Light stops, blind spots, and isospin violation in the MSSM
In the framework of the MSSM, we examine several simplified models where only a few superpartners are light. This allows us to study WIMP–nucleus scattering in terms of a handful of MSSM parameters and thereby scrutinize their impact on dark matter direct-detection experiments. Focusing on spin-independent WIMP–nucleon scattering, we derive simplified, analytic expressions for the Wilson coefficients associated with Higgs and squark exchange. We utilize these results to study the complementarity of constraints due to direct-detection, flavor, and collider experiments. We also identify parameter configurations that produce (almost) vanishing cross sections. In the proximity of these so-called blind spots, we find that the amount of isospin violation may be much larger than typically expected in the MSSM. This feature is a generic property of parameter regions where cross sections are suppressed, and highlights the importance of a careful analysis of the nucleon matrix elements and the associated hadronic uncertainties. This becomes especially relevant once the increased sensitivity of future direct-detection experiments corners the MSSM into these regions of parameter space.
Establishing the microscopic nature of Dark Matter (DM) is one of the central, open questions in cosmology and particle physics. In the context of cold nonbaryonic DM, the prevailing paradigm is based on weakly interacting massive particles (WIMPs), and extensive theoretical and experimental resources have been devoted towards identifying viable candidates and developing methods to detect them. One of the most studied WIMPs arises in the Minimal Supersymmetric Standard Model (MSSM), where an assumed -parity ensures that the lightest superpartner (LSP) is a stable neutralino composed of bino , wino , and Higgsino eigenstates. The mass of the LSP is expected to lie in the range of tens to hundreds of GeV.
In its general form, however, the MSSM contains more than parameters, most of which are tied to the hidden sector which breaks supersymmetry (SUSY) at some scale . Since these parameters are unknown a priori, it is necessary to restrict the dimensionality of the parameter space in order to obtain a predictive framework with which to undertake phenomenological analyses. One way to achieve this is to adopt a specific mechanism that describes high-scale SUSY-breaking in terms of a small number of parameters. For example, the constrained MSSM (CMSSM) with minimal supergravity Chamseddine et al. (1982); Nath et al. (1983); Barbieri et al. (1982); Hall et al. (1983) only involves five free parameters, but faces increasing tension Buchmueller et al. (2012a, b); Cabrera et al. (2013); Strege et al. (2013); Fowlie et al. (2012); Buchmueller et al. (2014a); Bechtle et al. (2012, 2013) with the non-observation of superpartners at the LHC experiments and other observables like the measured Higgs mass and anomalous magnetic moment of the muon. Alternatively, one can remain agnostic about the features of SUSY-breaking and incorporate data-driven constraints, as in e.g. the p(henomenological)MSSM Djouadi et al. (1998); Berger et al. (2009), where only 19 free parameters are used to capture the essential features of weak-scale SUSY.
In both approaches, long computational chains involving spectrum generators, the calculation of decay rates, or the DM relic abundance are typically required in order to explore the relevant parameter space. This strategy has been used extensively for the CMSSM Buchmueller et al. (2012a, b); Cabrera et al. (2013); Strege et al. (2013); Fowlie et al. (2012); Buchmueller et al. (2014a); Bechtle et al. (2012, 2013) and pMSSM Boehm et al. (2013); Grothaus et al. (2013); Han et al. (2013); Strege et al. (2014) to analyze –nucleon scattering and impose limits from current DM direct-detection experiments such as SuperCDMS Agnese et al. (2014), XENON100 Aprile et al. (2011), and LUX Akerib et al. (2014), as well as upcoming proposals like XENON1T Aprile (2013), LUX-ZEPLIN (LZ) Malling et al. (2011), and SuperCDMS SNOLAB Brink (2012). While these parameter scans allow one to gain useful information about the status of the theory in light of global fits, they generally hinder attempts to clearly identify which contributions associated with the underlying theory parameters can have the greatest impact on a signal of interest. An analytical understanding of the underlying parameter space can instead be obtained in the context of so-called simplified models, defined111For a definition of “simplified models” in the context of LHC searches, see Alwall et al. (2009); Alves et al. (2012); Barnard and Farmer (2014); Edelhäuser et al. (2014); Calibbi et al. (2014). Cheung et al. (2013) to be minimal theories of weak-scale SUSY where all but a handful of the superpartners relevant for DM phenomenology are decoupled from the spectrum.
For spin-independent (SI) –nucleon scattering, the choice of simplified model is guided by the dominant contributions to the cross section, namely, Higgs and squark exchange Goodman and Witten (1985); Griest (1988); Srednicki and Watkins (1989); Giudice and Roulet (1989); Shifman et al. (1978); Drees and Nojiri (1993a). To date, the focus has largely concerned the role of the Higgs sector, both in the decoupling limit where a single SM-like Higgs is present in the spectrum Cheung et al. (2013); Hisano et al. (2013), or in the more general case Huang and Wagner (2014); Anandakrishnan et al. (2015) where the heavier -even Higgs is included. This focus is chiefly motivated by the fact that current bounds on the masses of gluinos and (degenerate) squarks of the first two generations are larger than about 1 TeV Aad et al. (2014); Chatrchyan et al. (2014), and so their contribution to the SI cross section can be safely ignored.222However, for non-degenerate squarks Nir and Seiberg (1993); Nir and Raz (2002) the constraints from FCNCs are satisfied Crivellin and Davidkov (2010), and the collider bounds are significantly weakened Mahbubani et al. (2013). In this case, contributions from the first two generations could also be important for SI –nucleon scattering.
However, the decoupling of third-generation squarks—especially stops—upsets the main motivation behind the MSSM, namely, its ability to stabilize the electroweak scale 174 GeV against loop corrections in a technically natural fashion Dimopoulos and Giudice (1995); Giudice and Romanino (2004); Kitano and Nomura (2006); Perelstein and Spethmann (2007); Brust et al. (2012); Papucci et al. (2012). In other words, if naturalness is to remain a useful criterion with which to constrain the MSSM parameter space, then the spectrum should (minimally) include light stops and—due to invariance—a left-handed sbottom Barbieri and Giudice (1988); Dimopoulos and Giudice (1995); Cohen et al. (1996); Barbieri and Pappadopulo (2009). While the search for top and bottom squarks remains a primary focus of the LHC experiments, their impact on DM direct-detection limits has not been explored in detail.
The purpose of this paper is to present an analytical scheme which allows one to successively include those states which are most relevant for naturalness and DM direct detection. In particular, we consider a bino-like LSP and derive simplified, analytic expressions for the Wilson coefficients associated with SI scattering. We examine in detail the contributions from Higgs and third-generation squark exchange, and study the interplay of collider, flavor, and DM constraints. As in previous analyses of the Higgs sector Cheung et al. (2013); Hisano et al. (2013); Huang and Wagner (2014); Anandakrishnan et al. (2015), our scheme allows us to identify so-called blind spots in parameter space, where the SI cross section is strongly suppressed by either a particular set of parameters Hisano et al. (2013); Cheung et al. (2013), or destructive interference Huang and Wagner (2014); Anandakrishnan et al. (2015) in the scattering amplitude. This effect was first identified numerically through a scan of the CMSSM parameter space Ellis et al. (2000a, 2001); Baer et al. (2007), while lower bounds on the –nucleon cross sections were first discussed in Mandic et al. (2000) for both the CMSSM and a generalized MSSM framework. A key feature of our analysis is that in the vicinity of blind spots, the amount of isospin violation may be much larger than typically expected for the MSSM Ellis et al. (2008).333Large isospin violation has been put forward as a mechanism to reconcile contradictory DM direct-detection signal claims and null observations Kurylov and Kamionkowski (2004); Giuliani (2005); Chang et al. (2010); Feng et al. (2011); Cirigliano et al. (2012, 2014). In order to account for isospin-violating effects originating from the nucleon matrix elements of the scalar quark currents, we use the formalism developed in Crivellin et al. (2014a), which provides an accurate determination of the hadronic uncertainties.
The paper is organized as follows. In Sec. II we establish our notation for –nucleus scattering and comment on the differing treatments Ellis et al. (2008); Bélanger et al. (2014); Crivellin et al. (2014a) of the nucleon scalar matrix elements found in the literature. The leading MSSM contributions to the Wilson coefficients are then identified using a systematic expansion in which generates simplified analytic expressions for the Wilson coefficients associated with Higgs and squark exchange. Section III examines four simplified models, where, driven by naturalness, we successively include the most relevant particles as active degrees of freedom. In each case, we discuss the conditions for blind spots and examine the amount of isospin violation allowed by current and projected limits from SI DM scattering. Our analysis shows that the absence of DM signals pushes the MSSM into regions of parameter space where isospin-violating effects are likely to become relevant.
Ii Theoretical Preliminaries
ii.1 Spin-independent neutralino–nucleus cross section: scalar matrix elements
We start by providing some definitions for the elastic scattering of the lightest neutralino off a species of nucleus , where and denote the atomic and mass numbers respectively. Typically, the dependence of the cross section on the small momentum transfer is assumed to be described by nuclear form factors. At zero momentum transfer and for one-body currents only, the cross section for is given by
Here, is the reduced mass of the – system, while and are effective (zero-momentum) SI couplings of the LSP to the proton and neutron respectively.
For nucleons , the – couplings are defined by
where is the Wilson coefficient of the scalar operator with running quark mass , and
The coefficients can be interpreted as the fraction of the nucleon mass generated by the respective quark scalar current and are often referred to as nucleon scalar couplings. In the framework adopted in (3), the heavy quarks are integrated out, so that, via the trace anomaly Adler et al. (1977); Minkowski (1976); Nielsen (1977); Collins et al. (1977) of the QCD energy-momentum tensor, their scalar coefficients can be expressed in terms of the light-quark ones Shifman et al. (1978). As shown by Drees and Nojiri Drees and Nojiri (1993b), this procedure fails if the squarks are sufficiently light, and in Sec. II we discuss the necessary modifications to (2) which account for the exact one-loop result.
We note that (3) holds at leading order in : in the case of the charm quark, this may not be sufficiently accurate, so that either higher-order corrections Kryjevski (2004); Vecchi (2013); Hill and Solon (2015) or a non-perturbative determination on the lattice could become mandatory. Similarly, corrections to the single-nucleon picture underlying (1) in the form of two-nucleon currents can be systematically taken into account using effective field theory Prézeau et al. (2003); Cirigliano et al. (2012, 2014); Hoferichter et al. (2015a). In this paper, we use (1) and (3) to investigate the amount of isospin violation that can be generated within several simplified models, given the hadronic uncertainties of the single-nucleon coefficients for the light quarks .
Traditionally, the scalar matrix elements of the light quarks have been determined from a combination of chiral perturbation theory (PT) and phenomenological input inferred from the pion–nucleon -term and the hadron mass spectrum Ellis et al. (2000a); Corsetti and Nath (2001); Ellis et al. (2008); Cirigliano et al. (2012). A central feature of this approach is that the up- and down-quark coefficients are reconstructed from two three-flavor quantities: the so-called strangeness content of the nucleon
and another parameter
that is related to isospin violation. As a result, the inherent uncertainties of PT (typically of order ) propagate to the two-flavor sector. Furthermore, is usually extracted from a leading-order fit to baryon masses Cheng (1989), and this compounds the problem of obtaining reliable uncertainty estimates.
For the up- and down-quark coefficients , these problems can be circumvented by using the two-flavor theory PT directly, thus avoiding the three-flavor expansion in the first place Crivellin et al. (2014a). Starting from the PT expansion of the nucleon mass at third chiral order and including the effects due to strong isospin violation, one finds
where the -term is defined as , averaged over proton and neutron, , and
is taken from Colangelo et al. (2011).
For the present work, one particularly important aspect of the PT approach Crivellin et al. (2014a) is that isospin violation can be rigorously accounted for, including uncertainty estimates. This aspect can be nicely illustrated by considering the differences
wherein the terms from (II.1) cancel.444The chiral expansion of the nucleon mass difference is known to have a large chiral logarithm at fourth order, with coefficient Frink and Meißner (2004); Tiburzi and Walker-Loud (2006). We have checked that including this logarithm in the analysis leads to changes of the well within the uncertainties given in (II.1). Using the PT approach, these differences are overestimated by roughly a factor of , as in e.g. Bélanger et al. (2014):
Alternatively, one could introduce further measures of isospin violation like and (motivated by the quark-model picture of the nucleon), but these combinations depend on the specific value of . In the isospin-conserving limit, all up- and down-coefficients obtained from the chiral expansion of the nucleon mass become equal , so that the relations and are fulfilled.
Ultimately, the quantities relevant for the direct-detection cross section are the parameters defined in (2), after multiplication by the Wilson coefficients and summing over quark flavors. In particular, the cross section (1) may be rewritten as
so that the departure of from unity emerges as a convenient measure of isospin violation. In this context, care has to be taken in interpreting the limits on the WIMP–nucleon cross section given by experimental collaborations. Indeed, these are generally extracted via the relation
where is the reduced mass of the –nucleon system. We stress that in (11) can be identified with the SI –proton cross section only under the assumption . If isospin-violating effects are large, it is natural to compare against the –nucleus cross section (1) directly, and (11) indicates how the experimental limits are to be rescaled.
In general, the effects of isospin violation depend on the target nucleus. For the mass range - GeV considered in most of our analysis (Sec. III), the strongest limits on SI –nucleon scattering are currently set by LUX Akerib et al. (2014). In the context of isospin violation, this prompts us to focus on the projected reach of upcoming xenon-based experiments like XENON1T Aprile (2013) and LZ Malling et al. (2011). However, this raises the question whether other experiments like SuperCDMS SNOLAB Brink (2012) (based on germanium) can be used to place complementary constraints on . To quantify the difference between xenon-based constraints and other nuclei, consider the dependence of the ratio
normalized such that in the isospin-conserving limit. For SI scattering off argon and germanium, the result is shown in Fig. 1, where we observe a maximum difference of around 10% for much larger or smaller than unity. In the simplified models considered in Sec. III, the difference between and is generated entirely by SM quantities, so the improved limits offered by e.g. SuperCDMS SNOLAB are limited to the percent level. Moreover, the location of blind spots is determined by the condition , so neither the blind spot nor the uncertainty on depends significantly on the atomic/mass numbers of the nuclear target.
The crucial input quantity is not yet precisely determined: in Sec. III we show the dependence on this parameter explicitly in the case of generic Higgs exchange, and later fix its central value to for illustrative purposes. The need for a precise determination of has triggered many ongoing efforts, including lattice-QCD calculations at (nearly) physical values of the pion mass; see Young (2012); Kronfeld (2012); Junnarkar and Walker-Loud (2013); Bélanger et al. (2014) for a compilation of recent results and improved phenomenological analyses. The challenge in the phenomenological approach, i.e. extracting from scattering, lies in controlling the required analytic continuation of the isoscalar amplitude into the unphysical region Cheng and Dashen (1971), which might even be sensitive to isospin-breaking corrections Hoferichter et al. (2009, 2010). This analytic continuation can be stabilized with the help of the low-energy data that have become available in recent years thanks to accurate pionic-atom measurements Gotta et al. (2008); Strauch et al. (2011), leading to a precise extraction of the scattering lengths Baru et al. (2011a, b). A systematic analysis of scattering based on this input as well as constraints from unitarity, analyticity, and crossing symmetry along the lines of Ditsche et al. (2012); Hoferichter et al. (2012); Hoferichter et al. (2015b) will help clarify the situation concerning the phenomenological determination of Gasser et al. (1991); Pavan et al. (2002); Alarcón et al. (2012).
In our numerical analysis, we compare three different methods used to determine the scalar couplings and their uncertainties:
Method 2: Corresponds to the traditional PT approach Ellis et al. (2000a); Corsetti and Nath (2001); Ellis et al. (2008); Cirigliano et al. (2012), where and are determined via the three-flavor quantities and . In this approach, the strange-quark scalar matrix element is defined via
where Colangelo et al. (2011), the strangeness content is taken from the relation , with Borasoy and Meißner (1997), and is extracted from leading-order fits to the baryon mass spectrum Cheng (1989). This approach introduces uncertainties that are difficult to quantify and is sensitive to the precise value of . The range covering the determinations discussed above Young (2012); Kronfeld (2012); Gasser et al. (1991); Pavan et al. (2002) translates to . Even for moderate values of the -term, large values have been inferred in this way. Such large values are incompatible with recent lattice calculations, which provide a more reliable determination of (see (13) for a recent average). A determination of the uncertainty bands arising from this approach requires us to attach an error to , which, as argued before, is impossible to quantify reliably. Therefore, based on general expectations for the convergence pattern in PT, we simply attach to a error.555This is consistent with an analysis Shanahan et al. (2012) of the quark-mass dependence of octet baryons.
Method 3: Corresponds to the implementation in micrOMEGAs-4.1.2 Bélanger et al. (2014) and follows the traditional approach in Method 2, but with (14) inverted so that is a function of . With lattice QCD input (13) for , this method has reduced uncertainties compared with Method 2, but suffers from the fact that still depend on the three-flavor quantities and .
ii.2 Simplified expressions for spin-independent scattering of bino-like dark matter
Let us now derive analytic expressions for SI –nucleon scattering in the MSSM. We first review the complete expressions due to tree-level Higgs and squark exchange, and then simplify them by expanding in powers of . For light third-generation squarks, a procedure Bélanger et al. (2009) to extend our results to include the one-loop corrections Drees and Nojiri (1993b) is discussed below.
The lightest neutralino is a linear combination of , , and interaction eigenstates,
while the neutralino mass matrix is given by
Here () are the soft SUSY-breaking masses of the bino (wino), is the Higgsino mass parameter, and are the two Higgs vacuum expectation values, whose ratio is denoted by . Note that while can be rendered real and positive by an appropriate phase redefinition of the Higgs fields, and are in general complex if the gluino mass is assumed to be real (as is standard convention). The neutralino mixing in (15) is determined by the unitary matrix which diagonalizes Rosiek (1995):
In the squark sector, the squared masses are eigenvalues of the matrices in flavor/chirality space,
Here, the soft SUSY-breaking squark masses are and , are complex Yukawa matrices, is the CKM matrix, and we have assumed flavor universality for the trilinear -terms. We also use and for sine and cosine, so that , , etc. The weak neutral-current couplings
are defined in terms of the third component of weak isospin , electric charge , and . A unitary transformation
gives the physical basis with diagonal squark mass matrices, where we adopt the convention to order the states in increasing mass. We have also defined the super-CKM basis in (18) as the one with diagonal (and in general, complex) Yukawa couplings Crivellin and Nierste (2009).
We have now introduced the necessary ingredients to discuss –quark scattering in the MSSM. The tree-level contributions to the Wilson coefficients are the diagrams shown in Fig. 2. For -conserving neutralino interactions, these amplitudes were calculated long ago in Goodman and Witten (1985); Griest (1988); Srednicki and Watkins (1989); Giudice and Roulet (1989); Shifman et al. (1978), and extended in Falk et al. (1999) to include -violating effects.
In our conventions, the contributions to due to squark exchange in the - and -channels at zero momentum transfer read666We have for generation indices and for squark mass eigenstates. To recover the expressions in Falk et al. (1999), one needs to make the identification etc.
where there is no sum over , and a pole mass enters in the squark propagator. Since we work at , the running quark masses must also be evaluated at this scale. For the Higgs-exchange contribution we have777In principle, the -odd Higgs can contribute to SI –quark scattering if or the Yukawa couplings are allowed to be complex. We exclude this possibility in our numerical analysis since we take real and the real part of after threshold corrections are included.
where and . We assume that GeV is the mass of the Higgs-like resonance found at the LHC Aad et al. (2012); Chatrchyan et al. (2012) and . In general, the and couplings appearing in (21) and (22) are complicated expressions involving the mixing matrices and . Thus the SI cross section is typically determined numerically. However, it is known Arnowitt and Nath (1996); Hofer et al. (2009); Crivellin et al. (2011a) that one can obtain analytic results by diagonalizing perturbatively888For real and , the exact diagonalization of is known Guchait (1993); El Kheishen et al. (1992); Barger et al. (1994), although the resulting formulae are not simple. in powers of . For complex and , one finds to leading order
where is the phase of . Note that the presence of a pole at has no physical meaning: it is a consequence of the fact that we assume a bino-like LSP and used non-degenerate perturbation theory to diagonalize the neutralino mass matrix.
In Appendix B, we show how (23) can be used to simplify (21) and (22) if flavor-violating effects are neglected,999Flavor violation in DM direct detection is strongly suppressed since the effect can only enter via double flavor changes which are experimentally known to be small. Furthermore, the effect of flavor off-diagonal entries can be largely absorbed by a change of the physical squark masses. while allowing for non-universal -terms and squark masses. The resulting expressions read
where the squark mixing is defined as
while the squark propagators are
and and are the upper and lower diagonal components of the squark (mass) matrices in (18). In deriving (26-27), we have imposed conservation so that the neutralino mass parameters and are real. We also take and so that both signs of are allowed. Expressions for in the -violating case are provided in Appendix B.
The simplified expressions in (24-25) are valid provided the squarks are sufficiently heavy, i.e. if . This requirement is not met for light third-generation squarks, and thus (24-25) must be corrected to account for the one-loop result Drees and Nojiri (1993b). To do so, we follow the prescription adopted in Bélanger et al. (2009) and replace all tree-level squark propagators
in terms of a linear combination of one-loop functions ,101010The term proportional to in (A5) of Bélanger et al. (2009) is missing a factor of .
whose form is given in Appendix B of Drees and Nojiri (1993b). In the heavy squark limit, the function agrees with to leading order in .
We have made use of the tree-level relation in order to obtain (24-27). For down quarks, however, this relation can be modified by one-loop graphs which induce an effective coupling between and the neutral component of . These corrections Hall et al. (1994); Carena et al. (1994, 2000); Hofer et al. (2009); Crivellin et al. (2011a) are non-decoupling and enhanced by a factor of .111111In principle, large -terms can also change the values of significantly Banks (1988); Borzumati et al. (1999); Crivellin and Girrbach (2010); Crivellin (2011); Crivellin et al. (2011a). However, this effect drops out in the Higgs–quark–quark couplings where the effective (physical) mass enters. Furthermore, since we assume flavor-universal -terms in our numerical analysis, the effect cannot be very large without violating vacuum stability bounds. For example, the gluino contribution at one-loop modifies the tree-level relation so that
Since is proportional to , we can account for (32) by a simple rescaling of the Wilson coefficients
where we include corrections Hall et al. (1994); Carena et al. (1994, 2000); Hofer et al. (2009); Crivellin et al. (2011a) beyond the gluino loop (32). These threshold corrections feature in our analysis of heavy Higgs and sbottom contributions (Sec. III) to the SI amplitude. Note that corrections to the light Higgs coupling cancel in the relation .
Iii Simplified models: blind spots and isospin violation
We now apply our analytic results (24-27) to four simplified models; each motivated by the following experimental and naturalness considerations. Firstly, the ATLAS Aad et al. (2012) and CMS Chatrchyan et al. (2012) experiments at the LHC have discovered a Higgs boson with SM-like properties and a mass below the upper bound 135 GeV of the MSSM. Secondly, a natural resolution of the gauge hierarchy problem requires several conditions Barbieri and Giudice (1988); Dimopoulos and Giudice (1995); Cohen et al. (1996) to be met:
In order to cancel the top-quark correction to the Higgs mass parameter , top squarks must be light with masses in the sub-TeV range;
The gluino mass must be around a TeV in order to prevent radiative corrections driving the stop masses too heavy;
Light Higgsinos must be present in the spectrum so that tree-level electroweak symmetry breaking implies that is satisfied.
It has also been observed Katz et al. (2014) that naturalness constrains the additional Higgs bosons to not be too heavy. Barring the gluino, the current experimental bounds on the masses of the above particles are rather weak. In contrast, the mass of the gluino and squarks of the first two generations are constrained to lie above 1 TeV. Therefore, naturalness prompts us to consider the simplified models shown in Fig. 3, where we start from a minimal, light particle spectrum necessary to have bino-like DM scattering [model (A)] and successively include as active degrees of freedom those particles which are (a) required to be light by naturalness, and (b) relevant for DM direct detection. Note that due to invariance, the models (C-D) involving two light stops always require a light sbottom in the spectrum. (Only if there is a single, mostly right-handed stop, can sbottoms be decoupled.)
In general, a bino-like LSP produces a DM relic density that is too large in most of the parameter space considered in Sec. III. However, the overproduction of bino-like DM in the MSSM can be diluted by either -channel resonance exchange involving , or – co-annihilation with a sfermion that is nearly degenerate in mass with .121212See e.g. Han et al. (2013) for a detailed analysis of these effects in the pMSSM. Both mechanisms Griest and Seckel (1991) increase the annihilation cross section before thermal freeze-out and can produce the observed relic abundance. In each of the models shown in Fig. 3, the relic density constraint may be satisfied by either mechanism or, if necessary, by extending the spectrum to include a tau slepton which generates additional co-annihilations Ellis et al. (1998, 2000b). Since the mass can be tuned without affecting naturalness or DM direct detection, we do not consider the DM relic density constraint in our subsequent analysis.
Similarly, we do not consider the constraint from the anomalous magnetic moment of the muon , whose world average is dominated by the Brookhaven measurement Bennett et al. (2006). The resulting value deviates from the SM prediction by –, depending on the details of the evaluation of the hadronic contributions Jegerlehner and Nyffeler (2009); Prades et al. (2009). Recent developments in the evaluation of the SM prediction include: the QED calculation has been carried out at -loop accuracy Aoyama et al. (2012), after the Higgs discovery Aad et al. (2012); Chatrchyan et al. (2012) the electroweak contribution is complete at two-loop order Gnendiger et al. (2013); Czarnecki et al. (2003), and hadronic corrections have been considered at third order in the fine-structure constant Kurz et al. (2014); Colangelo et al. (2014a). Although an improved determination of the leading hadronic contribution, hadronic vacuum polarization, mainly requires improved data input, see Jegerlehner and Nyffeler (2009); Davier et al. (2011); Hagiwara et al. (2011); Blum et al. (2013), the uncertainties in the subleading hadronic-light-by-light contribution have been notoriously difficult to estimate due to substantial model dependence Jegerlehner and Nyffeler (2009); Prades et al. (2009); Benayoun et al. (2014). Recently, data-driven techniques have been put forward to reduce the model dependence based on dispersion relations Colangelo et al. (2014b, c); Pauk and Vanderhaeghen (2014); Colangelo et al. (2015), and a first lattice calculation has become available Blum et al. (2015). All these efforts are motivated by two new experiments, at FNAL Grange et al. (2015) and J-PARC Saito (2012), which each aim at improving the measurement by a factor of and thus help clarify the origin of the discrepancy between experiment and the SM prediction.
Should the discrepancy persist, an explanation within the MSSM is possible provided certain assumptions are made about the SUSY parameters entering the smuon, chargino, and neutralino mass matrices. If these parameters are all equal to , then a positive contribution to requires since the dominant one-loop amplitude scales approximately with ; see e.g. the review Stöckinger (2007) and references therein. In the blind spot regions where , this condition would require us to relax the assumption that . However, the requirement does not necessarily apply if the SUSY mass parameters are non-degenerate. For example, it has been shown Cho et al. (2011); Fargnoli et al. (2013, 2014) that a positive contribution to can arise if , in which case and must have opposite sign. The key point is that neither the sign of nor the smuon masses are relevant for our analysis of SI scattering, so it would be possible to account for the experimental value of by a suitable choice of these parameters. Furthermore, the discrepancy could also be explained by large terms Borzumati et al. (1999); Crivellin et al. (2011b); Endo et al. (2013) not correlated with DM scattering.
We conclude this section by anticipating a key result of our analysis: isospin-violating effects can be magnified in the proximity of blind spots, where the SI direct-detection cross section lies below the lower bounds set by the irreducible neutrino background. For these parameter-space configurations, the SI amplitude itself becomes tiny and hence more susceptible to small variations in the input quantities, such as corrections from isospin breaking. In particular, the ratio of proton and neutron SI cross sections becomes very sensitive to the values of the scalar matrix elements and their uncertainties ,
so that the overall uncertainty on can become large near blind spots where . In each of the four simplified models (A-D), we examine the amount of isospin violation associated with the three methods of Sec. II.1.
iii.1 SM-like Higgs exchange
We begin by considering the minimal particle spectrum for which an observable SI cross section is possible. From the rightmost diagram in Fig. 2, it seems reasonable to conclude that the SM-like Higgs and the bino-like LSP is sufficient in this case. However, in the limit , (26) and (27) become
and thus the scattering amplitude decouples with the Higgsino mass . It follows that a measurable cross section due to Higgs exchange implies the presence of light Higgsinos in the spectrum, thereby satisfying one of the minimal naturalness requirements. Although this feature does not prevent the reintroduction of fine-tuning in the MSSM altogether, it becomes relevant in our subsequent analysis where light stops are added to the spectrum.
To compare (36) to data, we first note that vanishes when
and thus a blind spot arises in the SI cross section provided is negative. The prospects for constraining this feature (37) have been extensively analyzed Cheung et al. (2013) for –nucleon scattering. To examine isospin violation, however, we need limits on –nucleus cross sections, so we use (11) in order to constrain the relevant parameter space.
Let us first consider the limits associated with (36) when the scalar matrix elements of Method 1 are employed. In Fig. 4, we update the results from Cheung et al. (2013) and show constraints for various values of in the plane from current and upcoming xenon experiments. For , we find that only a narrow strip is excluded by the existing limits from LUX Akerib et al. (2014), while the projected reach from XENON1T Aprile (2013) and LZ Malling et al. (2011) will probe most of the naturalness-preferred region where is of order . As is increased, the term in is suppressed, thereby weakening the direct detection limits. If no signal is seen at LZ, then the allowed parameter space is focused towards and values of GeV. In the region and for small , the naturalness-preferred values of occur at and are concentrated near the blind spot. Although the irreducible neutrino background make this region difficult to probe experimentally, larger values of decrease the blind spot slope, so that natural values of become allowed for - GeV.
By taking a slice through the plane, we can also extract the limits due to a small mass splitting GeV between the bino and Higgsinos. This choice is motivated by the current CMS results CMS-PAS-SUS-12-019 (2014) on same-flavor opposite-sign dilepton searches. Here CMS sees a deviation which can be explained by a heavier neutralino decaying to a lighter one. Fig. 5 shows the resulting constraints, where we plot the SI cross sections as a function of the bino mass. For , the limits from LUX are stringent, with values below GeV excluded. The strength of these limits is due to an enhancement in the amplitude (36) from both a nearly degenerate denominator and lack of interference in the numerator terms. For , there are no constraints from LUX, although XENON1T and LZ will exclude the whole parameter space in the absence of a DM signal.
We now examine the hadronic uncertainties associated with each of the three methods discussed in Sec. II.1. For exchange, the Wilson coefficient (36) is independent of quark flavor, so the SI amplitude (2) factorizes
Evidently, the resulting SI cross section is sensitive to the value of , with a dramatic effect observed Giedt et al. (2009) on the regions of excluded parameter space when the typically large value of Method 3 is replaced with much smaller determinations (13) from the lattice. We emphasize that this sensitivity is also present in any analysis of isospin violation, where is the quantity of interest. For the present discussion, (38) implies that the ratio
is independent of , and thus isospin violation is entirely determined by hadronic quantities. In Fig. 6 we compare the uncertainties on as a function of . For Methods 1 and 3, we find stability across a large range of values, with isospin violation allowed at around the five and ten percent level respectively. As noted in Crivellin et al. (2014b), this stability is due to the fact that the constant term of in (38) dominates the remainder whenever is fixed by lattice input. In contrast, the PT formalism of Method 2 produces a strong dependence of on , which in turn affects . From Fig. 6, isospin violation greater than is allowed, in marked contrast to the precision of Method 1. This example clearly demonstrates the huge uncertainties associated with Method 2, which, however, is still used in the literature Buchmueller et al. (2014a, b).
iii.2 Light and heavy Higgs exchange
Let us now extend model (A) to include the heavy Higgs bosons [model (B) in Fig. 3]. The inclusion of these additional degrees of freedom is motivated by naturalness Katz et al. (2014), however, only contributes to the SI cross section (Fig. 2).
From our simplified expressions (26-27), we see that the couplings to up and down quarks differ by a factor of , but are identical131313Up to threshold corrections (34), which enhance by tens of percent at large . Their inclusion does not have a large impact on the numerical analysis. among different generations . As a result, the SI amplitude may be expressed as
collect the scalar coefficients associated with the up- and down-type Wilson coefficients. A blind spot occurs if the condition
is satisfied, and the resulting suppression of the SI cross section has been identified numerically Ellis et al. (2000a, 2001); Baer et al. (2007); Anandakrishnan et al. (2015) and further studied analytically Huang and Wagner (2014). In the latter case, an explicit formula Huang and Wagner (2014) for the blind spot can be found for moderate to large values of and :
In effect, (42) has been recast as an interference condition between the and amplitudes; a feature which has important consequences for isospin violation in the MSSM. As with exchange, negative values of are required in order to generate the blind spot. However, note that in the vicinity of (37), the first term in (43) is suppressed, so in some cases the contribution from exchange may dominate the scattering amplitude Huang and Wagner (2014).