Flavored dark matter beyond Minimal Flavor Violation
We study the interplay of flavor and dark matter phenomenology for models of flavored dark matter interacting with quarks. We allow an arbitrary flavor structure in the coupling of dark matter with quarks. This coupling is assumed to be the only new source of violation of the Standard Model flavor symmetry extended by a associated with the dark matter. We call this ansatz Dark Minimal Flavor Violation (DMFV) and highlight its various implications, including an unbroken discrete symmetry that can stabilize the dark matter. As an illustration we study a Dirac fermionic dark matter which transforms as triplet under , and is a singlet under the Standard Model. The dark matter couples to right-handed down-type quarks via a colored scalar mediator with a coupling . We identify a number of “flavor-safe” scenarios for the structure of which are beyond Minimal Flavor Violation. For dark matter and collider phenomenology we focus on the well-motivated case of -flavored dark matter. The combined flavor and dark matter constraints on the parameter space of turn out to be interesting intersections of the individual ones. LHC constraints on simplified models of squarks and sbottoms can be adapted to our case, and monojet searches can be relevant if the spectrum is compressed.
Keywords:Flavor, Dark Matter, Beyond the Standard Model
Dark matter (DM) provides a strong connection between the two phenomenologically rich arenas: particle astrophysics and beyond Standard Model (SM) physics. While the existence of DM is part of the standard model of cosmology, its particle physics origins are largely unknown. The WIMP (weakly interacting massive particle) miracle however provides a tantalizing hint that DM is associated with new physics (NP) at the weak scale, and such candidates should be accessible to various ongoing experiments. Signals at these experiments depend strongly on the nature of interactions of the DM with SM fields, and are less sensitive to other details of the model. This motivates the study of simplified models, which minimally extend the SM to include couplings of DM particles with the SM. Each simplified model can then capture the dark matter phenomenology of a wide range of models.
Once we consider different classes of simplified models, one new category of models arises in analogy with SM flavor: flavored DM Kile:2011mn (); Kamenik:2011nb (); Batell:2011tc (); Agrawal:2011ze (); Batell:2013zwa (); Kile:2013ola (); Lopez-Honorez:2013wla (); Kumar:2013hfa (); Zhang:2012da (). In this setup DM particles come in multiple copies, and have a non-trivial flavor structure in their couplings with quarks and leptons.111An alternative scenario in which dark matter arises from a discrete symmetry in the neutrino sector has been considered in Hirsch:2010ru (); Boucenna:2011tj (). This framework does show up in a very specific way in supersymmetric models as sneutrino DM models Ibanez:1983kw (); Ellis:1983ew (); Hagelin:1984wv (); Goodman:1984dc (); Freese:1985qw (); Falk:1994es (); MarchRussell:2009aq (), but clearly there are more general possibilities.
This class of models is constrained, like other DM models, by both indirect and direct detection DM experiments as well as collider searches. The relevant schematic interaction responsible for these signatures is shown in the left panel of figure 1. Additionally precision flavor experiments have to be taken into account due to the flavor violation introduced by the dark sector. Schematically this contribution is displayed in the right panel of figure 1, adding a new class of diagrams to the well-studied DM-SM interaction.
Flavored dark matter models can have significantly distinct phenomenology. For indirect detection experiments, the spectrum of photons and leptons arising from DM annihilation depends on the relative annihilation into various final states. For example, it was shown that a DM candidate annihilating exclusively to -quarks provides a good fit to the spectrum of excess photons observed in a recent analysis of Fermi-LAT data from the galactic center Daylan:2014rsa (); Berlin:2014tja (); Agrawal:2014una (); Izaguirre:2014vva (); Boehm:2014hva (); Ipek:2014gua (); Kong:2014haa (); Ko:2014gha (); Boehm:2014bia (); Abdullah:2014lla (); Ghosh:2014pwa (); Martin:2014sxa (); Berlin:2014pya (); Basak:2014sza (); Modak:2013jya () . Direct detection predictions for scattering vary widely depending upon whether the ambient DM particles couple to the first generation quarks directly or not. The absence of direct detection signals so far then point to the possibility of suppression of such a coupling, which can be achieved through either a loop suppression or a small mixing angle. Collider searches for DM are also sensitive to the DM couplings to various quark flavors, both in terms of the DM production cross section, as well as the flavor pattern of visible final states which can be produced in association with the DM.
While some of these effects have been explored, the study of flavor phenomenology has largely been restricted to elaborate models such as the MSSM. Previous analyses often assume for simplicity universality or minimal flavor violation (MFV) Chivukula:1987py (); Hall:1990ac (); Buras:2000dm (); D'Ambrosio:2002ex (); Buras:2003jf (), so that flavor changing neutral current (FCNC) effects are automatically suppressed. On one hand this is welcome due to the good agreement of the flavor data with the SM prediction, but on the other hand interesting effects in the flavor sector are eliminated.
In this paper we abandon the MFV principle and consider instead a general flavor violating coupling of DM particles with quarks. DM is introduced as a triplet under a new global flavor symmetry . While in our analysis the coupling matrix (denoted by ) is taken to be completely general, we make one simplifying assumption that turns out to be helpful in various respects. We impose that is the only new source of flavor breaking, in addition to the SM Yukawa couplings. As this assumption generalizes the MFV principle to the DM sector, we call it Dark Minimal Flavor Violation (DMFV). We will point out the following features of DMFV:
The DMFV framework, while bearing some conceptual similarity to MFV, goes well beyond the latter framework, as it allows for large FCNC effects. The structure of needs to be determined from the available constraints.
The DMFV ansatz naturally preserves a residual symmetry, which guarantees the stability of the DM particle.
DMFV significantly reduces the number of new parameters in the Lagrangian, as the DM mass term must be flavor conserving up to corrections of the form .
DMFV guarantees “flavor-safety” of the UV complete theory. It is therefore sufficient to identify flavor-safe scenarios for the structure of within the simplified model framework.
In the phenomenological part of our paper we will restrict ourselves to the study of the simplest version of DMFV, which we refer to as the minimal DMFV (mDMFV) model in order to distinguish it from the more general framework. The DM is taken to be a Dirac fermion , interacting with the right-handed down-type quarks via the coupling
with a scalar mediator . While leaving the question of a possible UV completion unanswered, this study captures the most important phenomenological effects accessible to current experiments. Our studies extend the existing literature on the phenomenology of flavored DM in the following ways:
We go beyond the simple MFV hypothesis that automatically suppresses all flavor effects to an acceptable level. Instead we study the implications of a completely general coupling matrix , embedded in the DMFV ansatz, and derive its structure from the experimental constraints.
We consider a large number of relevant precision observables which can potentially be affected by the mDMFV model. These are in particular the constraints from meson-antimeson mixing, radiative and rare and decays, electroweak precision observables and electric dipole moments.
From the analysis of meson-antimeson mixing observables we identify a number of “flavor-safe” scenarios for the structure of . These scenarios will be useful for future studies of flavored DM models beyond MFV, as they can be imposed simply and render detailed re-analyses unnecessary.
Subsequently we perform a simultaneous analysis of flavor and DM constraints, such as the relic abundance from thermal freeze-out, and direct detection data from LUX Akerib:2013tjd (). While restricting ourselves to the phenomenologically interesting case of -flavored DM, we consider several mass hierarchies in the dark sector, i. e. large and small splittings between the DM particle and the heavier flavors.
We reveal a non-trivial interplay of the complementary flavor and DM constraints, such that the combined constraint on the parameter space of turns out to be interesting intersections of the individual ones. This result underlines the importance of taking into account the various constraints simultaneously.
We point out a cancellation between various mDMFV one-loop contributions (photon penguin and box diagram) to the WIMP-nucleon scattering, occurring for a certain range of coupling parameters. As the photon penguin is only present for scattering off protons, while the box diagram contributes to proton and neutron scattering cross-sections, this cancellation provides a possible realization of Xenophobic DM Feng:2011vu (); Feng:2013vod ().
We review the constraints from collider searches on the mDMFV model with -flavored DM. The most stringent bounds are placed by searches for bottom squark pair production, constraining the parameter space of the model up to a mediator mass . Monojet searches can be important for very compressed spectra, or for very heavy such that its direct production is suppressed.
Our paper is organized as follows. In section 2 we introduce the concept of Dark Minimal Flavor Violation (DMFV), and describe the minimal model realizing this hypothesis, the mDMFV model. Section 3 deals with the implications of the DMFV hypothesis that are valid beyond the minimal model. In section 4 we provide the formalism for a detailed study of the constraints from meson anti-meson mixing on the mDMFV model. We also consider potential new contributions to radiative and rare and decays, electroweak precision observables and electric dipole moments and find all of these observables to be SM-like. Section 5 is devoted to a detailed numerical analysis of the constraints on the coupling matrix arising from meson-antimeson mixing. We identify a number of “flavor-safe” scenarios for the coupling matrix . In section 6 we provide a comprehensive summary of the results of the numerical flavor analysis and the different scenarios emerging for the analysis of DM constraints. In section 7 we study the DM phenomenology of the mDMFV model, considering both the relic abundance constraint from thermal freeze-out and the emerging WIMP-nucleon cross section observed in direct detection experiments. A combined numerical analysis of flavor and DM constraints is performed in section 8, studying the various possible mass hierarchies in turn. In section 9 we estimate the constraints on the mDMFV model from the LHC, stemming in particular from monojet searches and searches for supersymmetric bottom squarks. We also mention some new signatures for these models. In section 10 we summarize our results. Some technical details are relegated to the appendices.
2 Flavored dark matter beyond MFV – a minimal model
We consider a setup where DM transforms in the fundamental representation of a new flavor symmetry , in analogy with the SM flavor symmetry. While we posit this symmetry as an ansatz, it will be an interesting future direction to study possible UV completions (e.g. extended Grand Unified Theories) which incorporate this structure.
We assume that the global
flavor symmetry is broken only by the SM Yukawa couplings , and the DM-quark coupling . This ansatz generalizes the MFV hypothesis D'Ambrosio:2002ex (); Buras:2000dm (); Buras:2003jf (); Chivukula:1987py (); Hall:1990ac () to include an extra symmetry under which the DM field transforms, and an additional Yukawa coupling . We refer to this assumption as Dark Minimal Flavor Violation (DMFV). Depending on the type of quark to which the DM couples, different classes of DMFV can be defined, see appendix A for details.
In what follows we restrict ourselves to the coupling of to right-handed down-type quarks via a scalar mediator . While the DM particle is a gauge singlet, the mediator has to carry color and hypercharge. This helps to keep the model simple, since no further electroweak structure is required when assuming the new particles to be singlets under . Further the choice of down-type quarks ensures to have an effect in relevant flavor observables such as and meson mixing and well-measured rare decays.
The most general renormalizable Lagrangian including the minimal field content is then given by
with the symmetry transformation properties summarized in table 1. Note that the flavor symmetry in the DM sector guarantees that at the Lagrangian level all three DM flavors have the same mass , although they acquire a small splitting from higher order DMFV corrections. In what follows we refer to this model as the minimal DMFV (mDMFV) model.
The mDMFV model has some similarities to simplified models of supersymmetry and should be understood in an analogous manner. In contrast to the SUSY case however in mDMFV the flavor charge is carried by the DM fermions and not by the scalar mediator. Further we assume to be a Dirac fermion (a Majorana mass term would violate the symmetry), while in the minimal SUSY models the gauginos are Majorana.
We stress that the DMFV ansatz, in contrast to the MFV ansatz, potentially allows for large flavor violating effects. A careful analysis of FCNC constraints is therefore necessary.
3 Implications of the Dark Minimal Flavor Violation hypothesis
In the present section we consider the consequences of the DMFV ansatz. We stress that these implications go beyond the simple mDMFV model introduced in section 2 and hold in any scenario with the same DMFV flavor symmetry breaking pattern.
3.1 New flavor violating parameters and a convenient parametrization for
In the DMFV setup the flavor symmetry in the quark sector is broken only by the SM Yukawa couplings , and the DM-quark coupling . In a first step the SM flavor symmetry can be used to remove unphysical parameters from the SM Yukawas. They can be parametrized as usual in terms of the six quark masses and the CKM matrix, signaling the misalignment between and .
In the second step we remove unphysical parameters from the coupling matrix . Being an arbitrary complex matrix, it contains at first 9 real parameters and 9 complex phases. Some of them can be removed by making use of the DM flavor symmetry .
We start by parametrizing in terms of a singular value decomposition
where is a diagonal matrix with real and positive entries, and and are unitary matrices. Note that and are not uniquely defined, as is invariant under the diagonal rephasing
We use this freedom to reduce the number of phases in to three. Then has 9 real parameters and 9 phases in the parametrization (4). We can now use the invariance to fully remove the unitary matrix . Consequently we are left with the matrix
It contains nine parameters: three non-negative elements of , and three mixing angles and three CP violating phases in . Note that the mixing angles are restricted to the range in order to avoid a double-counting of parameter space. This choice ensures that each DM flavor couples dominantly to the quark of the same generation. For instance we can refer to as -flavored DM.
A convenient parametrization for has been derived in Blanke:2006xr () in the context of the Littlest Higgs model with T-parity. It can be written as
where and . Performing the product one obtains the expression
Finally it turns out to be convenient to parametrize the diagonal matrix as
The first parametrization is useful for the analysis of DM and collider constraints. The second parametrization instead is better suited for the flavor analysis, since it quantifies the deviations from a flavor universal coupling.
3.2 Non-DMFV contributions and dark matter stability
The flavor structure of the SM is accidental — there exist no other gauge-invariant operators beyond the Yukawa terms at the renormalizable level. It is then worth asking if the DMFV ansatz can also arise naturally in an analogous way. In this section we study how generic the DMFV ansatz is from a UV point-of-view. We stress however that the goal of this work is merely to study the novel phenomenology arising from this ansatz, and a complete UV model is beyond the current scope. We merely study the corrections to the ansatz to the extent that they can affect low energy phenomenology, particularly DM decay.
Interestingly, in the exact DMFV limit, all operators inducing decay of the triplet are forbidden, even at the non-renormalizable level. In analogy to the stability of DM in the MFV case Batell:2011tc () it can straightforwardly be shown – see appendix B for details – that the flavor symmetry (2) broken only by the Yukawa couplings , and , together with imply an unbroken symmetry. It is then natural to impose this symmetry as exact, under which only the new particles and are charged, and transform as
This symmetry prevents the decay of any of these states into SM particles only, and therefore renders the lightest state stable.
We now estimate the size of non-DMFV effects that can arise. We imagine a UV scale, , above which the DM flavor symmetry is unbroken. While this scale could in principle be associated with the SM flavor scale as well, for simplicity we assume that the SM flavor structure is generated at a higher scale. Generically, we expect all operators allowed by symmetries to be generated at the scale . The most important contributions at low energy arise from relevant and marginal operators.
The relevant operator
is the leading operator that is generated. It maximally violates the DMFV ansatz, while preserving the . This operator can be prevented from being generated at the scale if the mass of the DM fermions is generated at a lower scale, through a flavor-blind sector. Note the analogy to flavor-blind SUSY breaking, which yields MFV. It is an open question whether such a scenario can be achieved simply in this framework. We assume henceforth that this operator is negligible.
At the marginal level, we generate the following two operators,
which are lepton and baryon number violating respectively. These are prohibited by the discrete symmetry, however.
We see that additional discrete symmetries can be imposed in order to prevent the DM from decaying, and preventing non-DMFV contributions at the renormalizable level. Higher dimensional non-DMFV operators can then only modify other aspects of phenomenology, which are not as severely constrained. Therefore, for a reasonably high scale , these operators are not expected to alter the phenomenology appreciably.
3.3 Mass splitting in the dark sector
As noted in section 2 the DMFV hypothesis ensures that to leading order in the coupling , the masses for different DM particles are equal. There are three potential sources for splittings.
Firstly, there can be contributions to the mass matrix directly violating DMFV. We assume that such contributions are absent. Secondly, higher dimensional DMFV-violating contributions can still induce splittings, but these are expected to be suppressed by the heavy scale where DMFV is broken.
An unavoidable contribution is through the renormalization group running, where a universal mass at the high scale is renormalized by the presence of the DM coupling at low scales. Generically, it is also possible that there is a DMFV preserving contribution at tree level. If present, this would be the largest contribution to the DM splittings. Of course, the pattern of splittings generated by the running and by such threshold effects is identical, since both cases are consistent with DMFV.
The splittings are given by
where summation is not implied in the last term. Here is a real coefficient whose value depends on the details of the model. If the contribution to the mass matrix arises at tree level, then is expected to be an number. On the other hand, the contribution from running is schematically given by
where is the dark flavor scale noted above.
The DMFV expansion above is only valid if higher order corrections are parametrically suppressed. In order to ensure convergence, in what follows we will assume .
4 Constraints from flavor and precision observables
In this section we study all relevant constraints from flavor observables on the mDMFV model. We start the analysis of the well-measured and strongly constraining observables from meson anti-meson mixing, followed by relevant rare decays. Finally we take a brief look at electroweak precision tests and electric dipole moments. While we will find that processes significantly shape the structure of a phenomenologically viable coupling matrix , effects of other flavor observables are negligible.
We restrict ourselves to providing the formulae directly relevant for our study, a more detailed description of relevant techniques and necessary formulae for the study of processes reaching from effective Hamiltonian to flavor observables can be found for instance in Blanke:2011ry (). A recent comprehensive review can be found in Buras:2013ooa ().
4.1 Constraints from meson anti-meson mixing
In the mDMFV model new contributions to processes arise first at the one loop level. The relevant box diagram is shown in figure 2 for the case of mixing. Evaluating this diagram we obtain the following contribution to the effective Hamiltonian:
with , and the loop function can be found in appendix C. As the new particles and couple only to right-handed down-type quarks, the only effective operator which receives new contributions is
i. e. the chirality-flipped counterpart of the SM operator.
The mass splittings among the fields constitutes a higher order correction in the DMFV expansion, which we assume to be small. Thus we can take the limit of equal masses in (16). The effective Hamiltonian then simplifies to
where . The loop function can be found in appendix C. We also defined
The mDMFV contribution of the DM sector to the off-diagonal element of the mass matrix can then be obtained from
The parameter summarizes the corrections from the renormalization group running from the weak scale down to the scale GeV, where the lattice calculations are performed, as well as the corrections due to the matching of the full theory to the effective theory calculated within the SM.
By parametrizing the NLO corrections by , we make two approximations. We neglect the running from the NP scale to the scale , as well as the difference in the matching conditions between the SM and the NP scenario studied here. In order to estimate the error associated to our approach, it is useful to compare our case with the discussion of the 331 models in Buras:2012dp (). In the latter framework the inclusion of the next-to-leading order (NLO) corrections amounts to a few percent correction to the size of the NP contribution. We expect similar conclusions to hold also in our case, in particular since in the MSSM the NLO corrections to the Wilson coefficient have been found to be small Virto:2009wm ().
In an analogous manner we find
where we define
In passing we note that the mDMFV model, coupling only to down-type quarks, does not contribute to meson observables at the one loop level.
4.2 Radiative and rare and decays
We now turn our attention to radiative and rare decays, starting with the electromagnetic dipole operator generating the transition.
The effective Hamiltonian describing the decay can be written as
where we omitted the tree level and chromomagnetic dipole operators that contribute to via renormalization group mixing. We use the normalization
In the SM the Wilson coefficient is strongly suppressed due to the chiral structure of weak interactions:
Conversely the DM in our scenario couples only to right-handed SM fermions – therefore the only relevant new contribution arises in the chirality-flipped Wilson coefficient . The relevant diagrams are analogous to the ones depicting the gluino contribution in supersymmetric models, replacing the gluino by the DM particles and the squarks by the scalar mediator , and keeping only the coupling to right-handed SM quarks. Correcting for the different coupling and taking into account that are QCD singlets while the gluino is a color octet, we can straightforwardly obtain the result for from Bertolini:1990if (); Cho:1996we (). Adjusting eq. (A.5) of Cho:1996we () to our model, we find
With the size of the new contribution can be estimated as
Comparing this result to the constraints on the size of NP contributions, see e. g. figure 2 in Altmannshofer:2013foa (), we see that the effect on the electromagnetic dipole operators generated in the present model is completely negligible. This is very welcome in view of the good agreement of with the data.
Contributions to the four-fermion operators mediating transitions like or can generally be split into tree level and one loop box and penguin diagrams. In the mDMFV model new tree level diagrams are forbidden by the residual symmetry (see appendix B), while box diagrams are not generated since the new particles and do not couple to leptons. We are hence left with potential contributions to the and photon penguins. An explicit calculation shows that the penguin contribution vanishes. This can be explained by the chiral structure of our model with the new particles coupling only to right-handed quarks, and is also confirmed by adapting the SUSY results of Cho:1996we () to our scenario. The photon penguin contribution is non-zero, however numerically small, as known from supersymmetric models Cho:1996we (); Altmannshofer:2013foa ().
In summary we are left with completely SM-like rare decays like , and . Consequently the mDMFV model does not ameliorate the tension in the data.
A bit more care is however required in the case of semileptonic decays with neutrinos in the final state, such as or . Since the neutrinos escape detection, the experimental signatures are and respectively. Consequently also the decays222We denote by the lightest flavor which is stable and provides the DM. and , mediated by a tree level exchange, will contribute to the measured branching ratio if the decay is kinematically allowed. See Kamenik:2011vy () for a detailed discussion. In order to avoid these potentially stringent constraints, in the remainder of our analysis we will assume and therefore well outside the kinematically allowed region for these decays.
4.3 Electroweak precision tests and electric dipole moments
Besides the flavor violating and decays discussed above, the flavor conserving electroweak precision constraints and the bounds on electric dipole moments also put strong constraints on many NP models. In this section we consider these observables within the mDMFV model.
We start by considering electroweak precision observables. Due to the residual symmetry corrections from the mDMFV model can arise only at the loop level and are therefore suppressed by a loop factor . Furthermore the mDMFV model introduces no new doublets, and only carries hypercharge. Consequently the contributions to electroweak precision observables receive an additional suppression by . Together with the loop factor and the scale above the electroweak scale we conclude that all new contributions to electroweak precision observables are safely small.
Similarly we also find no significant new contribution to electric dipole moments. The reasons are as follows. Due to the chiral structure of the mDMFV model with new particles coupling only to right-handed down-type quarks no EDM is generated at the one loop level. At the two loop level a Barr-Zee type diagram Barr:1990vd () with running in the loop exists – however its CP-violating phase is zero because the coupling is real.
5 Flavor pre-analysis of possible structures for the DM-quark coupling
We are now prepared to study the allowed regions of parameter space from flavor observables as well as correlations between different parameters of the coupling matrix .
5.1 Strategy of the numerical analysis
In order to determine the constraints from observables on the mDMFV model, we use the latest New Physics Fit results of the model-independent NP fit presented by the UTfit collaboration Bona:2005eu (). To this end we define
where is the full mixing amplitude containing both SM and mDMFV contributions. Furthermore
These six parameters are constrained by a global fit of the NP amplitude to the available tree level and data Bona:2005eu (); Bona:2007vi (). In order to be conservative we impose the resulting constraints at the level, see table 2 for a summary. In the case of we allow for a uncertainty in order to capture the poorly known long distance effects. For consistency we set the CKM parameters to their central values obtained in the UTfit fit. All other input parameters are set to their central values listed in table 3 of Buras:2013dea ().
Altogether, we have the following new parameters relevant for flavor violating decays:
Our goal is to obtain a clear picture of patterns in the coupling matrix that are implied by the available data. To this end we fix the flavor conserving parameters , and to the values
The impact of varying these parameters can be estimated from the functional dependence of (), which is roughly given by
Note that is a monotonically decreasing function varying from 1 to 1/3 over the range .
The constraints then translate directly into constraints on the values of , and , as shown in figure 3. We observe that the strongest constraints come from mixing, and in particular the CP-violating parameter , which forces the phase of to be very close to unless . The physics constraints are less stringent and in particular do not yield a specific pattern for the phases of . The weakest constraints are found in the system.
This pattern of allowed deviations from the SM is not specific to the mDMFV model, but can be found in all models with a generic NP flavor structure that do not induce the chirally enhanced left-right operators, like the Littlest Higgs model with T-parity analyzed in detail in Hubisz:2005bd (); Blanke:2006sb (); Blanke:2009am (). It is a direct consequence of the CKM hierarchies that determine the size of effects within the SM, as well as the theoretical uncertainties involved. Note that e. g. in Randall-Sundrum (RS) models with bulk fermions Csaki:2008zd (); Blanke:2008zb (); Bauer:2009cf () and left-right models Zhang:2007da (); Blanke:2011ry (), the strong enhancement of the left-right operators in the kaon system makes the constraints even more severe.
5.2 “Flavor-safe” scenarios for the structure of
We now analyze the structure of the coupling matrix that is implied by the constraints. To this end we show in figure 4 the allowed points in the , and spaces, respectively.
We observe that the allowed points fall into five distinct scenarios for the structure of , which we discuss in some detail in the following. In order to analytically understand the scenarios, we recall the parametrization of in terms of three two-flavor rotation matrices and a diagonal matrix :
universality scenario (black):
In this case so that . Since flavour violating effects are governed by the off-diagonal elements of , the constraints are trivially fulfilled for arbitrary and there are no FCNC effects beyond the SM.
If the first two generations of DM fermions are quasi-degenerate, then – as seen from the blue points in figure 4 – the mixing angle can be generic while have to be small. This can be understood by taking the limit , in which the mixing matrix becomes non-physical, and we are left with
It is easy to see that in order to fully suppress flavor violating effects we need and therefore .
In the case , shown by the red points in figure 4, the first and third DM flavor are quasi-degenerate, and consequently is unconstrained. In order to suppress the remaining flavor violating effects both and have to be small.
Finally if , the second and third DM flavor are quasi degenerate. Consequently the mixing angle is arbitrary, while and have to be small. This scenario is shown by the green points in figure 4.
small mixing scenario (yellow): arbitrary
Finally if does not exhibit any degeneracies, then FCNC effects have to be suppressed by the smallness of all three mixing angles . This scenario, shown by the yellow points in figure 4, corresponds to a diagonal but non-degenerate coupling matrix .
In order to quantify the allowed size of deviations from the degeneracy scenarios discussed above, we show in figure 5 the mixing angles as a function of the deviation from the corresponding degeneracy line. We observe that the constraint on and mixings are comparable with the former being somewhat stronger, while the constraint on the mixing angle is significantly weaker. This is a direct consequence of the allowed sizes of NP effects in the various meson systems, see figure 3.
5.3 A note on flavor safety of the UV completion
FCNC processes are known to be sensitive to NP at very high scales. It is therefore questionable whether a study of the simplified mDMFV model is sufficient to capture all relevant effects.
Following the DMFV principle we can write any contribution from the UV completion in terms of higher-dimensional operators that are suppressed by powers of the UV scale and made formally invariant under the flavor group (2) by insertion of the appropriate combination of spurion fields and . The leading contribution to the effective Hamiltonian is then
where is an coefficient that is common to all three meson systems. Comparing this to the new contribution generated first at the one loop level in the simplified model (see (16)), that can schematically be written as
we observe that both contributions carry the same flavor structure. Furthermore the UV contribution is suppressed with respect to the simplified model one if
Therefore, NP close to the mass of does not change the flavor phenomenology as long as it respects the DMFV hypothesis. Generic flavor violation needs to be suppressed by a much higher scale TeV Isidori:2010kg ().
5.4 Recovering the MFV limit in the structures for
Earlier studies of flavored DM have been restricted to the MFV framework in order to be safe from undesired effects in flavor observables. The flavor-safe scenarios identified above are more general than the MFV ansatz. It is also worthwhile to study how MFV can be recovered in the DMFV framework.
Let us first consider the case where is identified with . The MFV hypothesis then requires that takes the schematic form
where is an arbitrary coefficient. The matrix is diagonal in the down quark mass basis. In particular is proportional to the down-type quark masses. Considering that can be approximated by , MFV must be close to the 12-degeneracy. Additionally MFV requires
where is an arbitrary coefficient. The same expansion for is obtained when inserting (41) into the DMFV expansion (14), so that MFV in this case is consistent with the DMFV hypothesis. As and are diagonal in the same basis, all three flavor mixing angles are zero. Thus the MFV limit can be recovered as a very specific subset of parameter space, (determined by the specific choice of ) close to the 12-degeneracy line with all mixing angles zero.
If instead is identified with , then
where as before is an arbitrary coefficient. We can see immediately that the mass splittings are not directly correlated with the matrix, as was the case for DMFV (14) with the separate U(3) symmetry.
Finally identifying with , we have
where and are arbitrary coefficients. Again we observe that the pattern of splittings in are not directly correlated with the matrix.
In summary we find that if is assumed to transform under then the MFV limit can be recovered as a small subset of the scenarios for , with an approximate 12-degeneracy and all mixing angles identically zero. On the other hand, the MFV limits for transforming under or have mass splitting patterns which are not correlated with the coupling matrix , and therefore to capture these cases one needs to consider additional contributions to the mass splitting in (14).
6 From the flavor pre-analysis to dark matter scenarios
Our flavor pre-analysis shows that a generic coupling matrix leads to unacceptably large corrections to observables. We have identified a number of non-trivial scenarios for the structure of for which flavor violating effects are efficiently suppressed:
Universality scenario: all elements of the diagonal matrix equal and arbitrary flavor mixing angles.
-degeneracy scenarios (): , arbitrary and the other mixing angles small.
Small mixing scenario: small mixing angles and arbitrary .
While these scenarios have been identified in a scan with fixed flavor conserving parameters , and , we stress that these structures for also remain valid for different choices of parameters. Furthermore, even though our analysis has been performed within the simplified framework of the mDMFV model, the identified scenarios for remain flavor-safe in non-minimal versions of DMFV also. Thus they provide a useful framework for future study of the phenomenology of DMFV models – employing any of these scenarios for the structure of efficiently evades all FCNC constraints, without the need for an involved study of the latter.
|scenario||specification||lightest DM particle|
|universal scenario ()||-||all hierarchies possible|
|12 degeneracy ()|
|13 degeneracy ()|
|23 degeneracy ()||or|
|small mixing scenario||-||all hierarchies possible|
In table 3 we summarize the flavor-safe scenarios for and their implications for the mass pattern in the DM sector. It is clear that flavor constraints do not impose a specific mass hierarchy on the dark sector, i. e. from the point of FCNC constraints any dark flavor can be the lightest. Note that an exact degeneracy of two flavors is unnatural, since in the case of universal it is violated by the presence of at higher orders in the DMFV expansion. We therefore assume that the observed DM is composed of a single flavor, while the decay of the heavier states is fast enough to have happened in the early universe. We refer the reader to appendix D for an estimate of the life-time of the heavier states.
However not all DM flavors are equally motivated from the point of DM and collider phenomenology. DM that couples dominantly to first generation quarks, like -flavored DM, is strongly constrained by the direct detection experiments. If the DM relic density is assumed to arise from thermal freeze-out in the early universe, the relic abundance condition is in severe tension with the experimental constraints. We will therefore not consider the case of -flavored DM further.
As far as direct detection constraints are concerned, - and -flavored DM are on equal footing. Interestingly the same holds, at least qualitatively, also for the flavor phenomenology – as we have seen in figure 5 the amount of flavor violation allowed by constraints is almost symmetric under the exchange of the second and third generation, .
The case is however different for collider phenomenology. While pair production of the mediator and its subsequent decay will dominantly produce light jets and missing energy in the -flavored case, in the case of -flavored DM the large coupling to the quark will give rise to -jet signatures in a significant fraction of the events. Since events with -jets are much more easily distinguished from the QCD background, the collider phenomenology of -flavored DM is at the same time more constraining (in particular concerning the bound on the mediator mass) and also more promising, as quite distinctive signatures arise.
A further motivation for -flavored DM comes from indirect detection. Recently it has been shown that a 35 GeV provides a good fit to the excess -rays observed at the galactic center Agrawal:2014una ().
Therefore, in the rest of our analysis we restrict ourselves to the case of -flavored DM, i. e. . We also assume that the DM relic abundance is set by the thermal freeze-out condition, so that has to be large. Due to the strong constraints on the first generation coupling from direct detection and collider data, we deduce that . Consequently in order to ensure the correct mass hierarchy, we have .
We are then left with the following scenarios for DM freeze-out:
single flavor freeze-out: The - and -flavored states are split from the -flavored DM by at least 10%.
two flavor freeze-out:
13-degeneracy – and are quasi-degenerate, while is split
23-degeneracy – and are quasi-degenerate, while is split
three flavor freeze-out: All three states are quasi-degenerate. Such a scenario can either be achieved by a quasi universal coupling matrix , or if the DMFV expansion parameter is loop-suppressed, .
In our numerical analysis we will study all of these scenarios in turn.
7 Phenomenology of -flavored dark matter
In this section, we study the constraints arising from requiring the DM to be a thermal relic and from direct detection experiments. We note that the relic abundance constraints may be potentially relaxed in the presence of other particles in the dark sector.
The presence of multiple flavors can affect the DM freeze-out significantly. This occurs when the mass splitting between different flavors of DM is much smaller than the freeze-out temperature, (). For mass splittings much bigger than this scale, the freeze-out follows the standard WIMP paradigm.
We show that in the range of parameter space we consider, the heavier DM flavors decay before big bang nucleosynthesis (BBN) (see appendix D). The direct detection constraints then depend sensitively on the couplings of the lightest flavor of DM. In particular, when the lightest dark flavor couples appreciably to the first generation quarks, it gives rise to a very large direct detection signal. If this contribution is suppressed, the dominant contribution then arises at 1-loop level, which is seen to be within the reach of present and future direct detection experiments.
7.1 Relic abundance
We will consider two different qualitative regimes. When their masses are nearly degenerate, then all DM flavors are present during freeze-out and can be treated together. Otherwise only the lightest flavor of DM remains in the thermal bath.
We start with a single flavor freeze-out. The dominant annihilation during freeze-out occurs in the lowest partial wave. In this limit
where we have ignored the masses of the final state quarks.
The relic abundance is determined by solving the Boltzmann equation for the DM number density at late times. For a Dirac fermion, it is useful to convert the annihilation cross section into an effective cross section Griest:1990kh (); Servant:2002aq ().
which is approximately required to be Steigman:2012nb ()
in order to produce the correct relic abundance of DM.
Next we consider the case, where the mass splitting between the DM flavors is much smaller than the temperature at freeze-out. Consequently, we have to take into account the co-annihilation between different flavors. We assume that flavor changing (but DM number preserving) interactions are fast during the epoch of DM freeze-out. The rate for these processes is enhanced over the DM annihilations—which are approximately in thermal equilibrium—by a large Boltzmann factor (). Thus, this approximation is valid as long as any individual cross sections are not suppressed enough to overwhelm this factor.
Then the Boltzmann equation for freeze-out has a very similar form to the single DM case, and can be solved in exactly the same way. The relic abundance is in fact relatively insensitive to the change in the number of DM species, changing by only about 5% when other parameters are kept fixed. In the limit of small splitting, the effective cross section is well approximated by Griest:1990kh (),
The co-annihilation cross section can be derived by modifying equation (47), e.g.
Note that the splitting between DM masses is in this case negligible ().
If only two states are nearly degenerate, then a two flavor freeze-out occurs. The corresponding formulae can be straightforwardly obtained from the above results.
7.2 Direct detection
For -flavored DM, the direct detection scattering arises either through mixing, or at one loop.
We focus on the spin-independent contribution to the WIMP-nucleus scattering. The reported experimental bounds are translated to the WIMP-nucleon cross section, which can be written as
where is the reduced mass of the WIMP-nucleon system, and are the mass and atomic numbers of the nucleus respectively, and and parametrize DM coupling to neutrons and protons. The relevant processes are shown in figure 7.
For direct detection, we can safely work in the effective theory with the integrated out. The Lorentz structure of the four-fermion operator generated after performing the Fierz transformation is given by
There are three contributions to :
These contributions are individually given as follows:
-channel at tree-level:
In presence of significant mixing in the matrix, this is the dominant contribution to direct detection:
The spin-independent part in equation (53) arises from the matrix element of the quark vector current bilinear in the nucleons. Thus, only the valence -quark contributes.
One-loop photon exchange:
The interaction of DM with nucleons via photon exchange is conveniently parametrized as the electromagnetic form factors of the DM coupling with those of the nucleus. In particular, the scattering cross sections arise from charge-charge, dipole-charge and dipole-dipole interactions Agrawal:2011ze (). In the region of interest, the charge-charge interactions dominate, leading to
in the leading-log approximation.
The tree-level contribution, being flavor violating, constrains the mixing to be small. The LUX experiment Akerib:2013tjd () is sensitive to even the loop level scattering cross sections for WIMP DM. These contributions are present even in the absence of flavor violation. The box and the photon loop diagrams are seen to destructively interfere.
8 Combined numerical analysis of flavor and dark matter constraints
Having all relevant formulae for the DM phenomenology in hand, we are now ready to perform a combined numerical analysis of both DM and flavor constraints. We restrict ourselves to the phenomenologically most interesting case of -flavored DM, and study in turn the scenarios identified in section 6.
The DM mass is allowed to vary in the phenomenologically interesting region . We also assume in order to suppress the coupling to the first generation, in order to cope with the strong direct detection and collider constraints. Convergence of the DMFV expansion is ensured by requiring . Finally, since corrections to are unavoidably generated at the one loop level, we take . In summary,
8.1 Single flavor freeze-out
If the masses of the heavier flavors are sufficiently split from the DM mass