CTPU-PTC-21-33

Lepto-axiogenesis in minimal SUSY KSVZ model

Junichiro Kawamuraa,b1, Stuart Rabyc2,

aCenter for Theoretical Physics of the Universe, Institute for Basic Science, Daejeon,34126, Korea

bDepartment of Physics, Keio University, Yokohama 223-8522, JapancDepartment of Physics, Ohio State University, Columbus, Ohio 43210, USA

Abstract

We study the lepto-axiogenesis scenario in the minimal supersymmetric KSVZaxion model. Only one Peccei-Quinn (PQ) field and vector-like fields are introducedbesides the MSSM with the type-I see-saw mechanism. The PQ field is stabilizedby the radiative correction induced by the Yukawa couplings with the vector-likefields introduced in the KSVZ model. We develop a way to follow the dynamics ofthe PQ field, in particular we found a semi-analytical solution which describes therotational motion under the logarithmic potential with including the thermalizationeffect via the gluon scattering which preserves the PQ symmetry. Based on thesolution, we studied the baryon asymmetry, the effective number of neutrino, andthe dark matter density composed of the axion and the neutralino. We foundthat the baryon asymmetry is successfully explained when the mass of PQ field isO(106 GeV) (O(105 GeV)) with the power of the PQ breaking term being 10 (8).

1jkawa@ibs.re.kr2raby.1@osu.edu

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Contents

1 Introduction 1

2 PQ field dynamics 22.1 Scalar potential and initial condition . . . . . . . . . . . . . . . . . . . . . 22.2 PQ dynamics when H & mP . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 PQ dynamics at H mP . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Energy evolution and thermalization . . . . . . . . . . . . . . . . . . . . . 12

3 Cosmology 133.1 Thermal history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Baryon asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Dark matter density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Axion density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.2 LSP density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Summary 28

1 Introduction

The rotational motion of a complex scalar field is considered to be a possible source forthe baryon asymmetry of the universe, as originally considered in the Affleck-Dine (AD)baryogenesis scenario [1,2]. Flatness of a potential is key to generating a sufficient amountof asymmetry, which is naturally explained by a flat direction in the scalar potential of theMinimal Supersymmetric Standard Model (MSSM) in the original AD scenario. Recently,it was proposed in Ref. [3] that the asymmetry can also originate from the rotational mo-tion of the Peccei-Quinn (PQ) field introduced to solve the strong CP problem [4, 5]. Inthis so-called axiogenesis scenario, the PQ asymmetry is generated by the rotational mo-tion, and then it is readily converted to a baryon asymmetry through sphaleron processesand perturbative interactions. In particular, the conversion can be efficiently inducedthrough the Weinberg operator [6] for neutrino masses which violates lepton number, andthis scenario is known as lepto-axiogenesis [7].

In this paper, we study the lepto-axiogenesis scenario in the minimal supersymmetricKSVZ axion model. The model has only one PQ field and it has Yukawa couplings tovector-like fields, so that the QCD anomaly is induced to solve the strong CP problem.The PQ field is stabilized at its minimum by the potential induced radiatively through theYukawa couplings to the vector-like fields [8]. Here, we consider supersymmetry (SUSY)to ensure that the scalar potential is almost flat along the PQ field direction. In theminimal model, the PQ field is thermalized only by gluon scattering [9–11]. This is theminimal possibility for lepto-axiogenesis with the KSVZ mechanism [12, 13]. The modelhas already been studied in Ref. [7]. In that paper the authors evaluated the dynamics ofthe PQ field using analytical approximations based on conservation laws, whereas, in thispaper, we examine the scenario by following the dynamics explicitly using a numericalevaluation based on the equation of motions. In this way, we can calculate the cosmologicalobservables by directly solving the evolution equations of energy and number densities.Further, our way of calculation can be applied for any combinations of energy densities.We find that the radiation and PQ field energies are comparable for substantially longtimes in a wide parameter space. In such a case, the estimations based on simply radiationdomination (RD) or matter domination (MD) may not be applied. Assuming the valueof the reheating temperature after inflation, we can calculate the gravitino density, andhence the density of the lightest SUSY particle (LSP). We shall also discuss the darkmatter (DM) composed of the axion and the LSP.

The rest of this paper is organized as follows. A way to follow the PQ field dynamicswith rotational motion is shown in Section 2. In Section 3, the cosmological implicationsof the dynamics are discussed. The paper is summarized in Section 4.

1

2 PQ field dynamics

2.1 Scalar potential and initial condition

We shall study the dynamics of the complex PQ field, P , whose scalar potential is givenby

V = m2P

(log|P |2v2P

− 1

)|P |2 +m2

Pv2P + λ2 |P |2(n−1)

M2(n−3)p

+APMn−3

p

(P n + h.c.) + VH , (2.1)

where the Hubble induced potential is assumed to be

VH = −cHH2 |P |2 , (2.2)

with cH > 0. The constant term m2Pv

2P is introduced so that the potential energy in the

vacuum is vanishing. This potential is motivated by SUSY models with a superpotential,

WP = yPΨΨ + λP n

Mn−3p

, (2.3)

where Ψ, Ψ are vector-like superfields, such that the mixed anomaly of the PQ symmetryand SU(3)C is induced 1. In this work, we assume the Yukawa coupling constant y to beO (1), so that the radiative PQ breaking [8] is realized by the logarithmic SUSY breakingmass in the first term of Eq. (2.1). In addition, the self-coupling of the PQ field is inducedby the SUSY breaking effect parametrized by AP . Since this self-coupling violates thePQ symmetry and hence induces a mass term for the QCD axion, the power n should besufficiently large such that

∆θ ∼ 1064−10n ×(

AP1 PeV

)( vP108 GeV

)n< 10−10, (2.4)

to be consistent with the measurement of neutron EDM [15–17]. Thus, n ≥ 8 is requiredfor the axion quality if vP & O (108 GeV).

We shall consider a scenario where the PQ field starts to move when the reheatingprocess after inflation ends, i.e. P = 0 at T = Ti, where Ti is the reheating temperature.The initial location of the PQ field is assumed to be at the minimum of the potentialwithout the SUSY breaking effects,

P = Pi :=

(cHH

2iM

2n−6p

(n− 1)λ2

) 12n−4

eiθi , (2.5)

1If the QCD anomaly is vanishing, the axion is not the QCD axion, but axion-like particle. This opensup new possibilities [14], e.g. wider range of the decay constant, but this is beyond the scope of thispaper.

2

where Hi is the Hubble constant at the initial time. The initial angle θi may be random,since the potential does not depend on the angle unless the A-term is sizable. The initialvalue is estimated as 2

|Pi| ∼(π2g∗(Ti)cH90(n− 1)λ2

T 4i M

2n−8p

) 12n−4

∼ 6.5× 1016 GeV ×(

Ti1012 GeV

) 2n−2

. (2.6)

Here, we assume that the radiation energy dominates the universe. The initial valueincreases slightly as Ti increases and/or the power n is smaller. We assume this initialvalue even if Hi . mP and the point is not the minimum of the potential. This situationcan be achieved if the PQ fields starts to move before the reheating ends 3. With thelarge value of |Pi|, the thermal-log effects [18], Vth ∼ α2

sT4 log(|P |2 /T 2), will be negligible

throughout the dynamics of the PQ field since the matter energy scales as a−3, while thethermal potential scales as a−4, with a is a scale factor 4. Note that the vector-like fieldsare heavier than the temperature throughout the dynamics as discussed in Section 2.3,and hence the thermal mass correction is absent.

2.2 PQ dynamics when H & mP

The PQ field is kicked by the A-term in the early stage when the amplitude is large. Theequation of motion is given by

P + 3HP +∂V

∂P ∗= 0, (2.7)

∂V

∂P ∗=

(m2P log

|P |2v2P

− cHH2 + (n− 1)λ2 |P |2(n−2)

M2n−6p

)P + n

APMn−3

p

P ∗(n−1). (2.8)

It is convenient to introduce a dimensionless variable

u := loga

ai, (2.9)

where a is the scale factor of the universe and ai is its initial value at t = ti.We parametrize P as

P =:S√2eiθ =: |Pi|χeiθ, (2.10)

2For the numerical values in this section, we take g∗(Ti) = gMSSM∗ = 228.75, cH = λ = 1 and n = 10,

although we keep this in the analytical expressions. We assume g∗(T ) = gMSSM∗ throughout the paper

except discussions about ∆Neff and the DM in Section 3.3.3In this case, however, the velocity P at T = Ti is not zero. We neglect this effect for simplicity and

take P = 0 at T = Ti, since it depends on details of the inflation.4The dynamics will be affected if P−1∂Vth/∂P ∼ α2

sT4/ |P |2 is not negligible compared with m2

P −cHH

2.

3

where S, χ, θ are real functions of u. Here, S and θ are respectively the radial and angulardirections of the PQ field P . The evolution equations for χ and θ are given by

χ′′ +

(3 +

H ′

H

)χ′ −

(cH + θ′2

)χ (2.11)

+χ

H2

[m2P

(log|Pi|2 χ2

v2P

)+ cHH

2i χ

2n−4 +H2i aPχ

n−2 cosnθ

]= 0,

θ′′ +

(3 +

H ′

H

)θ′ + 2θ′

χ′

χ− H2

i

H2aPχ

n−2 sinnθ = 0, (2.12)

where

aP :=A

Hi

√cHn2

(n− 1)λ2. (2.13)

Here, χ′ etc. denotes the derivative with respect to u. The Hubble parameter H is givenby

3M2pH

2 = ρR + ρP , ρR =π2g∗30

T 4i e−4u, ρP =

∣∣∣P ∣∣∣2 + V. (2.14)

It is convenient to introduce the notation,

y0 = χ, y1 =H

Hi

e3uχ′, y2 = θ, y3 =H

Hi

e3uθ′, (2.15)

so that the evolution equations for ya (a = 0, 1, 2, 3),

y′0 =Hi

He−3uy1, y′2 =

Hi

He−3uy3, (2.16)

y′1 =Hi

Hy0

[e−3uy2

3 + e3u

cHH2

H2i

−(m2P

H2i

log|Pi|2 y2

0

v2P

+ cHy2n−40 + aPy

n−20 cosny2

)],

y′3 =Hi

H

(−2e−3uy1y3

y0

+ e3uaPyn−20 sinny2

),

are independent of H ′. With this parametrization,∣∣∣P ∣∣∣2 = H2i |Pi|2 e−6u

(y2

1 + y20y

23

), V = V (|Pi| y0e

iy2). (2.17)

We numerically solve Eq. (2.16) up to H2 = 10−4 × m2P , where the Hubble constant

becomes negligible compared to the mass, and P oscillates around its minimum very fastper unit Hubble time. We define the value of u at this time as u1, i.e. H(u1) := 0.01×mP .

A result of numerical evaluation is shown in Fig. 1. The blue curves are the resultsof numerical evaluation, and the yellow lines in the right panels show the approximate

4

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2ReP [GeV] ×1017

−0.5

0.0

0.5

1.0

1.5

2.0

ImP

[GeV

]

×1016

0 1 2 3 4 5u

1029

1031

1033

|P|2 [G

eV2]

Numerical

Approx.

0 1 2 3 4 5u

10−2

100

102

θ/mP

Numerical

Approx.

Figure 1: The dynamics of the PQ field when the parameters are given by θi = 20/π, Ti =1013 GeV, vP = 108 GeV and mP = 106 GeV. The left panel is the plot on (ReP, ImP )plane. The upper (lower) right panel shows |P |2 (θ/mP ). The blue lines are the numericalsolution, and the yellow lines are the approximate solution.

solution. Before the rotation around the minimum starts at u = uosc where H = mP , thePQ field locates around the minimum of the potential, 5

P ∼(cHH

2M2n−6p

(n− 1)λ2

) 12n−4

eiθ. (2.18)

The value |P |2 at u < uosc is given by this equation. Its value at u > uosc and θ areextrapolated from the approximate solution at u > u1 derived in the next section. Theapproximate solution roughly agrees with the numerical solutions 6.

We see that the PQ field is kicked around P ∼ (8+1.5i)×1016 GeV, and then starts torotate around the minimum. The PQ number density, nPQ := i(P ∗P −P ∗P ), is generatedby this kick. The evolution equation of nPQ is given by

nPQ + 3HnPQ = 2nAPMn−3

p

|P |n sinnθ. (2.19)

Solving this, the generated PQ number density is given by

nPQ = e−3u

∫ u

0

du′∆nPQ(u′), ∆nPQ(u) := 2nAPMn−3

p

e3u |P |nH

sinnθ. (2.20)

The scaling of the integrated function in Eq. (2.20), ∆nPQ are as follows. Before theoscillation starts, u < uosc, ∆nPQ ∝ e3uH2/(n−2), so the exponent is positive for n &

5Note, u1 > uosc.6Note, however, that the solution at u > u1 uses the result of the numerical evaluation at u < u1, so

this is not a fully analytical result.

5

8, whatever the dominant energy. While after the rotation starts, u > uosc, ∆nPQ ∝e3u |P |n /H ∝ e3(2−n)u/2/H has the negative exponent with assuming |P | ∝ e−3/2u 7. ThePQ charge yield, YPQ := nPQ/s, where s is the entropy density, generated around u ∼ uosc

is estimated as

Yosc := YPQ(uosc) ∼n2

6n− 20

(cH

(n− 1)λ2

) n2n−4

(90

π2g∗

) 14

AP

(Mn−6

p

m3n−10P

) 12n−4

sin (nθosc) ,

∼ 350×(

1 PeV

mP

)5/4(AP

1 PeV

)sin (nθosc) , (2.21)

where we used Eq. (2.18) to estimate |P | at u = uosc and n = 10 is chosen in the secondline. Here, radiation energy domination is assumed. If Hi < mP , the kick effect isdominated by the initial time. Hence, the PQ number density and yield are given byformally replacing mP → Hi, Tosc → Ti and θosc → θi. The estimation Eq. (2.21) tendsto overestimate its value, because the value of |P | is typically smaller than the estimationEq. (2.18), as we see from Fig. 1. Since YPQ depends on |P |n, even small deviations makeslarge differences due to the large power n. Thus, we need to numerically evaluate YPQ bysolving the evolution equation. In fact, we will see that YPQ ∼ O (10) is obtained fromthe numerical evaluation for sufficiently large Ti.

The numerical evaluation becomes ineffective for later times due to the extremelyrapid oscillation per Hubble time. Thus we invoke a semi-analytical solution to evaluatethe dynamics for u > u1.

2.3 PQ dynamics at H mP

At u > u1, H mP , the higher-dimensional terms are negligible. The evolution equationsof S and θ are given by

S − θ2S + 3HS +m2PS log

S2

2v2P

= −ΓS, (2.22)

θS + 2θS + 3HθS = 0. (2.23)

Here, we phenomenologically introduce the decay term on the right-hand side of Eq. (2.22),so that the radial direction loses its energy via thermalization. As discussed later, weshall consider that the PQ field is dominantly thermalized via gluon scattering and thethermalization rate is proportional to T 3/S2 [9–11]. We introduce the thermalization termonly to the radial direction since this thermalization process preserves the PQ symmetry.From Eq. (2.23), the PQ number density nPQ = i(P ∗P − P ∗P ) = θS2 is conserved upto the Hubble expansion, i.e. nPQ + 3HnPQ = 0, and a non-zero term on the right-sidewould violate the PQ symmetry.

7We will derive this in the next section.

6

Equations (2.22) and (2.23) are equivalent to

ψ′′ +

(3 +

H ′

H

)ψ′ +

m2P

H2ψ log |ψ|2 = − Γ

H

|ψ|′|ψ| ψ, (2.24)

where ψ := P/vP . ψ and ψ′ are related to χ, θ as

ψ1 =|Pi|vP

χ1eiθ1 , ψ′1 =

|Pi|vP

(χ′1 + iθ′χ1) eiθ1 , (2.25)

where F1 := F (u1) for F = ψ, χ, θ and their derivatives.We can derive the approximate solution for the evolution equation Eq. (2.24). We

consider the ansatz

ψ = eΩ++iB+ + eΩ−−iB− , (2.26)

where Ω± and B± are real positive functions, and then we introduce real functions f±,

B′± =:mP

Hf±. (2.27)

We assume that f± and Ω± do not grow as fast as H−1. Neglecting the terms not enhancedby mP/H,[(

log |ψ|2 − f 2+

) m2P

H2+ i(f ′+ + 3f+ + 2Ω′+f+

) mP

H

]eΩ++iB+ (2.28)

+

[(log |ψ|2 − f 2

−) m2

P

H2− i(f ′− + 3f− + 2Ω′−f−

) mP

H

]eΩ−−iB−

= −i mPΓ

2H2 |ψ|2(f+ + f−)

×e2Ω−

(1− e−2i(B++B−)

)eΩ++iB+ − e2Ω+

(1− e2i(B++B−)

)eΩ−−iB−

.

Taking the real and imaginary parts of the coefficients of eΩ±±iB± ,

log |ψ|2 − f 2± =

Γ

2mP

f+ + f−

|ψ|2e2Ω∓ sin 2φ, (2.29)

f ′± + 3f± + 2Ω′±f± = − Γ

2H

f+ + f−

|ψ|2e2Ω∓ (1− cos 2φ) , (2.30)

where φ := B+ + B−. We define Ω± =: Ω ± ∆/2, and a real function γ whose firstderivative is given by

Γ

H=:

2e2Ω+γ′

|ψ|2=

γ′e∆

cosh ∆ + cosφ. (2.31)

7

We factorize the oscillating part (and e∆) of the thermalization rate from the other parts.Since the oscillating motion is very fast, ∼ mP/H, we replace the φ dependent parts bytheir averaged values, e.g.⟨

log |ψ|2⟩φ

:=1

2π

∫ π

−πdφ log |ψ|2 = 2Ω + ∆, (2.32)

and similarly,⟨sin 2φ

(cosh ∆ + cosφ)2

⟩φ

= 0,

⟨1− cos 2φ

(cosh ∆ + cosφ)2

⟩φ

= 2 (coth ∆− 1) . (2.33)

Then Eqs. (2.30) and (2.29) are arranged to

∆′ = γ′, f ′ + 3f + 2Ω′f = −f (log(sinh ∆))′ , f 2 = 2Ω + ∆, (2.34)

where f := f+ = f−. The solutions for f and Ω are given by

f =

√w

2, Ω =

w

4− ∆

2, w :=W

(C2w

(1 + coth ∆

2

)2

e−6(u−u1)

), (2.35)

where W(z) is the Lambert function, which satisfies W(z)eW(z) = z. The approximatebehavior of the Lambert function is given by

W(z) ∼

z − z2 +O (z3) z < e−1

log z − log log z +O(

log log z

log z

)z & 3

. (2.36)

Since the Lambert function does not grow exponentially, this solution meets the assump-tion. The function ∆ is a solution for the differential equation,

∆′ =|ψ|2

2ew/2Γ

H, C∆ := ∆(u1). (2.37)

Note that the right-hand side is a function of ∆. We solve this equation numericallytogether with the evolution equations of the energy densities discussed in the next section.Qualitatively, ∆ ' C∆ for Γ H, while ∆→∞ for Γ & H.

Altogether, the approximate solution is given by

ψ = ew/4(eiB+ + e−∆−iB−

), (2.38)

with

B± = C± +

∫ u

u1

du′mP

H

√w

2. (2.39)

8

Here, C±, C∆ and Cw are arbitrary real constants obtained from the integrations; to bedetermined by the initial condition at u = u1. With this solution, the PQ charge is givenby 8

nPQ =√

2mPv2PCwe

−3(u−u1) +O(Γv2

P

), (2.40)

so the constant Cw is determined from the PQ charge,

Cw =nPQ(u1)√

2v2PmP

=H1√2mP

× i(ψ′∗1 ψ1 − ψ′1ψ∗1). (2.41)

The other constants C∆ and C± are determined from

C∆ = log

∣∣∣∣∣∣∣∣ψ′1 +

(3w1

2(1 + w1)+ i

mP

H1

√w1

2

)ψ1

ψ′1 +

(3w1

2(1 + w1)− imP

H1

√w1

2

)ψ1

∣∣∣∣∣∣∣∣ , (2.42)

and

C± = ±Arg

(∓i[ψ′1 +

(3w1

2(1 + w1)± imP

H1

√w1

2

)ψ1

]), (2.43)

where Γ/H is neglected at u = u1. Here we assume that the PQ number density is positiveas is necessary to produce the correct baryon asymmetry.

The values of YPQ(u1) and C∆ on (vP , Ti) plane are shown in Fig. 2. In this figure,n = 10, θi = 20/π and mP = 106 GeV. The value of Cw is determined from YPQ(u1)through Eq. (2.41). The white lines in the left (right) panel shows H/mP (ρiR/ρ

iP ). We

see that YPQ(u1) ∼ O (10) at Ti & 1012 GeV where Hi & mP . YPQ(u1) is larger forsmaller Ti due to the smaller entropy. The value of C∆ is O (0.1) at Ti & 1012 GeV andthe largest value is about 0.4. For lower Ti, C∆ is O (0.01), but is not smaller than 0.01.From the right panel, ρiR = ρiP at Ti ∼ 1011.3 GeV, hence the PQ field energy dominatesthe universe from the beginning below this line.

The form of the solution Eq. (2.26) can be understand as a linear combination of thepositively rotating mode eiB+ and the negatively rotating mode e−iB− . While Γ H and∆ . O (1), both positive and negative rotational motions exist and hence the motion iselliptic. The minimum radius per rotation is given by 9

Smin√2

= vP ew/4(1− e−∆

). (2.44)

8O(Γv2

P

)term is proportional to Γ |ψ|2 sinφ, so it is vanishing after averaging and negligible.

9If the masses of vector-like fields become lighter than the temperature, i.e. mVL = ySmin/√

2 < T ,the thermal effects from the vector-like particles should be taken into account [19–21]. However, we didnot find any point in the parameter space where this happens during the dynamics.

9

8.0 8.5 9.0log10 vP/GeV

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

log 1

0Ti/

GeV

log10 YPQ(u1)

Hi/mP = 100

Hi/mP = 10

Hi/mP = 1

Hi/mP = 0.1

8.0 8.5 9.0log10 vP/GeV

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

log 1

0Ti/

GeV

log10C∆

ρiR/ρiP = 102

ρiR/ρiP = 1

ρiR/ρiP = 10−2

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

−1.5

−1.3

−1.1

−0.9

−0.7

−0.5

−0.3

Figure 2: Values of YPQ(u1) (left) and C∆ (right), where n = 10, θi = π/20 and mP = 106

GeV. Hi/mP (ρiR/ρiP ) is shown by the white lines in the left (right) panel.

Note, the motion is circular for larger ∆, whereas it is more elliptic for smaller ∆. Inparticular, the PQ field is simply oscillating along one direction if ∆ 1. This canhappen if the kick effect via the A-term is not sufficiently large due to the too smallinitial amplitude. However, the produced baryon asymmetry may be too small, so we arenot interested in such a case. After the thermalization, Γ & H and ∆→∞, the negativerotation ceases and hence the motion becomes circular.

We shall calculate the averaged values of the various quantities based on the solution.The averaged values of |P |2 and |P |2 are given by

⟨|P |2

⟩= v2

P ew/2(1 + e−2∆

), ρP :=

⟨∣∣∣P ∣∣∣2⟩ = v2Pm

2Pwe

w/2 1 + e−2∆

2+O

(v2PH

2, v2PΓ2

).

(2.45)

Hereafter, we omit the negligible contributions of O (v2PH

2, v2PΓ2) appeared in the deriva-

tives. The potential energy is given by

ρV := 〈V 〉 = m2Pv

2P

[ew/2

(1 + e−2∆

) (w2− tanh ∆

)+ 1], (2.46)

where

〈cosφ log (cosh ∆ + cosφ)〉φ = e−∆ (2.47)

is used. Thus, the total energy of the PQ field is given by

ρP := ρP + ρV = m2Pv

2P

[ew/2

(1 + e−2∆

)(w − tanh ∆) + 1

]. (2.48)

10

The PQ field energy is vanishing by the red-shift, w → 0 and the thermalization ∆→∞,since ρP → e−2∆ when w → 0. The energies of the radial and rotational motions arerespectively given by

ρS :=

⟨1

2S2

⟩= v2

Pm2Pwe

w/2e−2∆, (2.49)

ρθ :=

⟨1

2θ2S2

⟩=

⟨n2

PQ

2S2

⟩= v2

Pm2Pwe

w/2 1− e−2∆

2, (2.50)

so we see that ρP = ρS + ρθ. Qualitatively, the kinetic energy of radial motion is lost bythe thermalization, while the rotational energy is slightly increased due to the shrinkingof the radius. In particular, most of the kinetic energy is lost by the thermalization whenC∆ 1 and the motion is highly elliptic. The ratio of the energy after to before thethermalization is given by

ρP |∆→∞ρP |∆=C∆

=

√w∆→∞

w∆=C∆

cothC∆, (2.51)

so only a fraction of the energy is lost if C∆ is not so small, while most of the energy islost if C∆ 1. Finally, the angular velocity is given by⟨

θ⟩

=

⟨nPQ

2v2P |ψ|2

⟩=nPQ

4v2P

e−w/2+∆

⟨1

cosh ∆ + cosφ

⟩φ

= mP

√w

2. (2.52)

where 〈(cosh ∆ + cosφ)−1〉φ = 1/ sinh ∆. This is consistent with the result obtained inRef. [22] derived assuming PQ conservation.

Altogether, when the PQ field is away from the minimum, i.e. z & 3,

S2 ∝ ew/2 ∝ a−3, ρP ∝ wew/2 ∝ a−3, θ ∝ √w ∝ a0, (2.53)

while, when the PQ field reaches its minimum value, i.e. z ∼ 0,

S2 ∝ ew/2 ∝ a0, ρP ∝ wew/2 ∝ a−6, θ ∝ √w ∝ a−3. (2.54)

Thus our solution describes both matter oscillation and kination. The PQ field energybecomes kination-like around

uK := u1 +1

3logCw, (2.55)

where z = 1. Here, ∆→∞ at this time is assumed so that the thermalization completessuccessfully. These averaged values are used in the calculations of various observablesnext section.

11

2.4 Energy evolution and thermalization

The rest of the work needed to numerically calculate the PQ field dynamics is to deter-mine ∆. In the minimal KSVZ model, the saxion is dominantly thermalized by gluonscattering [9–11] and the decay to axions [23] whose rates are respectively given by

Γ := Γg + Γa, Γg = bgT 3

S2, Γa =

m3P

32πS2, (2.56)

where bg = 1.0× 10−5 in our numerical analysis. The evolution equations for the energydensities are given by 10

ρP + 6H∣∣∣P ∣∣∣2 = − (Γg + Γa) S

2, (2.57)

ρR + 4HρR = + ΓgS2, (2.58)

ρa + 4Hρa = + ΓaS2, (2.59)

with

3M2pH

2 = ρP + ρR + ρa. (2.60)

Here, ρa is the radiation energy density of the axion produced from the decay. We usedEqs. (2.22) and (2.23) to derive Eq. (2.57). As discussed before, only the kinetic energyof the radial direction is converted to radiation energy. The PQ field energy ρP is givenby Eq. (2.48).

We numerically solve the evolution equations of the energy densities together withthat of ∆ given by Eq. (2.37). Defining XR and Xa as

ρR = ρ1RXRe

−4(u−u1), ρa = ρ1RXae

−4(u−u1), (2.61)

with ρ1R := π2g∗T

41 /30. Averaging the right-hand sides over the rotation, the equations

for ∆, XR and Xa are given by

∆′ =e−w/2

4v2PH

(bgX

3/4R T 3

1 e−3(u−u1) +

m3P

32π

), (2.62)

X ′R =bgm

2PT

31

2ρ1RH

X3/4R eu−u1

we−∆

sinh ∆, (2.63)

X ′a =m5P

64πρ1RH

e4(u−u1) we−∆

sinh ∆, (2.64)

and the initial conditions are

∆(u1) = C∆, XR(u1) = 1, Xa(u1) = 0. (2.65)

10In our analysis, the dissipation effect of the axion [21], whose rate is given by ∼ bgm2PT/S

2, isomitted, since it might be negligible [24].

12

Here we use ⟨sin2 φ

(cosh ∆ + cosφ)2

⟩φ

= coth ∆− 1. (2.66)

We numerically solve Eqs. (2.62), (2.63) and (2.64), so that the evolution of ∆ and theradiation energies are determined. For the numerical evaluation, we require that ∆(uK) >10 for the completion of the thermalization.

3 Cosmology

3.1 Thermal history

In this section, we discuss the thermal history in the scenario based on the analyticalsolutions. The numerical results obtained by solving the equations derived in the previoussections will be shown later. Since the initial amplitude is large, the PQ field may dominatethe universe at some time. When the MD starts, ρP = ρR, the temperature is given by11

TM ∼4

3

√wM2

cothC∆ ·mPYosc (3.1)

∼ 9× 107 GeV ×(√

wM cothC∆

10

)( mP

106 GeV

)(Yosc

10

),

There is no MD era if this temperature is lower than the temperature when the PQ fieldenergy becomes kination-like, i.e.

TM < Tie−uK =

(45v2

PmP√2π2g∗Yosc

) 13

(3.2)

∼ 2× 106 GeV ×( vP

108 GeV

) 23( mP

1 PeV

) 13

(10

Yosc

) 13

.

The MD starts at some time if this condition is not satisfied. Once the PQ field dominatesthe energy density of the universe, it should be thermalized predominantly by gluonscattering. If, instead, the PQ field is thermalized by the decay to axions, it contributesto the effective number of neutrinos, ∆Neff , which is severely constrained by currentobservation. Thus we assume that the thermalization is dominated by gluon scatteringin the following analytical estimation.

We can understand the thermalization process by solving Eqs. (2.62) and (2.63) withneglecting the decay to axions. Assuming the thermalization rate is negligible at u = uM ,

11We used

ρP (uM ) ∼ m2P v

2PwMe

wM/2(1 + e−2∆) =

√wM

2mPnPQ coth ∆

with assuming wM := w(uM ) 1.

13

∆′ 1, so ∆ ' C∆ is the constant at this time. The solution for the XR when ∆′ 1is given by

X1/4R = 1 + β

(e

52

(u−uM ) − 1), β :=

9√

10bgw14Mw

34Ne−C∆

2π3g3/2∗ sinhC∆

m2PMp

T 3M

. (3.3)

In the analytical analysis, we assume that the thermalization is ineffective when the MDstarts, and thus β 1. If this is not true, the discussion in this section is not applicable,and the numerical evaluation is necessary. Hence, the thermalization becomes effectiveand the non-adiabatic (NA) era starts at u = uN with

uN = uM −2

5log β, (3.4)

where βe5(uN−uM )/2 = 1. This is about a time when the NA era starts, because theevolution is adiabatic while XR = 1 and X

1/4R ∼ βe5/2(u−uM ) > 1 at u > uN . uN < uK

is necessary for successful thermalization, since the thermalization rate drops faster thanthe Hubble constant at u > uK . At u & uN , the scaling of the temperature becomes

T ∝ X1/4R e−u ∝ e3/2u.

We can estimate the scale when the thermalization completes, uth, by equating ρSwithout the thermalization effect and the radiation energy in the NA era,

ρS(uth)|∆=C∆= ρR(uth) ∼ ρR(uM)e−4(uN−uM )e6(uth−uN ). (3.5)

This equation means that the kinetic energy stored in the radial direction converts toradiation energy. In the second equality, we assume the scaling of the temperature T ∝e−u (e3u/2) before (after) u = uN . From this relation, we obtain the thermalizationtemperature as follows. The left-hand side of Eq. (3.5) is given by

ρS(uth)|∆=C∆=

√wth

2mPnPQ(uM)e−3(uth−uN )e−3(uN−uM ) e−C∆

2 sinhC∆

, (3.6)

with wth = w(uth). While the right-hand side of Eq. (3.5) is

ρR(uM)e−4(uN−uM )e6(uth−uN ) = ρP (uM)e−4(uN−uM )e6(uth−uN ) (3.7)

∼√wM2mPnPQ(uM) cothC∆e

−4(uN−uM )e6(uth−uN ).

Equating them, we get

e9(uth−uN ) =

√wth

wM

e−C∆

2 coshC∆

euN−uM . (3.8)

Then given the scaling laws of the temperature assumed in Eq. (3.6), this becomes(Tth

TN

)6

=

√wth

wM

e−C∆

2 coshC∆

TMTN

. (3.9)

14

Thus, we arrive at

T 6th =

√wth

wM

e−C∆

2 coshC∆

TMT5N . (3.10)

Finally, using Eq. (3.4), we obtain

Tth =

(√wth

wM

e−C∆

2 coshC∆

β2

) 16

TM (3.11)

∼ 4× 107 GeV ×(√

wthw3N

700

e−3C∆ tanhC∆

sinh2C∆

) 16 (

bg10−5

) 13(

228.75

g∗

) 12 ( mP

106 GeV

) 23.

The dilution factor is given by

D :=saft

sbef

=

(ρR(u)

ρ1Re−4(u−u1)

) 34

∣∣∣∣∣uuth

= XR(u uth)34 , (3.12)

with saft (sbef) the entropy after (before) dilution by the thermalization. In the numericalevaluation, we evaluate this directly at u = uK + 2. This is estimated as

D ∼ T 3th

T 3Me−3(uth−uM )

=T 5

th

T 5N

=

√wth

wM

e−C∆

2 coshC∆

TMTth

(3.13)

∼ 10×(w

5/2th e−3C∆ coshC∆

105 × w3/2N sinh4C∆

) 16 ( g∗

228.75

) 12

(10−5

bg

) 13(Yosc

50

)( mP

106 GeV

) 13,

where the scalings of T used to derive Eq. (3.6) are assumed again. Thus the dilutionfactor is about O (10) for Yosc ∼ O (10), while it can be larger for lower Ti where Yosc islarger, see Fig. 2. After the successful thermalization, the PQ field reaches its minimumvalue when the temperature is

TK := T (uK) ∼(

45v2PmP√

2π2g∗YPQ

) 13

, (3.14)

where YPQ is the PQ number yield after the dilution. The value of TK is given by in theright-hand side of Eq. (3.2) by replacing Yosc → YPQ. The kination domination (KD) eraexists if ρP (uK) > ρR(uK). This condition is approximately given by

1 >1215v2

P

32π2g∗Ω3Y 4PQm

2P

∼ 3×(

5

YPQ

)4(228.75

g∗

)( vP108 GeV

)2(

106 GeV

mP

)2

, (3.15)

where Ω := eW(1)/2(W(1)− 1) + 1 ∼ 0.43. The KD era is absent for typical values of theparameters, although it could happen for a certain parameter set. No KD means thatradiation becomes the dominant form of energy after thermalization.

15

8 10 12 14 16

24

26

28

30

32

34

36

38

40

log 1

0ρ/G

eV4

ρP

ρR

ρS

ρa

u=uL

u=uM

u=uR

2

u=u

th

u=uK

8 10 12 14 16u

1

2

〈T5/H

S2〉[

GeV

2]

×1025

6 8 10 12 14

24

26

28

30

32

34

36

38

40

log 1

0ρ/G

eV4

ρP

ρR

ρS

ρa

u=uL

u=u

th

u=uK

6 8 10 12 14u

0

2

4〈T

5/H

S2〉[

GeV

2]

×1023

Figure 3: The evolution of the energy densities of the PQ field (blue solid), radiation(red), radial direction (blue dashed) and axion (green dotted). n = 10, θi = π/20,Ti = 1013 GeV, vP = 108 GeV and mP = 106 GeV in the left panel, and n = 8, θi = π/16,Ti = 1012 GeV, vP = 108 GeV and mP = 105 GeV in the right panel. The lower panelsshow 〈T 5/HS2〉 which is integrated to calculate the baryon asymmetry.

The axion contribution to the effective number of neutrinos is given by [25]

∆Neff =43

7

(10.75

g∗(Tth)

) 13 ρaρR

∣∣∣∣uuth

. (3.16)

This is estimated as

∆Neff '43

7

(10.75

g∗(Tth)

) 13 Γa

Γg

∣∣∣∣T=Tth

=43

7

(10.75

g∗(Tth)

) 13 m3

P

32πbgT 3th

(3.17)

= 0.28×(

228.75

g∗(Tth)

) 13(

10−5

bg

)(mP/Tth

0.05

)3

.

Thus, ∆Neff will be suppressed if Tth is an order of magnitude larger than mP . Thecurrent limit is ∆Neff < 0.3 [26].

Figure 3 shows the evolution of energy densities at the benchmark points 12. Theevolution equation of ∆ and the energy densities are given by Eqs. (2.57), (2.58) and (2.59).

12Note, uN is not shown in the figure since it is not calculated in the numerical analysis. uN isintroduced for the analytical estimation.

16

8.0 8.5 9.0

11

12

13

14

log 1

0Ti/

GeV

log10 Tth/ GeV

noth

erm

aliz

atio

n

8.0 8.5 9.0

11

12

13

14log10 TM/Tth

noth

erm

aliz

atio

n8.0 8.5 9.0

log10 vP/GeV

11

12

13

14

log 1

0Ti/

GeV

log10D

noth

erm

aliz

atio

n

8.0 8.5 9.0log10 vP/GeV

11

12

13

14log10 ρ

KP /ρ

KR

noth

erm

aliz

atio

n

6.9

7.0

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

1.4

1.8

2.2

2.6

3.0

3.4

3.8

0

1

2

3

4

5

−2.1

−1.8

−1.5

−1.2

−0.9

−0.6

−0.3

0.0

0.3

Figure 4: The values of Tth (top-left), TM/Tth (top-right), D (bottom-left) and ρKP /ρKR

(bottom-right) in the case of n = 10, θi = π/20, and mP = 106 GeV.

In the figure, uL is the scale when the lepton number violating interaction becomes outof equilibrium as discussed in the next section and uR2 is the scale when the second RDstarts after the MD starts. On the left panel, n = 10, θi = π/20, Ti = 1013 GeV, vP = 108

GeV and mP = 106 GeV. The MD starts at u ∼ 9.3, then the thermalization starts atu ∼ 11 and it ends at u ∼ 13.5. The dilution factor is 6.00 at this point. We see thatthe kinetic energy of the radial direction, ρS, is converted to the radiation energy throughthe thermalization process, so only a fraction of the PQ field energy is lost. The PQ fieldenergy becomes kination-like at u ∼ 15.8, but the dominant energy at this time is theradiation energy, since the radiation energy starts to dominate during the thermalizationprocess. On the right panel, n = 8, θi = π/16, Ti = 1012 GeV, vP = 108 GeV andmP = 105 GeV. In this case, the radiation energy dominates the universe throughout thehistory because of the smaller amplitude of the PQ field due to smaller n.

17

Figure 4 shows the values of Tth (top-left), TM/Tth (top-right), D (bottom-left) andρKP /ρ

KR (bottom-right) calculated by the numerical calculation when n = 10, θi = π/20,

and mP = 106 GeV. From the top-left panel, the thermalization temperature, where ∆ =10, is about 7× 107 GeV as expected from our estimation Eq. (3.11). It should be notedthat this is the temperature when the thermalization completes, and the thermalizationitself occurs slightly before this time. We can see this from Fig. 3, the radiation energyis non-adiabatic at 11.5 . u . 13 and is before u = uth ∼ 13.5. Hence, e.g. ∆Neff isdominantly produced around u . 13 and the typical temperature is effectively higher thanTth. The thermalization does not complete in the dark gray region at vP & 109.2 GeV,i.e. ∆(uK) < 10 in this region. The PQ field reaches its minimum value too early, beforethermalization completes, if vP is too large. In the top-right panel, the ratio of TM to Tth

is shown. The ratio is at most about 100 for Ti & 1012 GeV, and, in this case, the MDera does not last such a long time. While the MD era can be longer for Ti . 1012 GeVand Hi < mP . The dilution factor D is shown in the bottom-left panel. D ∼ O (10)reflecting the short MD era at T & 1012 GeV, while it can be larger for lower Ti. Theratio of the PQ field to radiation energy at u = uK is shown in the bottom-right panel.The PQ field energy is smaller than the radiation energy throughout the figure, except forthe small region where vP . 108 GeV and Ti ∼ 1012.3 GeV. Although it is not the leadingenergy, the kination energy and the radiation energy are comparable at Ti & 1012 GeV.This is because the ratio TM/Tth is small and the PQ field and radiation energies are stillcomparable when thermalization occurs.

In our numerical analysis, we focus on the two parameter sets,

• n = 10, θi = π/20, mP = 106 GeV

• n = 8, θi = π/16, mP = 105 GeV

with varying vP and Ti. We name the former (latter) scenario as n = 10 (n = 8) scenario.In both cases, we take cH = λ = 1 and AP = mP .

3.2 Baryon asymmetry

The PQ number density produced from the rotational motion is converted to the baryonasymmetry via sphaleron processes and MSSM interactions [3,7]. In the lepto-axiogenesisscenario, the generated lepton chiral asymmetry and Higgs number asymmetries are con-verted into the B−L asymmetry via the Weinberg operator (LHu)

2, which can be obtainedby integrating out Majorana neutrinos in the type-I see-saw mechanism [27–30]. The B−Lasymmetry is finally converted to the baryon asymmetry via the weak sphaleron process.The Boltzmann equation for the B−L asymmetry reads

nB−L + 3HnB−L = cB−LθT2ΓL. (3.18)

Here, ΓL is a rate of conversion from the Higgs asymmetry and/or lepton asymmetry toB−L asymmetry via the Weinberg operator and is given by

ΓL '1

4π3

m2ν

v4H

T 3. (3.19)

18

m2ν :=

∑i=1,2,3m

2νi

is the sum of neutrino masses squared. This should be in the range

0.0025 eV2 . m2ν . 0.03 eV2 to be consistent with the neutrino oscillation data [31–33]

and the cosmological bound∑mνi < 0.26 eV [26]. The constant cB−L depends on

the scattering rates of MSSM particles. The baryon asymmetry per entropy density,YB := nB/s, is given by

YB = YPQ ×cBm

2ν

4π3v4H

∫ u

uL

du′T 5

HS2(3.20)

where cB := 10/31×cB−L in the MSSM [34]. Here, uL is when the lepton number violatingprocess goes out of equilibrium. The value of cB is O (0.01− 0.1) depending on the detailsof the couplings [7]. Hence, we consider 3×10−5 eV2 < cBm

2ν < 3×10−3 eV2 as a suitable

range. In our analysis, we assume that the Yukawa interactions, yνN1LHu , are in thermalequilibrium when T > MN1 , with MN1 the mass of the lightest Majorana neutrino, as inthe strong washout regime of thermal leptogenesis [35,36]. This is particularly motivatedby the Yukawa texture, Yν ∼ Yu, in Grand Unified Theories [37–39], where Yu is theYukawa coupling matrix for up quarks. We shall take MN1 = 109 GeV for concreteness, souL = log(Ti/109 GeV) 13. The baryon asymmetry is evaluated by numerically integratingthe averaged function, 〈T 5/HS2〉, from uL to uK+2. The evolution of 〈T 5/HS2〉 is shownin the lower panels of Fig. 3.

Let us estimate the baryon asymmetry analytically. If there is no MD, the integrationcan be done with neglecting the variation of ∆ in w 14

∫ u

uL

du′⟨T 5

HS2

⟩∼(

90

π2g∗

) 32 mPMp

36YPQ

(√wL2

(3 + wL)−√w(u)

2(3 + w(u))

), (3.21)

where wL := w(uL). Since w(u) wL, at sufficiently late times, the baryon asymmetryis given by

Y RDB ∼ cBm

2νmPMp

144π3v4H

(90

π2g∗

) 32√wL2

(3 + wL) (3.22)

∼ 1× 10−10 ×(

cBm2ν

0.003 eV2

)( mP

105 GeV

),

where wL = 20 is assumed in the second equality. The observed value, Y obsB ∼ 8.7 ×

10−11 [26], can be explained for mP & O (105 GeV).If the MD exists, the integration during the period of RD is given by Eq. (3.21) with

formally replacing u → uM . The second term may not be negligible in this case. This

13With this assumption, the strong sphaleron is always in equilibrium at u > uL.14We use the integration formula∫

dx√W (ae−bx) = −2

√w

3b(3 + w) , w =W

(ae−bx

).

with a, b constants.

19

8.00 8.25 8.50 8.75 9.00log10 vP/GeV

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

log 1

0Ti/

GeV

cBm2ν × 10−3 [eV2]

ξa > 1

ξa =0.25

ξa =0.50

∆Neff = 0.02

∆Neff = 0.01

noth

erm

aliz

atio

n

0.0

0.6

1.2

1.8

2.4

3.0

8.00 8.25 8.50 8.75 9.00log10 vP/GeV

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

log 1

0Ti/

GeV

cBm2ν × 10−3 [eV2]

ξa > 1

ξa = 0.25

ξa = 0.50

cBm2ν > 3× 10−3 eV2

noth

erm

aliz

atio

n

0.0

0.6

1.2

1.8

2.4

3.0

Figure 5: The values of cBm2ν to explain the baryon asymmetry in the n = 10 (n = 8)

scenario on the left (right) panel. The thermalization does not complete in the dark grayregion. ∆Neff = 0.02 (0.01) on the black dashed (dotted) line. The fraction of the axiondensity, ξa := Ωa/ΩDM, is 0.75, 0.50, 0.25 on the brown solid, dot-dashed, dashed lines,respectively, when NDW = 1. The axion is overproduced in the brown region. On theright panel, there is no MD above the white line and cBm

2ν > 0.003 eV2 in the purple

region.

contribution is diluted by the entropy production from the thermalization. NeglectingO (1) variations of w and tanh ∆, 〈T 5/HS2〉 scales as e12u and e−u/2 during the adiabaticand the NA era, respectively 15. Hence, the baryon asymmetry is predominantly producedwhen the thermalization ends. We can see this from the lower-left panel of Fig. 3. Thusthe baryon asymmetry produced after an epoch of MD is estimated as

Y MDB ∼ YPQ

cBm2ν

4π3v4H

T 5th

HthS2th

∼ cBm2νMp

32π3v4H

(90

π2g∗

) 32 (wth

2

) 14

√3mPTth

YPQ

(3.23)

∼ 1× 10−10 ×(

cBm2ν

0.003 eV2

)(wth

5

) 14( mP

106 GeV

) 12

(Tth

107 GeV

) 12(

10

YPQ

) 12

.

Thus, in this case, mP & O (106 GeV) may be necessary to explain the baryon asymmetry.Figure 5 shows values of cBm

2ν to explain the baryon asymmetry based on the numerical

calculation. The meanings of the regions and lines are explained in the caption. The resultof the n = 10 (n = 8) scenario is shown in the left (right) panel. In the n = 10 scenario, theMD era exists at some time, and hence the baryon asymmetry is produced predominantlyat the end of thermalization. The baryon asymmetry is explained at vP . 109.2 GeV

15The scaling-low becomes e15/2u if the RD starts during the thermalization. The asymmetry is pro-duced at the end of thermalization also in this case, and hence the estimation here is not changed.

20

with cBm2ν ∼ (3−12)× 10−4 eV2. On the black dashed (dotted) line, ∆Neff = 0.02 (0.01)

and hence inside the dashed line is accessible in future observations [40]. The brown linesare the fraction of the axion density in the case of NDW = 1. The calculation of axiondensity is shown in the next section. For NDW > 1, the density is divided by NDW. Inthe n = 8 scenario, there is no MD era above the white line since the amplitude of thePQ field is smaller due to the smaller value of n. We note that vP & O (109 GeV) maycause the axion quality problem from Eq. (2.4), but this region is already incompatiblewith successful thermalization. The baryon asymmetry can be explained when cBm

2ν ∼

(0.9−2.4) × 10−3 eV2, but it is unacceptably large for cBm2ν > 0.003 eV2 in the purple

region, roughly below the white line where a MD era exists. The smaller mP = 105 GeVis enough to explain the baryon asymmetry due to the absence of the MD era.

3.3 Dark matter density

In this model, the axion φ and the LSP χ contribute to the DM density, i.e.

ΩDM = Ωa + Ωχ =: (ξa + ξχ)ΩDM. (3.24)

We assume that the observed DM density, ΩobsDMh

2 = 0.12 [26], is explained by thesetwo particles. Since the reheating temperature Ti needs to be large in order to explainthe baryon asymmetry, the LSP will be predominantly produced by the decay of grav-itinos. We shall discuss the conditions necessary to explain the DM density and theirconsequences for DM searches; especially for indirect detection.

3.3.1 Axion density

The scalar potential for the axion φ := NDWfaθ, with the axion decay constant, fa =√2vP/NDW, is given by

Va = f 2ama(T )2

(1− cos

φ

fa

), (3.25)

and the axion mass is approximately given by

ma(T ) = 6 meV

(109 GeV

fa

)×

1 T ≤ ΛQCD(ΛQCD

T

)pT ≥ ΛQCD

, (3.26)

where ΛQCD = 150 MeV is the QCD scale. Here, we take p = 4 motivated by the diluteinstanton gas approximation [41]. We also define the axion mass at zero temperature,ma0 := ma(T = 0). The axion is produced by the kinetic misalignment mechanism(KMM) [42] if the kinetic energy of the axion, N2

DWf2a θ

2/2, is higher than the barrier ofthe axion potential, 2f 2

ama(T )2, when the oscillation of the axion starts, 3H(T∗) = ma(T∗),

21

8.00 8.25 8.50 8.75 9.00log10 vP/GeV

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

log 1

0Ti/

GeV

KD

YPQ

D = 10

D = 1000

YP

Q/Y

c=

8

noth

erm

aliz

atio

n

0.0

1.5

3.0

4.5

6.0

7.5

9.0

10.5

8.00 8.25 8.50 8.75 9.00log10 vP/GeV

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

log 1

0Ti/

GeV

YPQ

cBm2ν > 3× 10−3 eV2

no MD

noth

erm

aliz

atio

n

0.0

1.5

3.0

4.5

6.0

7.5

9.0

10.5

Figure 6: YPQ after dilution in the n = 10 (n = 8) scenario on the left (right) panel. Thesolid (dashed) white line is D = 1000 (10) and the yellow line is YPQ/Yc = 8 on the leftpanel. The KD era exists in the white line on the left panel, and the MD era exists belowthe white line on the right panel.

where T∗ is the temperature at this time. This condition is satisfied if

YPQ =2θv2

P

s> Yc := NDW ×

9

2

(10

π2g∗(T∗)

) 512

(f 12a

ma0Λ4QCDM

7p

) 16

(3.27)

= 0.07×NDW ×(

80

g∗(T∗)

) 512(

fa109 GeV

) 136

.

If this condition is satisfied, the axion can overcome the potential and thus the timing tostart the oscillation around a minimum is delayed. Otherwise, the axion will be producedby the conventional misalignment mechanism [43–45]. Thus, the axion density is givenby

Ωah2 =

s0h2

ρc(3.28)

×

N−1DWCama0YPQ ∼ 0.1×N−1

DW ×(

108 GeV

fa

)(YPQ

3

), YPQ > Yc,

9

4

(10

π2g∗

) 512

(m5a0f 12a

Λ4QCDM

7p

) 16

θ2∗ ∼ 0.11× θ2

∗

(80

g∗

) 512(

fa1011.8 GeV

) 76

, YPQ < Yc,

where ρc/s0h2 = 3.6×10−9 GeV. Here, θ∗ is the value of θ when the oscillation starts. The

constant Ca ' 2 is determined from the numerical calculation for the delay of oscillationby the kinetic energy [42]. The axion density is proportional to the PQ yield, YPQ, in theKMM.

22

8.00 8.25 8.50 8.75 9.00log10 vP/GeV

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

log 1

0Ti/

GeV

log10 Y3/2

noth

erm

aliz

atio

n

−13.7

−12.5

−11.3

−10.1

−8.9

−7.7

8.00 8.25 8.50 8.75 9.00log10 vP/GeV

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

log 1

0Ti/

GeV

log10 Y3/2

cBm2ν > 3× 10−3 eV2

noth

erm

aliz

atio

n

−13.7

−12.5

−11.3

−10.1

−8.9

−7.7

Figure 7: Y3/2 after dilution in the n = 10 (n = 8) scenario on the left (right) panel.

The value of YPQ in our model is shown in Fig. 6. In both cases, YPQ is O (1), henceYPQ > Yc is satisfied and the axion density is close to the DM density when NDWfa ∼108 GeV. The axion density, in the case of NDW = 1, is shown in Fig. 5. There areregions of parameter space where Ωa > ΩDM and the maximum value is about ξa ∼ 2 forNDW = 1. However Ωa < ΩDM for NDW > 1 in most of the parameter space.

3.3.2 LSP density

The neutralino will be the LSP in our scenario and will be predominantly produced fromthe late-time decay of the gravitino. Since we consider a sizable A-term, to producea sizable PQ asymmetry, we expect that SUSY breaking is gravity mediated. Hence,the gravitino mass may be at O (mP ) or heavier. In this case, the gravitino is abun-dantly thermally produced in the early universe due to the high reheating temperature& 1010 GeV [46]. The Boltzmann equation for the gravitino number density n3/2 is givenby

n3/2 + 3Hn3/2 = C3/2, (3.29)

where the collision term is given by [47,48],

C3/2(T ) =3ζ(3)T 6

16π3M2p

∑a=1,2,3

cag2a(T )

(1 +

M2a (T )

3m23/2

)log

kaga(T )

. (3.30)

Here, a = 1, 2, 3 is for the U(1)Y , SU(2)L and SU(3)C of the SM. The constants are ca =(11, 27, 72) and ka = (1.266, 1.312, 1.271). ga and Ma are the gauge coupling constantsand gaugino masses. In our numerical analysis, we directly solve this equation assumingMa m3/2. If the radiation energy dominates the universe when T = Ti, the gravitino

23

is produced at T = Ti and hence the gravitino yield is approximately given by [48],

Y3/2 :=n3/2

s' C3/2(Ti)×D−1

(90

π2g∗

) 32 Mp

4T 5i

(3.31)

∼ 2.0× 10−10 ×(

Ti/D

1012 GeV

).

If this is not the case, the gravitino is produced at the end of thermalization. This isparticularly important when the dilution factor is very large. Figure 7 shows the values ofY3/2, and the gravitino yield is O (10−14−10−8) depending on the reheating temperatureand the dilution factor.

The gravitino decay width to MSSM particles is given by [49],

Γ3/2 =193m3

3/2

384πM2p

, (3.32)

so the decay temperature of the gravitino reads

T3/2 '(

10

g∗

) 14

√193m3

3/2

384π2Mp

(3.33)

∼ 0.1 GeV ×(

10

g∗(T3/2)

) 14 ( m3/2

106 GeV

) 32.

This is typically lower than the temperate at the freeze-out of the neutralino LSP, Tf ∼mχ/20 16. In our analysis, we assume that the axino mass is heavier than O

(m3/2/16π2

),

and hence the axino decays before the freeze-out of the neutralino [52]. Thus the neu-tralino LSP produced from axinos does not affect the LSP density in the current uni-verse 17.

The neutralino is copiously produced from gravitino decay after its thermal freeze-out.Then it will annihilate if the rate is sufficiently large. The Boltzmann equation is givenby

nχ + 3Hnχ = −n2χ〈σv〉χ. (3.34)

The thermally averaged annihilation rates for the wino and higgsino are respectively givenby [57,58],

〈σv〉W '8πα2

2

M22

(1− xW )3/2

(2− xW )2, (3.35)

〈σv〉H '8πα2

2

µ2

((1− xW )3/2

16(2− xW )2+

(1− xZ)3/2

32c4W (2− xZ)2

), (3.36)

16For m3/2 & 106 GeV assumed in this paper, the gravitino decyas well before the big bang nucleosyn-thesis [46,50,51].

17See e.g. Refs [53–56] for more discussions about the axino.

24

where xV = m2V /m

2χ (V = W,Z) and cW = cos θW with the weak angle θW . Here M2

and µ are the wino and higgsino mass parameters, respectively. Solving Eq. (3.34), theinverse of the neutralino yield Yχ := nχ/s, is given by

Y −1χ ' Yχ(T3/2)−1 +

√8π2g∗(T3/2)

45MpT3/2〈σv〉χ, (3.37)

where Yχ(T3/2) is the neutralino yield at T = T3/2. Yχ(T3/2) ' Y3/2, assuming that thethermally produced neutralino abundance is negligible. However, this is not the casefor binos. The annihilation is effective when the gravitino yield is large, i.e. the decaytemperature is higher and the annihilation rate is larger.

When the annihilation is ineffective, the neutralino density from the gravitino decayis given by

Ω3/2χ h2 ∼ s0h

2

ρcmχY3/2 ∼ 0.11×

( mχ

2 GeV

)( Ti/D

1012 GeV

). (3.38)

The neutralino should be lighter than O (1 GeV) in order to avoid overproduction forTi/D & O (1012 GeV). This might be the case for the bino LSP, but the bino LSP will beoverproduced thermally in this case since neither the co-annihilation via nearly degeneratesleptons nor an enhanced annihilation rate via the mixing with the wino/higgsino [59]are available. Thus the annihilation must be effective, so that the LSP density is notoverproduced for Ti/D & O (1012 GeV). For smaller Ti/D, the LSP production from thegravitino decay become negligible and the LSP density is governed by the usual thermalfreeze-out.

If the annihilation is effective and the second term in Eq. (3.37) dominates, the DMdensity is given by

ΩDMh2 ∼ ξ−1

χ Ω3/2χ h2 ∼ s0h

2

ρc

√216

193

(10

g∗

) 14 mχξχ√

m33/2Mp〈σv〉eff

χ

(3.39)

∼ 0.12×(

10

g∗

) 14(

107 GeV

m3/2

) 32 ( mχ

170 GeV

)( ξχ0.1

)(10−26 cm3/s

〈σv〉effχ

),

where 〈σv〉effχ := ξ2

χ〈σv〉χ is the effective annihilation cross section constrained by theindirect detections for the DM. Thus the DM density can be explained if the gravitinomass isO (107 GeV) which is one or two orders of magnitude larger thanmP , and 〈σv〉eff

χ ∼10−26 cm3/s which can be realized by the O (100 GeV) wino and higgsino.

The indirect detection for DM can probe the annihilation process originating from DMrich environments, such as Dwarf Spheroidal Galaxies (dSphs), the Galactic Center and soon [60]. The most relevant limits come from Fermi-LAT [61] and AMS-02 [62] which searchfor gamma ray fluxes from dSphs and anti-proton fluxes, respectively. Figure 8 shows theeffective annihilation cross section with different values of ξχ in the wino (higgsino) LSPcases on the left (right) panel. The gravitino mass is chosen such that the DM density is

25

250 500 750 1000 1250 150010−27

10−26

10−25

10−24

10−23

ξ2 χ〈σv〉[

cm3/s

]

Fermi-LAT

AMS-02

wino DM

100 200 300 400 500 600 70010−27

10−26

10−25

10−24

10−23

Fermi-LAT

AMS-02

higgsino DM

ξχ = 0.10

ξχ = 0.20

ξχ = 0.30

ξχ = 0.40

ξχ = 0.50

ξχ = 0.60

ξχ = 0.70

ξχ = 0.80

ξχ = 0.90

ξχ = 1.00

250 500 750 1000 1250 1500M2 [GeV]

106

107

108

m3/

2[G

eV]

100 200 300 400 500 600 700µ [GeV]

106

107

108

Figure 8: Indirect detection constraints. The green lines show the predictions of the LSPsproduced from the gravitino decay. The gravitino mass is chosen such that ξχ := Ωχ/ΩDM

is explained. The red (blue) line is central values of AMS-02 (Fermi-LAT) constraint.The gravitino masses are shown in the lower panels. The gray line is T3/2 = Tf = mχ/20,so the neutralinos are produced by the usual freeze-out mechanism above this line.

explained for a given LSP mass, and its value is shown in the lower panels. We explorethe LSP masses up to T3/2 < Tf ∼ mχ/20, so that the LSP is produced from the gravitinodecay. This limit is shown by the gray line on the lower panels. For heavier massesT3/2 > Tf , the LSP are produced by the usual thermal freeze-out which has been studiedextensively in the literature [63–65]. The red (blue) line is the upper bound from theFermi-LAT (AMS-02) experiment on the cross section obtained in Ref. [66]. Althoughthe central limits from AMS-02 is much stronger, this could be weaker significantly dueto the propagation uncertainties. We find that the effective annihilation cross-section canbe as low as 2× 10−27 cm3/s for ξχ ∼ 0.1 and mχ ∼ 100 GeV. These light LSPs will beprobed by the future experiments such as the CTA experiment [67,68].

Table 1 shows values of various quantities at the benchmark points 18. At all thepoints, the baryon asymmetry is explained with reasonable values of cBm

2ν and ∆Neff is

smaller than the current limit. We consider the pure wino and higgsino LSP scenarios,and the values of the LSP and gravitino masses are shown in the last six columns. Thesemasses are chosen, so that the DM density is explained for the given axion densitiesand 〈σv〉eff ∼ O (10−26 cm3/s). At the point (C), the LSP is predominantly producedby thermal freeze-out since the gravitino yield is small. Here, we assume the typicalmasses [64, 65] for the thermal wino and higgsino as the referenced values. Note thatthe co-annihilation with charginos is important for the freeze-out. At the other points,the LSP is predominantly produced from gravitino decay, so m3/2 ∼ O (107 GeV) and

18A model with n = 10 and NDW = 4, assumed in the points (B) and (C), is realized in a Pati-Salamunification with non-anomalous ZR

4 × Z5 symmetry [69].

26

Table 1: Values of various quantities at the benchmark points.

point (A) point (B) point (C) point (D)

n 10 10 10 8

NDW 1 4 4 1

mP [GeV] 1.0000× 106 1.0000× 106 1.0000× 106 1.0000× 105

fa [GeV] 1.7804× 108 1.0000× 108 1.0000× 108 1.7804× 108

Ti [GeV] 1.0000× 1013 1.0000× 1013 1.0000× 1011 1.0000× 1012

cBm2ν [GeV2] 1.0581× 10−21 1.2929× 10−21 1.4518× 10−21 3.7825× 10−21

∆Neff 7.2175× 10−3 8.1775× 10−3 2.0087× 10−3 1.0646× 10−5

D 5.8146 5.6425 2.2186× 103 1.0697

YPQ 4.8015 4.8735 1.0080 5.7087

Y3/2 3.6757× 10−10 3.7880× 10−10 2.4831× 10−14 1.9484× 10−10

ξχ 0.2509 0.6616 0.9300 0.1093

M2 [GeV] 7.8946× 102 1.3748× 103 2.8000× 103 2.2986× 102

m3/2 [GeV] 3.9811× 107 6.3096× 107 − 6.3096× 106

〈σv〉eff [cm3/s] 8.3976× 10−27 1.9326× 10−26 9.2192× 10−27 1.7640× 10−26

µ [GeV] 2.3621× 102 4.7004× 102 1.0000× 103 1.2617× 102

m3/2 [GeV] 1.5849× 107 3.1623× 107 − 1.0000× 107

〈σv〉eff [cm3/s] 9.9940× 10−27 1.8602× 10−26 9.5176× 10−27 4.7915× 10−27

mχ ∼ O (100−1000 GeV).The wino and higgsino above 100 GeV are not excluded by the LEP experiment [70],

but these can be tested by the LHC and future colliders. If the LSP is purely wino orhiggsino, the searches for disappearing tracks are available [71–73] and the current limitsare 660 (210) GeV for the wino (higgsino) LSP [74]. Hence, the benchmark (D) in Table 1is excluded for both cases of wino and higgsino LSP. The limits are relaxed if the LSP isa mixture and the lifetime of the chargino is shorter. This can be more easily achievedin the higgsino LSP case, when the mass difference between the chargino and the LSPis O (1 GeV). The O (1 GeV) mass difference can be achieved if e.g. the wino mass isat sub-TeV [75]. For the mixed LSP case [59], however, the direct searches for DM givestronger constraints, and hence this case would be tested by near future observations.The detailed study about the constraints from the LHC and direct searches are beyondthe scope of this paper.

27

4 Summary

In this paper, we studied the lepto-axiogenesis scenario in the minimal SUSY KSVZ axionmodel with the type-I see-saw mechanism. We developed a way to follow the PQ fielddynamics from the beginning of the rotation to the approach to the minimum. Whilethe rotation is not too fast, we can directly follow the dynamics of the PQ field bysolving Eq. (2.16). The evaluation becomes, however, not efficient for later times due tothe extremely fast rotation, hence we trace the dynamics by averaging over the rotationbased on the ansatz Eq. (2.26), and the solution is given by Eq. (2.38). We find thethermalization is well described by ∆ which represents the ellipticity of the rotationalmotion. The evolution of ∆, together with the radiation energies, can be calculated bysolving Eqs. (2.62), (2.63) and (2.64). Based on the solution obtained by directly solvingthe equations of motion, we can evaluate the averaged values of the amplitude of the PQfield, energy densities, angular velocity and so on. The solutions and evolution equationssolved numerically do not rely on the form of the dominant energy density of the universe.Thus our solution is applicable for the case when the radiation and the PQ field energy arecomparable, such as the case shown in the left panel of Fig. 3. Although we focus on theminimal KSVZ model for illustration, a similar analysis could be applied for models withmore PQ fields and/or different thermalization mechanisms which appear in e.g. DFSZmodel [76, 77].

We studied the baryon asymmetry, ∆Neff and the DM density based on the PQ fielddynamics. When n = 10, the matter domination era always appears and the baryonasymmetry is predominantly produced at the end of the thermalization. The soft mass forthe PQ field mP , should be O (106 GeV) in this case to explain the correct amount of thebaryon asymmetry. Although there is matter domination, the produced PQ asymmetry orgravitino yield are not diluted significantly since the dilution factor is up to O (10) whenTi & 1012 GeV and Hi > mP . Thus the gravitino mass tends to be produced abundantlyand the annihilation of the LSP should be effective in order to avoid the overproductionof the LSP. The DM density is explained if the gravitino mass is O (107 GeV) and thewino (higgsino) mass is O (103 GeV) (O (100 GeV)). The light neutralino DM will betested by the future indirect detection experiments, such as CTA. We also find that theLSP is predominantly produced from thermal freeze-out for lower Ti and the sufficientlylarge initial amplitudes. Since there is a parameter space where ∆Neff ∼ O (0.01), thefuture experiment would be able to probe this scenario. In addition, kination energy canbe a dominant or sizable component of the total energy, and hence this could be seen ingravitational wave spectrum [22,78].

When n = 8, the matter domination epoch is absent for a sufficiently large initialtemperature, and hence mP ∼ O (105 GeV) can explain the baryon asymmetry. Thefavored mass range for the gravitino and neutralino are similar to the case of n = 10.

28

Acknowledgment

The work of J.K. is supported in part by the Institute for Basic Science (IBS-R018-D1),and the Grant-in-Aid for Scientific Research from the Ministry of Education, Science,Sports and Culture (MEXT), Japan No. 18K13534. The work of S.R.is supported in partby the Department of Energy (DOE) under Award No. DE-SC0011726.

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