Renormalization Group Effects in Dark Matter Interactions

We present a renormalization-group (RG) analysis of dark matter interactions with the standard model, where dark matter is allowed to be a component of an electroweak multiplet, and has a mass at or below the electroweak scale. We consider, in addition to the gauge interactions, the complete set of effective operators for dark matter interactions with the standard model above the weak scale, up to and including mass dimension six. We calculate the RG evolution of these operators from the high scale Lambda down to the weak scale, and perform the matching to the tower of effective theories below the weak scale. We also summarize the RG evolution below the weak scale and the matching to the nonrelativistic nuclear interactions. We present several numerical examples and show that in certain cases the dark matter - nucleus scattering rate can change by orders of magnitude when the electroweak running is included.


I. INTRODUCTION
For a large class of dark matter (DM) models, the physics of direct detection experiments can be described using Effective Field Theory (EFT) . There are several scales that enter the problem: the DM mass, m χ , the scale of the mediators, Λ, through which the DM interacts with the visible sector, and, finally, the standard model (SM) scales -the masses of the SM particles and the scale of strong interactions, Λ QCD . The EFT description of DM direct detection is appropriate as long as the mediators are heavier than a few hundred MeV, i.e., above the typical momentum exchange in direct detection experiments. Furthermore, the EFT description is necessary in order to consistently treat the hadronic physics in the scattering of DM on nuclei.
In the present manuscript we are interested in the connection between the DM theory at the mediator scale, Λ, and the EFT describing DM direct detection. To do so one needs to run through a tower of EFTs that connects the UV scale Λ with the nuclear scale. We where m Z = 91.1876 GeV is the Z-boson mass. Fig. 1 depicts the resulting tower of EFTs.
At a particular scale µ the appropriate EFT is constructed from the relevant propagating performed [18][19][20][21]. This is then used in a chiral EFT approach to nuclear forces together with the nuclear response functions to obtain the hadronic matrix element for each of the DM-nucleon interaction operators [9-13, 23, 34, 58, 59].
The EFT operators mix under the renormalization-group (RG) evolution when going from Λ to m Z , from m Z to m b , etc. The primary purpose of this paper is to calculate the anomalous dimensions for the RG running from Λ to m Z for the case of Dirac fermion DM in an arbitrary electroweak multiplet. This RG running can be phenomenologically important since it can mix operators that are velocity suppressed in the nonrelativistic limit with operators that are not velocity suppressed (see Refs. [22,[60][61][62][63][64][65][66][67] for further examples of relevant loop corrections in DM interactions). In addition, we also perform the rest of the running and matching down to the nuclear level and give several numerical examples.
The possibility that DM is part of an electroweak multiplet is allowed by direct detection constraints as long as DM does not couple to the Z boson at tree level (for instance, this is the case if the DM multiplet has odd dimensionality and does not carry hypercharge). The exchanges of W, Z, h bosons with a quark current then generate a contribution to DM-nucleon scattering at one-loop and two-loop level, see Fig. 2 and Refs. [68,69]. Since these contributions are loop-suppressed and result in either a chirality flip or spin-dependent scattering, it is quite possible that the leading contribution is due to exchanges of heavy mediators. This is illustrated in Fig. 3 where we show for several non-renormalizable interactions at which values of the mediator mass, Λ equal , the non-renormalizable and renormalizable contributions to scattering on Xenon are equal. For mediators lighter than Λ equal the scattering rates are dominated by the non-renormalizable interactions. Even if the mediators are very heavy, many orders of magnitude heavier than the weak scale, they can still give the leading effect in spin-independent scattering. Furthermore, the operators that lead to velocity-suppressed contributions, such as vector-axial interactions, are only poorly constrained. A mixing into velocity unsuppressed, coherently enhanced operators at one-loop, two-loop, or potentially even three-loop can therefore still be the leading contribution to the scattering rate. This motivates both the use of the complete tower of EFTs and the calculation of the leadinglogarithmic effects captured by RG running.
In our analysis we cover both the case of DM with electroweak-scale mass, m χ ∼ m Z , and light DM, m χ m Z . Note that we do not require DM to be a thermal relic, and therefore allow for a large range of DM masses and interactions. Above the electroweak scale we limit our analysis to a basis of operators with mass dimension five and six, and work to one-loop order for the anomalous dimensions. The matching corrections are calculated at tree level, except for the cases where one-loop contributions can be numerically important, for instance, if the matching generates gluonic operators. The subsequent RG evolution below the electroweak scale has been described in detail in Refs. [16,21]; see also Ref. [23] for a computer code that implements the running numerically. Several interesting cases are left for future work, such as the case of several DM multiplets, the case of scalar DM, the case of very heavy DM, m χ m Z , as well as the analysis of higher dimension operators.
The paper is organized as follows. In Section II we give the complete basis of dimension-

II. EFFECTIVE LAGRANGIAN ABOVE THE ELECTROWEAK SCALE
We extend the SM by a single Dirac fermion, a Z 2 -odd electroweak multiplet of dimension d χ = 2I χ + 1, hypercharge Y χ , and mass m χ , whose electrically neutral component is the DM. Here I χ is the weak isospin of the DM multiplet (see App. A for our conventions). Oneloop electroweak corrections split the multiplet components, so that the charged particles are heavier than DM and decay in the early universe (see Refs. [7,70]). In the numerical examples in Section V we set Y χ = 0, so that the phenomenologically dangerous tree-level vectorial Z couplings are absent. In the calculation of anomalous dimension in this section we do, however, keep the Y χ dependence, as this may prove useful in other applications of our results.
Within our set-up there are two types of DM interactions with the visible sector: either through the exchanges of SM particles, or through new states -the mediators. In general both of these contributions are present. Our default assumption is that DM has electroweak scale mass, while the mediators are much heavier, with masses of order Λ m Z . We thus have the following hierarchy of scales, where q ∼ O(100 MeV) is the typical momentum transfer in DM scattering on nuclei. We will also discuss the case of light DM, m χ m Z .
When one considers processes at energy scales below the mass of the mediators, µ < Λ, the mediators can be integrated out. The effective DM Lagrangian, valid for Λ > µ > m Z , is given by where the superscripts denote the dimensionality of the operators in the Lagrangian. The renormalizable part of the effective Lagrangian is, for a Dirac-fermion DM multiplet, The covariant derivative comprises the interactions with the electroweak gauge bosons W a µ and B µ ; see App. A for further details on our notation. For µ m Z ∼ m χ the mass parameter m χ can effectively be set to zero.
The non-renormalizable terms in the effective Lagrangian (3), arise from integrating out the mediators. Depending on the mediator model it is possible that only L (5) χ or only L (6) χ are generated, but in general both will be present. We truncate the expansion at dimension six since most mediator models generate nonzero Wilson coefficients, a , in at least one of the two effective Lagrangians, L (5) χ , L (6) χ (for exceptions where the first contributions arise only at dimension seven, see, e.g., Ref. [71,72]; the complete basis at dimension seven has been presented in Ref. [26]). When writing the basis we assume that there is a conserved global dark U (1) D quantum number, which forbids currents of the form χ c Γχ, where χ c is the charge-conjugated DM field, and Γ denotes a generic string of Dirac matrices. (This assumption is to be relaxed in a follow-up work, where we plan to extend our analysis to the case of Majorana fermions and more than one multiplet.)

A. Dimension-five operator basis
The CP-conserving dimension-five operators are while the CP-odd operators have an extra insertion of γ 5 , Here and below, H is the SM Higgs doublet, and the SU (2) generatorsτ a , τ a are defined in App. A. All non-displayed SU (2) (and, below, also color) indices are assumed to be contracted within the brackets. Note that if χ is a SU (2) singlet, the operators Q 4 , Q 6 , and Q   At dimension six there are many more operators. We do not consider flavor-violating operators, keeping our discussion minimal. For each SM fermion generation, i = 1, 2, 3, there are then eight operators that are products of DM currents and quark currents, Here Q L denotes the left-handed quark doublet, and u R , d R the right-handed up-and downtype quark singlets, respectively. The analogous operators involving lepton currents are where L L denotes the left-handed lepton doublet, and R the right-handed down-type lepton singlet. Finally, there are four dimension-six operators involving Higgs currents, Q The Higgs currents are defined in terms of hermitian combinations of the covariant derivatives, Additional operators with covariant derivatives acting on the DM fields vanish via the DM equations of motion, up to total derivatives.
As in the case of dimension-five operators, the basis simplifies if DM is a SU (2) singlet. In this case, the operators Q 9,i , Q 12,i , Q 15 , and Q (6) 17 vanish and should be dropped from the basis.

III. RENORMALIZATION GROUP RUNNING
The RG running proceeds through several sequential steps, Λ → µ EW → µ b → µ c , with matching thresholds at the electroweak scale, µ EW , the bottom-quark mass scale, µ b , and the charm-quark mass scale, µ c . We first review briefly each of the steps, and then give the details in this and the subsequent section.
Running from the mediator scale, Λ, to the EW scale, µ EW ∼ m Z ∼ m χ , results in the mixing of the operators in the effective DM Lagrangian, Eq. (3). We perform the calculation of the RG running using dimensional regularization in d = 4 − 2 dimensions. Following the conventions in Ref. [73], we define the anomalous dimension matrix γ by where C is a vector of Wilson coefficients 2 , and the superscript T denotes matrix transposition. The anomalous dimension matrix receives a number of different contributions that we treat separately, so that Here, we defined α f ≡ y 2 f /4π and α λ ≡ λ/4π, where y f is the Yukawa coupling of the fermion f and λ the Higgs quartic coupling (for normalizations see Appendix A), while the other parameters are defined in terms of the gauge couplings in the usual way, α i ≡ g 2 i /4π. The ellipsis denotes higher-order contributions. Note that the anomalous dimension above the EW scale does not depend on the QCD coupling constant, since DM does not carry color, while all the DM-quark operators in (10)-(13) contain conserved quark currents in the limit of zero quark masses. The situation is different below the EW scale.
The solution to the RG evolution equation (21) gives the Wilson coefficients at any scale where U (µ, Λ) is the evolution operator from Λ to µ, obtained by solving (21), or equivalently with the initial condition U (Λ, Λ) = 1. The leading-order RG evolution effectively sums the terms of the form α n i log n (Λ/µ EW ) to all orders. Since some of the anomalous dimensions are large, we count α i log(Λ/µ EW ) ∼ O(1). We work to leading-logarithmic order and thus include all terms that are O(1). This means that the matching conditions are calculated to the same order, i.e., are obtained at tree level. Matching is done at one-loop, if the tree level contribution vanishes and the one-loop contribution is numerically important, for details see below.
The first matching arises at the EW scale, µ EW ∼ m Z , where one integrates out the top quark, Higgs, W and Z. For µ < µ EW the propagating degrees of freedom are then the photon, the gluons, n f = 5 quark flavors, and the leptons. The RG running in the five-flavor theory is given by the anomalous dimension matrix γ [5] . It receives QCD and electromagnetic contributions, so that at one loop order, where α s and α are the strong and electromagnetic coupling constants. At µ b ∼ m b the bottom quark is integrated out. The resulting four flavor EFT has as the propagating degrees of freedom the photon, gluons, leptons, and n f = 4 flavors of quarks. It is valid for where µ c ∼ m c is the scale at which the charm quark and the τ lepton are integrated out. The running from Λ down to the scale µ had ∼ 2 GeV, where the hadronic matrix elements are evaluated, can thus formally be written as Here U [n f ] (µ, µ ) are the evolution operators from µ to µ in a theory with n f quark flavors that satisfy an evolution equation similar to (24), In the numerics we take µ had ∼ µ c ∼ 2 GeV, and thus set U [3] (µ had , µ c ) = For the computation of the anomalous dimensions we used two independent automated setups. In the first, the amplitudes were generated using qgraf [74] and the anomalous dimensions were computed using the computer algebra system form [75]. The second setup relied on Mathematica packages: the Feynman rules were generated using FeynRules [76], the amplitudes with FeynArts [77], and the anomalous dimensions were computed using FormCalc [78].

A. Mixing of dimension five operators
We start by providing the anomalous dimension matrices for mixing of the CP conserving dimension-five operators Q 1 , . . . , Q 8 , defined in Eqs. (6)- (9). For the column of the dimension-five Wilson coefficients entering the RG evolution equation (23) we use the rescaled Wilson coefficients 3 , C 4 , The explicit factors of α 1,2 /2π = g 2 1,2 /8π 2 in C ensure that the anomalous dimension matrices γ The corresponding rescaled operators are also denoted by a prime and read while Q The anomalous dimension for Q 1 , . . . , Q 1 , . . . , Q 4 , and the CP odd operators, Q 5 , . . . , Q 8 , while there is no mixing between the two blocks. The QCD anomalous dimensions vanish, since all fields are color neutral. The remaining one-loop anomalous dimensions for the Q 1···4 ×Q 1 , Q  Here J χ = I χ (I χ + 1), with d χ = 2I χ + 1 the dimensionality of the DM electroweak multiplet, and Y χ its hypercharge. The β functions for the gauge couplings g 1 and g 2 are given by respectively, where N χ is the number of DM multiplets in the representation I χ (we will mostly take N χ = 1). The anomalous dimension matrices for the CP-odd operators Q 5 , . . . , Q are also given by the same matrices, [γ , i = 1, 2, y, λ, as required by the fact that CP breaking is not probed by the relevant one-loop diagrams.
The anomalous dimensions γ 1,2 , into the Higgs current operators, Q 3,4 , arise from the diagrams in Fig. 6. These mixing contributions vanish for singlet DM (Y χ = J χ = 0). This is true also for the mixing of Q and Q 4 : They arise from the Higgs wave function renormalization, and in the case of γ After running from µ ∼ Λ to µ ∼ m Z , we revert the rescaling of the Wilson coefficients, i.e., 3 , C corresponding to our original definition of operators in Eqs. (6)- (9). We use the unprimed Wilson coefficients for determining the matching conditions in Sec. IV.

B. Mixing of dimension six operators
We turn next to the anomalous dimensions for the dimension-six operators. Counting the three SM fermion generations and keeping only flavor-diagonal fermion currents, there are 46 operators in total that couple DM with the SM. We work in the limit of flavor conservation which simplifies the structure of the anomalous dimensions.
A technical remark is in order. To project the one-loop matrix elements onto our operator basis within the context of dimensional regularization, we have to manipulate Dirac γ matrices in d = 4 dimensions. Strictly speaking, this requires the extension of the operator basis by evanescent operators. However, the one-loop anomalous dimensions are not affected by the choice of the evanescent operator basis, and we can effectively use four-dimensional Dirac algebra [79].
We start with the mixing among the operators that are products of DM and quark currents, Q 1,i , . . . , Q 8,i , Eqs. (10)-(13), within the same quark generation. The corresponding 8 × 8 block of the anomalous dimension matrix is given by 1,i···8,i ×Q for the part of the anomalous dimension matrix proportional to g 2 1 , while the part of the anomalous dimension proportional to g 2 2 is Both of the anomalous dimension matrices are diagonal in flavor indices. As far as the U (1) gauge interaction is concerned, for Y χ = 0 the operators Q 1,i , . . . , Q 14,i and Q 15 , . . . , Q 18 due to the B µ , W a µ penguin insertion (right-handed quarks and leptons can also be on the external lines).
vanish. That this is not the case can be understood as the result of a non-multiplicative renormalization, allowed for U (1) gauge groups; see Ref. [80]. Similar arguments apply for the QED anomalous dimensions discussed in Sec. IV C.
The Feynman diagrams that lead to nonzero entries in the two matrices are given in 1,i and Q (6) 5,i , while these operators do not mix with any of the remaining operators. The same contributions also mix Q 3,i and Q       The penguin insertions, Fig. 8, also lead to mixing between operators involving quark currents of different generations. The corresponding anomalous dimensions are given, for for the part of the anomalous dimension matrix proportional to g 2 1 , while the part of the anomalous dimension proportional to g 2 2 has the following non-zero entries for i = j All the other entries vanish.
We turn next to the 6 × 6 block of the anomalous dimension matrix that describes the mixing of the lepton operators Q 9,i , . . . , Q 14,i , Eqs. (14)- (16), among themselves, giving and The latter two anomalous dimension matrices are straightforward modifications of the ones for the DM-quark operators in (39) Penguin-type insertions lead to mixing between different generations also for leptons, 10,i ,Q 11,j )×(Q 10,i ,Q 13,i ,Q 14,j )×(Q 13,i ,Q and All the other entries vanish.
A very interesting effect of the one-loop RG running is that the penguin diagrams mix the operators with quark-and operators with lepton currents. This is shown in Fig. 8 where the two quark lines coming from the EFT operator are contracted into a loop, while the emission of a B converts this into a lepton current. Conversely, an operator with a leptonic current can be converted to a DM-quark operator at one-loop. The corresponding mixing of the quark operators Q 1 , . . . , Q 8 into the lepton operators Q 9 , . . . , Q 14 is given by the following 8 × 6 block of γ (0) 1 , now for arbitrary generation indices i, j The corresponding block of the γ (0) 2 matrix has only two nonzero entries, while the remaining entries in this 8 × 6 block of γ 14 , into the quark operators, Q 1 , . . . , Q 8 , is given for arbitrary generation indices i, j by the following 6×8 block of the γ The corresponding 6 × 8 block of the γ 2 anomalous matrix has only two nonzero entries, 18 operators at one loop. These diagrams are in addition to the set of diagrams in Fig. 5 which, at dimension six, include an additional derivative w.r.t. to the dimension-five operators.
The mixing of DM-quark and DM-lepton operators, Eqs. (47)-(50), has important phenomenological consequences. One implication is that, in any theory where one introduces DM-quark interactions, one-loop mixing will generate DM-lepton interactions. The converse is also true: a theory of purely "leptophilic" DM is impossible. An interaction between DM and leptons will lead to an interaction between DM and quarks via one-loop mixing. Note that the mixing is nonzero irrespective of whether or not DM carries any electroweak charge.
Penguin insertions will also generate DM-quark and DM-lepton interactions, when initially only the pure DM operators (Eq. (19)) are present; see App. C.
Finally, we move to the mixing of dimension-six operators with Higgs currents, 18 , Eqs. (17)- (18). We start with the 4 × 4 blocks of the anomalous dimension matrices that give the mixing of these operators among themselves, 15···18 ×Q The relevant diagrams are shown in Figs. 5 and 9. The renormalization induced by these contributions is multiplicative and does not lead to mixing of the DM-Higgs operators.
In addition there is mixing of the operators with quark and lepton currents into the Higgs-current operators and vice versa (see Fig. 8). The resulting mixing of the DMquark operators, Q 1,i , . . . , Q 8,i , and the lepton operators Q 9,i , . . . , Q 14,i into the DM-Higgs operators, Q 15 , . . . , Q 18 , are given by the following 8 × 4 and 6 × 4 blocks in the γ (0) 1 anomalous dimension matrix (i = 1, 2, 3), respectively, and by the corresponding 8 × 4 and 6 × 4 blocks in the γ (0) 2 anomalous dimension matrix, which, however, only have two nonzero entries each, The mixing of the DM-Higgs operators, Q 15 , . . . , Q 18 , into the DM-quark operators, Q 14 , is given by and respectively, for the corresponding blocks of γ Note that both the mixing of DM-quark and DM-lepton operators into the DM-Higgs ones and vice versa is present even if the DM does not carry any electroweak charge.
For the third-generation DM-quark operators, Q 1,3 , . . . Q 8,3 , there is also the renormalization due to the Yukawa interaction with the Higgs (we neglect all the Yukawa interactions except with the third fermion generation and the charm Yukawa), giving and The off-diagonal entries in Eq.
15 , . . . , Q 18 , and as well as to the mixing of the DM-Higgs operators into the DM-third-generation quark operators, The Yukawa interactions also renormalize the Higgs operators themselves due to the renormalization of the Higgs fields, giving = diag(6, 6, 6, 6) .
Finally, we also present the anomalous dimensions due to the tau Yukawa coupling, leading to mixing among the four-fermion operators, mixing of four-fermion into Higgs operators, Higgs operators into four-fermion operators, and diagonal self mixing of the Higgs operators due to Higgs field renormalization, The results given in this section are sufficient if one works to leading-log accuracy without resummation of the logarithms. However, the set of operators Q 1,i , . . . , Q

IV. MATCHING TO EFT BELOW THE WEAK SCALE
The running from the mediator scale, µ ∼ Λ, down to the weak scale, µ ∼ m Z , is described by the evolution operator U (µ EW , Λ) in Eq. (26). The relevant anomalous dimension matrix γ, appearing in Eq. (24), was presented in Sections III A and III B. The next step is to calculate the matching onto a five-flavor theory at µ ∼ v EW by integrating out the top quark, the Higgs and W, Z gauge bosons. This gives the matrix M EW→ (5) in the evolution equation (26). Since we are interested in the elastic direct detection scattering we can, below the electroweak scale, ignore all the charged components of the χ multiplet. From now on χ will thus denote only the neutral component of the DM electroweak multiplet.
After the matching at µ ∼ m Z we arrive at the n f = 5 effective Lagrangian which we organize in terms of the dimensionality of the operators, In the matching we keep all the numerically leading terms. The leading contributions from dimension-five (∝ 1/Λ) and dimension-six operators (∝ 1/Λ 2 ), Eq. (3) Before proceeding we remark that both the DM mass, m χ , and the DM field, χ, get shifted by the Higgs vacuum expectation value due to the contributions from the Q 3,4 operators, Eq. (7), and from the Q The redefinition of the χ field is a simple chiral rotation, χ = exp i 2 γ 5 φ χ, with (see also Ref. [82]) while the new mass term is The field redefinition also changes the operators Q 1 , . . . , Q 8 in Eqs. (6)-(9) and the corresponding Wilson coefficients, C From now on we will assume that the above field and mass redefinitions have been performed and drop the primes on the Wilson coefficients, the DM fields, and the DM mass.

A. Light dark matter
In the case of light DM, m χ m Z , we can use relativistic DM fields to construct the effective theory below the weak scale. The effective Lagrangians containing operators of dimensionality d in Eq. (69) are given by where we introduced the dimensionful Wilson coefficientsĈ a | n f =5 in order to simplify the notation. They are suppressed by inverse powers of the NP scale Λ and/or the top, W , where F µν is the electromagnetic field strength tensor. The operator Q Here f denotes any quark, f = u, d, s, c, b, or charged lepton flavor, f = e, µ, τ . We find it convenient to express the operators in terms of (axial-)vector and (pseudo-)scalar currents, which have definite non-relativistic limits. Operators with neutrinos are not needed for our purposes as they do not run below the EW scale.
In the effective Lagrangian Eq. (69)  The relevant dimension-seven operators involving the DM and gluon fields are given by where G µν = 1 2 ε µνρσ G ρσ and a = 1, . . . , 8 are the color indices. The strong coupling constant α s is defined in the five-flavor scheme. The normalization reflects the fact that these operators are typically generated at one-loop level. Note that Q 2 and Q 3 are CP odd. There are also four scalar operators with f denoting any quark (f = u, d, s, c, b) or charged lepton flavor (f = e, µ, τ ). The definitions of Q 5,f , . . . , Q 8,f include an explicit power of the corresponding quark or lepton mass. This reflects the leading contributions to their Wilson coefficients, see below.
In the remainder of the subsection we give the results of the matching at µ EW ∼ m Z . We start with the dimension-five operators where the contributions come from W and B dipole operators above m Z after rotating the EW gauge eigenstates into the mass eigenstates after EWSB: 3Ĉ Equation (82) also receives a one-loop contribution from dimension-four gauge interactions, denoted by the ellipsis, proportional to the hypercharge of the DM multiplet. We omit this contribution here since a non-zero hypercharge leads to a tree-level Z exchange with nuclei which is excluded by direct detection experiments.
For the dimension-six operators we start with the operators with external quark legs.
The contributions from dimension-six UV operators with external quark legs arê 17 + C where i is a generation index (u 1 ≡ u, u 2 ≡ c and d 1 ≡ d, d 2 ≡ s, d 3 ≡ b) and the upper(lower) signs apply for up(down) quarks. For each of the Wilson coefficients the last 1/Λ 2 -suppressed term is due to Z exchange, shown in Fig. 11 (left). For a DM multiplet with nonzero hypercharge Y χ , Z exchange due to the renormalizable gauge coupling (4), see 4,u i (d i ) | n f =5 receives a contribution, denoted by the ellipsis, from gauge interactions at one-loop (see Fig. 2) that does not vanish for Y χ = 0. This requires a two-loop matching calculation with m χ kept parametrically small, which is beyond the scope of present paper. In the numerical evaluations we thus use the results from Ref. [68], that were obtained assuming that m χ is not much smaller than m Z .
Similarly we find for the dimension-six operators with leptons ( 1 ≡ e, 2 ≡ µ, 3 ≡ τ ) on the external legŝ 17 + C As before, Z-boson exchange due to the renormalizable gauge coupling (4) leads to the additional contributionsĈ 4, i | n f =5 receives a one-loop contribution from gauge interaction that does not vanish for Y χ = 0, see Ref. [68].
The dimension-seven operators receive contributions from both the renormalizable electroweak interactions of the DM multiplet as well as from the higher dimension operators.
For the gluonic operators Q 1,2 the higher dimension UV operators give a contribution after h t g χ g χ Figure 12: Matching contributions to dimension-seven effective operators involving gluons for µ < m Z that arise from integrating out the top quark.
integrating out the top quark at one loop, see Fig. 12. We then havê andĈ (7) 3(4) | n f =5 = 0. Note that the loop factor is already included in the definition of the operators Q (7) i . The explicit top-quark mass dependence drops out because we expand to leading (quadratic) order in the small external momenta. This limit is equivalent to the limit of heavy top mass in on-shell Higgs decays to two photons or gluons, where the non-decoupling of chiral fermions is a familiar result. The ellipsis denotes the two-loop contributions from renormalizable electroweak interactions, see Ref. [68].
For scalar operators we havê 8,f | n f =5 = 0. The right diagram in Fig. 11 shows the tree-level contributions from higher dimension operators. Here, the ellipsis denotes the one-loop "Higgs penguin" contribution from gauge interactions (see Fig. 2 and Ref. [68]).

B. Electroweak scale dark matter
The case that the DM mass is comparable to the electroweak scale, m χ ∼ O(m Z ), needs to be treated separately. In this case we integrate out at the electroweak scale, in addition to the top quark, the Higgs and the W, Z bosons, also the high-momentum fluctuations of the DM field. In this way we arrive at the Heavy Dark Matter Effective Theory (HDMET).
The HDMET is an effective theory the describes the scattering of a heavy DM, where the momenta exchanges are much smaller than the DM mass, q m χ . The HDMET uses the HQET (Heavy Quark Effective Theory) formalism [83][84][85] applied to DM direct detection physics [8]. The result is an effective theory where terms are organized as an expansion in 1/m χ . In practice the HDMET description is not necessary if one performs electroweak matching only at tree level. However, some of the one-loop matching corrections are important phenomenologically and need to be kept, requiring the use of HDMET.
The construction of the requisite terms in HDMET has been presented in Ref. [20]. Here, we just collect the main results and refer the interested reader to the literature for details.
The HDMET Lagrangian is given by 4 Here, χ v denotes only the neutral component of the DM electroweak multiplet, i.e., only the DM state. The first term is the LO HDMET Lagrangian and describes an infinitely heavy DM particle, and contains no explicit dependence on m χ . The O(1/m χ ) term is fixed by reparametrization invariance [87], with ellipsis denoting terms of higher order in the 1/m χ expansion.
The effective Lagrangians L The only difference is that we now denote explicitly at which order in 1/m χ the operators enter, such thatĈ For very heavy DM, m χ m Z , the DM mass is integrated out before the weak gauge bosons [8,16,81,86], giving where at tree level c W = c B = 1, and the ellipsis denotes terms of higher order in 1/m χ , as well as the 1/Λ suppressed interactions. The covariant derivative contains the W a µ and B µ gauge fields, so that in the infinite mass limit the DM multiplet, χ v , acts as a static source of the electroweak gauge fields.
We also need the following two subleading operators since the presence of the photon pole in the interaction of the magnetic dipole with the nuclear current requires that we go to the second order in the expansion of the DM tensor current. We defined σ µν χ v , and S µ = γ µ ⊥ γ 5 /2 is the spin operator, while v µ = (1, 0 ) is the velocity label of the nonrelativistic DM field (cf. Ref. [20]).
At tree-level we havê where the equalities get corrections at loop level. Again,Ĉ 1,2 , are given in Eqs. (82), (83). 5 Note that the Λ m Z ∼ m χ limit reduces the set of HDMET operators that are generated. For instance, at dimension seven the operator (χ v χ v )G aµν G a µν arises in the matching, but not the operator (χ v χ v )v µ v ν G a µρ G aρ ν . The latter would arise from the dimension-nine UV operator (χ∂ µ ∂ ν χ)G a µρ G aρ ν and is thus m 2 χ /Λ 2 suppressed. In contrast, for m χ ∼ Λ the two operators are of the same size, and thus both arise in the matching to HDMET at scale µ ∼ Λ (see, e.g., the discussion of twist-two operators in Ref. [88]).
The dimension-six operators of LO in 1/m χ are Q (6,0) In addition, we need the following d − m = 6 operators that are 1/m χ suppressed Q (7,1) where our convention is that the derivatives act only within the brackets or on the nearest bracket. The Q and in addition C (7,1) Note that the equalities denoted by "tree" are only valid for tree-level matching, while the remaining relations are valid to all orders due to reparametrization invariance. The light DM Wilson coefficients C The top-quark loop contributions to the gluonic operators, Eq. (109) and (110), are the same as in Eq. (92), so that The Wilson coefficientsĈ The Wilson coefficients for the scalar operators arê 3,4 in Eq. (7) and Q give the same matching conditions as in the case of light DM, Eqs. (93) and (94). Note that within this subsection, the full (unexpanded) results of Ref. [68] should be used.

C. RG running below the electroweak scale
The matching at µ ∼ µ EW is followed by the QCD and QED RG running from µ EW to µ c ∼ µ had ∼ 2 GeV. The five-flavor theory below µ EW is matched onto the four-flavor theory at the bottom quark threshold, µ b , and then onto the three-flavor theory at the charm quark threshold, µ c ; see Eq. (26). There is no running in the three flavor basis because of our choice of scales, µ c = µ had . This RG evolution was discussed in detail in Ref. [16]. For completeness and convenience we convert the results of Ref.
[16] to our notation. (See also Ref. [23] for a computer implementation of the RG evolution, as well as Ref. [22] for the case of vector mediators.) QCD running. Since the vector currents are conserved,Ĉ 1,q andĈ 1···4 ×Q where q runs over active quark flavors and C F = 4/3.
2,f and vice versa. The required anomalous dimensions can be found in Ref. [61], and are 1,f ,Q 2,f ,Q where Q f is the electric charges of the SM fermion f , while N f c = 1(3), if f is a lepton (quark). In analogy with Eq. (22), we use the notation where the ellipsis denotes higher orders.
while at µ = µ c ,Ĉ 5,c(6,c) | n f =4 (µ c ) , C such that the effects of the heavy quarks appear, at low energies, as additional contributions to the gluonic operators, Eqs. (78)- (79). All the other Wilson coefficients cross the thresholds continuously,Ĉ

D. DM interactions with nucleons and nuclei
The final step in the RG evolution is the matching at µ ∼ µ had onto an effective theory contains 14 operators with up to two derivatives which are needed to describe the chirally leading interactions. The momentum-independent nonrelativistic operators are while the relevant subset of momentum-dependent operators consists of with N = p, n. We use the conventions of [21,23], so that where p 1(2) and k 1(2) are the incoming (outgoing) nucleon and DM three-momenta, respectively.
The coefficients of the two momentum-independent operators (121) are, schematically, At leading chiral order one also has the contributions from the operators with two derivatives, O N 5,6 , whose coefficients are The sums in Eqs.
The coefficients c N 7 and c N 8 arise from vector×axial and axial×vector parts of the operators Q 8,f , respectively, while all of these contribute to c N 9 . Since these operators are momentum (velocity) suppressed, they will give subleading contributions to the scattering rates, unless the leading contributions (to c N 1 from the vector×vector parts, and to c N 4,6 from axial×axial parts) cancel. In the next section we will discuss in more detail how realistic this is. The operators O N 7 , . . . , O N 9 , also receive contributions from Q 5,6 induce an electric dipole moment for DM, giving a 1/q 2 -enhanced direct detection scattering rate [92].
If the EFT above the weak scale is extended to mass dimension seven, then also the nonrelativistic operators, [26]. They arise from dimension-seven operators with tensor DM currents and from interactions of DM with the GG current. The scaling estimates for the corresponding coefficients are c N setting the dimensionless Wilson coefficients to unity. Having obtained the coefficients in the effective Lagrangian for DM scattering on nonrelativistic nucleons, Eq. (120), the final step is to calculate the DM-nucleus scattering cross section [9,10,12], Here, E R is the recoil energy of the nucleus, m A the mass of the nucleus, j A its spin, and v the initial DM velocity in the lab frame. The kinematic factors contain the c N i coefficients, where v ⊥ T is defined as in Eq. (125), but with the nucleus replacing the nucleon. The sum in (133) is over isospin, τ, τ = 0, 1, so that The nuclear response functions depend on | q | and have the approximate scaling (see, e.g.,

Fig. 2 in [93])
The W Σ , W Σ , W ∆ , and W ∆Σ response functions depend strongly on the detailed properties of nuclei, for instance, whether or not they have an un-paired nucleon in the outer shell.
Here W Σ ,Σ measure the spin content of the nucleus, W ∆ the average angular momentum in the nucleus, and W ∆Σ the interference of the two. Their sizes can thus differ drastically between different isotopes of the same element.
The W M response function encodes the coherent scattering enhancement, O(A 2 ), where A is the atomic mass number. This is achieved in the long-wavelength limit, q → 0, where DM scatters coherently on the whole nucleus, for instance, due to the O N 1 contact interaction. The coherent scattering due to O N 5 is O(q 2 v 2 T ) suppressed. However, since its coefficient is 1/q 2 enhanced, the corresponding contribution is of leading order [21]. The contributions due to O N 8,11 , though coherently enhanced, are at the same time velocity suppressed.

Magnetic or electric dipole operators
The DM magnetic dipole operators, Q 1,2 ∼ (χσ µν χ){B µν , W µν }, Eq. (6), induce both spin-dependent and spin-independent interactions. These give parametrically similar contributions to the DM-nucleus scattering cross section. Schematically, , but no coherent enhancement. Which term dominates then depends on the details of the nuclear structure for the nuclei in the target [21].
The DM electric dipole operators, Q 5,6 ∼ (χσ µν γ 5 χ){B µν , W µν }, Eq. (8), match onto the nuclear operator O p 11 ∼ i q · S χ . This leads to coherently enhanced scattering independent of the nuclear spin, with the 1/| q | 2 pole only partially cancelled, Compared to the magnetic dipole operators, the bounds on the NP scale Λ for electric dipole interactions of DM are thus more stringent by a factor of order m χ /| q |.

Operators with Higgs scalar currents
The operators Q Both of these match onto the nuclear operator O N 1 ∼ 1 χ 1 N , giving a coherently enhanced spin-independent cross section, where in the last term we kept the numerically important factor 2/27.
The operators with pseudoscalar DM current, Q where the DM pseudoscalar current gives an O(q) suppressed operator, O N 11 ∼ iq · S χ . The resulting DM-nucleus scattering cross section is still coherently enhanced, but suppressed

Operators with DM vector current and with quark vector or axial-vector currents
We focus next on the operators Q 1,i , . . . , Q 4,i ∼ (χγ µ χ){q L γ µ q L , q R γ µ q R }, Eqs. (10)- (13). Barring cancellations, the leading contribution is due to the vector×vector part of the operators, (χγ µ χ)(qγ µ q). For couplings to the first generation quarks this leads to coherently enhanced spin-independent scattering, The estimate is different, if DM only couples to quarks of the second or third generation. For these the nuclear matrix element of the vector current vanishes, and the leading contribution comes from closing the quarks in a loop, exchanging a photon with the up-or down-quark vector currents. This also results in a spin-independent scattering, with a cross 1q,...,4q 2 A 2 (2 nd and 3 rd generation quarks) .
In addition there are subleading contributions from matching onto higher dimension operators with gluons, as well as spin-dependent, velocity-suppressed scattering from the axial currents.
The situation is qualitatively different if the UV physics is such that at µ ∼ Λ it projects the Q The two contributions are comparable for | q | ∼ 0.1m N and m χ ∼ O(100 GeV). If the vector×axial-vector operators involve only the heavy quarks, q = c, b, t, the scattering cross section is further severely suppressed by the small contributions of the heavy quarks to the nucleon spin (see Sec. V A 4 below and Ref. [25] for a more detailed discussion).
Note that the spin-dependent scattering in Eq. (147) is suppressed by v 2 T ∼ q 2 /m 2 χ ∼ 10 −6 . There is no such suppression for the spin-independent cross section, Eq. (145), which is, in addition, enhanced by the coherence factor A 2 . This means that the Wilson coefficients contributing to the quark vector currents at the scale µ m Z need to cancel to the level if the spin-dependent scattering is to be the dominant DM-nucleus interaction. Perfect cancellation at all scales is impossible to arrange, since the contributions come from operators in different representations of the SM gauge group, χγ µ χ Q L γ µ Q L ), . Even if one engineers the Wilson coefficients of these operators such that the vector currents are zero at one scale, a small amount of running will make them nonzero at a different scale. The required cancellation is numerically of threeloop order, so that even the radiative corrections may need to be canceled by fine tuning in order for the spin-dependent scattering to be the leading effect.
If the operator (χγ µ γ 5 χ)(qγ µ γ 5 q) involves light quarks, q = u, d, s, this results in a spindependent cross section (not displaying explicitly the suppression for strange quark due to its small axial charge, ∆s = −0.031(5) [21,[94][95][96][97]), If the operator (χγ µ γ 5 χ)(qγ µ γ 5 q) involves only the heavy quarks, q = t, b, c, the scattering cross section is generally very small. The axial charges of charm and bottom quarks are tiny and poorly determined. Ref. [98] obtained ∆c ≈ −5 · 10 −4 , ∆b ≈ −5 · 10 −5 , with probably at least a factor of two uncertainty on these estimates. Despite this, for heavy quark axial-axial interactions the heavy quark axial charges still dominate the cross section over the contributions from mixing induced couplings to light quarks, discussed in the next section (see also Ref. [25]).
We focus next on the limiting case where at µ ∼ m Z only the axial-vector×vector operators, (χγ µ γ 5 χ)(qγ µ q), are generated. For q = u, d, s these match on two nonrelativistic operators with one derivative, O N 8 , O N 9 . Both lead to momentum suppressed incoherent scattering, with O N 8 giving rise, in addition, to spin-independent scattering that is coherently enhanced, but velocity suppressed, The two contributions are of parametrically similar size for heavy nuclei, A ∼ O(100), in which case | v T |A ∼ | q|/m N . Which of the two contributions dominates then depends on the details of the nuclear structure for the particular isotope.
For (χγ µ γ 5 χ)(qγ µ q) with q = t, b, c, the leading contribution comes from closing the heavy quark loop, exchanging a photon with the up-or down-quark vector current. The cross section is suppressed with respect to Eq. (149) by an additional factor of (α/4π) 2 , There is also a contribution from matching onto higher dimension operators with gluons, which is expected to be at most of similar size.
In general the sum of Q 5,i , . . . , Q 8,i operators matches onto both (χγ µ γ 5 χ)(qγ µ γ 5 q) and (χγ µ γ 5 χ)(qγ µ q) operators at µ = µ EW , giving a cross section that is a sum of Eqs. (148) and (149). The spin-dependent scattering in Eq. (148) is parametrically the largest. Since the parametric enhancement is not large, however, this expectation does depend on the target material, and spin-independent scattering could be equally important.

Operators with Higgs vector currents
The operators Q 8,i that we discussed before, where a more general structure of DM couplings to quarks was allowed.
For the operators Q (6) 15,16 ∼ (χγ µ χ)(H † D µ H), the dominant contribution comes from a quark vector current, giving a coherently enhanced, spin-independent scattering cross section The Z-boson exchange at µ ∼ m Z also generates the χγ µ χ qγ µ γ 5 q operator. This leads to momentum-suppressed, spin-dependent scattering that is always subleading.
On the other hand, for the operators with DM axial-vector currents, Q 17,18 ∼ (χγ µ γ 5 χ)(H † D µ H), one needs to keep both the spin-dependent and spin-independent scattering contributions. The induced axial-vector×vector and axial-vector×axial-vector inter-actions lead to a cross section that scales as the sum of Eqs. (148) and (149), From scaling considerations, spin-dependent scattering is expected to be dominant in nuclei with an unpaired nucleon that is not in an s-shell. But even then the spin-independent scattering contributions may need to be included, depending on the nucleus. An example is dicsussed in Section V B 1.

B. Inclusion of RG running
The modifications due to RG running can significantly impact the cross section predictions. We will show several examples where the RG running effects are particularly large.
While the sizes and patterns of the induced corrections does depend on the electroweak charges of DM, the effects themselves are not "optional". They are due to SM particles in the loops, and are thus always present.
Consider, for instance, SU (2) L -singlet DM, where all mixing proportional to g 2 vanishes, as can be seen by inspecting γ (0) 2 . Another example is DM that is hypercharge neutral, Y χ = 0, for which all the mixings due to B µ exchanges with the DM line vanish. However, in both cases there is still mixing due to the running of the non-conserved SM currents. For instance, for DM that is a complete SM singlet the main mixing is induced by the top-quark Yukawa interaction. This case has been discussed in detail in the literature [22,60,61] (see also Ref. [25] for the discussion of weak-mixing effects below the weak scale).
Here, we will use our general results from Sec. III and apply them to the simplest nontrivial example of DM with electroweak charges -a Dirac fermion multiplet that is hypercharge neutral, Y χ = 0, and an electroweak triplet, I χ = 1. The choice Y χ = 0 is imposed on us by the phenomelogical requirement that DM (the neutral component of the multiplet) should not couple to the Z boson at tree level, in order to avoid a too large direct detection scattering scattering cross section.
We will illustrate the effects of RG running for several different choices of nonrenormalizable DM interactions, taking I χ = 1 as an example. The scattering rates then receive two types of contributions. First, there are contributions from higher dimension operators. These vanish in the limit Λ → ∞. However, for I χ = 0 there are also contributions from renormalizable electroweak interactions that are independent of Λ. The leading contributions of this type are due to the small "Higgs penguin", the one-loop and two-loop contributions shown in Fig. 2. They lead to coherently enhanced scattering of parametric size (for I χ = 1, using right diagram in Fig. 2) where A 2 = O(10 4 ) for scattering on Xenon, and A 2 = O(10 2 ) for scattering on Fluorine.
If the target nucleus has non-zero spin, the W box shown in Fig. 2 gives an additional contribution scaling as These scalings omit a proportionality factor that depends on the DM mass and velocity, the recoil energy, and the detailed structure of the nucleus. In our numerical evaluations we use the exact results from Ref. [68] (for heavy DM see also [7,27,81]); the ratio of scaling 1,i and Q 5,i it is possible to construct interactions of DM with only up-or downquark currents separately. This would, however, require a nonzero DM hypercharge, Y χ = 0, which is phenomenologically not viable for Dirac fermion DM.
That is, we assume that DM couples to the SM through renormalizable weak interactions (I χ = 1, Y χ = 0) and, in addition, through the dimension 6 effective operator − Q 6,3 + Q 7,3 + Q the RG running from Λ to µ ∼ M W generates the Wilson coefficients (keeping only the linear logarithmic term) 6,3 ,Q 6,3 ,Q In deriving Eqs. (158) and (159) 18 . The mixing into Q (6) 18 is due to the top-quark Yukawa interaction of the Higgs. It is present whether or not DM is part of an electroweak multiplet, i.e., even if DM is an electroweak singlet.
In Fig. 14 we compare the predicted rates for scattering on Xenon and Fluorine, obtained with (blue lines) and without (red lines) RG evolution. In the case of no RG evolution the scattering is almost entirely due to the contribution of renormalizable weak interactions. 18 leads to a cross section that scales roughly as while the cross section induced by the mixing into Q 1,3 , scales roughly as In shows that the contributions are of the same size.
As already mentioned, the scattering on Fluorine is dominated by the Q 18 -induced contributions, Eq. (160), where the leading term comes from spin-dependent scattering. Inspection of the Σ , Σ response functions, Ref. [9], shows that spin-dependent scattering on Fluorine is about ten times larger than for 129 Xe, while the other Xenon isotopes give negligible contributions. In Fig. 14 we weighted the contributions according to the natural abundance of Xenon isotopes, giving an additional roughly five-fold suppression of the spin-dependent rate for Xenon. Consequently, the effect of the RG evolution is large only for scattering on fluorine (right panel of Fig. 14).

Vector -axial-vector (first generation)
Next, we assume that at Λ = 1 TeV the only nonzero Wilson coefficients are − C This leads to spin-dependent scattering rate that scales roughly as 2,1 (Λ) gets modified by the two-step mixing in the RG evolution to (we neglect numerically subleading contributions) 2,1 ,Q The mixing contributions for the other two Wilson coefficients, C 3,1 and C resulting in the enhanced scattering rate, as shown in Fig. 15.
It is important to realize that it is not sufficient to use the first-order-expanded solution to the RG equations, as the effect arises only at the second order in the mixing. While the effect corresponds to a two-loop correction in the "full theory", our method automatically captures the leading-logarithmic part of it.

Vector -axial-vector (third generation)
Finally, let us consider an initial condition − C 2,3 (Λ) = C 3,3 (Λ) = C 4,4 (Λ) , so that DM couples to the third generation of quarks through the operator − Q 2,3 + Q 3,3 + Q This axial-vector current has a vanishingly small nuclear matrix element, see Sec. V A. Without mixing, the leading contribution to the scattering rate is thus due to the renormalizable gauge interactions, Eqs. (153) and (154).
The largest contribution comes, however, from the mixing. At one loop the top-quark Yukawa interactions induce mixing of Q 2,3 and Q 3,3 into Q 16 with anomalous dimensions 6 and −6, respectively, see Eq. (60). The contributions add up for the axial-vector quark current, giving, for the initial condition (167), 2,3 ,Q 3,3 ,Q The above result takes into account the cancelations of contributions due to the actual values of the anomalous dimensions, and neglects numerically subleading terms. (The twostep mixing effect, described in the previous subsection, is still present, but subleading.) The operator Q 16 leads to a vector-vector interaction after integrating out the Z boson, cf. Eq. (84), giving a coherently enhanced scattering cross section of parametric size which is several orders of magnitude larger than the Higgs-penguin induced one. This is illustrated in Figure 15. Included in the numerics is the additional enhancement of the cross section by the resummation of leading QCD logarithms below the weak scale (see Ref. [25] for details). Note that the mixing induced effect, Eq. (170), is independent of the weak isospin of DM and is present even for SM-singlet DM [60].

VI. SUMMARY AND CONCLUSIONS
In this article we presented a Renormalization Group (RG) analysis of Dark Matter (DM) interactions with the SM mediated by higher dimension operators, up to and including dimension six. We calculated the one-loop RG evolution of these operators, for the case of Dirac fermion DM, from the high scale Λ down to the weak scale, and the matching to the tower of effective theories below the weak scale, distinguishing the two cases, m χ ∼ m Z and m χ m Z . We allow for DM to be part of an electroweak multiplet. Here we collect the conventions that we use in the paper. Our convention for the Lorentz vectors is p µ = (p 0 , p ), p µ = (p 0 , − p ), while for the completely antisymmetric Levi-Civita tensor µνρσ we use the convention 0123 = +1. The field-strength tensors are The SU (3), SU (2), and U (1) coupling constants are g s , g 2 , and g 1 , while f abc , abc are the completely antisymmetric SU (3) and SU (2) structure constants, respectively.
The covariant derivative acting on a fermion f is, in our convention, with T a ,τ a the generators of SU (3) and SU (2), respectively, and Y f the hypercharge of fermion f . Specializing to the DM fields we thus have with Y χ the DM hypercharge. The SU (2) generatorsτ a for a general representation of weak isospin I χ can be chosen as with k, l running over the values −I χ , −I χ + 1, . . . , I χ − 1, I χ .
The Higgs Lagrangian in terms of the complex Higgs doublet H reads In the calculation of the anomalous dimensions above the electroweak scale the Higgs mass term can be neglected as it does not affect the UV divergences.
The Yukawa interactions are given by with k, l the generation indices, while the charge-conjugated Higgs field is given byH = iσ 2 H * . In the calculation of the electroweak mixing we neglect the up, down, strange, electron, and muon Yukawa couplings.
We further complement the Lagrangian involving the matter fields with a gauge-fixing term. It is most convenient to perform the calculation in a background-field gauge; the gauge-fixing Lagrangian can be taken in analogy to the case of QCD [99,100]. We use a generalized R ξ gauge, with gauge fixing term [101] and checked explicitly the ξ W and ξ B gauge-parameter independence of our results.
After electroweak symmetry breaking we use the mass eigenbasis for the gauge bosons, where c w ≡ cos θ w = g 2 / g 2 1 + g 2 2 , s w ≡ sin θ w . The electric charge is given by e = g 1 g 2 The electric charge of the components of the DM multiplet is given by the Gell-Mann Nishijima relation, Q χ =τ 3 + Y χ /2. Definingτ ± =τ 1 ± iτ 2 , we can write the covariant derivative (A5) in terms of the broken fields as The Higgs doublet field after EWSB is given by where G + (x) and G 0 (x) are the pseudo-Goldstone fields.

Numerical inputs for the electroweak running
The parameters used in our numerics for the electroweak RG evolution areĝ 1 ,ĝ 2 ,ĝ 3 ,ŷ c , where the hat denotes the values in the MS scheme at scale M Z . All the numerical inputs are taken from Ref. [102]. Our strategy to determine the initial values at scale µ = M Z is as follows. We use the values sin 2 θ w (M Z ) = 0.23122(4), α −1 (M Z ) = 127.955 (10) to determineĝ 1 andĝ 2 directly via the relation In this way we findĝ 1 = 0.36,ĝ 2 = 0.65. The strong couplingĝ 3 = 1.22 is determined from α s (M Z ) = 0.1181 (11).
To determineŷ τ we use m τ = 1.77686(12) GeV, and the relations Note that G F = 1.1663787 (6)  Ref. [103], but with renamed operators) where the operators involving only quark fields are The operators involving only lepton fields can be chosen as The mixed quark-lepton operators are while the Higgs-fermion operators can be taken as The remaining operator, involving only Higgs fields, is The mixing of the SM sector into the DM-SM sector proceeds via penguin insertions.
The nonzero results for the four-fermion operators are, for i = j, 1...7,ii ×Q 12...17,ii ×Q for mixing into operators with quark currents, and γ (0) 1 S for mixing into operators with lepton currents.
The mixing proportional to g 2 has only a few non-vanishing entries, given by 1,ii Q All the other entries are zero.
The nonzero mixings of the operators involving Higgs currents are given by 18,i Q as well as by 19,i ···S 21,i ×Q 2,i ···Q 23,i ,S 24,i ×Q 10,i ,Q 18,i Q 23,i Q 2,i -Q 4,i into S 2,ij -S 7,ij is given by 2,i ,S 3,i ,S 2,i ,S 3,i ,S 2,i ,S 3,i ,S 4,i ,S while the remaining entries are zero. The mixing of Q 5,i -Q 18 into S 1,ij -S 7,ij vanishes. The mixing of Q 17,ij is given by 2,i ,S 2,i ,S The mixing of Q 1,i -Q 4,i into S 12,ji -S 17,ji , with reversed indices, vanishes. The mixing of Q 1,i -Q (6) 4,i into S (6) 19,i -S (6) 21,i is given by 2,i ,S The mixing of Q (6) 10,i , Q 11,i into the operators S 8,ij -S (6) 10,ij is given by 10,i ,S 10,i ,S 11,i ,S The mixing of Q 10,i , Q 11,i into the operators S 12,ij -S 17,ij is given by where the relevant operators are given in Eq. (19) (recall that we neglect the mixing of operators within the dark sector).
The mixing of DM-SM operators into DM operators is given by
Several additional operators, vanishing due to the e.o.m. for the DM fields, are needed to project all one-loop Greens functions with insertions of the operators in Eqs. (6), (8) and Eqs. (17), (18), respectively: two dimension-five operators, P