations of the PQ symmetry. The 5D model also explains the Standard Model fermion mass hierarchy and flavor structure, while al- lowing for either Dirac or Majorana neutrinos. This is done by introducing right-handed neutrinos charged under the PQ symmetry. In the Majorana neutrino case, the origin of the Majorana mass scale is associated with explicit, Planck-scale PQ symmetry violation on the UV brane. Nevertheless, by localizing the right-handed neutrino profiles towards the IR brane, hierarchically small sterile neutrino masses can be generated, offering a mechanism to naturally extend the applicability of the seesaw mechanism to much lower mass scales. These light sterile neutrino states may be observable in astrophysical or laboratory experiments (see e.g. [36]). In the Dirac case, tiny effective 4D neutrino Yukawa couplings arise from the exponentially small overlap between the left and right-handed neutrino profiles. The axion and neutrino profile structure leads to specific predictions for the axion–neutrino couplings; however, these are well below current experimental limits and would require a substantial improvement in experimental sensi- tivity to be probed. The holographic dual 4D description suggests some new strong dynamics with accidental PQ and lepton number symmetries that confines at an intermediate scale 10 & 10 GeV and gives rise to a composite axion and composite sterile neutrinos. The light sterile neutrinos result from the suppressed transmission of the explicit lepton FIG. 3: The axion–neutrino couplings for both the number breaking to the composite sector, similar to the Dirac and Majorana neutrino cases. The top figure setup considered in [24]. It would be interesting to con- shows the active–active axial-vector coupling, cA, as a ν struct the underlying 4D theory (along the lines studied function of the lightest active neutrino mass. The solid in [37]). Nevertheless, the 5D model provides a complete (open) shapes denote the Dirac (Majorana) couplings. framework that connects axion and neutrino physics to The diagonal coupling is (cA) (orange square), while ν 33 the Standard Model flavor structure and further moti- the off-diagonal couplings are (cA) (green circle), ν 12 vates ongoing experimental searches for axions and sterile (cA) (red triangle), and (cA) (blue diamond). The ν 13 ν 23 neutrinos. bottom figure shows the active–sterile axial-vector A coupling (cν )3j (j = 4, 5, 6) in the Majorana case as a function of the sterile neutrino mass. The lightest, next-to-lightest and heaviest sterile neutrinos are Acknowledgments denoted by diamonds (brown), triangles (purple) and circles (orange), respectively. The work of P.C. is supported by the Australian Gov- ernment through the Australian Research Council. The work of T.G. and M.N. is supported in part by the De- CONCLUSION partment of Energy under Grant DE-SC0011842 at the University of Minnesota, and T.G. is also supported by the Simons Foundation. T.G. acknowledges the Aspen We have presented a 5D model that simultaneously Center for Physics which is supported by the National addresses several problems associated with the Standard Science Foundation grant PHY-1607611, where part of Model flavor structure, neutrinos and axion physics. The this work was done. VEV of a 5D complex scalar field spontaneously breaks the Peccei-Quinn symmetry, giving rise to the axion as a Nambu-Goldstone boson with an axion decay constant determined by the IR scale. This setup automatically addresses the axion quality problem by suppressing the ∗ [email protected] axion profile near the UV brane, where there can be ex- † [email protected] 8

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