IPMU22-0051 From axion quality and naturalness problems to a high-quality ZNQCD relaxion

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IPMU22-0051

From axion quality and naturalness problems to a

high-quality ZNQCD relaxion

Abhishek Banerjee ID ,1, ∗Joshua Eby ID ,2, †and Gilad Perez ID 1, ‡

1Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 761001, Israel

2Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

(Dated: June 13, 2023)

We highlight general issues associated with quality and naturalness problems in theories of light

QCD-axions, axion-like particles, and relaxions. We show the presence of Planck-suppressed opera-

tors generically lead to scalar coupling of axions with the Standard model. We present a new class

of ZNQCD relaxion models that can address both the QCD relaxion CP problem as well as its

quality problem. This new class of models also leads to interesting experimental signatures, which

can be searched for at the precision frontier.

I. INTRODUCTION

The Standard Model of particle physics (SM) is an ex-

tremely successful yet incomplete description of nature.

It cannot account for the observed neutrino masses and

mixings, the matter anti-matter asymmetry, and the ori-

gin of Dark Matter (DM). Even within the framework

of the SM, we have the Higgs-hierarchy and the Strong

CP problems. On top of that, the eﬀect of gravity is ex-

pected to be signiﬁcant at the Planck scale despite the

lack of knowledge about its quantum nature. In partic-

ular, quantum gravity is expected to violate global sym-

metries in the UV, implying the existence of symmetry-

breaking operators suppressed by powers of the Planck

mass MPl = 2.4×1018 GeV in the framework of eﬀective

ﬁeld theory (EFT). For an axion ﬁeld Φ with a global

Peccei-Quinn (PQ) symmetry [1], one for instance ex-

pects, among others, operators of the form

L ⊃ 1

2cNΦN+ h.c.

Pl O(I.1)

where Nis an integer, cNis a dimensionless EFT parame-

ter, and Ois any dimension-four operator consistent with

the unbroken gauge symmetries. Expanding Φ = f ei ϕ/f ,

this Lagrangian generates a shift-symmetric potential of

the form

V∆=|cN|∆Ncos N ϕ

f+βO,(I.2)

where ∆ ≡f/MPl and β= arg(cN) is an arbitrary phase

which is generically O(1). Note that, if CP is not broken

by gravity then β= 0. The dimension Nof the PQ-

breaking operator in Eqs. (I.1-I.2) is dictated by the

unbroken gauge symmetries of the underlying theory.

The leading contribution to PQ-breaking arises from a

constant operator multiplied by M4

Pl to match the di-

mension. This deﬁnition ﬁxes N > 4 so that these

∗abhishek.banerjee@weizmann.ac.il

†joshaeby@gmail.com

‡gilad.perez@weizmann.ac.il

operators are suppressed in the limit MPl → ∞ (see

e.g. [2, 3]). This implies a contribution V∆=

|cN|∆NM4

Pl cos (Nϕ/f +β) to the scalar ﬁeld theory. If

this ﬁeld is identiﬁed with the QCD axion [4–9], then the

coeﬃcient |cN|∆Nin Eq. (I.2) cannot be too large or else

it will spoil the solution to the strong CP problem; this is

the so-called axion quality problem [10–12], and it can be

solved by either (a) ﬁne-tuning, e.g. taking |cN| ≪ 1, (b)

taking fvery small (which is constrained by measure-

ments of axion couplings to matter), or (c) forbidding

operators of dimension Nup to some large value, for ex-

ample by imposing some unbroken gauge symmetry (e.g.

ZN).

We describe the constraints on these operators in

greater detail in the subsections below.

II. AXION PHENOMENOLOGY

A. Axion-like particles and naturalness

A general axion-like particle (ALP) which is not cou-

pled to QCD does not exhibit a quality problem related to

the vacuum structure (see next section), and therefore it

might seem that the presence of Planck-suppressed oper-

ators would be harmless. However, these same operators

can induce large contributions to the ALP mass, leading

to a ﬁne-tuning problem.

Planck-suppressed operators can also generate ALP

couplings to the SM scalar operators, which we discuss in

Section II E. In the absence of any CP violation, ALPs in-

teract with the SM scalar operators quadratically at the

leading order, whereas if gravity does not respect CP, i.e.

for β̸= 0 in Eq. (I.2), these interactions are generated at

linear order.

An ALP is deﬁned by its mass mand coupling with

the SM pseudoscalar operators. These couplings are as-

sociated with an energy scale, which we will identify with

f. To analyse the eﬀect of Planck-suppressed operators,

we consider an ALP potential of

VALP(ϕ) = −m2f2cos ϕ

f,(II.1)

arXiv:2210.05690v3 [hep-ph] 11 Jun 2023

which deﬁnes the ALP mass m. However, the second

derivative of the potential induced by Planck-suppressed

operators in Eq. (I.2) is

V′′

∆(ϕ) = |cN|∆N−2M2

Pl N2cos Nϕ

f+β.(II.2)

Therefore, at leading order in ϕ/f ≪1, we have a bare

contribution to the mass m2and a correction of order

δm2≃ |cN|cos β∆N−2N2M2

Pl .(II.3)

Such corrections satisfy δm2≪m2only if



|cN|cos β∆N−2N2M2

m2

=

|cN|cos β∆NN2M4

m2f2≪1.

(II.4)

Assuming cN∼cos β∼1, one can translate the

inequality (II.4) into an upper bound on fas a func-

tion of Nin order to have negligible ﬁne-tuning of the

ALP mass. We illustrate these limits for ALP masses

m= 1,10−7,10−14 eV using the red, blue, and green

dotted lines (respectively) in Figure 1.

B. QCD axion quality and naturalness

QCD axions [4–9] exhibit a quality problem when the

contribution of Planck-suppressed operators contribute

signiﬁcantly to a shift in the low-energy vacuum of the

potential [10–12]. At low energy, QCD axions have a

potential of the form [13, 14]

Va(ϕ) = −Λ3

QCD(mu+md)

×s1−2z

(1 + z)21−cos ϕ

f+¯

θ,(II.5)

where z=mu/md≃0.485 [15, 16] is the ratio of up and

down quark masses, ΛQCD =⟨¯qq⟩1/3is the QCD scale

deﬁned by the quark condensate, and ¯

θis the eﬀective

CP violating angle. At leading order in z≪1 (and

ignoring an irrelevant constant), we have

Va(ϕ)≃ −Λ4

acos ϕ

f+¯

θ,(II.6)

where for simplicity we deﬁne Λa= (Λ3

QCDmu)1/4≃

84 MeV.

In the presence of the leading Planck-suppressed op-

erator, one can ﬁnd the minimum of the QCD-axion po-

tential as

0 = V′(⟨ϕ⟩)

=|cN|∆NN M4

Pl sin (Nϵ +β′)+Λ4

asin ϵ

≈ |cN|∆NN M4

Pl sin β′+ Λ4

aϵ, (II.7)

where ϵ≡ ⟨ϕ⟩/f −¯

θand β′≡β−N¯

θwhich is generically

O(1). Non-observation of the neutron electric dipole mo-

ment (EDM) implies that |ϵ|≲10−10 (see e.g. [17, 18]),

so in the last step we have expanded in small ϵ, Nϵ ≪1.

In order to not spoil the QCD axion solution to the strong

CP problem, one must require

|ϵ|=

|cN|sin β′∆NNM4

Λ4

a

≲10−10.(II.8)

At leading order in Nthis gives (for cN≃sin β′≃1)

N≳log 10−10(Λa/MPl)4

log (∆) =201

19 −log (f/1010 GeV) ,

(II.9)

see also [19]. So for PQ quality to be preserved, one

needs to forbid operators with N≲10 (13) for f= 1010

(1012) GeV. A simple way to do this is with a gauged ZN

symmetry (see Section II C).

The inequality of (II.8) is illustrated by the black solid

line in Figure 1. Comparing the QCD case to an ALP

where m2f2≃Λ4

a, we observe a natural suppression of

10−10Nin the ALP naturalness condition in Eq. (II.4),

relative to Eq. (II.8). Further, ALPs can populate a

wider space of values for mand f, allowing for more free-

dom in parameter inputs. Still, it is intriguing that the

requirement of natural ALP mass given in (II.4) is nearly

as restrictive as the quality problem for QCD axions.

As we point out above, generically scalar ﬁelds acquire

large mass corrections from Planck-suppressed operators.

Therefore in principle there is another constraint on the

quality of the QCD axion, arising from ﬁne-tuning of the

axion mass, though this is always weaker than the con-

straint above (this was also pointed out in [19]). Finally,

note that in principle one could satisfy Eq. (II.8) even

at small Nby tuning the EFT coeﬃcient |cN| ≪ 1 or

the phase parameter |β′|=|β−N¯

θ| ≪ 1. However, this

quickly leads to a ﬁne-tuning as bad as (or worse than)

the original strong CP problem.

C. High-quality, natural ZNQCD Axion

It was shown in [20] that an extended sector with N

copies of the SM, related by a ZNsymmetry, can lead

to a QCD-like axion of mass much smaller than that of

canonical QCD axion, due to additional suppression by

∼zNin the eﬀective QCD scale. This idea was fur-

ther investigated in [3] and shown to simultaneously ad-

mit a viable ultralight axion DM candidate [21]. If this

ZNsymmetry is gauged, it can protect the theory from

Planck-suppressed operators in Eq. (I.1).

Let us consider Ncopies of the SM which are related

to each other by a ZNsymmetry which is non-linearly

realized by the axion ﬁeld ϕ, as

ZN: SMk→SMk+1(mod N)(II.10)

ϕ→ϕ+2πk

Nf , (II.11)

0 10 20 30 40

104

108

1012

1016

FIG. 1: The value of fmax for a given operator dimen-

sion Nwhich satisﬁes the quality problem constraint

for ordinary QCD (Eq. (II.8), black solid) or ZNQCD

(Eq. (II.15), black dashed), compared to fmax to satisfy

δm/m ≪1 in ALP case for m= 1,10−7,10−14 eV (Eq.

(II.4), red, blue, and green dotted curves, respectively).

with k= 0,··· , N −1. The most general ZNsymmetric

Lagrangian1can be written as

N−1

k=0 LSMk+αs

8πϕ

f+¯

θ+2πk

NGk˜

Gk.(II.12)

The axion will receive contributions from all the Nsec-

tors; the combined potential can be written as

Vtot(ϕ) =

N−1

k=0

Vϕ

f+¯

θ+2πk

N,(II.13)

where, the axion potential in each sector is

V(x) = −Λ3

QCDmup1 + z2+ 2zcos x ,

as shown in Eq. (II.5).

At low energies, this theory diﬀers from the generic

QCD case because the eﬀective QCD scale is shifted.

This is apparent in the eﬀective potential of the theory

[3] (see Eq. (2.30)):

VN(ϕ)≃ −r1−z2

π N (−z)N−1Λ3

QCDmucos N ϕ

f+¯

θ.

(II.14)

The requirement V′(ϕ) = V′

∆(ϕ) + V′

N(ϕ) = 0 implies

|ϵ|=

|cN|sin β′∆NM4

Λ4

κ

≲10−10,(II.15)

where κ≡zN−1p(1 −z2)/(π N).

1Note that there could also be portal couplings between sectors,

though we postpone discussion of this to Section III D.

The ZNaxion case of Refs. [3, 20, 21] is illustrated by

the black dashed line in Figure 1. The symmetry provides

a mechanism for suppressing operators up to some large

Nrelative to the vanilla QCD case; however, at any given

N, the inequality (II.15) has a natural enhancement of

order 1/κ ≫1 relative to the minimal QCD axion (c.f.

Eq. (II.8)).

D. Challenges associated with the QCD relaxion

idea

The relaxion framework, proposed in [22] provides a

new insight on the hierarchy problem, which does not

require TeV-scale new physics, but rather implies a non-

trivial cosmological evolution of the Higgs mass. The

original relaxion model was based on the QCD axion

model [22]2.

However, as the back-reaction and the rolling potential

are sequestered, the relaxion stopping point corresponds

to sizeable phase, and generically cannot be set to zero.

It was noticed in the original paper [22] as well. Further-

more, as was shown in [27], and further derived below

for the QCD-relaxion model, the peculiar nature of the

relaxion dynamics implies that the relaxion stops at a

highly non-generic point in the ﬁeld space. At this point,

the mass is parametrically suppressed, and the phase is

predicted to be very close to π/2, a mechanism dubbed

the relaxed relaxion. In [28] a solution was proposed to

this problem; however, it required non-classical evolution

of the relaxion and thus, led to further problems associ-

ated with the measure problem [29].

In addition to that, a successful relaxation of the Higgs

mass requires large hierarchy between the scales of the

rolling potential and the back-reaction potential [30] and

thus, the relaxion setup rely on a carefully designed po-

tential derived from the clockwork mechanism [31–34],

which is based on a U(1) global symmetry. The resulting

construction suﬀers from a fairly severe quality problem,

unless the relaxion is rather heavy [35]. In Section III,

we propose a new construction that addresses both of the

above challenges.

E. Axion/ALP couplings from unknown Planck

physics

As mentioned previously, the Planck-suppressed PQ-

breaking operators in Eq. (I.2) give rise to SM couplings.

This is, as we discuss below, due to the fact that the ad-

ditional terms may be misaligned in phase relative to the

terms induced by the IR QCD instantons. In the pres-

ence of CP violation, the resulting couplings can be linear

2See [23] this for a possible generalisation of the back-reaction po-

tential, and [24–26] for non-inﬂationary relaxation mechanism.

in the ﬁeld, whereas if CP is conserved the leading cou-

plings are quadratic. In addition to that, the QCD axion

always induces a scalar interaction with the nucleons at

the quadratic order of the axion ﬁeld [36].

For low-energy phenomenology, we consider

ALP/axion interaction with the electrons, photons,

or gluons; the Lagrangian of such interactions can be

written as

L ⊃ ϕ

MPl "d(1)

meme¯ee +d(1)

4F2+d(1)

gβ(g)

2gG2#

+ϕ2

2M2

Pl "d(2)

meme¯ee +d(2)

4F2+d(2)

gβ(g)

2gG2#.(II.16)

where, eis the electron ﬁeld, F2=Fµν Fµν ,G2=

2Tr(Gµν Gµν ), Fµν (Gµν ) is the electromagnetic (QCD)

ﬁeld strength. Also, gis the QCD gauge coupling and

β(g) is the beta function. Such couplings can be searched

for via the equivalence principle violations and/or ﬁfth

forces experiments [37–43], or oscillation of fundamental

constants (for a review, see for example [44]; for propos-

als, see [27, 45–54]; and for experiments providing bounds

on oscillations see [55–66]). Note that, one can also con-

sider ALP/axion interaction with mq¯qq, where q=u, d

denotes the light quarks; see e.g. [65] for bounds on such

couplings.

To see how the above interactions are generated from

Eq. (I.2), one can expand the cosine part up to quadratic

order to ﬁnd

cos N ϕ

f+β= cos β−sin βNϕ

f−cos β

2Nϕ

f2

+··· .

(II.17)

Comparing Eqs. (I.2) and (II.17), we can easily identify

d(1)

X=|cN|Nsin β∆N−1, d(2)

X=|cN|N2cos β∆N−2,

(II.18)

for X=me, α, g, which we will refer to as the quality

couplings of the theory (due to their possible connection

with the quality problem). As discussed before, if gravity

respects CP, then β= 0 and thus, there is no linear scalar

coupling between ALP and SM. However, the quadratic

interactions are present both for the CP-violating and

CP-conserving cases.

Experimental searches for equivalence principle viola-

tions and ﬁfth forces [38–42] have led to stringent con-

straints on light scalars with couplings dXas above. In

particular, for the linear gluon coupling d(1)

g≲10−3

(10−6) for all particle masses m≲10−6(10−14) eV

(see [42, 65] and refs. therein), for the linear electron

coupling d(1)

me≲1 (10−2) for m≲10−6(10−14) eV, and

for the linear photon coupling d(1)

α≲10−1(10−4) for

m≲10−6(10−14 eV) (see [43] and refs. therein). Con-

straints on the quadratic couplings are weaker, but as we

shall see below, still relevant.

One can also search for these couplings through direct

detection of oscillation of fundamental constants from the

oscillation of the bosonic DM ﬁeld [45, 67]. This variation

is characterized at leading order by

δX

X0≃d(j)

ϕj

Pl ≃d(j)

j√2ρDM

m MPl j

,(II.19)

where ρDM is the density of DM in the vicinity of the

experiment, and j= 1 (2) for linear (quadratic) coupling

to ϕ. The typical value for the local density is ρlocal = 0.4

GeV/cm3, though it can be larger if the ﬁeld becomes

bound to the Earth or Sun [68, 69]. Substituting Eq.

(II.18), we can write the above equations in a compact

form

δX(ϕ)

X0≃Nj∆N−j

j√2ρDM

m MPl j

≃











2×10−18 N∆N−110−13 eV

mrρDM

ρlocal

(for j= 1)

2×10−36 N2∆N−210−13 eV

m2ρDM

ρlocal

(for j= 2)

(II.20)

for X=me, α, g, where we have taken |cN| ≃ sin β≃

cos β≃1. For comparison, present experimental sensi-

tivity to δme/me,0is at the level of 10−16 for microwave

clocks, but somewhat higher for molecular clocks with

some prospect to improve to 10−21 in the coming years;

for the α-coupling, current optical clock searches can

achieve 10−18, and a nuclear clock could potentially reach

10−23 (see [44] and references therein). See [65] for a dis-

cussion about the precision probes related to the gluons

and quarks couplings.

For QCD axions, owing to the suppression required to

resolve the quality problem, direct searches for quality

couplings is challenging. The linear coupling (j= 1)

term in Eq. (II.20) gives

δX(ϕ)

X0QCD ∼10−98 10−13 eV

mrρDM

ρlocal

(II.21)

for f= 1010 GeV (N= 10), and even smaller for f=

1012 GeV (N= 13) and/or for quadratic couplings (j=

2).

The scale of these couplings is exceedingly small, even

for ALPs. For quadratic couplings (j= 2), there is a

simple expression for the coupling of Eq. (II.18) such

that it satisﬁes the condition δm ≪mof Eq. (II.4):

d(2)

X≪m2

= 10−56 m

eV 2,(II.22)

which is far out of reach of experimental searches for

the foreseeable future. For linear couplings (j= 1), the

condition is more complicated but can be written as

d(1)

X≪m2

tan β∆

N,(II.23)

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IPMU22-0051Fromaxionqualityandnaturalnessproblemstoahigh-qualityZNQCDrelaxionAbhishekBanerjeeID,1,∗JoshuaEbyID,2,†andGiladPerezID1,‡1DepartmentofParticlePhysicsandAstrophysics,WeizmannInstituteofScience,Rehovot761001,Israel2KavliIPMU(WPI),UTIAS,TheUniversityofTokyo,Kashiwa,Chiba277-8583,Japan(Dated:...

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