Thermal axion production in the primordial quark-gluon plasma

Peter Graf and Frank Daniel Steffen

Max-Planck-Institut fur Physik, Fohringer Ring 6, D–80805 Munich, Germany(Received 3 September 2010; published 13 April 2011)

We calculate the rate for thermal production of axions via scattering of quarks and gluons in the

primordial quark-gluon plasma. To obtain a finite result in a gauge-invariant way that is consistent to

leading order in the strong gauge coupling, we use systematic field theoretical methods such as hard

thermal loop resummation and the Braaten-Yuan prescription. The thermally produced yield, the

decoupling temperature, and the density parameter are computed for axions with a mass below

10 meV. In this regime, with a Peccei-Quinn scale above 6� 108 GeV, the associated axion population

can still be relativistic today and can coexist with the axion cold dark matter condensate.

DOI: 10.1103/PhysRevD.83.075011 PACS numbers: 14.80.Va, 95.30.Cq, 95.35.+d, 98.80.Cq

I. INTRODUCTION

If the Peccei-Quinn (PQ) mechanism is the explanationof the strongCP problem, axions will pervade the Universeas an extremely weakly interacting light particle species. Infact, an axion condensate is still one of the most compel-ling explanations of the cold dark matter in our Universe[1,2]. While such a condensate would form at temperaturesT & 1 GeV, additional populations of axions can originatefrom processes at much higher temperatures. Even if thereheating temperature TR after inflation is such that axionswere never in thermal equilibrium with the primordialplasma, they can be produced efficiently via scattering ofquarks and gluons. Here we calculate for the first time theassociated thermal production rate consistent to leadingorder in the strong gauge coupling gs in a gauge-invariantway. The result allows us to compute the associated relicabundance and to estimate the critical TR value belowwhich our considerations are relevant. For a higher valueof TR, one will face the case in which axions were inthermal equilibrium with the primordial plasma beforedecoupling at the temperature TD as a hot thermal relic[1,3]. The obtained critical TR value can then be identifiedwith the axion decoupling temperature TD.

We focus on the model-independent axion (a) interac-tions with gluons given by the Lagrangian1

L a ¼ g2s32�2fPQ

aGb��

~Gb��; (1)

with the gluon field strength tensor Gb��, its dual ~Gb

�� ¼�����G

b��=2, and the scale fPQ at which the PQ symmetry

is broken spontaneously. Numerous laboratory, astro-physical, and cosmological studies point to

fPQ * 6� 108 GeV; (2)

which implies that axions are stable on cosmological timescales [4,5]. Considering this fPQ range, we can neglect

axion production via �� ! �a in the primordial hot had-ronic gas [6,7]. Moreover, Primakoff processes such ase�� ! e�a or q� ! qa are not taken into account.These processes depend on the axion model and involvethe electromagnetic coupling instead of the strong one,such that they are usually far less efficient in the earlyUniverse [8].We assume a standard thermal history and refer to TR as

the initial temperature of the radiation-dominated epoch.While inflation models can point to TR well above1010 GeV, we focus on the case TR < fPQ such that no

PQ symmetry restoration takes place after inflation.Related studies exist. The decoupling of axions out of

thermal equilibrium with the primordial quark-gluonplasma (QGP) was considered in Refs. [1,3]. While thesame QCD processes are relevant, our study treats thethermal production of axions that were never in thermalequilibrium. Moreover, we use hard thermal loop (HTL)resummation [9] and the Braaten-Yuan prescription [10],which allow for a systematic gauge-invariant treatment ofscreening effects in the QGP. In fact, that prescription wasintroduced on the example of axion production in a hotQED plasma [10]; see also Ref. [11].

II. THERMAL PRODUCTION RATE

Let us calculate the thermal production rate of axionswith energies E * T in the hot QGP. The relevant 2 ! 2scattering processes involving (1) are shown in Fig. 1. Thecorresponding squared matrix elements are listed in Table I,where s ¼ ðP1 þ P2Þ2 and t ¼ ðP1 � P3Þ2 with P1, P2, P3,and P referring to the particles in the given order. Workingin the limit T � mi, the masses of all particles involvedhave been neglected. Sums over initial and final spins havebeen performed. For quarks, the contribution of a singlechirality is given. The results obtained for processes A andC point to potential IR divergences associated with theexchange of soft (massless) gluons in the t channel and uchannel. Here screening effects of the plasma become

1The relation of fPQ to the vacuum expectation value h�i thatbreaks the Uð1ÞPQ symmetry depends on the axion model and theassociated domain wall number N: fPQ / h�i=N; cf. [1,2] andreferences therein.

PHYSICAL REVIEW D 83, 075011 (2011)

1550-7998=2011=83(7)=075011(5) 075011-1 � 2011 American Physical Society

relevant. To account for such effects, the QCD Debye mass

mD ¼ ffiffiffi

3p

mg, with mg ¼ gsTffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Nc þ ðnf=2Þq

=3 for Nc ¼ 3

colors and nf ¼ 6 flavors, was used in Ref. [3]. In contrast,

our calculation relies on HTL resummation [9,10] whichtreats screening effects more systematically.

Following Ref. [10], we introduce a momentum scalekcut such that gsT � kcut � T in the weak coupling limitgs � 1. This separates soft gluons with momentum trans-fer of order gsT from hard gluons with momentum transferof order T. By summing the respective soft and hardcontributions, the finite rate for thermal production ofaxions with E * T is obtained in leading order in gs,

EdWa

d3p¼ E

dWa

d3p

�

�

�

�

�

�

�

�softþE

dWa

d3p

�

�

�

�

�

�

�

�hard; (3)

which is independent of kcut; cf. (5) and (7) below.In the region with k < kcut, we obtain the soft contribu-

tion from the imaginary part of the thermal axion self-energy with the ultraviolet cutoff kcut,

EdWa

d3p

�

�

�

�

�

�

�

�soft¼ � fBðEÞ

ð2�Þ3 Im�aðEþ i�; ~pÞjk<kcut (4)

¼EfBðEÞ3m2

gg4sðN2

c�1ÞT8192�8f2PQ

�

ln

�

k2cutm2

g

�

�1:379

�

(5)

with the equilibrium phase space density for bosons (fer-mions) fBðFÞðEÞ ¼ ½expðE=TÞ � 1��1. Our derivation of

(5) follows Ref. [10]. The leading order contribution toIm�a for k < kcut and E * T comes from the Feynmandiagram shown in Fig. 2. Because of E * T, only one ofthe two gluons can have a soft momentum. Thus only oneeffective HTL-resummed gluon propagator is needed.In the region with k > kcut, bare gluon propagators can

be used since kcut provides an IR cutoff. From the resultsgiven in Table I weighted with appropriate multiplicities,statistical factors, and phase space densities, we then ob-tain the (angle-averaged) hard contribution

EdWa

d3p

�

�

�

�

�

�

�

�hard

¼ 1

2ð2�Þ3Z d�p

4�

Z

�

Y

3

j¼1

d3pj

ð2�Þ32Ej

�

�ð2�Þ44ðP1þP2�P3�PÞ�ðk�kcutÞ�X

f1ðE1Þf2ðE2Þ½1�f3ðE3Þ�jM1þ2!3þaj2 (6)

¼ Eg6sðN2

c � 1Þ512�7f2PQ

�

nffBðEÞT3

48�lnð2Þ þ

�

Nc þnf2

�

fBðEÞT3

48�

��

ln

�

T2

k2cut

�

þ 17

3� 2�þ 2 0ð2Þ

ð2Þ�

þ NcðIð1ÞBBB � Ið3ÞBBBÞ þ nfðIð1ÞFBF þ Ið3ÞFFBÞ�

(7)

TABLE I. Squared matrix elements for axion (a) production intwo-body processes involving quarks of a single chirality (qi)and gluons (ga) in the high-temperature limit, T � mi, with theSUðNcÞ color matrices fabc and Ta

ji. Sums over initial and final

state spins have been performed.

Label i Process i jMij2=ð g6s128�4f2

PQ

ÞA ga þ gb ! gc þ a �4 ðs2þstþt2Þ2

stðsþtÞ jfabcj2B qi þ �qj ! ga þ a ð2t2s þ 2tþ sÞjTa

jij2C qi þ ga ! qj þ a ð� 2s2

t � 2s� tÞjTajij2

FIG. 1. The 2 ! 2 processes for axion production in the QGP.Process C also exists with antiquarks �qi;j replacing qi;j.

FIG. 2. Leading contribution to the axion self-energy for softgluon momentum transfer and hard axion energy. The blob onthe gluon line denotes the HTL-resummed gluon propagator.

PETER GRAF AND FRANK DANIEL STEFFEN PHYSICAL REVIEW D 83, 075011 (2011)

075011-2

with Euler’s constant �, Riemann’s zeta function ðzÞ,Ið1ÞBBBðFBFÞ ¼

1

32�3

Z 1

0dE3

Z EþE3

0dE1 ln

�jE1�E3jE3

�

��

��ðE1�E3Þ d

dE1

�

fBBBðFBFÞE22

E2ðE2

1þE23Þ�

þ�ðE3�E1Þ d

dE1

½fBBBðFBFÞðE21þE2

3Þ�

þ�ðE�E1Þ d

dE1

�

fBBBðFBFÞ�

E21E

22

E2�E2

3

���

;

(8)

Ið3ÞBBBðFFBÞ ¼1

32�3

Z 1

0dE3

Z EþE3

0dE2fBBBðFFBÞ

��

�ðE� E3Þ E21E

23

E2ðE3 þ EÞ þ�ðE3 � EÞ E22

E3 þ E

þ ½�ðE3 � EÞ�ðE2 � E3Þ ��ðE� E3Þ�ðE3 � E2Þ�� E2 � E3

E2½E2ðE3 � EÞ � E3ðE3 þ EÞ�

�

; (9)

fBBB;FBF;FFB ¼ f1ðE1Þf2ðE2Þ½1� f3ðE3Þ�: (10)

The sum in (6) is over all axion production processes 1þ2 ! 3þ a viable with (1). The colored particles 1–3 werein thermal equilibrium at the relevant times. Performingthe calculation in the rest frame of the plasma, fi are thusdescribed by fF=B depending on the respective spins.

Shorthand notation (10) indicates the corresponding com-binations, where þ (�) accounts for Bose enhancement(Pauli blocking) when particle 3 is a boson (fermion). Withany initial axion population diluted away by inflation and Twell below TD so that axions are not in thermal equilib-rium, the axion phase space density fa is negligible incomparison to fF=B. Thereby, axion disappearance reac-

tions (/ fa) are neglected as well as the respective Boseenhancement (1þ fa � 1). Details on the methods appliedto obtain our results (7)–(9) can be found in Refs. [11–13].

III. RELIC AXION ABUNDANCE

We now calculate the thermally produced (TP) axionyield YTP

a ¼ na=s, where na is the corresponding axionnumber density and s the entropy density. For T suffi-ciently below TD, the evolution of the thermally producedna with cosmic time t is governed by the Boltzmannequation

dnadt

þ 3Hna ¼Z

d3pdWa

d3p¼ Wa: (11)

HereH is the Hubble expansion rate, and the collision termis the integrated thermal production rate

Wa ¼ ð3Þg6sT6

64�7f2PQ

�

ln

�

T2

m2g

�

þ 0:406

�

: (12)

Assuming conservation of entropy per comoving volumeelement, (11) can be written as dYTP

a =dt ¼ Wa=s. Sincethermal axion production proceeds basically during the hotradiation-dominated epoch, i.e., well above the tempera-ture of radiation-matter equality Tmat¼rad, one can changevariables from cosmic time t to temperature T accordingly.With initial temperature TR at which YTP

a ðTRÞ ¼ 0, the relicaxion yield today is given by

YTPa � YTP

a ðTmat¼radÞ ¼Z TR

Tmat¼rad

dTWaðTÞ

TsðTÞHðTÞ¼ 18:6g6s ln

�

1:501

gs

��

1010 GeV

fPQ

�

2�

TR

1010 GeV

�

: (13)

This result is shown by the diagonal lines in Fig. 3 forcosmological scenarios with different TR values rangingfrom 104 to 1012 GeV. Here we use gs ¼ gsðTRÞ as de-scribed by the one-loop renormalization group evolution [5]

gsðTRÞ ¼�

g�2s ðMZÞ þ

11Nc � 2nf

24�2ln

�

TR

MZ

���1=2; (14)

where g2sðMZÞ=ð4�Þ ¼ 0:1172 at MZ ¼ 91:188 GeV.Since the methods [9,10] allowing for the gauge-invariantderivation require gs � 1, (13) is most reliable for TR �104 GeV. Ifgs is too large, unphysical negative values of (3)are encountered for E & T that lead to an artificial suppres-sion of (13) as indicated by the logarithmic factor; cf. Sec. 3in Ref. [14], where a similar behavior is discussed in moredetail. Nevertheless, we think that the applied methods

FIG. 3. The relic axion yield today originating from thermalprocesses in the primordial plasma for cosmological scenarioscharacterized by different TR values covering the range from 104

to 1012 GeV. The dotted, dash-dotted, dashed, and solid lines areobtained for fPQ ¼ 109, 1010, 1011, and 1012 GeV.

THERMAL AXION PRODUCTION IN THE PRIMORDIAL . . . PHYSICAL REVIEW D 83, 075011 (2011)

075011-3

presently still allow for the most reliable treatment sincethey respect gauge invariance.

Note that (13) is only valid when axion disappearanceprocesses can be neglected. In scenarios in which TR

exceeds TD, this is not justified since there has been anearly period in which axions were in thermal equilibrium.In this period, their production and annihilation proceededat equal rates. Thereafter, they decoupled as hot thermalrelics at TD, where all standard model particles are effec-tively massless. The present yield of those thermal relicaxions is then given by Y

eqa ¼ n

eqa =s � 2:6� 10�3. In

Fig. 3 this value is indicated by the horizontal lines.In fact, the thermally produced yield cannot exceed Yeq

a .In scenarios with TR such that (13) turns out to be close toor greater than Yeq

a , disappearance processes have to betaken into account. The resulting axion yield from thermalprocesses will then respect Y

eqa as the upper limit. For

example, for fPQ ¼ 109 GeV, this yield would show a

dependence on the reheating temperature TR that is verysimilar to the one shown by the dotted line in Fig. 3. Theonly difference will be a smooth transition instead of thekink at YTP

a ¼ Yeqa .

IV. AXION DECOUPLING TEMPERATURE

The kinks in Fig. 3 indicate the critical TR value whichseparates scenarios with thermal relic axions from those inwhich axions have never been in thermal equilibrium.Thus, for a given fPQ, this critical TR value allows us to

extract an estimate of the axion decoupling temperatureTD. We find that our numerical results are well describedby

TD � 9:6� 106 GeV

�

fPQ1010 GeV

�

2:246: (15)

In a previous study [3], the decoupling of axions that werein thermal equilibrium in the QGP was calculated. Whenfollowing [3] but including (14), we find that the tempera-ture at which the axion yield from thermal processesstarted to differ by more than 5% from Yeq

a agrees basicallywith (15). The axion interaction rate � equals H already attemperatures about a factor 4 below (15) which, however,amounts to a different definition of TD.

V. AXION DENSITY PARAMETER

Since thermally produced axions basically have a ther-mal spectrum also, we find that the density parameter fromthermal processes in the primordial plasma can be de-scribed approximately by

�TP=eqa h2 ’

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

hpa;0i2 þm2a

q

YTP=eqa sðT0Þh2=�c (16)

with the present average momentum hpa;0i ¼ 2:701Ta;0

given by the present axion temperature Ta;0 ¼ 0:332T0 ’0:08 meV, where T0 ’ 0:235 meV is the present cosmicmicrowave background temperature, h ’ 0:7 is Hubble’s

constant in units of 100 km=Mpc=s, and �c=½sðT0Þh2� ¼3:6 eV. A comparison of Ta;0 with the axion mass ma ’0:6 meVð1010 GeV=fPQÞ shows that this axion population

is still relativistic today for fPQ * 1011 GeV.In Fig. 4 the solid, dashed, and dash-dotted lines show

�TP=eqa h2 for TR ¼ 106, 107, and 108 GeV, respectively. In

the fPQ region to the right (left) of the respective kink, in

which TR < TD (TR > TD) holds, �TPðeqÞa h2 applies, which

behaves as / f�3PQ (f�1

PQ ) for ma � Ta;0 and as / f�2PQ (f0PQ)

for ma � Ta;0. The gray dotted curve shows �eqa h2 for

higher TR with TR > TD and also indicates an upper limiton the thermally produced axion density. Observationof those axions will be extremely challenging. Even�

eqa h2 stays well below the cold dark matter density

�CDMh2 ’ 0:1 (gray horizontal bar) and also below the

photon density ��h2 ’ 2:5� 10�5 (gray thin line) [5] in

the allowed fPQ range (2). There, the current hot dark

matter limits are also safely respected [15].In cosmological settings with TR > TD, axions produced

nonthermally before axion decoupling (e.g., in inflatondecays) will also be thermalized, resulting in �eq

a h2. Theaxion condensate from the misalignment mechanism, how-ever, is not affected—independent of the hierarchy be-tween TR and TD—since thermal axion production in theQGP is negligible at T & 1 GeV. Thus, the associated

density �MISa h2 0:15�2i ðfPQ=1012 GeVÞ7=6 [1,2,16] can

108

109

1010

1011

101210

-10

10-8

10-6

10-4

10-2

1

FIG. 4. The axion density parameter from thermal processesfor TR ¼ 106 GeV (solid line), 107 GeV (dashed line), and108 GeV (dash-dotted line) and the one from the misalignmentmechanism for �i ¼ 1, 0.1, and 0.01 (dotted line). The densityparameters for thermal relic axions, photons, and cold darkmatter are indicated, respectively, by the gray dotted line(�

eqa h2), the gray thin line (��h

2), and the gray horizontal bar

(�CDMh2).

PETER GRAF AND FRANK DANIEL STEFFEN PHYSICAL REVIEW D 83, 075011 (2011)

075011-4

coexist with�TP=eqa h2 and is governed by the misalignment

angle �i as illustrated by the dotted lines in Fig. 4. Thereby,the combination of the axion cold dark matter condensate

with the axions from thermal processes, �ah2 ¼

�MISa h2 þ�

TP=eqa h2, gives the analog of a Lee-Weinberg

curve. Taking into account the relation between fPQ and

ma, this is exactly the type of curve that can be inferredfrom Fig. 4. Here our calculation of thermal axion

production in the QGP allows us to cover, for the firsttime, also cosmological settings with TR < TD.

ACKNOWLEDGMENTS

We are grateful to Thomas Hahn, Josef Pradler, GeorgRaffelt, and Javier Redondo for valuable discussions. Thisresearch was partially supported by the Cluster ofExcellence ‘‘Origin and Structure of the Universe.’’

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