HEP-BNU-2022-0002 Axion-like Dark Matter from the Type-II Seesaw Mechanism Wei Chao1Mingjie Jin1yHai-Jun Li1zand Ying-Quan Peng1x

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HEP-BNU-2022-0002

Axion-like Dark Matter from the Type-II Seesaw Mechanism

Wei Chao,1, ∗Mingjie Jin,1, †Hai-Jun Li,1, ‡and Ying-Quan Peng1, §

1Center for Advanced Quantum Studies, Department of Physics,

Beijing Normal University, Beijing 100875, China

Although axion-like particles (ALPs) are popular dark matter candidates, their mass generation

mechanisms as well as cosmic thermal evolutions are still unclear. In this letter, we propose a

new mass generation mechanism of ALP during the electroweak phase transition in the presence

of the type-II seesaw mechanism. As ALP gets mass uniquely at the electroweak scale, there is a

cutoﬀ scale on the ALP oscillation temperature irrelevant to the speciﬁc mass of ALP, which is a

distinctive feature of this scenario. The ALP couples to the active neutrinos, leaving the matter

eﬀect of neutrino oscillations in a dense ALP environment as a smoking gun. As a by-product, the

recent W-boson mass anomaly observed by the CDF collaboration is also quoted by the TeV-scale

type-II seesaw. We explain three kinds of new physics phenomena are with one stroke.

Introduction.— Various cosmological observations

have conﬁrmed the existence of cold dark matter (DM),

which accounts for about 26.8% [1] of the cosmic energy

budget. However, the particle nature of DM still elude

us. Axion [2–5] is one of the most popular DM candidates

motivated by addressing the strong CP problem, with its

mass induced by the QCD instanton and its relic abun-

dance arising from the misalignment mechanism [6–11],

which drives the coherent oscillation of axion ﬁeld around

the minimum of the eﬀective potential. Couplings of the

axion to the standard model (SM) particles are model-

dependent and there are three general types of QCD ax-

ion models, PQWW [2,3], KSVZ [12,13], and DFSZ

[14,15], of which the PQWW axion is excluded by the

beam-dump experiments [16–18] and other axion mod-

els can be detected via their couplings to photons or SM

fermions.

To relax property constraints to the QCD axions, more

general classes of axion-like particle (ALP) DM mod-

els [19–32] are proposed, with the mass ranging from

10−22 eV to O(1) GeV [9,31], where the lower bound

is from the fuzzy DM [33] and the upper bound is from

the LHC limits. The mass generation mechanism as well

as the relic abundance of axion-like DM are blurred and

indistinct since people usually pay more attention to the

detection signal of ALP in various experiments via its

coupling to photon [34–45], a/faFe

F, where ais the ALP

ﬁeld and fais the ALP decay constant. It should be men-

tioned that the mass generation mechanism of the ALP is

highly correlated with its interactions with the SM parti-

cles. So one cannot simply ignore these facts and directly

apply the strategy of searching for QCD axion to detect

the ALP. This issue has been concerned recently and sev-

eral novel approaches have been proposed to address the

relic abundance of the light scalar DM, such as the ther-

mal misalignment mechanism [46,47], which supposes a

feeble coupling between the DM and thermal fermions.

These attempts provide novel insights to the origin of

ALP in the early Universe.

In this letter, we propose a new mechanism of gener-

ating the ALP mass during the electroweak phase tran-

sition with the help of a Higgs triplet ∆ with Y= 1,

which is the seesaw particle in the type-II seesaw mecha-

nism [48–53]. Active neutrinos get Majorana mass as ∆

develops a tiny but non-zero vacuum expectation value

(VEV). We explicitly show that an ALP, which is the

Goldstone boson arising from the spontaneous break-

ing of them global U(1)Lsymmetry, can get tiny mass

through the quartic coupling with the Higgs triplet and

the SM Higgs doublet Φ whenever the global lepton num-

ber is explicitly broken by the term µΦTiτ2∆†Φ + h.c.In

such a scenario, symmetries break sequently: the U(1)L

ﬁrst breaks at high energy scale resulting a massless ALP

serving as dark energy, then electroweak symmetry is

spontaneously broken leading the mass generation of the

ALP, which begins to oscillate as its mass is comparable

with the Hubble parameter. We derive the relic density

of ALP by investing its thermal evolution and solving its

equation of motion (EOM) analytically. To further in-

vestigate its signal, we explicitly derive the interactions

between ALP and SM particles, which arise from the

mixing of ALP with other CP-even particles. We argue

that neutrino oscillations in certain speciﬁc environment

may be a smoking gun. As a by-product, we show that

the recent W-boson mass anomaly observed by the CDF

collaboration [54–64] can be addressed in the same model

without conﬂicting with the LHC constraints.

Framework.— We assume a complex scalar singlet S

carries two units of lepton number charge and the U(1)L

is spontaneously broken at high temperature when Sgets

VEV. Besides, the type-II seesaw mechanism is required

for the origin of neutrino mass and Scouples to the Higgs

triplet ∆ and the SM Higgs doublet Φ via the quartic

interaction with a real coupling. The most general scalar

potential is

V(S, Φ,∆) =V(Φ,∆) −µ2

S(S†S) + λ6(S†S)2

+λ7(S†S)(Φ†Φ) + λ8(S†S)Tr(∆†∆)

+µΦTiτ2∆†Φ + λSΦTiτ2∆†Φ+h.c. ,

(1)

where V(Φ,∆) is the most general potential for the type-

II seesaw mechanism given in the Supplemental Mate-

arXiv:2210.13233v1 [hep-ph] 24 Oct 2022

rial. The quartic couplings λ7,8are relevant for the ther-

mal mass of S. It is obvious that Smay get non-zero

VEV in the early Universe by assuming the small quar-

tic couplings, which is consistent with experimental ob-

servations [65–68], leaving the CP-odd component of S

as ALP. ALP is massless at the early time until the tem-

perature drops down to the electroweak scale at which

both Φ and ∆ get non-zero VEVs. Then ALP acquires

a tiny mass double suppressed by the VEV of the Higgs

triplet and the tiny lepton-number-violating parameter

µ, which should be naturally small accorded to the nat-

uralness principle of t’Hooft [69].

To analytically derive the mass of ALP, the Φ, ∆, and

Scan be parametrized as

Φ = "φ+

vφ+φ+iχ

√2#,∆ = "∆+

√2∆++

∆0−∆+

√2#, S =vs+ ˜s+i˜a

√2,(2)

where ∆0= (v∆+δ+iη)/√2 being the neutral compo-

nent of the Higgs triplet, the vφ,v∆, and vsare the VEVs

of Φ, ∆, and S, respectively. After the electroweak sym-

metry breaking (EWSB), the remaining physical scalars

are as follows, two charged scalar pairs H±± and H±,

two CP-odd scalars Aand a, and three CP-even scalars

h,H, and s, whose masses may be obtained by unitary

transformations to their squared mass matrices. The de-

tailed procedures of diagonalization of all the scalar mass

matrices are given in the Supplemental Material. Then

the ALP mass in the CP-odd sector can be written as

a'√2µv2

φv∆(v2

φ+ 4v2

∆)

2v2

φ(v2

∆+v2

s)+8v2

∆v2

.(3)

In the limits v2

∆/v2

φ1 and v2

∆/v2

s1, one has

a'µv2

φv∆/(√2v2

s), which is double suppressed by the

parameters v∆and µin the type-II seesaw mechanism.

ALP DM.— As discussed above, the ALP gets a tiny

but non-zero mass via the type-II seesaw mechanism dur-

ing the electroweak phase transition at the critical tem-

perature TC'160 GeV [70]. Neglecting the radiative

corrections, the temperature-dependent ALP mass can

be written as

a(T) = 









µv2

φ(T)v∆(T)

√2f2

, T ≤TC

0, T > TC

(4)

where fa=vs,vφ(T) and v∆(T) are the temperature-

dependent VEVs of the SM Higgs and Higgs triplet, re-

spectively. The EOM of the homogeneous ALP ﬁeld a

(a≡θfa) in the FRW Universe can be written as [6–8]

θ+ 3H(T)˙

θ+m2

a(T)θ= 0 ,(5)

where the dot denotes the derivative with the respect to

time, and H(T)≡˙

R/R is the Hubble parameter in terms

of the scale factor R. In the radiation-dominated epoch,

ma>maC

ma<maC

Oscillation

Time

Energy Scale

ma=0

ma=maC(1.079×10-4eV)

T=TC

3H(t)

Late time

Early time

FIG. 1. The evolution of the energy scales for ALP mass ma

(blue line) and the Hubble parameter (red line) as a function

of the time. Three cases of maare shown for comparisons.

The green intersections represent the temperatures when the

oscillation begins. The vertical dashed line represents the

critical temperature (TC).

we have H(T)=1/(2t)=1.66pg∗(T)T2/mpl , where g∗

is the eﬀective number of the degrees of freedom, and

mpl = 1.221 ×1019 GeV being the Planck mass. The ini-

tial conditions are taken as θ(ti) = qhθ2

a,iiand ˙

θ(ti) = 0,

where the angle brackets denote the initial misalignment

angle θ(ti) averaged over [−π, π) [10]. The value of hθ2

a,ii

depends on whether the U(1)Lbreaking occurs before

the inﬂation ends or after the inﬂation [10,30].

In general, the ALP becomes dynamical and starts to

oscillate when ma(Tosc)=3H(Tosc) [9–11], where Tosc is

the oscillation temperature. Before the EWSB, the ALP

is massless and the angle θremains a constant with the

initial value θ(t) = θ(ti). Therefore, there is an upper

bound on the oscillation temperature Tmax

osc ≡TC, which

leads to the existence of a critical mass

maC= 1.079 ×10−4eV .(6)

The oscillation temperature can be divided into two cases

Tosc =(T∗, ma< maC

TC, ma≥maC

(7)

where T∗is derived from the condition ma= 3H(T∗).

Eq. (7) implies that the traditional oscillation condition

is only available to the case ma< maC. For ma≥maC,

the oscillation temperature is always equal to the critical

temperature TC, as shown in Fig. 1. Note that we use

the parameter 3Hinstead of the Hubble parameter Hto

better show the critical point given by Eq. (7).

We now investigate the evolution of the ALP, which

is frozen at the initial value by the Hubble friction at

early times (3H > ma) and behaves as dark energy. As

the temperature Tof the Universe drops to Tosc given

by Eq. (7), the ALP starts to oscillate with damped am-

plitude, and its energy density scales as R−3, which is

Analytical

Numerical

0.5 5 50 500

-2

-1

T[GeV]

ma=2×10-8eV

T=T*

ma=2×10-4eV

T=TC

FIG. 2. The analytical (solid red) and numerical (dashed

blue) evolution of θas a function of Tfor two benchmark

ALP masses ma< maC(ma= 2 ×10−8eV) and ma> maC

(ma= 2 ×10−4eV). The vertical dashed lines correspond to

the oscillation temperatures.

similar with the ordinary matter [9,10], until the angle

θoscillates around the potential minimum of the ALP

at the late time. The evolution of θcan be described

by the analytical solution of EOM in the radiation domi-

nated Universe when H > HE∼10−28 eV [9,71], where

HEis the Hubble rate at the matter-radiation equality

in ΛCDM. The exact analytical expression is given in

Sec. B of the Supplemental Material. Alternatively, we

can also numerically solve Eq. (5) with the given initial

values. Here we consider the post-inﬂationary scenario

and take the initial value as θ(ti) = π/√3 [10,30]. The

analytical and numerical results are shown in Fig. 2with

the two benchmark ALP masses. We ﬁnd that the nu-

merical results of the evolution are consistent with the

analytical ones.

The energy density of ALP is ρa(t) =

θ2(t)f2

a/2 + m2

a(T)θ2(t)f2

a/2. Since the ratio of

ALP number density to the entropy density is

conserved, the ALP energy density at the present

can be written as ρa(T0)'ρa(Rosc) (Rosc/R)3=

1/2ma(Tosc)ma(T0)f2

aθ2

a,is(T0)/s(Tosc) [9–11],

where T0is the CMB temperature at present, and

s= 2π2g∗sT3/45 is the entropy density with g∗sthe

relativistic degrees of freedom of the entropy. The

ALP mass is almost temperature-independent, which

indicates ma(Tosc) = ma(T0) = ma, so the ALP energy

density at present is

ρa(T0)'1

2m2

af2

aθ2

a,ig∗s(T0)

g∗s(Tosc)T0

Tosc 3

.(8)

The relic density of ALP at present is deﬁned as Ωah2=

(ρa(T0)/ρc,0)h2[9,10], where ρc,0≡3m2

plH2

0/(8π) is the

critical energy density, T0= 2.4×10−4eV, and g∗s(T0) =

3.94 [72]. Combining these parameters with Eq. (8), the

10-910-810-710-610-510-410-310-20.1 1 10 102

10-2

0.1

ma[eV]

Ωah2

fa=2.5×1010 GeV

fa=1.0×1012 GeV

fa=3.0×1012 GeV

fa=1.0×1013 GeV

Case I: ma<maC

Ωah2=0.12

ma=maC

Case II: ma>maC

FIG. 3. The relic density Ωah2as a function of mafor

various fa. The vertical dotted line represents the criti-

cal mass (maC). The initial misalignment angle is taken as

θ(ti) = π/√3. The gray region is excluded by the overabun-

dance of DM.

relic density of ALP can be estimated as

Ωah2=











0.056 θ2

a,i3.94

g∗s(T∗)g∗(T∗)

3.36 3

×fa

1013 GeV 2ma

10−7eV 1

2, ma< maC

0.0146 θ2

a,ifa

1010 GeV 2ma

10−2eV 2

ma≥maC

(9)

Since the initial misalignment angle θ2

a,i1/2∼ O(1),

the relic density is almost determined by the decay con-

stant faand its mass ma. In Fig. 3, we show the relic

density Ωah2as a function of mawith the four bench-

mark values of fa∼ O(1010 −1013) GeV. The verti-

cal black dotted line represents the critical mass maC,

on two sides of which the ALP density evolve diﬀer-

ently. We ﬁnd that there exists the allowed parameter

space that may address the observed DM relic abun-

dance, Ωah2'0.12 [1,72].

ALP interactions.— Now we investigate interac-

tions of the ALP with ordinary matters including the

Higgs and active neutrinos. ALP may couple to the SM

Higgs as well as active neutrinos in forms λhaahaa and

νC

Liaλaνν νL+ h.c.with the couplings

λhaa :−λU11V13V23fa+1

2λU21V2

13fa,

λaνν :V23mν/v∆,

(10)

where Uij , Vij (i, j = 1,2,3) are the orthogonal matrices

diagonalizing scalar matrices given in the Supplemental

Material, and mνis the neutrino mass matrix in the ﬂa-

vor basis. The complete interactions of the ALP are

listed in Table IV of the Supplement Material. Given

vacuum

axion star

0.01 0.05 0.10 0.50 1

0.0

0.1

0.2

0.3

0.4

0.5

E[GeV]

P(νe⟶νμ)

FIG. 4. The transition probability P(νe→νµ) as a func-

tion of the neutrino energy E. The red and blue lines stand

for the neutrino oscillations in vacuum and dense axion stars,

respectively. We take ma= 10−5eV, fa= 1012 GeV, and

v∆= 1 MeV for axion, and take Mdense

a= 13.6Mand

Rdense

a= 45.2 km for the dense axion star [74]. The three-

ﬂavor oscillation parameters are taken from Ref. [75].

that the SM Higgs decays into two ALPs (h→aa), the

constraint of Higgs invisible decay from the LHC set an

upper bound on the coupling λhaa <1.536 GeV [73]. We

have checked that the coupling predicted by this model

always satisfy this constraint.

The interaction of ALP with active neutrinos may

cause matter eﬀect in neutrino oscillations. Since ALP is

the classical ﬁeld, the eﬀective potential can be directly

written as Veﬀ =i√2ρaV23m−1

av−1

∆cos(mat)νC

LmννL+

h.c., which contributes an eﬀective mass to active neutri-

nos and can be diagnonalized by the same unitary trans-

formation as that in vacuum. In this case, the three-

ﬂavor neutrino oscillation amplitude can be written as

Aα→β=X

Uβi b

U∗

αi exp −im2

2E1 + ρaV2

av2

∆

+ρaV2

23 cos 2max

2xm3

av2

∆,

(11)

where b

Uαi is the matrix element of the PMNS matrix

[76,77], α, β ={e, µ, τ},i={1,2,3}, and miis the

mass of the i-th neutrino mass eigenstate. Notice that

Eq. (11) is same as the formula of neutrino oscillation in

the vacuum up to the factor in the bracket.

We ﬁnd that it is diﬃcult to probe this matter eﬀect

with a ﬁxed v∆in vacuum, because of the low DM en-

ergy density ρaand the super-small suppression factor

V23. The matter eﬀect induced by this ALP-neutrino

interaction becomes important only if the active neutri-

nos propagate in a dense celestial body, such as an axion

star performed in [74,78]. As an illustration, we show

in Fig. 4the neutrino oscillation probability P(νe→νµ)

as a function of the neutrino energy in an axion star by

setting ρdense

a= 6.97×1019 g m−3[74], which corresponds

200 500 1000 2000

80.30

80.35

80.40

80.45

80.50

80.55

80.60

mL[GeV]

mW[GeV]

CDF

η=2002GeV2

η=1502GeV2

η=1002GeV2

FIG. 5. The mWas a function of the lightest triplet-like Higgs

mL. Here we set µ= 10−5GeV, v∆= 1 MeV, vs= 1013 GeV,

and ms= 1000 GeV. Three typical values of ηare selected

for comparisons. The dashed and solid lines correspond to

the cases of η < 0 and η > 0, respectively. The dashed green

line and the gray region represent the SM prediction [80] and

the recent 2σbound set by CDF [54], respectively.

to the axion star of mass Mdense

a= 13.6Mand radius

Rdense

a= 45.2 km. In Fig. 4, the matter eﬀect induced by

a dense axion star makes the neutrino oscillation spec-

trum diﬀerent from that in the vacuum.

Wmass anomaly.— Now we calculate the deviation

of W-boson mass from the SM prediction at one-loop

level within the framework of this model. In general, the

expression of the W-boson mass mWcan be parameter-

ized as [79,80]

W=m2

2"1 + s1−4παem

√2GFm2

(1 + ∆r)#,(12)

where GFis the Fermi constant, αem is the ﬁne-structure

constant, and ∆r= ∆αem −c2

W/s2

W∆ρloop + ∆rrem. The

explicit expression of ∆ris given in the Supplemental

Material.

Both the Higgs triplet [81–92] and the scalar singlet

[68,93] may contribute to ∆rand thus to the W-boson

mass. Given that vsis much larger than vφand v∆, it is

reasonable to expect that the mixing angles α2and α3

in CP-even sector are approximately zero (α2, α3'0)

as the scalar singlet is nearly decoupled from the other

scalar ﬁelds. The remaining α1can be written as

α1'arctan "vφv∆(λ4+λ5)−2M2

∆v∆/vφ

mh2−M2

∆#.(13)

Here we take the coupling λ4= 0 [82,94] for simplic-

ity. The M2

∆and λ5correlated with the splitting of

triplet mass spectrum are obtained from the limits of

∆/v2

φ, v2

∆/v2

s1 as

λ5'4m2

A−4m2

H+

φ'4m2

H+−4m2

H++

,(14)

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HEP-BNU-2022-0002Axion-likeDarkMatterfromtheType-IISeesawMechanismWeiChao,1,MingjieJin,1,yHai-JunLi,1,zandYing-QuanPeng1,x1CenterforAdvancedQuantumStudies,DepartmentofPhysics,BeijingNormalUniversity,Beijing100875,ChinaAlthoughaxion-likeparticles(ALPs)arepopulardarkmattercandidates,theirmassgenerati...

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