arXiv:1110.2491v2 [hep-ph] 28 Dec 2011
Preprint typeset in JHEP style - HYPER VERSION
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: email@example.com, firstname.lastname@example.org, email@example.com
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
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
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. The first of these is solved
by the introduction of softly broken weak scale supersymmetry (SUSY) into the theory
(which receives some indirect support from the measured values of gauge couplings at
LEP and from global fits to precision electroweak data), while the second problem
is solved by the introduction of a global U(1)PQPeccei-Quinn (PQ) symmetry broken by
QCD anomalies, which requires the existence of an (“invisible”) axion[6, 7], with mass
expected in the micro-eV or below range. 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-
2axino ˜ a(x), in addition to the usual pseudoscalar axion field a(x):
ˆ a =s + ia
+ i√2¯θ˜ aL+ i¯θθLFa,(1.2)
in 4-component spinor notation.
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. The axion, saxion and axino couplings
to matter depend on the PQ breaking scale fa1, which is required fa? 109GeV by stellar
cooling calculations. 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), whilst right-hand neutrino states are expected to exist near the GUT
scale according to the elegant see-saw mechanism for neutrino mass. 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 to match the
gravitino?G or possibly the superpartner of a right-handed neutrino. 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. .
– 1 –
WMAP-measured cold DM abundance:
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.
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, giving rise to mixed axion/axino
(a˜ a) CDM. In supergravity theories however, the axino mass is expected to lie at the
weak scale, 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. Detailed calculations of the relic abundance of mixed a?Z1
The standard calculation of the neutralino Yield Ystd
tralino number density and s is the entropy density) gives
where g∗(Tfr) is the number of active degrees of freedom at temperature T = Tfr, where
In a recent paper, Choi et al. presented a semi-analytic approach for estimating
to cases where the thermally averaged neutralino annihilation cross section times relative
CDM were performed in Ref. , where formulae for the neutralino and axion abundances
(where n? Z1is the neu-
fr= m? Z1/ln[
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]
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
Γ˜ aMP/?π2g∗(T˜ a
– 2 –
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  and  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  and , 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), followed by late time saxion decay, may inject considerable entropy
into the early universe, thus diluting all relics present at the saxion decay temper-
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
• The treatments of  and  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  and  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
– 3 –
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.  and , 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.  and .
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 multiplets3, 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. 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:
˙ ni+ 3Hni= Si−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
3Here, we assume standard rates for thermal axino production as calculated in the literature[21, 29, 30].
In Ref. , 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.
– 4 –
For obtaining the various unstable particle widths, we calculate Γ˜ afrom the ˜ a → ˜ gg,
?Ziγ and?ZiZ partial widths as presented in Ref. . For gravitino decays, we adopt the
s → gg and s → ˜ g˜ g decays as presented in Ref. . We note here that in the DFSZ
model, it is also possible to have s → hh decays and possibly s → aa decays. We neglect
these latter two cases, so that our results apply to the supersymmetrized KSVZ model,
where the gg and ˜ g˜ g final states should dominate.
Once the total and partial widths are known, we can easily compute the required
gravitino widths as presented in Ref. . For the saxion width, we include Γsfrom the
BR(˜ a,?Z1) = 1, BR(s,?Z1) = 2 ×Γ(s → ˜ g˜ g)
The factor 2 in BR(s,?Z1) takes care of the multiplicity of neutralinos for each saxion
that s → ˜ g˜ g plays a crucial role in the PQMSSM dark matter cosmology.
Finally, we assume that the branching ratios for computing the energy injection into
the thermal bath from unstable particle decays are given by:
, BR(?G,?Z1) = 1(A.15)
cascade decay. While the s → gg decay width is always dominant, we showed in Sec. 3
BR(˜ a,X) = BR(s,X) = BR(?G,X) = 1.(A.16)
Although some of the decay energy is lost into neutralinos (except for s → gg decays),
we assume that in the final product of the cascade decay of axinos, saxions and gravitinos
most of the initial energy has been converted into radiation, so Eq. A.16 consists in a good
 For a review, see R. D. Peccei, Lect. Notes Phys. 741 (2008) 3 [arXiv:hep-ph/0607268].
 H. Baer and X. Tata, Weak Scale Supersymmetry: From Superfields to Scattering Events,
(Cambridge University Press, 2006).
 S. Dimopoulos, S. Raby and F. Wilczek, Phys. Rev. D 24 (1981) 1681; U. Amaldi, W. de
Boer and H. Furstenau, Phys. Lett. B 260 (1991) 447; J. Ellis, S. Kelley and D. V.
Nanopoulos, Phys. Lett. B 260 (1991) 131; P. Langacker and Luo, Phys. Rev. D 44 (1991)
 S. Heinemeyer, W. Hollik, D. Stockinger, A. M. Weber and G. Weiglein, J. High Energy
Phys. 0608 (2006) 052.
 R. Peccei and H. Quinn, Phys. Rev. Lett. 38 (1977) 1440 and Phys. Rev. D 16 (1977) 1791;
S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 40 (1978) 279.
 J. E. Kim, Phys. Rev. Lett. 43 (1979) 103; M. A. Shifman, A. Vainstein and V. I. Zakharov,
Nucl. Phys. B 166 (1980) 493.
 M. Dine, W. Fischler and M. Srednicki, Phys. Lett. B 104 (1981) 199; A. P. Zhitnitskii, Sov.
J. Nucl. 31 (1980) 260.
– 23 –
 For a recent review, see P. Sikivie, hep-ph/0509198; M. Turner, Phys. Rept. 197 (1990) 67; J.
E. Kim, Phys. Rept. 150 (1987) 1; J. E. Kim and G. Carosi, Rev. Mod. Phys. 82 (2010) 557.
 For recent reviews of axino dark matter, see F. Steffen, Eur. Phys. J. C 59 (2009) 557; L.
Covi and J. E. Kim, New J. Phys. 11 (2009) 105003.
 D. Dicus, E. Kolb, V. Teplitz and R. Wagoner, Phys. Rev. D 18 (1978) 1829 and Phys. Rev.
D 22 (1980) 839; for a review, see G. Raffeldt, hep-ph/0611350.
 L. F. Abbott and P. Sikivie, Phys. Lett. B 120 (1983) 133; J. Preskill, M. Wise and F.
Wilczek, Phys. Lett. B 120 (1983) 127; M. Dine and W. Fischler, Phys. Lett. B 120 (1983)
137; M. Turner, Phys. Rev. D 33 (1986) 889.
 P. Sikivie, Phys. Lett. B 695 (2011) 22; O. Erken, P. Sikivie, H. Tam and Q. Yang,
 E. J. Chun, H. B. Kim and D. H. Lyth, Phys. Rev. D 62 (2000) 125001; E. J. Chun,
H. B. Kim, K. Kohri and D. H. Lyth, J. High Energy Phys. 0803 (2008) 061; S. Kim,
W. I. Park and E. D. Stewart, J. High Energy Phys. 0807 (2009) 015.
 T. Asaka, K. Ishiwata and T. Moroi, Phys. Rev. D 73 (2006) 051301 and Phys. Rev. D 75
 M. Kawasaki, K. Kohri, T. Moroi and A. Yotsuyanagi, Phys. Rev. D 78 (2008) 065011.
 M. Gell-Mann, P. Ramond and R. Slansky, in Supergravity, Proceedings of the Workshop,
Stony Brook, NY 1979 (North-Holland, Amsterdam); T. Yanagida, KEK Report No. 79-18,
1979; R. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912.
 J. Ellis and K. Olive, Phys. Lett. B 514 (2001) 114; H. Baer and A. Box, Eur. Phys. J. C 68
 E. Komatsu et al. (WMAP collaboration), arXiv:1001.4538 (2010).
 H. Baer, A. Box and H. Summy, J. High Energy Phys. 1010 (2010) 023.
 K. Rajagopal, M. Turner and F. Wilczek, Nucl. Phys. B 358 (1991) 447.
 L. Covi, J. E. Kim and L. Roszkowski, Phys. Rev. Lett. 82 (1999) 4180; L. Covi, H. B. Kim,
J. E. Kim and L. Roszkowski, J. High Energy Phys. 0105 (2001) 033.
 H. Baer, A. Box and H. Summy, J. High Energy Phys. 0908 (2009) 080.
 P. Moxhay and K. Yamamoto, Phys. Lett. B 151 (1985) 363; E. Chun and A. Lukas, Phys.
Lett. B 357 (1995) 43.
 K-Y. Choi, J. E. Kim, H. M. Lee and O. Seto, Phys. Rev. D 77 (2008) 123501.
 For a review, see G. Jungman, K. Griest, M. Kamionkowski, Phys. Rept. 267 (1996) 195.
 H. Baer, A. Lessa, S. Rajagopalan and W. Sreethawong, JCAP1106 (2011) 031.
 M. Turner, Phys. Rev. D 28 (1983) 1243.
 K. J. Bae, K. Choi and S. H. Im, J. High Energy Phys. 1108 (2011) 065.
 P. Graf and F. Steffen, Phys. Rev. D 83 (2011) 070511.
 A. Strumia, J. High Energy Phys. 1006 (2010) 036.
 P. Svrcek and E. Witten, J. High Energy Phys. 0606 (2006) 051.
– 24 –
 E. Kolb and M. Turner, The Early Universe, Addison-Wesley Pub. (1990).
 K. Kohri, T. Moroi and A. Yotsuyanagi, Phys. Rev. D 73 (2006) 123511.
 R. H. Cyburt, J. Ellis, B. D. Fields and K. A. Olive, Phys. Rev. D 67 (2003) 103521; R. H.
Cyburt, J. Ellis, B. D. Fields, F. Luo, K. Olive and V. Spanos, JCAP0910 (2009) 021.
 M. Kawasaki, K. Kohri and T. Moroi, Phys. Lett. B 625 (2005) 7 and Phys. Rev. D 71
(2005) 083502; K. Kohri, T. Moroi and A. Yotsuyanagi, Phys. Rev. D 73 (2006) 123511; for
an update, see M. Kawasaki, K. Kohri, T. Moroi and A. Yotsuyanagi, Phys. Rev. D 78
 K. Jedamzik, Phys. Rev. D 70 (2004) 063524 and Phys. Rev. D 74 (2006) 103509.
 ISAJET, by H. Baer, F. Paige, S. Protopopescu and X. Tata, hep-ph/0312045; see also
H. Baer, J. Ferrandis, S. Kraml and W. Porod, Phys. Rev. D 73 (2006) 015010.
 H. Baer, C. Balazs and A.Belyaev, J. High Energy Phys. 0203 (2002) 042.
 L. Visinelli and P. Gondolo, Phys. Rev. D 80 (2009) 035024.
 M. Drees and M. Nojiri, Phys. Rev. D 47 (1993) 376; H. Baer and M. Brhlik, Phys. Rev. D
57 (1998) 567; H. Baer, M. Brhlik, M. Diaz, J. Ferrandis, P. Mercadante, P. Quintana and
X. Tata, Phys. Rev. D 63 (2001) 015007; J. Ellis, T. Falk, G. Ganis, K. Olive and
M. Srednicki, Phys. Lett. B 510 (2001) 236; L. Roszkowski, R. Ruiz de Austri and T. Nihei,
J. High Energy Phys. 0108 (2001) 024; A. Djouadi, M. Drees and J. L. Kneur, J. High
Energy Phys. 0108 (2001) 055; A. Lahanas and V. Spanos, Eur. Phys. J. C 23 (2002) 185.
 L. Randall and R. Sundrum, Nucl. Phys. B 557 (1999) 79; G. Giudice, M. Luty, H.
Murayama and R. Rattazzi, J. High Energy Phys. 12 (1998) 027.
 H. Baer, S. de Alwis, K. Givens, S. Rajagopalan and H. Summy, J. High Energy Phys. 1005
 G. Lazarides, C. Panagiotakapolous and Q. Shafi, Phys. Lett. B 192 (1987) 323;G. Lazarides,
R. Schaefer, D. Seckel and Q. Shafi, Nucl. Phys. B 346 (1990) 193;J. McDonald, Phys. Rev.
D 43 (1991) 1063; C. Pallis, Astropart. Phys. 21 (2004) 689.
 J. E. Kim, Phys. Rev. Lett. 67 (1991) 3465.
 M. Kawasaki, T. Moroi and T. Yanagida, Phys. Lett. B 383 (1996) 313.
 K. Choi, E. J. Chun and J. E. Kim, Phys. Lett. B 403 (1997) 209.
 T. Banks, M. Dine and M. Graesser, Phys. Rev. D 68 (2003) 075011.
 P. Fox, A. Pierce and S. Thomas, hep-th/0409059 (2004).
 B. Acharya, K. Bobkov and P. Kumar, J. High Energy Phys. 1011 (2010) 105.
 J. Hasenkamp and J. Kersten, Phys. Rev. D 82 (2010) 115029.
 M. Kawasaki, N. Kitajima and K. Nakayama, Phys. Rev. D 83 (2011) 123521.
 H. Baer and A. Lessa, J. High Energy Phys. 1106 (2011) 027.
 H. Baer, S. Kraml, A. Lessa and S. Sekmen, JCAP1104 (2011) 039.
 M. Kawasaki, K. Nakayama and M. Senami, JCAP0803 (2008) 009.
– 25 –
 K. L. Chan, U. Chattopadhyay and P. Nath, Phys. Rev. D 58 (1998) 096004; J. Feng,
K. Matchev and T. Moroi, Phys. Rev. Lett. 84 (2000) 2322 and Phys. Rev. D 61 (2000)
075005; J. Feng, K. Matchev and F. Wilczek, Phys. Lett. B 482 (2000) 388; J. Feng and F.
Wilczek, Phys. Lett. B 631 (2005) 170; see also H. Baer, C. H. Chen, F. Paige and X. Tata,
Phys. Rev. D 52 (1995) 2746 and Phys. Rev. D 53 (1996) 6241; H. Baer, C. H. Chen,
M. Drees, F. Paige and X. Tata, Phys. Rev. D 59 (1999) 055014; for a model-independent
approach, see H. Baer, T. Krupovnickas, S. Profumo and P. Ullio, J. High Energy Phys.
0510 (2005) 020.
 H. Baer and J. O’Farrill, JCAP 0404 (2004) 005; H. Baer, A. Belyaev, T. Krupovnickas and
J. O’Farrill, JCAP 0408 (2004) 005.
 T. Moroi and L. Randall, Nucl. Phys. B 570 (2000) 455.
 G. Gelmini and P. Gondolo, Phys. Rev. D 74 (2006) 023510; G. Gelmini, P. Gondolo, A.
Soldatenko and C. Yaguna, Phys. Rev. D 74 (2006) 083514; G. Gelmini, P. Gondolo, A.
Soldatenko and C. Yaguna, Phys. Rev. D 76 (2007) 015010.
 B. Acharya, K. Bobkov, G. Kane, P. Kumar and J. Shao, Phys. Rev. D 76 (2007) 126010
and Phys. Rev. D 78 (2008) 065038; B. Acharya, P. Kumar, K. Bobkov, G. Kane, J. Shao
and S. Watson, J. High Energy Phys. 0806 (2008) 064.
 H. Baer, R. Dermisek, S. Rajagopalan and H. Summy, JCAP1007 (2010) 014.
 K. Y. Choi, L. Covi, J. E. Kim and L. Roszkowski, arXiv:1108.2282 (2011).
 J. Pradler and F. Steffen, Phys. Lett. B 648 (2007) 224; see also M. Bolz, A. Brandenburg
and W. Buchmuller, Nucl. Phys. B 606 (2001) 518; J. Pradler and F. Steffen, Phys. Rev. D
75 (2007) 023509; V. S. Rychkov and A. Strumia, Phys. Rev. D 75 (2007) 075011.
– 26 –