Weak mixing below the weak scale in dark-matter direct detection

If dark matter couples predominantly to the axial-vector currents with heavy quarks, the leading contribution to dark-matter scattering on nuclei is either due to one-loop weak corrections or due to the heavy-quark axial charges of the nucleons. We calculate the effects of Higgs and weak gauge-boson exchanges for dark matter coupling to heavy-quark axial-vector currents in an effective theory below the weak scale. By explicit computation, we show that the leading-logarithmic QCD corrections are important, and thus resum them to all orders using the renormalization group.


I. INTRODUCTION
A useful approach to describe the results of Dark Matter (DM) direct-detection experiments is to relate them to an Effective Field Theory (EFT) of DM coupling to quarks, gluons, leptons, and photons [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. In this EFT, the level of suppression of DM interactions with the Standard Model (SM) depends on the mass dimension of the interaction operators, i.e., the higher the mass dimension the more suppressed the operator is. The mass dimension of operators is thus the organizing principle in capturing the phenomenologically most relevant effects, which is why in phenomenological analyses one keeps all relevant terms up to some mass dimension, d. An important question is, at which value of d one can truncate the expansion. The obvious choice would be to keep all operators of dimension five and six, and a subset of dimension-seven operators that do not involve derivatives, as in this case one covers most of the UV models of DM.
In this work, we show that the leading contribution to the scattering cross section originates from double insertions of dimension-six operators if the DM interaction is predominantly due to DM vector currents coupling to heavy-quark axial-vector currents. This effectively means that in such a case it is necessary to extend the EFT to include operators of mass dimension eight. That such corrections are important was first pointed out in Refs. [10,11], with the phenomenological implications further discussed in [17]. We improve on the analysis of Ref. [11] in two ways: i) we clarify how to consistently include the doubleinsertion contributions in the EFT framework, ii) we also perform the resummation of the QCD corrections at leading-logarithmic accuracy. Moreover, the generality of our approach covers also the case of non-singlet DM in the theory above the electroweak scale.
The paper is structured as follows. In Sections II-VI we derive our results for the case of Dirac-fermion DM. These are then extended to the case of Majorana-fermion DM and to the case of scalar DM in Section VII. In Section II we first show that the electroweak corrections have to be included if DM couples only to vector or axial-vector currents with heavy quarks.
The weak interactions below the weak scale are encoded in an effective Lagrangian, which is introduced in Section III. Section IV contains our results for the anomalous dimensions controlling the operator mixing, while the renormalization-group evolution is given in Section V. In Section VI we show how our results connect to the physics above the electroweak scale.
Section VIII contains conclusions, while Appendix A collects some unphysical operators entering in intermediate steps of our calculation.

II. THE IMPORTANCE OF WEAK CORRECTIONS FOR AXIAL CURRENTS
We start by considering the DM EFT valid below the electroweak scale, µ b < µ < µ ew , for Dirac-fermion DM when five quark flavors are active, The sums run over the dimensions of the operators, d, and the operator labels, a. The operators are multiplied by dimensionless Wilson coefficients, C (d) a , and the appropriate powers of the mediator mass scale, Λ. Since we are interested in the theory below the electroweak scale, any interactions with the top quark, W , Z bosons, and the Higgs are integrated out and are part of the Wilson coefficients C (d) a . In this work, we focus on dimension-six operators, namely where f can be any of the SM fermions apart from the top quark. Our dimension counting follows Refs. [16,18], such that scalar four-fermion operators are considered to be dimension seven, i.e., we assume they originate from a Higgs field insertion above the electroweak scale.
As we show below, a proper description of DM scattering on nuclei due to dimension-six operators requires including corrections from QED and the weak interactions. By contrast, such corrections are always subleading for dimension-five and dimension-seven operators.
The basis of dimension-five operators, which couple DM to photons, can be found, e.g., in Refs. [16,18], while the full basis of dimension-seven operators was derived in Ref. [19].
If only a single operator in Eqs.
(2)-(3) contributes, the cross section for DM-nucleus scattering can be written as where a ] is an "effective scattering amplitude". It is a product of the scattering amplitude, the nuclear response functions [13,14,[20][21][22][23], and all the relevant kinematic factors.
a ] in three different limits: i) in the limit of only strong interactions, ii) including QED corrections, and iii) also including corrections from weak interactions.
i) Switching off QED and weak interactions, the effective scattering amplitudes for dimension-six operators have the following parametric sizes (see Ref. [18]):  are therefore modified to 2,q ], (10) while the parametric estimates for the operators with quark axial-vector currents, i.e., Eqs. (7) where s w is the sine of the weak mixing angle. The proportionality to the square of the heavy-quark mass m c(b) -necessary for dimensional reasons -can be deduced from the fact that it is the only relevant mass scale in the regime µ c(b) < µ < µ ew . For Q 3,c(b) these contributions dominate over the axial charge contribution, Eq. Refs. [9,11,18]. In the present work, we calculate the logarithmically enhanced contributions due to the weak interactions. They arise, via double insertions, at second order in the dimension-six effective interactions, cf. Eq. (11). Accordingly, they can mix into dimensioneight operators, which, therefore, also have to be included.
It turns out that the weak corrections are numerically irrelevant for operators coupling DM to light quarks at tree level. Since the weak interactions do not conserve parity, they can lift the velocity suppression in the matrix elements of Q (6) 3,q through the mixing into the coherently enhanced operator Q 1,q . However, the resulting relative enhancement of order A/v T ∼ 10 5 is not enough to compensate for the large suppression of the weak corrections, of order α/(4πs 2 w )(m q /m Z ) 2 10 −9 (m q /100 MeV) 2 . The weak corrections are also much less important for the dimension-five and dimensionseven operators coupling DM to the SM fields [18,19]. Most of these operators have a nonzero nucleon matrix element already without including electroweak corrections, in which case the latter only give subleading corrections. This is the case for the operators coupling DM to gluons or photons, for pseudoscalar currents with light quarks, and for scalar quark currents, including the ones with heavy bottom and charm quarks. In the special case where DM couples only to pseudoscalar heavy-quark currents the nuclear matrix elements vanish.
This remains true also after one-loop electroweak corrections are included.
In the next two sections, we will obtain the leading-logarithmic expressions for the electroweak contributions in Eq. (11) and also resum the QCD corrections by performing the RG running from the weak scale, µ ew ∼ O(m Z ), to the hadronic scale, µ had ∼ O(2 GeV), where we match to the nonrelativistic theory.

III. STANDARD MODEL WEAK EFFECTIVE LAGRANGIAN
The SM interactions below the weak scale are described by an effective Lagrangian, obtained by integrating out the top quark and the Z, W , and Higgs bosons at the scale µ ew ∼ m Z . In this section we focus on quark interactions. We discuss leptons in Section VI.
We can neglect any operators involving flavor-changing neutral currents as well as terms suppressed by off-diagonal Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The only necessary operators are where G F is the Fermi constant and D (6) a are dimensionless Wilson coefficients. The sums run over all light quarks, q, q = u, d, s, c, b, and the labels of the operators with two different quark flavors (q = q ) 2,qq = (qγ µ γ 5 q) (q γ µ γ 5 q ) , 3,qq = (qγ µ γ 5 q) (q γ µ q ) , O O 5,qq = (qγ µ γ 5 T a q) (q γ µ γ 5 T a q ) , O 6,qq = (qγ µ γ 5 T a q) (q γ µ T a q ) , and a single quark flavor, 2,q = (qγ µ γ 5 q) (qγ µ γ 5 q) , Here, T a are the SU (3) c generators normalised as Tr(T a T b ) = 1 2 δ ab . As seen from the above operator basis, there are fewer linearly independent operators with a single quark than with two different quarks. The reason is that Fierz identities relate operators, like for instance the counterpart of O (6) qq with four equal quark fields, to the operators O (6) i,q with i = 1, . . . , 4. One way of implementing the Fierz relations is to project Green's functions onto the basis that includes so-called Fierz-evanescent operators, like E q 7 and E q 8 in Eq. (A2) of Appendix A, that vanish due to Fierz identities. SM operators with scalar or tensor currents do not contribute in our calculation. This is most easily seen by inspecting their chiral and Lorentz structure, neglecting operators with derivatives (see below).
Integrating out the W and the Z bosons at tree level gives the following values for the Wilson coefficients at µ ew 4,qq = D and 2,q = 2s 2 w c 2 w a 2 q , D Here, s w ≡ sin θ w , c w ≡ cos θ w , with θ w the weak mixing angle, while I 3 q is the third component of the weak isospin for the corresponding left-handed quark, i.e., I 3 q = 1/2 for q = u, c and I 3 q = −1/2 for q = d, s, b. The CKM matrix, V qq , will be set to unity unless specified otherwise, while the vector and axial-vector couplings of the Z boson to the quarks where Q q is the electric charge of the corresponding quark. Note that D (6) i,qq ≡ D (6) i,q q for i = 1, 2, 4, 5, since the corresponding operators are symmetric under q ↔ q . Figure 2: A generic Feynman diagram with a double insertion of dimension-six operators, leading to the mixing into dimension-eight operators.

IV. OPERATOR MIXING AND ANOMALOUS DIMENSIONS
We are now ready to derive the leading contributions to the DM-nucleon scattering rates for the case that, at the weak scale, DM interacts with the visible sector only through the dimension-six operators Q where For future convenience, we defined the operators including two inverse powers of the strong where we set µ ew = m Z . This equation shows that the operators with derivatives, for instance, (χγ µ χ)∂ 2 (qγ µ q), can be neglected because their effect on the scattering rates is not enhanced by the large ratio m 2 b,c /m 2 q . Furthermore, the set of operators in Eqs. (23)-(24) is closed under RG running up to mass-dimension eight, if we keep only terms proportional to two powers of the bottom-or charm-quark mass in the RG evolution. At higher orders in QCD the purely electroweak expression Eq. (25) gets corrected by terms of the order of m Z , these terms can amount to O(1) corrections. In the following we resum these large QCD logarithms to leading-logarithmic order.
In Eq. (26), we have already made use of the fact that the QCD anomalous dimensions of the operators Q 1 The only exception occurs when the values of the Wilson coefficients at the weak scale conspire to exactly cancel the divergence, so that the sum of the double-insertion diagrams is finite. This scenario is not fine tuned if it is protected by a symmetry. A example of from the SM is the charm-quark contribution to the parameter K , where the GIM mechanism associated with the approximate flavor symmetry of the SM serves to cancel all divergences. We call the analogous mechanism for DM the "judicious operator equality GIM", in short "Joe-GIM" mechanism. For Joe-GIM DM there is no mixing of dimension-six operators into dimension-eight operators below the weak scale [29]. The leading contributions to the dimension-eight operators are then obtained by a finite one-loop matching calculation at the heavy-quark scales.
The RG evolution of the dimension-six Wilson coefficients is determined by a RG equation that is linear in the Wilson coefficients, On the other hand, the running of the dimension-eight Wilson coefficients receives two contributions. In addition to the running of the m 2 q /g 2 s prefactor, encoded inγ, there are contributions from double insertions of dimension-six operators, see Fig. 2. This leads to a RG equation that is quadratic in dimension-six Wilson coefficients [33,34], To leading order in the strong coupling constant the rank-three anomalous dimension tensor γ ab,c [33,34] is given byγ Next we provide the explicit values for the anomalous dimensions. In our notation, the anomalous dimensions are expanded in powers of α s , with γ (n) ∝ (α s /4π) n , and similarly forγ andγ.
We start by giving the results for the mixing of the dimension-eight operators coupling DM to quarks, Eqs. (23) and (24). This mixing is encoded in theγ andγ anomalous dimensions. We obtain theγ from the poles of the double insertions, Fig. 2. The only nonzero entries leading to mixing into operators with light-quark currents arê 3,q q ,Q 3,q ;Q 3,q q ,Q 4,q ;Q 3,q ;Q 4,q ;Q The remaining contribution to the RG running of the dimension-eight operators is entirely due to the m 2 q /g 2 s prefactors in the definition of the operators, namelỹ with C F = 4/3 for QCD, and N f the number of active quark flavors. The RG running of the dimension-six operators in the SM weak effective Lagrangian is due to one-loop gluon exchange diagrams, see Fig. 3. Since the corresponding anomalous dimension matrix γ has many entries, we split the result into several blocks.
The anomalous dimension matrix in the subsector spanned by the operators in Eqs. (16)- 1,q , O 2,q , O 3,q , O Note that, at one-loop, there is no mixing into operators with a different quark flavor.
The anomalous dimensions describing the mixing of the same operators, O 1,q , O 2,q , O 3,q , O into the operators 2,qq , O 3,qq , O 3,q q , O 4,qq , O 5,qq , O 6,qq , O 6,q q , read The anomalous dimension matrix in the subsector spanned by the operators in Eqs. (13)- 1,qq , O 2,qq , O 3,qq , O 3,q q , O 4,qq , O 5,qq , O 6,qq , O 6,q q , reads The part of the anomalous dimension matrix mixing the operators, O 1,qq , O 2,qq , O 3,qq , O 3,q q , O 4,qq , O 5,qq , O 6,qq , O 6,q q , into the same operators, but with different quark flavor structure, q = q , O 1,qq , O 2,qq , O 3,qq , O 3,q q , O 4,qq , O 5,qq , O 6,qq , O is Finally, the part of the anomalous dimension matrix mixing the operators O 1,qq , O 2,qq , O 3,qq , O 3,q q , O 4,qq , O 5,qq , O 6,qq , O 1,q , O 2,q , O 3,q , O has only two nonzero entries, All the remaining entries in γ vanish. We extracted the anomalous dimensions from the off-shell renormalization of Green's functions with appropriate external states. We checked explicitly that our results are gauge-parameter independent. In Appendix do not run, thus C b (µ ew ). The RG running for the remaining Wilson coefficients is controlled by the RG equations in Eqs. (28) and (29), which we combine into a single expression, Here, we defined a vector of Wilson coefficients as and absorbed the (scale-independent) Wilson coefficients C a into the effective anomalousdimension matrix Since the C a Wilson coefficients are RG invariant, the tensor product effectively transforms the rank-three tensorγ ab,c into an equivalent matrix, C (6) ·γ, with all its entries constant, that is equivalent to the tensor for the purpose of RG running. This has the advantage that one can use the standard methods for single insertions to solve the RG equations.
The RG evolution proceeds in multiple steps. The first step is the matching of the (complete or effective) theory of DM interactions above the weak scale onto the five-flavor EFT. This matching computation yields the initial conditions for C (6) a (µ ew ) and C  We discuss the latter case in Section VI. For the RG evolution below the electroweak scale one also needs the coefficients D (6) a (µ ew ). The SM contributions to the tree-level initial conditions for D (6) a (µ ew ) are provided in Eqs. (18)- (20). The second step is to evolve C Here the µ b(c) denote the threshold scale at which the bottom(charm)-quark is removed from the theory. In our numerical analysis we will use µ b = 4.18 GeV and µ c = 2 GeV. At leading-logarithmic order, there are no non-trivial matching corrections at the bottom-and charm-quark thresholds, and we simply have This means that we can switch to the EFT with four active quark flavors by simply dropping all operators in Eq. (26) that involve a bottom-quark field, and to the EFT with three active quark flavors by simply dropping all operators with charm-quark fields. The leading-order matching at µ q ∼ m q comes with a small uncertainty due to the choice of matching scale that is of order log(µ q /m q ). This is formally of higher order in the RG-improved perturbation theory. The uncertainty is canceled in a calculation at next-to-leading-logarithmic order by finite threshold corrections at the respective threshold scale.
This is a good point to pause and compare our results with the literature. The RG

evolution of the operators in Eqs. (2)-(3) below the electroweak scale has been studied in
Ref. [11], which effectively resummed the large logarithms log(µ had /µ ew ) to all orders in the Secondly, such a resummation is not consistent within the EFT framework. Since there are no Higgs-boson exchanges in the EFT below the weak scale, the scheme-dependence of the anomalous dimensions and the residual matching scale dependence at the heavy-quark thresholds is not consistently canceled by higher-orders, leading to unphysical results.
Continuing with our analysis, the final step is to match the three-flavor EFT onto the EFT with nonrelativistic neutrons and protons that is then used to predict the scattering rates for DM on nuclei using nuclear response functions. The matching for the dimensioneight contributions proceeds in exactly the same way as described in Refs. [14,16] for the operators up to dimension seven. In practice, this means that we obtain the following contributions to the nonrelativistic coefficients (see Refs. [14,16,18,20,22]), and similarly for neutrons, with p → n. The quark masses and the strong-coupling constant in these expressions should be evaluated in the three-flavor theory at the same scale as the nuclear response functions, i.e., µ had = 2 GeV. The ellipses denote the contributions from dimension-six interactions proportional to C (6) a as well as the contributions due to dimensionfive and dimension-seven operators, which can be found in Eqs. (17)-(24) of Ref. [16].
The strong coupling α s appears in Eqs. (51)-(56) as a consequence of the 1/g 2 s prefactor in the definition of the dimension-eight operators in Eqs. (23)- (24). When expanding the resummed results in the strong coupling constant, the α s cancels in the leading expressions, and we find The quark masses in these expressions should be evaluated at the weak scale, m q = m q (m Z ), while µ q is the scale at which the q quark is integrated out. We have provided the SM Wilson coefficients, D 2,qq and D 3,q q , in Eqs. (18) and (19). The expanded equations clearly illustrate that the leading terms are of electroweak origin, and thus of O(α 0 s ), while the corrections due to QCD resummation start at O(α s ). These were obtained using the one-loop QCD running to evolve the MS quark masses 3,c C 4,c C 3,c (31% for the result without resummation), as there is an up to 10% cancellation between the D 3,cu and D 3,cd contributions with respect to the case of unit CKM matrix. For all other cases the error due to setting the CKM matrix to unity is less than 10%.
Finally, we compare the contributions to DM scattering originating from electroweak corrections as opposed to the intrinsic charm and bottom axial charges. For the case of intrinsic: 4,c + ∆b C 4,c + 0.2C We see that for the bottom quarks the weak contribution, Eq. (58), is comparable to the contribution from the intrinsic bottom axial charge, while for charm quarks the contribution due to the intrinsic charm axial charge dominates.
For vector-axial-vector interactions (C 3,c = 0, C 3,b = 0) we have weak: 3,c − 3.0 C intrinsic: 3,c + ∆b C 3,c + 0.1C 3,c + ∆b C 3,c + 0.9C where in the last equality we set m χ = 100 GeV. The effective scattering amplitude is parametrically given by where v T ∼ 10 −3 , q/m N 0.1, and A ∼ 100 for heavy nuclei. The loop-induced weak contributions thus dominate the scattering rates of weak scale DM.

VI. CONNECTING TO THE PHYSICS ABOVE THE WEAK SCALE
We now describe how to apply and extend our results for the case in which there is a separation between the mediator scale and the electroweak scale, i.e., if Λ m Z . In this case, the effective Lagrangian valid above the weak scale is with the operators Q where the index i = 1, 2, 3 labels the generation, and τ a = σ a /2, with the Pauli matrices σ a . If χ is an electroweak singlet, the operators Q 1,i and Q 6,3 , and C 8,3 , and such that they satisfy the relation Y χ C We first derive the leading electroweak contribution to DM-nucleon scattering rates for this case and then discuss the case in which DM couples only to charm axial currents.
To this end, we fist assume that the initial conditions at µ Λ satisfy Eq. (74). At scales µ ew < µ < Λ, the operators Q (6) a,i mix at one-loop via the SM Yukawa interactions into the Higgs-current operators [10,11,38] and a similar set of operators with theτ a ⊗τ a structure (above, . This mixing is generated by "electroweak fish" diagrams, see Fig. 6 (left), and induces at µ ew m Z C 16 (m Z ) v 2 /Λ 2 . The factor v 2 originates from the two Higgs fields relaxing to their vacuum expectation values and the factor G F from integrating out the Z boson.
Since the one-loop RG running from Λ to µ ew , Eq. (76), followed by tree-level matching at µ ew , induces interactions proportional to y 2 b v 2 , it is convenient to match such corrections to initial conditions of the dimension-eight operators in Eqs. (24). 2 For the pattern of initial conditions in Eq. (74), we then find that the Wilson coefficient of the dimension-eight operator Q 2 Here, we have decided to ascribe the tree-level Z exchange contribution from the matching at µ ew to dimension-eight four-fermion operators. Alternatively, we could have absorbed also this contribution into the Wilson coefficients of dimension-six operators. This choice would have the unattractive property of having the parametric suppression of y 2 b v 2 G F = m 2 b G F hidden in the smallness of some of (the parts of) the Wilson coefficients C Again, we only show the parametric dependence, including loop factors, but omit O (1) factors, e.g., from the actual values of anomalous dimensions (for details see Ref. [38]). In particular, C a,3 (Λ) denotes a linear combination of the Wilson coefficients with a = 5, 6, 8. The subsequent RG evolution from µ ew to µ had proceeds as described in Section V, Eqs. (47)-(50). The only difference is that the initial conditions C      3,2 = 0 this means that just the (χγ µ χ)(bγ µ γ 5 b) or (χγ µ χ)(cγ µ γ 5 c) operators are generated. Such relations do not necessarily imply fine-tuning, as they can originate from the quantum number assignments for the mediators, DM, and quark fields in the UV theory. They do require the DM hypercharge Y χ to be nonzero. 3 This conclusion changes, if at µ Λ we also include dimension-eight operators of the form (χγ µ χ)(Q L Hγ µ HQ L ) alongside the dimension-six (χγ µ χ)(b R γ µ b R ) operators. In this case, it is possible to induce only the (χγ µ χ)(bγ µ γ 5 b) or (χγ µ χ)(cγ µ γ 5 c) operators even for DM with zero hypercharge (and thus without a renormalizable interaction to the Z boson). This, however, requires fine-tuning of dimension-six and dimension-eight contributions.
Note that the relation in Eq. (74) also requires DM to be part of an electroweak multiplet.
For singlet DM there is no operator Q (6) 5,i and so C (6) 5,i is trivially zero. Therefore, for singlet DM a coupling to an axial-vector bottom-quark current is always accompanied by couplings to top quarks. In this case our results get corrected by terms of order y 2 t log(µ ew /Λ) from the RG evolution above the electroweak scale due to top-Yukawa interactions [38].
Another phenomenologically interesting case is the one of DM coupling only to leptons at µ ew , i.e., through operators in Eqs.
2,f , Q 4,f , Q 2,f , and Q (8) 4,f should be multiplied by a factor of 1/2 to account for the additional Wick contractions (see, for instance, Ref. [18]). With these modifications, the coefficients of the nuclear effective theory are still given by Eqs. (51)-(56).

B. Scalar dark matter
The relevant set of operators for scalar DM is where ϕ * ↔ ∂ µ ϕ ≡ ϕ * ∂ µ ϕ − (∂ µ ϕ * )ϕ. These operators are part of the dimension-six effective Lagrangian for scalar DM, cf., Ref. [14], with C a the dimensionless Wilson coefficients. Note that we adopt the same notation for operators and Wilson coefficients in the case of scalar DM as we did for fermionic DM. No confusion should arise as this abuse of notation is restricted to this subsection.
Apart from having a different DM current, nothing changes in our calculations. Therefore, after defining the dimension-eight effective Lagrangian in the three-flavor theory as the additional contributions to the nuclear coefficients are given, for complex scalar DM, by (cf. Ref. [18]) For real scalar DM, the operators in Eq. (79) vanish. For completeness, we display also the dimension-eight contributions to the nuclear coefficients, expanded to leading order in the strong coupling constant, 2,q (m Z ) q=u,d,s

Evanescent operators
The one-loop mixing among the "physical" operators is not affected by the definition of evanescent operators, i.e., operators that are required to project one-loop Green's functions in d = 4 − 2 dimensions but vanish in d = 4. Indeed, our one-loop results could also have been obtained by performing the Dirac algebra in d = 4 instead off in non-integer dimensions. Since i) this no longer possible at next-to-leading order computations and ii) we use dimensional regularization to extract the poles of loop integrals, we find it convenient Here, N c = 3 denotes the number of colors. The operators E q 7 and E q 8 are Fierz-evanenscent operators, i.e., they vanish due to Fierz identities and not due to special d = 4 relations of the Dirac algebra.
6,qq . (A4) The four-fermion pieces of these e.o.m.-vanishing operators contribute to the same amplitudes as the physical four-fermion operators. Therefore, the mixing of physical operators