Power Corrections to the Universal Heavy WIMP-Nucleon Cross Section

WIMP-nucleon scattering is analyzed at order $1/M$ in Heavy WIMP Effective Theory. The $1/M$ power corrections, where $M\gg m_W$ is the WIMP mass, distinguish between different underlying UV models with the same universal limit and their impact on direct detection rates can be enhanced relative to naive expectations due to generic amplitude-level cancellations at leading order. The necessary one- and two-loop matching calculations onto the low-energy effective theory for WIMP interactions with Standard Model quarks and gluons are performed for the case of an electroweak SU(2) triplet WIMP, considering both the cases of elementary fermions and composite scalars. The low-velocity WIMP-nucleon scattering cross section is evaluated and compared with current experimental limits and projected future sensitivities. Our results provide the most robust prediction for electroweak triplet Majorana fermion dark matter direct detection rates; for this case, a cancellation between two sources of power corrections yields a small total $1/M$ correction, and a total cross section close to the universal limit for $M \gtrsim {\rm few} \times 100\,{\rm GeV}$. For the SU(2) composite scalar, the $1/M$ corrections introduce dependence on underlying strong dynamics. Using a leading chiral logarithm evaluation, the total $1/M$ correction has a larger magnitude and uncertainty than in the fermionic case, with a sign that further suppresses the total cross section. These examples provide definite targets for future direct detection experiments and motivate large scale detectors capable of probing to the neutrino floor in the TeV mass regime.

The heavy WIMP regime is also challenging for direct detection prospects. First, since the abundance of astrophysical dark matter particles for a given local energy density scales inversely as the particle mass, WIMPs are less abundant and detection rates for a given cross section are smaller. Second, as the mass spectrum of new physics states becomes stretched above the weak scale, the absence of accessible intermediate states forbids the simplest higgs-mediated interactions of WIMPs with nucleons, causing cross sections to be smaller.
However, although the interaction rates between WIMPs and nucleons may become smaller, they also become more certain. Heavy WIMP symmetry emerges in the limit that the WIMP mass, M , is large compared to the electroweak scale, i.e., M m W . Scattering cross sections become universal for given WIMP gauge quantum numbers, independent of the detailed UV physics [29,30]. For example, the cross section in this limit is independent of whether the particle is scalar or fermion, composite or fundamental. This universality provides robust sensitivity targets for ambitious next generation direct detection experiments, and will be key to interpreting any confirmed signal.
In previous work, two of the authors (RJH and MPS) analyzed the universal heavy WIMP limit for WIMP-nucleon scattering [29][30][31][32]. In this limit a generic amplitude-level cancellation [29,30,33] was shown to suppress the low-velocity WIMP-nucleon cross section to the level of ∼ 10 −47 cm 2 for wino-like WIMPs (i.e., self-conjugate electroweak triplets), and higgsino-like cross sections to an even smaller value. It is natural to ask whether in the presence of such cancellations, formally subleading effects can become numerically relevant beyond naive dimensional estimates. For example, focusing on the electroweak triplet case, the cancellation results in a total amplitude whose magnitude is ∼ 20% the size of the component subamplitudes [32], and a WIMP-nucleon cross section that is therefore suppressed by more than an order of magnitude. For TeV scale WIMPs, corrections of order m W /M could potentially enter at a similar numerical level. Here we analyze such 1/M power corrections, and quantify the corresponding violations of heavy WIMP universality.
The remainder of the paper is structured as follows. Section 2 extends Heavy WIMP Effective Theory (HWET) to incorporate 1/M power corrections, and Sec. 3 matches to the low energy effective theory after integrating out weak-scale particles. Section 4 computes the low-velocity scattering cross section of WIMPs on nucleons. Section 5 provides a summary and outlook.
2 Heavy WIMP Effective Theory at order 1/M Heavy particle effective theory can be used to analyze Standard Model (SM) extensions consisting of electroweak multiplets whose mass M is large compared to SM particle masses, M m W . Additional heavy multiplets, of mass M , may be integrated out for generic mass splitting M − M = O(M ). The special case M −M = O(m W ) requires that the additional multiplet appear explicitly in the HWET [30] Here we focus on a single multiplet of self-conjugate heavy particle fields with arbitrary spin, transforming under irreducible representations of electroweak SU (2) W ⇥ U (1) Y . Where a specific representation is required, we illustrate with an electroweak triplet. Working through order 1/M , the gauge-and Lorentz-invariant lagrangian in the one-heavy-particle sector (i.e., bilinear in h v ) is [29] where the timelike unit vector v µ defines the heavy WIMP velocity, µ⌫ t a is the field strength, and D µ ? = D µ v µ v · D. The heavy particle field h v satisfies projection relations as discussed in detail in Ref. [35]; for example, a fermionic heavy particle field obeys / vh v = h v . The self-conjugate condition is enforced in the e↵ective theory by requiring invariance of the lagrangian under where h c v denotes charge conjugation. For an irreducible representation of a self-conjugate field, we necessarily have zero hypercharge and integer isospin. The interactions labeled by c W 1 and c W 2 are present for the fermionic case. They contribute only to spin-dependent interactions at low velocity and will be ignored in the following.
The coe cient, 1/2, of the kinetic term D 2 ? /M in Eq. (1) is fixed by relativistic invariance [35,36]. The residual mass, m in Eq. (1), may be chosen for convenience. In a theory without electroweak symmetry breaking, taking m = 0 would enforce that M is the physical particle (pole) mass. For matching calculations at the electroweak scale, it is convenient to choose m = c H h|H| 2 i/M to cancel the mass contribution from electroweak symmetry breaking.
The parameter c H encodes ultraviolet physics above the scale M , and can be determined by a matching computation between a specified UV theory and HWET, described by Eq. (1). As an example, let Here we focus on a single multiplet of self-conjugate heavy particle fields with arbitrary spin, transforming under irreducible representations of electroweak SU (2) W × U (1) Y . Where a specific representation is required, we illustrate with an electroweak triplet.
Working through order 1/M , the gauge-and Lorentz-invariant lagrangian in the one-heavy-particle sector (i.e., bilinear in h v ) is [29] where the timelike unit vector v µ defines the heavy WIMP velocity, µν t a is the field strength, and D µ ⊥ = D µ − v µ v · D. The heavy particle field h v satisfies projection relations as discussed in detail in Ref. [35]; for example, a fermionic heavy particle field obeys / vh v = h v . The self-conjugate condition is enforced in the effective theory by requiring invariance of the lagrangian under where h c v denotes charge conjugation. For an irreducible representation of a self-conjugate field, we necessarily have zero hypercharge and integer isospin. The interactions labeled by c W 1 and c W 2 are present for the fermionic case. They contribute only to spin-dependent interactions at low velocity and will be ignored in the following.
The coefficient, −1/2, of the kinetic term D 2 ⊥ /M in Eq. (1) is fixed by relativistic invariance [35,36]. The residual mass, δm in Eq. (1), may be chosen for convenience. In a theory without electroweak symmetry breaking, taking δm = 0 would enforce that M is the physical particle (pole) mass. For matching calculations at the electroweak scale, it is convenient to choose δm = c H |H| 2 /M to cancel the mass contribution from electroweak symmetry breaking.
The parameter c H encodes ultraviolet physics above the scale M , and can be determined by a matching computation between a specified UV theory and HWET, described by Eq. (1). As an example, let us consider the case where the UV theory is given by the SM and an electroweak triplet of Majorana fermions. Matching onto HWET is illustrated in Fig. 1. The matching can be performed in the electroweak symmetric theory. After expanding in the Higgs mass parameter, the EFT diagrams are scaleless but dimensionful and thus vanish in dimensional regularization. Evaluation of the full theory diagrams yields the matching condition, As a simple renormalizable extension of this case, consider an additional electroweak multiplet transforming with higgsino quantum numbers (SU(2) W doublet, hypercharge Y = 1/2) with mass M D . For generic doublet-triplet mass splitting, where κ is the renormalizable trilinear coupling between the triplet and doublet fermions and the SM Higgs field [30,31]. As expected, when M D /M T → ∞, the result (4) reduces to the pure triplet result (3).
As an example involving scalar versus fermionic WIMP, consider the pseudo-Goldstone bosons that emerge from a QCD-like SM extension with vector-like SU(2) W couplings to underlying fermions [37,38]. Recall that the lightest such states form an electroweak triplet, regardless of the fermionic SU(2) W representation, and these "weakly interacting stable pions" are stabilized by a discrete symmetry (the unbroken analog of Standard Model G parity) [38]. The matching is again illustrated in Fig. 1, where now the full theory diagrams involve relativistic scalars, and also a counterterm four point function between the WIMP and SM Higgs field. The one-loop diagrams are UV divergent as a function of the cutoff Λ h representing the new strong interaction scale. The divergence is cancelled by the counterterm contribution. For the composite theory under consideration, the divergence corresponds to a logarithmically enhanced term in the matching. Taking this "chiral" logarithm as an estimate, we have where the ellipsis denotes O(1) terms that are not logarithmically enhanced. The last equality corresponds to a chiral symmetry breaking mass M induced by SU(2) W radiative corrections: [38]. The precise matching condition could in principle be computed using strong interaction methods in the chosen UV theory.
The cases (3), (4), and (5) establish the range of c H encountered in a variety of weakly coupled UV models, involving fermions and scalars, composite and elementary particles, and both pure-state and multi-component models. Before investigating the impact of these differences on direct detection cross sections, let us perform the remaining step of matching HWET onto effective QCD operators.

Effective Theory Below the Weak Scale
The scale separation m W Λ QCD , is exploited by matching onto a heavy particle effective theory for the relevant electrically neutral component of the WIMP, interacting with five flavor QCD: This matching step is common to different UV realizations of the electroweak triplet WIMP. In Eq. (6), h v is the neutral WIMP, and the spin-0 and spin-2 QCD operators for quarks and gluons are given by where d = 4 2✏ is the spacetime dimension, D ⌘ ! D D , and curly braces denote symmetrization, The ellipsis in Eq. (6) denotes higher dimension operators suppressed by ⇤ QCD /m W , and spin-dependent operators.
By restricting to dimension seven operators in Eq. (6), we are neglecting contributions suppressed by additional powers of ⇤ 2 low energy /m 2 W , where ⇤ low energy denotes any scale below m W (e.g., m b , or ⇤ QCD ). However, we will account for corrections of order m W /M in the coe cient functions appearing in Eq. (6) in our analysis of HWET power corrections. This power counting is appropriate for dark matter masses in the few hundred GeV to TeV range, a focus for current and next generation direct detection experiments.
We now proceed to match the theory (1) to the theory (6). By integrating out weak scale particles (the Higgs boson, electroweak gauge bosons, and the top quark), we obtain a solution for the twelve e↵ective theory coe cients (c   for S = 0, 2, leaving six independent coe cients. In the following, we denote generic up-and down-type quarks in five-flavor QCD by U and D, respectively, and an arbitrary quark flavor by q. Feynman diagrams contributing to the matching at O(1/M ) for the quark and gluon coe cients are shown in Figs. 2 and 3, respectively. Diagrams for gluon operators contain an additional loop compared to diagrams for quark operators. However, owing to the large gluon matrix elements of the nucleons, these 5 Figure 2: Diagrams contributing to 1/M quark matching, with the same notation as in Fig. 1. Diagrams with crossed W lines are not displayed. Figure 2: Diagrams contributing to 1/M quark matching, with the same notation as in Fig. 1. Diagrams with crossed W lines are not displayed.
where d = 4 2✏ is the spacetime dimension, D ⌘ ! D D , and curly braces denote symmetrization, The ellipsis in Eq. (6) denotes higher dimension operators suppressed by ⇤ QCD /m W , and spin-dependent operators.
By restricting to dimension seven operators in Eq. (6), we are neglecting contributions suppressed by additional powers of ⇤ 2 low energy /m 2 W , where ⇤ low energy denotes any scale below m W (e.g., m b , or ⇤ QCD ). However, we will account for corrections of order m W /M in the coe cient functions appearing in Eq. (6) in our analysis of HWET power corrections. This power counting is appropriate for dark matter masses in the few hundred GeV to TeV range, a focus for current and next generation direct detection experiments.
We now proceed to match the theory (1) to the theory (6). By integrating out weak scale particles (the Higgs boson, electroweak gauge bosons, and the top quark), we obtain a solution for the twelve e↵ective theory coe cients (c   for S = 0, 2, leaving six independent coe cients. In the following, we denote generic up-and down-type quarks in five-flavor QCD by U and D, respectively, and an arbitrary quark flavor by q. Feynman diagrams contributing to the matching at O(1/M ) for the quark and gluon coe cients are shown in Figs. 2 and 3, respectively. Diagrams for gluon operators contain an additional loop compared to diagrams for quark operators. However, owing to the large gluon matrix elements of the nucleons, these where

and curly braces denote symmetrization,
The ellipsis in Eq. (6) denotes higher dimension operators suppressed by Λ QCD /m W , and spin-dependent operators.
By restricting to dimension seven operators in Eq. (6), we are neglecting contributions suppressed by additional powers of Λ 2 low−energy /m 2 W , where Λ low−energy denotes any scale below m W (e.g., m b , or Λ QCD ). However, we will account for corrections of order m W /M in the coefficient functions appearing in Eq. (6) in our analysis of HWET power corrections. This power counting is appropriate for dark matter masses in the few hundred GeV to TeV range, a focus for current and next generation direct detection experiments.
We now proceed to match the theory (1) to the theory (6). By integrating out weak scale particles (the Higgs boson, electroweak gauge bosons, and the top quark), we obtain a solution for the twelve effective theory coefficients (c for S = 0, 2, leaving six independent coefficients. In the following, we denote generic up-and down-type quarks in five-flavor QCD by U and D, respectively, and an arbitrary quark flavor by q. Feynman diagrams contributing to the matching at O(1/M ) for the quark and gluon coefficients are shown in Figs. 2 and 3, respectively. Diagrams for gluon operators contain an additional loop compared to diagrams for quark operators. However, owing to the large gluon matrix elements of the nucleons, these operators are numerically of similar size, or dominant. We compute each of the operator coefficients in Eq. (6) to leading order in electroweak couplings, and hence we neglect one-loop diagrams involving c H for quark matching and two-loop diagrams involving c H for gluon matching. The impact of higher order contributions is estimated in the numerical analysis by varying the factorization scale. The techniques for electroweak scale matching detailed in Ref. [31] can be applied to the present calculation. We describe some pertinent details here. Compared to the leading power analysis considered in Ref. [31], computation of the 1/M corrections requires an extended master integral basis, and different components of the electroweak polarization tensor for the background field gluon matching.
In performing the gluon matching, it is convenient to distinguish between amplitudes with one or two bosons exchanged in the t-channel. One-boson exchange amplitudes are shown in the top row of Fig. 3, while two-boson exchange amplitudes are shown in the bottom row. The one-boson exchange amplitudes factorize into the one-boson exchange amplitudes for quark matching (top row of Fig. 2) times the quark loop, and contribute only to the scalar coefficient. For the two-boson exchange amplitudes, we employ electroweak polarization tensors, Π µν , induced by a loop of quarks in a background field of external gluons [31,39,40]. The temporal components, v µ v ν Π µν , are sufficient for the leading power analysis, while for the 1/M corrections we require also the spatial components; these may be extracted from Ref. [31]. The renormalization of Wilson coefficients for the quark and gluon operators is discussed in Ref. [32].
From the sum of one and two loop diagrams in Figs. 2 and 3, we obtain the final results for coefficients renormalized in the MS scheme: Here Li 2 (z) ≡ ∞ k=1 z k /k 2 is the polylogarithm of order 2. We also introduce the shorthand notation c i = (πα 2 2 /m 3 W )ĉ i for the effective operator coefficients, x i = m i /m W for masses expressed in units of m W , subscripts U and D denote arbitrary up-type (u, c or t) or down-type (d, s or b) quarks, respectively (so that the Kronecker delta, δ Db , is equal to unity for D = b and vanishes for D = d, s), and N = 2 is the number of massless Standard Model generations. The leading power results, represented by M → ∞ in Eq. (8), were obtained in Ref. [29]. 2 Let us remark that our results (8) obey the correct formal limit at small x t : [29] where c(n f = 6) denotes the coefficient in six-flavor QCD computed with three massless generations (i.e., m t m W ). [42] for c (2) g disagrees with the corresponding results in Ref. [29] and Eq. (8). We note that the expression for c (2) g in Ref.
[42] does not have the correct m t → 0 limit.

Cross sections
Let us consider the standard benchmark process for direct detection: the zero velocity limit of (spinindependent) WIMP-nucleon scattering. The cross section is determined by the spin-0 and spin-2 matrix elements, M In order to evaluate the hadronic matrix elements using available low energy inputs, the five flavor QCD theory must be matched to the appropriate three or four flavor theory, accounting for heavy quark threshold matching corrections and renormalization group evolution from electroweak to hadronic scales. Details of this matching can be found in Ref. [32]. For the spin-0 matrix elements, we match to the three flavor theory with NNNLO QCD corrections, 4 and following Ref. [32] make the default scale choices µ t = (m t + m W )/2 = 126 GeV, µ b = 4.75 GeV, µ c = 1.4 GeV, and µ 0 = 1.2 GeV. For the spin-2 matrix elements, we use NLO running and matching, and check that our evaluation is consistent with an evaluation at the weak scale, in the five flavor theory. The impact of higher order perturbative QCD corrections is estimated by varying factorization scales m 2 There are additional uncertainties associated with the hadronic form factors that characterize the overlap between the nucleon states and the quark and gluon operators. We employ the form factor central values and uncertainties from Ref. [32], which were adapted from Refs. [43][44][45][46] (see also Ref. [47]). Errors from all sources are added in quadrature to obtain the total cross section error.
cross sections for scattering on protons or neutrons are identical: 5 where m r = m p M/(m p + M ) ≈ m p is the reduced mass of the WIMP-nucleon system. In Fig. 4 The numerical evaluation (12) exhibits the partial cancellation of the universal M → ∞ result. For the Majorana fermion case, where c H = −3α 2 2 , the m W /M power correction also exhibits a surprising cancellation. The impact of neglected higher-order power corrections is estimated by including an uncertainty in the tensor amplitude as M 8 WIMP mass, the dependence of the cross section on the Higgs coupling c H differentiates the fermion and scalar cases. Figure 4 compares to existing limits from LUX [49], XENON1T [50], and PandaX-II [51], 6 and to projected sensitivities for the Xenon based experiments XENONnT [53], LZ [54], and DARWIN [55], and the Argon based experiment DEAP-3600 [52]. Also shown is the "discovery limit" for both Xenon and Argon due to neutrino backgrounds, taken from Ref. [48].

Summary
The scattering of atomic nuclei from approximately static sources of electroweak SU(2) is a well posed but intricate field theory problem that finds application in the search for WIMP dark matter in our local halo. LHC bounds have pushed the scale of new physics into a regime of large mass where direct detection is more challenging; however at the same time, universal predictions emerge in this regime and provide well-defined targets for next generation searches.
Generic amplitude level cancellations imply a potentially enhanced sensitivity of direct detection rate predictions to naively power suppressed interactions. In this paper we considered the general framework to analyze these power corrections, and analyzed the canonical case of a self-conjugate electroweak-triplet WIMP through order 1/M . Owing to heavy particle universality, the leading cross section prediction is identical whether such a WIMP is fermion or scalar, elementary or composite, and whether the WIMP is accompanied by other, heavier, particles in the Standard Model extension. Power corrections differentiate these scenarios, as illustrated in Fig. 4 for the benchmark low-velocity WIMP-nucleon cross section. For the elementary fermion case, two contributions to the power correction largely cancel, resulting in a small deviation from the universal M → ∞ limit. Our result represents the most complete calculation of the cross section for wino-like dark matter in the TeV regime. A standard thermal cosmology, consistent with the observed dark matter abundance, predicts M ∼ 2 − 3 TeV for such electroweak charged WIMPs [56][57][58][59][60]. The elementary Majorana fermion case involves no free parameters, and a prediction M ≈ 2.9 TeV is obtained after careful accounting for nonperturbative enhancements [61]. For the scalar case, the precise annihilation cross section, and hence cosmological mass constraint, depends on internal structure. At the TeV mass scales indicated by cosmological arguments, the predicted WIMP-nucleus scattering rate is comparable to the rate for neutrino-induced backgrounds. This cross section benchmark motivates very large scale detectors, and techniques to understand and probe into the so-called neutrino floor [62].
A number of investigations are suggested by our results. Besides its computational power, the heavy WIMP expansion provides an excellent classification scheme for WIMP direct detection in the increasingly important heavy WIMP regime. The SU(2) triplet (i.e., wino-like) case represents a canonical benchmark. Other quantum numbers such as the higgsino-like case may be similarly investigated. The proximity of the triplet cross section in Fig. 4 to the neutrino floor makes the precise WIMP mass of particular interest. For the composite scalar case, new nonperturbative physics enters in two key places: the Higgs coupling parameter c H that determines the size of the direct detection cross section; and the annihilation process that determines the cosmological mass constraint within a specified cosmological model. This physics could be accessed by lattice field theory [63] and/or chiral lagrangian analysis for the new strongly coupled sector. Nuclear effects such as two-body correlations could potentially have differing impacts on the spin-0 and spin-2 operators in Eq. (6). Like the 1/M corrections, the existence of a severe cancellation in the leading cross section can potentially enhance the impact of such naively subleading effects. Existing estimates for such nuclear effects, focused on the spin-0 sector, indicate a small impact relative to other uncertainties [64][65][66], however a more systematic analysis is warranted.