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arXiv:1110.2491v2 [hep-ph] 28 Dec 2011

Preprint typeset in JHEP style - HYPER VERSION

OU-HEP-111006

Coupled Boltzmann calculation of

mixed axion/neutralino cold dark matter

production in the early universe

Howard Baera, Andre Lessaband Warintorn Sreethawonga

aDept. of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA

bInstituto de F´ ısica, Universidade de S˜ ao Paulo, S˜ ao Paulo - SP, Brazil

E-mail: baer@nhn.ou.edu, lessa@fma.if.usp.br, wstan@nhn.ou.edu

Abstract: We calculate the relic abundance of mixed axion/neutralino cold dark matter

which arises in R-parity conserving supersymmetric (SUSY) models wherein the strong

CP problem is solved by the Peccei-Quinn (PQ) mechanism with a concommitant ax-

ion/saxion/axino supermultiplet. By numerically solving the coupled Boltzmann equa-

tions, we include the combined effects of 1. thermal axino production with cascade decays

to a neutralino LSP, 2. thermal saxion production and production via coherent oscillations

along with cascade decays and entropy injection, 3. thermal neutralino production and

re-annihilation after both axino and saxion decays, 4. gravitino production and decay and

5. axion production both thermally and via oscillations. For SUSY models with too high a

standard neutralino thermal abundance, we find the combined effect of SUSY PQ particles

is not enough to lower the neutralino abundance down to its measured value, while at

the same time respecting bounds on late-decaying neutral particles from BBN. However,

models with a standard neutralino underabundance can now be allowed with either neu-

tralino or axion domination of dark matter, and furthermore, these models can allow the

PQ breaking scale fato be pushed up into the 1014−1015GeV range, which is where it is

typically expected to be in string theory models.

Keywords: Supersymmetry Phenomenology, Supersymmetric Standard Model, Dark

Matter, Axions.

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1. Introduction

The Standard Model (SM) of particle physics is beset by two afflictions: 1. in the scalar

(Higgs) sector of the theory, quadratic divergences require large fine-tunings of electroweak

parameters which depend on the scale Λ below which the SM is regarded as the correct

effective field theory of nature and 2. in the QCD sector of the theory, the Lagrangian

term

¯θ

32π2FAµν˜Fµν

required by ’tHooft’s solution to the U(1)Aproblem is constrained to a value¯θ ? 10−10

to gain accord with measurements of the neutron EDM[1]. The first of these is solved

by the introduction of softly broken weak scale supersymmetry (SUSY) into the theory[2]

(which receives some indirect support from the measured values of gauge couplings at

LEP[3] and from global fits to precision electroweak data[4]), while the second problem

is solved by the introduction of a global U(1)PQPeccei-Quinn (PQ) symmetry broken by

QCD anomalies[5], which requires the existence of an (“invisible”) axion[6, 7], with mass

expected in the micro-eV or below range[8]. Solving both problems simultaneously requires

supersymmetrization of the SM (usually via the Minimal Supersymmetric Standard Model,

or MSSM) along with the introduction of an axion supermultiplet ˆ a into the theory. The ˆ a

supermultiplet contains an R-parity-even spin-0 saxion field s(x) along with an R-parity-

odd spin-1

2axino ˜ a(x), in addition to the usual pseudoscalar axion field a(x):

L ∋

A

(1.1)

ˆ a =s + ia

√2

+ i√2¯θ˜ aL+ i¯θθLFa, (1.2)

in 4-component spinor notation[2].

In such a theory, it is expected that SM superpartner particles with weak scale masses

should emerge, along with a weak scale saxion, whilst the axino mass is more model de-

pendent, with m˜ a∼ keV-TeV being expected[9]. The axion, saxion and axino couplings

to matter depend on the PQ breaking scale fa1, which is required fa? 109GeV by stellar

cooling calculations[10]. The axion is often considered as a very appealing dark matter

(DM) candidate[11, 12].2

In the MSSM, DM candidates include the lightest neutralino?Z1(a WIMP), the spin-3

matter is tightly constrained and disfavored by the standard picture of Big Bang nucle-

osynthesis (BBN)[15], whilst right-hand neutrino states are expected to exist near the GUT

scale according to the elegant see-saw mechanism for neutrino mass[16]. Many authors thus

expect dark matter to be comprised of the SUSY neutralinos, a natural WIMP candidate

which is motivated by the so-called “WIMP miracle”. However, detailed analyses show

that neutralino dark matter requires a rather high degree of fine-tuning[17] to match the

2

gravitino?G or possibly the superpartner of a right-handed neutrino[14]. Gravitino dark

1Throughout this work we omit the number of generations factor N, which appear along with the PQ

scale, fa/N, in the DSFZ model and in the KSVZ model with more than one heavy quark generation. All

our results can then be trivially generalized replacing fa by fa/N.

2For a somewhat different axion/axino scenario, see Ref. [13].

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WMAP-measured cold DM abundance[18]:

ΩDMh2= 0.1123 ± 0.0035 at 68% CL. (1.3)

In fact, the measured abundance lies in the most improbable locus of values of neutralino

relic density as predicted by general scans over SUSY model parameter space[19].

The PQ-extended Minimal Supersymmetric Standard Model (PQMSSM) offers addi-

tional possibilities to describe the dark matter content of the universe. In the PQMSSM,

the axino may play the role of stable lightest SUSY partner (LSP)[20, 21], while the quasi-

stable axion may also constitute a component of DM[22], giving rise to mixed axion/axino

(a˜ a) CDM. In supergravity theories however, the axino mass is expected to lie at the

weak scale[23], so that the neutralino remains as LSP, and the possibility occurs for mixed

axion/neutralino (a?Z1) CDM.

the relic abundance of neutralinos in the mixed a?Z1CDM scenario. This approach applies

velocity ?σv? is approximately constant with temperature, as occurs for a wino-like or

higgsino-like neutralino[25]. Detailed calculations of the relic abundance of mixed a?Z1

were presented.

The standard calculation of the neutralino Yield Ystd

tralino number density and s is the entropy density) gives

?90/π2g∗(Tfr)?1/2

4?σv?MPTfr

where g∗(Tfr) is the number of active degrees of freedom at temperature T = Tfr, where

3√5?σv?MPm3/2

π5/2T1/2

In a recent paper, Choi et al.[24] presented a semi-analytic approach for estimating

to cases where the thermally averaged neutralino annihilation cross section times relative

CDM were performed in Ref. [26], where formulae for the neutralino and axion abundances

? Z1

≡

n?

Z1

s

(where n? Z1is the neu-

Ystd

?Z1

=

, (1.4)

Tstd

fr= m? Z1/ln[

? Z1

frg1/2

∗ (Tfr)

]. (1.5)

is the freeze-out temperature and MP is the reduced Planck mass.

If instead axinos are thermally produced (TP) at a large rate at re-heat temperature

TRafter inflation, then they cascade decay to (stable) neutralinos at decay temperature

?

and can boost the neutralino abundance. The late-time injection of neutralinos into the

cosmic soup at temperatures T˜ a

D< Tfrmay cause a neutralino re-annihilation effect such

that the neutralino Yield is instead given by[24, 26]

?90/π2g∗(T˜ a

Since T˜ a

Dis typically in the MeV-GeV range, i.e. well below Tfr∼ m? Z1/20, the neutralino

abundance after re-annihilation can be highly enhanced relative to the standard cosmo-

logical picture. In addition, one must fold into the relic abundance the axion contribution

T˜ a

D=

Γ˜ aMP/?π2g∗(T˜ a

D)/90?1/4, (1.6)

Yre−ann

?Z1

|T=T˜ a

D≃

D)?1/2

D

4?σv?MPT˜ a

. (1.7)

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arising from coherent axion field oscillations beginning at axion oscillation temperature

Ta∼ 1 GeV.

An additional complication comes from entropy production from axino decay after

Tfr(which may dilute the neutralino abundance) or after Ta(which may dilute the axion

abundance). This may occur in the case where axinos temporarily dominate the energy

density of the universe. Depending on the PQ parameters of the PQMSSM model (fa, m˜ a,

initial axion misalignment angle θi and TR), the dark matter abundance may be either

neutralino- or axion-dominated. In fact, cases may occur where the DM relic abundance

is shared comparably between the two. In the latter case, it might be possible to directly

detect relic neutralino WIMP particles as well as relic axions!

While the semi-analytic treatment of Ref’s [24] and [26] provides a broad portrait of

the mixed a?Z1CDM picture, a number of important features have been neglected. These

• For bino-like neutralinos, ?σv? ∼ a + bT2where a ∼ 0 since we mainly have p-wave

annihilation cross sections. In this case, ?σv? is no longer independent of temperature,

and the simple formulae 1.4 and 1.7 are no longer valid.

include the following.

• In Ref’s [24] and [26], the effects of saxion production and decay in the early universe

are neglected. In fact, saxion thermal production or production via coherent oscilla-

tions (CO)[27], followed by late time saxion decay, may inject considerable entropy

into the early universe, thus diluting all relics present at the saxion decay temper-

ature Ts

D. Saxions may also add to the neutralino abundance via decays such as

s → ˜ g˜ g, followed by gluino cascade decays. There exists the possibility of saxion and

axino co-domination of the universe. In this case, there might be a second neutralino

re-annihilation taking place at Ts

D.

• The treatments of [24] and [26] invoke the “sudden decay” approximation for late-

decaying axinos, whereas in fact the decay process is a continuous one proceeding in

time until the decaying species is highly depleted (all have decayed).

• The treatments of [24] and [26] largely ignore the effect of gravitino production and

decay in the early universe.

To include the above effects into a calculation of the mixed a?Z1relic abundance, one

a full solution of the coupled Boltzmann equations which govern various abundances of

neutralinos, axinos, axions, saxions, gravitinos and radiation.

Toward this end, in Sec. 2 we present a simplified set of coupled Boltzmann equations,

which we use to calculate the relic abundance of mixed axion/neutralino dark matter.

More details about the approximations made and each term present in our equations are

discussed in Appendix A.

In Sec.3, we present various numerical results for the mixed a?Z1 CDM scenario

saxion field, adjusting the parameters of the PQMSSM can only increase the neutralino

must go beyond the semi-analytic treatment presented in Ref’s [24, 26], and proceed with

using the full set of Boltzmann equations. We find that, even after the inclusion of the

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abundance, and not decrease it, while at the same time respecting bounds on late-decaying

neutral particles from BBN. This result is the same as found in Refs. [24] and [26], but

now corresponds to a much stronger statement, since the saxion entropy injection had

been neglected in the previous works. Furthermore, our results also apply to models with

bino-like neutralinos, which could not be studied in the semi-analytical framework used in

Refs. [24] and [26].

Since the neutralino abundance can be only enhanced in the PQMSSM, in models such

as mSUGRA, those points which are excluded by a standard overabundance of neutralinos

are still excluded in the PQMSSM! This rather strong conclusion does depend on at least

three assumptions: 1. that thermal axino production rates are not suppressed by low-lying

PQ-charged matter multiplets[28]3, 2. that saxion decay is dominated by gluon and gluino

pairs and 3. that the assumed saxion field strength s(x) ≡ θsfais of order the PQ-breaking

scale fa, i.e. that θs∼ 1.

We also examine several cases with a standard underabundance of neutralino dark

matter. In these cases, again the neutralino abundance is only increased (if BBN constraints

are respected). Thus, adjustment of PQMSSM parameters can bring models with an

underabundance of neutralinos into accord with the measured DM relic density. In these

cases, the DM abundance tends to be neutralino-dominated. Also, in these cases, solutions

exist where the PQ scale fais either near its lower range, or where fais much closer to

MGUT, with fa∼ 1014GeV typically allowed. This is much closer to the scale of fawhich is

thought to arise from string theory[31]. In Sec. 5, we present a summary and conclusions.

2. Mixed axion/neutralino abundance from coupled Boltzmann equations

Here, we present a brief description of our procedure to calculate the relic abundance of

mixed a?Z1CDM in the PQMSSM. A more detailed discussion is left to Appendix A.

2.1 Boltzmann equations

The general Boltzmann equation for the number density of a particle species can be gener-

ically written as[32]:

˙ ni+ 3Hni= Si−1

γiΓini

(2.1)

where Sirepresents a source term, Γiis the decay width and γiis the relativistic dilation

factor to take into account the suppressed decays of relativistic particles. To describe the

thermal production of a particle species i as well as its decoupling from the radiation fluid

and the non-thermal production coming from other particles decays, we include in Sithe

following terms:

Si= −[n2

i− (neq

i(T))2]?σv?i(T) +

?

j

BR(j,i)Γjnj

γj

(2.2)

3Here, we assume standard rates for thermal axino production as calculated in the literature[21, 29, 30].

In Ref. [28], it has been shown that if PQ-charged matter multipletsˆΦ exist well below the PQ breaking

scale fa, then axino production is suppressed by factors of mΦ/TR.

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