Bottom quark contribution to spin-dependent dark matter detection

We investigate a previously overlooked bottom quark contribution to the spin-dependent cross section for Dark Matter(DM) scattering from the nucleon. While the mechanism is relevant to any supersymmetric extension of the Standard Model, for illustrative purposes we explore the consequences within the framework of the Minimal Supersymmetric Standard Model(MSSM). We study two cases, namely those where the DM is predominantly Gaugino or Higgsino. In both cases, there is a substantial, viable region in parameter space ($m_{\tilde{b}} - m_\chi \lesssim \mathcal{O}(100)$ GeV) in which the bottom contribution becomes important. We show that a relatively large contribution from the bottom quark is consistent with constraints from spin-independent DM searches, as well as some incidental model dependent constraints.


I. INTRODUCTION
The existence of non-baryonic Dark Matter(DM) has been established by many astronomical observations [1,2]. Amongst the many candidates for DM, the socalled Weakly Interacting Massive Particles(WIMPs), which would have a mass in the range O(1) GeV −O(1) TeV, are one of the most attractive. These particles would only interact with Standard Model(SM) particles through weak interactions (and gravity), in order to yield a DM relic density consistent with measurement Ω DM h 2 = 0.1199 ± 0.0027 [3].
Direct detection of DM relies on observing the recoil energy after scattering from normal matter through weak interactions. Several DM direct detection experiments have claimed a possible excess, namely DAMA [4], Co-GeNT [5],CRESST [6] and CDMS [7]. On the other hand, these results are challenged by the absence of signals at XENON100 [8] and LUX [9], as well as CDMSlite [10] in the light DM region. The coherent, spin-independent(SI) interaction between a DM particle, generically labelled χ, and a nucleus is proportional to the nucleon number. Because of the relatively heavy nuclei chosen for most of the above mentioned experiments, both the observed excess and stringent exclusion limits are based on SI χ − p scattering.
As for spin-dependent(SD) DM detection [11], in a simple shell model the spin of the nucleus is that of a single, unpaired nucleon. As a consequence, the matrix element for SD χ-nucleus scattering will be roughly comparable with that for SI χ-nucleon scattering, with no enhancement by the nucleon number. As a result, the current DM direct searches place only very loose bounds on the SD cross section [12][13][14].
In the standard calculation of SD DM-nucleon scattering the heavy quark contribution is usually neglected. That is, only the contributions from ∆u, ∆d and ∆s are * E-mail: jinmian.li@adelaide.edu.au included. However, as explained in the context of the proton weak charge [15], the usual decoupling of heavy quarks through the Appelquist-Carrazone theorem [16] does not apply to quantities influenced by the U(1) axial anomaly [17][18][19][20][21]. In that case, rather than being suppressed by inverse powers of the heavy quark mass, the suppression is only logarithmic. These logarithmic corrections were studied in considerable detail by Bass et al. in Refs. [15,22,23], at both leading and next-to-leading order. As we shall explain here, there are interesting scenarios of supersymmetry(SUSY), generally involving a relatively light sbottom, where the logarithmic radiative correction involving the b-quark that is further enhanced by resonant effect may make a significant contribution to SD DM-nucleon scattering.
Indeed, SUSY [24,25] is widely believed to provide the most promising explanation for new physics beyond SM. In SUSY models with R-parity conservation, the lightest supersymmetric particle(LSP) is stable and can become a DM candidate. On the other hand, both the LHC SUSY searches [26,27] and naturalness arguments [28,29] suggest that only the third generation supersymmetric quarks(squarks) can be light. In Ref. [30], it has been argued that an sbottom with a mass as light as ∼ O(15) GeV might still be consistent with current searches. In other models, such as the simplified model framework [31] and flavored DM models [32,33], the DM can only couple to the bottom quark, as motivated by the recent DM indirect signals [34]. Studying the bottom quark contribution to the DM-nucleon SD cross section is crucial in models of this type.
In this work, we focus on the Minimal Supersymmetric Standard Model (MSSM) with a relatively light sbottom, showing when and how the bottom contribution becomes important. When the DM is Wino, there is no coupling between DM and the Z-boson and only squark mediated processes can contribute to χ-nucleon scattering. We investigate the parameter space where the sbottom contribution is comparable to, or larger than, the first generation squark contribution. When the DM is Higgsino, the first two generation squark mediated processes are greatly suppressed by their small Yukawa couplings. However, the Higgsino can couple to the Z-boson. The constructive and destructive interference effects between Z and sbottom (b) mediated processes are discussed in detail for a number of variations on the structure of the neutralino.
Any sbottom mediated process that contributes to the SD scattering cross section can also contribute to SI scattering. We consider the stringent LUX constraint on SI DM detection for light sbottom scenarios of interest. A relatively large SD bottom contribution can indeed be found, while maintaining consistency with the LUX constraint. We also consider several model dependent constraints from collider searches. We stress that our conclusion has implications beyond the MSSM, which is used here purely for purposes of illustration.
This paper is organised as follows. In Sec. II, we explain the theoretical framework for the calculation of the SD DM-nucleon scattering cross section. Sec. III discusses the bottom contribution for Wino and Higgsino DM. The corresponding SI detection and LHC constraints on the light sbottom scenario are considered in Sec. IV and Sec. V. In Sec. VI we present some concluding remarks.

II. EFFECTIVE INTERACTION FOR SPIN-DEPENDENT DM-NUCLEON SCATTERING IN MSSM
Given a general effective Lagrangian the spin-dependent χ−nucleon scattering cross section can be written as where a p,n = q d q ∆q p,n .
The factors ∆q p,n parameterise the corresponding quark spin content of the nucleon: where s µ is the nucleon spin. The preferred values of the light quark contributions in the proton and neutron are: where the strange quark contribution is motivated by a recent lattice QCD calculation [35]. In the MSSM at tree level, there are two processes which can contribute to the effective Lagrangian. The corresponding Feynman diagrams are given in Fig. 1.
From those processes, we are able to calculate the coefficients of the effective Lagrangian from the renormalisable Lagrangian below: The corresponding couplings in the MSSM are written as [36] We consider first the Z boson mediated amplitude in the non-relativistic limit: so that the effective coupling d q in Eq. (1) is: Next, for theq mediated process we find: In this case the effective coupling in Eq. (1) is: Note that the tree level effective coupling d q is only reliable when mq − m χ is significantly larger than m q . Some discussions regarding the precision of the tree level approximation are given in Appendix A. And we have also checked that the result calculated from Eq. (16) matches well with numerical tool micrOMEGAs for light flavor quark.

III. LIGHT SBOTTOM CONTRIBUTION
For most processes of physical interest the Appelquist-Carrazone theorem tells us that heavy quark contributions are suppressed by order 1/m 2 Q . However, as explained in the introduction, because of the U (1) axial anomaly, the heavy quark contributions to spindependent quantities are only logarithmically suppressed. The particular case where this has been explored in great detail is the neutral weak charge of the proton. Without heavy quarks this is just ∆u − ∆d − ∆s, which has been used to infer values of ∆s. However, for a precise determination one must include the radiative corrections involving heavy quark loops which enter at order 1/ ln m Q . For example, one finds a LO correction from the b-quark equal to [15]: We note that Eq. (17) is second order in the strong coupling at the b mass, as is evident in the residual 5-flavor factor 6/23 appearing there. However, the regularisation of the triangle diagram leads to a logarithm in m b in the numerator which has been used to cancel the logarithm in one factor ofα b . The logarithmic radiative correction ∆ is around 2 order of magnitude below the ∆ (p) u . We will show later that with further enhancement from resonant effect the contribution from ∆ Provided that the difference between the sbottom mass and that of the DM candidate is significantly larger than the mass of the b-quark, theq propagator in Fig. 1 can be effectively factored out, leaving the familiar triangle diagram which involves the U (1) axial anomaly. In this case the bottom contribution to the axial charge of the target proton can be taken directly from Eq. (17). We shall take the running couplingα b = 0.2. As a result, for the Z-mediated process, the contribution of ∆ can only change the result by a factor of ( ∆ which is clearly very small. On the other hand, the term involving ∆ can give a significant contribution to the spin-dependent χ-nucleon cross section whenb is relatively close in mass to the DM candidate, i.e. with resonant enhancement. For simplicity we study the cases where the DM particle is either pure Wino or pure Higgsino. The corresponding couplings are: 2mW cos β m b . So, the cross section can be written as where we have assumed the gauge eigenstate limit and only the sbottom mediated process is contributing. By fixing m χ at either 10 or 100 GeV and taking Z L b = 1 and tan β = 40, Z 31 N = 1 √ 2 for Wino and Higgsino DM, respectively, we can calculate the corresponding cross section as a function of mb. The result is shown in Fig. 2. From Fig. 2 we see that the sbottom can give a very large contribution when the mass splitting mb − m χ is 100 GeV.
A. Comparison with the contribution from the first generation squarks First, we study the simpler case where the DM is gaugino. In this case there is no coupling between the Z boson and DM and only the squark mediated process can contribute to the SD interaction. In this subsection, we investigate the extent to which the sbottom should be lighter than first generation squark, so that they at least have comparable cross sections. In the following we consider the sum of the contributions of all first generation squarks (ũ L ,ũ R ,d L ,d R ), with their masses taken to be degenerate for simplicity.
Assuming that the DM is either pure Wino or Bino, the ratio of the corresponding SD cross section for sbottom to the sum of the contributions from all the first generation squarks can be calculated as The corresponding contours of σb −χ SD = σq u,d −χ SD are shown in Fig. 3. The case of Wino DM is more interesting than that of Bino DM because of its larger g 2 coupling. In this case, for 1.5 TeV first generation squarks and O(100) GeV DM, an sbottom lighter than about 200 GeV gives a larger cross section than the first generation squarks. On the other hand, for Bino DM, a much lighter sbottom(∼ 110 GeV) is required -too light for the present calculation to be reliable.

B. Contribution coherent with that of the Z-boson
When the DM is predominantly Higgsino, the first two generation squark mediated processes are greatly suppressed by their small Yukawa couplings. Its couplings to the Z boson and sbottom are dependent on the mixing between the two Higgsino states.
Firstly, we briefly discuss the Higgsino mixing in the MSSM. In the basis (B,W ,H 0 d ,H 0 u ), the neutralino mass matrix is given by: From the mass matrix we conclude that if For a few TeV gaugino and a few hundred GeV Higgsino, ∆NH 0 . Then, the contribution from the Z boson mediated process can be estimated by which is ∼ 10 −6 pb. From Fig. 2, we conclude that this corresponds to mb ∼ 150 GeV for m χ ∼ 100 GeV.
To have a closer look at the coherent effects of Z boson and sbottom mediated processes, we have chosen the decoupled Wino/Bino limit, with the Higgsino DM mixing: where a 2 + b 2 = 1 and b = 1.01a, as argued previously. This corresponds to a cross section for the Z mediated process of order ∼ 3 × 10 −6 pb. In this region, theb mediated process may also give a competitive contribution. As a result, the DM will have opposite sign coherent effects for the proton and neutron: This makes the detailed consequences for SD DM scattering from real nuclei [37] potentially very complex. We show the importance of the b-quark contribution through its coherent effects between Z mediated and sbottom mediated processes in Fig. 4. There we have marked out the compressed spectrum region (∆(mb − m χ ) 20 GeV) where our tree level calculation cannot be considered reliable. According to the Eq. (31) with assumed Z 41 N = 1.01Z 31 N , the first term in the parenthesis is positive for proton and negative for neutron, while the second term is always negative because ∆ As a result, the interference terms for proton and neutron are constructive and destructive, respectively. We find that for the DM mass around O(10 − 100) GeV an sbottom with mass 300 GeV can make a non-negligible contribution. In some specific regions, the corresponding cross section for the Z-mediated process may even be enhanced or reduced by several orders of magnitude.

IV. SPIN-INDEPENDENT DM DETECTION CONSTRAINT FROM LUX
The same process shown in the right panel of Fig. 1, which can give rise to an enhancement of the spindependent scattering cross section, can also contribute to spin-independent scattering. As a result, the very stringent spin-independent DM search bound from LUX [9] may already exclude some of the parameter region found to be of interest here.
We start with the following effective Lagrangian [38][39][40][41]: where χ is the DM field, m χ its mass and the twist-2 operator: The corresponding spin-independent scattering cross section of DM with a proton can be written as The light quark parameters f Tq are defined by and f TG = 1 − q=u,d,s f Tq . Recent lattice simulations give [42][43][44][45]: The second moments of the parton distribution functions(PDFs) can be used to evaluate the matrix element of O q µν : which from the CTEQ PDFs [46] yields at the Z boson mass scale. Using a similar technique to that used in calculating the SD effective coefficient, d q , above, we can find the corresponding effective coefficient for the spin-independent case. Based on the renormalizable Lagrangian Eq. (6), we have As a result the twist-2 operator, O q µν , gives a much larger contribution than f q in most cases. For Higgsino DM, where a q = b q , the second term of f q can easily become dominant. However in this case it is negative, so a cancellation between f q and g q may happen in some of the parameter regions.
We first consider the pure Wino DM case, with onlỹ b L mediated scattering. From Eq. (36), we have The tree level calculation for the spin-independent and spin-dependent cross sections is shown in Fig. 5. We conclude from the figure that mb − m χ 50 GeV is required to evade the spin-independent bound from LUX for Wino DM. It has to be noted that the pole at mb = m b + m χ , for SI tree level results, will not show up when the full NLO effects are taken into account [47]. We have checked that our results fit the numerical results from micrOMEGAs [48,49] quite well, outside the pole region. GeV, respectively.
Next, we discuss the more interesting case where the DM is predominantly Higgsino. As discussed above, in this case the relatively large spin dependent cross section from the sbottom mediated process can interfere coherently with the Z mediated process, leading to very different SD scattering rates for protons and neutrons.
The SI DM-proton effective coupling is The corresponding tree level calculation for the spindependent and spin-independent cross sections is shown in Fig. 6. From that figure we see that a small component of left-handed sbottom is favoured in order to evade the LUX bound. When the left-handed sbottom component is relatively large, a large SD cross section may also be consistent with the LUX experiment if there is a cancellation in σ SI .

V. MODEL DEPENDENT CONSTRAINTS AND A GENERAL ARGUMENT
We have presented a representative study of the potential importance of the bottom quark contribution to DM spin-dependent detection within the framework of the MSSM. This particular contribution has hitherto been overlooked. However, in a realistic model such as MSSM, there are many other incidental constraints. We will briefly outline how these may be evaded, while keeping our discussion as general as possible in this section. LEP placed a very stringent bound on the chargino mass (mH ± (W ± ) > 92.4(91.9) GeV) [50]. Because for either Wino and Higgsino DM there is a charged partner (chargino), which has very similar mass with the DM, we cannot have Wino and Higgsino DM of m χ 90 GeV in a typical MSSM framework.
As for m χ 100 GeV, on the other hand, it will be constrained by LHC sbottom searches [51,52] and monojet search [53], since we usually need a relatively light sbottom to enhance the bottom quark spin dependent contribution. The corresponding LHC exclusion bounds and spin-dependent χ − p scattering cross section in the mb − m χ plane are shown in Fig. 7. To generate this figure we have used Eq. (21) and (22), where only the sbottom mediated process is considered. The contours of σ χ−p SD show the condition when the sbottom mediated contribution is half the size of Z mediated process for Wino DM and a typical Higgsino DM candidate with Z 41 N = 1.01Z 31 N . This figure suggests that a large portion of parameter space is excluded by the LHC sbottom searches.
However, there are several ways to avoid these constraints: • For m χ 100 GeV, we can work in a more general framework, where the dark matter does not have any charged partners. Its couplings to the Z boson andb may be of the same order; e.g. the simpli- fied model framework [31] or flavored dark matter models [32,33].
• For m χ 100 GeV, if the charged Higgsino decays into DM and a relatively long lived particle, with lifetime of O(10−100) cm, similar to Ref. [54]. As a result, the reconstructed track will not point to the interaction point and would therefore be unlikely to be considered a "good" track. In this case, the LEP constraints on charginos can be evaded. The light sbottom constraint can also be evaded by tuning appropriate mixing -see e.g. Refs. [30,55].
• For m χ 100 GeV, if the sbottom is decayed in more complicate modes other thanb → bχ, the corresponding LHC bound on sbottom mass can be loosened.
• We can also work with heavier DM, e.g. m χ = 300 GeV, for example, as shown in Fig. 8. In this case, mb 350 GeV is consistent with the LHC searches, while the sbottom mediated process can give a significant contribution to spin-dependent scattering.

VI. CONCLUSION
In this work, we have demonstrated the potential importance of the bottom quark contribution to the DM spin-dependent cross section due to the axial anomaly and resonant enhancement, which has hitherto been overlooked. Even though our calculation was carried out within the framework of the MSSM, the general conclusion will be relevant to any models with similar particle content, since the only relevant ingredients are χ,b and the Z-boson.
In the MSSM, we calculated the bottom quark contribution to spin-dependent χ − p scattering. Firstly, we considered Gaugino DM, where there is no coupling between the Z-boson and DM. Assuming m χ = 100 GeV and degenerate first generation squarks with a mass of 1.5 TeV, we found that an sbottom of mass mb 200 GeV can give rise to a larger spin-dependent cross section for Wino DM. By contrast, for Bino DM a much lighter sbottom mass (mb ∼ 110 GeV) is required to give a competitive cross section. As for Higgsino DM, the first generation squark contributions are suppressed by their small Yukawa couplings. However, the Z-boson mediated process does contribute. For a given Higgsino mixing of DM the sbottom mediated process may interfere either constructively or destructively with Z-boson mediated processes, with different signs for protons and neutrons. For a typical mixing of Higgsino DM with mass around O(10 − 100) GeV, we find that an sbottom of mass below 300 GeV can have non-negligible effects.
The squark mediated process that gives rise to an increase in spin-dependent DM scattering can also contribute to the spin-independent cross section. Our calculation shows that ∆(mb − m χ ) 50 GeV is required to evade the LUX constraint for Wino DM, while for Higgsino DM, either a small component of left-handed sbottom or a large cancellation in σ SI is needed. Some incidental model dependent constraints from LEP and the LHC are considered as well. Those constraints, however, can be evaded in more general theoretical frameworks.
As pointed out earlier, our tree level results may break down as the sbottom and DM masses become degenerate (∆(mb − m χ ) 20 GeV). We leave the higher order calculation for this small region for future work. Finally, while the calculations for the top quark case will be more complicated because there is no clear separation of mass scales for interesting ranges of DM mass, there is a clear need to investigate the role of the axial anomaly for that case too. Finally, after integrating out the l µ , we will get a simple mq and p χ dependence of Γ ab µαβ : On the other hand, the mq dependence of the tree level effective coupling can be factored out as (A6) So we can define the ratio Taking the non-relativistic limit for the DM momentum, i.e. p χ = (m χ , 0, 0, 0), m = m b and mq = m ∆ + m χ , we solve the ratio numerically. The results are shown in the upper panel of Fig. 10. From the figure we find that the ratio tends to a constant in the heavy squark region for given m χ , which means the tree level description is accurate. While in the region of small mass splitting, the tree level results deviate from the full loop calculation considerably by a amount.
The range of m ∆ that permits the tree level approximation in required precision P can be solved by using the inequality Ratio(m ∆ )/Ratio(m ∆ = 500) > (1 − P ) (A8) at each given DM mass. In the lower panel of Fig. 10, we show the m ∆ region in which the tree level approximation matches the loop level result within 20% and 50% precision, respectively. For example, when m χ ∼ 10 GeV, m ∆ 20 GeV is sufficient to guarantee that the tree level approximation is accurate within 20%.