June 2011

Mirror & hidden sector dark matter

in the light of new CoGeNT data

R. Foot^{1}^{1}1
E-mail address: rfoot@unimelb.edu.au

ARC Centre of Excellence for Particle Physics at the Terascale,

School of Physics, University of Melbourne,

Victoria 3010 Australia

The CoGeNT collaboration has recently made available new
data collected over a period of 15 months. In addition to
more accurately measuring the spectrum of nuclear recoil candidate events they
have announced evidence for an annual modulation signal.
We examine the implications of these new results
within the context of mirror/hidden sector dark matter models. We find that the new CoGeNT
data can be explained within this framework with parameter space consistent with
the DAMA annual modulation signal, and the null results of the other experiments.
We also point out that the CoGeNT spectrum at low energies is observed to obey
which suggests that dark matter interacts via
Rutherford scattering
rather than the more commonly assumed contact (four-fermion) interaction.

The CoGeNT experiment operating in the Soudan Underground Laboratory has been searching for light dark matter interactions with a low energy threshold P-type Point Contact germanium detector[1, 2]. They have obtained a low energy spectrum which is not readily explainable in terms of known background sources, and is consistent with elastic scattering of light dark matter particles[3, 4, 5]. Recently, 15 months of data has been analyzed[2] which greatly improves the measurement of the low energy spectrum and appears to be annually modulated with a phase consistent with dark matter expectations[6]. This further strengthens the dark matter interpretation of the CoGeNT low energy spectrum.

The recent CoGeNT results also reinforce the long standing observation of the DAMA collaboration of an annual modulation signal in their detector[7, 8]. The DAMA signal is extremely impressive with statistical significance of sigma with phase and period in agreement with the dark matter expectations to high accuracy. Attempts to interpret the DAMA signal in terms of a hypothetical background have become more and more implausible, and there is ample reason to be confident that DAMA and now CoGeNT have observed dark matter.

The positive results of DAMA and CoGeNT together with the null results of very sensitive, but higher threshold experiments such as CDMS[9] and XENON100[10] suggest that dark matter is light ( GeV). Having dark matter light, has long been known to alleviate the tension between DAMA and the higher threshold experiments[11] (see also ref.[12]). However, the sensitivity of the higher threshold experiments has got to the point where there is now some tension between DAMA/CoGeNT and e.g. XENON10. XENON100, CDMS/Si, CDMS/Ge when interpreted in terms of standard WIMPs even if they are light[13]. Another issue is that the allowed parameter regions for the DAMA and CoGeNT signals, although close together do not significantly overlap in the standard WIMP framework[5]. It turns out that both of these difficulties can be resolved if dark matter is not only light, but assumed to be multi-component and self interacting.

A generic example of this[14] is dark matter from a hidden sector, which contains an unbroken gauge interaction which is mixed with the standard via renormalizable kinetic mixing interaction:[15]

(1) |

where is the standard gauge boson field strength tensor, and is the field strength tensor for the hidden sector . This interaction enables hidden sector charged particles (of charge ) to couple to ordinary photons with electric charge . We consider the case where the hidden sector contains two (or more) stable charged dark matter particles, and with masses and [ and can be fermionic or bosonic]. Under the standard assumptions of a dark halo forming an isothermal sphere the condition of hydrostatic equilibrium relates the temperature of the particles to the galactic rotational velocity, :

(2) |

where is the mean mass of the particles in the galactic halo. We have assumed that the self interactions mediated by the unbroken gauge interactions are sufficiently strong so that they thermalize the hidden sector particles, and . The interaction length is typically much less than a parsec[16] and the dark matter particles form a pressure-supported halo. The dark matter particles are then described by a Maxwellian distribution with where

(3) |

With the assumptions that and that the abundance of is much less than , we have that . The narrow velocity dispersion (recall ) can greatly reduce the rate of dark matter interactions in higher threshold experiments such as XENON100 whilst still explaining the signals in the lower threshold DAMA and CoGeNT experiments.

While generic hidden sector models are interesting in their own right and have been
studied in some detail (see e.g. ref.[17]), I consider
mirror dark matter as the best motivated example of such a multi-component
self-interacting theory.
Recall, mirror dark matter posits that the inferred dark matter in the Universe arises from
a hidden sector which is an exact copy of the standard model sector[18] (for a review
and more complete list of references see ref.[19])^{2}^{2}2
Note that successful big bang nucleosynthesis and successful
large scale structure requires effectively asymmetric initial
conditions in the early Universe, and .
See ref.[20] for further discussions..
That is,
a spectrum of dark matter particles of known masses are predicted: (with
etc).
The galactic halo is then presumed to be composed predominately of a spherically distributed
self interacting mirror particle plasma comprising these particles[21].
Kinetic mixing of the and its mirror counterpart
allows ordinary and mirror particles to interact with each other and can thereby
explain the direct detection experiments[3, 4, 11].
The simplest scenario involves kinetic mixing induced
elastic (Rutherford) scattering of the dominant mirror metal component, , off target nuclei.
[The and components are too light to give a signal above the DAMA/CoGeNT energy threshold].
Previous work[3, 4] (see also ref.[11, 14]) has shown that such
elastic scattering can explain the
normalization and energy dependence of the DAMA annual
modulation amplitude and also the initial (56 day) CoGeNT spectrum
consistently with the null results of the other experiments,
and yields a measurement of and :

(4) |

where is the halo mass fraction of the species and is the proton mass. The measured value of is consistent with , which by analogy with the ordinary matter sector would be the naive expectation. Taking a range for , , suggests that could realistically range from to . Kinetic mixing in this range is consistent with laboratory and astrophysical constraints[22] and has a number of fascinating applications[22, 19, 23]. Early Universe cosmology, though, prefers[24] .

The interaction rate in experiments depends on the halo distribution function
and the interaction cross-section. The former is expected to be a Maxwellian distribution,
, with depending on , as discussed above. In the mirror
dark matter case, is expected to be around 1 GeV, but with significant uncertainties[3, 4].
Generally it has been found[3] that the dark matter detection experiments are relatively insensitive
to the precise value of (and hence ) so long as .
Kinetic mixing induced elastic Rutherford scattering is particularly natural in mirror/hidden sector models
as it arises from the renormalizable interaction, Eq.(1). In the present study
we again assume that it is the dominant interaction mechanism coupling ordinary and dark matter.
The cross-section for a dark matter particle of charge
to elastically scatter off an ordinary nucleus (presumed at rest with mass
and atomic numbers ) is given by[11]:^{3}^{3}3We employ natural units where
.

(5) |

where

(6) |

and is the form factor which takes into account the finite size of the nuclei. In the case where dark matter particles also have finite size, as in the mirror dark matter case, a form factor for those particles also needs to be included. [For elastic scattering of mirror nuclei, , of atomic number we must replace in the above cross-section formula]. A simple analytic expression for the form factor, which we adopt in our numerical work, is the one proposed by Helm[25, 26].

The event rate is given by:

(7) |

where is the number of target nuclei per kg of detector and is the number density of halo dark matter particles at the Earth’s location (we take ). Here is the velocity of the halo particles relative to the Earth and is the velocity of the Earth relative to the galactic halo. The integration limit, , is given by the kinematic relation:

(8) |

The halo distribution function in the reference frame of the Earth is given by, . The integral, Eq.(7), can easily be evaluated in terms of error functions[14, 26] and numerically solved.

To compare with the measured event rate, we must include detector resolution effects and overall detection efficiency (when the latter is not already included in the experimental results):

(9) |

where is the measured energy and describes the resolution. The measured energy is typically in keVee units (ionization/scintillation energy). For nuclear recoils in the absence of any channeling, , where is the relevant quenching factor. Channeled events, where target atoms travel down crystal axis and planes, have . In light of recent theoretical studies[27], we assume that the channeling fraction is negligible. It is of course still possible that channeling could play some role, which could modify the favoured regions of parameter space somewhat.

For this study we consider two of the simplest examples of multi-component
dark matter models. Following our earlier works[3, 4, 14, 11]
we consider mirror dark matter with a
dominant mirror metal component, , of
atomic number . In this case the electric charge of the dark matter particle is .
^{4}^{4}4
In our numerical work we allow
, to have non-integer values,
with . Since the realistic case will involve a spectrum of
elements, the effective mass can be non-integer.
The quantity is obtained from Eq.(3) with
GeV, which corresponds to a dominated halo,
, expected[28] for .
We also consider the more generic two component hidden sector dark matter model discussed above,
in which case is less constrained.

One can define a quantity and compare these theories with experiment.
We consider the reference point km/s which is
representative of recent measurements for the local rotational velocity[29].
The data we consider consists of (a) the CoGeNT energy spectrum: 31 bins of
width keVee given in the inset of figure 1 of ref.[2]. This spectrum
has already been corrected for efficiency and stripped of background components.
(b) The DAMA annual modulation energy spectrum in the energy range .
We have taken into account systematic uncertainties in energy scale by minimizing
over a variation in quenching factors, i.e.
for DAMA and
for CoGeNT.
The mirror dark matter candidate provides an excellent fit to the data,
with values of 23.1/29 for data set (a),
and 8.9/10 for data set (b).^{5}^{5}5
The low CoGeNT threshold of keVee potentially makes the experiment sensitive
to the component
via -electron scattering, which would be expected to lead
to a large rise in event rate at low energies[30].
The data is adequately fit by elastic scattering, with no evidence for an extra
scattering contribution. This
suggests that the halo component has a lower temperature than
the mirror nuclei component. Such a scenario is possible due to the inefficient energy transfer between
the light and much heavier mirror nuclei.
Favoured regions in the
plane can be obtained by evaluating contours corresponding to
(roughly C.L. allowed region).
In figure 1 we show the parameter regions favoured by
the data for the km/s reference point.
The favoured regions for the DAMA and CoGeNT signals are in as good an agreement
as one might expect given the
systematic uncertainties which we have not considered including the
fiducial bulk volume uncertainty in CoGeNT of and variation of within
its estimated uncertainty.

Figure 1: CoGeNT and DAMA C.L. favoured parameter () regions for the mirror dark matter model. The reference point km/s is assumed. Also shown are the exclusion curves evaluated from the null results of XENON100, CDMS/Si and CDMS/Ge.

Also displayed in figure 1 is the exclusion limits evaluated for the CDMS/Si[31],
CDMS/Ge[9] and XENON100[10] experiments^{6}^{6}6
There are also lower threshold analysis by XENON10[32] and CDMS[33]
collaborations. However when
systematic uncertainties are properly incorporated, neither analysis is capable of
excluding light dark matter explanations of the DAMA/CoGeNT signal[34]..
In computing these limits, we have conservatively
taken the energy thresholds of these experiments to be higher than the advertised values, to
allow for systematic uncertainties in energy calibration and quenching factor
^{7}^{7}7Within the mirror dark matter framework the higher threshold experiments such as CDMS/Ge
and XENON100 have an important role in probing the heavier component[35]..
We also show in figure 2a,b, the predicted results for each
data set for a particular parameter point near the global best fit, as well as a point
near the best fit for each data set considered separately.

Figure 2a: Mirror dark matter versus the CoGeNT spectrum. The solid line is for a point near the CoGeNT best fit [] while the dashed line is for a point near the global best fit [].

Figure 2b: Mirror dark matter versus the DAMA annual modulation spectrum. The solid line is for a point near the DAMA best fit [] while the dashed line is for the point near the global best fit considered in figure 2a [].

It is interesting to compare the 15 month CoGeNT favoured region, as shown in figure 1, with results for the same model obtained with the initial 56 days of data[3, 4]. The current favoured region is significantly reduced in size. CoGeNT data now feature an upper limit on GeV, which is also supported by the null results of XENON100 and CDMS/Si.

The CoGeNT collaboration report[2] evidence for an annual modulation signal in their data at about C.L. The amplitude of the modulation, averaged over , is measured to be roughly cpd/kg/keVee. This assumes the amplitude and phase are set to theoretical expectations, while a larger amplitude is preferred if the phase is left free. For the theories offered here, we find that the CoGeNT annual modulation amplitude (averaged over ) is typically around cpd/kg/keVee for the parameter region near the global best fit, and does not get above for any parameter point in the global C.L. favoured parameter region (for the reference point km/s). Thus we find an annual modulation somewhat below the CoGeNT central value. This difference, though, is not currently, statistically significant, representing only a 1.5-2 sigma downward fluctuation from the central value measured in the month-kg data sample. Obviously future data, especially the measurement of the energy spectrum of the annual modulation amplitude, will be important tests of the theories considered here.

Similar results hold for the more generic two component hidden sector dark matter model discussed earlier. For definiteness we have assumed the same value for (i.e. same value) as for in mirror dark matter. We have computed the as before, minimizing over systematic uncertainties in quenching factor. The best fit features values of 23.0/29 for the CoGeNT data set (a), and 9.2/10 for the DAMA data set (b). The parameter range favoured by the CoGeNT and DAMA data sets (a) and (b) discussed above is given in figure 3 for this case. Note that since the electric charge of is rather than the allowed region is shifted c.f. the mirror matter case: . We have also found that the model can fit the data for a wide range of values: .

The explanation of the DAMA/LIBRA and CoGeNT experiments considered here has a number of interesting features. As noted previously[4] the signals seen in these experiments arise predominately from dark matter particle interactions in the body of their Maxwellian velocity distribution rather than the tail (as in the model of ref.[12, 5]). Because of this, we do not have a great deal of freedom in modifying the predicted shape of the spectrum, and thus the agreement of the model with the spectrum observed by CoGeNT is a non-trivial test of the theory. In fact in the limit, the energy dependence of [Eq.(7)] follows exactly that of and is proportional to for and for . [Excepting here the energy dependence of the form factor which is relatively minor for keVee in germanium]. The dependence of follows directly from the masslessness of the exchanged photon in the Feynman diagram describing the interaction and is thus a distinctive feature of dark matter interacting via Rutherford scattering. For finite the behaviour is expected provided that is sufficiently small that , i.e. for

(10) |

For , km/s and GeV (the latter suggested by the fit to the DAMA annual modulation signal), we have

(11) |

for keVee. This prediction is impressively consistent with the observations as indicated in figure 4. CoGeNT’s spectrum falls off more rapidly than at keVee. This suggests the onset of the kinematic threshold at these energies and is the origin of the GeV upper limit indicated in figures 1,3.

Figure 3: CoGeNT and DAMA C.L. favoured parameter () regions for the generic hidden sector dark matter model discussed in the text. The reference point km/s is assumed. Also shown are the exclusion curves evaluated from the null results of the XENON100, CDMS/Si and CDMS/Ge experiments.

The dark matter explanation of CoGeNT’s spectrum offered here can be compared with the model of ref.[12, 5] which features WIMPS elastically scattering via a contact (four fermion) interaction rather than via Rutherford scattering. The contact interaction produces a flat (in ) cross-section, excepting the mild (at these energies) recoil energy dependence of the form factor. A rapidly falling spectrum would then only be expected if dark matter particles are so light that only particles in the tail of the halo velocity distribution can lead to recoils with enough energy to be observed. In such a scenario the shape of the spectrum necessarily depends very sensitively on . Only for GeV (and with standard assumptions)[2] can that model account for the observed spectrum energy dependence at low . However the energy dependence is accommodated, rather than explained, which is in contrast to the Rutherford scattering scenarios considered here.

Figure 4: Low energy CoGeNT spectrum compared with where (dashed line), (solid line) and (dotted line). The data clearly favour the case, which is expected in the mirror dark matter/hidden sector models considered here, and is characteristic of dark matter interacting via elastic Rutherford scattering.

Future experiments should be able to more clearly distinguish mirror/hidden sector models from other theoretical explanations - such as the one discussed in ref.[12, 5] and many others considered in recent literature - by e.g. precise measurements of the annual modulation energy spectrum. The mirror/hidden sector models predict a characteristic change in sign of the annual modulation amplitude at low energies (see figure 6 of ref.[4]). Distinguishing the mirror dark matter case from the more generic hidden sector model might prove more challenging. Whilst the two theories give essentially identical results for the DAMA/CoGeNT experiments mirror dark matter predicts a spectrum of particles of known masses. In particular the scattering of low mass components, on electrons and on target nuclei can ultimately be seen in very low threshold experiments. Higher mass sub-components, such as a or component would also be expected and should ultimately be observed if dark matter is of the mirror type.

In conclusion, we have examined mirror/hidden sector dark matter in the light of CoGeNT’s more precisely measured spectrum and annual modulation signal[2]. The CoGeNT spectrum is observed to obey at low energies which suggests that dark matter interacts via a massless or light mediator (Rutherford scattering) rather than the more commonly assumed contact (four-fermion) interaction. Such Rutherford scattering is a feature of mirror and more generic hidden sector dark matter models considered here and in previous works[3, 4, 14, 11]. We have found that such models provide an excellent fit to the data which is easily consistent with the null results of the sensitive but higher threshold experiments, such as CDMS and XENON100. The next generation P-type Point Contact detectors, including CoGeNT(C-4), MAJORANA, GERDA and CDEX should be able to provide a decisive test of these models by e.g. a precise measurement of the annual modulation energy spectrum. These and other experiments are awaited with interest.

Acknowledgments

This work was supported by the Australian Research Council.

## References

- [1] C. E. Aalseth et al. (CoGeNT Collaboration), Phys. Rev. Lett. 106: 131301 (2011) [arXiv:1002.4703].
- [2] C. E. Aalseth et al. (CoGeNT Collaboration), arXiv: 1106.0650.
- [3] R. Foot, Phys. Lett. B692: 65 (2010) [arXiv: 1004.1424].
- [4] R. Foot, Phys. Rev. D82: 095001 (2010) [arXiv: 1008.0685].
- [5] D. Hooper, J. I. Collar, J. Hall and D. McKinsey, Phys. Rev. D82: 123509 (2010) [arXiv: 1007.1005].
- [6] A. K. Drukier, K. Freese and D. N. Spergel, Phys. Rev. D33, 3495 (1986); K. Freese, J. A. Frieman and A. Gould, Phys. Rev. D37, 3388 (1988).
- [7] R. Bernabei et al. (DAMA Collaboration), Riv. Nuovo Cimento. 26, 1 (2003) [astro-ph/0307403]; Int. J. Mod. Phys. D13, 2127 (2004); Phys. Lett. B480, 23 (2000).
- [8] R. Bernabei et al. (DAMA Collaboration), Eur. Phys. J. C56: 333 (2008) [arXiv:0804.2741]; Eur. Phys. J. C67, 39 (2010) [arXiv: 1002.1028].
- [9] Z. Ahmed et al (CDMS Collaboration), Science 327: 1619 (2010) [arXiv: 0912.3592].
- [10] E. Aprile et al. (XENON100 Collaboration), arXiv: 1104.3121.
- [11] R. Foot, Phys. Rev. D69, 036001 (2004) [hep-ph/0308254]; astro-ph/0403043; Mod. Phys. Lett. A19, 1841 (2004) [astro-ph/0405362]; Phys. Rev. D74, 023514 (2006) [astro-ph/0510705].
- [12] P. Gondolo and G. Gelmini, Phys. Rev. D71: 123520 (2005) [hep-ph/0504010].
- [13] C. Savage, G. Gelmini, P. Gondolo and K. Freese, Phys. Rev. D83: 055002 (2011) [arXiv: 1006.0972].
- [14] R. Foot, Phys. Rev. D78, 043529 (2008) [arXiv: 0804.4518].
- [15] R. Foot and X-G. He, Phys. Lett. B267, 509 (1991).
- [16] R. Foot, Phys. Lett. B699, 230 (2011) [arXiv:1011.5078].
- [17] J. L. Feng, M. Kaplinghat, H. Tu and H-B. Yu, JCAP 0907, 004 (2009) [arXiv: 0905.3039].
- [18] R. Foot, H. Lew and R. R. Volkas, Phys. Lett. B272, 67 (1991); Mod. Phys. Lett. A7, 2567 (1992).
- [19] R. Foot, Int. J. Mod. Phys. D13, 2161 (2004) [astro-ph/0407623]; P. Ciarcelluti, Int. J. Mod. Phys. D19: 2151 (2010) [arXiv: 1102.5530].
- [20] H. M. Hodges, Phys. Rev. D47, 456 (1993); Z. Berezhiani, D. Comelli and F. L. Villante, Phys. Lett. B503, 362 (2001) [hep-ph/0008105]; L. Bento and Z. Berezhiani, Phys. Rev. Lett. 87, 231304 (2001) [hep-ph/0107281]; A. Yu. Ignatiev and R. R. Volkas, Phys. Rev. D68, 023518 (2003) [hep-ph/0304260]; R. Foot and R. R. Volkas, Phys. Rev. D68, 021304 (2003) [hep-ph/0304261]; Phys. Rev. D69, 123510 (2004) [hep-ph/0402267]; Z. Berezhiani, P. Ciarcelluti, D. Comelli and F. L. Villante, Int. J. Mod. Phys. D14, 107 (2005) [astro-ph/0312605]; P. Ciarcelluti, Int. J. Mod. Phys. D14, 187 (2005) [astro-ph/0409630]; Int. J. Mod. Phys. D14, 223 (2005) [astro-ph/0409633]. For pioneering work, see: S. I. Blinnikov and M. Yu. Khlopov, Sov. J. Nucl. Phys. 36, 472 (1981); Sov. Astron. 27, 371 (1983).
- [21] R. Foot and R. R. Volkas, Phys. Rev. D70, 123508 (2004) [astro-ph/0407522].
- [22] R. Foot, A. Yu. Ignatiev and R. R. Volkas, Phys. Lett. B503, 355 (2001) [arXiv: astro-ph/0011156]; R. Foot, Int. J. Mod. Phys. A19 3807 (2004) [astro-ph/0309330]; R. Foot and Z. K. Silagadze, Int. J. Mod. Phys. D14, 143 (2005) [astro-ph/0404515]; R. Foot, Phys. Lett. B699, 230 (2011) [arXiv:1011.5078]. See also, S. Davidson, S. Hannestad and G. Raffelt, JHEP 5, 3 (2000) [arXiv: hep-ph/0001179].
- [23] R. Foot and S. Mitra, Astropart. Phys. 19, 739 (2003) [astro-ph/0211067]; Phys. Lett. A315, 178 (2003) [cond-mat/0306561]; Phys. Lett. B558, 9 (2003) [astro-ph/0301229].
- [24] P. Ciarcelluti and R. Foot, Phys. Lett. B679, 278 (2009) [arXiv: 0809.4438].
- [25] R. H, Helm, Phys. Rev. 104, 1466 (1956).
- [26] J. D. Lewin and P. F. Smith, Astropart. Phys. 6, 87 (1996).
- [27] N. Bozorgnia, G.B. Gelmini and P. Gondolo, JCAP 1011:019 (2010) [arXiv: 1006.3110]; JCAP 1011:028 (2010) [arXiv: 1008.3676].
- [28] P. Ciarcelluti and R. Foot, Phys. Lett. B690, 462 (2010) [arXiv:1003.0880].
- [29] A. Brunthaler et al. arXiv: 1102.5350.
- [30] R. Foot, Phys. Rev. D80, 091701 (2009) [arXiv: 0909.3126].
- [31] J. P. Filippini, Ph.D thesis, 2008.
- [32] J. Angle et al. (XENON10 Collaboration), arXiv: 1104.3088.
- [33] Z. Ahmed et al. (CDMS Collaboration), Phys. Rev. Lett. 106: 131302 (2011) [arXiv: 1011.2482].
- [34] J. I. Collar, arXiv: 1010.5187; 1103.3481.
- [35] R. Foot, Phys. Rev. D81, 087302 (2010) [arXiv:1001.0096].