The Bino Variations: Effective Field Theory Methods for Dark Matter Direct Detection
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 two 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.
- I Introduction
- II Fixed Order Approach to WIMP-Nucleon Scattering
- III Effective Theory Approach to WIMP-Nucleon Scattering
- IV Phenomenology
- V Conclusions
- VI Appendix
- A Model
- B Hadronic Inputs
- C Running and Matching Matrices
- D Collection of Wilson Coefficients
Decades of technological advances and increased detector sizes have led to impressive projected sensitivities of on-going and future dark matter (DM) direct detection experiments Cushman:2013zza (); Akerib:2012ys (); Akerib:2015cja (); Aprile:2011dd (). For DM with mass GeV, the LUX-ZEPLIN (LZ) experiment is projected to reach cross sections as small as , tantalizingly close to the neutrino background, residing at cross sections an order of magnitude smaller. As these experiments extend their reach, they will push through a number of important benchmarks in the hunt for Weakly Interacting Massive Particles (WIMPs).
Current experiments are in fact already probing rates several orders of magnitude below “weak-scale” cross sections: constraints from LUX and Xenon100 reach as low as , while a simple estimate suggests that the spin-independent (SI) scattering cross section through the -boson is . The scattering of a WIMP on nucleon targets, however, depends strongly on its identity. While a scalar electroweak doublet has a large cross section through the -boson, Majorana fermions have no vector coupling, and the axial-vector interactions are either -suppressed or lead to spin-dependent (SD) scattering.
At tree-level, this leaves scattering through the Higgs boson as the process for leading SI interactions. For neutralinos, the size of the scattering through the Higgs boson depends on its electroweak composition. Triplet (“wino”), doublet (“higgsino”), and singlet (“bino”) states mix with each other, allowing the lightest stable neutral WIMP, , to couple to the Higgs at tree-level: . This gives rise to a typical scattering cross section . Thus, the currently running and next generation ton-scale experiments are probing tree-level “Higgs-interacting” massive particles.
Pure electroweak states (wino, Higgsino, or bino), however, do not couple to the Higgs at tree-level. For these cases, the evaluation of direct scattering of the lightest electrically-neutral state on nucleon targets requires the analysis of loop amplitudes at leading order. Assuming weak-scale mediators, a simple estimate of the scattering cross section is given by , where is the nucleon mass and GeV. The prospects for wino and Higgsino dark matter, however, are challenged by an accidental cancellation between amplitudes, leading to cross sections smaller by a few orders of magnitude Hisano:2011cs (); Hill:2011be (); Hill:2013hoa (); Hisano:2015rsa (). For the wino, the cross section was found to be , while for the Higgsino, the cancellation gives rise to an unreachably small scattering cross section. Nonetheless, it is remarkable that in some cases, while the tree-level cross section may be absent, ton-scale direct detection experiments are becoming sensitive to one-loop interactions.
Similar to the wino and Higgsino, bino scattering through the Higgs boson vanishes at tree-level. If heavy flavor squarks or sleptons are nearby in the spectrum, however, loop processes are induced. In this case, prospects for detection are improved through direct coupling to colored scalars. The interplay of a number of effects, such as power suppression if the new states are heavy compared to the electroweak scale, enhancement from on-shell-poles, and sizable mixing between colored scalars, could impact this. We assume that light flavor squarks and the Higgsino are decoupled from the low-energy spectrum since tree-level amplitudes would otherwise dominate over loops. To quantify the degree to which these must be decoupled, we show in Fig. 1 the SI cross section as a function of the Higgsino mass and the sdown mass , when the lightest supersymmetric particle (LSP) is a bino-like neutralino that interacts with the Standard Model (SM) Higgs and a right-handed down-squark (). Sufficient decoupling occurs when the leading order scattering rate in Fig. 1 drops below cm .
Processes relevant for one-loop bino scattering cross sections and related simplified models have already been considered in the literature Djouadi:2001kba (); Drees:1993bu (); Djouadi:2000ck (); Hisano:2010ct (); Hisano:2011um (); Gondolo:2013wwa (); Hisano:2015bma (); Berlin:2015ymu (); Ibarra:2015fqa (); Chang:2013oia (); Chang:2014tea (); Garny:2015wea (). At the same time, a great deal of effective field theory (EFT) machinery has recently been developed for systematically integrating out heavy particle thresholds and running Wilson coefficients to the low scales characteristic of the processes in direct detection experiments Hill:2014yka (); Hill:2014yxa (); D'Eramo:2014aba (). Our aim is to apply these techniques, focusing on QCD effects, to the case of bino DM where the SM is extended with a Majorana gauge singlet, and a few sfermions with the same quantum numbers as either left- or right-handed quarks or leptons.
We capture a number of effects that have been previously neglected. First, we are able to systematically incorporate the multiple scales involved in direct scattering, accounting for potentially large contributions, . Second, we are able to include additional states at low energies, beyond those of -flavor QCD. For example, when the mass difference between the bino and sbottom is much less than the weak scale, both are active degrees of freedom at low energies, and we use heavy particle techniques to describe their interactions with soft bottom quarks. Third, we are able to assess the uncertainties from both higher-order perturbative corrections and hadronic inputs.
In addition to incorporating renormalization group evolution (RGE), we also go beyond previous fixed-order computations that have focused on the parameter space for either purely left- or right-handed sfermions. We explore a larger part of the minimal supersymmetric standard model (MSSM) parameter space by considering the impact of mixing between left- and right-handed third generation squarks. We also perform a complete leading order matching at the weak scale, considering contributions such as the spin-2 gluon operator (significant when a sbottom is close in mass to the bino), and the anapole operator from photon exchange.
While we adopt the nomenclature and explicit couplings of the MSSM for definiteness, key components of our analysis, such as the results for loop amplitudes and RGE solutions, are generic, and can be readily applied to investigate the phenomenology of other models that incorporate interactions of DM with scalars charged under the SM. For example, many of the effects considered here may also be applied to the case of suppressed tree-level scattering (“blind-spots”), where loop corrections are necessary to meaningfully compare theory and experiment Cheung:2012qy (); Cheung:2013dua (); Huang:2014xua (); Crivellin:2015bva ().
The remainder of the paper is structured as follows. In Sec. II, we review the standard fixed-order approach in the literature for determining amplitudes for WIMP-nucleon scattering. This lays the groundwork for the effective theory framework described in Sec. III. There we discuss the factorization of the scattering amplitude into contributions from the relevant physical scales, and illustrate the techniques for matching, renormalization, and coefficient evolution by presenting three detailed examples of increasing intricacy: a bino coupled to (i) a right-handed stop, (ii) a heavier right-handed sbottom, and (iii) a nearly mass degenerate right-handed sbottom. The reader interested in the phenomenological results may go straight to Sec. IV, where we evaluate cross sections for models with stop, sbottom, and slepton mediators. The most promising case for detection is a bino interacting with a nearly degenerate right-handed sbottom: a bino as heavy as 10 TeV may be detected at LZ if the mass splitting is a few GeV. On the other hand, a bino nearly degenerate with a right-handed stop is only detectable above the neutrino background for masses below about GeV.
We collect the technical results in the appendices. In Appendix A, we set up our conventions for the sfermion mass matrices, as well as the DM-fermion-sfermion interactions. Appendices B and C contain the hadronic form factors and the running and matching matrices employed in our numerical analysis. In Appendix D, we present details of the Wilson coefficients for all relevant amplitudes, such as tree-level sbottom exchange, one-loop Higgs, , and exchange, one-loop diagrams involving charged electroweak gauge bosons, and one-loop contributions to the 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 results differ from previous literature.
Ii Fixed Order Approach to WIMP-Nucleon Scattering
Amplitudes for WIMP-nucleon scattering involve energy scales that span several orders of magnitude, ranging from the masses of the new particles and the mediating SM particles (), to the scales of heavy quark thresholds and of hadronic physics (), and the typical momentum transfers relevant for direct detection (). A standard approach in the DM literature is to determine these amplitudes at “fixed order,” treating this broad range of physical scales at a single scale. In this section, we review this matching procedure between the full theory of the SM and its extension, specified at high energies , and an EFT for WIMP-nucleon scattering, specified at low energies .
At high energies, , the basic interaction that we consider is of a single sfermion () with a bino LSP () and a SM fermion (), adopting the following notation:
The couplings are parametrized in terms of the SM hypercharge coupling and the sfermion mixing angles of Eqs. (35) and (A). To simplify the discussion in this section and the next, we illustrate general methods for the case where constitutes a single right-handed stop or sbottom and the corresponding top or bottom quark, and assuming the theory in Eq. (1) is defined at the weak scale . The impact (from RGE) of considering couplings defined at an even higher scale is illustrated in Sec. IV.2. Examples pertaining to mixed stops and sbottoms, and sleptons, are treated in a similar way, and we discuss them in Secs. IV.3 and IV.4.
The hadronic matrix elements necessary for describing WIMP-nucleon scattering are determined, e.g., from lattice measurements, at low energies , in a theory with three quark flavors. At these energies, an effective theory captures the interactions of the WIMP with the degrees of freedom of 3-flavor QCD. For the bino, a gauge-singlet Majorana fermion, a set of operators for low-velocity scattering is
where the relevant QCD currents are
with the gluon field strength and the spacetime dimensions. We adopt the notation and , and have neglected operators that lead to kinematically suppressed contributions. Leading order SI scattering is given by the scalar () and spin-2 () quark and gluon currents, while leading order SD scattering is given by the quark axial current (). We neglect the operator involving the quark vector current, which leads to SI scattering that is power-enhanced relative to the scalar and spin-2 contributions, but is velocity suppressed. We have reduced the operators to a linearly independent set; e.g., the operators and are redundant in the forward scattering limit. We ignore flavor non-diagonal operators, whose nucleon matrix elements have an additional weak-scale suppression relative to those considered. We will not be concerned here with operators involving leptons.
In the standard fixed-order approach, the full theory in Eq. (1) is matched onto the effective theory in Eq. (II), by integrating out the sfermion , the gauge bosons , the Higgs , the Goldstones , and the heavy quarks , altogether at a single scale. The matching condition for the case of a right-handed stop or sbottom (denoted as ) is shown in Fig. 2. The leading contributions to the quark and gluon coefficients are at and , respectively.
Once the Wilson coefficients are determined, the hadronic matrix elements are evaluated. We adopt the definitions and values from Sec. 4 of Ref. Hill:2014yxa () for the hadronic matrix elements of the QCD currents in Eq. (II). For completeness, we collect their definitions here:
where for proton or neutron, for quark or gluon, and the spin vector satisfies and , assuming non-relativistic normalization for the spinor .
The axial form factors, , are extracted from hyperon semileptonic decay, from scattering, or from observables of polarized deep inelastic scattering. The scalar quark form factors, , are extracted from lattice measurements, while the scalar gluon form factor is obtained through the leading order relation Shifman:1978zn ()
The quark and gluon spin-2 form factors, , are extracted from the second moment of parton distribution functions (PDFs). In Appendix B, we collect the values employed in our numerical analysis.
These nucleon matrix elements, together with the Wilson coefficients, define the SI and SD amplitudes
and, finally, the cross sections for SI and SD scattering on a nucleon target are obtained,
This is a straightforward strategy for determining WIMP-nucleon scattering cross sections, with, however, limitations that motivate a more thorough analysis. First, there are potentially large perturbative corrections, , inherent in treating a multiscale process at a single scale. For example, while the Wilson coefficients are determined at the weak scale employing , the leading order scalar gluon form factor in Eq. (5) is subject to sizable corrections due to the large size of . Second, determining higher order corrections in a fixed-order framework is difficult; e.g., at NLO two- or three-loop amplitudes are required. Theoretical control of perturbative corrections would allow us to estimate their numerical impact, and, in the event of a detection, to systematically improve predictions for WIMP-nucleon scattering. In the next section, we lay out the effective theory framework to deal with these issues head on.
Iii Effective Theory Approach to WIMP-Nucleon Scattering
As mentioned in the previous section, WIMP-nucleon scattering involves a multitude of physical scales, and the separation between the weak scale, , and the hadronic scale, , may lead to large uncertainties when employing the fixed-order framework. In this section, we discuss the “effective theory” approach, which factorizes the scattering amplitudes into contributions from different physical scales by constructing a sequence of EFTs from the weak scale down to the hadronic scale, and connecting them through RGE and matching. This allows for the separate analysis of perturbative corrections at each energy threshold and for the resummation of large logarithms, e.g., .
This framework is depicted in Fig. 3. To further elaborate on its general features, let us present the corresponding factorized amplitude, and briefly discuss its components in turn; a more detailed discussion is given in the subsections below. In the EFT approach, the scattering amplitude is determined as
where the renormalization scales , , , and correspond respectively to the weak scale , the bottom quark threshold , the charm quark threshold , and the hadronic scale , where nucleon matrix elements are defined. The vector collects the Wilson coefficients determined at the scale by integrating out weak scale degrees of freedom, and matching onto a theory with five quark flavors. The matrix implements coefficient running from down to , while the matrix implements coefficient matching across the bottom quark threshold, between the theory with five and four quark flavors. The matrices and are analogously defined, implementing running in 4-flavor QCD and matching across the charm quark threshold. Finally, the coefficients are run down to the hadronic scale in 3-flavor QCD, using , and the matrix elements are evaluated through multiplication of the (transposed) vector , which collects the form factors defined in Eq. (II).
Clearly, Eq. (8) has separation of scales, with components , , , and depending only on scales of a similar order. The logarithms in the amplitude are resummed through the RGE factors , and additional perturbative corrections to each component can be separately and systematically analyzed without having to evaluate the whole amplitude at higher loop order. Note that does not constitute a large logarithm, and hence integrating out the bottom and charm quarks at a single scale would suffice. Nonetheless, since is sizable, higher-order corrections may have significant impact, and we may conveniently employ known results for the matrices , , and to include them.
Note also that the PDFs relevant for the spin-2 matrix elements defined in Eq. (II) are available at a high-scale, e.g., GeV, and thus allows us to evaluate the amplitude without running down these Wilson coefficients to a low-scale. The running, however, would be relevant for relating the spin-2 current to low-energy effective DM-nucleon contact operators (see e.g., Refs. Fan:2010gt (); Fitzpatrick:2012ix ()), and for including the impact of multi-nucleon effects (see e.g., Refs. Prezeau:2003sv (); Cirigliano:2012pq ()). In the present analysis, we RG evolve all Wilson coefficients as a default, but have checked that our results are consistent, up to uncertainties, with an evaluation at the high scale. We find that the additional perturbative uncertainty from running the spin-2 coefficients increases the overall uncertainty by less than 10.
The factorization in Eq. (8) is a general result of our effective theory analysis, and in the following subsections we provide further details on each of its components. Section III.1 considers formalism for representing the relevant degrees of freedom in the low energy theory, and for matching at the weak scale . In Secs. III.2, III.3, and III.4, we go into explicit detail by applying the effective theory framework to three examples, classified according to the mass, , of the fermion partnered to the sfermion, and the mass splitting, , between the sfermion and bino. Case I considers and arbitrary , case II considers , and case III considers . These examples illustrate, in increasing complexity, the key ingredients of the effective theory framework. Case I goes through the basic computational pipeline involving the components , , , and of Eq. (8). Case II presents an example where nontrivial renormalization of the bare coefficients arises. Finally, for case III, a heavy sfermion field (denoted as in Fig. 3) is included in the low-energy theory to account for sfermion-bino interactions.
iii.1 Integrating out the Mass but Not the Particle
A key step in the effective theory approach involves integrating out weak scale degrees of freedom by matching onto a low energy theory of the bino and the quarks and gluons of 5-flavor QCD. In this procedure, the gauge, Higgs, and Goldstone bosons, as well as the stop and top, are integrated out. However, the bino, despite having a weak scale mass, , is not integrated out – the goal of calculating a WIMP-nucleon scattering cross section requires that it is kept in the low energy theory. Moreover, the same applies to a sbottom whose mass is close to that of the bino: despite , the sbottom should not be integrated out since the bottom quark is an active degree of freedom in the low energy theory and bino-sbottom interactions are thus allowed.
How do we integrate out the mass of a field without integrating out the field itself? The idea is simple and can be pictured by considering the following parametrization of the bino momentum at low energies: , where is a reference time-like unit vector and . The interactions of the heavy bino with the much lighter quarks and gluons of 5-flavor QCD involve only soft momenta of , while the large momentum component , corresponding to its mass, plays no role and can be integrated out. This procedure is formally done by going from a relativistic description of the field to a “heavy particle” description, order-by-order in the small parameter . The technique is called “heavy particle effective theory,” and is known from applications for heavy quark physics (for a review see, e.g., Ref. Manohar:2000dt ()).
We may pass from a relativistic to a heavy particle description for the bino (Majorana fermion) by making the field redefinition
where the spinors obey and . In terms of the momentum decomposition discussed above, the phase extracts the large momentum component . Upon introducing this field redefinition into the kinetic term , we find that the component has mass , and is thus integrated out, e.g., at tree-level by solving its equation of motion. The remaining component describes the heavy bino degree of freedom with the (canonically normalized) kinetic term , depending only on the soft momentum . The Majorana condition allows us to write the field redefinition (9) alternatively as
where charge conjugation is denoted by with the unitary and symmetric matrix obeying . This implies an invariance of the heavy particle Lagrangian for under the simultaneous transformations Kopp:2011gg (); Heinonen:2012km ()
Instead of introducing the field redefinition (9) into a basis of relativistic operators, we may also proceed in the spirit of effective theory, employing building blocks to directly write down low energy operators consistent with symmetries. For our low-energy theory, the building blocks are the usual relativistic degrees of freedom (quarks and gluons), the reference vector , and the heavy bino field . Thus, for a Majorana dark matter particle whose mass satisfies , the basis of operators describing its interactions with 5-flavor QCD is
where the ellipsis denotes higher dimension operators, and the relevant QCD currents are given in Eq. (II). Here, we have subtracted off the component of which vanishes between the heavy particle bilinear, defining . Alternatively, Eq. (III.1) is obtained by making the substitution (9) into the basis of operators in Eq. (II). We have introduced a conventional factor of on the left-hand side of Eq. (III.1) since the field redefinition (9) would otherwise lead to a factor of 2 discrepancy between the coefficients in Eqs. (II) and (III.1).
In the relativistic basis of Eq. (2), and are treated on equal footing, despite corresponding to operators whose mass dimensions differ by two, i.e., seven and nine, respectively. As a result, power counting is possible but not manifest (leading order SI scattering involves operators of dimension seven and nine), and it is less straightforward how the basis extends beyond leading order. In contrast, power counting is manifest in Eq. (III.1), and thus the operators relevant at each order are known without having first to evaluate the full theory amplitudes. In particular, leading order low-velocity SI (SD) scattering is obtained from dimension seven (six) operators, and subleading corrections can be systematically computed. In the remainder of the paper, when referring to Wilson coefficients, we assume the form given in Eq. (III.1).
Having discussed the formalism for incorporating both relativistic and heavy particle degrees of freedom at low energies, let us now turn to the computation of weak scale coefficients of Eq. (8). At the scale , we match the full relativistic theory of Eq. (1), with six quark flavors and a relativistic bino , onto the low energy theory of Eq. (III.1), with five quark flavors and a heavy particle bino . The full theory diagrams are computed using standard relativistic Feynman rules, while the effective theory diagrams are computed using the Feynman rules of Eq. (III.1).
This matching procedure determines the bare Wilson coefficients, and may involve loop contributions from the low energy effective theory. It is simplest to compute the full theory amplitudes setting all mass scales much lighter than the weak scale to zero, and regulating infrared divergences in dimensions. The weak scale coefficients then depend only on the weak scale masses , , , , , and , and are determined up to corrections of . Of course, for matching a full theory amplitude onto the scalar quark current of Eq. (II), the leading factor should be retained. In dimensional regularization, the loop integration measure has scaling dimension [mass] , and therefore any loop integral is dimensionful. A loop integral that has no mass scale to soak up this dimensionality must vanish by consistency. This is the well-known statement that scaleless integrals vanish in dimensional regularization. With light quark masses set to zero, the effective theory loop contributions are scaleless, and hence vanish. Alternatively, keeping light quark masses nonzero would regulate infrared divergences, but would require the computation of non-vanishing effective theory loop amplitudes. An explicit example involving such effective theory loop contributions will be presented in Sec. III.3.
The remaining poles in the bare coefficients are UV divergences of the low energy theory, and are renormalized accordingly. For a detailed discussion on the renormalization of the QCD currents in Eq. (II), we refer the reader to Sec. 3 of Ref. Hill:2014yxa (). Here, we will simply quote the results. At leading order in , the scalar and axial-vector coefficients are trivially renormalized, i.e., , while the spin-2 coefficients are renormalized as
where the sum runs over the active quark flavors, i.e., in 5-flavor QCD. The terms of the coefficients introduce a pole in that is cancelled by the pole in . Note that the nontrivial renormalization also requires the terms of the coefficients . We will see an explicit example of this renormalization in Sec. III.3 when is divergent due to gluons emitted from massless quarks.
As mentioned above, a sfermion that is nearly degenerate in mass with the bino should be a degree of freedom in the low energy theory if sfermion-bino interactions with light fermions are present. Hence, only the sfermion mass is integrated out (encoded in Wilson coefficients through the full theory amplitudes), and a heavy sfermion field is included at low energies. In particular, a so-called “heavy-light current” describes the interactions of the heavy bino with the heavy sfermion and light fermion. This is described in Sec. III.4.
Let us now move on to three cases that illustrate in explicit detail the general aspects of the EFT approach discussed above. Previous works have focused on fixed-order calculations Gondolo:2013wwa (); Ibarra:2015nca (); Ibarra:2015fqa (); Hisano:2010ct (); Djouadi:2001kba () or on the EFT treatment of the scalar gluon coupling Hisano:2015bma (). In the present analysis, we perform leading order matching onto the complete set of operators in Eq. (III.1), including contributions to quark operators from exchanges of electroweak bosons. For example, we find that the Higgs-exchange diagrams are numerically relevant, significantly improving the projected reach of LZ (e.g., compared to those found in Ref. Ibarra:2015nca ()). Moreover, the following subsections present a pedagogical discussion of the EFT framework, illustrating aspects such as matching and the infrared pole structure, and the application of the heavy-light current. The case of a sfermion nearly degenerate in mass with the bino discussed in Sec. III.4 is new and physically relevant.
iii.2 Case I: Right-Handed Stop
The simplest example arises when the mass of the fermion partnered to the sfermion is of order or greater than the weak scale, . Although this case broadly applies to many models, for concreteness, we will restrict to the case of a single right-handed stop () interacting with the bino () and a top quark (). Let us discuss in turn the ingredients , , , and of the factorization presented in Eq. (8).
Weak scale coefficients : The matching condition at the weak scale is shown in Fig. 4. The full theory amplitudes are computed using the Lagrangian in Eq. (1), while the effective theory amplitudes are computed using the Lagrangian in Eq. (III.1). The weak scale particles are highly virtual at low energies and are thus integrated out. Their effects are encoded into the Wilson coefficients of an effective theory describing a heavy bino interacting with the quarks and gluons of 5-flavor QCD.
The contributions to the quark and gluon coefficients begin at and , respectively. The -exchange diagrams contribute to the scalar coefficient , while the -exchange diagrams contribute to the axial-vector coefficient . The box diagrams exchanging or contribute to , , and . The explicit results for the relevant diagrams are collected in Eqs. (D.2), (69), (D.5), and (D.6). Working consistently at leading order, the gluon matching condition does not include contributions from effective theory diagrams involving loops of quarks since these are . Accordingly, we also drop the terms in the renormalization condition in Eq. (13), and thus all bare Wilson coefficients are trivially renormalized for this example, i.e., . We collect the renormalized Wilson coefficients in the vectors
where is representative of the five quark flavors, i.e., , and hence the vectors and have twelve and five components, respectively. The coefficients are collected into two vectors in anticipation of evaluating the SI and SD amplitudes separately.
Running and matching matrices and : For cases where the degrees of freedom below the weak scale are a gauge singlet (under ) DM particle and the quarks and gluons of -flavor QCD, the relevant matrices for running and matching are specified by loop-level matrix elements of the QCD currents in Eq. (II). We adopt the results from Tables 5 and 6 of Ref. Hill:2014yxa (), and collect their leading order forms in Appendix C for completeness. In practice, we work at leading log (LL) order. For the axial current, the corrections to coefficient evolution and threshold matching begin at ), and are therefore subleading Larin:1993tq (); Grozin:1998kf (); Grozin:2006xm (). In particular, this implies that the weak scale coefficients contribute to the amplitude, while may be neglected. Nonetheless, we will keep the discussion of weak scale coefficients general, including the determination of .
Nucleon matrix elements : Let us collect the nucleon matrix elements defined in Eq. (II) in the following vectors:
where is representative of the three light quark flavors, i.e., the vectors and have eight and three components, respectively. To be consistent with the higher order effects included in the running and matching matrices and , we must also include higher order corrections to the leading order gluon scalar matrix element of Eq. (5). From the nucleon mass sum rule that links the gluon and quark scalar form factors (see, e.g., Ref. Hill:2014yxa ()), we have
where with the QCD beta function, and is the quark mass anomalous dimension. In our numerical analysis, we include terms in and through (see Eq. (48)).
With all ingredients specified, we may now evaluate the amplitudes as in Eq. (8). The result can be expressed as
which when expanded takes the form in Eq. (II). The vectors contain the low energy coefficients properly mapped from the weak scale through the running and matching factors:
These vectors are defined as in Eq. (14) but with the light quarks () and gluon of 3-flavor QCD. In practice, we will not evolve the coefficients after integrating out the charm quark at , and hence we take . Finally, the cross section is determined as in Eq. (7). Note that Eq. (7) applies for a relativistic Majorana field , but is also valid for our heavy particle field , given the conventional factor of on the left-hand side of Eq. (III.1).
iii.3 Case II: Right-Handed Sbottom, Large Mass Splitting
An example similar to the previous one, but slightly more involved due to the interplay between quark and gluon coefficients, is when the mass of the fermion partnered to the sfermion is much lighter than the weak scale, , and the mass splitting between the sfermion and the bino is comparable to or greater than the weak scale, . Although the procedure described here applies to a wide variety of models, for definiteness, we focus on the case of a right-handed sbottom () interacting with the bino () and bottom quark (). Let us discuss in turn the ingredients , , , and of the factorization presented in Eq. (8).
Weak scale coefficients : The matching condition at the weak scale is shown in Fig. 5. As in the previous example, the full theory amplitudes are computed using the Lagrangian in Eq. (1), while the effective theory amplitudes are computed using the Lagrangian in Eq. (III.1). The weak scale particles are integrated out, and their effects are encoded in Wilson coefficients of the effective theory describing a heavy bino interacting with the quarks and gluons of 5-flavor QCD.
As in the previous example, the leading contributions to the coefficients are loop diagrams. What distinguishes this case is the presence of a tree-level, , contribution to the bottom quark coefficients and the associated loop-level, , effective theory contributions to the gluon coefficients . As discussed in Sec. III.1, we adopt the scheme where all mass scales much lighter than the weak scale (such as ) are set to zero, and employ dimensional regularization. The full theory contribution to is IR divergent due to gluons emitted off of a massless bottom quark. The effective theory contributions from a bottom quark loop, shown on the right-hand side of Fig. 5, are scaleless, and thus vanish. In the low energy theory, the remaining pole of the bare coefficient is regarded as an UV divergence that is renormalized according to Eq. (13). For illustration, we present the explicit pole structure of the contributions to the renormalized spin-2 gluon coefficient:
where () is the the full (effective) theory loop contribution appearing on the left (right) side of the gluon matching condition in Fig. 5, and the last term comes from the renormalization prescription of Eq. (13). We have omitted the label “bare” on the coefficients on the right-hand side, and expressed the vanishing effective theory contribution, , in terms of canceling UV and IR poles. Note the required consistency between (given in Eq. (LABEL:eq:HeavySbottomTree)) and the infrared pole of the full theory contribution (given in Eq. (D.6)) to yield a finite renormalized coefficient . The other coefficients and are simply renormalized as .
As before, we collect the renormalized Wilson coefficients in the vectors
where is representative of the five quark flavors, i.e., , so that these two vectors are 12 and 5 dimensional, respectively. Note that is non-zero only for . In general, -exchange contributes to the SD interaction , but when and the sbottom is purely right-handed, this amplitude vanishes at leading order in momentum transfer by gauge invariance (Eq. (70)). The loop diagram where the Higgs is radiated off the bottom quark also vanishes, while the one where the Higgs is radiated off the sbottom contributes to (Eq. (62)).
Running and matching matrices and , and nucleon matrix elements : Since the theory below the weak scale is again given by Eq. (III.1), the mapping of the weak scale coefficients to the hadronic scale is identical to the previous example of Sec. III.2. In particular, the components and implement RGE and matching across heavy quark thresholds, respectively, while applies nucleon matrix element form factors.
iii.4 Case III: Right-Handed Sbottom, Small Mass Splitting
Finally, we consider the case where both the mass of the fermion partnered to the sfermion and the mass splitting between the sfermion and the bino are much lighter than the weak scale, . For definiteness, we focus on the case of a right-handed sbottom () interacting with the bino () and bottom quark ().
In this example, the sbottom is not highly virtual at low energies since the small sbottom-bino mass splitting kinematically allows for sbottom-bino interactions through a soft bottom. Weak-scale physics is still integrated out by matching onto 5-flavor QCD, but both the bino and sbottom are kept as heavy fields in the effective theory (valid for ). The relevant interactions may be obtained from the full theory by introducing the field redefinition of Eq. (10) for the relativistic bino field , and
where for a right-handed sbottom . The residual mass term is given by the mass splitting , and the sbottom-bino coupling is the heavy particle version of Eq. (1). Physically, the heavy particle velocity, , is conserved in the scattering process. Thus, the sign in the kinetic term denotes a sbottom coming into the vertex, or by using integration by parts, an anti-sbottom coming out of the vertex. In contrast to the relativistic case where , the fields and can only be related through the invariance in Eq. (11). Hence, the two vertices above are the only ones that contribute to amplitudes involving as the initial and final state (e.g., there are no charge-reversed diagrams in Fig. 7). Note from the canonically normalized kinetic term that the heavy sbottom has scaling dimension (hence the factor of appearing in the field redefinition in Eq. (21) and in the sbottom-bino coupling). In the low energy theory the interactions of the heavy bino with the quarks and gluon of 5-flavor QCD are still described by Eq. (III.1).
The sbottom-bino interaction introduced in Eq. (22) can be viewed similarly to the so-called “heavy-light current” in applications for B-meson decays Shifman:1986sm (); Politzer:1988wp (); Politzer:1988bs (). In particular, its running due to QCD corrections from down to is significant, and we account for this when implementing the RGE down to the bottom quark threshold. Let us discuss in turn the ingredients , , , and of the factorization presented in Eq. (8).
Weak scale coefficients : The matching condition at the weak scale is shown in Fig. 6. The full theory amplitudes are computed using the Lagrangian in Eq. (1), while the effective theory amplitudes are computed using the Lagrangians in Eqs. (III.1) and (22). The coefficients are determined by the same loop diagrams of the previous two examples. Since we set all mass scales much lighter than the weak scale to zero, we are implicitly taking the limit of both the full theory and effective theory amplitudes. Of course, it is precisely in this limit that the relativistic and heavy particle Feynman rules match. Therefore, the full theory contribution from Eq. (1) and the effective theory contribution from Eq. (22) cancel in the gluon and bottom quark matching, yielding coefficients and that vanish up to corrections. As an explicit example, the relativistic sbottom propagator in the tree-level diagram is expanded as
where we have included a factor of for the crossed diagram, and used . Note that the above result matches the tree-level amplitude obtained from the Feynman rules of Eq. (22). In contrast, the usual expansion of the sbottom propagator in terms of local operators (corresponding to nonzero coefficients) is valid for . For the gluon matching, we find that the full theory amplitudes vanish at , which must be the case since the gluon coefficients scale as , but the only mass scale is (see Eqs. (D.6)-(85) for the explicit forms of the full theory gluon diagrams in the limit ). Similarly, the effective theory loop diagrams are scaleless, and hence vanish, as discussed in Sec. III.1 and in the example of Sec. III.3. In principle, setting introduces IR poles as in Sec. III.3, but in this case they appear at . Thus, with no spin-2 quark or gluon coefficients generated at , all bare Wilson coefficients are trivially renormalized, i.e., . Collecting the Wilson coefficients as in Eq. (20), up to corrections of , we find
Note that these two vectors, as in Eq. (20), are 12 and 5 dimensional for SI and SD, respectively. The coefficient is only non-zero for the four quark flavors , and is generated from integrating out the Higgs (corresponding to the full theory diagram where a Higgs is radiated off the sbottom, given in Eq. (63)). On the other hand, neither nor any of the spin-2 quark and gluon coefficients are generated at because the sbottom is kept in the low-energy effective theory below the weak scale. As in the previous case, the contributions from a Higgs radiated off a bottom quark and Z-exchange vanish in the chiral limit .
Running from down to : At leading order in , the only nonvanishing coefficients are those corresponding to the scale invariant current , and thus the coefficients in Eq. (24) do not evolve, i.e., . We must also account for the scale evolution of the sbottom-bino couplings in Eq. (22). The anomalous dimension of the current , with Dirac structure , is the same as that of the heavy-light current describing the interaction of a heavy quark with a light quark Shifman:1986sm (); Politzer:1988wp (); Politzer:1988bs (). It is independent of the Dirac structure , and is given by . The evolution of the coefficients is thus