The Quest for an Intermediate-Scale Accidental Axion and Further ALPs

43 Pages·2014·0.79 MB·English

DESY 14-020
The Quest for an Intermediate-Scale Accidental Axion and Further ALPs
A. G. Dias1,∗ A. C. B. Machado2,† C. C. Nishi1,‡ A. Ringwald3,§ and P. Vaudrevange4¶
1Universidade Federal do ABC - UFABC,
Santo Andr´e, Sa˜o Paulo, Brazil
2Instituto de F´ısica Teo´rica–Universidade Estadual Paulista,
R. Dr. Bento Teobaldo Ferraz 271, Barra Funda
Sa˜o Paulo - SP, 01140-070, Brazil
3Deutsches Elektronen-Synchrotron, DESY,
Notkestr. 85, 22607 Hamburg, Germany
4Excellence Cluster Universe,
Technische Universit¨at Mu¨nchen,
Boltzmannstr. 2, D-85748, Garching, Germany
The recent detection of the cosmic microwave background polarimeter experiment BICEP2 of
tensor fluctuations in the B-mode power spectrum basically excludes all plausible axion models
where its decay constant is above 1013GeV. Moreover, there are strong theoretical, astrophysical,
andcosmologicalmotivationsformodelsinvolving,inadditiontotheaxion,alsoaxion-likeparticles
(ALPs),withdecayconstantsintheintermediatescalerange,between109GeVand1013GeV. Here,
4
we present a general analysis of models with an axion and further ALPs and derive bounds on the
1
relative size of the axion and ALP photon (and electron) coupling. We discuss what we can learn
0
from measurements of the axion and ALP photon couplings about the fundamental parameters of
2
the underlying ultraviolet completion of the theory. For the latter we consider extensions of the

r Standard Model in which the axion and the ALP(s) appear as pseudo Nambu-Goldstone bosons
a
from the breaking of global chiral U(1) (Peccei-Quinn (PQ)) symmetries, occurring accidentally as
M
low energy remnants from exact discrete symmetries. In such models, the axion and the further
ALPareprotectedfromdisastrousexplicitsymmetrybreakingeffectsduetoPlanck-scalesuppressed
3
operators. ThescenariosconsideredexploitheavyrighthandedneutrinosgettingtheirmassviaPQ
2
symmetry breaking and thus explain the small mass of the active neutrinos via a seesaw relation

betweentheelectroweakandanintermediatePQsymmetrybreakingscale. Foranumberofexplicit
]
h models, we determine the parameters of the low-energy effective field theory describing the axion,
p the ALPs, and their interactions with photons and electrons, in terms of the input parameters,
- in particular the PQ symmetry breaking scales. We show that these models can accommodate
p
simultaneously an axion dark matter candidate, an ALP explaining the anomalous transparency
e
of the universe for γ-rays, and an ALP explaining the recently reported 3.55 keV gamma line
h
from galaxies and clusters of galaxies, if the respective decay constants are of intermediate scale.
[
Moreover, they do not suffer severely from the domain wall problem.

1
v
0
6
7 Contents
5
.
3 I. Introduction 2
0
4
II. Correlations between Axion and ALPs Couplings in Multi-Axion Models 6
1
:
v III. Axion and Further ALPs in Ad-Hoc Peccei-Quinn SM Extensions 11
i A. KSVZ-type models 11
X
B. DFSZ-type models 12
r
a C. Models relating the PQ scale with the seesaw neutrino scale 13
D. Photophilic models 14
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
§Electronicaddress: [email protected]
¶Electronicaddress: [email protected]
2
IV. Intermediate-Scale accidental axion and further ALPs from Field Theoretic Bottom-Up SM
Extensions 15
A. Z ⊗Z ⊗Z(cid:48) model with axion-ALP mixing 16
13 5 5
B. Z ⊗Z models with a photophilic ALP 22
n m
1. Z ⊗Z model with extended Higgs sector 23
11 9
2. Z ⊗Z model with minimal SM Higgs sector 25
11 9
3. Z ⊗Z model with minimal SM Higgs sector 27
11 7
C. Z ⊗Z ⊗Z model with two photophilic ALPs 27
11 9 7
V. Summary and Outlook 29
Acknowledgments 31
A. Axion and ALP Couplings to Gluons, Photons, and Electrons 31
B. Effects from Gravity through Planck-Scale Suppressed Operators 33
C. Domain-Wall Problem in the Models 36
References 38
I. INTRODUCTION
TheaxionAisastronglymotivatedveryweaklyinteractingultralightparticlebeyondtheStandardModel(SM).Its
existence is predicted in the course of an elegant solution to the strong CP problem [1–3], that is the non-observation
of flavor conserving CP violation originating from the theta-angle term in the Lagrangian of QCD,
α
L ⊃− s θ¯Ga G˜a,µν, (1)
QCD 8π µν
involving the gluon field strength Ga and its dual, G˜a,µν ≡ (cid:15)µνλρGa /2, with ε0123 = 1. This solution consists
µν λρ
in adding to the Standard Model a scalar field theory describing a (pseudo) Nambu-Goldstone boson arising from
the breaking of a global chiral (Peccei-Quinn (PQ)) U(1) symmetry: that is, the corresponding scalar field A(x)
PQ
satisfies a shift symmetry A(x)→A(x)+const. which is only violated by the anomalous coupling to gluons,
α A
L ⊃− s Ga G˜a,µν. (2)
8π f µν
A
In fact, the θ¯-term can then be eliminated by absorbing it into the axion field, A + θ¯f → A. Moreover, non-
A
perturbative QCD effects1 provide for a non-trivial potential for the shifted axion field A – minimized at zero expec-
tation value, (cid:104)A(cid:105) = 0, thus wiping out strong CP violation – and predict a mass for the particle excitation around
this minimum, the axion A,
√
m f z (cid:18)109GeV(cid:19)
m = π π (cid:39)6meV× . (3)
A f 1+z f
A A
in terms of the pion mass, m = 135 MeV, its decay constant, f ≈ 92 MeV, the ratio z = m /m ≈ 0.56 of up
π π u d
and down quark masses, and the decay constant f . For large f , the axion is an ultralight particle with very weak
A A
interactions with the Standard Model [4–7]. Its universal and phenomenologically most important interaction with
photons [8–11],
g α α m (cid:18)109GeV(cid:19) (cid:16) m (cid:17)
L ⊃− Aγ AF F˜µν; |g |∼ ∼ A ∼10−12 GeV−1 ∼10−12 GeV−1 A , (4)
4 µν Aγ 2πf 2πm f f 6meV
A π π A
1 Semiclassicalinstantonmethodsarenotreliabletocalculatethepotentialaccurately. Onehastousematchingtothelow-energychiral
Lagrangianinstead[2,8].
3
is tiny (see the yellow band in Fig. 1), since observations in astrophysics – in particular the observed duration of the
neutrino signal from supernova SN 1987A – require a large decay constant f (small mass m ) [18],
A A
f (cid:38)4×108GeV⇒m (cid:46)16meV. (5)
A A
A further strong motivation for the axion comes from the fact that, for sufficiently large decay constant f ,
A
109GeV(cid:46)f (cid:46)1013GeV⇒10−7eV(cid:46)m (cid:46)1meV, (6)
A A
it is a cold dark matter candidate, being non-thermally produced in the early universe by the vacuum-realignment
mechanism and, in some models and under certain circumstances, also via the decay of topological defects such as
axion strings and domain walls [19–27].
The upper bound on the decay constant in Eq. (6) follows from the recent discovery of the cosmic microwave
background polarimeter experiment BICEP2 of tensor fluctuations in the B-mode power spectrum [28]. This implies
a high value for the Hubble scale during inflation,
H (cid:39)1.1×1014GeV, (7)
I
which, together with constraints on isocurvature fluctuations [29–33], rule out plausible scenarios where inflation
occurs after PQ symmetry breaking [34–36], that is where
f >max{T ,T }, (8)
A GH max
with
H
T = I (9)
GH 2π
the Gibbons-Hawking temperature during inflation and
(cid:115)
(cid:114)
3
T =(cid:15) M H (10)
max eff 8π Pl I
the maximum thermalization temperature after reheating. Here M = 1.22 × 1019GeV is the Planck mass and
Pl
0 < (cid:15) < 1 is an efficiency parameter. Ways out of this conclusion have been put forward in Refs. [34, 35]. The
eff
pre-inflationary PQ symmetry breaking scenario would have allowed, in principle, much higher values of the decay
constantwithoutovershootingthedarkmatterabundance, byinvokingsmallvaluesoftheinitialmisalignmentangle.
AfterBICEP2, theplausibleaxionpossibilitieshavenarroweddowntoscenarioswithpost-inflationaryPQsymmetry
breaking. A conservative upper bound on the axion decay constant is then
H
f < I (cid:39)1.8×1013GeV, m >0.3 µeV. (11)
A 2π A
A more stringent upper bound can be obtained by requiring that the axion dark matter abundance generated via the
vacuum-realignment mechanism should not exceed the observed dark matter abundance, leading to [23–25, 31, 36]
f (cid:46)(1÷10)×1011GeV, m (cid:38)(6÷60) µeV. (12)
A A
To be on the very conservative side, the red region in Fig. 1 labelled“Axion DM”comprises still both cases, the
disfavored pre- and the favored post-inflationary PQ symmetry breaking. Two haloscope experiments (ADMX [37]
and ADMX-HF) are currently aiming at the direct detection of axion dark matter, based on microwave cavities in
the mass region 2×10−6eV (cid:46) m (cid:46) 2×10−5eV (see the green regions labelled“Haloscopes”in Fig. 1). Further
A
experiments have been proposed to extend this region on both ends of the spectrum [38–49].
There is also a theoretical motivation for the existence of additional axion-like particles (ALPs) (for reviews, see
[50–52]), emerging as pseudo Nambu-Goldstone bosons from the breaking of other global symmetries, such as lepton
number [53, 54] or family symmetries [55–57]. Most notably, the low-energy effective field theories emerging from
stringtheorygenericallycontaincandidatesfortheaxionpluspossiblyamultitudeofadditionalALPs. Indeed, when
compactifying the six extra spatial dimensions of string theory, Kaluza-Klein zero modes of antisymmetric form fields
–thelatterbelongingtothemasslessspectrumofthebosonicstringpropagatingintendimensions–appearasaxion
andfurtherALPcandidatesinthelow-energyeffectiveaction[58–63],theirnumberbeingdeterminedbythetopology
of the internal compactified manifold. Moreover, the axion and a multitude of additional ALP candidates may also
4
(cid:45)8
Massive Stars
(cid:45)10
SN1987A Γ(cid:45)Ray Burst ALPS(cid:45)II
IAXO
Γ(cid:45)Ray Transparency
(cid:45)12 (cid:68)
(cid:45)1 Quasar Polarization Axion
V
Ge (cid:64) Soft X(cid:45)Ray Excess from Coma opes CDM DM
gΓa (cid:45)14 (cid:200)(cid:200) Halosc ALP
Log10 Decaying
PIXIE PRISM m
(cid:45)16 fro
(cid:144) Axion Line
V
ke
(cid:45)18 55
3.
M
C D
ALP
(cid:45)20
(cid:45)40(cid:45)38(cid:45)36(cid:45)34(cid:45)32(cid:45)30(cid:45)28(cid:45)26(cid:45)24(cid:45)22(cid:45)20(cid:45)18(cid:45)16(cid:45)14(cid:45)12(cid:45)10 (cid:45)8 (cid:45)6 (cid:45)4 (cid:45)2 0 2 4
Log m eV
10 a
FIG.1: Predictionoftheaxion-photoncouplingversusitsmass(yellowband). Alsoshownareexcludedregionsarisingfromthe
(cid:64) (cid:68)
non-observationofananomalousenergylossofmassivestarsduetoaxionorALPemission[12],ofaγ-rayburst(incoincidence
with the observed neutrino burst) from SN 1987A due to conversion of an initial ALP burst in the galactic magnetic field, of
changes in quasar polarizations due to photon-ALP oscillations, and of dark matter axions or ALPs converted into photons
in microwave cavities placed in magnetic fields [13–17]. Axions and ALPs with parameters in the regions surrounded by the
red lines may constitute all or a part of cold dark matter (CDM), explain the cosmic γ-ray transparency, and the soft X-ray
excess from Coma. The green regions are the projected sensitivities of the light-shining-through-wall experiment ALPS-II, of
the helioscope IAXO, of the haloscopes ADMX and ADMX-HF, and of the PIXIE or PRISM cosmic microwave background
observatories.
arise as pseudo Nambu-Goldstone bosons from the breaking of accidental global U(1) symmetries that appear as low
energy remnants of exact discrete symmetries occurring in string compactifications [64–66]. Intriguingly, very light
and weakly coupled generic ALPs a , with decay constants in the same range as Eq. (6), share with the axion the
i
property of being cold dark matter candidates since they are also produced via the vacuum-realignment mechanism
[67–69]. In fact, their relic abundance, in the now strongly favored post-inflationary symmetry breaking scenario, is
roughly given by2
(cid:16)m (cid:17)1/2(cid:18) f (cid:19)2
Ω h2 ≈0.053× ai ai . (13)
ai eV 1011 GeV
Therefore, there isa strong motivationtosearchnotonlyfortheaxionA, butalso foradditionallightALPs a , for
i
which the low-energy effective Lagrangian describing their interactions with (the lightest Standard Model particles)
2 Theredlinelabelled“ALPCDM”inFig. 1correspondstotheALPphotoncoupling,whereanALP,accordingto(131),canaccountfor
allofthedarkmatterintheuniverse,assumingthatitcoupleswithstrengthgaiγ =α/(2πfai). Theregionbelowthislineisstrongly
disfavoredafterBICEP2byoverdensityconstraints[34,35].
5
photons can generically be written as
L ⊃ 1∂ A∂µA− 1m2A2− gAγ AF F˜µν +(cid:88)nax (cid:18)1∂ a ∂µa − 1m2 a2− gaiγ a F F˜µν(cid:19). (14)
2 µ 2 A 4 µν 2 µ i i 2 ai i 4 i µν
i=2
However, unlike the axion, which inherits many of its properties (m , g ), from non-perturbative QCD effects
A Aγ
associated with chiral symmetry breaking, the masses m and the photon couplings g of the additional ALPs are
ai aiγ
model dependent. Thus, there exists the possibility that ALPs are hierarchically more strongly coupled to photons
than axions with the same mass and therefore easier to detect.
Interestingly,thereareindicationsfromgamma-rayastronomy,whichmaypointtotheexistenceofatleastoneALP
beyondtheaxion. Gamma-rayspectrafromdistantactivegalacticnuclei(AGN)shouldshowanenergyandredshift-
dependent exponential attenuation, exp(−τ(E,z)), due to e+e− pair production off the extragalactic background
light (EBL) – the stellar and dust-reprocessed light accumulated during the cosmological evolution following the era
of re-ionization. The recent detection of this effect by Fermi [70] and H.E.S.S. [71] has allowed to constrain the
EBL density. At large optical depth, τ (cid:38) 2, however, there are hints that the Universe is anomalously transparent
to gamma-rays [72–76]. This may be explained by photon ↔ ALP oscillations: the conversion of gamma rays into
ALPsinthemagneticfieldsaroundAGNsorintheintergalacticmedium,followedbytheirunimpededtraveltowards
our galaxy and the consequent reconversion into photons in the galactic/intergalactic magnetic fields [77–81]. This
explanation requires a very light ALP, which couples to two photons with strength [82],
|g |(cid:38)10−12 GeV−1; m (cid:46)10−7 eV. (15)
aγ a
Note that the entire parameter region (15) has no overlap with the universal prediction of the axion, see Fig. 1.
In fact, an axion with m (cid:46) 10−7eV would have a photon coupling |g | (cid:46) 10−16GeV−1. Therefore, if this hint is
A Aγ
taken seriously, it points to the existence of an ALP beyond the axion.
Intriguingly, anobservedsoftX-rayexcessintheComaclustermayalsobeexplainedbytheconversionofacosmic
ALP background radiation (CABR) – produced by heavy moduli decay and corresponding to an effective number
(cid:52)N ∼0.5 of extra neutrinos species [83, 84] – into photons in the cluster magnetic field [85, 86]. This explanation
eff
requiresthatthespectrumoftheCABRispeakedinthesoft-keVregionandthattheALPcouplingandmasssatisfy
(cid:112)
|g |(cid:38)10−13 GeV−1 0.5/N ; m (cid:46)10−12 eV, (16)
aγ eff a
respectively,overlappingwiththeparameterrange(15)preferredbytheALPsolutionofthegamma-raytransparency
puzzle, as is apparent in Fig. 1.
Astrophysical bounds on ALPs arising from magnetised white dwarfs [87] and from the non-observation of a γ-ray
burstincoincidencewithneutrinosfromthesupernovaSN1987A[88,89]providelimitscloseto|g |(cid:46)10−11 GeV−1,
aγ
for masses m (cid:46) 10−7 eV and m (cid:46) 10−9 eV, respectively, and thus cut into the parameter range (15) preferred by
a a
the cosmic γ-ray transparency anomaly. Even stronger limits, |g | (cid:46) 2×10−13 GeV−1, for m (cid:46) 10−14 eV, have
Aγ A
been obtained by exploiting high-precision measurements of quasar polarizations [90, 91]. But there remains still a
sizeable region in ALP parameter space motivated by the above anomalies and at the same time consistent with all
astrophysical constraints, as can be seen in Fig. 1.
At small masses below 10−14 eV, a part of the region of interest in ALP parameter space will be probed indirectly
by the next generation of cosmic microwave background (CMB) observatories such as PIXIE [92] and PRISM [93]
(see the region labelled“PIXIE/PRISM”in Fig. 1, based on the assumption of an extragalactic magnetic field B of
nG size; the projected sensitivity scales with the magnetic field as B−1), because resonant photon-ALP oscillations
in primordial magnetic fields may lead to observable spectral distortions of the CMB [94–96].
A complementary part of the region of interest in parameter space will soon be probed by a pure laboratory exper-
iment: the light-shining-through-a-wall (LSW) [97] experiment Any-Light-Particle-Search II (ALPS-II) is designed to
detect photon–ALP–photon oscillations in the range [98] (see the green region labelled“ALPS-II”in Fig. 1)
|g |(cid:38)2×10−11 GeV−1; m (cid:46)10−4 eV. (17)
aγ a
Further experimental opportunities covering this region in ALP parameter space will open if the International Axion
Observatory (IAXO), a helioscope searching for solar axions and ALPs, is realized [99]. Its projected sensitivity is
|g |(cid:38)5×10−12 GeV−1; m (cid:46)10 meV. (18)
aγ a
The latter instrument has also the possibility to probe the possible coupling of the axion or further ALPs to
electrons,
(g ∂ A+g ∂ a)
L⊃ Ae µ ae µ e¯γµγ e, (19)
2m 5
e
6
via their solar production by atomic axio-recombination, axio-deexcitation, axio-Bremsstrahlung in electron-ion or
electron-electroncollisions, andComptonscattering[100]. Thisisofconsiderableinterestbecauseofhintsofanextra
stellar cooling mechanism not accounted by the Standard Model. In fact, the white dwarf (WD) luminosity function
seems to require a new energy-loss channel that can be interpreted in terms of losses due to sub-keV mass axions or
ALPs, with Yukawa couplings [101],
|g |≡|C |m /f ∼10−13 and/or |g |≡|C |m /f ∼10−13, (20)
Ae Ae e A ae ae e a
whicharewellintherangeexpectedforanintermediatescaleaxionorfurtherALPs(ifthemodel-dependentcouplings
C andC areoforderunity). Thesameparameterrangeispreferredtoexplaintheanomaloussizeoftheobserved
Ae ae
period decrease of the pulsating WDs G117-B15A and R548 by additional axion/ALP losses [102, 103]. A third
independent hint of anomalous stellar losses has recently been found in the red-giant branch of the globular cluster
M5,whichseemstobeextendedtolargerbrightnessthanexpectedwithintheStandardModel. Apossibleexplanation
ofthisobservationisthattheheliumcoresofredgiantsloseenergyinaxionsorfurtherALPswithelectroncouplings
of the same order as in Eq. (20) [104].
Very recently, two groups have found an unidentified X-ray line signal at 3.55 keV in the stacked spectrum of a
numberofgalaxyclusters[105]andtheAndromedagalaxy[106]. Itistemptingtoidentifythislinewiththeexpected
signal from two photon decay of 7.1 keV mass ALP dark matter [107–110]. To match the observed X-ray flux, but
allowing for the likely possibility, that the ALP dark matter makes only a fraction x ≡ρ /ρ of the total density
a a DM
of dark matter, the required lifetime and thus coupling of the ALP is [108]
τ = 64π =x ×(cid:0)4×1027−4×1028(cid:1) s ⇒ 3×10−18GeV−1(cid:18) 1 (cid:19)1/2 (cid:46)|g |(cid:46)10−12GeV−1(cid:18)10−10(cid:19)1/2 .
a g2 m3 a x aγ x
aγ a a a
(21)
Therefore, it is timely to have a close look onto ultraviolet extensions of the Standard Model featuring, apart from
an intermediate scale axion, also further ALPs and to investigate possible correlations between the low-energy axion
and ALP couplings. In fact, as we will show in Sec. II, such correlations inevitably occur if there are originally two
(or more) Nambu-Goldstone fields coupled to GG˜. The crucial determination of the decay constants and couplings of
axion-likefieldsfromoriginalhigh-scaletheoriestogluons, photons, andelectronswillbedoneinSecs. IIIandIV.In
the latter section, we construct particularly well-motivated ultraviolet completions of the Standard model featuring
accidental Peccei-Quinn symmetries and deduce their low-energy phenomenology. Finally, we summarize, conclude
and give an outlook for further investigations in Sec. V.
II. CORRELATIONS BETWEEN AXION AND ALPS COUPLINGS IN MULTI-AXION MODELS
Up to now, most phenomenological studies have considered just one particle type at a time: either the axion or
an ALP different from the axion, without taking into account possible relations between their low-energy parameters
(see, however, Refs. [65, 66, 111–113]). In this section, we will study, in multi-axion models, the phenomenologically
most important axion and ALP couplings to photons and electrons in depth. We will find that there are possibly
strong correlations in these couplings.
The most general low-energy effective Lagrangian of a model with n axion-like fields enjoying shift symmetries
ax
a(cid:48) →a(cid:48)+const.,i.e. realizinganon-linearrepresentationofn U(1) symmetries,couplingtogluonsandphotons,
i i ax PQi
with field strengths G and F, respectively, and to electrons, reads [63],
L = 1∂ a(cid:48)∂µa(cid:48) − αs (cid:32)(cid:88)nax C a(cid:48)i (cid:33)Gb G˜b,µν − α (cid:32)(cid:88)nax C a(cid:48)i (cid:33) F F˜µν + 1 (cid:32)(cid:88)nax C ∂µa(cid:48)i (cid:33) e¯γµγ e, (22)
2 µ i i 8π igf µν 8π iγf µν 2 ie f 5
a(cid:48) a(cid:48) a(cid:48)
i=1 i i=1 i i=1 i
where f are the decay constants of the axion-like fields a(cid:48). The anomaly coefficients, C and C , and the electron
a(cid:48) i ig iγ
i
coupling, C , are typically of order unity, see Secs. III and IV 3.
ie
The proper axion field A, that is the field which solves the strong CP problem, is the linear combination of the
axion-like fields in front of the GG˜ term in Eq. (22) [58],
A (cid:88)nax a(cid:48)
≡ C i . (23)
f igf
A a(cid:48)
i=1 i
3 Therecanbelargehierarchiesinthesecoefficientsinmulti-axionmodelsfromIIBstringfluxcompactifications[63].
7
Its particle excitation, the axion A, mixes with the pion, rendering it a pseudo Nambu-Goldstone boson, whose mass
has been given in Eq. (3). To this end, f has to be chosen such that the kinetic term of A is normalized canonically.
A
The remaining n −1 axion-like fields a , orthogonal to the axion A, are still massless Nambu-Goldstone bosons.
ax i
To be more explicit, let us specialize first to the phenomenologically well motivated two-axion model, n =2 (see
ax
theappendixofRef. [63]forafurtherexpositionofthemulti-axioncase)anddeferthediscussionofthemoregeneral
case to the end of this section. The properly normalized axion A and the additional ALP a are related to the original
axion-like fields a(cid:48) by
i
A a(cid:48) a(cid:48) a a(cid:48) a(cid:48)
=C 1 +C 2 , =−C 1 +C 2 , (24)
f 1gf 2gf f 2gf 1gf
A a(cid:48) a(cid:48) a a(cid:48) a(cid:48)
1 2 2 1
with normalization [66]
1 1 (cid:18)C (cid:19)2 (cid:18)C (cid:19)2
= = 1g + 2g . (25)
f2 f2 f f
A a a(cid:48)1 a(cid:48)2
The low-energy Lagrangian of this model, below the chiral symmetry breaking scale, is then
(cid:18) (cid:19)
L = (cid:88) 1∂ φ∂µφ− 1m2φ2− gφγ φF F˜µν + gφe ∂ φe¯γµγ e , (26)
2 µ 2 φ 4 µν 2m µ 5
e
φ=A,a
where
α (cid:32)(cid:18)f (cid:19)2 (cid:18)f (cid:19)2 2 4+z(cid:33)
g = A C C + A C C − , (27)
Aγ 2πf f 1g 1γ f 2g 2γ 3 1+z
A a(cid:48) a(cid:48)
1 2
m (cid:32)(cid:18)f (cid:19)2 (cid:18)f (cid:19)2 (cid:33)
g = e A C C + A C C , (28)
Ae f f 1g 1e f 2g 2e
A a(cid:48) a(cid:48)
1 2
α f
g = A (C C −C C ), (29)
aγ 2πf f 1g 2γ 1γ 2g
a(cid:48) a(cid:48)
1 2
m f
g = e A (C C −C C ). (30)
ae f f 1g 2e 1e 2g
a(cid:48) a(cid:48)
1 2
As in a single axion model [2, 8–11], the axion A has a universal, model-independent contribution to its coupling to
the photon (the last term in Eq. (27)) which arises from its mixing with the pion (see, e.g., appendix of Ref. [63]).
The parameters in the above expressions for the couplings are however redundant, because f is constrained by
A
Eq. (25). The physics is more obvious if one expresses the transformations in Eq. (24) in terms of mixing angles [65],
A=a(cid:48) cosδ+a(cid:48) sinδ, a=−a(cid:48) sinδ+a(cid:48) cosδ, (31)
1 2 1 2
with
cosδ =C fA, sinδ =C fA, tanδ = C2g fa(cid:48)1. (32)
1gf 2gf C f
a(cid:48) a(cid:48) 1g a(cid:48)
1 2 2
In terms of this parametrization, the couplings can then be written as
(cid:18) (cid:19)
α C C 2 4+z
g = 1γ cos2δ+ 2γ sin2δ− , (33)
Aγ 2πf C C 3 1+z
A 1g 2g
(cid:18) (cid:19)
m C C
g = e 1e cos2δ+ 2e sin2δ , (34)
Ae f C C
A 1g 2g
(cid:18) (cid:19) (cid:18) (cid:19)
α C C α C C tanδ
g = 2γ − 1γ sinδcosδ = 2γ − 1γ , (35)
aγ 2πf C C 2πf C C 1+tan2δ
A 2g 1g A 2g 1g
(cid:18) (cid:19) (cid:18) (cid:19)
m C C m C C tanδ
g = e 2e − 1e sinδcosδ = e 2e − 1e . (36)
ae f C C f C C 1+tan2δ
A 2g 1g A 2g 1g
Effectively, the couplings depend on the dimensionful parameter f , on the dimensionless ratios C /C , C /C ,
A 1γ 1g 2γ 2g
C /C , C /C , and on the mixing angle δ.
1e 1g 2e 2g
8
Apart from the model-independent contribution due to the mixing with the pion, the photon coupling of the axion
has a slight, order one, model dependence due to the ratio of the anomaly coefficients C /C , C /C , and the
1γ 1g 2γ 2g
mixing angle, cf. Eq. (33). On the other hand, the electron coupling of the axion A is very much model dependent:
a non-vanishing g requires that at least one of the original axion-like fields has a non-zero coupling to the electron,
Ae
C (cid:54)=0 for i=1 and/or 2, cf. Eq. (34).
ie
This strong model dependence is shared by the photon and electron couplings of the ALP a: here one needs that
at least one of the original axion-like fields has a non-zero coupling to the photon (electron), C (cid:54)=0 for i=1 and/or
iγ
2 (C (cid:54)= 0 for i = 1 and/or 2), otherwise g (g ) vanishes, cf. Eq. (35) (Eq. (36)). In this context it is also
ie aγ ae
important to mention that eventual explicit mass terms, m2 a(cid:48)2/2, for axion-like fields arising from breaking of the
a(cid:48) i
i
Peccei-Quinn symmetries, e.g. by higher dimensional operators suppressed by some high scale (see next section), will
have negligible effects on the photon and electron couplings of the axion A and the ALP a, as long as the masses are
much smaller than the dynamically generated axion mass, m (cid:28)m (cf., e.g., appendix of Ref. [63]).
a(cid:48) A
i
We conclude that in the case that the axion A and the ALP a consist of an appreciable mixture of the original
axion-like fields, i.e. as long as |tanδ| is of order unity, the couplings are all determined by f and are therefore
A
expected to be approximately of the same size for for the axion A and the ALP a, up to order one factors. A
hierarchical difference between the axion and the ALP coupling can possibly only arrive in the situation where the
axion (and correspondingly also the ALP) originates essentially only from one axion-like field, which occurs for δ ≈0
(π/2), meaning that cosδ ≈ 1 (sinδ = 1), i.e. that a(cid:48) (a(cid:48)) constitutes the axion. In fact, concentrating without loss
1 2
of generality on the former case, in which
f1A ≈ |Cfa1(cid:48)g|, |tanδ|=(cid:12)(cid:12)(cid:12)(cid:12)CC21gg ffaa(cid:48)1(cid:48) (cid:12)(cid:12)(cid:12)(cid:12)≈(cid:12)(cid:12)(cid:12)(cid:12)C2gffaA(cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:28)1, (37)
1 2 2
the sizes of the axion and ALP couplings appear to decouple from each other,
(cid:18) (cid:19) (cid:18) (cid:19)
α C 2 4+z αC C 2 4+z
g ≈ 1γ − ≈ 1g 1γ − , (38)
Aγ 2πf C 3 1+z 2πf C 3 1+z
A 1g a(cid:48) 1g
1
m C m
g ≈ e 1e ≈ e C , (39)
Ae f C f 1e
A 1g a(cid:48)
1
g ≈ α (cid:18)C2γ − C1γ(cid:19)C2g fa(cid:48)1 ≈ α (cid:18)C −C C2g(cid:19), (40)
aγ 2πf C C C f 2πf 2γ 1γ C
A 2g 1g 1g a(cid:48) a(cid:48) 1g
2 2
g ≈ me (cid:18)C2e − C1e(cid:19)C2g fa(cid:48)1 ≈ me (cid:18)C −C C2g(cid:19), (41)
ae f C C C f f 2e 1e C
A 2g 1g 1g a(cid:48) a(cid:48) 1g
2 2
the former (latter) being determined by f (f ). However, as long as C (cid:54)= 0, the ALP couplings still can not be
a(cid:48) a(cid:48) 2g
1 2
hierarchically larger than the axion couplings, since the above relations imply an upper bound on the ratio
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(cid:12)(cid:12)gaγ(cid:12)(cid:12)(cid:12)∼(cid:12)(cid:12)(cid:12)C2γ −C1γ CC21gg (cid:12)(cid:12)(cid:12) fA (cid:28)(cid:12)(cid:12)(cid:12) CC22γg − CC11γg (cid:12)(cid:12)(cid:12). (42)
(cid:12)gAγ(cid:12) (cid:12)(cid:12) CC11γg − 23 41++zz (cid:12)(cid:12)fa(cid:48)2 (cid:12)(cid:12)CC11γg −1.95(cid:12)(cid:12)
Atruehierarchyinthecouplings,inparticularanALP-photoncouplingmuchlargerthananaxion-photoncoupling,
is only possible if there is no mixing at all, C ≡0, but still C (cid:54)=0, i.e. if the ALP a(cid:48) is photophilic4. In this case
2g 2γ 2
we get
(cid:18) (cid:19)
α C 2 4+z
g = 1γ − , (43)
Aγ 2πf C 3 1+z
A 1g
m C
g = e 1e, (44)
Ae f C
A 1g
α
g = C , (45)
aγ 2πf 2γ
a(cid:48)
2
m
g = e C , (46)
ae f 2e
a(cid:48)
2
4 For this conclusion, we exclude the possibility of accidental cancellation between model dependent contributions and the universal
contributiontotheaxion-photoncouplinginEq.(27).
9
such that e.g. |g | can very well be in the range suggested by the ALP explanation of the cosmic γ-ray transparency
aγ
puzzle and be accessible to the next generation of laboratory experiments,
10−10 GeV−1 (cid:38)|g |(cid:38)10−12 GeV−1; m (cid:46)10−7 eV, (47)
aγ a
provided that f /|C is in the range,
a(cid:48) 2g
2
107GeV(cid:46)f /|C |(cid:46)109GeV, (48)
a(cid:48) 2γ
2
while at the same time |g | can be right in the cosmic axion window, provided that f =f /|C | is in the range
Aγ A a(cid:48) 1g
1
1013GeV(cid:38)f (cid:38)109GeV⇒10−7eV(cid:46)m (cid:46)1meV⇒10−16 GeV−1 (cid:46)|g |(cid:46)10−12 GeV−1. (49)
A A Aγ
Wethereforeconcludethatinmodelswithmultipleaxion-likefieldsitisofveryhighphenomenologicalrelevanceto
knowwhetherseveralofthemcoupletoGG˜ simultaneously. Thisquestionwillbeconsideredinthefollowingsections.
Let us end this section by noting that even further insight into the general relation between the axion and ALP
couplings can be obtained by purely geometrical means. To this end, we rewrite Eqs.(27) and (29) as
(cid:18) (cid:19)
α y·z α
g = y −C = z(yˆ·zˆ)−g(0).
Aγ 2π y2 0 2π Aγ
(50)
(cid:18) (cid:19)
α y z −y z α
g = 1 2 2 1 = z((cid:15) yˆzˆ ),
aγ 2π y 2π ij i j
where
(cid:115)
(cid:18)C C (cid:19) (cid:18)C (cid:19)2 (cid:18)C (cid:19)2 1
y=(y ,y )≡ 1g, 2g , yˆ =(yˆ ,yˆ )=y/y, y =|y|= 1g + 2g ≡ ,
1 2 f f 1 2 f f f
a(cid:48) a(cid:48) a(cid:48) a(cid:48) A
1 2 1 2 (51)
(cid:115)
(cid:18)C C (cid:19) (cid:18)C (cid:19)2 (cid:18)C (cid:19)2 1
z=(z ,z )≡ 1γ, 2γ , , zˆ=(zˆ ,zˆ )=z/z, z =|z|= 1γ + 2γ ≡ .
1 2 f f 1 2 f f f
a(cid:48) a(cid:48) a(cid:48) a(cid:48) γ
1 2 1 2
and
α α 24+z α
g(0) ≡ C = ≈ ×1.95. (52)
Aγ 2πf 0 2πf 31+z 2πf
A A A
Thus yˆ points into the direction of the axion, A(x) = yˆa(cid:48)(x), whereas zˆ points into the direction of an ALP that
i i
couples to photons (the orthogonal direction decouples from photons).
Plane geometry ensures us that we can write then
g =g cosΘ−g(0), g =g sinΘ, (53)
Aγ γmax Aγ aγ γmax
where
α
g ≡ , (54)
γmax 2πf
γ
and Θ is the angle between yˆ,zˆ, from yˆ to zˆ,
(yˆ·zˆ)=cosΘ, ((cid:15) yˆzˆ )=sinΘ. (55)
ij i j
This implies
(cid:16) (cid:17)2
g +g(0) +(g )2 =g2 , (56)
Aγ Aγ aγ γmax
see Fig.2. In particular, the ALP-photon coupling is constrained by
|g |≤g , (57)
aγ γmax
10
g
aγ
(−g(0),0) Θ
Aγ
g
Aγ
FIG.2: Theaxion-photoncouplingg andtheALP-photoncouplingg areconstrainedtolieonacirclewithorigin(−g(0),0)
Aγ aγ Aγ
and radius g , see Eq. (56). The case g(0) ≤g of Eq. (58) is depicted.
γmax Aγ γmax
while the axion-photon coupling satisfies
g(0)−g ≤|g |≤g(0)+g , for g(0) >g ;
Aγ γmax Aγ Aγ γmax Aγ γmax
(58)
0≤|g |≤g(0)+g , for g(0) ≤g .
Aγ Aγ γmax Aγ γmax
If we take the hierarchy f (cid:29)f , with order one coefficients C ,C (cid:54)=0, we realize that the vectors in Eqs.(51)
a(cid:48) a(cid:48) ig iγ
1 2
are both approximately aligned along (0,±1). This situation leads to Θ=0 or π and hence to
|g |≈|g(0)∓g |, |g |≈0. (59)
Aγ Aγ γmax aγ
Then f ∼f and |g |(cid:28)|g |, barring accidental cancellations.
A γ aγ Aγ
Theoppositehierarchy,|g |(cid:29)|g |,ismoredifficulttobeobtainedwithoutfinetuningandrequiresg(0) ≤g ,
aγ Aγ Aγ γmax
or, equivalently, f ≤ f /C (cid:39) f /1.95. Analyzing the various possibilities, the only way we can achieve such a
γ A 0 A
hierarchy from hierarchical scales f is to require one of the axion-like fields to be photophilic.
a(cid:48)
i
Notice that Eq.(56) can be directly generalized for the case of n >1 ALPs, besides the axion. The constraint
ALP
generalizes to
(cid:16)g +g(0)(cid:17)2+(cid:88)nax (g )2 =g2 , (60)
Aγ Aγ aiγ γmax
i=2
where Eq.(54) still defines g with the generalization
γmax
(cid:118)
1 ≡(cid:117)(cid:117)(cid:116)(cid:88)nax (cid:32)Ciγ(cid:33)2, (61)
f f
γ a(cid:48)
i=0 i
where n =n +1 is the number of axion-like fields. The couplings in this case are given by
ax ALP
α (cid:16) y (cid:17)
g = z zˆ·yˆ− C .
Aγ 2π z 0
(62)
α
g = z(zˆ·u ),
aiγ 2π i
where u are orthonormal basis vectors spanning the space orthogonal to yˆ.
i
An analogous constraint for the electron couplings can be obtained by introducing the angle Θ between the vector
e
y in (51), pointing in the axion direction, and the vector
(cid:18) (cid:19)
C C
z ≡ 1e, 2e , (63)
e f f
a(cid:48) a(cid:48)
1 2