Natural axion model from flavour

Salvador Centelles Chuliá,1, ∗ Christian Döring,2, † Werner Rodejohann,2, ‡ and Ulises J. Saldaña-Salazar2, §

1AHEP Group, Institut de F́ısica Corpuscular – C.S.I.C./Universitat de València, Parc Cient́ıfic de Paterna.

C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia), SPAIN2Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, GERMANY

We explore a common symmetrical origin for two long standing problems in particle physics: the

strong CP and the fermion mass hierarchy problems. The Peccei–Quinn mechanism solves the former

one with an anomalous global U(1)PQ symmetry. Here we investigate how this U(1)PQ could at the

same time explain the fermion mass hierarchy. We work in the context of a four-Higgs-doublet model

which explains all quark and charged fermion masses with natural, i.e. order 1, Yukawa couplings.

Moreover, the axion of the model constitutes a viable dark matter candidate and neutrino masses

are incorporated via the standard type-I seesaw mechanism. A simple extension of the model allows

for Dirac neutrinos.

I. INTRODUCTION

Though having been very successful, the standard model of particle physics (SM) lacks the explanation for a

number of experimental observations that strongly point towards, amongst other things, the existence of dark

matter (DM) and a non-zero neutrino mass. Furthermore the SM poses so called fine-tuning problems in which

symmetry-allowed parameters have to be assigned very small numbers in order to fit the experimental results.

One of such problems is the strong CP problem. The non-trivial topological structure of the QCD vacuum leads

to a CP-violating total derivative term θ GµνG̃µν in the QCD Lagrangian with the gluon field strength tensor

Gµν , its dual, and the CP violating angle θ. Naturally1 one expects such parameters to be of O(1). Measurements

of the neutron electric dipole moment, however, limit the angle from above to be |θ| < 10−10, thus hinting tosome new physics that explains such smallness.

Among other proposals, the introduction of a light pseudo-scalar field [3, 4], called axion [5] is the most popular

solution to the problem. Hereby θ is promoted from being a parameter to a dynamical complex pseudo-scalar

field, coming along with a QCD anomalous global U(1)PQ symmetry, called Peccei–Quinn (PQ) symmetry.

The breaking of this symmetry at high scales of order (109 − 1015) GeV leads to a pseudo Goldstone boson,the axion a(x), and in consequence to the presence of the CP-violating term aGµνG̃

µν . Hence θ is replaced by

the dynamical field a and its vacuum expectation value (VEV) can be chosen to zero, 〈a〉 = 0, such that the CPviolating term vanishes, and thus dynamically explains the smallness of the value.

Axions are typically produced in the early Universe via non-thermal mechanisms such as vacuum realignment

[6–8] or string decay [9–20]. These mechanisms lead to a high population of non-relativistic axions. Together

with their weak interactions, besides solving the strong CP-problem, axions fulfill the criteria for cold DM. One of

∗ salcen@ific.uv.es† christian.doering@mpi-hd.mpg.de‡ werner.rodejohann@mpi-hd.mpg.de§ ulises.saldana@mpi-hd.mpg.de1 Typically, in the literature, the term ’natural’ has two distinct meanings. In the first one a small parameter is said to be natural

when setting it to zero leads to a bigger symmetry of the Lagrangian, i.e. it has a symmetry protection [1]. In turn, this implies

that renormalization corrections to this parameter will be proportional to itself. Examples of this behaviour in the SM include the

strong CP-violating parameter, protected by CP symmetry, and the small fermionic Yukawas, protected by chiral symmetry. In

the second meaning [2], which we use throughout this work, we say that a dimensionless parameter is natural when it is of O(1).An abnormally small parameter may be a hint towards a hidden physical mechanism that explains its smallness.

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the two important scales of the cosmological history of axion DM is the temperature at which the PQ symmetry

breaks. One has to distinguish whether the scale breaks during or after inflation, which leads to different allowed

mass parameter ranges for the axion. The second important scale is the QCD scale, around which the axion

acquires a mass due to non-perturbative effects.

The allowed ranges for the axion mass, in order to have DM, depend strongly on the production mechanism

and whether the PQ symmetry breaks before or after inflation. Taking all mechanisms into account opens a

wide allowed range for ma ≤ 10−2 eV [21]. Experimental studies like ADAMX [22] and CAST [23] have alreadyreleased limits and further searches are on going or planned for the future, e.g. ABRACADABRA [24], KLASH

[25] or MADMAX [26].

Axion models fall into two major categories, the KSVZ [27, 28] and the DFSZ [29, 30] model. KSVZ models

require the introduction of new vectorlike quarks while in DFSZ models the Higgs sector is extended leaving

the fermion sector untouched. The model presented in this paper falls into the DFSZ category by introducing

additional Higgs doublets. For more details of the PQ mechanism see the reviews [21, 31–34].

Another fine tuning problem that the SM faces appears in the fermionic Yukawa sector. From a naturalness

point of view, one expects all fermions (i.e. the three generations of up-type quarks qu, down-type quarks qd,

charged leptons ` and neutrinos ν) to have masses of the order of the electroweak (EW) scale (ΛEW = 174 GeV)

if Yukawas are of O(1). However, when experimentally measured, various puzzling and apparently unrelatedpatterns appear, in particular:

i) mt ∼ ΛEW � mα (α = b, τ, c, s, µ, d, u, e);

ii) {mb,mτ ,mc} ∼ 10−2 ΛEW;

iii) {mµ,ms} ∼ 10−3 ΛEW;

iv) {md,mu,me} ∼ 10−5 ΛEW.

When enlisted in this way, the patterns seem to suggest four different mass scales. Adding small active neutrino

masses adds another one.

The pattern can either be understood by Yukawas of different order and a fixed VEV or by Yukawas of similar

order and different order VEVs. If there is a Higgs doublet for each set of fermions in items i)−iv) above, natural(order 1) Yukawa couplings are possible when the associated Higgs doublets feature a sharp VEV hierarchy. The

focus of attention is then shifted from a hierarchy in the Yukawa sector to a hierarchy in the VEVs of the doublets.

In Ref. [35] this idea was realized. The original particle content contained in particular an enlarged scalar

sector with four Higgs doublets and an additional Z′2 × Z′′2 × Z′′′2 symmetry. It is then by the sequential soft-breaking of the aforementioned symmetry that four different mass scales originate. To overcome the appearance

of tree-level flavour-changing-neutral-currents (FCNCs) the singular alignment ansatz as described in Ref. [35]

was implemented. Neutrino masses, which by itself correspond to yet another scale in addition to the ones in

i)− iv), were generated via a type-I seesaw mechanism in which the mass scale of set iv) was identified with theDirac mass matrix.

Interestingly, these two ideas can be combined into one unified symmetric explanation for both problems. In

other words, the axion is not only a viable solution to the strong CP and the DM problems, but it may also be

employed as a solution to the fermion mass hierarchy problem, see [36? –43].

Here we focus on the 4 Higgs Doublet Model (4HDM) of [35]. Note that the Abelian nature of the symmetrical

group Z32 could naturally originate from a global U(1) symmetry, suggesting a connection to the origin of the

axion. This observation motivates us to try to find a common symmetrical origin for the previously discussed

3

problems (strong CP, DM, and mass hierarchy). In this paper, we explore under what conditions we could make

such a statement. Note that, compared to Refs. [40, 41, 44], where the Froggatt–Nielsen mechanism [45] plays

the dominant role as a solution to the fermion mass hierarchy puzzle, we instead make use of a renormalizable

mechanism for the same target.

The paper is organised as follows. In Sec. II a 4HDM in the spirit of [35] is introduced, in which each fermion

set i) − iv) couples to only one Higgs doublet. This feature is enforced thanks to a global U(1) symmetry. Wedescribe how this global U(1) is spontaneously broken by two scalar singlets A and χ, which along with the

spontaneous electroweak symmetry breaking generates the required VEV hierarchy in a natural way. In Sec. III

we identify the U(1) global anomalous symmetry with the PQ symmetry and flesh out the phenomenological

details of the axion of our model. In particular, we compute the coupling between the axion and photons and

show that our model has a non-universal coupling of axions and charged fermions. This can imply the existence

of axion-mediated FCNCs which is a potential experimental signal from the model. However, the axion-mediated

FCNCs can also be taken to zero with a judicious choice of parameters.

A benchmark numerical scenario showing the main features of the model and satisfying the main experimental

constraints has been relegated to Appendix A. Finally, in Appendix B we describe an alternative scenario with

Dirac neutrinos.

II. A 4 HIGGS DOUBLET MODEL WITH ORDER ONE YUKAWA COUPLINGS

A. General Structure of the Model

The central idea of the 4HDM model in [35] is to explain the mass hierarchy of the fermions by dividing them

into four mass classes and assigning each of them to a single Higgs doublet:

• ({mt};φt),

• ({mb,mτ ,mc};φb),

• ({mµ,ms};φµ),

• ({md,mu,me};φd),

where Yukawa couplings between a Higgs and its assigned fermions are of O(1) and we define the scalar doubletcomponents as

φα =

(φ+α

(vα +ρα√2)e

iaα√2vα

), (α = t, b, µ, d), (1)

with v2t +v2b +v

2µ+v

2d = (174 GeV)

2. The four Higgs doublets must have a sharp VEV hierarchy, which in return

gives different order of magnitude masses to each of the four sets of fermions. It is possible to subsequently

induce the VEVs from the next-highest VEV, thereby explaining their hierarchy. This VEV hierarchy can be

achieved with a careful choice of U(1) charges and field inventory, as it can be seen in Tab. I. Note that the

fermion content of the model is exactly the SM, and only the scalar sector has been enlarged.

The presence of an additional scalar singlet (χ), besides the PQ breaking one (A), is justified as follows.

Once φt acquires a VEV from spontaneous EW-symmetry breaking, the VEVs of the other Higgs doublets are

4

Fields SU(2)L × U(1)Y U(1)PQ

Lep

tons

Li (2,−1/2) lνR,i (1, 0) 0

eR (1,−1) 2lµR (1,−1) 2l − kτR (1,−1) 2l − 2k

Quark

sQi (2, 1/6) q

uR (1, 2/3) q − lcR (1, 2/3) q − l + 2ktR (1, 2/3) q − l + 3kdR (1,−1/3) q + lsR (1,−1/3) q + l − kbR (1,−1/3) q + l − 2k

Sca

lars

φt (2, 1/2) 3k − lφb (2, 1/2) 2k − lφµ (2, 1/2) k − lφd (2, 1/2) −lχ (1, 0) k

A (1, 0) k/2

TABLE I. Particle content and symmetry transformations under the EW gauge group and U(1)PQ symmetry. All the scalars are

color singlets while only the quarks transform non-trivially under SU(3)C . The Lagrangian and charges have been checked using

Sym2Int [46].

sequentially induced thanks to a cubic interaction with the singlet scalar χ, triggering a cascade in which each

Higgs field induces a VEV for the next lower hierarchical level, vt ↪→ vb(vt) ↪→ vµ(vb) ↪→ vd(vµ), such thatv2t � v2b � v2µ � v2d. Unless there is extreme tuning, the VEV of χ is required to be not too far away fromthe weak scale, which in turn implies that its Goldstone boson associated to the broken global symmetry cannot

be the axion. Instead, another scalar A needs to be introduced, which will be associated to the axion. It also

couples to χ, and in fact induces a VEV to it.

B. Fermion Sector

We now give the Lagrangian which satisfies the desired properties and in parallel derive the PQ charges, already

shown in Tab. I, that generate it. Starting from the Yukawa sector of the model we have, for the up and down

quarks

−Lu =Yu,iQi φ̃d uR + Yc,iQi φ̃b cR + Yt,iQi φ̃t tR + h.c., (2)−Ld =Yd,iQi φd dR + Ys,iQi φµ sR + Yb,iQi φb bR + h.c., (3)

where i = 1, 2, 3 sums over the three generations of left-handed quark doublets Q. Note how each quark only

receives a single contribution to its mass coming from one Higgs doublet. In the following, we denote the U(1)-

global charges by X such that, e.g. X (Qi) ≡ XQi . From this Yukawa Lagrangrian we can immediately find some

5

initial conditions for the PQ charges:

XQ1 = XQ2 = XQ3 = q, (4)

while for the right-handed quarks we obtain

XuR = q + Xφd , XdR = q −Xφd ,XcR = q + Xφb , XsR = q −Xφµ ,XtR = q + Xφt , XbR = q −Xφb .

(5)

Analogously, the charged lepton Yukawa couplings are given by

− Le = Ye,i Li φd eR + Yµ,i Li φµ µR + Yτ,i Li φb τR + h.c., (6)

which leads to the following conditions for the PQ charges:

XL1 = XL2 = XL3 = l,XeR = l −Xφd , XµR = l −Xφµ , XτR = l −Xφb .

(7)

The pieces of the charged Yukawa Lagrangian can be written in matrix form, which explicitly shows the wanted

features of the model:

LQ =(Q1 Q2 Q3

)Yu,1 φ̃d Yc,1 φ̃b Yt,1 φ̃tYu,2 φ̃d Yc,2 φ̃b Yt,2 φ̃tYu,3 φ̃d Yc,3 φ̃b Yt,3 φ̃t

uRcRtR

+ (Q1 Q2 Q3)Yd,1 φd Ys,1 φµ Yb,1 φbYd,2 φd Ys,2 φµ Yb,2 φbYd,3 φd Ys,3 φµ Yb,3 φb

dRsRbR

+ h.c.

−Le =(L1 L2 L3

)Ye,1 φd Yµ,1 φµ Yτ,1 φbYe,2 φd Yµ,2 φµ Yτ,2 φbYe,3 φd Yµ,3 φµ Yτ,3 φb

eRµRτR

+ h.c., (8)where LQ = −(Lu + Ld).

In the neutrino sector, we can easily implement a Majorana type-I seesaw [47–51] and choose the Dirac term

to be given by the VEV of φd:

− Lν = Yν,ij Li φ̃d νR,j +Mij

2νcR,i νR,j + h.c. (9)

Here vd is the smallest VEV of the model. Eq. (9) implies for the PQ charges:

XνR,1 = XνR,2 = XνR,3 = 0 and Xφd = −l. (10)

In return, by virtue of the seesaw formula, the light neutrino mass matrix is

mν = −v2dYνM−1YTν . (11)

In the ’standard’ seesaw case, if we take the Yukawas to be of order 1 and the neutrino mass scale to be 0.1

eV then the mediator has to have a mass of M ∼ 1014 GeV. However, in our model the neutrino masses aregiven by vd ∼ 0.001 GeV instead of the SM VEV. In this case, the mass of the mediator is given by M ∼ 104GeV = 10 TeV. As shown in [35] these values still escape the experimental constraints coming from colliders

and lepton flavour violation processes [52, 53]. Let us now note that by relaxing the Yukawas to be of order 0.1

would lead to a mediator mass of just 100 GeV. Therefore, this model has a range for observable lepton flavour

6

violation and collider physics in certain regions of the parameter space. Notice further that the identification

of Yukawa interactions between neutrinos and φd is not required and could have been done with any of the

other three scalar doublets without any further consequence for the main features of the model. A detailed type

I-seesaw phenomenological study is out of the scope of this work and has been studied extensively in the literature.

As a side note, we remark that if the right-handed neutrino charges are taken to be different from zero then

the second term in Eq. (9) would be forbidden and thus neutrinos would be Dirac particles. Their small mass

could then be explained by a fifth Higgs doublet whose VEV should be much smaller than the other four, in the

same spirit as the explanation to the charged fermion hierarchy. See Appendix B for more details.

The total Lagrangian made of Eqs. (8) and (9) is the same fermionic Lagrangian as the one shown in [35] i.e.

each fermion set receives its mass by the Higgs associated to it. That is, {t}, {b, τ, c}, {µ, s} and {d, u, e} acquiretheir mass by φt, φb, φµ, φd, respectively. As we will show below, the mass hierarchy of such sets is explained

by a hierarchy in the VEVs of the different Higgs doublets and order 1 Yukawa couplings. In particular, and

as an example, all masses can be accommodated with order 1 Yukawas if the VEVs are taken to be vt = 174

GeV, vb ∼ 1 GeV, vµ ∼ 0.1 GeV, and vd ∼ 0.001 GeV. A typical problem arising in models with multipleHiggs doublets is the presence of tree-level flavour changing neutral currents. A solution was suggested in [35]

by introducing what was called singular alignment (see Ref. [54] for the same ansatz under a different approach).

Moreover, this alignment allows us to set aside the problem of mixing as it is automatically decoupled from the

problem of mass. This can be understood as all individual Yukawa matrices are simultaneously diagonalised

when moving to the mass basis. We then assume that this mechanism is realized here as well. The next step is

to illustrate how the U(1) model acquires the necessary VEV hierarchy, and how the new scalar singlets χ and

A help in that, and how they are connected to the usual axion physics.

C. Scalar Sector

We now turn our attention to the scalar sector of the model. For the time being, let us study it with the four

Higgs doublets and only χ out of the two singlet scalars. The scalar Lagrangian can be divided into four different

pieces,

Lscalar ⊃ LS1 + LS2 + LS3 + Lkin. (12)

Here Lkin contains the corresponding kinetic terms of all scalar fields. The first term groups all the quadraticand quartic terms where a given field and its complex conjugate always appear together:

−LS1 =∑

a∈{t,b,µ,d}

[µ2a φ

†a φa +

λa2

(φ†a φa)2

]+ µ2χχ

†χ +λχ2

(χ†χ)2 +∑

a∈{t,b,µ,d}

λaχ (φ†a φa)χ

†χ

+∑

a,b∈{t,b,µ,d}; a>b

[λab1(φ

†a φa)(φ

†b φb) + λab2(φ

†a φb)(φ

†b φa)

]. (13)

We note that LS1 has an accidental U(1)5 symmetry, one U(1) per scalar field. If the scalar Lagrangian werejust formed by LS1 then the model would have five Goldstone bosons. However, we add the following termsto our Lagrangian in order to reduce the symmetry of the model U(1)5 → U(1)Y × U(1)PQ and therefore, by

7

construction, only two Goldstone bosons remain. These extra terms are given by

−LS2 = λtbbµ(φ†tφb)(φ†µφb) + λtbµd1(φ†tφb)(φ†dφµ) + λtbµd2(φ†tφµ)(φ

†dφb) + λbµµd(φ

†bφµ)(φ

†dφµ) + h.c., (14)

−LS3 = −κtbχ (φ†tφb)χ − κbµχ (φ†bφµ)χ − κµdχ (φ†µφd)χ + λtµχχ (φ†tφµ)χ

2 + λbdχχ (φ†bφd)χ

2 + h.c., (15)

where we have assumed a CP symmetrical scalar potential by considering real couplings. Now, note that LS2breaks U(1)5 → U(1)Y × U(1)PQ × U(1)χ and LS3 breaks it further into U(1)Y × U(1)PQ. While LS2 has nofurther consequences of interest to us, it will be automatically generated if LS3 is allowed by the symmetries ofthe model. On the other hand, the dimensionful couplings of LS3 are the relevant sources of the ‘VEV cascade’and thus key to our construction, as we show later.

After spontaneous EW symmetry breaking it can be shown that the mass matrix for the CP-even scalars

has rank 5, while the mass matrix for CP-odd scalars has rank 3, i.e. it has two massless eigenstates, the two

Goldstone bosons mentioned before. We come back to this point in Sec. III.

Note that in order to allow the κ terms in LS3, the charges of two consecutive Higgs doublets must be separatedby a constant charge which is at the same time the charge of χ. With this in mind, and with Eq. (10), we find

the last conditions for our PQ charges:

Xχ = k, Xφd = −l, (16)Xφµ = k − l, Xφb = 2k − l, Xφt = 3k − l. (17)

We can now express all the PQ charges in terms of q, l, and k, which are the charges of the quark doublet, the

lepton doublet, and the singlet scalar χ, respectively. This information is already shown in Tab. I.

Let us now focus on the term LS3 in Eq. (15) and assume that

µ2t < 0, µ2b > 0, µ

2µ > 0, and µ

2d > 0, (18)

such that, in the absence of Eqs. (14) and (15), only φt develops a VEV, vt 6= 0, while all others remain zero.Moreover, the singlet scalar χ also has a VEV, for which the axion is responsible (as discussed below). Under

these features, Eq. (15) induces small and hierarchical VEVs, as obtained from the minimum conditions

∂(LS1 + LS2 + LS3)∂ρα

∣∣∣∣∣min

= 0, (α = t, b, µ, d), (19)

where ρα is defined in Eq. (1), and from which one correspondingly obtains an intricate system of polynomial

equations:

vt

µ2t + λtv2t + λtχv2χ + ∑k=b,µ,d

λ̃tkv2k

+ λtbbµvµv2b + λ̃tbµdvbvµvd + vχ(λtµχχvµ − vbκtbχ) = 0,vb

µ2b + λbv2b + λbχv2χ + ∑k=t,µ,d

λ̃bkv2k

+ λbµµdv2µvd + 2λtbbµvtvbvµ − (κtbχvt + κbµχvµ)vχ + λbdχχvdv2χ = 0,vµ

µ2µ + λµv2µ + λµχv2χ + ∑k=t,b,d

λ̃kµv2k

+ vt(λtbbµv2b + λtµχχv2χ) + λ̃tbµdvtvbvd + 2λbµµdvbvµvd − (κbµχvb + κµdχvd)vχ = 0,vd

µ2d + λdv2d + λdχv2χ + ∑k=t,b,µ

λ̃kdv2k

+ λbµµdvbv2µ + λbdχχvbv2χ + λ̃tbµdvtvbvµ − κµdχvµvχ = 0,(20)

8

where, in general, for arbitrary values there is no guaranteed solution and we have defined λ̃x = λx1 + λx2. We

find that by imposing the conditions κijχvχ < (O(10− 100) GeV)2 and all dimensionless couplings appearing inEqs. (14) and (15) to be smaller than 10−1, we guarantee that each Higgs field induces a VEV for the next lower

hierarchical level. Furthermore, both conditions make also possible to directly apply the results of Ref. [35] to

our case, see Appendix A for further details. In return, the VEVs can be nicely approximated by

vt '√−µ2tλt

, vb 'κtbχvχvt

µ2b + (λtb1 + λtb2)v2t + λbχv

2χ

,

vµ 'κbµχvχvb

µ2µ + (λtµ1 + λtµ2)v2t + λµχv

2χ

, vd 'κµdχvχvµ

µ2d + (λtd1 + λtd2)v2t + λdχv

2χ

.

(21)

Addition of the second singlet scalar, whose CP-odd component is to be later identified with the dominant

contribution to the axion, brings about the last piece of information for the full scalar Lagrangian, Lscalar =LS1 + LS2 + LS3 + LA + Lkin:

−LA1 = µ2AA†A +λA2

(A†A)2 +∑

a∈{t,b,µ,d}

[λaA (φ

†a φa)A

†A]

+ λχA(χ†χ) (A†A)

−LA2 = −κAAχ(AAχ† + λtbAφ†tφbA2 + λbµA φ†bφµA2 + λµdAφ†µφdA2 + h.c.LA = LA1 + LA2 . (22)

A major consequence of the previous equation2 is that, through the minimization conditions

∂Lscalar∂Re(A)

∣∣∣∣∣min

= 0 and∂Lscalar∂Re(χ)

∣∣∣∣∣min

= 0, (23)

which imply

vA(−2vχκAAχ + λAv2A + λχAv2χ + µ2A) = 0,

vχ

µ2χ + λχv2χ + λχAv2A + ∑k=t,b,µ,d

λkχv2k

− ∑j=tb,bµ,µd

κjχv2j − κAAχv2A + 2(λbdχχvbvd + λtµχχvtvµ)vχ = 0,

(24)

it shows how A induces a small VEV to χ;

vA '√−µ

2A

λAand vχ '

κAAχv2A

µ2χ + λχAv2A

. (25)

Here we have assumed µ2A < 0 and µ2χ > 0. Realize the induced hierarchical nature of the VEVs, vA � vχ. For

example, for vA ∼ 1012 GeV and µχ ∼ 107 GeV, we obtain vχ ' κAAχ = O(1 − 10) GeV. Moreover, to avoidlarge loop corrections and PQ-scale contributions to the scalar masses and VEVs we fine-tune the mixing between

the singlet scalar A and the doublet scalar fields such that it is negligible. This is the well known hierarchy

problem which in the DFSZ model always appears, as well as in any model with a scale much higher than the EW.

2 With the introduction of a sixth scalar, the previous discussion on accidental symmetries changes as follows: The self-conjugated

scalar terms LS1 + LA1 have an accidental U(1)6 symmetry. The presence of the non-Hermitian terms LS2 +LS3 +LA2 explicitlybreaks it, U(1)6 → U(1)Y × U(1)PQ.

9

breaking Electroweak breaking

timehigh PQ-scale"waiting" until EW breaking

cascade mechanism

: implies: acquiresallowing crosstalking

FIG. 1. Schematic view of the development of the radial field components of the model. At high scales, the new global

U(1)PQ symmetry spontaneously breaks. The radial component (solid line) of A then acquires a VEV while the phase

becomes a massless Goldstone boson (GB), i.e. the axion (not shown here). The radial component, however, induces a

VEV for the radial component of χ. Until EW symmetry breaking this has no impact whatsoever, but as soon as the

radial component of φt acquires a VEV by spontaneous EW symmetry breaking, a cascade is triggered. Together, vχ and

vt imply a VEV for vb and so on. Due to the scalar terms, explicitly breaking U(1)6 → U(1)Y × U(1)PQ, there are only

two massless GBs at the end of the chain, instead of six, which are a mixture of the CP-odd components of all the scalar

fields.

As a conclusion for this section, let us point out the importance of the global symmetry U(1)PQ. Thanks to

the symmetry breaking pattern of the scalars one can naturally obtain a sharp VEV hierarchy between the four

Higgs doublets. At the same time, the U(1) structure of the fermions combined with this sharp VEV hierarchy

leads to the observed fermion mass hierarchy with order 1 Yukawas. The whole mechanism is summarised in

Figure 1 and an explicit realization is offered in Appendix A. Note that no predictions for the fermion mixings

can be made, since we are making use of an Abelian symmetry and assuming singularly aligned Yukawa matrices.

In what follows, we study the consequences of identifying the global symmetry U(1) with the PQ symmetry

and the Goldstone boson with the axion, which is the main point of this paper.

III. THE PECCEI–QUINN SECTOR

Having introduced our model and explained the cascade mechanism, this section is dedicated to the phase of the

PQ-breaking field A, which we identify with the axion. We start by explicitly deriving the conserved current

from the PQ-symmetry using Noether’s theorem which is given by

jµPQ =∂L

∂(∂µψi)δψi(x) +

∂L∂(∂µϕj)

δϕj(x)→ jµPQ = iα(Xψi ψ̄iγµψi + ϕ†jXϕj∂µϕj

)+ h.c., (26)

where L is the full Lagrangian, ψi and ϕi run over all the fermion and scalar fields, respectively, and Xψi and Xϕiare the PQ charges of the field in the subscript; δψi = iαXψiψi, δϕi = iαXϕiψi, α is the infinitesimal parameterof the continuous transformation and a sum over the indices i and j is understood. By definition only the phases

of the Higgs fields and the PQ breaking scalar contribute to the axion. Hence we will now drop the fermionic

fields, the charged Higgs and the radial components of the neutral Higgses in jµPQ. Then we have

φk ⊃ vkeiak√2 vk

(0

1

), (k ∈ {t, b, µ, d}), χ ⊃ vχe

iaχ√2 vχ , A ⊃ vAei

aA√2 vA , (27)

which upon substitution in Eq. (26) immediately yields

jµPQ|a = viXi∂µai, (28)

where we have removed a constant global factor −√

2α. Now, note that by defining

a =1

va

∑i

Xiviai and v2a =∑i

X 2i v2i ≈ X 2Av2A, (29)

10

we obtain jµPQ|a = va∂µa and 〈0|jµPQ|a|a〉 = ivapµ, as expected for a massless Goldstone boson and thus a canbe identified with the axion.

This kind of models for the axion fall under the category of DFSZ models. Specifically, since we use four

Higgs doublets, it would fall under the category of DFSZ-IV [33]. This category considers the maximum allowed

number of Higgs doublets, nH , contributing to the ratio between the QCD and QED anomalies, which is nH = 9

(one per charged fermion). Hence, our model is the most minimal realization of a DFSZ-IV theory where each

flavour, of a given fermion species, couples to a different scalar doublet whose VEV judiciously represents one of

four groups of similar masses. We will now flesh out the phenomenological implications of the axion of our model.

In order to avoid a dangerous kinetic mixing between the axion and the Z boson, see Ref. [21], we must impose

the orthogonality between the PQ and the hypercharge currents, i.e.

jµPQ|a jYµ |a = 0 ⇒∑

i∈{t,b,µ,d,χ,A}

YiXiv2i = 0 , (30)

with i running for all the scalars in the model. Since the singlets χ and A do not carry hypercharge and all

doublets have the same hypercharge, Eq. (30) simplifies to∑i∈{t,b,µ,d}

Xiv2i = 0 ⇒ (3k − l) v2t + (2k − l) v2b + (k − l) v2µ − l v2d = 0 , (31)

with i running only for the four scalar doublets. Thanks to the sharp hierarchy between the VEVs the equation

can be solved by taking k ' l/3, since vt dominates3.

The choice of charges leads to a number of triangular anomalies. First, the fermion U(1)PQ charges given by

Tab. I are anomalous with respect to the colour group. In particular, we have computed the coefficients for all

the possible anomalies using Susyno [55]. Since the gauge structure is the same as in the SM and our model

does not feature new fermions, the gauge group anomalies are cancelled like in the standard case. However, all

the triangular anomalies related to U(1)PQ and the gauge group will be non-zero. We give special attention to

U(1)2EM × U(1)PQ, U(1)EM × U(1)2PQ, and SU(3)2C × U(1)PQ. The non-trivial vacuum of SU(3)C will generateinstanton effects which will lead to CP conservation in the QCD vacuum, like in the usual PQ mechanism. The

four vertices generated by these anomalies are shown in Fig. 2.

We now compute the ratio between the QCD anomaly and the QED anomaly, typically called E/N in the

literature. This ratio is given by

E

N=

2

3+ 2

∑i

((XeL,i −XeR,i) + (XuL,i −XuR,i)

)∑i

((XdL,i −XdR,i) + (XuL,i −XuR,i)

) = 83, (34)

where i runs over the three generations of fermions. The first equality is completely general (as long as the fermion

content with non-trivial gauge charges is the same as in the SM) while the second equality is the particular value

of our model. This result is irrespective of the choice of the charges q, l, and k. It is a common result in axion

3 For a more general scenario, where the four VEVs are arbitrary, the orthogonality condition would require:

Xt = s2βts2βbs2βµc

2βbc2βµ , Xb = s

2βbs2βµc

2βtc2βµ , Xµ = −s

2βµc2βtc

2βb, Xd = −c2βtc

2βbc2βµ , (32)

where the angles are defined by

sinβt =

√v2b + v

2µ + v

2d

v2t + v2b + v

2µ + v

2d

, sinβb =

√v2µ + v

2d

v2b + v2µ + v

2d

, sinβµ =

√v2d

v2µ + v2d

. (33)

11

FIG. 2. Triangular diagrams associated with the axion in our model. Here a, γ, and g denote the axion, the photon,

and the gluon, respectively, while ψi denotes all the fermions and qi the six quarks. The first panel comes from the

SU(3)2C × U(1)PQ anomaly and generates instanton effects implying CP conservation in the strong sector. The secondpanel comes from the U(1)2EM × U(1)PQ anomaly and allows a decay of an axion into two photons and thus a detectionpossibility, suppressed by 1/fa. The third and fourth panels depicts the vertices aaγ and aaa which are suppressed by

1/f2a and 1/f3a , respectively.

models in which, like ours, the U(1) charges in the down quarks and charged leptons imply coupling to the same

scalar doublets,

XdL,i −XdR,i = XeL,i −XeR,i , (i = 1, 2, 3) , (35)

although there are other possibilities. For example, a simpler model with just two Higgs doublets and universal

PQ charges for each fermion type could have either E/N = 8/3 or E/N = 2/3, depending on which Higgs couples

to the electron. Additionally, it is necessary to remark here that E/N = 8/3 is a shared feature amongst axion

models where the fermion mass hierarchy is explained through the PQ symmetry group, e.g. see Refs. [41, 56–61].

Explicitly, E/N = 8/3, is a consequence of any theory where the down-quark and charged lepton mass hierarchy

is explained by the same scalars irrespective of the up-quarks, which is very reasonable as they have similar

masses, md,i = O(1)me,i (i = 1, 2, 3).As mentioned before, axions can be a major component of the DM density and serve due to their properties

as cold DM4. In order to have the axion to be DM and to determine the amount it contributes, one has to make

specific choices. In particular, if the PQ breaking occurs before inflation, topological defects get inflated away

and their contribution to axion production is negligible. In this case, the dominant axion production mechanism

is the misalignment mechanism. Then, the axion DM abundance will depend on the initial misalignment angle θi.

However, these issues depend on the phenomenological and cosmological scenarios and the concrete realization of

model parameters which go beyond the scope of this work. Since our model is just a realization of a DFSZ-type

4 Thermally produced axions could constitute hot DM [62].

12

of model, we directly apply general results from earlier studies of this class. Hence the only requirement for the

axion in our model being cold DM is to have a mass mDMa ≤ 10−2 eV in order to fall in the range of any of theallowed scenarios. However, note that having multi-component DM is also a viable possibility [63–73] and the

axion could be responsible for only a fraction of the DM density, thus relaxing the constraints. For more details

on axion DM we refer the reader to [62].

Following [31–33], we can compute the mass of the axion in our model

ma ≈ 57.(6)(4)µeV(

1011 GeV

fa

)= mDMa

(1011GeV

fa

), (36)

where we have defined mDMa = 57.6µeV as the highest mass of the axion that leads to a correct one-component

DM relic density, assuming the axion is the main component of dark matter, and fa = va/(2N) is the axion decay

constant. This happens when fa ≈ 1011 GeV. Remember that the constraint for axion dark matter is fa > 1011GeV [74]. Note that the only model-dependence of this formula is via fa. The effective coupling between the

axion and the photon, which will depend on E/N = 8/3 in our model, can also be computed and reads

gaγγ ≈αEM2πfa

(E

N− 1.92(4)

)≈ 8.67× 10−4 1

fa= gDMaγγ

(1011 GeV

fa

), (37)

where we have defined gDMaγγ = 8.67×10−15 GeV−1 as the maximum coupling between the axion and two photonsin a model with axion DM and E/N = 8/3. Combining Eqs. (36) and (37) leads to a simple relation between

ma and gaγγ , which is shown graphically in Fig. 3 including the present and future experimental constraints:

gaγγ =

(gDMaγγmDMa

)ma . (38)

Although major parts of the axion DM parameter space with E/N = 8/3 are still out of reach of current

experiments, exciting times are ahead of us, with experiments like MADMAX [26], ADMX [22], and the use

of topological insulators [75] having enough sensitivity to probe this region in the future. Also note that this

conclusion holds for any axion model with the same ratio of color and electromagnetic anomalies.

Analogously, we could compute the coupling constant of the vertex aaγ and the vertex aaa, both shown in

Fig. 2, this time driven by the anomalies U(1)2PQ × U(1)EM and U(1)3PQ, respectively. However, these processeswill be suppressed by 1/f2a and 1/f

3a , respectively, and thus are expected to be extremely small.

As a final remark, let us point out the current situation of our model on FCNCs mediated by the axion.

Following the derivation in Ref. [21], we can write an effective interaction between the axion and fermions in the

form

Laψ =∂µa

2fa

[ψ̄iγ

µ(CVψij − CAψijγ5

)ψj], (39)

where ψi determines the fermion type (up quarks, down quarks, charged leptons, and neutrinos) and a sum over

the indices i, j, which run from 1 to 3, is understood. The coupling constants are given in general by

CV,Aψij =1

2N

(Uψ†L XψLU

ψL ±Uψ†R XψRUψR

)ij, (40)

with UψL(R) are the left (right) unitary diagonalization matrices of each fermion type ψ and XψL,R is a diagonal

matrix with the PQ charges of the type with the given chirality. Given our choice of U(1) charges, as shown in

Table I, we find that XψL is a matrix proportional to the identity and thus the equation simplifies to

CV,Aψij =1

2N

(XψL I±Uψ†R XψRUψR

)ij. (41)

13

HESS

10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101

ma [eV]

10−16

10−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

10−6

|g aγ|[

GeV−

1 ]

RB

F+

UF

AD

MX

HA

YS

TA

C

IAXOIAXO

MA

DM

AX

AB

RA

CA

DA

BR

A

Top

olo

gica

lIn

sula

tors

KL

AS

H

Opt

ical

Hal

osco

pe

Perseus

SN-γ

Fermi

Tele

scop

es

Horizontal branchHorizontal branch

Solar νSolar ν

CAST

LSW

Cold Dark Matter

E/N=

8/3

FIG. 3. Axion exclusion plot from [76] with the derived E/N = 8/3 of this model. Shown is the axion-photon coupling

versus the axion mass. The colored areas are excluded regions while transparent colored regions will be tested by future

experiments [22–26, 75, 77–89]. Also shown in grey the mass region for axion-DM.

Moreover, realize that

XψR = g(q, l)I ±

0 0 00 αk 00 0 (α+ 1)k

, (42)where g(q, l) ≡ {q − l, q + l, l}, {+,−,−}, and α = {2, 1, 1} correspond to the up-quarks, down-quarks, andcharged leptons, respectively. Now, notice that it is only k which may induce flavour-violating couplings. Their

possible presence, however, can be removed by a judicious choice of UψR which is not constrained by fermionic

mixing as its counterpart, UψL. This freedom is a direct a consequence of having applied singular alignment to

the model.

IV. CONCLUSIONS

We have presented a model where the strong CP, DM, neutrino masses, and fermion mass hierarchy problems find

a common symmetrical origin in the Peccei–Quinn global and Abelian symmetry group, U(1)PQ. The construction

allows all dimensionless couplings in the EW and strong sector to be O(1). We have required the introductionof two gauge singlet and complex scalars and three Higgs doublets apart from the SM one. Additionally, small

14

neutrino masses find their origin in a standard type-I seesaw mechanism by including three heavy right-handed

neutrinos, whose charge under PQ is assumed to be trivial. Relaxing this assumption can lead to Dirac neutrinos,

whose small mass can be explained in the same spirit as the charged fermion mass hierarchy by the addition of

a fifth Higgs doublet, φν , with a very small VEV vφν ∼ 1 eV.To avoid tree-level FCNCs we have applied the singular alignment ansatz in the Yukawa sector. Moreover,

since this ansatz provides a linear realization of the principle of minimal flavour violation, the appearance of

FCNCs at the loop-level poses no risk. Two singlet scalars are required, as otherwise, the large PQ scale induces

unacceptable amounts of fine-tuning in the model.

The ratio between the colour and electromagnetic anomalies is predicted to be E/N = 8/3, irrespective of the

explicit U(1)PQ charge assignments of the model. This value originates from allocating the same scalar doublets

to couple to the down-quarks and charged leptons. Our approach is the most minimal realization of a DFSZ-IV

model wherein all charged fermions couple to a different Higgs doublet. Moreover, non-universal charges for each

fermion family leads to the possibility of appearance of axion-mediated FCNCs suppressed by 1/fa. However,

this FCNCs can be taken to vanish with a judicious choice of unconstrained parameters, thanks to the singular

alignment ansatz.

The interplay between Higgs and axion physics plays a major role in distinguishing our model from other axion

models. The study of the SM-like Higgs couplings to the other fermions will already be sufficient to test some of

our predictions.

In conclusion, we present a possible solution to two fine tuning problems in the SM, namely, the flavour and

strong CP problems, by combining them in a single unified symmetrical framework with natural dimensionless

parameters. In turn, the model is compatible with the parameter space in which the axion is the Dark Matter,

thus solving a third issue of the Standard Model.

ACKNOWLEDGMENTS

S.C.C. would like to thank the Max-Planck-Institut für Kernphysik in Heidelberg for hospitality during his

visit, where this work was initiated. The work of S.C.C. is supported by the Spanish grants SEV-2014-0398,

FPA2017-85216-P (AEI/FEDER, UE), PROMETEO/2018/165 (Generalitat Valenciana), Red Consolider Mul-

tiDark FPA2017-90566-REDC and BES-2016-076643. The work of W.R. is supported by the DFG with grant

RO 2516/7-1 in the Heisenberg program. U.J.S.S. acknowledges support from CONACYT (México).

Appendix A: Benchmark scenario

As a proof of principle, we provide an explicit realization of the VEV cascade and show its implications. First of

all, notice that when the set of dimensionless couplings

C = {λtbbµ, λtbµd1, λtbµd2, λbµµd, λtµχχ, λbdχχ, λtA, λbA, λµA, λdA, λtbA, λbµA, λµdA} (A1)

is set to zero, C → 0, and after integrating out the singlet scalars, χ and A, we recover the original model ofRef. [35] with all of its couplings of the type Zab vanishing,

{Ztb, Ztµ, Ztd, Zbµ, Zbd, Zµd} = 0. (A2)

15

This allows us to know which part of the parameter space will closely resemble the original model.

Let us now, step by step, argue the following considerations:

• To avoid the hierarchy problem we set to zero those couplings inducing very large mixing between the axionand the Higgs doublets:

{λtA, λbA, λµA, λdA, λtbA, λbµA, λµdA} = 0. (A3)

• To avoid unnecessary interference during the VEV cascade mechanism we set an upper bound to thefollowing couplings:

{λtbbµ, λtbµd1, λtbµd2, λbµµd, λtµχχ, λbdχχ} ≤ 10−1. (A4)

One may allow couplings to be order 1, but the smallest VEV, namely vd, will suffer from non-negligible

contributions and thus Eq. (21) will not longer be completely valid.

• Bounding from below the scalar potential requires some necessary conditions which nevertheless are notsufficient:

λt,b,µ,d,χ,A ≥ 0, λtb1 ≥ −√λtλb, λtµ1 ≥ −

√λtλµ, λtd1 ≥ −

√λtλd,

λbµ1 ≥ −√λbλµ, λbd1 ≥ −

√λbλd, λµd1 ≥ −

√λµλd, λχA ≥ −

√λχλA,

λtχ ≥ −√λtλχ, λbχ ≥ −

√λbλχ, λµχ ≥ −

√λµλχ, λdχ ≥ −

√λdλχ.

(A5)

• To ensure unitarity and perturbativity bounds we set the limits

0 < λt,b,µ,d,χ,A . 2, −4 . (λij1 + λij2) . 2, |λij1| . 3, |λij2| . 3, (A6)

where ij = {tb, tµ, td, bµ, bd, µd}. These limits were obtained using the K-matrix formalism [90–92].

• We consider |µA| � |µχ| which is sufficient to guarantee that to a very good degree of approximationvχ ∼ κAAχ holds.

• To create the right hierarchical structure among the EW-VEVs we assume

{κAAχ, κtbχ, κbµχ, κµdχ} ∈ O(1− 100) GeV. (A7)

However, as the off-diagonal entries of the scalar mass matrices depend on the combination vχκijχ =

κAAχκijχ to avoid large mixing, we require these two kind of couplings to behave inversely proportional to

each other, κAAχ ∝ 1/κijχ.

• To guarantee that the lightest scalar, to be associated to the SM-Higgs, mainly comes from φt we considerthe explicit values:

|µt| = 88.5 GeV and λt = 0.26, (A8)

which imply mh = 125 GeV. Realize that as long as φt is the initial field acquiring a VEV, such that

vt � vb � vµ � vd and vχ ∼ κAAχ, then Eq. (A8) is the only set of values giving rise to mh, mimickingthose already appearing in the SM. That is, there is no ambiguity on which neutral scalar becomes the

lightest one.

16

• Any charged scalar mass should obey the constraint [93]:

mH± > 80 GeV. (A9)

• Measured couplings of fermions with the lightest neutral scalar should lie in the range:

κt = 1.02+0.19−0.15, κb = 0.91

+0.17−0.16, κτ = 0.93± 0.13, κµ = 0.72+0.50−0.72, (A10)

obtained from combined fits of data taken at√s = 13 TeV [94]. The coupling modifiers, κj , are defined

such that for a given production process or decay mode one has κ2j = σj/σSM or κ2j = Γj/ΓSM, respectively.

This parametrisation allows to conclude that in the SM scenario the coupling modifier is always positive

and equal to unity, κj = 1. We merge the bottom and tau lepton coupling modifiers to κ̃b = 0.92+0.15−0.14 as

the bottom quark, tau lepton, and charm quark couple to the same Higgs doublet.

• We only select those scenarios where the VEVs imply Yukawa couplings (in the mass basis) not larger orequal than one, yf ≤ 1. The notion of O(1) Yukawa couplings is, in fact, ambiguous. Here we use the rangeyf ∈ [0.1, 1], meaning:

vt = 174 GeV, vb ∈ [mb, 10mc], vµ ∈ [mµ, 10ms], vd ∈ [md, 10me]. (A11)

• The mass difference among the heavy scalars is taken to be less than 50 GeV to guarantee negligiblecontributions (. 10−4) to the ρ-parameter,

∆ρ = 0.0005± 0.0005 (±0.0009), (A12)

that is, it should be consistent with the maximum allowed deviation from the SM expectation [95]. The

first and second uncertainty corresponds to the choice of fixing the oblique parameter U to zero or not

within the multi-parameter fit. For further details please refer to Ref. [35].

• For the analysis, we consider the following ranges for the µb,µ,d parameters

µb ∈ (125, 1000] GeV, µµ ∈ (125, 1000] GeV, µd ∈ (125, 1000] GeV, (A13)

while for those ones in charge of inducing the cascade mechanism,

κtbχ ∈ [0.5, 500] GeV, κbµχ ∈ [0.5, 500] GeV, κµdχ ∈ [0.5, 500] GeV. (A14)

• We want to add here that as we are assuming singular alignment, there are no tree-level FCNCs. Further-more, as already discussed in Ref. [35], the ansatz of singular alignment corresponds to a linear realization

of the principle of minimal flavour violation [96], which also helps to understand that the appearance of

FCNCs at loop level will still be sufficiently suppressed respecting all experimental bounds [97].

We denote the scalar mass eigenstates by

CP-even neutral scalars: {h,H01 , H02 , H03},CP-odd neutral scalars: {A01, A02, A03},

Electrically charged scalars: {H±1 , H±2 , H±3 },

17

200 300 400 500 600 700

200

300

400

500

600

700

mH10 [GeV]

mH1± [Ge

V]

ξhb

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

300 400 500 600 700 800 900300

400

500

600

700

800

900

mH20 [GeV]mH2± [Ge

V]

ξhμ

-0.50

0.5

1.0

1.5

2.0

500 600 700 800 900 1000 1100

500

600

700

800

900

1000

1100

mH30 [GeV]

mH3± [Ge

V]

ξhd

-4-3-2-10

1

2

3

4

FIG. 4. Correlation plots of the scalar masses. In each figure we have contour plotted the value of the effective couplings,

in the allowed 3σ ranges, between the fermions and the SM-like Higgs. In all cases ξth = 1, mh = 125 GeV, and equal

plots can be found by replacing H0k → A0k. The chaotic behaviour in the couplings can be associated to the large amountof parameters.

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FIG. 5. Linear correlations among the neutral and heavy scalar masses.

where we consider h to describe the lightest neutral scalar with SM-like behaviour, mh = 125 GeV. On the other

hand, we parametrize the deviations from the SM-couplings between the fermions and the SM-like Higgs by ξfh ,

as implied from

− Ly ⊃∑f

mf(246 GeV)

ξfh f̄fh, (A15)

where for ξfh = 1 one recovers the SM case.

Figures 4 and 5 show the correlation among the heavy scalar masses fulfilling all our aforementioned criteria.

The linearity that can be seen among them mainly originates from the necessity to satisfy the contributions

to the ρ-parameter. In all cases, the lightest CP-even neutral scalar is mh = 125 GeV, the top quark couples

with ξth = 1, and the masses of the CP-even components of the gauge singlet scalars are at the PQ scale,

mχ,mA ∼ 1012 GeV.

Appendix B: Dirac neutrinos from PQ symmetry

In Sec. II, we chose XνR = 0 in order to allow for a simple type-I Majorana seesaw. This is indeed a minimalisticand elegant approach to explain the smallness of neutrino masses without the need to invoke unnaturally small

18

Yukawa parameters. However, here we are precisely exploring a mechanism that gives a Dirac mass to all fermions

with natural Yukawas. Moreover, in recent times the topic of Dirac neutrino masses is getting more and more

attention [98? –113], fueled by the as of yet non-observation of neutrinoless double beta decay. It has also

been shown that the PQ symmetry can protect the Dirac nature of neutrinos [114–116]. Therefore, let us briefly

expose how one could extend this mechanism to neutrinos too. We relax the assumption of trivially charged

right-handed neutrinos, i.e. XνR 6= 0 and add a new Higgs doublet φν whose small VEV will give small neutrinomasses. Then, Eqs. (9) and (10) get replaced by

− Lν = Yν,ij Li φ̃ν νR,j + h.c., (B1)

and

XνR,1 = XνR,2 = XνR,3 = Xφν + l, (B2)

where all the parameters are defined in a straightforward way following the same notation as Eqs. (13), (14),

and (15). The same mechanism that gives a hierarchy to the other four Higgs doublets will work here. In order

to have this ’cascade mechanism’ to get an appropriate value for the VEV of φν , we need to include the term

κdνχ(φ†νφd)χ, thus replacing Eq. (16) by

Xχ = k, Xφd = n, (B3)Xφν = n− k, Xφµ = n+ k, Xφb = n+ 2k, Xφt = n+ 3k, (B4)

and obtaining a VEV given by

vν 'κdνχvχvd

µ2ν + (λtν1 + λtν2)v2t + λνχv

2χ

, (B5)

which can satisfy vν ∼ 0.1 eV by taking, for example, vχ = 1 GeV, vd = 0.001 GeV, κdνχ = 10 GeV andµ2ν = (10

4 GeV)2 � v2t . Note that now we have one extra degree of freedom for the charges of the model, sincewe do not impose XνR,i = 0. However, this new freedom does not modify E/N = 8/3 since neutrinos are gaugesinglets.

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BrickRed Natural axion model from flavourAbstractI IntroductionII A 4 Higgs Doublet Model with Order One Yukawa CouplingsA General Structure of the ModelB Fermion SectorC Scalar Sector

III The Peccei–Quinn sectorIV Conclusions AcknowledgmentsA Benchmark scenarioB Dirac neutrinos from PQ symmetry References