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Title Bino variations: Effective field theory methods for dark matter direct detection

Permalink https://escholarship.org/uc/item/137270zv

Journal Physical Review D, 93(9)

ISSN 2470-0010

Authors Berlin, A Robertson, DS Solon, MP et al.

Publication Date 2016-05-10

DOI 10.1103/PhysRevD.93.095008

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California PHYSICAL REVIEW D 93, 095008 (2016) Bino variations: Effective field theory methods for dark matter direct detection

Asher Berlin,1 Denis S. Robertson,2,3,4 Mikhail P. Solon,2,3 and Kathryn M. Zurek2,3 1Department of Physics, Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA 2Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94709, USA 3Berkeley Center for Theoretical Physics, University of California, Berkeley, California 94709, USA 4Instituto de Física, Universidade de São, Paulo R. do Matão, 187, São Paulo, São 05508-900, Brazil (Received 7 December 2015; published 10 May 2016) We apply effective field theory methods to compute bino-nucleon scattering, in the case where tree-level interactions are suppressed and the leading contribution is at loop order via heavy flavor squarks or sleptons. We find that leading log corrections to fixed-order calculations can increase the bino mass reach of direct detection experiments by a factor of 2 in some models. These effects are particularly large for the bino-sbottom coannihilation region, where bino dark matter as heavy as 5–10 TeV may be detected by near future experiments. For the case of stop- and selectron-loop mediated scattering, an experiment reaching the neutrino background will probe thermal binos as heavy as 500 and 300 GeV, respectively. We present three key examples that illustrate in detail the framework for determining weak scale coefficients, and for mapping onto a low-energy theory at hadronic scales, through a sequence of effective theories and renormalization group evolution. For the case of a squark degenerate with the bino, we extend the framework to include a squark degree of freedom at low energies using heavy particle effective theory, thus accounting for large logarithms through a “heavy-light current.” Benchmark predictions for scattering cross sections are evaluated, including complete leading order matching onto quark and gluon operators, and a systematic treatment of perturbative and hadronic uncertainties.

DOI: 10.1103/PhysRevD.93.095008

I. INTRODUCTION (“wino”), doublet (‘Higgsino”), and singlet (“bino”) states mix with each other, allowing the lightest stable neutral Decades of technological advances and increased detec- WIMP, χ, to couple to the Higgs at tree level: λχhχχ¯ . tor sizes have led to impressive projected sensitivities of This gives rise to a typical scattering cross section ongoing and future dark matter (DM) direct detection λ 2 2 4 σ ∼ χ 10−45 2 experiments [1–4]. For DM with mass 10 − 10 GeV, SI ð0.1Þ cm . Thus, the currently running and next the LUX-ZEPLIN (LZ) experiment is projected to reach generation ton-scale experiments are probing tree-level σ ∼ 10−47–10−48 2 “ ” cross sections as small as SI cm , tantaliz- Higgs-interacting massive particles. ingly close to the neutrino background, residing at cross Pure electroweak states (wino, Higgsino, or bino), sections an order of magnitude smaller. As these experi- however, do not couple to the Higgs at tree level. For ments extend their reach, they will push through a number these cases, the evaluation of direct scattering of the lightest of important benchmarks in the hunt for weakly interacting electrically neutral state on nucleon targets requires the massive particles (WIMPs). analysis of loop amplitudes at leading order. Assuming Current experiments are in fact already probing rates weak-scale mediators, a simple estimate of the scattering σ ∼ α4 4 6 ∼ 10−46 2 several orders of magnitude below “weak-scale” cross cross section is given by SI wmN=mweak cm , ∼ 100 sections: constraints from LUX and Xenon100 reach as where mN is the nucleon mass and mweak GeV. The σ ∼ 10−45 2 low as SI cm , while a simple estimate suggests prospects for wino and Higgsino dark matter, however, are that the spin-independent (SI) scattering cross section challenged by an accidental cancellation between ampli- σ ∼ 10−39 2 through the Z boson is SI cm . The scattering of tudes, leading to cross sections smaller by a few orders of a WIMP on nucleon targets, however, depends strongly on magnitude [5–8]. For the wino, the cross section was found σ ∼ 10−47 2 its identity. While a scalar electroweak doublet has a large to be SI cm , while for the Higgsino, the can- cross section through the Z boson, Majorana fermions have cellation gives rise to an unreachably small scattering cross no vector coupling, and the axial-vector interactions either section. Nonetheless, it is remarkable that in some cases, are v2 suppressed or lead to spin-dependent (SD) scattering. while the tree-level cross section may be absent, ton-scale At tree level, this leaves scattering through the Higgs direct detection experiments are becoming sensitive to one- boson as the process for leading SI interactions. For loop interactions. neutralinos, the size of the scattering through the Higgs Similar to the wino and Higgsino, bino scattering boson depends on its electroweak composition. Triplet through the Higgs boson vanishes at tree level. If heavy

2470-0010=2016=93(9)=095008(29) 095008-1 © 2016 American Physical Society BERLIN, ROBERTSON, SOLON, and ZUREK PHYSICAL REVIEW D 93, 095008 (2016)

Tree Level, M1 500 GeV, tan 5 3 the multiple scales involved in direct scattering, accounting 10 ∼α mt for potentially large contributions, s log 1 GeV. Second, 49 49 we are able to include additional states at low energies, beyond those of nf-flavor QCD. For example, when the 48 mass difference between the bino and sbottom is much less

GeV than the weak scale, both are active degrees of freedom at

1 47 low energies, and we use heavy particle techniques to

M describe their interactions with soft bottom quarks. Third,

R 48 we are able to assess the uncertainties from both higher- d

m 2 46 order perturbative corrections and hadronic inputs. 10 In addition to incorporating renormalization group

47 evolution (RGE), we also go beyond previous fixed-order computations that have focused on the parameter space for 102 103 104 either purely left- or right-handed sfermions. We explore a

M1 GeV larger part of the minimal supersymmetric standard model (MSSM) parameter space by considering the impact of FIG. 1. SI nucleon cross sections from tree-level Higgs and mixing between left- and right-handed third generation squark exchange in the Higgsino and sdown mass plane for a bino squarks. We also perform a complete leading order match- mass of M1 ¼ 500 GeV and tan β ¼ 5. The labeled contours ing at the weak scale, considering contributions such as the σ 2 correspond to values of log10ð SI=cm Þ, while the vertical black spin-2 gluon operator (significant when a sbottom is close dashed line denotes the precise value of μ at which the lightest ’ in mass to the bino), and the anapole operator from photon neutralino s coupling to the Higgs vanishes at tree level. exchange. While we adopt the nomenclature and explicit couplings flavor squarks or sleptons are nearby in the spectrum, of the MSSM for definiteness, key components of our however, loop processes are induced. In this case, prospects analysis, such as the results for loop amplitudes and RGE for detection are improved through direct coupling to solutions, are generic and can be readily applied to inves- colored scalars. The interplay of a number of effects, such tigate the phenomenology of other models that incorporate as power suppression if the new states are heavy compared interactions of DM with scalars charged under the SM. For to the electroweak scale, enhancement from on-shell poles, example, many of the effects considered here may also be and sizable mixing between colored scalars, could impact applied to the case of suppressed tree-level scattering (“blind this. We assume that light flavor squarks and the Higgsino spots”), where loop corrections are necessary to mean- are decoupled from the low-energy spectrum since tree- ingfully compare theory and experiment [24–27]. level amplitudes would otherwise dominate over loops. To The remainder of the paper is structured as follows. In quantify the degree to which these must be decoupled, Sec. II, we review the standard fixed-order approach in the we show in Fig. 1 the SI cross section as a function of literature for determining amplitudes for WIMP-nucleon the Higgsino mass μ and the sdown mass m~ , when the dR scattering. This lays the groundwork for the effective theory lightest supersymmetric particle (LSP) is a binolike neu- framework described in Sec. III. There we discuss the tralino that interacts with the Standard Model (SM) Higgs factorization of the scattering amplitude into contributions ~ and a right-handed down squark (dR). Sufficient decou- from the relevant physical scales and illustrate the tech- pling occurs when the leading order scattering rate in Fig. 1 niques for matching, renormalization, and coefficient σ ∼ 10−49 2 drops below SI cm . evolution by presenting three detailed examples of increas- Processes relevant for one-loop bino scattering cross ing intricacy: a bino coupled to (i) a right-handed stop, (ii) a sections and related simplified models have already been heavier right-handed sbottom, and (iii) a nearly mass considered in the literature [9–20]. At the same time, a great degenerate right-handed sbottom. The reader interested deal of effective field theory (EFT) machinery has recently in the phenomenological results may go straight to been developed for systematically integrating out heavy Sec. IV, where we evaluate cross sections for models with particle thresholds and running Wilson coefficients to the stop, sbottom, and slepton mediators. The most promising low scales characteristic of the processes in direct detection case for detection is a bino interacting with a nearly experiments [21–23]. Our aim is to apply these techniques, degenerate right-handed sbottom: a bino as heavy as focusing on QCD effects, to the case of bino DM where the 10 TeV may be detected at LZ if the mass splitting is a SM is extended with a Majorana gauge singlet, and a few few GeV. On the other hand, a bino nearly degenerate with sfermions with the same quantum numbers as either left- or a right-handed stop is only detectable above the neutrino right-handed quarks or leptons. background for masses below about 500 GeV. We capture a number of effects that have been previously We collect the technical results in the Appendixes. In neglected. First, we are able to systematically incorporate Appendix A, we set up our conventions for the sfermion

095008-2 BINO VARIATIONS: EFFECTIVE FIELD THEORY … PHYSICAL REVIEW D 93, 095008 (2016) X mass matrices, as well as the DM-fermion-sfermion inter- L ð0Þχχ¯ ð0Þ ð1Þχγ¯ γ5χ ð1Þμ actions. Appendixes B and C contain the hadronic form ¼ cq Oq þ cq μ Oq factors and the running and matching matrices employed in q¼u;d;s 2 our numerical analysis. In Appendix D, we present details ð Þ cq χ¯ ∂ ∂ χ ð2Þμν þ 2 i μi ν Oq of the Wilson coefficients for all relevant amplitudes, such mχ as tree-level sbottom exchange, one-loop Higgs, Z, and γ ð2Þ exchange, one-loop diagrams involving charged electro- ð0Þχχ¯ ð0Þ cg χ¯ ∂ ∂ χ ð2Þμν þ cg Og þ 2 i μi ν Og ; ð2Þ weak gauge bosons, and one-loop contributions to the mχ gluon coefficients. We compute these keeping all fermion and sfermion masses explicit, and allowing for left-right sfermion mixing. We note for each diagram where our where the relevant QCD currents are results differ from previous literature. ð0Þ ð1Þμ μ 5 Oq ¼ mqqq;¯ Oq ¼ q¯γ γ q; II. FIXED ORDER APPROACH TO 1 1 ð2Þμν fμ νg μν WIMP-NUCLEON SCATTERING O ¼ q¯ γ iD− − g iD− q; q 2 d Amplitudes for WIMP-nucleon scattering involve energy 1 ð0Þ A 2 ð2Þμν Aμλ Aν μν A 2 scales that span several orders of magnitude, ranging from Og ¼ðGμνÞ ;Og ¼ −G G λ þ g ðGαβÞ ; the masses of the new particles and the mediating SM d particles (≳100 GeV) to the scales of heavy quark thresh- ð3Þ olds and of hadronic physics (≳1 GeV), and the typical momentum transfers relevant for direct detection (∼MeV). A 4 − 2ϵ A standard approach in the DM literature is to determine with Gμν the gluon field strength and d ¼ the these amplitudes at “fixed order,” treating this broad range of spacetime dimensions. We adopt the notation D− ≡ D~ − physical scales at a single scale. In this section, we review D⃖ and AfμBνg ≡ ðAμBν þ AνBμÞ=2, and have neglected this matching procedure between the full theory of the SM operators that lead to kinematically suppressed contribu- and its extension, specified at high energies E ≳ 100 GeV, tions. Leading order SI scattering is given by the scalar 0 2 μν and an EFT for WIMP-nucleon scattering, specified at low (Oð Þ) and spin-2 (Oð Þ ) quark and gluon currents, while energies E ≳ 1 GeV. q;g q;g ≳ 100 leading order SD scattering is given by the quark axial At high energies, E GeV, the basic interaction ð1Þμ μ 5 current (O ). We neglect the operator χγ¯ γ χq¯γμq that we consider is of a single sfermion (f~) with a bino LSP q (χ) and a SM fermion (f), adopting the following notation: involving the quark vector current, which leads to SI scattering that is power enhanced relative to the scalar 5 and spin-2 contributions, but is velocity suppressed. We L ⊃ f~ f¯ ðα þ β γ Þχ þ H:c: ð1Þ f f have reduced the operators to a linearly independent set; χ¯ ∂ γ χ ð2Þμν χ¯ ∂ ∂ χ ð2Þμν The couplings αf, βf are parametrized in terms of the SM e.g., the operators i μ ν Oq;g and i μi ν Oq;g are hypercharge coupling g0 and the sfermion mixing angles of redundant in the forward scattering limit. We ignore flavor Eqs. (A1) and (A7). To simplify the discussion in this nondiagonal operators, whose nucleon matrix elements section and the next, we illustrate general methods for the have an additional weak-scale suppression relative to those case where f~ constitutes a single right-handed stop or considered. We will not be concerned here with operators sbottom and f the corresponding top or bottom quark, involving leptons. assuming the theory in Eq. (1) is defined at the weak scale In the standard fixed-order approach, the full theory in ∼100 GeV. The impact (from RGE) of considering cou- Eq. (1) is matched onto the effective theory in Eq. (2),by ~ plings defined at an even higher scale is illustrated in integrating out the sfermion f, the gauge bosons Z, W , Sec. IV B. Examples pertaining to mixed stops and sbot- the Higgs h, the Goldstones G, G , and the heavy quarks toms, and sleptons, are treated in a similar way, and we t, b, c, altogether at a single scale. The matching condition discuss them in Secs. IV C and IV D. for the case of a right-handed stop or sbottom (denoted as ~ The hadronic matrix elements necessary for describing f) is shown in Fig. 2. The leading contributions to the 2 WIMP-nucleon scattering are determined, e.g., from lattice quark and gluon coefficients are at OðαwÞ and OðαwαsÞ, measurements, at low energies E ∼ 1 GeV, in a theory with respectively. three quark flavors. At these energies, an effective theory Once the Wilson coefficients are determined, the had- captures the interactions of the WIMP with the degrees of ronic matrix elements are evaluated. We adopt the defi- freedom of 3-flavor QCD. For the bino, a gauge-singlet nitions and values from Sec. IVof Ref. [22] for the hadronic Majorana fermion, a set of operators for low-velocity matrix elements of the QCD currents in Eq. (3).For scattering, is completeness, we collect their definitions here:

095008-3 BERLIN, ROBERTSON, SOLON, and ZUREK PHYSICAL REVIEW D 93, 095008 (2016)

FIG. 2. Matching conditions for a fixed-order calculation. Charge-reversed diagrams are not shown. Here, f~ denotes a right-handed stop or sbottom, and q refers to the quarks of 3-flavor QCD. In the bottom line, the ellipsis denotes similar diagrams where the insertion of the gluon legs vary (see Appendix D6).

ð0Þ ≡ ð0Þ and, finally, the cross sections for SI and SD scattering on a hNjOq jNi mNfq;N; nucleon target are obtained, −9α μ 0 0 sð Þ ð Þ ð Þ 2 hNjOg ðμÞjNi ≡ m f ðμÞ; 4 mχm 8π N g;N σ N M 2 SI ¼ j SI;Nj ; ð1Þμ μ ð1Þ π mχ þ mN hNðkÞjOq ðμÞjNðkÞi ≡ s f ðμÞ; q;N 2 12 mχmN 2 2 μν 1 1 2 σ ¼ jM j : ð7Þ ð Þ μ ≡ μ ν − 2 μν ð Þ μ SD π SD;N hNðkÞjOi ð ÞjNðkÞi k k mNg fi;Nð Þ; mχ þ mN mN 4 This is a straightforward strategy for determining ð4Þ WIMP-nucleon scattering cross sections, with, however, limitations that motivate a more thorough analysis. where N ¼ p, n for proton or neutron, i ¼ q, g for quark First, there are potentially large perturbative corrections, μ ¯ γμγ5 or gluon, and the spin vector s ¼ uðkÞ uðkÞ satisfies ∼α log mt , inherent in treating a multiscale process at a 0 2 −1 s 1 GeV k · s ¼ and s ¼ , assuming nonrelativistic normali- single scale. For example, while the Wilson coefficients are zation for the spinor uðkÞ. α ∼100 ð1Þ determined at the weak scale employing sð GeVÞ, The axial form factors, fq;N, are extracted from hyperon the leading order scalar gluon form factor in Eq. (5) is ν semileptonic decay, from p scattering, or from observ- subject to sizable corrections due to the large size of ables of polarized deep inelastic scattering. The scalar αsð∼1 GeVÞ. Second, determining higher order corrections ð0Þ quark form factors, fq;N, are extracted from lattice mea- in a fixed-order framework is difficult; e.g., at next-to- surements, while the scalar gluon form factor is obtained leading order (NLO) two- or three-loop amplitudes are through the leading order relation [28] required. Theoretical control of perturbative corrections would allow us to estimate their numerical impact, and in X 0 0 the event of a detection, to systematically improve pre- fð Þ ¼ 1 − fð Þ þ Oðα Þ: ð5Þ g;N q;N s dictions for WIMP-nucleon scattering. In the next section, q¼u;d;s we lay out the effective theory framework to deal with these issues head on. ð2Þ ð2Þ The quark and gluon spin-2 form factors, fq;N, fg;N, are extracted from the second moment of parton distribution III. EFFECTIVE THEORY APPROACH TO functions (PDFs). In Appendix B, we collect the values WIMP-NUCLEON SCATTERING employed in our numerical analysis. These nucleon matrix elements, together with the Wilson As mentioned in the previous section, WIMP-nucleon coefficients, define the SI and SD amplitudes scattering involves a multitude of physical scales, and the separation between the weak scale, ∼100 GeV, and the X 3 hadronic scale, ∼1 GeV, may lead to large uncertainties M ð0Þ ð0Þ ð2Þ ð2Þ SI;N ¼ mN fq;Ncq þ 4 fq;Ncq when employing the fixed-order framework. In this section, q¼u;d;s “ ” we discuss the effective theory approach, which factor- 8π 3 izes the scattering amplitudes into contributions from − ð0Þ ð0Þ ð2Þ ð2Þ fg;Ncg þ fg;Ncg ; different physical scales by constructing a sequence of 9αs 4 X EFTs from the weak scale down to the hadronic scale and M ð1Þ ð1Þ SD;N ¼ fq;Ncq ; ð6Þ connecting them through RGE and matching. This allows q¼u;d;s for the separate analysis of perturbative corrections at each

095008-4 BINO VARIATIONS: EFFECTIVE FIELD THEORY … PHYSICAL REVIEW D 93, 095008 (2016)

flavors. The matrices Rðμc; μbÞ and MðμcÞ are analogously defined, implementing running in 4-flavor QCD and match- ing across the charm quark threshold. Finally, the coeffi- cients are run down to the hadronic scale in 3-flavor QCD, using Rðμ0; μcÞ, and the matrix elements are evaluated T through multiplication of the (transposed) vector f ðμ0Þ, which collects the form factors fq;g defined in Eq. (4). Clearly, Eq. (8) has separation of scales, with compo- nents cðμtÞ, MðμbÞ, MðμcÞ, and fðμ0Þ depending only on scales of a similar order. The logarithms in the amplitude are resummed through the RGE factors R, and additional perturbative corrections to each component can be sepa- rately and systematically analyzed without having to evaluate the whole amplitude at higher loop order. Note that α log mb does not constitute a large logarithm, and s mc hence integrating out the bottom and charm quarks at a single scale would suffice. Nonetheless, since αsð1 GeVÞ is FIG. 3. In the fixed-order approach (left), the full theory is sizable, higher-order corrections may have significant directly matched onto the low-energy theory with 3-flavor QCD. impact, and we may conveniently employ known results In the effective theory approach (right), the full theory is matched for the matrices MðμbÞ, MðμcÞ, and R to include them. onto the low-energy theory with 3-flavor QCD by systematically Note also that the PDFs relevant for the spin-2 matrix passing through a sequence of effective theories defined at the elements defined in Eq. (4) are available at a high scale, weak scale (μ ∼ m ), the bottom mass scale (μ ∼ m ), the charm t t b b e.g., Oð100Þ GeV, and thus allows us to evaluate the scale (μc ∼ mc), and the hadronic scale (μ0 ∼ 1 GeV). The matching and running between these effective theories are amplitude without running down these Wilson coefficients discussed in the main text. If the mass splitting between a to a low scale. The running, however, would be relevant for ~ relating the spin-2 current to low-energy effective DM- sbottom (bR) and the bino is much smaller than the weak scale, then the effective theory setup is modified to include a heavy nucleon contact operators (see, e.g., Refs. [29,30]) and for ~ including the impact of multinucleon effects (see, e.g., sbottom field bR;v, accounting for sbottom-bino interactions at low energies. The subscript v denotes a heavy particle field as Refs. [31,32]). In the present analysis, we RG evolve all defined through the field redefinitions in Eqs. (9) and (21). Wilson coefficients as a default but have checked that our results are consistent, up to uncertainties, with an evalu- ation at the high scale. We find that the additional energy threshold and for the resummation of large loga- perturbative uncertainty from running the spin-2 coeffi- rithms, e.g., ∼α log mt . s 1 GeV cients increases the overall uncertainty by less than 10%. This framework is depicted in Fig. 3. To further elaborate The factorization in Eq. (8) is a general result of our on its general features, let us present the corresponding effective theory analysis, and in the following subsections factorized amplitude, and briefly discuss its components in we provide further details on each of its components. turn; a more detailed discussion is given in the subsections Section III A considers formalism for representing the below. In the EFT approach, the scattering amplitude is relevant degrees of freedom in the low-energy theory and determined as for matching at the weak scale μt ∼ mt. In Secs. III B, III C, T and III D, we go into explicit detail by applying the effective M ¼ f ðμ0ÞRðμ0;μ ÞMðμ ÞRðμ ;μ ÞMðμ ÞRðμ ;μ Þcðμ Þ; c c c b b b t t theory framework to three examples, classified according to ð8Þ the mass, mf, of the fermion partnered to the sfermion, and the mass splitting, δ~ ¼ m~ − mχ, between the sfermion and μ μ μ μ f f where the renormalization scales t, b, c, and 0 ≳ μ δ bino. Case I considers mf t and arbitrary f~, case II correspond, respectively, to the weak scale ∼mt, the bottom considers mf ≪ μt ≲ δ~ , and case III considers δ~ , mf ≪ μt. quark threshold ∼mb, the charm quark threshold ∼mc, and f f the hadronic scale ∼1 GeV, where nucleon matrix elements These examples illustrate, in increasing complexity, the key ingredients of the effective theory framework. Case I goes are defined. The vector cðμtÞ collects the Wilson coef- through the basic computational pipeline involving the ficients determined at the scale μt by integrating out weak scale degrees of freedom and matching onto a theory with components c, R, M, and f of Eq. (8). Case II presents an example where nontrivial renormalization of the bare five quark flavors. The matrix Rðμb; μtÞ implements coef- ficient running from μ down to μ , while the matrix Mðμ Þ coefficients arises. Finally, for case III, a heavy sfermion t b b ~ ~ implements coefficient matching across the bottom quark field fv (denoted as bR;v in Fig. 3) is included in the low- threshold, between the theory with five and four quark energy theory to account for sfermion-bino interactions.

095008-5 BERLIN, ROBERTSON, SOLON, and ZUREK PHYSICAL REVIEW D 93, 095008 (2016) A. Integrating out the mass but not the particle This implies an invariance of the heavy particle Lagrangian χ A key step in the effective theory approach involves for v under the simultaneous transformations [34,35] integrating out weak scale degrees of freedom by matching v → −v; χ → χc: 11 onto a low-energy theory of the bino χ and the quarks and v v ð Þ gluons of 5-flavor QCD. In this procedure, the gauge, This invariance and the form of the field redefinition in Higgs, and Goldstone bosons, as well as the stop and top, Eq. (10) will be useful in Sec. III D for considering the are integrated out. However, the bino, despite having a ≳ 100 — interactions of a heavy bino with a heavy sbottom. weak scale mass, mχ GeV, is not integrated out the Instead of introducing the field redefinition (9) into a goal of calculating a WIMP-nucleon scattering cross basis of relativistic operators, we may also proceed in the section requires that it is kept in the low-energy theory. spirit of effective theory, employing building blocks to Moreover, the same applies to a sbottom whose mass is directly write down low-energy operators consistent with ≈ ≳ 100 close to that of the bino: despite mb~ mχ GeV, the symmetries. For our low-energy theory, the building blocks sbottom should not be integrated out since the bottom are the usual relativistic degrees of freedom (quarks and quark is an active degree of freedom in the low-energy gluons), the reference vector vμ, and the heavy bino field theory and bino-sbottom interactions are thus allowed. χv. Thus, for a Majorana dark matter particle whose mass How do we integrate out the mass of a field without satisfies mχ ≫ mb, the basis of operators describing its integrating out the field itself? The idea is simple and can be interactions with 5-flavor QCD is pictured by considering the following parametrization of μ μ μ X the bino momentum at low energies: p ¼ mχv þ k , ð0Þ ð0Þ ð1Þ ⊥ 5 ð1Þμ μ Lχ =2 ¼ fcq χ¯ χ Oq þ cq χ¯ γμ γ χ Oq where v is a reference timelike unit vector and v v v v v μ μ q¼u;d;s;c;b k ≪ mχv . The interactions of the heavy bino with the ð2Þχ¯ χ ð2Þμν ð0Þχ¯ χ ð0Þ much lighter quarks and gluons of 5-flavor QCD involve þ cq v vvμvνOq gþcg v vOg μ only soft momenta of Oðk Þ, while the large momentum ð2Þ ð2Þμν μ þ cg χ¯vχvvμvνOg þ; ð12Þ component mχv , corresponding to its mass, plays no role and can be integrated out. This procedure is formally where the ellipsis denotes higher dimension operators, done by going from a relativistic description of the field and the relevant QCD currents are given in Eq. (3). to a “heavy particle” description, order by order in the 5 Here, we have subtracted off the component of γμγ which small parameter jkj=mχ. The technique is called vanishes between the heavy particle bilinear, defining “heavy particle effective theory,” and it is known from ⊥ μ μ γμ ¼ γ − v v. Alternatively, Eq. (12) is obtained by applications for heavy quark physics (for a review see, making the substitution (9) into the basis of operators in e.g., Ref. [33]). Eq. (2). We have introduced a conventional factor of 1=2 on We may pass from a relativistic to a heavy particle the left-hand side of Eq. (12) since the field redefinition (9) description for the bino (Majorana fermion) by making the would otherwise lead to a factor of 2 discrepancy between field redefinition the coefficients in Eqs. (2) and (12). ffiffiffi ð0Þ ð2Þ p In the relativistic basis of Eq. (2), cq and cq are treated χ 2 −imχ v·x χ ¼ e ð v þ XvÞ; ð9Þ on equal footing, despite corresponding to operators whose mass dimensions differ by two, i.e., seven and nine, where the spinors obey vχ ¼ χ and vX ¼ −X . In terms v v v v respectively. As a result, power counting is possible but of the momentum decomposition discussed above, the not manifest (leading order SI scattering involves operators phase e−imχ v·x extracts the large momentum component μ of dimensions seven and nine), and it is less straightforward mχv . Upon introducing this field redefinition into the how the basis extends beyond leading order. In contrast, 1 χ¯ ∂ − χ kinetic term 2 ði mχÞ , we find that the component Xv power counting is manifest in Eq. (12), and thus the has mass 2mχ and is thus integrated out, e.g., at tree level by operators relevant at each order are known without having solving its equation of motion. The remaining component first to evaluate the full theory amplitudes. In particular, χv describes the heavy bino degree of freedom with the leading order low-velocity SI (SD) scattering is obtained (canonically normalized) kinetic term χ¯viv · ∂χv, depend- from dimension seven (six) operators, and subleading μ ing only on the soft momentum k . The Majorana condition corrections can be systematically computed. In the remain- χ ¼ χc allows us to write the field redefinition (9) alter- der of the paper, when referring to Wilson coefficients, we natively as assume the form given in Eq. (12). pffiffiffi Having discussed the formalism for incorporating both imχ v·x c c χ ¼ 2e ðχv þ XvÞ; ð10Þ relativistic and heavy particle degrees of freedom at low energies, let us now turn to the computation of weak scale ψ c Cψ where charge conjugation is denoted by ¼ with the coefficients cðμtÞ of Eq. (8). At the scale μt ∼ mt, we match unitary and symmetric matrix C obeying C†γμC ¼ −γμ. the full relativistic theory of Eq. (1), with six quark flavors

095008-6 BINO VARIATIONS: EFFECTIVE FIELD THEORY … PHYSICAL REVIEW D 93, 095008 (2016) and a relativistic bino χ, onto the low-energy theory of low-energy theory if sfermion-bino interactions with light Eq. (12), with five quark flavors and a heavy particle bino fermions are present. Hence, only the sfermion mass is χv. The full theory diagrams are computed using standard integrated out (encoded in Wilson coefficients through the relativistic Feynman rules, while the effective theory full theory amplitudes), and a heavy sfermion field is diagrams are computed using the Feynman rules included at low energies. In particular, a so-called “heavy- of Eq. (12). light current” describes the interactions of the heavy bino This matching procedure determines the bare Wilson with the heavy sfermion and light fermion. This is coefficients and may involve loop contributions from the described in Sec. III D. low-energy effective theory. It is simplest to compute the Let us now move on to three cases that illustrate in full theory amplitudes setting all mass scales much lighter explicit detail the general aspects of the EFT approach than the weak scale to zero, and regulating infrared discussed above. Previous works have focused on fixed- divergences in 4 − 2ϵ dimensions. The weak scale coef- order calculations [9,12,14,17,36] or on the EFT treatment ficients cðμtÞ then depend only on the weak scale masses of the scalar gluon coupling [15]. In the present analysis, mW, mZ, mh, mt, mχ, and mf~ , and are determined up to we perform leading order matching onto the complete set of corrections of Oðmb=mtÞ. Of course, for matching a full operators in Eq. (12), including contributions to quark 0 theory amplitude onto the scalar quark current Oð Þ of operators from exchanges of electroweak bosons. For q example, we find that the Higgs-exchange diagrams are Eq. (2), the leading m factor should be retained. In q numerically relevant, significantly improving the projected dimensional regularization, the loop integration measure 4−2ϵ reach of LZ (e.g., compared to those found in Ref. [36]). has scaling dimension ½mass , and therefore any loop Moreover, the following subsections present a pedagogical integral is dimensionful. A loop integral that has no mass discussion of the EFT framework, illustrating aspects such scale to soak up this dimensionality must vanish by as matching and the infrared pole structure, and the consistency. This is the well-known statement that scaleless application of the heavy-light current. The case of a integrals vanish in dimensional regularization. With light sfermion nearly degenerate in mass with the bino discussed quark masses set to zero, the effective theory loop con- in Sec. III D is new and physically relevant. tributions are scaleless, and hence vanish. Alternatively, keeping light quark masses nonzero would regulate infrared B. Case I: Right-handed stop divergences, but would require the computation of non- vanishing effective theory loop amplitudes. An explicit The simplest example arises when the mass of the example involving such effective theory loop contributions fermion partnered to the sfermion is of order or is greater ≳ μ will be presented in Sec. III C. than the weak scale, mf t. Although this case broadly The remaining 1=ϵ poles in the bare coefficients are UV applies to many models, for concreteness, we will restrict to ~ divergences of the low-energy theory and are renormalized the case of a single right-handed stop (tR) interacting with accordingly. For a detailed discussion on the renormaliza- the bino (χ) and a top quark (t). Let us discuss in turn the tion of the QCD currents in Eq. (3), we refer the reader to ingredients c, R, M, and f of the factorization presented Sec. III of Ref. [22]. Here, we will simply quote the results. in Eq. (8). At leading order in αs, the scalar and axial-vector coef- μ ficients are trivially renormalized, i.e., cðμÞ¼cbare, while 1. Weak scale coefficients cð tÞ the spin-2 coefficients are renormalized as The matching condition at the weak scale μt ∼ mt is shown in Fig. 4. The full theory amplitudes are computed ð2Þ ð2Þbare cq ðμÞ¼cq þ OðαsÞ; using the Lagrangian in Eq. (1), while the effective theory X amplitudes are computed using the Lagrangian in Eq. (12). 2 1 α 2 bare 2 bare cð Þ μ s cð Þ cð Þ O α2 ; 13 ~ g ð Þ¼ ϵ 6π q þ g þ ð s Þ ð Þ The weak scale particles W , Z, h, G , t, tR are highly q virtual at low energies and are thus integrated out. Their effects are encoded into the Wilson coefficients of an where the sum runs over the active quark flavors, i.e., effective theory describing a heavy bino χ interacting with O ϵ0 v q ¼ u, d, s, c, b in 5-flavor QCD. The ð Þ terms of the the quarks and gluons of 5-flavor QCD. ð2Þbare ð2Þ coefficients cq introduce a 1=ϵ pole in cg ðμÞ that is The contributions to the quark and gluon coefficients ð2Þbare O α2 O α α canceled by the 1=ϵ pole in cg . Note that the nontrivial begin at ð wÞ and ð w sÞ, respectively. The h-exchange 1 ð0Þ renormalization also requires the Oðϵ Þ terms of the diagrams contribute to the scalar coefficient cq , while the ð2Þbare Z-exchange diagrams contribute to the axial-vector coef- coefficients cq . We will see an explicit example of ð1Þ ð2Þbare ficient c . The box diagrams exchanging W or G this renormalization in Sec. III C when cg is divergent q ð0Þ ð1Þ ð2Þ due to gluons emitted from massless quarks. contribute to cb , cb , and cb . The explicit results for the As mentioned above, a sfermion that is nearly degenerate relevant diagrams are collected in Eqs. (D7), (D15), (D27), in mass with the bino should be a degree of freedom in the and (D33). Working consistently at leading order, the gluon

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FIG. 4. Weak scale matching conditions for the case of a right-handed stop. Crossed and charge-reversed diagrams are not shown. Here, q refers to the quarks of 5-flavor QCD. In the bottom line, the ellipsis denotes similar diagrams where the insertion of the gluon legs vary (see Appendix D6). Single (double) lines correspond to relativistic (heavy particle theory) fields. We have omitted the label “bare” on the coefficients on the right-hand side. matching condition does not include contributions from 0 −8π 0 T μ ð Þ ð Þ μ f SI;Nð 0Þ¼mN fq;N; fg;Nð 0Þ; effective theory diagrams involving loops of quarks since 9αsðμ0Þ O α2 α O α2 α these are ð w sÞ. Accordingly, we also drop the ð w sÞ 3 3 ð2Þ μ ð2Þ μ terms in the renormalization condition in Eq. (13), and thus 4 fq;Nð 0Þ; 4 fg;Nð 0Þ ; all bare Wilson coefficients are trivially renormalized for μ bare T μ ð1Þ μ this example, i.e., cq;gð tÞ¼cq;g . We collect the renor- f SD;Nð 0Þ¼ffq;Nð 0Þg; ð15Þ malized Wilson coefficients in the vectors ð0;1;2Þ where fq;N is representative of the three light quark flavors; i.e., the vectors f SI;N and f SD;N have eight and three T μ ð0Þ μ ð0Þ μ ð2Þ μ ð2Þ μ T μ cSIð tÞ¼fcq ð tÞ;cg ð tÞ;cq ð tÞ;cg ð tÞg; cSDð tÞ components, respectively. To be consistent with the higher ð1Þ μ order effects included in the running and matching matrices ¼fcq ð tÞg; ð14Þ R and M, we must also include higher order corrections to the leading order gluon scalar matrix element of Eq. (5).

ð0;1;2Þ From the nucleon mass sum rule that links the gluon and where cq is representative of the five quark flavors, i.e., quark scalar form factors (see, e.g., Ref. [22]), we have q ¼ u, d, s, c, b, and hence the vectors cSI and cSD have 12 and 5 components, respectively. The coefficients are −α μ 9 X ð0Þ μ sð Þ 1 − 1 − γ μ ð0Þ collected into two vectors in anticipation of evaluating fg;Nð Þ¼ ð mð ÞÞ fq;N ; 4π β~ðμÞ the SI and SD amplitudes separately. q¼u;d;s ð16Þ 2. Running and matching matrices R and M ~ where β ¼ β=gs with β the QCD beta function, and γm is For cases where the degrees of freedom below the weak the quark mass anomalous dimension. In our numerical scale are a gauge singlet [under SUð3Þ × Uð1Þ ]DM c EM analysis, we include terms in β~ and γ through Oðα Þ particle and the quarks and gluons of n -flavor QCD, the m s f [see Eq. (B4)]. relevant matrices for running and matching are specified by With all ingredients specified, we may now evaluate the loop-level matrix elements of the QCD currents in Eq. (3). amplitudes as in Eq. (8). The result can be expressed as We adopt the results from Tables 5 and 6 of Ref. [22] and collect their leading order forms in Appendix C for T M ¼ f ðμ0Þc ðμ0Þ; completeness. In practice, we work at leading log (LL) SI;N SI;N SI M T μ μ order. For the axial current, the corrections to coefficient SD;N ¼ f SD;Nð 0ÞcSDð 0Þ; ð17Þ 2 evolution and threshold matching begin at Oðαs Þ, and are therefore subleading [37–39]. In particular, this implies that which when expanded takes the form in Eq. (6). The ð1Þ vectors c ðμ0Þ contain the low-energy coefficients the weak scale coefficients c contribute to the ampli- SI;SD u;d;s properly mapped from the weak scale through the running ð1Þ tude, while cc;b may be neglected. Nonetheless, we will and matching factors, keep the discussion of weak scale coefficients cðμtÞ 1 ð Þ cðμ0Þ¼Rðμ0; μcÞMðμcÞRðμc; μbÞMðμbÞRðμb; μtÞcðμtÞ: general, including the determination of cc;b. ð18Þ μ 3. Nucleon matrix elements fð 0Þ These vectors are defined as in Eq. (14) but with the light Let us collect the nucleon matrix elements defined in quarks (u, d, s) and gluon of 3-flavor QCD. In practice, we Eq. (4) in the following vectors: will not evolve the coefficients after integrating out the

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FIG. 5. Weak scale matching conditions for the case of a right-handed sbottom that is much heavier than the bino. Crossed and charge- reversed diagrams are not shown. In the full theory diagrams, q0 refers to u, d, s, c. The ellipsis denotes similar diagrams where the insertion of the gluon legs vary (see Appendix D6). Single (double) lines correspond to relativistic (heavy particle theory) fields. We have omitted the label “bare” on the coefficients on the right-hand side.

charm quark at μc, and hence we take μ0 ¼ μc. Finally, the discussed in Sec. III A, we adopt the scheme where all mass cross section is determined as in Eq. (7). Note that Eq. (7) scales much lighter than the weak scale (such as mb) are set applies for a relativistic Majorana field χ, but is also valid to zero, and employ dimensional regularization. The full χ ð2Þbare for our heavy particle field v, given the conventional factor theory contribution to c is IR divergent due to gluons 1 2 g of = on the left-hand side of Eq. (12). emitted off of a massless bottom quark. The effective theory contributions from a bottom quark loop, shown on C. Case II: Right-handed sbottom, large mass splitting the right-hand side of Fig. 5, are scaleless, and thus vanish. An example similar to the previous one, but slightly In the low-energy theory, the remaining 1=ϵ pole of the bare more involved due to the interplay between quark and coefficient is regarded as an UV divergence that is gluon coefficients, is when the mass of the fermion renormalized according to Eq. (13). For illustration, we partnered to the sfermion is much lighter than the weak present the explicit pole structure of the contributions to the scale, mf ≪ mt, and the mass splitting between the renormalized spin-2 gluon coefficient, sfermion and the bino is comparable to or greater than δ − ≳ the weak scale, f~ ¼ mf~ mχ mt. Although the pro- 2 2 2 2 α 1 cð Þ μ cð ÞFT − cð ÞEFT cð Þ s O α2 cedure described here applies to a wide variety of models, g ð Þ¼ g g þ b 6π ϵ þ ð s Þ UV for definiteness, we focus on the case of a right-handed " # 0 ~ −αsα mχ 1 sbottom (bR) interacting with the bino (χ) and bottom quark ¼ 2 2 2 ϵ þ finite (b). Let us discuss in turn the ingredients c, R, M, and f of 27ðm~ − mχÞ IR bR the factorization presented in Eq. (8). 2 α 1 1 2 α 1 − cð Þ s − cð Þ s O α2 ; b 6π ϵ ϵ þ b 6π ϵ þ ð sÞ 1. Weak scale coefficients cðμtÞ UV IR UV ð19Þ The matching condition at the weak scale μt ∼ mt is shown in Fig. 5. As in the previous example, the full theory amplitudes are computed using the Lagrangian in Eq. (1), ð2ÞFT ð2ÞEFT where cg (cg ) is the full (effective) theory loop while the effective theory amplitudes are computed using contribution appearing on the left (right) side of the gluon the Lagrangian in Eq. (12). The weak scale particles W , Z, matching condition in Fig. 5, and the last term comes from ~ h, G , t, bR are integrated out, and their effects are encoded the renormalization prescription of Eq. (13). We have in Wilson coefficients of the effective theory describing a omitted the label “bare” on the coefficients on the right- χ heavy bino v interacting with the quarks and gluons of 5- hand side and expressed the vanishing effective theory flavor QCD. ð2ÞEFT contribution, cg , in terms of canceling UV and IR As in the previous example, the leading contributions to ð2Þ ð0;1;2Þ 2 O α poles. Note the required consistency between cb [given in the coefficients cu;d;s;c are ð wÞ loop diagrams. What distinguishes this case is the presence of a tree-level, Eq. (D3)] and the infrared pole of the full theory con- ð2ÞFT OðαwÞ, contribution to the bottom quark coefficients tribution cg [given in Eq. (D34)] to yield a finite ð0;1;2Þ O α α ð2Þ μ cb and the associated loop-level, ð w sÞ, effective renormalized coefficient cg ð Þ. The other coefficients ð0;2Þ ð0;1Þ ð0Þ bare theory contributions to the gluon coefficients cg .As cq and cg are simply renormalized as cðμÞ¼c .

095008-9 BERLIN, ROBERTSON, SOLON, and ZUREK PHYSICAL REVIEW D 93, 095008 (2016) ~ ~ As before, we collect the renormalized Wilson coeffi- L ⊃ b ð−iv · D − δ~ Þb R;v bR R;v cients in the vectors 1 ~ ¯ α β γ5 χ þ pffiffiffiffiffiffi bR;vbð b þ b Þ v þ H:c:; ð22Þ T μ ð0Þ μ ð0Þ μ ð2Þ μ ð2Þ μ mχ cSIð tÞ¼fcq ð tÞ;cg ð tÞ;cq ð tÞ;cg ð tÞg; T ð1Þ pffiffiffi c ðμ Þ¼fcq g; ð20Þ 0 SD t where for a right-handed sbottom αb ¼ −βb ¼ −g =3 2. ð0;1;2Þ The residual mass term is given by the mass splitting where cq is representative of the five quark flavors, i.e., δ~ ¼ m~ − mχ ≪ mt, and the sbottom-bino coupling is q ¼ u, d, s, c, b, so that these two vectors are 12 and 5 bR bR 2 the heavy particle version of Eq. (1). Physically, the ð Þ μ dimensional, respectively. Note that cq ð tÞ is nonzero heavy particle velocity, vμ, is conserved in the scattering only for q ¼ b. In general, Z exchange contributes to process. Thus, the sign in the kinetic term denotes a ð1Þ the SD interaction cq , but when mb ¼ 0 and the sbottom sbottom coming into the vertex, or by using integration is purely right handed, this amplitude vanishes at by parts, an antisbottom coming out of the vertex. In c leading order in momentum transfer by gauge invariance contrast to the relativistic case where χ ¼ χ , the fields χv c [Eq. (D16)]. The loop diagram where the Higgs is radiated and χv can only be related through the invariance in off the bottom quark also vanishes, while the one where Eq. (11). Hence, the two vertices above are the only ones the Higgs is radiated off the sbottom contributes to that contribute to amplitudes involving χv as the initial ð0Þ cq [Eq. (D8)]. and final states (e.g., there are no charge-reversed diagrams in Fig. 7 below). Note from the canonically 2. Running and matching matrices R and M, normalized kinetic term that the heavy sbottom hasffiffiffiffiffiffi 3 2 1 p and nucleon matrix elements fðμ0Þ scaling dimension = [hence the factor of = mχ Since the theory below the weak scale is again given by appearing in the field redefinition in Eq. (21) and in Eq. (12), the mapping of the weak scale coefficients to the the sbottom-bino coupling]. In the low-energy theory the hadronic scale is identical to the previous example of interactions of the heavy bino with the quarks and gluon Sec. III B. In particular, the components R and M imple- of 5-flavor QCD are still described by Eq. (12). ment RGE and matching across heavy quark thresholds, The sbottom-bino interaction introduced in Eq. (22) “ respectively, while f applies nucleon matrix element form can be viewed similarly to the so-called heavy-light ” – factors. current in applications for B-meson decays [40 42].In particular, its running due to QCD corrections from μ ∼ μ ∼ D. Case III: Right-handed sbottom, small mass splitting t mt down to b mb is significant, and we account for this when implementing the RGE down to the Finally, we consider the case where both the mass of the bottom quark threshold. Let us discuss in turn the fermion partnered to the sfermion and the mass splitting ingredients c, R, M, and f of the factorization presented between the sfermion and the bino are much lighter than δ ≪ in Eq. (8). the weak scale, f~ , mf mt. For definiteness, we focus on the case of a right-handed sbottom (b~ ) interacting with the R μ bino (χ) and bottom quark (b). 1. Weak scale coefficients cð tÞ In this example, the sbottom is not highly virtual at low The matching condition at the weak scale μt ∼ mt is energies since the small sbottom-bino mass splitting shown in Fig. 6. The full theory amplitudes are computed kinematically allows for sbottom-bino interactions through using the Lagrangian in Eq. (1), while the effective theory a soft bottom. Weak-scale physics is still integrated amplitudes are computed using the Lagrangians in out by matching onto 5-flavor QCD, but both the ð0;1;2Þ Eqs. (12) and (22). The coefficients cu;d;s;c are determined bino and sbottom are kept as heavy fields in the effective 2 by the same OðαwÞ loop diagrams of the previous two theory (valid for m~ , mχ ≫ mb). The relevant interactions bR examples. Since we set all mass scales much lighter than may be obtained from the full theory by introducing the weak scale to zero, we are implicitly taking the m , the field redefinition of Eq. (10) for the relativistic bino b δ~ ≪ mt limit of both the full theory and effective theory field χ, and bR amplitudes. Of course, it is precisely in this limit that the 1 relativistic and heavy particle Feynman rules match. ~ pffiffiffiffiffiffiffiffiffi −imχ v·x ~ bR ¼ e bR;v ð21Þ Therefore, the full theory contribution from Eq. (1) and 2mχ the effective theory contribution from Eq. (22) cancel in the gluon and bottom quark matching, yielding coefficients for the relativistic sbottom field b~ . The field X from R v ð0;2Þ ð0;1;2Þ cg and c that vanish up to Oðm =mχ; δ~ =mχÞ Eq. (10) is again integrated out, and upon employing the b b bR invariance described in Eq. (11) for heavy self-conjugate corrections. As an explicit example, the relativistic sbottom fields, we obtain propagator in the tree-level diagram is expanded as

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FIG. 6. Weak scale matching conditions for the case of a right-handed sbottom that is nearly degenerate with the bino. Crossed and charge-reversed diagrams are not shown. In the full theory diagrams, q0 refers to u, d, s, c. In the bottom line, the ellipsis denotes similar diagrams where the insertion of the gluon legs vary (see Appendixes D6and D7). Single (double) lines correspond to relativistic (heavy particle) fields. We have omitted the label “bare” on the coefficients on the right-hand side.

2 2 O δ generated from integrating out the Higgs [corresponding to 2 2 ¼ 2 þ ð ~ =mχÞ − − − 2 δ bR the full theory diagram where a Higgs is radiated off the ðk pÞ m~ mb ðmχ b~ þ p · kÞ bR R ð0Þ 1 sbottom, given in Eq. (D9)]. On the other hand, neither cb O δ ¼ þ ð b~ =mχ;mb=mχÞ; nor any of the spin-2 quark and gluon coefficients are mχð−v · k − δ~ Þ R bR generated at Oð1=mχÞ because the sbottom is kept in the ð23Þ low-energy effective theory below the weak scale. As in the previous case, the contributions from a Higgs radiated off a where we have included a factor of 2 for the crossed diagram bottom quark and Z exchange vanish in the chiral μ μ 0 and used p ¼ mχv . Note that the above result matches the limit mb ¼ . tree-level amplitude obtained from the Feynman rules of μ μ Eq. (22). In contrast, the usual expansion of the sbottom 2. Running from t down to b propagator in terms of local operators (corresponding to At leading order in 1=mχ, the only nonvanishing coef- ð0;1;2Þ ≪ ficients are those corresponding to the scale invariant current nonzero cb coefficients) is valid for mb, mχ mb~ .For R ð0Þ the gluon matching, we find that the full theory amplitudes Oq ¼ mqqq¯ , and thus the coefficients in Eq. (24) do not O 1 μ μ μ μ μ vanish at ð =mχÞ, which must be the case since the gluon evolve, i.e., cSIð bÞ¼RSIð b; tÞcSIð tÞ¼cSIð tÞ.We coefficients scale as ½mass−3, but the only mass scale is must also account for the scale evolution of the sbottom- ∼ bino couplings α , β in Eq. (22). The anomalous dimension mχ mb~ [see Eqs. (D30) and (D31) for the explicit forms of b b R γ ~ ¯Γχ Γ the full theory gluon diagrams in the limit mb ¼ δ~ ¼ 0]. of the current bR;vb v, with Dirac structure , is the bR ¯ Γ Similarly, the effective theory loop diagrams are scaleless, same as that of the heavy-light current Qv q describing and hence vanish, as discussed in Sec. III A and in the the interaction of a heavy quark Qv with a light quark q – Γ example of Sec. III C. In principle, setting m ¼ 0 intro- [40 42]. It is independent of the Dirac structure and is b γ −α π duces IR poles as in Sec. III C, but in this case they appear at given by ¼ s= . The evolution of the coefficients 3 c ¼ α , β is thus Oð1=mχÞ. Thus, with no spin-2 quark or gluon coefficients b b generated at Oð1=mχÞ, all bare Wilson coefficients are α ðμ Þ 2=β0 trivially renormalized, i.e., cðμÞ¼cbare. Collecting the cðμ Þ¼cðμ Þ s b ; ð25Þ b t α ðμ Þ Wilson coefficients as in Eq. (20), up to corrections of s t O δ ðmb=mχ; b~ =mχÞ, we find R where β0 ¼ 11 − 2nf=3 ¼ 23=3. This completely specifies the theory at the scale μb, given by the Lagrangians in T μ ð0Þ μ 0 0 0 T μ 0 cSIð tÞ¼fcq ð tÞ; ; ; g; cSDð tÞ¼f g: ð24Þ Eqs. (12) and (22).

Note that these two vectors, as in Eq. (20), are 12 and 5 3. Matching at μb ð0Þ dimensional for SI and SD, respectively. The coefficient cq The matching condition at the bottom quark threshold μb is only nonzero for the four quark flavors q ¼ u, d, s, c and is is shown in Fig. 7. The diagrams on the left are computed in

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FIG. 7. Matching condition at the bottom quark threshold for a heavy particle effective theory of a right-handed sbottom that is nearly degenerate with the bino. Single (double) lines correspond to relativistic (heavy particle theory) fields. The ellipsis denotes similar diagrams where the insertion of the gluon legs vary (see Appendix D7). the theory above the threshold using Eqs. (12) and (22), dimension operators relevant for Majorana DM-nucleon while the diagrams on the right are computed in the theory scattering. In Secs. IV C and IV D, we present fixed-order below the threshold using Eq. (12) but with four active calculations involving left-right mixed stops and sbottoms ~ ~ ~ quark flavors. Since the q ¼ u, d, s, c sectors of the (t1;2, b1;2), and right-handed charged sleptons (lR), two theories are identical, the only consequence of the respectively. matching is to integrate out the bottom and the heavy sbottom, encoding their effects into the scalar and spin-2 A. Right-handed stop gluon coefficients. At this threshold, the mass scales mb δ We begin with the simple example of bino-nucleon and b~ are kept nonzero. It is straightforward to modify the matrix M μ in Appendix C to include the contribution scattering induced through interactions with a right-handed ð bÞ ~ from the heavy sbottom loop. Collecting the Wilson stop (tR). Note that a fixed-order calculation of this model coefficients as in Eq. (24), up to corrections of was presented in Ref. [36]. We go beyond this calculation by performing the complete leading order matching at the Oðmb=mχ; δ~ =mχÞ, we find bR weak scale, and a leading log analysis as described in 0 0 2 Sec. III B. cT μ cð Þ μ ;cð Þ μ ; 0;cð Þ μ ; SIð bÞ¼f q0 ð tÞ g ð bÞ g ð bÞg Constraints from LHC searches for direct production of T μ 0 stops are ameliorated in the limit of compressed stop cSDð bÞ¼f g; ð26Þ spectra (although see Ref. [43]). For example, monojet ð0Þ searches at a 14 TeV high-luminosity LHC can only where cq0 is representative of the four quark flavors, i.e., 0 exclude binos lighter than 500 GeV [44]. At the same q ¼ u, d, s, c. Note that these vectors are 10 and 4 time, approximate degeneracy avoids power suppression dimensional, respectively, instead of 12 and 5 dimensional of the amplitudes for bino-nucleon scattering, enhancing as in Eqs. (20) and (24). Here, integrating out the sbottom μ the prospects for direct detection. In light of this, we and bottom quark at the threshold b contributes to the focus on the optimistic scenario that the mass splitting, scalar and spin-2 gluon coefficients, while the spin-2 quark δ~ ¼ m~ − mχ, is much less than the weak scale, and coefficient is only generated at higher order. The analytic tR tR hence barring corrections of Oðδ~ =m Þ, we set m~ ¼ mχ forms of the gluon coefficients are given in Eq. (D41). tR t tR when determining weak scale matching coefficients. The resulting SI and SD cross sections per nucleon for 4. Running and matching matrices R and M, μ scattering on a xenon target are shown in Fig. 8. Varying and nucleon matrix elements fð 0Þ tan β would only affect these results at the level of a few Below the bottom quark threshold, the theory is given by percent. For SI scattering, we present a comparison of the the Lagrangian in Eq. (12) with four quark flavors, and leading order (LO) rate determined from the fixed order thus, for the remaining analysis down to 3-flavor QCD, we analysis described in Sec. II, and the “LO þ LL” rate employ the same components R, M, and f of the previous determined from the leading log EFT analysis described in two examples. Sec. III. The LO prediction includes the uncertainty from hadronic inputs, while the LO þ LL prediction also IV. PHENOMENOLOGY includes the perturbative uncertainty (added in quadrature), obtained from the variation of renormalization scales μ , μ , This section explores the phenomenology of several t b and μc, within the ranges given in Table I. For larger bino scenarios for bino DM in the MSSM. Sections IVA masses (∼1 TeV), the LL corrections enhance the rate by a and IV B focus on the specific examples of a right-handed 2 factor of a few (∼3), due in part to Oðα ðμ Þα Þ corrections stop (t~ ) and right-handed sbottom (b~ ), respectively. In s b w R R that are included in the EFT analysis, but are formally these sections, the matching and running prescription higher order in the fixed order approach. In particular, these identically follows Sec. III. In particular, as shown in are one-loop Higgs exchange diagrams that contribute to Eq. (12), our computational scheme follows that of ð0Þ Refs. [6,7,21,22], employing a matching procedure that cg at two-loop. While both quark and gluon weak scale ð0Þ 2 includes the leading order contributions for the lowest coefficients scale as cq;g ∼ 1=v mχ, where v is the SM

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FIG. 8. Left: The spin-independent cross section (per nucleon) for the case of a right-handed stop in the optimistic limit that its mass is nearly degenerate with that of the bino, mχ . For comparison, we show both the fixed-order result (“LO,” blue area) and the leading log result from the effective theory analysis (“LO þ LL,” red area). The thickness of the bands corresponds to the combined hadronic input and perturbative uncertainties. The grey dashed lines show the projected sensitivity of the LZ experiment and the neutrino background. Right: The spin-dependent cross section (per neutron) for the case of a right-handed stop in the optimistic limit that its mass is nearly degenerate with that of the bino, mχ. The thickness of the band corresponds to hadronic input uncertainties.

Higgs vacuum expectation value, the Higgs exchange B. Right-handed sbottom mχ contributions are enhanced due to a log m factor. The We now examine the direct detection prospects when the t ~ contribution from the spin-2 gluon amplitude is subdomi- bino interacts with a pure right-handed sbottom (bR). We nant. For SD scattering, we only consider the LO rate consider the two cases described in Sec. III, depending on determined from the fixed order analysis described in whether the mass splitting, δ~ ¼ m~ − mχ, is of order the bR bR Sec. II, since corrections to coefficient running and match- weak scale or much smaller. For the complete description O α2 ing enter at ð sÞ. While the analysis in Ref. [36] reported of the matching and running procedure, we refer the reader destructive interference between the Higgs and gluon to Secs. III C and III D for the large and small splitting diagrams of Fig. 4, we find no such interference and thus cases, respectively. Assuming that the squark correction to obtain substantially larger rates in Fig. 8. For more details, the SM Higgs gluon fusion amplitude is proportional to see Eqs. (D10) and (D11). μ v=m2 (where μ is the dimensionful trilinear squark- ∼200 q q~ q Although LZ will probe bino masses below GeV, squark-Higgs coupling) and that current LHC Higgs Higgs coupling measurements sensitive to deviations in the measurements in the gluon fusion channel constrain stops gluon fusion rate already exclude this region after Run 1 of to be heavier than ∼300 GeV, the rescaled limit for the LHC [36]. Future direct detection experiments pro- sbottoms approaches roughly ∼50 GeV in the large tan β jected to reach SI cross sections close to the neutrino ∼600 limit. Thus, throughout this section, we consider bino and background will probe bino masses lighter than GeV. sbottom masses greater than 100 GeV. Furthermore, without an enhancement from coherent scat- tering, the SD rate from Z exchange is below the neutrino background for masses ≳200 GeV. Note that in order to 1. Large mass splitting achieve the observed relic abundance from thermal freeze- We begin with the case where the sbottom is significantly out through coannihilation, the bino-stop mass splitting heavier than the bino, δ~ ∼ 100 GeV. The resulting SI bR varies between 30 and 40 GeV for sub-TeV bino dark cross sections per nucleon for scattering on a xenon target matter and gradually reaches sub-GeV splitting for dark are shown in Fig. 9. On the left panel, we include for matter mass above 2 TeV [36,45]. comparison predictions for both the LO and LO þ LL rates, as determined by the fixed order and EFT analyses, TABLE I. Numerical values used for the variation of renorm- respectively. Perturbative and hadronic uncertainties are alization scales of Fig. 3. calculated as in Sec. IVA. For this large mass splitting case, Scale Central Range the leading log corrections yield a slight enhancement of ffiffiffi ffiffiffi O 50% μ 2 126 p p ð Þ. On the right panel of Fig. 9, we show the SI cross t ðmW þ mtÞ= ¼ GeV ðm = 2;m 2Þ W pffiffiffi tpffiffiffi section as a function of mχ for values of the sbottom-bino μ m ¼ 4.75 GeV 2 2 b b ðmb= ;mb Þ mass splitting in the range 50–100 GeV. The rate is μ m ¼ 1.4 GeV (1 GeV, 2 GeV) c c dominated by the bino’s scalar coupling to gluons.

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FIG. 9. Left: The spin-independent cross section (per nucleon) for the case of a right-handed sbottom and a sbottom-bino mass splitting that is comparable to the weak scale (δ~ ¼ 100 GeV). For comparison, we show both the fixed-order result (LO, blue area) and bR the leading log result from the effective theory analysis (LO þ LL, red area). The thickness of the bands corresponds to combined theoretical and hadronic uncertainties. The gray dashed line shows the point at which the irreducible neutrino background should be relevant. Right: The spin-independent nucleon cross sections as a function of mχ for various values of the sbottom-bino mass splitting in GeV (white boxes). The calculation is performed using the full LO þ LL framework. The width of the bands corresponds to the combined theoretical and hadronic uncertainties.

Depending on the particular value of δ~ , the LZ experi- sections per nucleon for scattering on a xenon target are bR ment will probe light binos up to a few hundreds of GeV. shown in Fig. 10. On the left panel, we include predictions for both the LO and LO þ LL rates, as determined by the fixed order and EFTanalyses, respectively. Perturbative and 2. Small mass splitting hadronic uncertainties are calculated as in Sec. IVA. For this Let us now consider the degenerate case, δ~ ≪ 100 GeV, case, the rate receives large contributions from both the bR where the sbottom is kept as an active degree of freedom scalar and spin-2 gluon couplings. −3 below the weak scale. The explicit matching and running For small relative mass splittings (δ~ =mχ ≲ 10 ) bR prescription is detailed in Sec. III D. The resulting SI cross the enhancement from LL corrections has significant

FIG. 10. Left: The spin-independent cross section (per nucleon) for the case of a right-handed sbottom and a sbottom-bino mass splitting that is much less than the weak scale (δ~ ¼ 5 GeV). For comparison, we show both the fixed-order result (LO, blue area) and bR the leading log result from the effective theory analysis (LO þ LL, red area). We also illustrate the impact of including the running of the αf and βf coefficients of Eq. (1) from the scale μχ ∼ mχ (LO þ LLχ , green area). The thickness of the bands corresponds to combined hadronic and theoretical uncertainties. The gray dashed lines show the projected reach of the LZ experiment and the point at which the irreducible neutrino background should be relevant. Right: The spin-independent nucleon cross sections for various values of the sbottom-bino mass splitting in GeV (white boxes). The calculation is performed using the full LO þ LL framework. The width of the bands corresponds to the combined theoretical and hadronic uncertainties.

095008-14 BINO VARIATIONS: EFFECTIVE FIELD THEORY … PHYSICAL REVIEW D 93, 095008 (2016) implications for predicting the discovery potential of future Higgs exchange and gluon diagrams when calculating the ð0Þ experiments. For instance, while the fixed-order approach SI cross section. In calculating the Wilson coefficients cq , predicts that bino DM as heavy as ∼4 TeV has a scattering ð0Þ ð2Þ cg , and cg , we substitute the expressions for the rate above the neutrino background, the complete calculation interactions in Appendix A into the general results of ∼7 extends the reach up to TeV. In general, incorporating Appendix D. Note that mixing allows for the presence of the running of the weak scale Wilson coefficients down to additional states, resulting in new diagrams where multiple ∼3–4 the hadronic scale results in an overall factor of in the squarks are present in the same loop. Although nonzero, final cross section. As described in Sec. III D, a significant SD nucleon couplings are found to be subdominant ~ ¯χ portion of this enhancement is tied to the RGE of the bRb throughout the parameter space that we consider and are heavy-light current of Eq. (25), which alone rescales the therefore omitted from the discussion below. 24=23 fixed-order cross section by ½αsðmbÞ=αsðmtÞ ≈ 2. The lightest neutralino is assumed to be dominantly In the right panel of Fig. 10, we show the SI cross binolike. For this to hold true, the Higgsino mass parameter section for various choices of the small mass splitting δ~ . μ 10 bR is fixed at ¼ TeV, and we refrain from considering Here, δ~ ¼ 0 corresponds to a sbottom-bino mass splitting ≡ bR bino masses (mχ M1) much larger than 1 TeV. In this that is much smaller than the mass of the bottom quark. section, μ denotes the Higgsino mass parameter, not to be Bino DM with mass up to 3–20 TeV will remain above the confused with a renormalization scale. The other gaugino neutrino background for δ~ ≈ 10− 0 GeV, respectively. masses are assumed to be completely decoupled from the bR Interestingly, such small mass splittings are also needed low-energy spectrum. Two parameters independently gov- for standard freeze-out through sbottom coannihilation, and ern the mixing in the stop and sbottom sectors, hence LZ and future experiments will be sensitive to thermal bino DM in the multi-TeV mass range. Xt ≡ At − μ cot β;Xb ≡ Ab − μ tan β; ð27Þ In the analysis in Secs. II and III, we assumed, for pffiffiffi 1=2 definiteness, that the full theory described in Eq. (1) was where Xt;b ¼ 6m~ ~ ≡ ð6m ~ m~ ~ Þ corresponds μ ∼ 100 t;b Q3 tR;bR defined at the weak scale, t GeV. It is interesting to to maximal left-right mixing in the stop and sbottom consider the impact of additional RGE for cases where the sectors, respectively. This determines the A terms, At;b, full theory is defined at a higher scale, e.g., through imposing for a given value of μ and tan β. At every point in parameter theoretical constraints of specific ultraviolet completions or space, we will set the bino mass in terms of the physical observational constraints such as the relic abundance and squark masses, given in Eq. (A3), such that collider limits. For illustration, let us consider the running of the bino-sfermion-fermion couplings α and β of Eq. (1) f f mχ ¼ Minðm~ ;m~ Þ − δ~ ; ð28Þ t1;2 b1;2 q from a scale μχ ∼ mχ for the case of a sbottom nearly degenerate with the bino. The effective theory setup is which effectively defines the minimal mass splitting δq~ . similar to case III described in Sec. III D: at the scale μχ,we Since left-right mixing introduces several new degrees of match the full relativistic theory in Eq. (1) onto the heavy freedom compared to the models of the previous sections, particle effective theory in Eq. (22), and thus the running of α β we assume simplifying relations to reduce the size of the the b and b coefficients are again given by Eq. (25).Atthe parameter space. In particular, we focus on two different weak scale, the contributions from the Higgs exchange are O 1 2 schemes in parametrizing left-right mixing. In the first ð =mχÞ and can be neglected when working to leading scheme, we set the third-generation left and right soft 1 μ order in =mχ. Upon evolving down to the bottom scale b, squark masses and mixing parameters equal, the remaining analysis follows that of Sec. III D. The impact of the additional running is shown in the left panel of Fig. 10 m~ ≡ m ~ ¼ m~ ¼ m~ ;X≡ X ¼ X : ð29Þ q Q3 tR bR q t b as the green curve labeled LO þ LLχ. While the effect on the ∼60% cross section is only (the strong coupling asymptotes In the second scheme, we decouple the right-handed at high energies), the implied potential mass reach for an sbottom to 10 TeV and focus on left-right mixing in the experiment probing cross sections near the neutrino back- stop sector alone, ground may be increased by ∼1 TeV. ~ ≫ ~ ~ ~ ≠ ~ mb mQ3 ;mt ;mQ3 mt : ð30Þ C. Mixed squarks R R R Left-right mixing in the squark sector can affect the form The prospects for detecting bino-nucleon scattering of the cross sections considerably. In this section, we induced by its interactions with mixed stops and sbottoms present a fixed-order estimate for the bino-nucleon scatter- are shown in Fig. 11. The left and right panels employ the ing rate induced by interactions with mixed third generation parametrization of Eqs. (29) and (30), respectively. Here, squarks. Following the approach of Sec. II, we match we fix the squark-bino mass splitting of Eq. (28) to be directly to 3-flavor QCD and include contributions from δq~ ¼ 10 GeV, and tan β ¼ 5. We show the region currently

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FIG. 11. Results from a fixed-order calculation when the bino interacts with mixed stops and sbottoms using the parametrization of Eq. (29) (left) and Eq. (30) (right). The Higgsino mass is fixed to μ ¼ 10 TeV (not to be confused with a renormalization scale). The filled contours correspond to rates that are currently excluded by LUX (red area) or will be probed by future experiments like XENON1T (orange area) and LZ (yellow area). Also shown are regions with cross sections greater than the neutrino background (blue area). We do not consider bino masses lighter than 100 GeV or tachyonic squarks (both grey). For reference, we also show contours of 2 fixed bino mass in GeV (black dot-dashed lines) and the spin-independent nucleon cross section in units of log10ðσSI=cm Þ (green dashed lines).

~ excluded by LUX (red area), the projected reach of coupling to gluons through the exchange of t1;2 does not XENON1T (orange area) and LZ (yellow area), and the see the enhancement at Xq ¼ 0, and instead Higgs exchange parameter space with cross sections above the neutrino is the dominant process that involves stops. When jXq=mq~ j background (blue area). We do not consider values of is somewhat large, the Higgs-stop interaction grows and the 100 parameters where the bino is very light (mχ < GeV) or mass splitting between the two stops is several hundreds of the mass of one or more squarks is tachyonic (both in grey). GeV, effectively decoupling t~2. In this limit, we find that For both cases, left-right mixing tends to diminish the different behaviors emerge depending on the sign of X .In overall scattering rate, but for different reasons. For the left q panel, corresponding to Eq. (29), because of the small particular, for large and positive Xq, the two Higgs exchange ~ squark-bino mass splitting, the dominant scattering dia- diagrams where h is emitted off either an intermediate t1 or grams correspond to one-loop couplings to gluons through top quark (Fig. 15 below) interfere slightly, while for large ~ ~ the exchange of the light sbottoms b1 and b2 (see Fig. 19 and negative Xq this pair of diagrams tends to add coher- below). These diagrams add coherently when the degree of ently. Hence, even though large mixing stifles the contri- mixing is small, i.e., Xq ≪ mq~ . On the other hand, as soon as bution from sbottom-gluon diagrams, Higgs exchange via ~ ~ jXq=mq~ j ≳ 0.5, the diagrams involving b1 or b2 tend to virtual stops is able to somewhat lift this suppression for interfere deconstructively, vastly lowering the scattering large and negative Xq. This feature is clearly seen on the left- rate. This explains the sharp peak in the cross section hand side of Fig. 11, which shows less diminished scattering near Xq ¼ 0. rates near ð−Xq=mq~ Þ ∼ 2–3. For the right panel, corresponding to Eq. (30), larger mixing lowers the mass of the lightest stop relative to the D. Charged sleptons left-handed sbottom, which decouples the lightest sbottom In Secs. IVA–IV C, we presented examples where the from the bino for a fixed mass splitting, δ~ , and suppresses q fermion in Eq. (1) is either sufficiently heavy such that Higgs potential contributions from sbottom induced gluon cou- exchange is the primary scattering process or sufficiently plings. We find that, for these scenarios, squark mixing generally tends to reduce the reach of future direct detection light and colored such that coupling to gluons dominates the experiments. cross section. If the fermion is both light and uncolored, e.g., Mixing also strongly affects the stop sector. For brevity, a charged lepton (l), one must reconsider the processes that we focus the discussion on the left panel of Fig. 11, contribute to elastic nucleon scattering. In this section, we corresponding to Eq. (29); the behavior is similar for the will focus on the case where the bino (χ) interacts with a right panel, corresponding to the parametrization of single right-handed selectron (e~R) or stau (τ~R). Simplified Eq. (30). Because of the large mass of the top quark, models related to this scenario have been studied in [19,46].

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FIG. 12. Matching procedure for the case of a single right-handed slepton. Charge-reversed diagrams are not shown.

At leading order, SI scattering is given solely, to a good with finite lepton masses. In this scheme, no divergences χ approximation, by loop diagrams coupling to the electro- emerge, and hence cA is trivially renormalized. Explicit EM ν magnetic current, Jμ ≡∂ Fνμ, where Fμν is the photon forms for cA are given in Sec. D4. The proton matrix field strength. In particular, at dimension six, gauge element of the current corresponds to the counting operator invariance dictates that a Majorana fermion may only and is given by couple to the photon via the anapole operator, defined to μ 5 EM EM be χγ¯ γ χJμ . Therefore, at low energies, in place of hpðkÞjJμ ðμÞjpðkÞi ≡ eðμÞu¯ðkÞγμuðkÞ; ð33Þ Eq. (2), we consider the effective Lagrangian where the running of the electric coupling, eðμÞ, is the only μ 5 EM L ¼ cAχγ¯ γ χJμ : ð31Þ source of scale dependence. Unlike the scalar form factors in Sec. (4), the nucleon matrix element above is easily From the definition of the current, it is apparent that this evaluated at the weak scale, and hence a fixed-order interaction must vanish in the limit of zero momentum calculation suffices. After taking matrix elements, the SI transfer. This is also seen explicitly in the amplitude, where bino-proton cross section is then given by the contact interaction above leads to the effective form for the photon-amputated amplitude 2 σ e 2 SI ¼ 2π mpERcA; ð34Þ μ μ 2 μ 5 M ¼ 2cAu¯ðpfÞðq q − q γ Þγ uðpiÞ; ð32Þ where mp is the proton mass and ER is the recoil energy p p where i, f are the incoming and outgoing bino momenta, [46]. As a representative value we set ER ¼ 10 keV. uðpi;fÞ are the associated 4-component spinors, and q ≡ The reach in the SI cross section is shown in Fig. 13 for a pf − pi is the momentum transfer. The factor of 2 in the single right-handed selectron or stau. As the anapole above expression accounts for the Majorana nature of χ. Wilson coefficient is strongly enhanced when the lepton The prescription for matching Eq. (1) onto the anapole mass ml is much smaller than mχ, selectron mediated operator Eq. (31) is shown in Fig. 12. We regulate IR poles scattering benefits from large cross sections compared to

FIG. 13. Results from a fixed-order calculation for the case of SI bino-proton scattering mediated by either a right-handed selectron (left) or stau (right). The filled contours correspond to regions that will be probed by LZ (red area) or future direct detection experiments sensitive to rates above the irreducible neutrino background (blue area). For reference, we also show regions where the calculated relic abundance matches the observed dark matter density (black curve).

095008-17 BERLIN, ROBERTSON, SOLON, and ZUREK PHYSICAL REVIEW D 93, 095008 (2016) those mediated by a right-handed stau. Also in Fig. 13,we of a stop mediator, the mass reach is around 500 GeV, while overlay the region of parameter space where the for the case of a selectron mediator, the thermal mass reach relic abundance of χ matches the observed dark matter is around 300 GeV. density. Interactions relevant for annihilations and coanni- In the dark matter context, heavy particle effective hilations to SM particles are built in FeynRules [47] and theories are efficient for parametrizing unknown inter- implemented in micrOMEGAs [48]. Sommerfeld effects are actions of heavy (or nonrelativistic) DM particles with not included as photon exchange in the initial state is the SM degrees of freedom at a given energy scale, and for expected to only slightly alter the final calculated abun- factorizing amplitudes into contributions from the hard and dance [49,50]. While LZ will only be able to probe thermal soft modes of the process, necessary for resumming large coannihilating selectrons and binos for mχ ≲ 100 GeV, logarithms. These methods have been applied for inves- future direct detection experiments will be able to probe tigating universal behavior in the scattering of heavy selectron (stau) mediated scenarios for thermal bino masses WIMPs [6,7], the impact of large Sudakov logarithms mχ ≲ 300ð100Þ GeV. Left-right mixing introduces an addi- and Sommerfeld enhancement on the annihilation rate of tional slepton of opposite hypercharge and therefore tends heavy WIMPs [53–56], and the low-energy interactions of to diminish the overall scattering rate. DM with QCD and nucleons [22,29,30]. In this work, we applied heavy particle techniques to V. CONCLUSIONS three generic scenarios for bino scattering, depending on the mass hierarchy between the bino, the sfermion, and its We have presented EFT methods for computing direct partner fermion. In the first and second scenarios, where the detection rates, focusing on bino DM scattering through sfermion is integrated out of the theory below the weak loops mediated by heavy-flavor squarks or sleptons. In the scale, heavy particle theory was employed for writing the presence of large hierarchies between mass scales, such as basis of low-energy operators in Eq. (12). We have the weak and hadronic scales, large logarithms can sub- employed a matching procedure that includes the leading stantially contribute to the total scattering cross section. A order contributions for the lowest dimension operators sequence of effective theories, linked together by matching relevant for Majorana DM-nucleon scattering, including computations and renormalization group analysis, provides spin-2 couplings to gluons. Compared to the relativistic a systematically improvable framework for incorporating basis in Eq. (2), this basis has manifest power counting, and such contributions and assessing the impact of perturbative thus redundant or suppressed operators in the mχ ≫ m uncertainty. b limit are easily avoided. Nonetheless, the results for the Including these effects from running enhances the running and matching matrices, R and M, are properties of scattering cross section by a factor of ∼3–4 in some cases and significantly improves the DM mass reach of direct the QCD currents in Eq. (3) and can be applied regardless detection experiments. The specific sources of these effects of whether the DM is taken to be a relativistic or heavy vary for different models. For example, as explained in particle field. In the third scenario, where the sfermion is Sec. IVA, in our calculation for bino DM interacting with a kept as a degree of freedom below the weak scale, we used right-handed stop, leading log corrections increase the rate heavy particle theory to systematically separate the full theory amplitudes into contributions that either are encoded through the inclusion of OðαsÞ threshold terms for the scalar quark coefficient when evaluated near the hadronic in the coefficients of contact operators defined at the weak ~ ¯Γχ scale. Alternatively, in the fixed-order approach, these scale or are matched by the heavy-light current bR;vb v. contributions are formally higher order and are not The running of this current down to low energies is the included, highlighting one of the advantages of our scheme. dominant source of enhancement to the rate for bino On the other hand, in the case of bino DM coupled to a scattering mediated by a nearly degenerate squark. nearly degenerate right-handed sbottom, the mass reach In Sec. III, we focused on simple models with only a few increases from ∼4 to ∼7 TeV for an experiment such as LZ. parameters such that definite predictions can be made, and This is largely due to the fact that RG evolution signifi- radiative corrections become important not only for deter- cantly enhances the bino-sbottom-bottom interaction at low mining robust scattering rates but also for correlating differ- energies. Interestingly, if relic density constraints also ent constraints. Our analysis for nearly degenerate sleptons require such small mass splittings, this implies that much and mixed squarks is new, and within our simplifying of the coannihilation region may be constrained through assumptions, we find that mixing generally tends to reduce direct detection experiments. This motivates a careful the scattering rates. Furthermore, since these rates already investigation of the correlation between relic density and depend on several free parameters, we have not studied the direct detection observables, including, e.g., higher order impact of radiative corrections for this example. We still note QCD corrections, and a complete treatment of thermally that, similar to the models presented in Sec. III, it would be induced masses and Sommerfeld enhancement (see, e.g., interesting to consider leading log corrections for the mixed Refs. [36,45,51,52]). Assuming an experiment sensitive to case, since they may have substantial impact on the estimated cross sections close to the neutrino background, for the case reach of future experiments like XENON-1T and LZ.

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~ Aside from providing robust estimates of benchmark Above, mQ3 is the left-handed squark soft mass parameter, cross sections, employing EFT methods also allows for At;b are the soft trilinear couplings to the Higgs, tan β is the making the connection between parameters of a high scale ratio of the up and down type Higgs vacuum expectation theory to low-energy observables. We focused here on the values, μ is the supersymmetric Higgs mass parameter, θw starting point where the high-energy theory is defined at the m~ is the Weinberg angle, and f1 2 are the physical masses of weak scale. It is interesting to further consider the impact of ; the lightest, heaviest sfermion, respectively. These physical RGE from an even higher scale, where the parameters may tree-level masses, obtained by diagonalizing the sfermion be constrained by theoretical UV considerations or from mass matrix, are other phenomenological inputs such as collider limits and the DM relic density. Moreover, while we have focused 2 1 2 2 1 2 2 here on the effects from QCD corrections, previous studies m~ ¼ ðm ~ þ m~ Þþ m cos 2β þ mt t1;2 2 Q3 tR 4 Z have shown that electroweak corrections may also have 1 1 2 2 impact [23,57]. A complete picture of the complementarity ∓ 2 − 2 2 2β − 2θ ðm ~ m~ ÞþmZ cos sin w between DM observables, e.g., the correlation between 2 Q3 tR 4 3 1 parameters determined from relic density, collider limits, 2 2 μ β − 2 and direct detection, should incorporate the connection þ mt ð cot AtÞ ; between different scales in the physical processes. 2 1 2 2 1 2 2 m~ ¼ ðm ~ þ m~ Þ − m cos 2β þ m b1 2 2 Q3 b 4 Z b ACKNOWLEDGMENTS ; R 1 1 1 2 ∓ 2 − 2 − 2 2β − 2θ We thank Richard Hill and Jason Kumar for valuable ðm ~ m~ Þ mZ cos sin w 2 Q3 bR 4 3 discussions. A. B. is supported by the Kavli Institute 1 2 for Cosmological Physics at the University of Chicago þ m2ðμ tan β − A Þ2 ; ðA3Þ through Grant No. NSF PHY-1125897. D. R., M. S., and b b K. Z. are supported by the DoE under Contract No. DE- where m~ are the right-handed sfermion soft masses. AC02-05CH11231. D. R. is supported by the São Paulo fR Research Foundation (FAPESP). Note that we have chosen the sign convention for μ where the Higgsino contributions to the neutralino and μ −μ APPENDIX A: MODEL chargino mass matrices are given by þ and , respectively. Radiative corrections at one-loop can sig- Following the conventions in Ref. [58], we specify the nificantly alter the forms of the tree-level expressions masses of the bino/sfermion sector of the MSSM. The above [59–61]. For example, the correction to the bottom sfermion is assumed to be either a squark or a slepton. Yukawa can be parametrized in terms of a quantity Δb as The soft hypercharge gaugino mass parameter (M1) is taken to be positive so that no chiral field redefinitions are ffiffiffi mb yb → p ; ðA4Þ necessary to ensure the positivity of the physical bino 2v cos βð1 þ ΔbÞ mass. In particular, the bino and its physical mass will be χ denoted by and mχ. The mass matrices that relate the with the effect that in the sbottom mass matrix, Ab and ~ tan β are replaced by the effective parameters sfermion mass eigenstates (f1;2) to the gauge eigenstates ~ (fL;R) are given by β Ab β tan Ab;eff ¼ ; tan eff ¼ ; ðA5Þ ~ ~ 1 þ Δb 1 þ Δb f1 cos θf − sin θf f L : A1 ~ ¼ θ θ ~ ð Þ f2 sin f cos f fR as in Ref. [62]. Here we take the SM Higgs vacuum expectation value to be v ¼ 174 GeV. The trilinear ~ In the case that the sfermions are stops (t) or sbottoms coupling At may be defined similarly in the stop sector, (b~), the mixing angles are given explicitly by the tree-level for which the masses and Higgs interactions are inde- expressions pendent of tan β in the large tan β limit. From here on out, we will drop the “eff” subscript with the under- 2 2 2 1 2 2 2 m ~ þ mt þ m cos 2βð2 − 3 sin θwÞ − m~ standing that the squark masses and interactions are Q3 Z t1 tan θ ¼ ; defined in terms of these “effective” inputs at the weak t m ð−A þ μ cot βÞ t t scale. 2 2 2 2β − 1 1 2θ − 2 m ~ þ mb þ mZ cos ð 2 þ 3 sin wÞ m~ When dealing with sleptons, we will choose to ignore θ Q3 b1 tan b ¼ : intragenerational mixing since first and second generation mbð−Ab þ μ tan βÞ lepton masses are very small compared to the soft masses. ðA2Þ For the example of a single right-handed selectron (e~R),

095008-19 BERLIN, ROBERTSON, SOLON, and ZUREK PHYSICAL REVIEW D 93, 095008 (2016) its tree-level mass, m2 ≈ m2 − m2 cos 2βsin2θ , receives angles of Eq. (A1). We adopt the following notation for e~R e~1 Z w negligible corrections at one-loop and is essentially a free these interactions: X parameter controlled by the first generation right-handed L ⊃ ~ ¯ αðiÞ βðiÞγ5 χ fifð f þ f Þ þ H:c:; ðA6Þ slepton soft mass me~1 . ~ i¼1;2 The interactions of a pair of sfermions f1;2 with a bino LSP (χ) and SM fermion (f) are parametrized in terms of where the effective couplings for the stop/sbottom sector the SM hypercharge coupling g0 and the sfermion mixing are given by

1 −g0 1 1 −g0 1 2 −g0 1 αð Þ ≡ pffiffiffi cos θ þ 2 sin θ ; βð Þ ≡ pffiffiffi cos θ − 2 sin θ ; αð Þ ≡ pffiffiffi sin θ − 2 cos θ ; t 3 2 2 t t t 3 2 2 t t t 3 2 2 t t 2 −g0 1 1 −g0 1 1 −g0 1 βð Þ ≡ pffiffiffi sin θ þ 2 cos θ ; αð Þ ≡ pffiffiffi cos θ − sin θ ; βð Þ ≡ pffiffiffi cos θ þ sin θ ; t 3 2 2 t t b 3 2 2 b b b 3 2 2 b b 2 −g0 1 2 −g0 1 αð Þ ≡ pffiffiffi sin θ þ cos θ ; βð Þ ≡ pffiffiffi sin θ − cos θ : ðA7Þ b 3 2 2 b b b 3 2 2 b b pffiffiffi 0 The effective couplings for a single right-handed slepton are similarly defined, with αl ¼ −βl ¼ −g = 2. Although a pure bino possesses no tree-level interactions with the electroweak bosons of the SM, the sfermions and their associated SM fermion partners interact with the Z, photon, and SM Higgs (h) through terms that we parametrize as

L ⊃ μ v − a 2θ ~ †∂ ~ − ~ ∂ ~ † v a 2θ ~ †∂ ~ − ~ ∂ ~ † Z Z i½ðgf gf cos fÞðf1 μf1 f1 μf1Þþðgf þ gf cos fÞðf2 μf2 f2 μf2Þ a ~ † ~ ~ ~ † ~ † ~ ~ ~ † ¯ μ v a 5 − g sin 2θ ðf ∂μf2 − f1∂μf þ f ∂μf1 − f2∂μf Þ þ Zμfγ ðg þ g γ Þf; Xf f 1 2 2 1 f f L ⊃− μ ~ †∂ ~ − ~ ∂ ~ † − ¯γμ γ ieQfA ðfi μfi fi μfi Þ eQfAμf f; i¼1;2 X ðiÞ ~ † ~ ð12Þ ~ † ~ ~ ~ † mf ¯ L ⊃ ðμ hf f Þþμ hðf f2 þ f1f Þ − pffiffiffi hff; ðA8Þ h f i i f 1 2 2 i¼1;2 v where the effective parameters above for (s)tops, (s)bottoms, and (s)leptons are

−5e e −e e e e gv θ θ ;ga θ θ ;gv θ − θ ;ga θ θ ; t ¼ 12 tan w þ 4 cot w t ¼ 4 ðtan w þ cot wÞ b ¼ 12 tan w 4 cot w b ¼ 4 ðtan w þ cot wÞ e e −1 v 3 θ − θ a θ θ 2 3 −1 gl ¼ 4 ð tan w cot wÞ;gl ¼ 4 ðtan w þ cot wÞ;Qt ¼ = ;Qb ¼ 3 ;Ql ¼ ; ðA9Þ ffiffiffi p 2 1 − 2m 1 g v cos 2β μð Þ ¼ t m þ sin 2θ ðA − μ cot βÞ − pffiffiffi ½4sin2θ tan2θ þ cos2θ ð3 − tan2θ Þ; t v t 2 t t 6 2 t w t w ffiffiffi p 2 2 − 2m 1 g v cos 2β μð Þ ¼ t m − sin 2θ ðA − μ cot βÞ − pffiffiffi ½4cos2θ tan2θ þ sin2θ ð3 − tan2θ Þ; t v t 2 t t 6 2 t w t w 2 1 − 4 2θ 2θ 2β ð12Þ mffiffiffit g ð cos wÞsecffiffiffi wv cos μt ¼ p cos 2θtðAt − μ cot βÞþ p sin 2θt; 2v 12 2 ffiffiffi p 2 2 1 − 2m 1 g v cos 2βsec θ μð Þ ¼ b m þ sin 2θ ðA − μ tan βÞ þ pffiffiffi w ½3 þ cos 2θ ð1 þ 2 cos 2θ Þ; b v b 2 b b 12 2 b w ffiffiffi p 2 2 2 − 2m 1 g v cos 2βsec θ μð Þ ¼ b m − sin 2θ ðA − μ tan βÞ þ pffiffiffi w ½3 − cos 2θ ð1 þ 2 cos 2θ Þ; b v b 2 b b 12 2 b w 2 2β 1 2 2θ 2θ ð12Þ mffiffiffib g v cos ð þ cosffiffiffi wÞsec w μ ¼ p cos 2θbðAb − μ tan βÞþ p sin 2θb; ðA10Þ b 2v 12 2 2 174 such that g is the SUð Þw coupling, e is the electromagnetic coupling, v ¼ GeV, and we have worked in the alignment limit where the Higgs is SM-like.

095008-20 BINO VARIATIONS: EFFECTIVE FIELD THEORY … PHYSICAL REVIEW D 93, 095008 (2016) APPENDIX B: HADRONIC INPUTS TABLE II. Proton form factors for spin-2 operators at different values of μ. The neutron form factors follow from approximate In this section, we present the numerical values for the isospin symmetry (u ↔ d). hadronic form factors defined in Eq. (4). More detailed discussion on the determination of these quantities can be μ (GeV) ð2Þ μ ð2Þ μ ð2Þ μ ð2Þ μ fu;pð Þ fd;pð Þ fs;pð Þ fg;pð Þ found in Sec. IV of Ref. [22]. The up and down quark scalar form factors are deter- 1 0.404(9) 0.217(8) 0.024(4) 0.356(29) mined from the nucleon sigma terms, 1.4 0.370(8) 0.202(7) 0.030(4) 0.398(23) 2 0.346(7) 0.192(6) 0.034(3) 0.419(19) mu þ md ¯ Σπ ¼ hNjðuu¯ þ ddÞjNi¼44ð13Þ MeV; N 2 where neutron matrix elements follow from isospin ¯ symmetry (u ↔ d). Σ− ¼ðmd − muÞhNjðuu¯ − ddÞjNi¼2ð2Þ MeV; ðB1Þ Σ where the upper (lower) sign in − is for the proton APPENDIX C: RUNNING AND MATCHING (neutron) (see also Ref. [63]). For the strange quark, we use MATRICES 0 m fð Þ ¼ 40 20 MeV. The up and down quark scalar N s;N In this section, we present the analytic forms for form factors are then the leading order RGE and threshold matching matrices (R and M) appearing in Eq. (8). Since the scalar and spin-2 0 R Σπ 0 1 Σπ fð Þ ¼ ud N ð1 þ ξÞ;fð Þ ¼ N ð1 − ξÞ; operators do not mix with each other under RGE, in the basis u;N 1 R m d;N 1 R m þ ud N þ ud N of Eq. (14), these matrices have the block diagonal forms 1 þ R Σ− ξ ud 0 2 0 2 ¼ ; ðB2Þ R ¼fRð Þ; Rð Þg; M ¼fMð Þ; Mð Þg; ðC1Þ 1 − Rud 2ΣπN where RðSÞ and MðSÞ, for S ¼ 0, 2, are the running and where the ratios of quark masses are matching matrices for the scalar (S ¼ 0) and spin-2 (S ¼ 2) m m operators. Detailed discussion on the derivation of these R ≡ u ¼ 0.49 0.13;R≡ s ¼ 19.5 2.5: ðB3Þ ud m sd m quantities can be found in Sec. III of Ref. [22]. d d The running and heavy quark threshold matching for The gluon scalar form factor is determined from the next- the spin-1 axial-vector operators are trivial at leading α 1 to-leading order terms of Eq. (16). For our leading log order in s, and hence we take R ¼ M ¼ in evolving ð1Þ analysis, we take as default the coefficients cq . For scalar and spin-2 operators, the running matrix RðSÞ μ ; μ from a high scale (μ )toalow 2α μ X ð l hÞ h ð0Þ μ 1 − 1 sð 0Þ ð0Þ scale (μ ) in the basis ðu; d; s; …jgÞ with n flavors of quarks fg;Nð 0Þ¼ þ π fq;N: ðB4Þ l f q¼u;d;s has the form 0 1 α Note that, as long as s terms are consistently kept in the ðSÞ β~ μ γ μ μ B Rqg C functions ð 0Þ and mð 0Þ appearing in fð 0Þ and B . C B ðSÞ ðSÞ ðSÞ . C Rðμ0; μ Þ, the dependence on the low scale μ0 cancels in 1 − J . c ðSÞ μ μ B ðRqq Rqq0 Þþ Rqq0 C T μ μ μ R ð l; hÞ¼B C; the product f ð 0ÞRð 0; cÞ. We may thus simplify the B RðSÞ C μ μ ∼ @ qg A analysis by taking 0 ¼ c mc. ðSÞ ðSÞ ðSÞ Spin-2 form factors are derived from the second Rgq Rgq Rgg moments of parton distribution functions, C2 Z ð Þ 1 ð2Þ μ μ ¯ μ fq;Nð Þ¼ dxx½qðx; Þþqðx; Þ; where 1 and J are nf × nf matrices corresponding to the Z0 1 identity matrix and the matrix with all elements equal to unity, ð2Þ μ μ ðSÞ fg;Nð Þ¼ dxxgðx; Þ; ðB5Þ respectively. The elements Rij are specified in Table III.The 0 elements for the spin-2 operator involve the function where q x; μ , q¯ x; μ , and g x; μ are the quark, antiquark, 32=9þ2t=3 ð Þ ð Þ ð Þ 2 3−11 α ðμ Þ nf= and gluon parton distribution functions evaluated at the ≡ s l rðtÞ α μ : ðC3Þ scale μ, respectively. Table II lists values for renormaliza- sð hÞ μ 1 tion scales ¼ , 1.4, 2 GeV. Finally, for the spin-1 axial- For the scalar and spin-2 operators, the heavy quark (Q) vector form factors of the proton, we take threshold matching between nf þ 1 and nf-flavor QCD involves the ðn þ 1Þ × ðn þ 2Þ matrix MðSÞ, which is ð1Þ 0 75 8 ð1Þ −0 51 8 ð1Þ −0 15 8 f f fu;p ¼ . ð Þ;fd;p ¼ . ð Þ;fs;p ¼ . ð Þ; ðB6Þ given in the basis ðu; d; s; …jQjgÞ by

095008-21 BERLIN, ROBERTSON, SOLON, and ZUREK PHYSICAL REVIEW D 93, 095008 (2016)

TABLE III. Running matrices at leading order in αs for scalar presented in this appendix can easily be applied to other and spin-2 quark and gluon operators in nf-flavor QCD. Spin-2 models involving Majorana DM and charged scalars. In the operators are given in terms of the function rðtÞ [see Eq. (C3)]. case that two different sfermions can be present in the same loop, Latin subscripts are used to denote the fields. For Operator Running matrix example, we will often denote a single sfermion of a ð0Þ ð0Þ ð0Þ 1 ð0Þ 0 ~ Oq , Og Rqq ¼ , Rqq0 ¼ , generation as fi where i ¼ 1, 2. Alternatively, if the α μ α μ − 1 ð0Þ 16 sð lÞ= sð hÞ sfermion must be colored (as in the one-loop couplings Rqg ¼ 2 , ~ 3 nf − 11 to gluons), we will write qi to denote squarks. ð0Þ ð0Þ αsðμlÞ For most of this appendix, the Wilson coefficients are Rgq ¼ 0, Rgg ¼ αsðμhÞ presented as integrals over Feynman and Schwinger ð2Þ ð2Þ ð2Þ ð2Þ parameters, since they can be written in compact forms Oq , Og Rqq − R 0 ¼ rð0Þ, h qq i 1 16r n 3n that are easy to evaluate numerically. Of course, given ð2Þ ð fÞþ f − 0 R 0 ¼ rð Þ , certain assumptions for the couplings and mass spectrum, qq n 16 þ 3n f f these integrals can be evaluated to obtain analytic forms. 2 16½1 − rðn Þ Rð Þ ¼ f , For each diagram, in addition to the general model- qg 16 3n þ f independent result, we will provide limiting forms for 2 3½1 − rðn Þ 2 16 þ 3n rðn Þ ð Þ f ð Þ f f three distinct cases, namely a right-handed stop degenerate Rgq ¼ 16 3 , Rgg ¼ 16 3 þ nf þ nf with a bino (Secs. III B and IVA), a right-handed sbottom much heavier than the bino (Secs. III C and IV B 1), and a 0 1 right-handed sbottom nearly degenerate with the bino 1 0 0 B C (Secs. III D and IV B 2). B . . . C B . . . . C ðSÞ B C M ¼ B C; ðC4Þ 1. Tree level @ 1 0 0 A ðSÞ ðSÞ The simplest process that allows χ to scatter with nuclei 0 0 M Mgg gQ is the tree-level exchange of a squark. Here, we have in mind the exchange of a sbottom (as shown in Fig. 14) for ðSÞ with the elements Mij given in Table IV. matching onto 5-flavor QCD, although these results may be applied to first-generation squarks as well. Parametrizing APPENDIX D: COLLECTION OF WILSON the UV couplings as COEFFICIENTS L ⊃ ~ ¯ α βγ5 χ UV b b ð þ Þ þ H:c:; ðD1Þ In this appendix, we present the Wilson coefficients [in the notation of Eqs. (2) and (12)] that are obtained from and applying the appropriate Fierz transformations, we ≫ integrating out charged scalars as well as electroweak obtain the Wilson coefficients in the limit that mb~ mχ, vector and scalar bosons. We will begin with diagrams χ 2 2 α2 β2 that allow for to scatter with quarks at tree level and will ð0Þbare −ðα − β Þ mχð þ Þ then proceed to various loop-level processes in order of cb ¼ 2 2 þ 2 2 2 ; 4m ðm − mχÞ 8ðm − mχÞ increasing complexity. Throughout, we will parametrize the b b~ b~ α2 β2 Lagrangian governing the ultraviolet couplings (denoted as ð2Þbare mχð þ Þ L cb ¼ 2 2 2 : ðD2Þ UV) in a generic manner, although a simple mapping to the 2ðm~ − mχÞ MSSM can be performed by comparing to the particular b couplings of Appendix A. In this sense, the results The above expressions agree with those presented in Ref. [12]. For a right-handed sbottom, the coefficients TABLE IV. Heavy quark threshold matching matrices at lead- reduce to ing order in αs for scalar and spin-2 operators. The strong 0 coupling in the (nf þ 1)-flavor theory is denoted αs. mQ and μQ correspond to the mass of the heavy quark and the scale at which it is integrated out, respectively.

Operator Matching matrix 0 0 0 ð Þ ð Þ 0 −α ðμ Þ 0 Oq , Og ð Þ s Q ð Þ 1 MgQ ¼ 12π , Mgg ¼ 2 2 ð Þ ð Þ 2 α0 μ 2 Oq , Og Mð Þ ¼ s log Q , Mð Þ ¼ 1 gQ 3π m gg FIG. 14. Feynman diagram responsible for tree-level scattering. Q Crossed diagram not shown.

095008-22 BINO VARIATIONS: EFFECTIVE FIELD THEORY … PHYSICAL REVIEW D 93, 095008 (2016) 0 2 0 2 ð0Þbare ðg Þ mχ ð2Þbare ðg Þ mχ cb ¼ 2 2 2 ;cb ¼ 2 2 2 : 72ðm~ − mχÞ 18ðm~ − mχÞ bR bR ðD3Þ 2. Higgs exchange We now proceed to one-loop couplings to quarks. The simplest example of this process is through the t-channel exchange of a Higgs that is radiated off of an intermediate fermion or sfermion as shown in Fig. 15. For general FIG. 15. Feynman diagrams responsible for couplings to quarks sfermion mixing, the Higgs possesses both diagonal and ~ through Higgs exchange. Charge-reversed diagrams not shown. off-diagonal couplings to the sfermions f1 2 as discussed in ; 2 Appendix A. We will parametrize the interactions respon- 0 ðg0Þ cð Þbare g2 2β 2θ sible for this process as q ¼ 36π2 2 cos tan w mhmχ X 2 L ⊃ ~ ¯ αðiÞ βðiÞγ5 χ 1 2 mχ 3 1 2 2 UV ½fifð þ Þ þ H:c: 2β θ f f þ 2 mt log 2 2 þ 2 g cos tan w i mt v mχ X ðijÞ 2 2 −1=2 −1 2 2 1=2 μ † þð4mχ=m − 1Þ tan ð4mχ=m − 1Þ þ h hðf~ f~ þ H:c:Þþλhhff;¯ ðD4Þ t t 1 þ δ i j f 3 2 2 i≤j ij mt − 2 − mt 2 2β 2θ × 2 2 g cos tan w : ðD7Þ v mχ where the sum runs over a complete gauge multiplet of sfermions (e.g., for a single right-handed stop, i ¼ 1, while Similarly, for a nondegenerate right-handed sbottom, the for mixed left and right-handed stops and sbottoms i ¼ 1, general results simplify to 2, 3, 4). The bare scalar Wilson coefficient is given by − 0 2 2 2β 2θ ð0Þbare ðg Þ g cos tan w h X X cq ¼ 2 2 3 0 ðλ =m Þn 288π ð Þbare q q c mhmχ cq ¼ 2 2 Mij þ Mi ; ðD5Þ 8π m 2 h i≤j i m~ 2 − 2 − 2 bR × mχ ðm~ mχÞ log 2 2 ; ðD8Þ bR m − mχ ~ b~ where nc is the number of colors of fi. The contributions R Mij and Mi correspond to the left and right diagrams of while for a degenerate right-handed sbottom we find Fig. 15, respectively, and are given by 2 2 2 0 −ðg0Þ g cos 2βtan θ Z cð Þbare w : 1 μðijÞ 1 q ¼ 288π2 2 ðD9Þ ≡ h αðiÞαðjÞ 1 − mhmχ Mij 2 2 dx½ f f ðð xÞmχ þ mfÞ 1 þ δij m~ − m~ 0 fi fj Let us compare these results to limiting forms presented Δ βðiÞβðjÞ 1 − − i in Ref. [36]. For the case of a right-handed stop that is much þ f f ðð xÞmχ mfÞ log ; Δj heavier than the bino, the Wilson coefficient reduces to the Z approximate form 1 μ2 M ≡ −λhððαðiÞÞ2 − ðβðiÞÞ2Þ dxx log i f f f Δ 0 2 2 0 i ð0Þbare ðg Þ mt mχ Z cq ≈ ; ðD10Þ λh 1 12π2 2 2 2 f x ðiÞ 2 2 v mhmt~ dx α 1 − x mχ m R þ 2 Δ ½ð f Þ ðð Þ þ fÞ 0 i while for a right-handed stop that is degenerate with the − βðiÞ 2 1 − − 2 ð f Þ ðð xÞmχ mfÞ ; bino but much heavier than the top quark, 2 2 2 Δ ≡ − 1 − 0 2 2 2 2 i ðx Þðxmχ m~ Þþxmf: ðD6Þ 0 ðg Þ m log ðmχ=m Þ fi cð Þbare ≈ t t : q 24π2 2 2 ðD11Þ v mhmχ Above, we have dropped terms, such as UV poles, that vanish when summed over a complete gauge multiplet of Both Eqs. (D10) and (D11) agree with the limiting forms sfermions. Accordingly, the dependence on the renormal- presented in Ref. [36], up to an overall sign. On the other ization scale, μ, should also vanish, but we have kept it to hand, we agree with the full result in Ref. [17], and we also allow for a more compact form. check that our expressions are consistent with low-energy m~ mχ In the case of a degenerate right-handed stop ( tR ¼ ), Higgs theorems [64]. As a result, we find that Higgs the coefficient reduces to exchange adds constructively with the gluon diagrams of

095008-23 BERLIN, ROBERTSON, SOLON, and ZUREK PHYSICAL REVIEW D 93, 095008 (2016) and off-diagonal Z-sfermion couplings contribute. The UV couplings are parametrized as X L ⊃ ~ ¯ αðiÞ βðiÞγ5 χ UV ½fifð f þ f Þ þ H:c: i ðijÞ X ig~ f μ ~ †∂ ~ ~ †∂ ~ þ 1 δ Z ðfi μfj þ fj μfiÞþH:c: i≤j þ ij ¯ μ v a 5 þ Zμfγ ðg þ g γ Þf; ðD12Þ FIG. 16. Feynman diagrams responsible for couplings to quarks f f through Z exchange. Charge-reversed diagrams not shown. where the sum over i runs over a complete gauge multiplet of sfermions. Integrating out the Z in Fig. 16 generates the Sec. D6 when evaluating the scattering amplitude. axial-vector Wilson coefficient The phenomenological impact of this is discussed in a X X 1 g n Sec. IVA. cð Þbare q c M − M ; q ¼ 16π2 2 ij i ðD13Þ mZ i≤j i 3. Z exchange ~ Analogous to Higgs exchange, quark scattering can where nc is the number of colors of fi. The contributions also occur at one-loop order through the t-channel Mij and Mi correspond to the left and right diagrams of exchange of a Z boson. As shown in Fig. 16, both diagonal Fig. 16, respectively, and are given by

ðijÞ ðiÞ ðjÞ ðiÞ ðjÞ Z g~ ðα β þ β α Þ 2 1 μ2 μ2 ≡ f f f f f 1 Δ − Δ Mij þ 2 2 dx i log j log ; 1 þ δ m − m 0 Δ Δ ij f~ f~ i j i j Z Z 1 1 μ2 1 x ≡ a αðiÞ 2 βðiÞ 2 − 2 vαðiÞβðiÞ − a αðiÞ 2 1 − 2 Mi ½gfðð f Þ þð f Þ Þ gf f f dxx log þ dx fgf½ð f Þ ðð xÞmχ þ mfÞ 2 0 Δi 0 Δi βðiÞ 2 1 − − 2 2 vαðiÞβðiÞ 2 − 1 − 2 2 þð f Þ ðð xÞmχ mfÞ þ gf f f ðmf ð xÞ mχÞg; Δ ≡ − 1 2 − 2 2 i ðx Þðxmχ m~ Þþxmf: ðD14Þ fi

Here, we have dropped terms, such as divergent pieces, that scale, and when the sbottom is purely right handed, this vanish due to gauge invariance when summed over a corresponds to an enhanced chiral symmetry. In this case, complete multiplet of sfermions. Although the dependence the bino’s coupling to the Z is proportional to photon on the renormalization scale, μ, also drops out, we have exchange, which vanishes at zero-momentum transfer due kept it explicit above to allow for a more compact form. to gauge invariance (as shown in the next section). In the case of a degenerate right-handed stop (m~ mχ), the above result simplifies to tR ¼ 4. Photon exchange 0 2 a a 2 2 The bino’s interactions with charged scalars also generate g g g m mχ ð1Þbare ð Þ u q t 2 1 − 2 2 2 cq ¼ 2 2 2 log 2 þ ð mχ=mt Þ an effective coupling to the electromagnetic current 12π mχm mt EM ν Z Jμ ≡∂ Fνμ, where Fμν is the photon field strength. The 2 2 −1=2 −1 2 2 1=2 relevant Feynman diagrams are shown in Fig. 17.A × ð4mχ=m − 1Þ tan ð4mχ=m − 1Þ : t t Majorana fermion may couple to the photon via the anapole operator, defined in Eq. (31) of Sec. IV D. ðD15Þ The relevant interactions are parametrized as For a right-handed sbottom, we find L ⊃ ~ ¯ α βγ5 χ − μ ~ †∂ ~ UV ½f f ð þ Þ ieQfA f μf þ H:c: ð1Þbare ¯ μ cq ¼ 0: ðD16Þ − eQfAμfγ f; ðD17Þ

This can be understood in the following manner. We have set where e is the electromagnetic coupling constant and Qf is the bottom quark mass to zero when matching at the weak the electric charge of f (in units of e). Note that

095008-24 BINO VARIATIONS: EFFECTIVE FIELD THEORY … PHYSICAL REVIEW D 93, 095008 (2016)

FIG. 18. Box-type Feynman diagrams responsible for one-loop couplings to quarks in 5-flavor QCD. Crossed diagrams not FIG. 17. Feynman diagrams responsible for couplings to the shown. Here, G is the charged Goldstone. electromagnetic current. Charge-reversed diagrams not shown.

∂μ EM 0 Z Z conservation of the electromagnetic current ( Jμ ¼ ) X − αβ 1 1−x1 bare eQf implies that the photon cannot couple sfermions of different cA ¼ 2 dx1 dx2 8π 0 0 masses. Therefore, left-right sfermion mixing does not i affect the form of these interactions. In the limit that the x2ð2x2 þ x1 − 2Þ ð2x2 − 1Þð2x2 þ x1 − 1Þ × þ ~ ; fermion mass, mf, is much larger than thepffiffiffiffiffiffiffiffi typical momen- Δ 2Δ − 2 ∼ 50 tum transfer of scattering events ( q MeV), ðD21Þ we find with Z X 1 −nceQfαβ 3x − 2 bare 2 ≫− 2 2 2 2 cA ¼ 2 dx ðmf q Þ; Δ ≡ − 1 1 − 48π 0 Δ x1ðx1 Þmχ þ x1m~ þð x1Þmf i i f 2 ðD18Þ þ x2ðx1 þ x2 − 1Þq ; ~ 2 2 2 Δ ≡ x1ðx1 − 1Þmχ þð1 − x1Þm~ þ x1m ~ f f where nc is the number of colors of f, and 2 þ x2ðx1 þ x2 − 1Þq : ðD22Þ 2 2 2 Δ ≡ xðx − 1Þmχ þ xm þð1 − xÞm : ðD19Þ f~ f This contribution has already been presented in particular 5. Box diagrams limits. For example, in the limit that m , mχ ≪ m~ and f f In Sec. III B, box diagrams involving gauge and α −β λ 2 ¼ ¼ = , the above form reduces to Goldstone bosons contribute to one-loop couplings to bottom quarks. The relevant diagrams are shown in n eQ λ2 m2 cbare ¼ c f log f ; ðD20Þ Fig. 18. We parametrize the bino-stop interactions as A 96π2m2 m2 f~ f~ L ⊃ ~ ¯ α βγ5 χ which agrees with Ref. [46]. UV t t ð þ Þ þ H:c: ðD23Þ 2 ≲ − 2 However, in the limit that mf q , the light fermion cannot be integrated out, and instead of doing a Similar to the tree-level calculations of Appendix D1, simple matching to the local anapole operator, we keep obtaining the Wilson coefficients from the diagrams in 2 bare bare the full q dependence in cA . In this case, cA takes a Fig. 18 requires the application of Fierz transformations. more general form, Working in Feynman gauge, we find

2 Z Z 1 1−x1 ð0Þbare gw −ðα þ βÞ c dx1 dx2x2 1 − x1 − x2 3x1mχ α β 4m α − β b ¼ 16π2 ð Þ 2Δ2 ½ ð þ Þþ tð Þ 0 0 x1mχ α β α − β 2 þ Δ3 ½x1mχð þ Þþmtð Þ 2 Z Z λ 1 1−x1 − α − β G ð Þ dx1 dx2x2 1 − x1 − x2 3x1mχ α − β 4m α β þ 32π2 ð Þ 2Δ2 ½ ð Þþ tð þ Þ 0 0 x1mχ 2 x1mχ α − β m α β ; þ Δ3 ½ ð Þþ tð þ Þ ðD24Þ

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2 cð Þbare b Z Z 2 1 1− gwmχ x1 dx1 dx2x1x2 1 − x1 − x2 ¼ 4π2 ð Þ 0 0 1 1 α β 2 α β α − β 2 × 2Δ2 ð þ Þ þ Δ3 ½x1mχð þ Þþmtð Þ 2 Z Z λ 1 1−x1 Gmχ dx1 dx2x1x2 1 − x1 − x2 þ 8π2 ð Þ 0 0 1 2 1 2 α − β x1mχ α − β m α β ; × 2Δ2 ð Þ þ Δ3 ½ ð Þþ tð þ Þ

ðD25Þ FIG. 19. Full theory Feynman diagrams responsible for one- pffiffiffi loop couplings to gluons. Charge-reversed diagrams not shown. where gw ¼ −g=2 2, λG ¼ mt=2v, and we have defined

Δ ≡ − 1 2 2 2 1 − − 2 6. Gluon couplings in the full theory x1ðx1 Þmχ þ x1m~ þ x2mW þð x1 x2Þmt : t In addition to coupling to quarks, χ may scatter off ðD26Þ gluons at the one-loop level. The set of Feynman diagrams for this process is shown in Fig. 19. This calculation is mχ m~ In the case of a degenerate right-handed stop ( ¼ tR ), simplified by working with the Fock-Schwinger gauge in a the above form simplifies to background gluon field, where gauge invariance is made manifest by expressing the gluon field directly in terms 2 2 2 Z Z 0 1 1−x1 ð0Þbare g ðg Þ mt mχ of the field strength. However, this simplification comes at c ¼ dx1 dx2x1x2ðx1 þ x2 − 1Þ b 48π2 the cost of breaking translational invariance, and, as a 0 0 2 2 result, different forms of colored propagators are needed 1 1 x1mχ × − 1 þ ; for the nonreversed and charge-reversed diagrams. We refer 4Δ2m2 3Δ3 2m2 W Z W the reader to Refs. [12,22,65] for detailed discussions. We − 2 0 2 2 1 ð2Þbare g ðg Þ mt mχ have cross-checked our results by computing also in cb ¼ 2 dx1 36π 0 Feynman gauge. In this section, IR poles are regulated Z 4 − 2ϵ 1−x1 with dimensional regularization in d ¼ dimensions. × dx2x1x2ðx1 þ x2 − 1Þ Parametrizing the squark-bino interaction Lagrangian as 0 2 2 5 1 1 x1mχ L ⊃ q~ q¯ ðα þ β γ Þχ þ H:c:; ðD28Þ 1 ; UV q q × 4Δ2 2 þ Δ3 þ 2 2 mW mW 2 2 2 2 we find that the gluon Wilson coefficients corresponding to Δ ≡ x mχ þ x2m þð1 − x1 − x2Þm : ðD27Þ 1 W t each diagram (in Fock-Schwinger gauge) are given by

ð0Þbare ð2Þbare ð0Þbare ð2Þbare cg ðS1Þ¼cg ðS1Þ¼cg ðSFÞ¼cg ðSFÞ¼0; ðD29Þ

8 Z −α 2 1 3 > sm~ x − > q λðþÞ λð Þ 0 < 48π dx Δ3 ½x q mχ þ q mqðmq~ >mχ or mq > Þ ð0Þbare 0 cg ðS2Þ¼ ; > 5α λðþÞ :> s q 0 3 ðmq~ ¼ mχ and mq ¼ Þ 384π mχ 8 Z α 1 1 − 3 1 > s ð xÞx λðþÞ > dx 2 q mχ > 12π 0 Δ 2 <> 2 ð Þbare 1− − 2 cg ðS2Þ¼ x λðþÞ λð Þ 0 ; > þ Δ ðx q mχ þ q mqÞmχ ðmq~ >mχ or mq > Þ > > : λðþÞ 2 αs q 1 μ 3 ð þ log 2 þ 1Þðm~ ¼ mχ and m ¼ 0Þ 16π mχ ϵIR mχ q q

095008-26 BINO VARIATIONS: EFFECTIVE FIELD THEORY … PHYSICAL REVIEW D 93, 095008 (2016) 8 α R −1 2 − > s 1 ðx Þ λð Þ 3 − 2 3 2λðþÞ 3 − 1 2 > 128π 0 dx Δ2 f q mqðxð x Þþ Þþ q mχð x Þx > > 1 2 λð−Þ − 1 2 3 1 3 2 1 − 3 2 <> þ Δ ½3 q mqðx Þðmqð x þ Þþ mχð xÞx Þ 0 cð ÞbareðFÞ¼ − 1 λðþÞ − 1 4 2 2 6 − 1 − 2 9 7 ; ðD30Þ g > 3 q mχðx Þxð mχx ð x Þ mqð x þ ÞÞ > > 3 2 2 2 2 2 2 ðþÞ ð−Þ > þ 2 ðx − 1Þ x ðx mχ − m Þmχ½xλ mχ þ λ m g ðm > 0Þ :> Δ q q q q q 0 ðmq ¼ 0Þ 8 > α R −1 3 > smχ 1 ðx Þ x −λðþÞ 4 1 λðþÞ 2 2 3 − 1 > 16π 0 dx Δ2 f q ðx þ 3ÞþΔ ½ q ðmχx ð x Þ > > 1 2 2 ð−Þ > − m ð3x þ 5ÞÞ þ λ m mχxð3x − 2Þ > 6 q 3 q q > > 3 − 1 2 − 2 2 λðþÞ λð−Þ 0 > þ 2Δ2 ðx Þxðmq x mχÞmχ½x q mχ þ q mqg ðmq~ >mχ and mq > Þ > n h > ðþÞ 2 < −αsλq mχ 1 μ 3 2 2 2 2 þ log 2 2 þ ð Þbare 12π ðm −mχ Þ ϵ m −mχ 2 cg ðFÞ¼ q~ IR  q~ i ; ðD31Þ > 2 > ∂ − ∂ − ∂ 4 0; 4; mχ > þð 1 2 3Þ2F1 ; 2− 2 > mχ mq~ > o > 2 2 2 2 2 2 > ð mq~ þmχ Þ mq~ mq~ > − 5 log 2 2 þ 3 2 2 ðm~ >mχ and m ¼ 0Þ > mχ m −mχ mχ ðm −mχ Þ q q > q~ q~ > > −α λðþÞ 2 > s q 1 μ 1 : 3 ð þ log 2 − Þðm~ ¼ mχ and m ¼ 0Þ 8πmχ ϵIR mχ 2 q q

ðÞ where λq and Δ are defined to be

λðÞ ≡ α2 β2 Δ ≡ − 1 2 2 1 − 2 q q q; xðx Þmχ þ xmq~ þð xÞmq; ðD32Þ and ∂ið2F1Þ corresponds to differentiation of the hypergeometric function 2F1ða; b; c; dÞ in its ith argument. The above ð0Þ expressions for cg agree with the results presented in Ref. [12]. mχ m~ In the case of a degenerate right-handed stop ( ¼ tR ), the total contribution from the above expressions is 0 2 2 2 2 0 α ðg Þ mχ 3 m 2mχ mχ ð Þbare s − t − − 4 2 2 − 1 −1=2 −1 4 2 2 − 1 1=2 cg ¼ 2 2 2 2 2 2 ð mχ=mt Þ tan ð mχ=mt Þ ; 18πð4mχ − m Þ 4 12mχ 3m m t t t 2 2 2 4 6 α 0 1 mχ mχ 7mχ 8mχ ð2Þbare sðg Þ − 4 2 2 − 1 −2 cg ¼ 3 2 log 2 þ 2 4 þ 6 ð mχ=mt Þ 18πmχ mt mt mt mt 2 4 6 10mχ 92mχ 68mχ 1 − − 4 2 2 − 1 −5=2 −1 4 2 2 − 1 1=2 þ 2 þ 4 6 ð mχ=mt Þ tan ð mχ=mt Þ : ðD33Þ mt 3mt 3mt

Similarly, for a nondegenerate right-handed sbottom (mχ ≠ m~ ), we find bR

0 2 ð0Þbare −αsðg Þ mχ cg ¼ 2 2 2 ; 864π m~ ðm~ − mχÞ bR bR 2 4 2 − 3 2 4 2 − 2 2 2 2 2 −α ðg0Þ m~ mχ m~ mχ m~ α ðg0Þ mχ 1 μ ð Þbare s bR bR bR − s cg ¼ 3 2 2 þ 2 log 2 2 108π 2 2 2 ϵ þ log 2 2 216πmχ mχ − m~ mχ m~ − mχ ðm~ − mχÞ IR m~ − mχ bR bR bR bR 2 2 2 2 2 2 2 3 mχ ð m~ þ mχÞ m~ m~ ∂ − ∂ − ∂ 4 0; 4; − bR bR bR þ þð 1 2 3Þ2F1 ; 2 2 5 log 2 2 þ 3 2 2 : ðD34Þ 2 mχ − m~ mχ m~ − mχ mχðm~ − mχÞ bR bR bR

ð2Þbare Note that cg has an IR divergence in the full theory, which arises as a singularity in the integration over Feynman parameters. Upon performing weak scale matching, this is identified as an UV pole of the low-energy theory that is

095008-27 BERLIN, ROBERTSON, SOLON, and ZUREK PHYSICAL REVIEW D 93, 095008 (2016)

ð2Þbare renormalized according to Eq. (13). The contribution from cq , using Eq. (D3), cancels the divergence precisely, yielding the finite renormalized coefficient

2 4 2 − 3 2 4 2 − 2 2 2 2 2 −α ðg0Þ m~ mχ m~ mχ m~ α ðg0Þ mχ μ 3 ð Þ s bR bR bR − s cg ¼ 3 2 2 þ 2 log 2 2 2 2 2 log 2 2 þ 216πmχ mχ − m~ mχ m~ − mχ 108π ðm~ − mχÞ m~ − mχ 2 bR bR bR bR 2 2 2 2 2 2 2 mχ ð m~ þ mχÞ m~ m~ ∂ − ∂ − ∂ 4 0; 4; − bR bR bR þð 1 2 3Þ2F1 ; 2 2 5 log 2 2 þ 3 2 2 ; ðD35Þ mχ − m~ mχ m~ − mχ mχðm~ − mχÞ bR bR bR as explained in Eq. (19). iðp þ m Þ ð0Þ ≡ q iS ðpÞ 2 2 : ðD38Þ p − mq 7. Gluon couplings in heavy particle theory A A In Sec. III D, we discussed the modified matching The tensor GαμGβν can be projected onto the scalar and prescription when dealing with a light quark and nearly spin-2 gluon currents defined in Eq. (3) as1 degenerate squark (m , δ~ ≪ m ). In this section, we will q q t 1 present the gluon calculation using the Fock-Schwinger A A ð0Þ GαμGβν ¼ ðgαβgμν − gανgβμÞOg gauge in the framework of heavy particle theory. We have dðd − 1Þ checked our results in the Feynman gauge. The relevant 1 ð2Þ ð2Þ þ ð−gαβO μν − gμνO leading order Feynman diagram is shown in Fig. 20.Twoof d − 2 g gαβ the four possible gluon diagrams vanish exactly in Fock- ð2Þ ð2Þ Schwinger gauge as in Appendix D6. These are the heavy þ gανOgβμ þ gβμOgανÞþ; ðD39Þ particle theory equivalents of diagrams “S1” and “SF” in Fig. 19. Furthermore, the diagram involving the 4-point where the ellipsis denotes higher spin tensor con- q~ − q~ − g − g vertex is found to be subleading in 1=mχ, and tributions. therefore, only the diagram of Fig. 20 contributes at leading Using the top line of Eq. (D39) in Eq. (D37) leads to order. Parametrizing the heavy particle Lagrangian as Z − 2 πα d λðþÞ λð Þ 2 ð0Þbare i smq d l q mql þ q l 1 cg ¼ d 2 2 4 ; L ⊃ ~ ¯ α β γ5 χ dmχ ð2πÞ ðl − m Þ ðv · l þ δ~ Þ HPT pffiffiffiffiffiffi qvqð q þ q Þ v þ H:c:; ðD36Þ q q mχ ðD40Þ the diagram in Fig. 20 is written in the Fock-Schwinger ðÞ 2 2 ð2Þ where λ ≡ ðα β Þ. The coefficient c is similarly gauge as q q q g evaluated by using the bottom line of Eq. (D39) in Z −πα d ∂ ∂ Eq. (D37). We find s A A d l i iM ¼ GαμGβν 2 2π d − − δ ∂ ∂ Z − mχ ð Þ v · l q~ k1μ k2ν α m ∞ λð Þ ð0Þbare s q q − x 2λðþÞ λð−Þ ¯ α − β γ5 ð0Þ γα ð0Þ − γβ cg ¼ dx 2 3 ð q mq þ q xÞ ; × ½uðkÞð q q ÞiS ðlÞ iS ðl k1Þ 96πmχ 0 Δ 4Δ 0 5 ð Þ − − α β γ Z ðþÞ × iS ðl k1 k2Þð q þ q ÞuðkÞjk1 2 0; ðD37Þ α ∞ λ ; ¼ ð2Þbare s q x x 10λðþÞ 2 cg ¼ dx 2 þ 3 ð q mq 16πmχ 0 3Δ 48Δ where k is the residual bino momentum (p ¼ mχv þ k), ðþÞ 2 ð−Þ and we have defined the free quark propagator þ 3λq x þ 8λq m xÞ q 2 3x − 4m2 − x2 2λð Þm λðþÞx ; þ 128Δ4 ð q Þð q q þ q Þ ðD41Þ

where x is a dimensionful Schwinger parameter and 1 2 2 Δ ≡ 4 x þ δq~ x þ mq. FIG. 20. Leading order Feynman diagram responsible for one- loop couplings to gluons in heavy particle theory. Single (double) 1Note that the last two terms of the second line of Eq. (D39) lines correspond to relativistic (heavy particle theory) fields. differ by a sign from those of Eq. (50) in [12].

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