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

Journal of High Energy Physics, Feb 2018

Joachim Brod, Benjamin Grinstein, Emmanuel Stamou, Jure Zupan

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

HJE mixing below the weak scale in dark-matter Joachim Brod 0 1 3 6 7 Benjamin Grinstein 0 1 3 4 7 Emmanuel Stamou 0 1 3 5 7 Jure Zupan 0 1 2 3 7 0 5640 S Ellis Ave , Chicago, IL, 60637 U.S.A 1 9500 Gilman Dr. , La Jolla, CA, 92093 U.S.A 2 Department of Physics, University of Cincinnati 3 Otto-Hahn-Str. 4, Dortmund, D-44221 Germany 4 Department of Physics, University of California-San Diego , USA 5 Enrico Fermi Institute, University of Chicago 6 Fakultat fur Physik, TU Dortmund 7 400 Geology/Physics Bldg. , Cincinnati, Ohio, 45221 U.S.A 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 e ects of Higgs and weak gauge-boson exchanges for dark matter coupling to heavy-quark axial-vector currents in an e ective 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. Beyond Standard Model; Perturbative QCD; Renormalization Group; Re- - Weak 1 Introduction 2 3 4 5 6 7 8 Majorana and scalar dark matter 7.1 7.2 Majorana dark matter Scalar dark matter Conclusions A Unphysical operators A.1 Evanescent operators A.2 E.o.m.-vanishing operators Standard Model weak e ective Lagrangian Operator mixing and anomalous dimensions Renormalization group evolution 5.1 Numerical analysis and the impact of resummation Connecting to the physics above the weak scale 1 Introduction A useful approach to describe the results of Dark Matter (DM) direct-detection experiments is to relate them to an E ective Field Theory (EFT) of DM coupling to quarks, gluons, leptons, and photons [1{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 e ects, 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 ve and 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 double-insertion 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 2{6 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 7. In section 2 we rst show that the electroweak correcoperators entering in intermediate steps of our calculation. 2 The importance of weak corrections for axial currents We start by considering the DM EFT valid below the electroweak scale, b < < ew, for Dirac-fermion DM when ve quark avors are active, L = X Ca a;d (d) d 4 Q(ad) : (2.1) (2.2) (2.3) The sums run over the dimensions of the operators, d, and the operator labels, a. The operators are multiplied by dimensionless Wilson coe cients, Ca(d), and the appropriate powers of the mediator mass scale, . 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 coe cients Ca(d). In this work, we focus on dimension-six operators, namely Q(16;f) = ( Q(36;f) = ( )(f )(f f ) ; tude, the nuclear response functions [13, 14, 20{23], and all the relevant kinematic factors. We estimate A[Q(ad)] in three di erent limits: i) in the limit of only strong interactions, ii) including QED corrections, and iii) also including corrections from weak interactions. HJEP02(18)74 i) Switching o QED and weak interactions, the e ective scattering amplitudes for dimension-six operators have the following parametric sizes (see ref. [18]): A[Q(16;u)(d)] A[Q(26;u)(d)] A[Q(36;u)(d)] A ; 10 3 is the typical DM velocity in the laboratory frame, q is the typical momentum exchange, q=mN . 0:1, where mN is the nucleon mass, and A is the nuclear mass number (for heavy nuclei A 102). The approximate expressions for the e ective scattering amplitudes in eqs. (2.5){(2.8) include the parametric O(A) coherent enhancement of the spin-independent nuclear response function, WM (q), while all the other response functions were counted as O( 1 ). The vector and axial form factors at zero recoil are O( 1 ) for u; d quarks. For the strange, charm and bottom quarks the vector form factors vanish. The axial charge for the strange quark is reasonably well known, s = factor of two uncertainty on these estimates. Due to the non-relativistic nature of the problem and the sizes of the nuclear matrix elements, there are large hierarchies between the e ective scattering amplitudes. For light quarks this hierarchy can be as large as vT =A χ Q(a6,q)′ χ χ f γ Q(a6,q)′ q ′ Ob(,6q)q′ heavy quarks into DM vector interactions with light quarks. Here, f = u; d; s; c; b; e; ; can denote any of the quarks or charged leptons. see eq. (3.1) in the next section. A double insertion of one four-fermion operator from LeSM and one from L , see gure 2, induces the additional contributions to A[Q(36;c)(b)] 4 s2w mmc2(2Zb) A ; 4 s2w mmc2(2Zb) max vT A; q mN ; (2.11) (6) where sw is the sine of the weak mixing angle. The proportionality to the square of the heavy-quark mass mc(b) | necessary for dimensional reasons | can be deduced from the fact that it is the only relevant mass scale in the regime c(b) < < ew. For Q3;c(b) these contributions dominate over the axial charge contribution, eq. (2.7), { 4 { (6) by several orders of magnitude, while for Q4;c(b) the electroweak corrections are either comparable or smaller than in eq. (2.8). More details follow in the next sections. The above estimates show that QED and weak corrections are essential to capture the leading contributions for the dimension-six operators in eqs. (2.2){(2.3) that involve heavy quarks. The same type of QED and weak radiative corrections also induce the leading e ective amplitudes for the scattering on nucleons when the DM couples, at tree level, only to leptons. The logarithmically enhanced QED contributions are known, see for instance 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 e ective interactions, Accordingly, they can mix into dimension-eight 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(36;q) through the mixing into the coherently enhanced operator Q(16;q). However, the resulting relative enhancement of order A=vT sion of the weak corrections, of order 105 is not enough to compensate for the large suppres =(4 s2w)(mq=mZ )2 . 10 9(mq=100 MeV)2. The weak corrections are also much less important for the dimension- ve and dimension-seven operators coupling DM to the SM elds [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. (2.11) and also resum the QCD corrections by performing the RG running from the weak scale, ew O(mZ ), to the hadronic scale, had O(2 GeV), where we match to the nonrelativistic theory. 3 Standard Model weak e ective Lagrangian The SM interactions below the weak scale are described by an e ective Lagrangian, obtained by integrating out the top quark and the Z, W , and Higgs bosons at the scale ew mZ . In this section we focus on quark interactions. We discuss leptons in section 6. We can neglect any operators involving avor-changing neutral currents as well as terms suppressed by o -diagonal Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The only necessary operators are LeSM p 2GF 8 <X :q6=q0 2 1 X 4 2 i=1;2;4;5 Di;qq0 Oi(;6q)q0 + (6) X i=3;6 3 Di;qq0 Oi;qq0 5 + X (6) (6) X q i=1;:::;4 (6) Di;q Oi;q (6)= ; 9 ; (3.1) { 5 { Here, T a are the SU(3)c generators normalised as Tr(T aT b) = 12 ab. As seen from the above operator basis, there are fewer linearly independent operators with a single quark than with two di erent quarks. The reason is that Fierz identities relate operators, like for instance the counterpart of Oq(6q)0 with four equal quark elds, to the operators Oi;q (6) 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 E7q and E8 q in eq. (A.2) 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 coe cients at ew (6) D1;qq0 = 4s2wc2wvqvq0 + (6) D3;qq0 = 4s2wc2waqvq0 I j q 3 I j q 3 6 6 3 Iq0 j 3 Iq0 j jVqq0 j2 ; jVqq0 j2 ; and (6) D2;qq0 = 4s2wc2waqaq0 + I j q 3 3 Iq0 j 6 jVqq0 j2 ; (6) D4;qq0 = D5;qq0 = (6) (6) 3 D6;qq0 = jIq Iq30 jjVqq0 j2 ; where GF is the Fermi constant and Da run over all light quarks, q; q0 = u; d; s; c; b, and the labels of the operators with two di erent (6) are dimensionless Wilson coe cients. The sums quark avors (q 6= q0) and a single quark avor, O1(6;q) = (q q) (q q) ; O3(6;q) = (q D1(6;q) = 2s2wc2wvq2 ; D2(6;q) = 2s2wc2waq2 ; D3(6;q) = 4s2wc2wvqaq ; D4(6;q) = 0 : for q = u; c and Iq3 = the quarks are encoded in Here, sw sin w, cw cos w, with w the weak mixing angle, while Iq3 is the third component of the weak isospin for the corresponding left-handed quark, i.e., I q3 = 1=2 1=2 for q = d; s; b. The CKM matrix, Vqq0 , will be set to unity unless speci ed otherwise, while the vector and axial-vector couplings of the Z boson to vq I 3 q 2s2wQq 2swcw ; aq I 3 q 2swcw ; where Qq is the electric charge of the corresponding quark. Note that Di;qq0 i = 1; 2; 4; 5, since the corresponding operators are symmetric under q $ q0. (6) { 6 { (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) Di(;6q)0q for HJEP02(18)74 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(36;q) or Q(46;q), with q = b; c. To properly describe all the leading DM interactions we need to extend the dimension-six e ective Lagrangian L , eq. (2.1), to include the following dimension-eight operators where Q(18;q) = Q(38;q) = mq2 g2 ( s mq2 g2 ( s L p 2GF 2 )(q )(q q) ; 5q) ; X For future convenience, we de ned the operators including two inverse powers of the strong coupling constant. Even if the Wilson coe cient of the dimension-eight operators are zero at ew, they are generated below the electroweak scale from a double insertion of one of the dimension-six operators in L in eq. (2.1) and one of the dimension-six operator from LeSM in eq. (3.1), see gure 2.1 The logarithmic part of the running from ew to had gives where we set ew = mZ . This equation shows that the operators with derivatives, for instance, ( q), can be neglected because their e ect on the scattering rates is not enhanced by the large ratio mb2;c=mq2. Furthermore, the set of operators in eqs. (4.2){(4.3) 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. (4.4) gets corrected by terms of the order of sn 1 logn(mb(c)=mZ ). Since mb(c) mZ , these terms can amount to O( 1 ) corrections. In the following we resum these large QCD logarithms to leading-logarithmic order. The techniques for the calculation of leading-logarithmic QCD corrections with double insertions are standard [29{36]. We rst replace the bare Wilson coe cients in 1The only exception occurs when the values of the Wilson coe cients at the weak scale conspire to exactly cancel the divergence, so that the sum of the double-insertion diagrams is nite. This scenario is the parameter K, where the GIM mechanism associated with the approximate avor 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 nite one-loop matching calculation at the heavy-quark scales. { 7 { The compound indices a, b, c, run over both the operator labels and quark- avor indices. In eq. (4.5), we have already made use of the fact that the QCD anomalous dimensions of the operators Q(a6) in eqs. (2.2){(2.3) are zero, and have not introduced the corresponding renormalization constants. In dimension regularization around d = 4 2 space-time dimensions, the renormalization constants admit a double expansion in the strong coupling constant and HJEP02(18)74 Zab = ab + 4 s X k=0;1 1k Za(1b;k) + O( s2) ; and similarly for Z~ and Z^. equation that is linear in the Wilson coe cients, The RG evolution of the dimension-six Wilson coe cients is determined by a RG eqs. (2.1), (3.1), and (4.1) with their renormalized counterparts. The corresponding effective Lagrangian reads Le = 1 2 X a p 2GF 2 = ( 1 ) + (2) + ; { 8 { (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) d d Da(6)( ) = X b Db(6)( ) ba ; with = d log Z d log : On the other hand, the running of the dimension-eight Wilson coe cients receives two contributions. In addition to the running of the mq2=gs2 prefactor, encoded in ~, there are contributions from double insertions of dimension-six operators, see gure 2. This leads to a RG equation that is quadratic in dimension-six Wilson coe cients [33, 34], dd Ca(8)( ) = X b Cb(8)( )~ba + X Db(6)( )Cc(6)( )^bc;a : bc To leading order in the strong coupling constant the rank-three anomalous dimension tensor ^ab;c [33, 34] is given by ^ab;c = 2 s Z^a(1b;;c1) + O( s2) : Next we provide the explicit values for the anomalous dimensions. In our notation, the anomalous dimensions are expanded in powers of s , with (n) / ( s=4 )n, and similarly for ~ and ^. We start by giving the results for the mixing of the dimension-eight operators coupling DM to quarks, eqs. (4.2) and (4.3). This mixing is encoded in the ~ and ^ anomalous q ′ q q ′ q q ′ q q ′ q q of four-fermion operators. The poles of QCD penguin diagrams a ect the mixing via the e.o.m.vanishing operators de ned in appendix A. dimensions. We obtain the ^ from the poles of the double insertions, gure 2. The only nonzero entries leading to mixing into operators with light-quark currents are ^O(13()6;q)0q;Q(36;q)0 ;Q1;q ( 1 ) ( 1 ) (8) = ^O3(6;q)0q;Q(46;q)0 ;Q2;q (8) = ^O2(6;q)q0 ;Q(36;q)0 ;Q3;q (8) = ^O2(6;q)q0 ;Q(46;q)0 ;Q4;q (8) = ( 1 ) The remaining contribution to the RG running of the dimension-eight operators is entirely due to the mq2=gs2 prefactors in the de nition of the operators, namely ~ab = 2( m(0) with CF = 4=3 for QCD, and Nf the number of active quark avors. The RG running of the dimension-six operators in the SM weak e ective Lagrangian is due to one-loop gluon exchange diagrams, see gure 3. Since the corresponding anomalous dimension matrix has many entries, we split the result into several blocks. The anomalous dimension matrix in the subsector spanned by the operators in 16Nc mq2 4 mq20 s reads into the operators Note that, at one-loop, there is no mixing into operators with a di erent quark avor. The anomalous dimensions describing the mixing of the same operators, O1;q ; O2;q ; O3;q ; O4(6;q) ; (6) (6) (6) ( 1 ) = 4 0 B B B C C : A The anomalous dimension matrix in the subsector spanned by the operators in read eqs. (3.2){(3.4), reads is into the operators has only two nonzero entries, ( 1 ) = 4 s B B 0 B B B Finally, the part of the anomalous dimension matrix mixing the operators 0 12 0 0 5 0 0 19 3 (6) (6) (6) (6) 8 3 8 3 9 12 0 0 0 5 0 0 9 0 0 0 0 0 12 23 3 5 (6) (6) 4 3 4 3 ( 1 ) = 4 00 0 0 01 B B B0 0 0 0C B0 0 0 0C C C s BBB00 00 00 04CCC : B C 12 0 0 0 0 0 5 1 C C C C C C C C C C C C C C CC : The part of the anomalous dimension matrix mixing the operators, into the same operators, but with di erent quark avor structure, q00 6= q0, (6) O1;qq0 (6) (6) ; O2;qq0 ; O3;qq0 ; O3;q0q ; O4;qq0 ; O5;qq0 ; O6;qq0 ; O6;q0q ; (6) (6) (6) (6) ; (6) (6) ( 1 ) = diag 0; 0; 0; 0; ; 0; ; 0 : (6) (6) (6) O1;qq0 ; O2;qq0 ; O3;qq0 ; O3;q0q ; O4;qq0 ; O5;qq0 ; O6;qq0 ; O6;q0q (6) (6) (6) (6) (6) O1;q ; O2;q ; O3;q ; O4;q (6) All the remaining entries in vanish. We extracted the anomalous dimensions from the o -shell renormalization of Green's functions with appropriate external states. We checked explicitly that our results are gauge-parameter independent. In appendix A, we list the evanescent and e.o.m.-vanishing operators that enter at intermediate stages of the computation. 5 Renormalization group evolution With the anomalous dimensions of section 4, we can now compute the Wilson coe cients at had not run, thus Cb(6)( had) = Cb(6)( ew). The RG running for the remaining Wilson coe cients 2 GeV in terms of initial conditions at the weak scale. The Wilson coe cients Cb(6) do is controlled by the RG equations in eqs. (4.7) and (4.8), which we combine into a single Here, we de ned a vector of Wilson coe cients as d d Ca( ) = X Cb( ) bea : b C ( ) D C (6)( ) (8)( ) ! ; (5.1) (5.2) (5.3) and absorbed the (scale-independent) Wilson coe cients Ca(6) into the e ective anomalousdimension matrix e 0 C (6) ^ ~ ! with (C(6) ^)ba X c Cc(6)^bc;a : Since the Ca(6) Wilson coe cients are RG invariant, the tensor product e ectively transforms the rank-three tensor ^ab;c into an equivalent matrix, C(6) ^, 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 rst step is the matching of the (6) (complete or e ective) theory of DM interactions above the weak scale onto the ve- avor EFT. This matching computation yields the initial conditions for Ca(6)( ew) and Ca(8)( ew) at the weak scale. At leading-logarithmic order it su ces to perform the matching at M ew at tree-level. If the mediators have weak-scale masses, we obtain Ca(6)( ew) 6= 0 and Ca(8)( ew) = 0. If the mediators are much heavier than the weak scale, with masses of order mZ , the RG running above the electroweak scale can induce nonzero Ca(8)( ew) log( ew=M ). We discuss the latter case in section 6. For the RG evolution below the electroweak scale one also needs the coe cients Da(6)( ew). The SM contributions to the tree-level initial conditions for Da ( ew) are provided in eqs. (3.7){(3.9). The second step is to evolve Ca ( ) and Da(6)( ) from the electroweak scale to lower (8) scales according to eq. (5.1). The RG evolution is in a theory with Nf = 5 quark avors, when b ew, with Nf = 4, when c b, and with Nf = 3, when had c. Here the b(c) denote the threshold scale at which the bottom(charm)-quark is removed from the theory. In our numerical analysis we will use b = 4:18 GeV and c = 2 GeV. At leading-logarithmic order, there are no non-trivial matching corrections at the bottomand charm-quark thresholds, and we simply have Ca(d)( b) Nf =4 = Ca(d)( b) Nf =5 ; Ca(d)( c) Nf =3 = Ca(d)( c) Nf =4 : (5.4) This means that we can switch to the EFT with four active quark avors by simply dropping all operators in eq. (4.5) that involve a bottom-quark eld, and to the EFT with three active quark avors by simply dropping all operators with charm-quark elds. The leading-order matching at q0 mq0 comes with a small uncertainty due to the choice of matching scale that is of order log( q0 =mq0 ). This is formally of higher order in the RG-improved perturbation theory. The uncertainty is canceled in a calculation at next-toleading-logarithmic order by nite 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. (2.2){(2.3) below the electroweak scale has been studied in ref. [11], which e ectively resummed the large logarithms log( had= ew) to all orders in the Yukawa couplings. Such a resummation is problematic for two reasons. Firstly, it does not take into account that the RG evolution stops at the heavy-quark thresholds, below which the Wilson coe cients are RG invariant. (Below the heavy-quark thresholds, there are no double insertions with heavy quarks and the running of the factor mq2=gs2 is precisely canceled by the running of the Wilson coe cients of the dimension-eight operators.) This introduces a spurious scale dependence of the Wilson coe cients in the three- avor EFT, of the order of jlog( had=mb(c))j . 50%, that is not canceled by the hadronic matrix elements. 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 nal step is to match the three- avor 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 coe cients (see refs. [14, 16, 18, 20, 22]), HJEP02(18)74 c1p = c4p = c6p = c7p = 1 s p 2 4 1 s 4 1 1 s p 2GF 2 X (5.5) (5.6) (5.7) (5.8) c8p = c9p = 2 2 2GF 2 2 q=u;d;s X and similarly for neutrons, with p ! n. The quark masses and the strong-coupling constant in these expressions should be evaluated in the three- avor theory at the same scale as the nuclear response functions, i.e., had = 2 GeV. The ellipses denote the contributions from dimension-six interactions proportional to Ca to dimension- ve and dimension-seven operators, which can be found in eqs. (17){(24) (6) as well as the contributions due s appears in eqs. (5.5){(5.10) as a consequence of the 1=gs2 prefactor in the de nition of the dimension-eight operators in eqs. (4.2){(4.3). When expanding the resummed results in the strong coupling constant, the s cancels in the leading expressions, and we nd p c1 ' p c4 ' p c6 ' p c7 ' c8p = c9p = 3 p 2 2GF 2 12 2 p 2GF 2 3 p 2 2GF 2 6 p 2 2GF 2 6 p 2GF 6 p 2GF 2 2 2 2 X The quark masses in these expressions should be evaluated at the weak scale, mq0 = mq0 (mZ ), while q0 is the scale at which the q0 quark is integrated out. We have provided the SM Wilson coe cients, D2;qq0 and D3;q0q, in eqs. (3.7) and (3.8). The expanded equations clearly illustrate that the leading terms are of electroweak origin, and thus of O( s0), while the corrections due to QCD resummation start at O( s). 0− 0− × − × 0− × − × the scattering of DM on protons, taking only one of the Wilson coe cients nonzero, setting it to Ca (d) = 1= 2, with = 1 TeV. Hatched: the results without QCD resummation, eqs. (5.11){(5.16). 5.1 Numerical analysis and the impact of resummation In gures 4 and 5 we show two numerical examples that illustrate the relative importance of the above results. We set = 1 TeV and switch on a single dimension-six Wilson coe cient, C3;c , C4;c , C3(d;b), or C4(d;b) at a time, setting it to Ca (d) (d) (d) = 1. Figure 4 shows the resulting nonrelativistic couplings for SM scattering on protons, cip. The magenta columns are the full results, including QCD resummation. The hatched columns give the results without the QCD resummation from eqs. (5.11){(5.16). Figure 5 shows the corresponding results for DM couplings to neutrons, cin. In these examples we set had = c = 2 GeV and used the following quark masses at = mZ , mu(mZ ) = 1:4 MeV ; mc(mZ ) = 0:78 GeV ; md(mZ ) = 3:1 MeV ; mb(mZ ) = 3:1 GeV : ms(mZ ) = 63 MeV ; These were obtained using the one-loop QCD running to evolve the MS quark masses mu;d;s(2 GeV) and mc(b)(mc(b)), taken from ref. [37], to the common scale = mZ . For the nuclear coe cients that depend on the DM mass and/or the momentum transfer, we choose m = 100 GeV and a momentum transfer of q = 50 MeV. As seen from gures 4 and 5, the resummation of QCD logarithms enhances cip(n) by approximately 10% to 50% depending on the speci c case. The typical enhancement is O(30%). In the numerics we have set the CKM matrix element to unity, thus ignoring all avor changing transitions. This is a very good approximation for operators with 10− 10− × − × DM scattering on neutrons cn 4 cn 6 cn 7 × − × 0− 0− 8 × − × × − × 0− × 1 × − × − × bottom quarks. For charm quarks the e ect of avor o -diagonal CKM matrix elements is more important, yet still subleading. If we including the o -diagonal terms in eqs. (3.7){ (3.8), then the largest correction reaches 16% for c1p induced from C3(6;c) (31% for the result without resummation), as there is an up to 10% cancellation between the D3;cu and D3;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 axial-vector-axial-vector interactions (C4(6;c) 6= 0, C4(6;b) 6= 0) we have weak: intrinsic: p c4 ' p c4 ' 1 2 0:02 C4(6;c) 0:26 C4(6;b) 10 3; 4 2 c C4(6;c) + b C4(6;b) 1 2 2 C4(6;c) + 0:2C4(6;b) We see that for the bottom quarks the weak contribution, eq. (5.12), 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 (C3(6;c) 6= 0, C3(6;b) 6= 0) we have weak: intrinsic: p c1 ' p c7 ' p c9 ' 1 2 0:4 C3(6;c) 2 2 2 mN 2 m c C3(6;c) + b C3(6;b) c C3(6;c) + b C3(6;b) 1 2 C3(6;c) + 0:1C3(6;b) 10 3; 1 2 9 C3(6;c) + 0:9C3(6;b) (5.17) (5.18) (5.19) (5.20) (5.21) where in the last equality we set m = 100 GeV. The e ective scattering amplitude is parametrically given by A Ac1p + vT c7p + q mN 9 p c ; where vT 10 3, q=mN . 0:1, and A 100 for heavy nuclei. The loop-induced weak contributions thus dominate the scattering rates of weak scale DM. 6 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 mZ . In this case, the e ective Lagrangian valid above the weak scale is L = X a;d Ca(d) d 4 Q(ad) ; with the operators Q(ad) manifestly invariant under the full SM gauge group. We focus on the dimension-six e ective interactions involving DM and quarks currents, analogous to those in eqs. (2.2){(2.3). For the case of Dirac-fermion DM in a generic SU(2)L representation with generators ~a and hypercharge Y , the basis of dimension-six operators is [38] Q(16;i) = ( Q(26;i) = ( Q(36;i) = ( Q(46;i) = ( ~a )(QiL aQiL) ; )(QiL )(uiR )(diR QiL) ; uiR) ; diR) ; Q(56;i) = ( Q(66;i) = ( Q(76;i) = ( Q(86;i) = ( 5 ~a )(QiL aQiL) ; 5 )(QiL 5 )(uiR 5 )(diR QiL) ; uiR) ; diR) ; where the index i = 1; 2; 3 labels the generation, and a = a=2, with the Pauli matrices a. If is an electroweak singlet, the operators Q(16;i) and Q(56;i) do not exist. Below the weak scale the above operators, Q(nd;i), match onto the operators Q(md;)f in eqs. (2.2){(2.3). For certain patterns of Wilson coe cients, DM couples only to bottom- and/or charmquark axial-vector currents. Such possibilities are the main focus of this work. For instance, ' DM couples (at the mediator scale mZ ) only to the axial bottom-quark current if the only nonzero Wilson coe cients are C5(6;3), C6(6;3), and C8(6;3), and such that they satisfy the relation Y C5(6;3) = 4C6(6;3) = 2C8(6;3) : We rst 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 st assume that the initial conditions at satisfy eq. (6.6). At scales ew < < , the operators Q(a6;i) mix at one-loop via the SM Yukawa interactions ' into the Higgs-current operators [10, 11, 38] Q(166) = ( )(HyiD$ H) ; Q(168) = ( 5 )(HyiD$ H) ; (5.22) (6.1) (6.2) (6.3) (6.4) (6.5) (6.6) (6.7) χ Q(a6,i) H † Z χ χ hH †i Higgs operators, Q(166). Right: tree-level diagram that generates the dimension-eight operators from the Higgs operators through matching at the weak scale. and a similar set of operators with the ~a a structure (above, D D D ). This mix ing is generated by \electroweak sh" diagrams, see gure 6 (left), and induces at ew ' mZ $ C1(66)(mZ ) y 2 16b 2 Ca(6;3)( ) log mZ : Here, we only show the parametric dependence and suppress O( 1 ) coe cients from the actual value of the anomalous dimensions. At energies close to the electroweak scale, at which the Higgs obtains its vacuum expectation value, the two operators in eq. (6.7) result in couplings of DM currents to the Z boson. Integrating out the Z boson at tree-level induces DM couplings to quarks, see gure 6 (right). The Higgs-current operators in eq. (6.7) therefore match, at ew, onto four-fermion operators of the ve- avor EFT that couple DM to quarks with an interaction GF C1(66)(mZ ) v2= 2 . The factor v2 originates from the two strength of parametric size out the Z boson. Higgs elds relaxing to their vacuum expectation values and the factor GF from integrating Since the one-loop RG running from to ew, eq. (6.8), followed by tree-level matching at ew, induces interactions proportional to yb2v2, it is convenient to match such corrections to initial conditions of the dimension-eight operators in eqs. (4.3).2 For the pattern of initial conditions in eq. (6.6), we then operator Q(48;b) at ew ' mZ is nd that the Wilson coe cient of the dimension-eight and the Wilson coe cient of the dimension-six operator Q(46;b) is C4(8;b)(mZ ) g 2 16s 2 Ca(6;3)( ) log mZ ; C4(6;b)(mZ ) Ca(6;3)( ) : (6.8) (6.9) (6.10) 2Here, we have decided to ascribe the tree-level Z exchange contribution from the matching at ew to dimension-eight four-fermion operators. Alternatively, we could have absorbed also this contribution into the Wilson coe cients of dimension-six operators. This choice would have the unattractive property of having the parametric suppression of yb2v2GF = mb2GF hidden in the smallness of some of (the parts of) the Wilson coe cients Ca(6;f) , thus making the ve- avor EFT less transparent. With our choice, the parametric suppression of mb2GF is factored out of the Wilson coe cients and is part of the mass-dimension counting of the dimension-eight operator. 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, Ca(6;3)( ) denotes a linear combination of the Wilson coe cients with a = 5; 6; 8. The subsequent RG evolution from ew to had proceeds as described in section 5, eqs. (5.1){(5.4). nonzero. For instance, in the result for the non-relativistic coe cient c4p in eq. (5.12), there is an additional contribution from C4(8;b)(mZ ) / log(mZ = ). If one neglects the QCD e ects, the two contributions amount to adding up two logarithmically enhanced The only di erence is that the initial conditions C4(8;b)(mZ ) are now terms with exactly the same prefactors. The net e ect is to replace log(mq0 =mZ ) with log(mq0 = ) = log(mq0 =mZ ) + log(mZ = ) in eqs. (5.11){(5.16). This is equivalent to simply calculating the electroweak sh diagram, with u; d; s fermions attached to the Z line, and keeping only the log(mb= )-enhanced part. An analogous analysis applies if the only nonzero Wilson coe cients satisfy Y C5(6;2) = 4C6(6;2) = 2C7(6;2), so that just the ( 5 )(c 5c) operator is generated (setting the CKM matrix to unity for simplicity). Similarly, if the only nonzero Wilson coe cients satisfy HJEP02(18)74 Y C1(6;3) = 4C2(6;3) = the ( 2C4(6;3) 6= 0 or Y C1(6;2) = 4C2(6;2) = 2C3(6;2) 6= 0 this means that just essarily imply ne-tuning, as they can originate from the quantum number assignments for the mediators, DM, and quark elds in the UV theory. They do require the DM hypercharge Y to be nonzero.3 This conclusion changes, if at ' we also include dimension-eight operators of the form ( )(QLH HQL) alongside the dimension-six bR) operators. In this case, it is possible to induce only the ( malizable interaction to the Z boson). This, however, requires ne-tuning of dimension-six and dimension-eight contributions. Note that the relation in eq. (6.6) also requires DM to be part of an electroweak multiplet. For singlet DM there is no operator Q(56;i) and so C5(6;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 yt2 log( ew= ) 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 ew, i.e., through operators in eqs. (2.2){(2.3) with f = e; ; . We can readily adapt our results to this case by replacing in eqs. (3.2){(3.6) the bottom- and charm-quarks with leptons. The new operators are either color-singlets or conserved QCD currents so that their anomalous dimensions vanish. The hadronic functions cip(n), controlling DM scattering on nuclei, are then given by eqs. (5.11){(5.16) after substituting q0 ! ` = ; , and dividing by the number of colors, Nc = 3, implicit in these equations. 3This may or may not lead to potentially dangerous renormalizable couplings of DM to the Z-boson. An example of the latter is a DM multiplet that is a pseudo-Dirac fermion (a Dirac fermion with an additional small Majorana mass term), such as an almost pure Higgsino in the MSSM. In this case, the lightest mass eigenstate, the Majorana-fermion DM, does not couple diagonally to the Z boson at tree level. cross section. Strictly speaking, only the lepton can be integrated out when matching to the threeavor theory, and the analogy to the heavy-quark case breaks down for the muon, which will appear in the low-energy theory as an active degree of freedom. Since the muon mass is larger than the typical momentum transfer in the direct detection experiments, one might expect that keeping only the local contributions via eqs. (3.2){(3.6) is still a good For couplings to electrons, ` = e, we expect non-local contributions as well as dimension-eight operators with derivatives, which we have not considered here, to contribute to the scattering on nuclei at approximately the same order. Hence, keeping only the local contribution in eqs. (3.2){(3.6) is not expected to lead to a good estimate of the Majorana and scalar dark matter So far we focused on DM that is a Dirac fermion. However, the RG results discussed in this work do not depend on the precise form of the DM current. We can, therefore, generalize our results to the case of Majorana and scalar DM. Majorana dark matter Our results apply for Majorana DM with only small modi cations. It is su cient to drop from the operator basis the operators Q(16;f) and Q(36;f) in eqs. (2.2){(2.3) and likewise the operators Q(18;q) and Q(38;q) in eqs. (4.2){(4.3). The Lagrangian terms proportional to the remaining operators, Q2;f , Q4;f , Q(28;f) , and Q(48;f) should be multiplied by a factor of 1=2 to (6) (6) account for the additional Wick contractions (see, for instance, ref. [18]). With these modi cations, the coe cients of the nuclear e ective theory are still given by eqs. (5.5){(5.10). 7.2 Scalar dark matter The relevant set of operators for scalar DM is Lagrangian for scalar DM, cf., ref. [14], $ Q(16;f) = ' i@ ' (f f ) ; $ Q(26;f) = ' i@ ' (f (@ ' )'. These operators are part of the dimension-six e ective L' = X Ca (6) 2 Q(a6) ; a with Ca (6) the dimensionless Wilson coe cients. Note that we adopt the same notation for operators and Wilson coe cients 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 di erent DM current, nothing changes in our calculations. Therefore, after de ning the dimension-eight e ective Lagrangian in the three- avor theory as L'(8) = p 2GF 2 X (7.1) (7.2) the additional contributions to the nuclear coe cients are given, for complex scalar DM, by (cf. ref. [18]) c 1N = c 7N = 1 s p 2 1 s p 2 2GF m' 2 2GF m' X For real scalar DM, the operators in eq. (7.1) vanish. For completeness, we display also the dimension-eight contributions to the nuclear coe cients, expanded to leading order in HJEP02(18)74 the strong coupling constant, c N 1 ' c N 7 ' 6m' 2 p 2GF 2 p X If DM couples only to bottom- or charm-quark axial-vector currents, the dominant contribution to DM scattering on nuclei is either due to one-loop electroweak corrections or due to the intrinsic bottom and charm axial charges of the nucleons. Below the weak scale the electroweak contributions are captured by double insertions of both the DM e ective Lagrangian and the SM weak e ective Lagrangian. These convert the heavy-quark currents to the currents with u; d, and s quarks that have nonzero nuclear matrix elements. In this paper we calculated the nonrelativistic couplings of DM to neutrons and protons that result from such electroweak corrections, including the resummation of the leading-logarithmic QCD corrections. The latter are numerically important, as they lead to O( 1 ) changes in the scattering rates on nuclei. Our results can be readily included in the general framework of EFT for DM direct detection, and will be implemented in a future version of the public code DirectDM [18]. Acknowledgments We thank Francesco D'Eramo for useful discussions, and especially Fady Bishara for checking several equations. JZ acknowledges support in part by the DOE grant DE-SC0011784. The research of BG was supported in part by the DOE Grant No. DE-SC0009919. Unphysical operators A 4 We extract the anomalous dimensions by renormalizing o -shell Green's functions in d = 2 dimensions. In some intermediate stages of the computation it is thus necessary to introduce some unphysical operators. The one-loop mixing among the \physical" operators is not a ected by the de nition 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 o 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 nd it convenient to also perform the Dirac algebra in non-integer dimensions. To project the d = 4 2 amplitudes we thus need to also include some evanescent operators in the basis. For completeness and future reference, we list below the ones entering the one-loop computations: HJEP02(18)74 and E1 qq0 = (q E2 qq0 = (q E3 E4 E5 E6 qq0 = (q qq0 = (q qq0 = (q qq0 = (q E1q = (q E2q = (q E3q = (q E4q = (q E5q = (q E6q = (q E7q = (q E8q = (q (A.1) O1q + O2 q 4O4q ; 1 Nc 1 Nc q O3 ; O1q + O2q + 4O4q ; (A.2) q) (q0 T aq) operators, i.e., they vanish due to Fierz identities and not due to special d = 4 relations of the Dirac algebra. A.2 E.o.m.-vanishing operators In our conventions the equation of motion (e.o.m.) for the gluon eld reads D G a gsf abcG ;c)Gb X qT a q ; q up to gauge- xing and ghost terms. The sum is over all active quark elds. Hence the following operators vanish via the e.o.m. 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Joachim Brod, Benjamin Grinstein, Emmanuel Stamou, Jure Zupan. Weak mixing below the weak scale in dark-matter direct detection, Journal of High Energy Physics, 2018, 174, DOI: 10.1007/JHEP02(2018)174