# Uncertainties in WIMP dark matter scattering revisited

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## Abstract

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 \(\langle N |{{{\bar{u}}}} u, {{{\bar{d}}}} d, {{{\bar{s}}}} s| N \rangle \), we also discuss the importances of heavy quark scalar densities \(\langle N |{{{\bar{c}}}} c, {{{\bar{b}}}} b, {{{\bar{t}}}} t| N \rangle \), 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 \(\langle N |{{{\bar{u}}}} u + {{{\bar{d}}}} d | N \rangle \) that is larger than has been assumed in some phenomenological analyses, and a range of \(\langle N |{{{\bar{s}}}} s| N \rangle \) that is smaller but compatible with earlier estimates. We also analyze the importance of the \({{{\mathcal {O}}}}(\alpha _s^3)\) 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.

## 1 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 \(\sigma ^p_{SI}\) approaching the neutrino floor [5, 6]. 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.

*i*. Rates for the first set of interactions \(\propto \alpha _{3i}\) are related to quark contributions to the nucleon mass: \(m_{q_i} \langle N | {{{\bar{q}}}}_i q_i | N \rangle \) and are independent of the nuclear spin, whereas rates for interactions \(\propto \alpha _{2i}\) are related to nucleonic matrix elements of axial currents in nucleons: \(\langle N |{{{\bar{q}}}}_i \gamma _\mu \gamma _5 q_i | N \rangle \), which are related to quark contributions to the nucleon spin. We discuss here the uncertainties in the matrix elements of the spin-independent interactions \(\propto \alpha _{3i}\), which are relatively important, as we shall see. The uncertainties in the spin-dependent interaction matrix elements \(\propto \alpha _{2i}\) are relatively small, as we discuss briefly towards the end of this paper.

Several approaches have been taken to estimating the \(\langle N | {{{\bar{q}}}}_i q_i | N \rangle \) matrix elements. One of the first was to use octet baryon mass differences and SU(3) symmetry to estimate the combination \(\sigma _0 \equiv \frac{1}{2} (m_u + m_d) \langle N | {{{\bar{u}}}} u + {{{\bar{d}}}} d - 2 {{{\bar{s}}}} s | N \rangle \), together with data on low-energy \(\pi N\) scattering to estimate the quantity \(\Sigma _{\pi N} \equiv \frac{1}{2} (m_u + m_d) \langle N | {{{\bar{u}}}} u + {{{\bar{d}}}} d | N \rangle \).^{1} As has been discussed in previous work, see, e.g., [9, 12, 13, 14, 15], combining these estimates of \(\sigma _0\) and \(\Sigma _{\pi N}\) led to relatively large estimates for \(\langle N | {{{\bar{s}}}} s | N \rangle \), 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 \(\Sigma _{\pi N}\) and relatively small estimates for \(\langle N | {{{\bar{s}}}} s | N \rangle \). The corresponding values of \(\sigma _0\) may be similar to the estimates made using baryon masses and SU(3), but some calculations correspond to significantly larger values of \(\sigma _0\). In parallel, there have been calculations using baryon chiral perturbation theory (B\(\chi \)PT) that may lead to much larger values of \(\sigma _0\), see, e.g., [16], close to the data-based estimates of \(\Sigma _{\pi N}\), which may also correspond to relatively small values of \(\langle N | {{{\bar{s}}}} s | N \rangle \).

In this paper we compile the lattice and other estimates of \(\Sigma _{\pi N}\) and \(\sigma _s \equiv m_s \langle N | {{{\bar{s}}}} s | N \rangle \) 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 \(\Sigma _{\pi N}\) is similar to values suggested previously, but with a larger uncertainty, while our combined estimate of \(\sigma _s\) 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 \(\sigma ^p_{SI}\) at a specific benchmark point in the focus-point region [17, 18, 19, 20] of the constrained minimal supersymmetric Standard Model (CMSSM) [21, 22, 23, 24, 25, 26, 27, 28, 29, 30], noting that other supersymmetric parameter sets exhibit similar trends. As we discuss, the uncertainties in \(\sigma ^p_{SI}\) related to light quark masses and \(\langle N | {{{\bar{u}}}} u | N \rangle / \langle N | {{{\bar{d}}}} d | N \rangle \) are significantly smaller than those associated with \(\sigma _s\). We also discuss the uncertainties in \(\sigma ^p_{SI}\) associated with the heavy quark matrix elements \(\langle N | {\bar{c}} c, {{{\bar{b}}}} b, {{{\bar{t}}}} t | N \rangle \). 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 \(\sigma _c \equiv m_c \langle N | {{{\bar{c}}}} c | N \rangle \) are available, and span a wide range that straddles the perturbative estimate. If the \({{{\mathcal {O}}}}(\alpha _s^3)\) perturbative estimates are used for all the heavy-quark matrix elements, as we advocate, the corresponding uncertainties in \(\sigma ^p_{SI}\) are small, but if the full range of numerical estimates of \(\sigma _c\) is considered the associated uncertainty is comparable to that associated with \(\sigma _s\).

The layout of this paper is as follows. In Sect. 2 we review the strong-interaction quantities entering the calculation of the spin-independent cross-section \(\sigma ^p_{SI}\). Inputs to the calculation of \(\sigma ^p_{SI}\) are discussed in Sect. 2.1, the individual uncertainties in the matrix elements of the densities of the light quarks *u*, *d*, *s* are discussed in Sect. 2.2, and their propagation into the calculation of \(\sigma ^p_{SI}\) are discussed in Sect. 2.3. Section 2.4 is dedicated to a discussion of the matrix elements of the heavy quarks *c*, *b*, *t*. Finally, Sect. 3 contains a brief discussion of the uncertainties in the calculation of the spin-dependent cross-section \(\sigma _{SD}\), and our conclusions are summarized in Sect. 4.

## 2 Spin-independent WIMP-nucleon scattering

### 2.1 Inputs to the matrix element calculation

*Z*and atomic number

*A*can be written as [1, 7, 8, 31, 32, 33, 34, 35, 36, 37, 38, 39]

*n*, and the quantities \(f^N_{T_q}\) are defined by

As we also discuss later, there are also calculations of \(f^N_{T_c}, f^N_{T_b}\) and \(f^N_{T_t}\) to \({{{\mathcal {O}}}}(\alpha _s^3)\) in perturbative QCD. These perturbative calculations are expected to be very reliable for \(f^N_{T_b}\) and \(f^N_{T_t}\), perhaps less so for \(f^N_{T_c}\). Therefore, we also estimate \(\sigma ^p_{SI}\) 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 \(f^N_{T_c}\). As we discuss later, we consider the \({{{\mathcal {O}}}}(\alpha _s^3)\) six-flavour approach to be the best available at the present time.

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).

*z*from the ratio

*y*are known. For example, the ratio \(B_u^p/B_d^p\) can be calculated from the QCD contribution to the proton-neutron mass difference:

*y*given by the central values of \(\Sigma _{\pi N}\) and \(\sigma _s\) in (24) and (25) below, respectively, we estimate

### 2.2 Uncertainties in \(\Sigma _{\pi N}\), \(\sigma _0\) and \(\sigma _s\)

^{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 [30], namely with \(m_{1/2} = 3000\) GeV, \(m_0 = 8200\) GeV, \(A_0 = 0\) GeV, \(\tan \beta = 10\), and \(\mu > 0\). At this point, the LSP is mainly a Higgsino with mass \(\simeq 1.1\) TeV whose relic density matches that determined by CMB experiments [50, 51]. We have verified that similar trends in the dependences on \(\Sigma _{\pi N}\) and \(\sigma _0\) 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.

The left panel of Fig. 1 shows the values of \(\sigma ^p_{SI}\) obtained as a function of \(\Sigma _{\pi N}\) for three indicative values of \(\sigma _0 = 20, 36, 50\) MeV, and the right panel of Fig. 1 shows the values of \(\sigma ^p_{SI}\) obtained as a function of \(\Sigma _{0}\) for the three indicative values \(\Sigma _{\pi N} =40, 50, 60\) MeV. In the two cases, representative uncertainties of 7 MeV were assumed in \(\sigma _0\) and \(\Sigma _{\pi N}\), respectively. In making these plots, we have assumed that \(B_s \ge 0\) and hence imposed the restriction \(\Sigma _{\pi N} \ge \sigma _0\). For the indicative values \(\Sigma _{\pi N} = 50 \pm 7\) MeV and \(\sigma _0 = 36 \pm 7 \) MeV, we find \(\sigma ^p_{SI} = (2.5 \pm 1.5) \times 10^{-9}\) pb using the three-flavour formula (5).

‘Legacy’ values of \(\Sigma _{\pi N}\) and \(\sigma _0\) quoted in [15] were \(\Sigma _{\pi N} = 64 (8)\) MeV from \(\pi \)-N scattering and \(\sigma _0 = 36 (7)\) MeV from octet baryon mass differences [52, 53, 54, 55, 56]. Subsequently, since lattice calculations have tended to yield lower values of \(\Sigma _{\pi N} \gtrsim 40\) MeV, and the MasterCode collaboration has been using the ‘compromise’ value \(\Sigma _{\pi N} = 50 (7)\) MeV when using the experimental limits on spin-independent dark matter scattering on nuclei in global fits to supersymmetric models (see, e.g., [42]). In combination with \(\sigma _0 = 36 (7)\) MeV [52, 53, 54, 55, 56], this yields \(\sigma _s = 192(136)\) MeV. Another global fitting group, the GAMBIT Collaboration (see, e.g., [57]), has, on the other hand, been using the smaller value \(\sigma _s = 43 (8)\) MeV, which is based on a compilation of lattice data made in 2011 [58], together with a larger value of \(\Sigma _{\pi N} = 58 (9)\) MeV. This combination corresponds to \(\sigma _0 = 55\) MeV, considerably larger than the estimate from octet baryon mass differences [52], but within the range argued in [16] to be consistent with B\(\chi \)PT.

Here we revisit the uncertainties in \(\Sigma _{\pi N}\) and \(\sigma _s\) based on the considerable effort during the last decade made since [15], using lattice and other techniques, to determine \(\Sigma _{\pi N}\) and \(\sigma _s\) [59]. Although most of these recent values have been obtained from lattice calculations, many have been based on the phenomenology of low-energy \(\pi \)-nucleon interactions, and some have made extensive use of chiral perturbation theory, often in combination with lattice techniques. As already commented in [60], and discussed in more detail below, there is tension between these various estimates, and the uncertainties are not purely statistical.

^{3}

Estimates of \(\Sigma _{\pi N}\) and \(\sigma _s\)

References | \(\Sigma _{\pi N}\) | Uncertainties | \(\sigma _s\) | Uncertainties | Method |
---|---|---|---|---|---|

[42] | 50 | 7 | 191 | 135 | Compilation |

[57] | 58 | 9 | 43 | 8 | Compilation |

[61] | 53 | \( 2^{+21}_{-7}\) | 21.7 | \({}^{+15.1}_{-13.4}\) | Lattice |

[62] | 59 | 7 | B\(\chi \)PT, \(\pi \) atoms | ||

[63] | 31 | \(3 \pm 4\) | 71 | \(34 \pm 59\) | Lattice |

[64] | 38 | 12 | 12 | \(^{+23}_{-16}\) | Lattice |

[65] | 40.9 | \(7.5 \pm 4.7\) | Lattice | ||

59.6 | \(5.1 \pm 6.9\) | Lattice | |||

[66] | 45 | 6 | 21 | 6 | Lattice |

[67] | 37 | \(8 \pm 6\) | Lattice | ||

[68] | 8.4 | \(14.1 \pm 15.0\) | Lattice | ||

21.6 | \(27.2 \pm 26.3\) | Lattice | |||

[16] | 16 | \(80 \pm 60\) | B\(\chi \)PT | ||

[69] | 43 | \(1\pm 6\) | 126 | \(24 \pm 54\) | Lattice/B\(\chi \)PT |

[70] | 43.2 | 10.3 | Lattice | ||

[71] | 44 | 12 | \(\pi \)N scattering | ||

[72] | 45 | 6 | \(\pi \)N scattering | ||

[73] | 49 | \(10 \pm 15\) | Lattice | ||

[74] | 32.8 | 31.0 | Lattice | ||

[75] | 52 | \(3 \pm 8\) | Lattice/B\(\chi \)PT | ||

41 | \( 5 \pm 4\) | Lattice/B\(\chi \)PT | |||

[76] | 33.3 | 6.2 | Lattice/B\(\chi \)PT | ||

[77] | 27 | \(27 \pm 4\) | Lattice/B\(\chi \)PT | ||

59.1 | \(1.9 \pm 3\) | \(\pi \) atoms | |||

[80] | 38 | \( 3 \pm 3\) | 105 | \(41 \pm 37\) | Lattice |

[81] | 45.9 | \( 7.4 \pm 2.8\) | 40.2 | \(11.7 \pm 3.5\) | Lattice |

[82] | 37.2 | \( 2.6^{+4.7}_{-2.9}\) | 41.1 | \(8.2^{+7.8}_{-5.8}\) | Lattice |

[49] | 35 | 6.1 | 34.7 | 12.2 | Lattice |

[83] | 8.5 | \(4.4 \pm 86.6\) | \(\pi \) atoms, \(\pi \)N scattering | ||

144.7 | \(4.6 \pm 45.9\) | \(\pi \) atoms, \(\pi \)N scattering | |||

[84] | 35.2 | 5.5 | 30.5 | 8.5 | B\(\chi \)PT |

[85] | 64.9 | \( 1.5 \pm 13.2\) | Lattice/B\(\chi \)PT | ||

[86] | 58 | 5 | \(\pi \)N scattering | ||

[87] | 50.3 | \(1.2 \pm 3.4\) | Lattice/B\(\chi \)PT | ||

48 | 38 | 15 | Lattice/B\(\chi \)PT | ||

[90] | 69 | 10 | B\(\chi \)PT | ||

This work | 46 | 11 | 35 | 16 | New compilation |

^{4}shown in the left panel of Fig. 3. As can be discerned from Fig. 2, the values of \(\Sigma _{\pi N}\) are broadly distributed between 40 and 60 MeV, and the ideogram exhibits 3 minor peaks, slightly favoring the lower part of the range.

^{5}The rescaled value is displayed in the left panel of Fig. 3 as a vertical pink bar, for comparison with the ideogram.

### 2.3 Uncertainties in the elastic scattering cross-section

The analogue of Fig. 1 in terms of \(\Sigma _{\pi N}\) and \(\sigma _s\) is shown in Fig. 6, where the values of \(\sigma ^p_{SI}\) are again calculated using the three-flavour expression (5). In the left panel, we see that for fixed \(\sigma _s\), there is no longer the large dependence of the cross section on \(\Sigma _{\pi N}\) that was seen in Fig. 1. The three bands (which overlap) correspond to \(\sigma _s = 30, 50\) and 100 MeV. In the right panel, there are three bands corresponding to fixed values of \(\Sigma _{\pi N} = 40, 50\) and 60 MeV that lie almost on top of each other, and one sees quite clearly the dependence of the cross section on \(\sigma _s\). Note that the thickness of the bands here are significantly narrower than those in Fig. 1. This is mainly due to the smaller uncertainty in \(\sigma _s\) when it is taken directly from Eq. (25) rather than derived indirectly using estimates of \(\sigma _0\). Using \(\sigma _s\) as an input into calculations of \(\Sigma _{\pi N}\) is therefore preferred.

We now show how the uncertainties in \(\Sigma _{\pi N}\) and \(\sigma _s\) 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.

*y*, and hence depends ultimately on the uncertainties in \(m_s/(m_u+m_d) = 13.75 \pm 0.15\) and \(\sigma _s\). The expression (27) takes into account the correlation in the uncertainties between the light and heavy quark contributions.

We have verified in the benchmark model assumed that the uncertainties in \(m_u/m_d\) and in \(m_s/m_d\) given in (14) contribute very small uncertainties to \(\sigma ^p_{SI}\), 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 \(\pm 2\)% level for \(1< z < 2\).^{6} We note that our benchmark point is taken from a supersymmetric theory and the scattering of the dark matter candidate in this model on a proton is dominated by the heavy quark content. It is quite possible that other dark matter candidates are more sensitive to the scattering off of light quarks and in that case, the uncertainty due to \(B^p_d/B^p_u\) and *z* is more important.

Using our values for \(\Sigma _{\pi N}\) (24) and \(\sigma _s\) (25) that are also given in the last line of Table 1, we find \(\sigma ^p_{SI} = (1.25 \pm 0.13) \times 10^{-9}\) 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 \(\Sigma _{\pi N}\) and \(\sigma _0\) used in [42] is due largely to the effective reduction in \(\sigma _s\). Moreover, the uncertainty in the cross section is a factor of 10 smaller. This reduction can be traced to using \(\sigma _s\) (and its uncertainty) directly from the recent calculations - as we recommend - rather than using the value inferred from (10) and the older values of \(\Sigma _{\pi N}\) and \(\sigma _0\).

### 2.4 Dependence on heavy quark matrix elements

In this section, we explore the sensitivity of \(\sigma ^p_{SI}\) 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.

*G*, with arbitrary normalizations and the following central values and errors: the maximum:

Values of the \(f^{N}_{T_{q,G}}\)

Nucleon | \(f^N_{T_u}\) | \(f^N_{T_d}\) | \(f^N_{T_s}\) | \(f^N_{T_G}\) | \(f^N_{T_c}\) | \(f^N_{T_b}\) | \(f^N_{T_t}\) |
---|---|---|---|---|---|---|---|

Proton | 0.018(5) | 0.027(7) | 0.037(17) | 0.917(19) | 0.078(2) | 0.072(2) | 0.069(1) |

Neutron | 0.013(3) | 0.040(10) | 0.037(17) | 0.910(20) | 0.078(2) | 0.071(2) | 0.068(2) |

Using the four-quark expression (7) and the first Gaussian for \(\sigma _c\) in (30), we find \(\sigma ^p_{SI} = (1.07 \pm 0.15) \times 10^{-9}\) pb, whereas the second Gaussian in (30) yields \(\sigma ^p_{SI} = (1.40 \pm 0.25) \times 10^{-9}\) 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 sum is over the four quarks *u*, *d*, *s*, *c*, \(2/27 \rightarrow 2/25\), and the second sum is over two quarks *b*, *t*.

^{7}:

*b*and

*t*quarks are expected to be more reliable, and yield [92]:

We consider the full six-flavour calculation using the estimates (24, 25, 33) and (35) to be the best approximation to the spin-independent WIMP scattering cross section currently available.

## 3 Spin-dependent WIMP-nucleon scattering

In the case of the cross section \(\sigma _{SD}\) for spin-dependent WIMP-nucleon scattering, the relevant matrix elements \(\langle N |{{{\bar{q}}}}_i \gamma _\mu \gamma _5 q_i | N \rangle \) are related to the corresponding quark contributions to the nucleon spin \(\Delta q_i\). The combination \(\Delta u - \Delta d = g_A = 1.27\), the axial-current matrix element in neutron \(\beta \)-decay, which is known quite precisely. We estimate the combination \(\Delta u + \Delta d - 2 \Delta s = 0.59\) using other octet baryon weak decay matrix elements and SU(3) symmetry. A third combination of the light-quark \(\Delta q_i\) can be determined from parity-violating asymmetries in polarized deep-inelastic electron- and muon-nucleon scattering [96], which indicate a small but non-zero negative value of \(\Delta s = - 0.09 \pm 0.03\) when combined with the above-mentioned estimated of \(\Delta u - \Delta d\) and \(\Delta u + \Delta d - 2 \Delta s\). Measurements of hadron production asymmetries in polarized deep-inelastic scattering do not support a non-zero value of \(\Delta s\). Nevertheless, this confusion in the estimates of the \(\Delta q_i\) generates only moderate uncertainty in the cross section for spin-dependent WIMP-nucleon scattering, \(\sigma _{SD}\).

For the CMSSM focus-point benchmark point introduced above, we find that the value \(\Delta s = - 0.09 \pm 0.03\) indicated by the parity-violating asymmetries in the total polarized deep-inelastic cross sections leads to \(\sigma ^p_{SD} = (9.4 \pm 0.8) \times 10^{-7}\) pb, whereas the choice \(\Delta s = 0 \pm 0.03\) would yield \(\sigma ^p_{SD} = (8.2 \pm 0.7) \times 10^{-7}\) pb. The uncertainty in the spin-dependent cross section is largely determined by the uncertainty in \(\Delta s\), and ignoring the uncertainty in \(\Delta _s\) would reduce the uncertainty in \(\sigma ^p_{SD}\) to \(\pm 0.2\). The corresponding cross-section for scattering off neutrons is \(\sigma ^p_{SD} = (7.1 \pm 0.7) \times 10^{-7}\) pb for \(\Delta s = - 0.09\). When \(\Delta s = 0\), 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.

## 4 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 \(\langle N | {{{\bar{q}}}} q | N \rangle \). We find a central value for the combination \(\Sigma _{\pi N}\) of *u* and *d* densities that is somewhat smaller than found in previous compilations [15, 42, 57], though with a larger uncertainty: \(\Sigma _{\pi N} = 46 \pm 11\) MeV. All determinations are compatible within the stated errors. We also find \(\sigma _s = m_s \langle N | {{{\bar{s}}}} s | N \rangle = 35 \pm 16\) MeV, which is again smaller than suggested in previous compilations, with an uncertainty that is significantly smaller than in [15, 42] but somewhat larger than in [57]. We find (for the benchmark supersymmetric model we have studied) that the uncertainty in \(\sigma _s\) is the largest single source of uncertainty in \(\sigma _{SI}\) when it is calculated using the leading-order three-flavour approximation (5) for the spin-independent scattering matrix element.^{8} The corresponding values of the \(f^N_{T_{u,d,s,G}}\) for scattering on protons and neutrons obtained assuming \(z = 1.49\) are shown in the first four columns of Table 3, where we include the uncertainties due to the *u*, *d*, *s* mass ratios. Note that although the spin-independent cross section is not particularly sensitive to *z*, the values of \(f^N_{T_{u,d}}\) do depend on *z*. However, for \(1< z < 2\), their central values vary within the 1\(-\sigma \) ranges quoted in Table 3. Specifically, for \(z = 1 (2)\), we find \(f^p_{T_u} = 0.015 \pm 0.004\) \((0.020 \pm 0.005)\) and \(f^p_{T_d} = 0.034 \pm 0.008\) \((0.023 \pm 0.006)\).

We have also considered the impact of recent calculations of the heavy-quark scalar density matrix elements \(\langle N | {{{\bar{c}}}} c, {{{\bar{b}}}} b, {{{\bar{t}}}} t | N \rangle \). The spread in lattice and phenomenological estimates of \(\sigma _c = m_c \langle N | {{{\bar{c}}}} c | N \rangle \) is quite large, and potentially a large source of uncertainty in \(\sigma _{SI}\). That said, the possible range of \(\sigma _c\) includes the value found to \({{{\mathcal {O}}}}(\alpha _s^3)\) in QCD perturbation theory. The values of the \(f^N_{T_{c,b,t}}\) that we find using the \({{{\mathcal {O}}}}(\alpha _s^3)\) perturbative calculations of \(\sigma _c, \sigma _b\) and \(\sigma _t\) are shown in the last three columns of Table 3. Using the full six-quark expression for \(\sigma ^p_{SI}\), 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 \(\mathcal{O}(\alpha _s^3)\) six-flavour calculation overlap: \(\sigma ^p_{SI} = (1.25 \pm 0.13) \times 10^{-9}\) pb (three quarks)vs \(\sigma ^p_{SI} = (1.38 \pm 0.17) \times 10^{-9}\) pb (six quarks). As already mentioned, we consider the latter, using the estimates (24, 25, 33) and (35), 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 \(\Sigma _{\pi N}\) and \(\sigma _s\) using first-principles lattice techniques as well as phenomenological inputs, recalling that these are the dominant sources of uncertainty in \(\sigma _{SI}\), if one accepts the perturbative calculation of \(\sigma _c\). We also look forward to more accurate lattice calculations of \(\sigma _c\), so as to check the accuracy of this perturbative calculation. Lattice calculations have made great progress over the past decade, but improvement is still desirable.

## Footnotes

- 1.
- 2.
We describe the propagation of these uncertainties in Sect. 2.3.

- 3.
We apologize in advance to authors whose work we have overlooked or misrepresented in compiling this Table, and welcome suggestions for its completion and improvement.

- 4.
The ideogram is constructed using the prescription of the PDG [44], and is a sum of Gaussians for each measurement with an area normalized to be \(1/\sigma _i\) where \(\sigma _i\) is the uncertainty in the measurement.

- 5.
This feature may reflect the existence of unidentified systematic uncertainties that affect different lattice methods and B\(\chi \)PT approaches in different ways.

- 6.
Over this range of

*z*, \(\sigma ^p_{SI}/\sigma ^n_{SI}\) varies between 1.00 and 0.94. - 7.
Here and in the rest of this section we give results for \(\sigma _{c,b,t}\) in the proton. Becasue of isospin violation, perturbative calculations of the central values in the neutron yield slightly different results, but these are indistinguishable within the uncertainties. We report results for the \(f^{p,n}_{T_{c,b,t}}\) separately in Table 3 below.

- 8.
As we noted above, the uncertainty due to \(B_d^p/B_u^p\) and

*z*may be more important in models where the spin-independent scattering occurs primarily off*u*and*d*quarks.

## Notes

### Acknowledgements

We would like to thank M. Hoferichter, F. Kahlhoefer, U. Meißner and M. Voloshin for useful discussions. The work of JE was supported partly by the United Kingdom STFC Grant ST/P000258/1 and partly by the Estonian Research Council via a Mobilitas Pluss grant. The work of N.N. was supported by the Grant-in-Aid for Scientific Research (No.17K14270). The work of K.A.O. was supported in part by DOE grant DE-SC0011842 at the University of Minnesota.

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