

Light Higgs Boson, Light Dark Matter and Gamma Rays

Vernon Barger Department of Physics, University of Wisconsin, Madison, WI 53706    Yu Gao Department of Physics, University of Wisconsin, Madison, WI 53706    Mathew McCaskey Department of Physics, University of Wisconsin, Madison, WI 53706    Gabe Shaughnessy Northwestern University, Department of Physics and Astronomy, Evanston, IL 60208 USA HEP Division, Argonne National Lab, Argonne IL 60439 USA
December 11, 2020

A light Higgs boson is preferred by and measurements. A complex scalar singlet addition to the Standard Model allows a better fit to these measurements through a new light singlet dominated state. It then predicts a light Dark Matter (DM) particle that can explain the signals of DM scattering from nuclei in the CoGeNT and DAMA/LIBRA experiments. Annihilations of this DM in the galactic halo, , lead to gamma rays that naturally improve a fit to the Fermi Large Area Telescope data in the central galactic regions. The associated light neutral Higgs boson may also be discovered at the Large Hadron Collider.

preprint: MAD-PH-10-1561 NUHEP-TH/10-09 ANL-HEP-PR-10-39

I Introduction

Our knowledge of the amount of Dark Matter (DM) in the Universe has moved from a qualitative to a precision level from measurements of Supernovae Ia, the Cosmic Microwave Background (CMB), radiation, Large Scale Structure (LLS) and the Hubble constant. The combined analysis of these datasets, including the WMAP 7-year CMB data, yields a mass density ratio, , of = 0.11090.0056 where = 0.710.025 is the Hubble constant in units of 100 km s Mpc Larson:2010gs . It is widely presumed that the DM is a stable, or nearly stable, elementary particle for which theoretical models provide many candidates. Simulations of LLS tell us that the DM must be cold to seed large scale structure. Two principal categories of cold DM particles are very light axions and Weakly Interacting Massive Particles (WIMPs). Searches for the axion of the Standard Model have set stringent limits.

Much experimental effort is being devoted to direct searches for WIMPs through their scattering from nuclei in underground detectors. These DM particles are very non-relativistic. The relative velocity of the WIMPs with respect to the nucleons is due to the motion of the Earth through the WIMP halo. The underground experiments for WIMP detection record ionization, light, and/or phonons/heat from events. The signals of nuclei recoiling from WIMP scattering can be distinguished from background events using the division of energy, timing and stopping power. A variety of detector materials have been employed, such as NaI, Ge, and Xe.

Both Spin Independent (SI) and Spin-Dependent (SD) scattering have been studied and the present experimental reaches are about a factor of 10 lower for SI scattering. From a theoretical vantage point, the experiments have reached interesting SI sensitivities.

Already eight years ago, the DAMA collaboration, with a NaI detector at Gran Sasso, reported an annual modulation of event rates as evidence for the SI scattering of WIMPs. Newer DAMA/LIBRA data confirmed their earlier finding Bernabei:2010mq . The statistical significance of the combined DAMA/LIBRA data is 8.2 sigma. The DAMA/LIBRA signal corresponds a DM cross-section/nucleon of order cm, at a WIMP mass of 5 GeV Feng:2008dz ; Feng:2008qn with a signal band that extends roughly linearly down to 510 cm at a mass of 50 GeV Bottino:2009km . The band can be shifted by channeling, but it has been recently argued that such effects are small Bozorgnia:2010xy .

The CoGeNT experiment, with a ultra-low noise Ge detector in the Soudan mine, reported a rising low energy spectrum that is unexplained by backgrounds. This has been interpreted as a DM signal with a SI cross-section/nucleon just below cm for of 7 to 12 GeV Andreas:2008xy ; Chang:2010yk ; Andreas:2010dz ; Essig:2010ye ; Graham:2010ca ; Aalseth:2010vx .

In recoil experiments the quenching factors and other detection efficiencies in the relevant keV range are subject to systematic uncertainties, so the boundary contours of the signal regions may be only approximate Hooper:2010uy . For the efficiency assumptions of Hooper:2010uy , the inferred DAMA/LIBRA and CoGeNT regions meet at a DM mass of 7 GeV, for which the DM SI cross section is approximately 210 cm.

The null results found by the XENON10 and XENON100 experiments Aprile:2010um are compatible with the DM signal favored by the overlap of DAMA/LIBRA and CoGeNT Hooper:2010uy after the uncertainties on the scintillation efficiencies of liquid Xenon are taken into account. Sorensen:2010hq . The XENON data exclude the DAMA/LIBRA allowed region above a DM mass of 10 GeV.

Recently, the CRESST collaboration released preliminary data from their 400 kg-d run with nine 300g CaWO crystal targets. With a signal region defined by a recoil energy between 10-40 keV and a background dominated by recoils, they estimate the total background to be events while they observe a total of 32 events cresst-talk:2010 . Such a signal event rate is consistent with a DM mass of GeV and a cross section  cresst-talk:2010 .

Taken together, these experimental results favor a DM candidate of mass near 10 GeV with a scattering cross section in the range.

The DM annihilations in the galaxy halo can be a source of energetic cosmic rays. At a non-relativistic cross-section of = 1 pb that gives the right relic density, a light DM with mass of order 10 GeV can produce gamma rays at a level that is detectable at the Fermi Gamma Ray Space Telescope (FGST) bib:fermi . FGST in its scan mode measures the gamma ray energy spectrum of the sky and a good agreement has been found with the expected astrophysics background. However, deviations in the gamma ray spectrum expected from a power law background parameterization has been seen in the FGST data near the galactic center  bib:centralregion ; Vitale:2009hr ; Dobler:2009xz and a DM contribution Goodenough:2009gk ; Cholis:2009gv was shown to improve the agreement with the FGST data. Ref. Goodenough:2009gk shows that a 30 GeV DM particle provides the best fit to the FGST gamma ray data around the galactic center.

In the electroweak sector, improvements in both the measurements and the SM calculations have reached the level of precision at which new physics contributions can be tested. The CDF and D0 collaborations have provided the World’s best measurements on the -boson and -quark masses which indirectly constrain new physics via loop corrections. The loop contributions are sensitive to extensions of the Higgs sector. In the SM there is a tension between the Higgs boson mass inferred from the electroweak precision observables (EWPO), which prefers a Higgs mass of  90 GeV, and the lower LEP2 experimental bound of 114 GeV. This tension can be alleviated by having a Higgs singlet that mixes with the SM Higgs doublet such that the their is a Higgs mass eigenstate that is below the LEP2 bound.

Thus, the DM signals point to a light DM particle and the EWPO measurements point to a light Higgs boson. A minimal extension of the SM that can provide both of these particles is the Complex scalar singlet extension of the Standard Model (CSM).

We previously showed that the SM with a complex singlet can be in good agreement with the CoGeNT and DAMA/LIBRA signals Barger:2008jx . In the study, we show that it also provides a good fit to the precision observables and and to FGST gamma ray data in central galactic regions.

The remainder of this paper is organized as follows: In Section II, we provide a brief overview of the complex scalar singlet model, while in Section III, we show that the scalar mass eigenstates provide a better fit to the observed and measurements by CDF and D0 than the SM can provide. We discuss how this model can match the observed gamma ray excesses toward the center of the galaxy while maintaining the SI measurements in Section IV. Finally, in Section V, we summarize and conclude.

Ii The Complex Scalar Singlet Model

A real scalar singlet Silveira:1985rk ; McDonald:1993ex ; Burgess:2000yq ; OConnell:2006wi ; Barger:2007im ; He:2009yd ; He:2010nt added to the SM can either mix with the SM Higgs boson or be a Dark Matter particle. The CSM allows both the mixing and a Dark Matter particle. Assuming CP-conservation and including only renormalizable terms, the scalar potential of the CSM is Barger:2008jx


where is the SM Higgs which obtains a vev GeV and is the complex singlet, with a vev . The term breaks a global symmetric potential, giving mass to the DM state. A non-zero avoids domain walls from an accidental symmetry.

When the real component of the complex singlet obtains a vev, the term in the potential initiates mixing between and . The resulting coupling strengths of these Higgs eigenstates to the SM fermions and weak bosons are multiplied by the factors


resulting in a reduction in the production rate of the states, thereby allowing the lightest state to evade present SM bounds. The mixing angle is given by the model parameters. The complex term in leads to a scalar field that is stable and is thus the DM candidate. The mass of the DM particle is determined by the parameters and :


We use the scan and its associated constraints from Ref. Barger:2008jx as a guide through the parameter space and associated observables.

Iii Precision and measurements

As noted above, global analyses of electroweak precision observables (EWPO) Erler:2010wa prefer a mass for the SM Higgs boson that is below the direct lower bound of 114 GeV from LEP2 experiments. Indeed, in the real singlet mixing case, an improvement to the oblique corrections can be found when the mass of the light state is reduced below the LEP SM Higgs limit Barger:2007im . Moreover, a heavier state is allowed, up to GeV at the 95% C.L. for maximal mixing. However, as the state becomes dominantly singlet, this limit tightens to the SM limit of GeV. As the CSM also provides a similar mixing scenario, an improved fit to the EWPO parameters is predicted in this model as well.

Recently, the -boson mass has been well measured by the CDF and D0 collaborations with about 1 fb of integrated luminosity to remarkable precision Tevatron:2009nu . The world average, which includes a combination with the LEP II Alcaraz:2006mx result, is


while the top quark mass measurement by the CDF and D0 Collaborations with about 5.6 fb of integrated luminosity TevatronElectroweakWorkingGroup:2010yx is


The -boson mass depends through radiative corrections on the top-quark mass, the -boson mass, the QED and QCD coupling constants, and the Higgs boson mass. While these dependences are complicated, one can arrive at a reasonably accurate expansion in terms of the relevant parameters. Using the expansion of the partial 3-loop calculation in Ref. Awramik:2003rn the dependences are




The coefficients are given by


The current measured experimental values of the SM parameters are and  Amsler:2008zzb . The value is composed of separate hadronic and leptonic contributions, with  Steinhauser:1998rq and  Amsler:2008zzb . This parameterization yields a value of that is accurate to 0.5 MeV for a SM Higgs boson masses up to 1 TeV Awramik:2003rn 111The quality of the description of degrades gradually as the mass increases..

The dependence of
Figure 1: The dependence of on for the SM (with above 114 GeV) and on for the complex singlet model for various values of with GeV. The values of and are fixed to their central values.

The -boson mass for the CSM with the SM content of Eq. 2 is approximated by


where the additional dependences on and are implicit 222This relation is approximate since the singlet contributions become non-trivial once the order of the calculation goes beyond 1-loop. However, since we use the precision measurements of and as a motivation for a light Higgs state, the precision of this calculation is not required to go to 3-loops.. In Fig. 1, we see the dependence of on for the SM (above GeV) and on for the CSM for varying values of with GeV. Generally, as the SM-content of the lightest Higgs increases for a given mass, the -boson mass increases, allowing a better fit to the measured value. Moreover, for increasing , the fits worsens.

Due to experimental constraints one cannot arbitrarily increase the SM-content of . For a given light Higgs mass, there is an upper bound from LEP2 on the amount of singlet-Higgs mixing through the measured limit of the coupling Sopczak:2001tg . As the Higgs mass decreases, its SM content must correspondingly decrease to suppress the production rate at LEP2, thereby mitigating the improvement of the -boson mass prediction.

Contours of the World’s average
Figure 2: Contours of the World’s average and at the and levels. The SM (red dashed line) shows consistency with the measurements at about . The two CSM illustrations (brown and blue dashed lines ) give an improved description of the data via the light Higgs state that has a singlet admixture characterized by the mixing angle .

The contours in Fig. 2 represent the and measurements at one and two sigma. Overlaid is the SM prediction which has an almost linear dependence in this narrow mass window. Additionally, we show two CSM predictions with a light that is predominantly composed of a SM. These points satisfy the various constraints from LEP2 experiments detailed in Ref. Barger:2008jx . This demonstrates that the CSM can describe the measured masses better than the SM by having a fraction of the Higgs contribution to the -boson mass come from the lighter singlet dominated state. Nonetheless, a light state will fit the measured -boson and top-quark masses better if it has a larger SM Higgs component, as the GeV and case shows.

The (a) maximal SM Higgs content of the lightest Higgs after LEP constraints are applied and (b) the predicted The (a) maximal SM Higgs content of the lightest Higgs after LEP constraints are applied and (b) the predicted
Figure 3: The (a) maximal SM Higgs content of the lightest Higgs after LEP constraints are applied and (b) the predicted -boson mass given in the maximal-mixing and no-mixing cases.

The allowed values of the SM Higgs content upper bounding curve for GeV are shown by the shaded region in Fig. 3a. These ranges of mixing satisfy various constraints from LEP2. The coupling is restricted below the SM expectation within the range 12-114 GeV, with the lower limit shown in Fig. 3a. Limits on nonstandard Higgs boson decay modes that arise in singlet models are also included Abbiendi:2002qp ; Schael:2010aw ; these may be probed further at the Large Hadron Collier (LHC) Barger:2006sk ; Carena:2007jk ; Cheung:2007sva ; Chang:2008cw . The invisible decay is constrained by the combined LEP2 analysis in the 90-120 GeV mass range delphi:2001xz , while a more dedicated analysis from the DELPHI collaboration probes masses down to 40 GeV Abdallah:2003ry , below which the SM component of the state is allowed to increase somewhat, in turn giving a better fit to the : see Fig. 3b 333However, it is likely that an analysis of the LEP2 data in the mass range 40 GeV would make the peak less prominent.. Above GeV, the SM line is shown; the band includes the dependence on the top-quark mass uncertainty. Below 114 GeV, the dependence of on (assuming GeV) for zero mixing is given by the flat blue band, whereas the maximal mixing allowed by the LEP2 constraints is represented by the purple band. Again, as the SM content of the lightest Higgs increases, the predicted -boson mass is closer to the experimentally measured value denoted by the horizontal pink band.

Thus we have shown that the -boson mass prediction within the content of the CSM, or more generally any model with additional singlet states that mix with the SM Higgs boson, can be in better agreement with the measured Tevatron and measurements and LEP2 constraints than the SM.

Iv Gamma Ray & Nucleon Recoil Signals

The Fermi Gamma Ray Space Telescope (FGST) Abdo:2010nz has measured energetic electron and gamma rays over large regions of the sky. Very good agreement of the data with galactic backgrounds is found except for gamma rays above 1 GeV from angular regions close to the center of the Milky Way. Significant deviations are found in central galactic regions bib:centralregion ; Vitale:2009hr from typical power law energy dependences expected from inverse Compton scattering and decays from astrophysical sources. A recent study has considered the annihilation of WIMPs with a 30 GeV mass as the source of excessive gamma ray in small angular regions () near the galactic center Goodenough:2009gk .

Annihilations of a light DM particle may explain the excess of diffuse GeV gamma rays near the galactic center (GC). The enhancement of the DM source emissivity with a cuspy DM profile can explain the DM signal enhancement near the GC. A light DM particle can provide a cross section of the requisite size.

We consider annihilations of the light DM particle in the CSM as the origin of the observed gamma ray excess. The differential gamma or electron flux is given by


where sums over the annihilation channels,  is the inclusive non-relativistic annihilation rate, AF is the annihilation fraction into mode and denotes the differential photon and spectra of each annihilation channel. The dark matter distribution for which we adopt the cuspy Einasto Navarro:2008kc profile,


where the local dark matter density . As the CSM has pure -channel annihilation via the Higgs bosons, the dominant annihilation channels are the massive fermions: with annihilation fractions that are proportional to the squares of their respective masses, .

The ‘boost factor’ (BF) in Eq. 10 normally refer to mechanisms that enhances the DM annihilation rate, such as Sommerfeld effect Hisano:2003ec ; Bovy:2009zs ; Lattanzi:2008qa of DM halos. However, in CSM the WIMP candidate has no excitation states or coupling to any light vector field, thus no major boost factor is expected to the non-relativistic annihilation cross-section, although there is some uncertainty associated with the choice of the DM halo distribution.

The total non-relativistic annihilation cross section is given by 444 Ignoring the loop-level channel which is at the percent level.


where sums over heavy leptons and quarks; the color factor is for quarks and 1 for leptons; are short-hand notations for the -channel propagator and can be approximated as


since are always off-resonance in the regions of interest. The couplings in Eq. 12 are parametrized by


To calculate the DM relic density, we used the MicrOmegas Belanger:2010gh package. The SI scattering cross section for the CSM is Barger:2010yn


where is the proton mass and is the sum of integrated parton distribution for gluon and quarks inside protons Ellis:2000ds . There is no spin-dependent scattering in the CSM.

The CSM population (all points) on the
Figure 4: The CSM population (all points) on the   vs plane that reproduces the relic density observed by WMAP7 and the XENON100 exclusion limit on . The CoGeNT allowed points are marked as blue dots (right). The points with 2733 GeV are marked a black ‘+’s (left) that better explains the FGST gamma ray spectra.

Fig. 4 illustrates the CSM population in the  versus plane that agree with relic density measurement from WMAP7 and XENON100 exclusion for less than 50 GeV. The relic density constraint leads to a generic   near 1 pb that produces a gamma ray signal comparable to the galactic background. At low , is small and  receives -channel enhancement from being close to . In this scan we allowed the CSM parameters to vary over the following ranges


Including CoGeNT and DAMA/LIBRA data, the WIMP mass is further confined to a small range of GeV Barger:2010yn .

To show-case the gamma ray predictions, we choose 10 GeV as typical of a CSM explanation of the CoGeNT and/or DAMA/LIBRA signals. We also consider 30 GeV, which has a SI cross section 1.410 cm that is below the XENON100 bound, as an example of a somewhat higher mass. The gamma signal consists of prompt photon emissivity given by Eq. 10 and radiation associated with electrons produced in the DM annihilations, via their Inverse Compton scattering.

We test the dark matter contribution with a joint analysis of FGST data Abdo:2009zk and gamma ray spectra in two large areas: (i) the ‘middle latitude’ with  Abdo:2010nz and (ii) the ‘central’ region with and  bib:centralregion that includes the galactic center. In this analysis we do not take in account of possible correlations of the data from the two separate angular areas; this could have an effect on the systematics but should not alter the overall conclusions. The sizable angular coverage should smear out background fluctuations that may exist in very small regions.

             (cms) (cm) AF() AF() AF()
0.571 125 0.066 0.35 51 10 14 2.910 3.110 0.87 0.05 0.07
0.97 162 -0.16 0.38 51 30 55 4.610 1.410 0.86 0.04 0.07
Table 1: Sample points for the CSM. The lower sits inside CoGeNT bound while the higher gives better agreement with Fermi gamma ray spectra. Both sample points are consistent with DM relic density and XENON100 constraints. The DM annihilation rate is largely determined by and masses.

The galactic background and the dark matter induced gamma ray and electron signals are numerically evaluated with GALPROP package Strong:1999sv ; Strong:2001gh . For the galactic diffuse background we assume power-law injection spectra for nuclei and for the astrophysical electron background, where was allowed to vary along with five other variables that parametrize the diffusion process, astrophysical electron background and the measured electron energy. See Ref. Barger:2009yt for the detailed numerical simulation. The total DM annihilation rate was also treated as a free parameter. The annihilation channel branchings for two sample DM masses are listed in Table. 1. Fig. 5 illustrates the gamma ray contributions from the individual annihilation channels. The gamma rays originate from neutral pions. The and -quarks yield softer gamma rays than the leptons.

Differential gamma ray spectra from individual channels for a 10 GeV
Figure 5: Differential gamma ray spectra from individual channels for a 10 GeV . Although sub-dominant to the channel, the channel produces more hard photons at energy fraction through copious decays
Best fit scenarios for 10 GeV and 30 GeV DM masses. The angular regions are noted in the panels. The gamma ray contribution from dark matter enhances the total flux in the range of 1
Figure 6: Best fit scenarios for 10 GeV and 30 GeV DM masses. The angular regions are noted in the panels. The gamma ray contribution from dark matter enhances the total flux in the range of 110 GeV where an excess over a power-law galactic diffuse background has been observed. The Fermi electron data in the bottom panel are included in the fits. The right bottom panel shows the goodness of fit versus dark matter contribution rate for 10 GeV and 30 GeV DM masses. The number of degrees of freedom (DOF) is 54.

The WIMP mass is an important parameter in fitting the excessive gamma rays in the central galactic regions. annihilation is dominated by the channel and the DM induced gamma ray spectrum peaks at . Gamma rays from a below 10 GeV are likely to miss the energy range of the excess between 1 and 10 GeV. We find that the light dark matter at 30 GeV greatly improves the fit to Fermi gamma ray spectrum from a 2 background-only agreement to less than 1. At 10 GeV or lower mass the DM induced gamma rays are relatively soft and lead to less effect in the fit to the FGST data. Fig. 6 shows the minimal reduced .

A good description of Fermi data gives


and we can extrapolate the  over the narrow mass windows, using the predicted given by Eq. 10. At very light , below GeV, the gamma ray spectra from DM annihilations are too soft to explain the photon excess above 1 GeV, but the model still gives a fit with for up to 6 pb. In this case, we choose = 1 pb as a non-boosted annihilation cross-section which is detectable by FGST. For the 30 GeV case we choose the best-fit annihilation cross-section. There are numerous parameter combinations in the CSM that give a  that explains the Fermi data with a natural boost factor .

A 30 GeV dark matter mass gives an improved fit to the excess in the gamma ray spectrum with a best-fit =310cms; this gives a boost factor above 3 that is necessary to reach the best fit gamma ray signal level.

 The ratio of the annihilation rate fit in Eq. 
Figure 7: The ratio of the annihilation rate fit in Eq. 17, , to the theoretical CSM model prediction, . A ratio less than 1 means the model predicts a gamma ray signal larger than that allowed by the FGST data. The dark and light blue dots represent separate CoGeNT allowed parameter regions that have a mass in the GeV and GeV ranges, respectively. Each mass range has distinct collider signatures Barger:2007im . The CSM points with near 30 GeV (black ‘+’s) are also shown. Those points provide an improved fit to the FGST data, albeit with a boost factor of the order . All points satisfy the constraints in Fig. 4.

Fig. 7 shows the non-relativistic  of annihilation at the parameter points that satisfy the XENON100 and relic density bounds. All the points in Fig. 47  pass the and measurement constraints with less than . We note that many points of the CSM parameter space naturally satisfy the constraints. Two regions of CMS parameter space Barger:2007im that are allowed by CoGeNT are shown as light and dark blue points. The light blue region satisfies and a range of GeV while varies between 9 and 15 GeV . In the dark blue region has a broader range of GeV and is far off-shell, . The DM mass in this region is also more concentrated near 8 GeV, allowing the invisible decay mode of the light Higgs to be open.

FGST will take data for 10 years. The improved statistics will significantly improve its sensitivity to a DM annihilation source of gamma rays. The contributions from a cross section of = 1 pb of a 10 GeV DM particle will produce a distinctive shape of the gamma ray energy spectrum that will allow its confirmation or exclusion.

V Summary

The Complex singlet extended Standard Model (CSM) has 3 scalar particles: two singlets and the neutral member of the SM Higgs doublet. The stable CP-odd singlet () is the dark matter and the CP-even singlet mixes with the SM Higgs boson. This model can provide a natural explanation of the possible CoGeNT and DAMA/LIBRA DM Spin Independent cross section signals with a DM mass in the GeV range. There is then a light Higgs state () with a mass in either the GeV or the GeV range that is predominantly singlet, thereby allowing the light to escape detection at LEP2. The heavier Higgs state () therefore has a predominantly SM doublet composition, with its couplings to SM particles universally reduced by a mixing factor and a mass range that is similar to that of the SM Higgs particle: GeV. In this scenario, we have shown the new light Higgs boson allows a better fit to the precision observables and than the SM provides. The mass is shifted to higher values than in the SM, in closer agreement with LEP2 and Tevatron measurements, improving from a nearly deviation in the SM to a deviation for GeV.

The model predicts a DM annihilation contribution with  = 1 pb to gamma rays that explains the observed structure in the 1 - 10 GeV energy distribution of the Fermi diffuse gamma ray observations over the two large areas of the galaxy. The gamma ray signals of the DM annihilations originate through the produced and -quarks and -leptons, through their subsequently decays to s. The cuspy Einasto DM halo distribution yields the relative DM rates of the central galactic and mid-latitude data. A even better fit to the high-latitude FGST data is obtained with 30 GeV, albeit beyond the DM mass range allowed by XENON100. Since the mass in the CSM is at most 60 GeV, the associated LHC Higgs boson signatures are potentially interesting: the can decay via SM modes, an invisible mode (), and cascade modes, .


We thank D. Hooper for helpful information about Fermi data. This work was supported in part by the U.S. Department of Energy Division of High Energy Physics under grants No. DE-FG02-95ER40896, DE-FG02-05ER41361, DE-FG02-08ER41531, DE-FG02-91ER40684 and Contract DE-AC02-06CH11357, by the Wisconsin Alumni Research Foundation, and by the National Science Foundation grant No. PHY-0503584.




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