Uncertainties in WIMP Dark Matter Scattering Revisited

We revisit the uncertainties in the calculation of spin-independent scattering matrix elements for the scattering of WIMP dark matter particles on nuclear matter. In addition to discussing the uncertainties due to limitations in our knowledge of the nucleonic matrix elements of the light quark scalar densities<N |{\bar u} u, {\bar d} d, {\bar s} s| N>, we also discuss the importances of heavy quark scalar densities<N |{\bar c} c, {\bar b} b, {\bar t} t| N>, and comment on uncertainties in quark mass ratios. We analyze estimates of the light-quark densities made over the past decade using lattice calculations and/or phenomenological inputs. We find an uncertainty in the combination<N |{\bar u} u + {\bar d} d | N>that is larger than has been assumed in some phenomenological analyses, and a range of<N |{\bar s} s| N>that is smaller but compatible with earlier estimates. We also analyze the importance of the {\cal O}(\alpha_s^3) calculations of the heavy-quark matrix elements that are now available, which provide an important refinement of the calculation of the spin-independent scattering cross section. We use for illustration a benchmark CMSSM point in the focus-point region that is compatible with the limits from LHC and other searches.


Introduction
Direct searches for the scattering of weakly-interacting massive particle (WIMP) dark matter [1] are proceeding apace, with regular increases in the experimental sensitivity [2][3][4] and plans for new experiments capable of probing spin-independent scattering cross-sections σ p SI approaching the neutrino floor [5]. It is important that these important experimental efforts be well served by theoretical efforts to minimize the uncertainties in calculations of event rates within specific models of WIMP dark matter. These include astrophysical uncertainties in the local density and velocity distribution of the WIMPs, uncertainties in the accuracy with which effective interaction operator coefficients can be calculated within a specific model, uncertainties in the matrix elements of these operators in hadronic targets, and uncertainties in nuclear structure effects. The focus of this paper is on the uncertainties in the matrix elements for scattering on nucleon targets.
There are classes of two dimension-6 four-fermion interactions that yield velocityindependent cross-sections for elastic WIMP-nucleon scattering, namely [6,7] where the sums are over the quark flavours i. Rates for the first set of interactions ∝ α 3i are related to quark contributions to the nucleon mass: m q i N |q i q i |N and are independent of the nuclear spin, whereas rates for interactions ∝ α 2i are related to nucleonic matrix elements of axial currents in nucleons: N |q i γ µ γ 5 q i |N , which are related to quark contributions to the nucleon spin. We discuss here the uncertainties in the matrix elements of the spin-independent interactions ∝ α 3i , which are relatively important, as we shall see. The uncertainties in the spin-dependent interaction matrix elements ∝ α 2i are relatively small, as we discuss briefly towards the end of this paper.
Several approaches have been taken to estimating the N |q i q i |N matrix elements. One of the first was to use octet baryon mass differences and SU(3) symmetry to estimate the combination σ 0 ≡ 1 2 (m u + m d ) N |ūu +dd − 2ss|N , together with data on low-energy πN scattering to estimate the quantity Σ πN ≡ 1 2 (m u + m d ) N |ūu +dd|N 1 . As has been discussed in previous work, see, e.g., [7][8][9][10], combining these estimates of σ 0 and Σ πN led to relatively large estimates for N |ss|N , though with large uncertainties.
As we discuss below in some detail, in recent years a large effort has been put into lattice calculations, which have yielded a range of values of Σ πN and relatively small estimates for N |ss|N . The corresponding values of σ 0 may be similar to the estimates made using baryon masses and SU(3), but some calculations correspond to significantly larger values of σ 0 . In parallel, there have been calculations using baryon chiral perturbation theory (BχPT) that may lead to much larger values of σ 0 , see, e.g., [11], close to the data-based estimates of Σ πN , which may also correspond to relatively small values of N |ss|N .
In this paper we compile the lattice and other estimates of Σ πN and σ s ≡ m s N |ss|N that have appeared over the past decade, and propose simple Gaussian representations of their values and uncertainties. These may be useful for analyses of the constraints that direct searches for dark matter via spin-independent scattering impose on specific models. Our combined estimate of Σ πN is similar to values suggested previously, but with a larger uncertainty, while our combined estimate of σ s is somewhat smaller than older estimates, though consistent with their uncertainties. We illustrate the results of our analysis with calculations of the spin-independent dark matter scattering cross section σ p SI at a specific benchmark point in the focus-point region [12] of the constrained minimal supersymmetric Standard Model (CMSSM) [13,14], noting that other supersymmetric parameter sets exhibit similar trends. As we discuss, the uncertainties in σ p SI related to light quark masses and N |ūu|N / N |dd|N are significantly smaller than those associated with σ s . We also discuss the uncertainties in σ p SI associated with the heavy quark matrix elements N |cc,bb,tt|N . The b and t quarks are sufficiently heavy that a perturbative treatment of their hadronic matrix elements is appropriate, but this is not so evident for the c quark. Some lattice and other numerical estimates of σ c ≡ m c N |cc|N are available, and span a wide range that straddles the perturbative estimate. If the O(α 3 s ) perturbative estimates are used for all the heavy-quark matrix elements, as we advocate, the corresponding uncertainties in σ p SI are small, but if the full range of numerical estimates of σ c is considered the associated uncertainty is comparable to that associated with σ s . The layout of this paper is as follows. In Section 2 we review the strong-interaction quantities entering the calculation of the spin-independent cross-section σ p SI . Inputs to the calculation of σ p SI are discussed in Subsection 2.1, the individual uncertainties in the matrix elements of the densities of the light quarks u, d, s are discussed in Subsection 2.2, and their propagation into the calculation of σ p SI are discussed in Subsection 2.3. Subsection 2.4 is dedicated to a discussion of the matrix elements of the heavy quarks c, b, t. Finally, Section 3 contains a brief discussion of the uncertainties in the calculation of the spin-dependent cross-section σ SD , and our conclusions are summarized in Section 4.

Inputs to the Matrix Element Calculation
At zero momentum transfer, and neglecting nuclear structure effects, the spin-independent cross-section for the elastic scattering of a generic WIMP on a nucleus with charge Z and atomic number A can be written as [1,6,[15][16][17][18][19][20] where m r is the reduced WIMP mass, for N = p or n, and the quantities f N Tq are defined by We recall that the quantities f N Tq (σ q ) are independent of renormalization scheme and scale, whereas the quantities m q and B N q appearing have cancelling scheme and scale dependences. The contributions of the heavy quarks q = c, b, t have often been treated by integrating them out and replacing them by the one-loop contributions due to scattering off gluons [21], so that where In our subsequent analysis, this is the first approach we use to calculate σ p,n SI . However, as discussed later in more detail, there are by now a number of lattice calculations of f N Tc = m c N |cc|N /m N = σ c /m N , so for comparison we also estimate σ p SI using the one-loop 4-flavour versions of (5) and (6) together with a numerical estimate of f N Tc that is based (mainly) on lattice calculations. As we also discuss later, there are also calculations of f N Tc , f N T b and f N Tt to O(α 3 s ) in perturbative QCD. These perturbative calculations are expected to be very reliable for f N T b and f N Tt , perhaps less so for f N Tc . Therefore, we also estimate σ p SI using these calculations in the full six-flavour formula (3), for comparison with the three-quark formula (5) and the four-quark formula (7) evaluated using the available numerical estimates of f N Tc . As we discuss later, we consider the O(α 3 s ) six-flavour approach to be the best available at the present time.
In order to evaluate (5) we need estimates of the matrix elements N |ūu,dd,ss|N , for which isospin invariance ensures that p|ūu|p = n|dd|n = B p u , p|dd|p = n|ūu|n = B p d and p|ss|p = n|ss|n = B p s . An expression for one combination of these quantities is provided by the pion-nucleon σ term which may be extracted phenomenologically from data on low-energy π-nucleon scattering or on pionic atoms. In order to determine B p s , (8) has often been used in combination with the quantity which can be extracted phenomenologically from the octet baryon mass splittings, taking into account corrections that can be calculated in baryonic chiral perturbation theory (BχPT). One then has which is often parameterized by However, alternatives to these phenomenological estimates are now provided by the many lattice calculations that are now available, as we discuss in Section 2.2 below.
In order to evaluate Σ πN , we also need values for the ratios of quark masses m u /m d and m s /m d . In the past [10,22], we have used the estimates from [23], whereas the Particle Data Group (PDG) now quotes the following lattice estimates [24]: The PDG lattice review also quotes the following absolute values of the light quark masses in the MS scheme at a renormalization scale of 2 GeV: In contrast, in its Summary Tables the PDG quotes the broader ranges [24] m u m d = 0.38 to 0.58, 2m s m u + m d = 27.3 (7), m s m d = 17 to 22 (15) and As we discuss below, the resulting elastic scattering cross sections we study are relatively insensitive to the values of the quark mass ratios and are completely insensitive to the absolute value of the quark mass (e.g. m d ). For definiteness, we use (14) for the value of the strange quark mass and (13) for the values of the quark mass ratios.
The quantities B u + B d and B s discussed above suffice to calculate the matrix elements for scattering off nuclei with equal numbers of protons and neutrons, but additional information is required to calculate the difference between the cross sections for scattering off protons and neutrons, or for the scattering off nuclei with general values of (A, Z), as seen in (2).
Another combination of the B p q was calculated [25] using octet baryon masses in the relation: The uncertainty associated with measurements of the octet baryon masses is very small, but the accuracy of the octet mass formula (17) could be called into question.
Alternatively, one can calculate z from the ratio once B p u /B p d and the value of y are known. For example, the ratio B p u /B p d can be calculated from the QCD contribution to the proton-neutron mass difference: in combination with (8): using the value (13) of m u /m d . The measured mass difference m p − m n = −1.29 MeV (with negligible uncertainty) and the electromagnetic contribution is estimated to be m p − m n | QED = 1.04 (11) MeV, leading to m p − m n | QCD = −2.33 (11) MeV. Inserting the central value of m u /m d from (13) and the central value of Σ πN from (24) below into (20) to evaluate B p d /B p u , and then using (18) with y given by the central values of Σ πN and σ s in (24) and (25) below, respectively, we estimate A similar value z = 1.258(81) was found independently in a lattice calculation in [26]. Conservatively, we consider the range 1 < z < 2 in our subsequent estimates.

Uncertainties in
The most important uncertainties in calculations of the spin-independent WIMP-nuclear scattering matrix elements are those in Σ πN and σ s . In the past, σ s has often been evaluated by combining phenomenological estimates of Σ πN (8) and σ 0 (9) using (10). The uncertainties in Σ πN and σ 0 translate into significant uncertainties in the spin-independent cross section [7-10] 2 , as illustrated in Fig. 1, where the three-flavour expression (5) has been used. Here and in the analysis that follows, we use a representative point in the CMSSM consistent with the limits from LHC and other searches [14], namely with m 1/2 = 3000 GeV, m 0 = 8200 GeV, A 0 = 0 GeV, tan β = 10, and µ > 0. At this point, the LSP is mainly a Higgsino with mass 1.1 TeV whose relic density matches that determined by CMB experiments [27]. We have verified that similar trends in the dependences on Σ πN and σ 0 arise at other representative points, namely a stop-coannihilation point and s-channel A/H funnel point, though the values of the elastic cross section is very different for these points.
Here we revisit the uncertainties in Σ πN and σ s based on the considerable effort during the last decade made since [10], using lattice and other techniques, to determine Σ πN and σ s [35]. Although most of these recent values have been obtained from lattice calculations, many have been based on the phenomenology of low-energy π-nucleon interactions, and some have made extensive use of chiral perturbation theory, often in combination with lattice techniques.
The left panel of Fig. 2 displays all the estimates of Σ πN that we use, and the right panel displays all the estimates of σ s that are included in our analysis. We have tried to make a complete selection of all the determinations of these quantities that have not been superseded by later analyses by strongly-overlapping research groups. In each case, we have indicated by colour coding the primary phenomenological technique used in the calculation, and we have also indicated the corresponding arXiv reference number. We also indicate by shaded bands in Fig. 2 the estimates of Σ πN and σ s that we make on the basis of this new compilation, using the prescription that we describe below. More details of the determinations we use, including their numerical values, are given in Table 1   We now discuss the combinations of these estimates using the procedures adopted by the Particle Data Group (PDG) in cases where the uncertainties are not simply statistical [24]. Assuming uncorrelated Gaussian probability distributions for each of the estimates of Σ πN shown in the left panel of Fig. 2, we first construct the ideogram shown in the left panel of Fig. 3. As can be discerned from Fig. 2, the values of Σ πN are broadly distributed between 40-60 MeV, and the ideogram exhibits 3 minor peaks, slightly favoring the lower part of the range.
A naive weighted mean of all 21 determinations of Σ πN yields Naive : where we have combined statistical and systematic uncertainties in quadrature and centred asymmetric errors. It is clear, however, that this naive estimate would be a poor representation of the ideogram. One option proposed by the PDG under such circumstances is to rescale the error so that the χ 2 /d.o.f.= 1. In this case, the required renormalization factor is 1.7, yielding Rescaled : However, this would also be a poor representation of the ideogram, in view of its serrated Estimates of Σ πN and σ s (in MeV units). The first two lines are from previous compilations. The following lines are from recent determinations and, where two errors are quoted, the first is statistical and the second systematic. As indicated, most of the determinations are based on lattice calculations, many use baryon chiral perturbation theory (BχPT), three use data on pionic atoms (π atoms) and five use low-energy πN scattering data. The last line is our new compilation. displayed in the left panel of Fig. 3 as a vertical pink bar, for comparison with the ideogram.
As an alternative, we present in the right panel of Fig. 3 a representation of the estimates as a single Gaussian with the same normalization as the ideogram, and with its central value and error chosen to reproduce the 95% CL range of the ideogram as closely as possible: Gaussian representation : Although this functional form is far from perfect, we consider it a simple but fair representation of current estimates of Σ πN .
In this case the distribution obtained from the numerical estimates is again not symmetric, though it has a single-peak structure, and the representation (25) may again be considered a simple but fair representation of of current estimates of σ s . lattice methods and BχPT approaches in different ways.

Uncertainties in the Elastic Scattering Cross-Section
We now discuss the combination of the uncertainties in σ πN and σ s in the calculation of σ p SI . A scatter plot of values of (Σ πN , σ s ) from references in which values of both quantities are quoted (see Table 1) is shown in the left panel of Fig. 5. The horizontal and vertical grey bands show the ranges of Σ πN and σ s that we estimate on the basis of our global analysis. The right panel of Fig. 5 displays a two-dimensional joint ideogram of Σ πN and σ s based on the determinations in Fig. 5. We have studied via a regression analysis whether these determinations exhibit any correlations between σ πN and σ s . We find a slope in the (σ πN , σ s ) plane of 0.49 ± 1.08, i.e., no significant correlation. In the rest of this analysis we assume that there is no correlation between the uncertainties in σ πN and σ s . The analogue of Fig. 1 in terms of Σ πN and σ s is shown in Fig. 6, where the values of σ p SI are again calculated using the three-flavour expression (5). In the left panel, we see that for fixed σ s , there is no longer the large dependence of the cross section on Σ πN that was seen in Fig. 1 We now show how the uncertainties in Σ πN and σ s discussed above can be propagated into the uncertainty in the elastic scattering cross section. As we have discussed, the distributions for the hadronic matrix elements are not Gaussian, but we have provided Gaussian approximations to those distributions in (24) and (25), which we propagate to the errors in the cross section. We describe our procedure for scattering on protons only, the neutron case being simply related by an isospin transformation.
The uncertainty in the elastic cross section on a proton is simply where Since we are using σ s and its uncertainty directly, calculating the uncertainty in f Ts = σ s /m p is straightforward: In the cases of the other light quarks, the expressions for f Tq are more complicated: , . (29) To obtain the uncertainties in f T u,d , we propagate the uncertainties in Σ πN , the light quark mass ratio m u /m d = 0.46 ± 0.05 (13), and in the ratio B d /B u given in Eq. (18). This depends on the uncertainty in y, and hence depends ultimately on the uncertainties in m s /(m u +m d ) = 13.75±0.15 and σ s . The expression (27) takes into account the correlation in the uncertainties between the light and heavy quark contributiions.
We have verified that the uncertainties in m u /m d and in m s /m d given in (14) contribute very small uncertainties to σ p SI , a few per mille and below one per mille respectively. The uncertainty due to B p d /B p u is also small, at the ±2% level for 1 < z < 2 5 . Accordingly, we do not discuss these further.
We display in Fig. 7 contours of σ p SI (in units of 10 −9 pb) in the (Σ πN , σ s ) plane calculated using the three-flavour expression (5), together with the two-dimensional 68% and 95% CL regions (∆χ 2 < 2.3 and 5.99, respectively) given by our Gaussian fits (24,25) to Σ πN and σ s , assuming that there is no correlation, as discussed above.
Using our values for Σ πN (24) and σ s (25) that are also given in the last line of Table  1, we find σ p SI = (1.25 ± 0.13) × 10 −9 pb when we use the three-flavour expression (5) for our CMSSM benchmark point. The decrease in the cross section (by a factor of 2) relative to what we would have calculated using the values of σ πN and σ 0 used in [22] is due largely to the effective reduction in σ s . Moreover, the uncertainty in the cross section is a factor of 10 smaller. This reduction can be traced to using σ s (and its uncertainty) directly from the recent calculations rather than using the value inferred from (10) and the older values of Σ πN and σ 0 .

Dependence on heavy quark matrix elements
In this Section, we explore the sensitivity of σ p SI to the heavy quark matrix elements, using first the four-quark version (7) of the cross-section formula, and then the full six-flavour version.
There have been several calculations of the charm quark contribution to the proton mass, σ c ≡ m c N |cc|N , most using lattice techniques, as shown in Fig. 8 and Table 2. There have also been some phenomenological estimates of σ c , as also shown there. We do not include in our global fit the last three estimates, which depend on particular hypotheses concerning the fraction of the proton momentum in the infinite-momentum frame that is  Figure 7: Contours of σ p SI (in units of 10 −9 pb) in the (Σ πN , σ s ) plane, with the twodimensional 68% and 95% CL regions (darker and lighter shading) given by our Gaussian fits (24,25) to Σ πN and σ s . carried bycc pairs [64]. In the absence of a clear criterion for choosing between these hypotheses, we do not use them. An ideogram of the estimates of σ c that we retain is shown in Fig. 9. Following the same steps as used previously for Σ πN and σ s , we find Naive : σ c = 50.2 ± 8.5 MeV , Rescaled : σ c = 50.2 ± 9.6 MeV .
However, it is clear that these are exceedingly poor representations of the ideogram shown in Fig. 9, which is highly asymmetric. We choose to represent this by summing a pair of Gaussians G, with arbitrary normalizations and the following central values and errors: the maximum: Gaussian representation : This representation is also shown in Fig. 9, and gives a very good representation of the estimates in Table 2.
Using the four-quark expression (7) and the first Gaussian for σ c in (30), we find σ p SI = (1.07 ± 0.15) × 10 −9 pb, whereas the second Gaussian in (30) yields σ p SI = (1.40 ± 0.25) × 10 −9 pb, reflecting its larger central value and error. We note that, in the computation of these uncertainties, Eq. (27) must be modified in a way similar to Eq. (7), namely the first  Calculations of σ c (in MeV units). Where two errors are quoted, the first is statistical and the second systematic. As indicated, most of the determinations are based on lattice calculations. The last group of three phenomenological estimates are not included in our global fit.
sum is over the four quarks u, d, s, c, 2/27 → 2/25, and the second sum is over two quarks b, t.
Alternatively, one could adopt the value for σ c taken from a perturbative QCD calculation using the one-loop contribution associated with the gluon contribution [21] so that Using our estimates (24,25) of Σ πN and σ s , we find f T G = 0.917 ± 0.019 and hence  Figure 9: Ideogram compiling lattice and phenomenological calculations of σ c and our representation as a sum of two Gaussians (30). Also shown is a vertical pink band from our evaluation of the O(α 3 s ) perturbative QCD calculation of [65], shown in (33).
which lies between are two Gaussian estimates of σ c . Not surprisingly, the cross section calculated this way, σ p SI = (1.25 ± 0.14) × 10 −9 pb, is almost identical to our 3-quark calculation using Σ πN and σ s . In fact, this approach does not distinguish between the contributions of the heavy quarks and would imply σ b = σ t = σ c at this order in perturbation theory. Using Eq.
Also shown in Fig. 9 is a vertical pink band corresponding to this evaluation of the O(α 3 s ) perturbative QCD calculation [65]. It has been argued (see [68] for a review) that there may be non-perturbatively-generated intrinsic charm in the nucleon, in which case the perturbative calculation leading to (33) would be inapplicable. Another potential source of difference is caused by higher-dimensional operators that are induced when the charmed quark is integrated out [65], which are suppressed only by the charmed quark mass and thus may give a significant contribution to the nucleon mass. The difference between (30) and (33) may serve as a measure of the uncertainty associated with these possibilities. be more reliable, and yield [65]: which, in combination with our estimates (24,25) of Σ πN and σ s , and f T G yield Using the O(α 3 s ) perturbative calculations of σ c (33) and of σ b,t (36) in a full six-flavour calculation, we find σ p SI = (1.38 ± 0.17) × 10 −9 pb. The origin of the increase from the three-flavour approximation to the full six-flavour calculation is primarily the fact that the more detailed perturbative treatment of the heavy quarks increases their contributions, particularly that of the charmed quark, by O(15)%.
We consider the full six-flavour calculation using the estimates (24,25,33) and (36) to be the best approximation to the spin-independent WIMP scattering cross section currently available.

Spin-Dependent WIMP-Nucleon Scattering
In the case of the cross section σ SD for spin-dependent WIMP-nucleon scattering, the relevant matrix elements N |q i γ µ γ 5 q i |N are related to the corresponding quark contributions to the nucleon spin ∆q i . The combination ∆u − ∆d = g A = 1.27, the axial-current matrix element in neutron β-decay, which is known quite precisely. We estimate the combination ∆u+∆d−2∆s = 0.59 using other octet baryon weak decay matrix elements and SU(3) symmetry. A third combination of the light-quark ∆q i can be determined from parity-violating asymmetries in polarized deep-inelastic electron-and muon-nucleon scattering [69], which indicate a small but non-zero negative value of ∆s = −0.09 ± 0.03 when combined with the above-mentioned estimated of ∆u − ∆d and ∆u + ∆d − 2∆s. Measurements of hadron production asymmetries in polarized deep-inelastic scattering do not support a non-zero value of ∆s. Nevertheless, this confusion in the estimates of the ∆q i generates only moderate uncertainty in the cross section for spin-dependent WIMP-nucleon scattering, σ SD .
For the CMSSM focus-point benchmark point introduced above, we find that the value ∆s = −0.09±0.03 indicated by the parity-violating asymmetries in the total polarized deepinelastic cross sections leads to σ p SD = (9.4 ± 0.8) × 10 −7 pb, whereas the choice ∆s = 0 would yield σ p SD = (8.2 ± 0.7) × 10 −7 pb. The corresponding cross-section for scattering off neutrons is σ p SD = (7.1±0.7)×10 −7 pb for ∆s = −0.09. When ∆s = 0, there is virtually no difference between the cross sections for scattering on protons and neutrons. We conclude that the uncertainties in spin-dependent WIMP-nucleon scattering are comparable to the current uncertainties in spin-independent WIMP-nucleon scattering that have been the main focus of this paper.

Conclusions
We have re-analyzed in this paper ingredients in the calculation of the cross section for the spin-independent scattering of a massive WIMP on a nucleon. Based on available recent calculations using lattice and other techniques, we have used the prescription of the PDG to discuss the uncertainties in the quark scalar densities N |qq|N . We find a central value for the combination Σ πN of u and d densities that is somewhat smaller than found in previous compilations [10,22,33], though with a larger uncertainty: Σ πN = 46±11 MeV. All determinations are compatible within the stated errors. We also find σ s = m s N |ss|N = 35 ± 16 MeV, which is again smaller than suggested in previous compilations, with an uncertainty that is significantly smaller than in [10,22] but somewhat larger than in [33]. We find that the uncertainty in σ s is the largest single source of uncertainty in σ SI when it is calculated using the leading-order three-flavour approximation (5) for the spin-independent scattering matrix element.
We have also considered the impact of recent calculations of the heavy-quark scalar density matrix elements N |cc,bb,tt|N . The spread in lattice and phenomenological estimates of σ c = m c N |cc|N is quite large, and potentially a large source of uncertainty in Σ πN . That said, the possible range of σ c includes the value found to O(α 3 s ) in QCD perturbation theory. Using the O(α 3 s ) perturbative values of σ c , σ b and σ t in the full six-quark expression for σ πN , we find an enhancement of the cross section compared to the leading-order three-flavour approximation that is about 10% for the CMSSM fixed-point benchmark point that we have studied. We note, however, that the uncertainties in the leading-order three-quark approximation and the O(α 3 s ) six-flavour calculation overlap: σ p SI = (1.25 ± 0.13) × 10 −9 pb (three quarks)vs σ p SI = (1.38 ± 0.17) × 10 −9 pb (six quarks). As already mentioned, we consider the latter, using the estimates (24,25,33) and (36), to be the best approximation to the spin-independent WIMP scattering cross section currently available. As also mentioned above, we consider the spin-dependent WIMP scattering cross section to be relatively well understood.
For the future, we look forward to further refinements of calculations of Σ πN and σ s using first-principles lattice techniques as well as phenomenological inputs, recalling that these are the dominant sources of uncertainty in σ SI , if one accepts the perturbative calculation of σ c . We also look forward to more accurate lattice calculations of σ c , so as to check the accuracy of this perturbation calculation. Lattice calculations have made great progress over the past decade, but improvement is still desirable.