Constraining the interaction strength between dark matter and visible matter: I. fermionic dark matter
Abstract
In this work we study the constraints on the dark matter interaction with the standard model particles, from the observations of dark matter relic density, the direct detection experiments of CDMS and XENON, and the indirect detection of the ratio by PAMELA. A model independent way is adopted in the study by constructing the effective interaction operators between dark matter and standard model particles. The most general 4fermion operators are investigated. We find that the constraints from different observations are complementary with each other. Especially the spin independent scattering gives very strong constraints for corresponding operators. In some cases the indirect detection of data can actually be more sensitive than the direct detection or relic density for light dark matter ( GeV).
pacs:
95.35.+d, 95.30.Cq, 95.85.RyI Introduction
The existence of a significant component of nonbaryonic dark matter (DM) in the Universe has been well confirmed by astrophysical observations Tegmark:2006az (); Komatsu:2008hk (); Komatsu:2010fb () in recent years, however the nature of this substance remains unclear. Since there is no candidate for DM in the Standard Model (SM) of particle physics, it implies the existence of new physics beyond the SM. Probe of the microscopic identity and properties of DM has become one of the key problems in particle physics and cosmology (for reviews of DM, see, for instance, kolbturner (); Jungman:1995df (); Bertone:2004pz (); Murayama:2007ek (); Feng:2010gw ()).
Among a large amount of theoretical models, a wellmotivated candidate for DM is the weakly interacting massive particle (WIMP). This WIMP must be stable, nonrelativistic, electrically neutral, and colorless. If the mass of WIMP is from a few GeV to TeV and the interaction strength is of the weak scale, they can naturally yield the observed relic density of DM, which is often referred to as the WIMP miracle Feng:2010gw (). A huge variety of new physics models trying to solve the problems of the SM at the weak scale can naturally contain WIMP candidates, such as the supersymmetric models Jungman:1995df (); Goldberg:1983nd (); Ellis:1983ew (); Kane:1993td (), extra dimensional models Kolb:1983fm (); Cheng:2002ej (); Hooper:2007qk (); Servant:2002aq (); Servant:2002hb (); Agashe:2004ci (); Agashe:2004bm (); Agashe:2007jb (), little Higgs models Cheng:2004yc (); Low:2004xc (); Birkedal:2006fz (); Freitas:2009jq (); Kim:2009dr (), leftright symmetric models Dolle:2007ce (); Guo:2008hy (); Guo:2008si (); Guo:2010vy (), and many other theoretical scenarios.
The above mentioned specific models are wellmotivated, however they still lack experimental support. We do not know whether nature really behaves like one of them or some other yet unconsidered theories. Moreover, in case the DM particle is the only new particle within the reach of LHC and other new particle species are much heavier than DM, it will be very difficult to tell which model the DM particle belongs to. Additionally, it is possible that the DM may be first observed by direct or indirect detection experiments. These early observations may only provide information about some general properties of the DM particle, and may not be able to distinguish the underlying theories. Therefore, the modelindependent studies of the DM phenomenology are particularly important for they may avoid theoretical bias Birkedal:2004xn (); Giuliani:2004uk (); Kurylov:2003ra (); Beltran:2008xg (); Cirelli:2008pk (); Shepherd:2009sa (). Recently there have been quite a few papers following such consideration and adopting a modelindependent way to study various phenomenologies related with DM Cao:2009uv (); Cao:2009uw (); Beltran:2010ww (); Fitzpatrick:2010em (); Goodman:2010yf (); Bai:2010hh (); Goodman:2010ku (); Goodman:2010qn (); Bell:2010ei (). Especially the relic density measured by WMAP Komatsu:2010fb (), direct detection from CDMS Ahmed:2009zw (), XENON Aprile:2010um () and possible collider signals from LHC are considered in these studies.
In this work, we first construct the general effective 4fermion interaction operators between DM particles and the SM particles, which extend the effective fermionic WIMP interactions given in Ref. Beltran:2008xg (). Here we focus on Dirac fermionic DM. Discussions on scalar and vector DM will be presented in companion papers. We then give updated constraints from the DM relic density within the 7year WMAP data Komatsu:2010fb () and the spinindependent WIMPnucleus elastic scattering searches by CDMS II Ahmed:2009zw () and XENON100 Aprile:2010um (), and compare our results with those in Ref. Beltran:2008xg (). In addition, we present new phenomenological constraints on these effective models from the spindependent WIMPnucleus elastic scattering searches by CDMS Akerib:2005za () and XENON Angle:2008we () and the cosmicray antiprotontoproton ratio by PAMELA Adriani:2010rc (). We find that the constraints from different kinds of experiments are rather comparable. Combination of these constraints provides more information of the effective models.
This paper is organized as follows. In Sec. II we briefly describe the effective DM models of various 4fermion interaction operators. In Sec. III, IV and V we explore the constraints on these models from the DM relic density, direct and indirect detection searches, respectively. In Sec. VI we discuss the validity region of effective theory and present the combined constraints on the effective coupling constants of these models. Sec. VII is the conclusion.
Ii Effective Models
We start with the case that DM is a single Dirac fermionic WIMP (). Instead of considering a WIMP candidate from a specific theoretical model, we study the phenomenologies in a modelindependent way by constructing effective interaction operators between and the SM particles. These interaction operators are constrained only by the requirements of Hermiticity, Lorentz invariance and CPT invariance.
To proceed, we make the following assumptions similar to those in Ref. Beltran:2008xg (): (1) The WIMP is the only new particle at the electroweak scale, and any new particle species other than the WIMP has a mass much larger than the WIMP. This implies that the WIMP’s thermal relic density is not affected by resonances or coannihilations, and this makes it possible to describe the interaction between the WIMPs and the standard model particles in terms of an effective field theory. (2) The WIMP only interacts with the standard model fermions through a 4fermion effective interaction, but not with other particles like gauge or Higgs bosons. This 4fermion effective interaction is assumed to be dominated by only one form (scalar, vector, etc.) of the set of 4fermion operators for simplicity. (3) The WIMP annihilation channels to the standard model fermionantifermion pairs dominate over other possible channels. In other words, the possible channels to final states that include gauge or Higgs bosons are assumed to be negligible for simplicity. However, it should be noted that the supersymmetric DM model actually cannot satisfy the three assumptions above simultaneously in order to give the correct relic density. Even so these assumptions are still useful for a general research.
The effective Lagrangian between two fermionic WIMPs ( and ) and two standard model fermions ( and ) is given by only one of the following expressions:
(1)  
(2)  
(3)  
(4)  
(5)  
(6)  
(7)  
(8)  
(9)  
(10)  
(11)  
(12)  
(13)  
(14) 
where the sum of is over all the standard model fermions, and the effective coupling constants are real numbers which have mass dimension of . The 4 chiral interaction operators, , , and , are just the combinations of the other operators mentioned above. Here we do not include the interaction operators involving derivative insertions into fermion bilinears, for they have higher momentum dimensions and may be safely ignored in the small momentum limit. Note that the alternative tensor interaction term in (10) has two other equivalent forms, i.e., .
Each form of the interaction operators listed above represents an effective model of the WIMP coupled to the standard model fermions. For each case, we can calculate the corresponding annihilation and scattering cross sections, which depend on the WIMP mass and the coupling constants . Associated with the recent results of the DM relic density, direct and indirect detection experiments, we can obtain the phenomenological constraints on . It would be interesting and meaningful to compare the constraints derived from different experiments.
It is worthwhile to note the symmetry properties of these operators under discrete C, P, and T transformations. The first 5 forms of the operators, , , , and are separately invariant under C, P, and T, while the transformation properties of the other operators are summarized in Table 1. The transformation properties of the operators under CP are the same as those under T, given that the coupling constants are realvalued numbers. Thus, all the operators are actually CPT invariant. If the future experiments indicate that there were some of the C, P, and T symmetries in the DM sector, we may use this table to concentrate on or exclude some interaction operators.
P  

C  
T 
Iii WIMP annihilation and relic density
In order to determine the relic density of WIMPs and the source function of cosmicray particles produced by the DM annihilation in the Galaxy, which is relevant to the DM indirect detection, we need to calculate the cross sections of WIMPantiWIMP annihilation to fermionantifermion pairs for each case listed in the last section. The result is given by
(15)  
(16)  
(17)  
(18)  
(19)  
(20)  
(21)  
(22)  
(23)  
(24)  
(25)  
where is the Mandelstam variable, is the WIMP mass, the sum is over the final state fermion species , and are the color factors, equal to 3 for quarks and 1 for leptons. The annihilation cross sections of the four chiral interactions, , , and , have the exactly same formula, so they can be denoted by a common symbol with the corresponding coupling constants denoted by . Note that Eqs.(15)–(17) agree exactly with Eqs.(6)–(8) in Ref. Beltran:2008xg (), while Eqs.(18),(19) are slightly different from Eqs.(9),(10) in Beltran:2008xg ().
In the very early Universe, the WIMPs were in thermal equilibrium. As the Universe expands, the WIMPs departed from thermal equilibrium when they were nonrelativistic, and finally froze out to yield a cold relic roughly when the annihilation rate dropped below the Hubble rate. This evolution process is described by the Boltzmann equation
(26) 
where is the Hubble rate with denoting the Planck mass, () is the number density of WIMPs (antiWIMPs), and the thermal average will be explained below. For Dirac fermions without particleantiparticle asymmetry, , and the total DM paritcle number density is Gondolo:1990dk (); Jungman:1995df (); Srednicki:1988ce (). At the early time, when the temperature , the WIMP number density was very close to its equilibrium value , and the annihilation rate per unit volume was much greater than the Hubble expansion rate per unit volume and enough to maintain the thermal equilibrium. However, as the temperature decreased below , the equilibrium number density was exponentially suppressed, , where is the number of degrees of freedom of a fermionic WIMP. Eventually, the annihilation rate became smaller than the expansion rate, and the WIMPs froze out of equilibrium.
The thermally averaged quantity should be treated carefully. As pointed out by Ref. Gondolo:1990dk (), the Møller velocity in Eq. (26) is defined by with subscripts 1 and 2 labeling the two initial DM particles and particle velocities (). The Møller velocity equals the relative velocity only when the collision is collinear, . Since the Boltzmann equation, Eq. (26), is expressed in the cosmic comoving frame Gondolo:1990dk (), in which the gas is at rest as a whole, the thermal average must be taken in this frame. Fortunately, even including relativistic effects, it has been shown Gondolo:1990dk () that , where and the righthand side is computed in the lab frame, in which one of the two initial particles is at rest. Thus, it is convenient to calculate the thermal average in the lab frame using the method described in Gondolo:1990dk ().
Cold DM requires that the freezeout of WIMPs occurred when they were nonrelativistic. In the nonrelativistic limit, we can parameterize ^{1}^{1}1Note that this expansion in powers of is equivalent to that in Ref. Gondolo:1990dk (): with . Since , one easily obtains the relation between these two expansions: , , etc., where . According to Srednicki:1988ce (); Gondolo:1990dk (), we then obtain with . Now let us compute the coefficients and in the effective models. Due to the common factor in Eqs. (15) – (25), must be expanded up to order to get the correct coefficients . In the lab frame, . Substituting this expansion of into Eqs. (15) – (25) and expanding in powers of up to order , we obtain
(27)  
(28)  
(29)  
(30)  
(31)  
(32)  
(33)  
(34)  
(35)  
(36)  
(37) 
from which, one can easily read off the corresponding thermally averaged quantities . Our results of Eqs. (30), (35), and (37) agree well with Eqs. (35) and (39) of Ref. Srednicki:1988ce (). Note that Eq.(27) is the same as Eq.(13) in Ref. Beltran:2008xg (), while the terms of Eqs.(28)–(30), and both the and terms of Eq.(31) are different from the corresponding terms of Eq.(14)–(17) in Beltran:2008xg (). In spite of these differences, they do not have much effect on the main results of Ref. Beltran:2008xg (). This is because the calculated relic density depends mainly on the leading term of , as we will see below.
Using the standard procedure kolbturner (); Jungman:1995df () to approximately solve the Boltzmann equation (26), we obtain a relic density of DM particles as
(38) 
where with being the freezeout temperature, is the total number of effectively relativistic degrees of freedom at freezeout, K Mather:1998gm () is the present CMB temperature. The freezeout temperature parameter can be evaluated by numerically solving the following equation:
(39) 
where is an order one parameter defined by the freezeout criterion and determined by matching the latetime and earlytime solutions. The precise value of is not so significant for the numerical solution of due to the logarithmic dependence in Eq. (39), and we take the usual value in the calculation. Noting that in Eqs. (38) and (39) depends on the temperature , we adopt the recent numerical result of in Ref. Coleman:2003hs () where the quarkhadron transition temperature is taken to be 200 MeV.
According to the observed DM relic density, Komatsu:2010fb (), we can estimate the relation between the effective coupling constants and the WIMP mass in each effective model of 4fermion interaction operators, as shown in Fig. 1. Two kinds of coupling constants are considered here. In the upper frame of Fig. 1, we show the results for the case when the effective couplings to all the standard model fermions are equal (universal couplings). In the lower frame of Fig. 1, we show the results for the case when the coupling constants are proportional to the fermion mass . In both cases, decreases as increases for fixed in each effective model. Besides, Fig. 1 has several interesting features:

In both cases, the 4 curves of vs. for the effective models of scalar, scalarpseudoscalar, axialvector and axialvectorvector interactions lie well above the other curves. This comes from the fact that , and are of order ; although the leading term of is of order , it is smaller by a factor of than the terms for other types of interactions.

In the case of , there is an obvious downward bend in the curve of vs. at about GeV in each effective model. This can be easily explained as follows. In the low velocity limit, the threshold for the annihilation channel is about . Since , the WIMP couples much more strongly to the top quark than to other fermions, and the corresponding channel gives a tremendous contribution to the total . This finally makes the curve to bend down.

In the case of universal couplings, there are 5 pairs of nearly identical curves, because in each pair their corresponding differ only by terms of and/or terms of . This feature is also noted in Ref. Beltran:2008xg (). These pairs are: (1) the curves for scalar and scalarpseudoscalar interactions, and we denote this approximate identity of the two curves by SSP for short here and henceforth; (2) PPS; (3) VVA; (4) TC; (5) AAV except for some small regions. From these approximate identities for pairs of curves, we see that the predicted relic density relies mainly on the leading term of in each model, especially on the term in the leading term.

In the case of , there are also 5 pairs of nearly identical curves except for some small regions. These pairs are the same as those in the above case, though deviations of the two nearly identical curves in each pair become large in this case.
In Fig. 2 we show the curves of vs. for fixed coupling constants in the models of scalar, pseudoscalar, vector, axialvector, tensor and alternative tensor interaction operators. Due to the nearly identities described above, we do not include the curves for scalarpseudoscalar, pseudoscalarscalar, vectoraxialvector, axialvectorvector and chiral interactions. Here we still consider the two kinds of coupling constants. In the upper left and upper right frames, we show the results when the coupling constants are proportional to the standard model fermion masses, , and various values of the couplings are taken: , , , and . This proportionality of the couplings to the fermion masses may come from Yukawa couplings of a Higgs mediated interaction or some other unknown underlying mechanism. In the remaining 6 frames, we show the results for the case when the effective couplings to all the standard model fermions are equal (universal couplings), and various values of the couplings are taken: , , , and . The curves in Fig. 2 bend down to more or less at about GeV, 4.20 GeV and 171.2 GeV, which exactly correspond to the masses of charm, bottom, top quarks, respectively.
Comparing with the results in Ref. Beltran:2008xg (), we observe that the curves in the first 6 frames of Fig. 2 are a little higher than those given by Ref. Beltran:2008xg (). This slight difference may be caused by the following reasons: (1) We use for Dirac fermionic WIMPs with the assumption of no particleantiparticle asymmetry. (2) Some formulas of we obtained differ slightly from those given by Beltran:2008xg (), as already described at the bottom of Eq.(37). (3) We use the effective degrees of freedom given by Coleman:2003hs ().
It is important to note that the results in Figs. 1 and 2 are found under the assumptions presented in Sec. II. If resonances, coannihilations, or annihilations to final states other than fermionantifermion pairs are significant, the actual curves in Figs. 1 and 2 will be significantly lower than these shown there, as pointed out in Beltran:2008xg ().
Iv Direct detection
In this section we discuss the direct detection constraints on the effective models of Eqs.(1)–(14). Direct detection experiments are designed to measure the recoil energy of the atomic nuclei when the WIMPs elastically scatter off them. The WIMPquark interactions in the effective models naturally induce the WIMPnucleon interactions, and the latter further induce the WIMPnucleus interactions. Such interactions may lead to the elastic scattering of the WIMPs with the nuclei, which may be detected at the direct detection experiments.
The velocity of the WIMP near the Earth is thought to be of the same order as the orbital velocity of the Sun, . Because of this small velocity, the momentum transfer in the WIMPnucleus scattering is considerably small compared to the masses of the WIMP and the nuclei. Thus all the WIMPnucleus cross sections can be calculated in the limit of zero momentum tranfer. In this limit the WIMPquark interaction operators, , , , , and , and their correspondingly induced WIMPnucleon interaction operators have no contribution to the WIMPnucleus cross sections and thus they are not sensitive to direct detection experiments. This is because in the zero momentum transfer limit some fermion bilinear operators become zero, for example, the operator vanishes, and the time component of and the space components of vanish as well. For more details on this issue, see Ref. Agrawal:2010fh ().
Among the remaining operators relevant to direct detection, the scalar and vector interaction operators, and , are referred to as spinindependent (SI) interactions, while the axialvector and tensor interaction, and , belong to spindependent (SD) interactions. And the chiral interaction operators are the combinations of SI and SD interactions. The SI interactions of all the nucleons add coherently in the target nucleus, and the corresponding WIMPnucleus cross section is proportional to the square of the atomic mass number of the nucleus. On the other hand, since the spins of nucleons in a nucleus tend to cancel in pairs, the SD interactions rely mainly on the spin content of one unpaired nucleon and the corresponding cross section is not enhanced for heavy nuclei.
We would like to illustrate the calculation in the effective model of scalar interaction operators. Eq.(1) can induce the effective Lagrangian for the WIMPnucleon couplings, which reads
(40) 
where the WIMP’s effective Fermi couplings to the nucleons (protons and neutrons), (), are related to the coupling constants to quarks by
(41) 
where the nucleon form factors are , , , , , Ellis:2000ds (), and
(42) 
From Eq.(40) and its induced WIMPnucleus interactions, it follows that the cross section for a WIMP () scattering elastically from a nucleus () in the zero momentum transfer limit is given by Belanger:2008sj ()
(43) 
where is the target nucleus mass. and are the numbers of protons and neutrons in the nucleus. The factor in the square bracket comes from the fact that a Dirac fermionic WIMP and its antiparticle are different Belanger:2008sj (). However, this factor is missing in Eq.(20) of Ref.Beltran:2008xg (). Indeed, such a factor does not exist in the expression for a selfconjugated WIMP such as a Majorana fermion, but it seems that a Dirac fermionic WIMP is considered in Ref.Beltran:2008xg (), otherwise the fermion bilinears and vanish for a Majorana fermion . Taking the special case when the nucleus is just the nucleon (proton or neutron) in Eq.(43), we obtain the WIMPnucleon cross section in the zero momentum transfer limit:
(44) 
Likewise, we compute the WIMPnucleus cross section in the effective models of vector, axialvector, tensor and chiral interaction operators, resulting in
(45)  