Baryogenesis and Dark Matter from Mesons
We present a new mechanism of Baryogenesis and dark matter production in which both the dark matter relic abundance and the baryon asymmetry arise from neutral meson oscillations and subsequent decays. This set-up is testable at hadron colliders and -factories. In the early Universe, decays of a long lived particle produce mesons and anti-mesons out of thermal equilibrium. These mesons/anti-mesons then undergo CP violating oscillations before quickly decaying into visible and dark sector particles. Dark matter will be charged under Baryon number so that the visible sector baryon asymmetry is produced without violating the total baryon number of the Universe. The produced baryon asymmetry will be directly related to the leptonic charge asymmetry in neutral decays; an experimental observable. Dark matter is stabilized by an unbroken discrete symmetry, and proton decay is simply evaded by kinematics. We will illustrate this mechanism with a model that is unconstrained by di-nucleon decay, does not require a high reheat temperature, and would have unique experimental signals – a positive leptonic asymmetry in meson decays, a new decay of mesons into a baryon and missing energy, and a new decay of -flavored baryons into mesons and missing energy. These three observables are testable at current and upcoming collider experiments, allowing for a distinct probe of this mechanism.
The Standard Model of Particle Physics (SM), while now tested to great precision, leaves many questions unanswered. At the forefront of the remaining mysteries is the quest for dark matter (DM); the gravitationally inferred but thus far undetected component of matter which makes up roughly 26% of the energy budget of the Universe Ade et al. (2016); Aghanim et al. (2018). Many models have been proposed to explain the nature of DM, and various possible production mechanisms to generate the the DM relic abundance – measured to be Aghanim et al. (2018) – have been proposed. However, experiments searching for DM have yet to shed light on its nature.
Another outstanding question may be stated as follows: why is the Universe filled with complex matter structures when the standard model of cosmology predicts a Universe born with equal parts matter and anti-matter? A dynamical mechanism, Baryogenesis, is required to generate the primordial matter-antimatter asymmetry; , inferred from measurements of the Cosmic Microwave Background (CMB) Ade et al. (2016); Aghanim et al. (2018) and Big Bang Nucleosynthesis (BBN) Cyburt et al. (2016); Tanabashi et al. (2018). A mechanism of Baryogenesis must satisfy the three Sakharov conditions Sakharov (1967); C and CP Violation (CPV), baryon number violation, and departure from thermal equilibrium.
It is interesting to consider models and mechanisms that simultaneously generate a baryon asymmetry and produce the DM abundance in the early Universe. For instance, in models of Asymmetric Dark Matter Nussinov (1985); Dodelson and Widrow (1990); Barr et al. (1990); Kaplan (1992); Farrar and Zaharijas (2006); Kaplan et al. (2009), DM carries a conserved charge just as baryons do. Most models of Baryogenesis and/or DM production involve very massive particles and high temperatures in the early Universe, making them impossible to test directly and in conflict with cosmologies requiring a low inflation or reheating scale.
In this work we present a new mechanism for Baryogenesis and DM production that is unconstrained by nucleon or dinucleon decay, accommodates a low reheating scale , and has distinctive experimental signals.
We will consider a scenario where -quarks and anti-quarks are produced by late, out of thermal equilibrium, decays of some heavy scalar field (which can be, for instance, the inflaton or a string modulus). The produced quarks hadronize to form neutral -mesons and anti-mesons which quickly undergo CP violating oscillations111For instance, the SM box diagrams that mediate the meson anti-meson oscillations contain CP violating phases due to the CKM matrix elements in the quark- vertices (see for instance Tanabashi et al. (2018) for a review). Additionally, models of new physics may introduce additional sources of CPV to the system Grossman et al. (1998)., and decay into a dark sector via a four Fermi operator i.e. a component of DM is assumed to be charged under baryon number. In this way the baryon number violation Sakharov condition is “relaxed” to an apparent violation of baryon number in the visible sector due to a sharing with the dark sector (in similar spirit to Agashe and Servant (2005); Davoudiasl et al. (2010)). The decay of mesons into baryons, mesons and missing energy would be a distinct signature of our mechanism that can be searched for at experiments such as Belle-II. Additionally, the operator allows us to circumvent constraints arising in models with baryon number violation.
We will show that the CPV required for Baryogenesis is directly related to an experimental observable in neutral meson decays – the leptonic charge asymmetry . Schematically,
where we sum over contributions from both and , and is the branching fraction of a meson into a baryon and DM (plus additional mesons ). Note that a positive value of will be required to generate the asymmetry. Given a model, the charge asymmetry can be directly computed from the parameters of the oscillation system (for instance see Tanabashi et al. (2018); Artuso et al. (2016) for reviews), and as such it is directly related to the CPV in the system. Meanwhile, is experimentally extracted from a combination of various analysis of LHCb and B-factories by examining the asymmetry in various decays Tanabashi et al. (2018).
The SM predictions for and Artuso et al. (2016); Lenz and Nierste (2011) are respectively a factor of 5 and 100 smaller than the current constraints on the leptonic asymmetry. Therefore, there is room for new physics to modify . We will see that since generating the baryon asymmetry in our set-up requires a positive charge asymmetry, there is a region of parameter space where we can get enough CPV from the SM prediction (which is positive) of alone to get (provided ). However, generically the rest of our parameter space will assume new physics. Note that there are many BSM models that allow for a substantial enlargement of the leptonic asymmetries of both and systems over the SM values (see e.g. Artuso et al. (2016); Botella et al. (2015) and references therein). Note that the flavorful models invoked to explain the recent -anomalies also induce sizable mixing in the system (see e.g. Altmannshofer et al. (2014); Celis et al. (2015); Gripaios et al. (2015); Bečirević, Damir and Fajfer, Svjetlana and Košnik, Nejc and Sumensari, Olcyr (2016)).
We summarize the key components of our set-up which will be further elaborated upon in the following sections:
A heavy scalar particle late decays out of thermal equilibrium to quarks and anti-quarks.
Since temperatures are low, a large fraction of these quarks will then hadronize into mesons and anti-mesons.
The neutral mesons undergo CP violating oscillations.
mesons decay into into the dark sector via an effective operator. This is achieved by assuming DM carries baryon number. In this way total baryon number is conserved.
Dark matter is assumed to be stabilized under a discrete symmetry, and proton and dinucleon decay are simply forbidden by kinematics.
Our set up is illustrated in Figure 1, and the details of a model that can generate such a process will be discussed below. This paper is organized as follows: in Section II we introduce a model that illustrates our mechanism for Baryogenesis and DM generation, this is accompanied by a discussion of the unique way in which this set-up realizes the Sakharov conditions. Next, in Section III we analyze the visible baryon asymmetry and DM production in the early Universe, by solving a set of Boltzmann equations, while remaining as agnostic as possible about the details of the dark sector. Our main results will be presented here. Next, in Section IV we discuss the various possible searches that could probe our model, and elaborate upon the collider, direct detection, and cosmological considerations that constrain our model. In Section V we outline the various possible dark sector dynamics. We conclude in Section VI.
Ii Baryogenesis and Dark Matter: Scenario and Ingredients
We now elaborate upon the details of our mechanism, and in particular highlight the unique way in which this proposal satisfies the Sakharov conditions for generating a baryon asymmetry. Afterwards we will present the details of an explicit model that will contain all the elements needed to minimally realize our mechanism of Baryogenesis and DM production.
ii.1 Cosmology and Sakharov Conditions
Key to our mechanism is the late production of -quarks and anti-quarks in the early Universe. To achieve this we assume that a massive, weakly coupled, long lived scalar particle dominates the energy density of the early Universe after inflation but prior to Big Bang nucleosynthesis. could be an Inflaton field, a string modulus, or some other particle resulting from preheating. is assumed to decay, out of thermal equilibrium to -quarks and anti-quarks. While our mechanism will work in scenarios in which is produced after inflation, for simplicity we will call the Inflaton. We only require that decays late enough so that the Universe is cool enough for the quarks to hadronize before they decay i.e.
The lower bound ensures that Baryogenesis completes prior to nucleosynthesis. Note that given a long lived scalar particle late quark production is rather generic – there is no obstruction to scenarios in which decays to a variety of heavy particles: particles mainly decay to massive particles, namely , quarks, and Higgs bosons. Therefore, particles produced by the late decays of will typically be either -quarks, or will have prompt decay modes with substantial branching fractions into -quarks. For definiteness we will simply assume that decays out of thermal equilibrium directly into and quarks.
The quarks, injected into the Universe at low temperatures, will mostly hadronize as mesons – , , and . Upon hadronization the neutral mesons will quickly undergo CP violating oscillations Tanabashi et al. (2018). Such CPV occurs in the SM (and is sizable in the systems), but could also be augmented by new physics. In this way a long lived scalar particle realizes, rather naturally, two of the Sakharov conditions – departure from thermal equilibrium and CPV. Interestingly, we will find a region in parameter space where our mechanism can work with just the CPV of the SM, contrary to the usual lore in which the CPV condition must come from beyond the SM physics.
Let us now address the remaining Sakharov condition: baryon number violation. While baryon number violation appears in the SM non-perturbatively Kuzmin et al. (1985), and is utilized in Leptogenesis models Fukugita and Yanagida (1986); Dolgov (1992); Buchmuller et al. (2005); Davidson et al. (2008); Drewes et al. (2018), the SM baryon number violation will be suppressed at the low temperatures we consider here (as it must to ensure the stability of ordinary matter). It is possible to engineer models that utilize low scale baryon number violation, but this usually requires an arguably less than elegant construction. For instance, in the setup of Ghalsasi et al. (2015); Aitken et al. (2017) baryon number violation occurred primarily in heavy flavor changing interactions so as to sufficiently suppress the di-nucleon decay rate, which required a very particular flavor structure. In the present work, we assume that DM is charged under baryon number, thereby allowing for the introduction of new baryon number conserving dark-SM interactions.
If the mesons, after oscillations, can quickly decay to DM (plus visible sector baryons), the CPV from oscillations will be transferred to the dark sector leading to a matter-antimatter asymmetry in both sectors. Critically, the total baryon number of the Universe, which is now shared by both visible and dark sectors, remains zero. In this way we have “relaxed” the baryon number violation Sakharov condition to an apparent Baryon number violation in the visible sector.
ii.2 An Explicit Model
We now present an explicit model which realizes our mechanism. Minimally, we introduce four new particles; a long lived weakly coupled massive scalar particle (discussed above), an unstable Dirac fermion carrying baryon number, and two stable DM particles – a Majorana fermion and a scalar baryon . All are assumed to be singlets under the SM gauge group. To generate effective interactions between the dark and visible sectors, we introduce a TeV mass, colored, electrically charged scalar particle . We assume a discrete symmetry to stabilize the DM. Table 1 summarizes the new fields (and their charge assignments) introduced in this model. Possible extensions to this minimal scenario will be considered in later sections.
Operators and Charges
To generate renormalizable interactions between the visible and dark sectors, we a assume a UV model similar to that of Ghalsasi et al. (2015); Aitken et al. (2017). We introduce a electrically charged, baryon number , color triplet scalar which can couple to SM quarks. Such a new particle is theoretically motivated, for instance could be a squark of a theory in which a linear combination of the SM baryon number and a symmetry is conserved Beauchesne et al. (2017). The details of the exact nature and origin of are not important for the present set-up. Additionally, we introduce a new neutral Dirac fermion carrying baryon number .
The renormalizable couplings between and allowed by the symmetries include222We have suppressed fermion indices for simplicity as there is a unique Lorentz and gauge invariant way to contract fields. In particular, the and are SU(2) singlet right handed Weyl fields. Under , the first term of Equation (2) is the fully anti-symmetric combination of three fields, which is gauge invariant. While the second term is a singlet.:
We take the mass of the colored scalar to be and integrate out the field for energies less than its mass, resulting in the following four fermion operator in the effective theory:
Other flavor structures may also be present but for simplicity we consider only the effects of the above couplings (see Appendix .4 for other possible operators). Assuming is sufficiently light, the operator of Equation (3) allows the -quark within to decay; , or equivalently , where parametrizes mesons or other additional SM particles. Critically, note that in Equation (3) is a operator, so that the operator in Equation (3) is baryon number conserving since carries a baryon number .
In this way our model allows for the symmetric out of thermal equilibrium production of mesons and anti-mesons in the early Universe, which subsequently undergo CP violating oscillations i.e. the rate for will differ from that of . After oscillating the mesons and anti-mesons decay via Equation (3) generating an asymmetry in visible baryon/anti-baryon and dark / particles (the decays themselves do not introduce additional sources of CPV), so that the total baryon asymmetry of the Universe is zero.
Since, no net baryon number is produced, this asymmetry could be erased if the particles decay back into visible anti-baryons. Such decays may proceed via a combination of the coupling in Equation (3) and weak loop interactions, and are kinematically allowed since to ensure the stability of neutron stars McKeen et al. (2018). To preserve the produced visible/dark baryon asymmetry, the particles should mainly decay into stable DM particles. This is easily achieved by minimally introducing a dark scalar baryon with baryon number , and a dark Majorana fermion . We further assume a discrete symmetry under which the dark particles transform as , and . Then the decay can be mediated by a renormalizable Yukawa operator:
which is allowed by the symmetries of our model. And in particular, the (in combination with kinematic constraints), will make the two dark particles, and , stable DM candidates.
In this way an equal and opposite baryon asymmetry to the visible sector is transferred to the dark sector, while simultaneously generating an abundance of stable DM particles. The fact that our mechanism proceeds through an operator that conserves baryon number alleviates the majority of current bounds that would otherwise be very constraining (and would require less than elegant model building tricks to evade). Furthermore, the decay of a -meson (both neutral and charged) into baryons, mesons and missing energy would yield a distinctive signal of our mechanism at B-factories and hadron colliders. An example of a meson decay process allowed by our model is illustrated in Figure 2.
Note that, as in neutrino systems, neutral meson oscillations will only occur in a coherent system. Additional interactions with the mesons can act to “measure” the system and decohere the oscillations Cirelli et al. (2012); Tulin et al. (2012), thereby suppressing the CPV and consequently diminishing the generated asymmetry. Spin-less mesons do not have a magnetic moment. However, due to their charge distribution, scattering of directly off mesons can still decohere the oscillations (see Appendix .1 for details). To avoid decoherece effects, the mesons must oscillate at a rate similar to or faster then the scattering in the early Universe.
Parameter Space and Constraints
To begin to explore the parameter space of our model we note that the particle masses must be subject to several constraints. For the decay to be kinematically allowed we have the following:
Note that there is also a kinematic upper bound on the mass of the such that it is light enough for the decay to be allowed. This bound depends on the specific process under consideration and final state visible sector hadrons produced, for instance in the example of Figure 2 it must be the case that . A comprehensive list of the possible decay processes and the corresponding constraint on the mass are itemized in Appendix .4.
As mentioned above, DM stability is ensured by the symmetry, and the following kinematic condition:
The mass of a dark particle charged under baryon number must be greater then the chemical potential of a baryon in a stable two solar mass neutron star McKeen et al. (2018). This leads to the following bound333Note that constraints on bosonic asymmetric DM from the black hole production in neutron stars McDermott et al. (2012) do not apply to our model. In particular, we can avoid accumulation of particles if they annihilate with a neutron into particles. Additionally, there can be repulsive self couplings which greatly raises the minimum number required to form a black hole.:
Additionally, the constraint (7) automatically ensures proton stability.
The corresponding restrictions on the range of particle masses, along with the rest of our model parameter space, is summarized in Table 2.
Dark Sector Considerations
Throughout this work we remain as model independent as possible regarding additional dark sector dynamics. Our only assumption is the existence of the dark sector particles , and . In general the dark sector could be much richer; containing a plethora of new particles and forces. Indeed, scenarios in which the DM is secluded in a rich dark sector are well motivated by top-down considerations (see for instance Essig et al. (2013) for a review). Additionally, there are practical reasons to expect (should our mechanism describe reality) a richer dark sector.
The ratio of DM to baryon energy density has been measured to be Aghanim et al. (2018). Therefore, for the case where is the lightest dark sector particle, it must be the case that . Since does not carry baryon number and decays completely, once all of the symmetric component annihilates away we will be left with: , implying that – inconsistent with the kinematics of mesons decays (). Introducing additional dark sector baryons can circumvent this problem.
For instance, imagine adding a stable dark sector state . We assume carries baryon number , and in general be given a charge assignment which allows for interactions (e.g. ). Then the condition that becomes: ). Interactions such as can then reduce the number density, such that in thermodynamic equilibrium we need only require that , while can be somewhat heavier. In principle may have fractional baryon number so that both decay kinematics and proton stability are not threatened.
Additionally, the visible baryon and anti-baryon products of the decay are strongly interacting, and as such generically annihilate in the early Universe leaving only a tiny excess of baryons which are asymmetric. Meanwhile, the and particles are weakly interacting and have masses in the few GeV range. Since, as given the CP violation is at most at the level of the DM will generically be overproduced in the early Universe unless the symmetric component of the DM undergoes additional number density reducing annihilations. One possible resolution is if the dark sector contained additional states, which interacted with the and allowing for annihilations to deplete the DM abundance so that the sum of the symmetric () and the antisymmetric () components match the observed DM density value.
We defer a discussion of specific models leading to the depletion of the symmetric DM component to Section V. In what follows, we simply assume a minimal dark particle content and consider the interplay between , , and via Equation (4), and account for additional possible dark sector interactions with a free parameter.
Iii Baryon Asymmetry and Dark Matter Production in the Early Universe
Using the explicit model of Sec. II.2, we now perform a quantitative computation of the relic baryon number and DM densities. We will show that it is indeed possible to produce enough CPV from B meson oscillations to explain the measured baryon asymmetry in the early Universe. Interestingly, there will be a region of parameter space where the positive SM asymmetry in oscillations is alone, without requiring new physics contributions, sufficient to generate the matter-antimatter asymmetry. Additionally, we will see that a large parameter space exists that can accommodate the measured DM abundance. To study the interplay between production, decay, annihilation and radiation in the era of interest we study the corresponding Boltzmann equations.
iii.1 Boltzmann Equations
The expected baryon asymmetry and DM abundance are calculated by solving Boltzmann equations that describe the number and energy density evolution of the relevant particles in the early Universe: the late decaying scalar , the dark particles , , and radiation (). The processes of hadronization, oscillations and decay happen very rapidly compared with the lifetime; therefore allowing for approximations that significantly simplify the Boltzmann equations. We justify these assumptions below and in Appendix .2.
Radiation and the Inflaton
First we describe the evolution of and its interplay with radiation. need not be the Inflaton, but for simplicity we assume that at times much earlier than , the energy density of the Universe was dominated by non-relativistic particles, and that all of the radiation and matter of the current Universe resulted from decays. Furthermore, the inflaton decay products are very rapidly converted into radiation, and as such the Hubble parameter during the era of interest is:
The Boltzmann equations describing the evolution of the inflaton number density and the radiation energy density read:
where the source terms on the right-hand side of (9) describe the decays which cause the number density of to decrease as energy is being dumped into radiation. Note that if we pick an initial time , then is small enough that there is no sensitivity to initial conditions and may set . In practice, we assume that at some high , was in thermal equilibrium with the plasma and that at some temperature it decouples; fixing the number density to . This number density serves as our the initial condition and is subsequently evolved using Equation (9). For numeric purposes, we assume that the scalar decouples at . We note that, as expected, our results will not be sensitive to the exact decoupling temperature provided i.e. when all the SM particles except the top, Higgs and Electroweak bosons are still relativistic.
The Boltzmann equation for the dark Majorana fermion , the main DM component in our model when , reads:
where we have assumed that the processes of / production, hadronization and decay to the dark sector (see Appendix .2), all happen very rapidly on times scales of interest i.e. the particle production and subsequent decay happens rapidly and completely and we need not track the abundance. Therefore, the second term on the right hand side of Equation (11) entirely accounts for the dark particle production via the decays , and so we have defined:
Here is the decay width, and is the inclusive branching ratio of mesons into a baryon plus DM.
The quarks and anti-quark within all flavors of mesons and anti-mesons (both neutral and charged and ), will contribute to the abundance via decays through the operators in Equations (3) and (4). Therefore, in Equation (11), we have implicitly set the branching fraction of into charged and neutral mesons: . Note that only the neutral mesons can undergo CP violating oscillations thereby contributing to the matter-antimatter asymmetry. Therefore, we should account for the branching fraction into mesons and anti-mesons when considering the asymmetry.
The first term on the right hand side of Equation (11) allows for additional interactions, whose presence we require to deplete the symmetric DM component as discussed above.
For the region in parameter space where , DM is composed of the scalar baryons and anti-baryons, and the DM relic abundance is found by solving for the symmetric component, namely:
Analogously to the Boltzmann equation describing the evolution, the second term on the right hand side of Equation (13) accounts for possible dark sector interactions and self-annihilations, while the first term describes dark particle production via decays. Again we assume the fermion decays instantaneously, and DM can be produced from the decay of both neutral and charged mesons and anti-mesons.
As previously discussed, DM generically tends to be overproduced in this set-up. Additional interactions are required to deplete the DM abundance in order to reproduce the observed value. Whether the DM is comprised primarily of or , the scattering term in the Boltzmann equations allows for the dark particle abundance to be depleted by annihilations into lighter species. In our model, the thermally averaged annihilation cross sections for the fermion and scalar will receive contributions from generated by the Yukawa coupling of Equation (see Appendix .3 for rates). This interaction will transform the heavier dark particle population into the lighter DM state. The annihilation term can, in general, receive contributions from additional interactions. Therefore, when solving the Boltzmann equations, we simply parametrize additional contributions to and by a free parameter. In Sec. V, we will outline a couple of concrete models that accommodate a depletion of the symmetric DM component.
We have derived Equation (13) by tracking the particle and anti-particle evolution of the complex scalar using the following Boltzmann equations:
where we sum over contributions from oscillations. Likewise,
Since the the and particles are produced via several combinations of meson/anti-meson oscillations and decays, we encapsulate the corresponding decay width difference in a quantity (defined explicitly below in Equation (17)), which is a measure of the CPV in the and systems. is weighted by a function describing decoherence effects – these will play a critical role in the evolution of the matter-antimatter asymmetry as we discuss below. For the symmetric DM component, the solution of Equation (13), the dependance on cancels off as expected.
Finally, note that Equations (13) and (11) hold in the regime where the two masses and are significantly different. For the case where coannihilations become important i.e. there will be rapid processes mediated by which will enforce a relation between and . Specifically, in the non-relativistic limit , so that the equilibrium abundance depends on the dark sector temperature. It is reasonable to consider a construction where , so that it is justified to set the equilibrium abundance of the heavier particle to zero. However, since coannihilations represents a very small branch in our parameter space, for simplicity and generality, we simply assume we are far from the regime where coannihilations effects are important so that we can solve Equations (11), (14) and (15) for the dark sector particle abundances.
|Inflaton width||Decay between|
|Dirac fermion mediator||3.3 GeV||Lower limit from|
|Majorana DM||1.0 and 1.8 GeV|
|Scalar DM||1.5 and 1.3 GeV||,|
|Br of||Tanabashi et al. (2018)|
|Lepton Asymmetry||Tanabashi et al. (2018)|
|Lepton Asymmetry||Tanabashi et al. (2018)|
|Annihilation Xsec for||Depends upon the channel Aghanim et al. (2018)|
|Annihilation Xsec for||Depends upon the channel Aghanim et al. (2018)|
where we must consider contributions from decays of the anti-quarks/quarks within both and mesons/anti-mesons: we take the branching fraction for the production of each meson to be and according to the latest estimates Tanabashi et al. (2018).
Interestingly, we see from integrating Equation (III.1) that the baryon asymmetry is fixed by the product – a measurable quantity at experiments. In particular, is defined as:
which is directly related to the CPV in oscillating neutral meson systems. Here and are taken to be final states that are accessible by the decay of / only. Note that as defined, Equation (17) corresponds to the semi-leptonic asymmetry (denoted by in the literature) in which the final state may be tagged. However, at low temperatures and in the limit when decoherence effects are small, this is effectively equivalent to the leptonic charge asymmetry for which one integrates over all times. Therefore, in the present work we will use the two interchangeably.
Maintaining the coherence of oscillation is crucial for generating the asymmetry; additional interactions with the mesons can act to “measure” the state of the meson and decohere the oscillation Cirelli et al. (2012); Tulin et al. (2012), thereby diminishing the CPV and so too the generated baryon asymmetry. mesons, despite being spin-less and charge-less particles, may have sizable interactions with electrons and positrons due to the ’s charge distribution. Electron/positron scattering , if faster than the oscillation, can spoil the coherence of the system. We have explicitly found that this interaction rate is two orders of magnitude lower than for a generic baryon Aitken et al. (2017), but for temperatures above the process occurs at a much higher rate than the meson oscillation and therefore precludes the CP violating oscillation. We refer the reader to Appendix .1 for the explicit calculation of the scattering process in the early Universe.
Generically, decoherence will be insignificant if oscillations occur at a rate similar or faster then the meson interaction. By comparing the rate with the oscillation length , we construct a step-like function (we have explicitly checked that a Heaviside function yields similar results) to model the loss of coherence of the oscillation system in the thermal plasma:
Even without numerically solving the Boltzmann equations, we can understand the need for additional interactions in the dark sector . From Equations (11) and (13), we see that the DM abundance is sourced by ; the greater the value of this branching fraction, the more DM is generated. From Equation (III.1), we see that the asymmetry also depends on this parameter but weighted by a small number; . Therefore, generically a region of parameter space that produces the observed baryon asymmetry will overproduce DM, and we require additional interactions with the DM to deplete this symmetric component and reproduce .
iii.2 Numerics and Parameters
We use Mathematica Inc. () to numerically integrate the set of Boltzmann Equations (9), (10), (11), (13), and (III.1) subject to the constraint Equation (8). To simplify the numerics it is useful to use the temperature as the evolution variable instead of time. Conservation of energy yields the following relation Scherrer and Turner (1988); Hannestad (2004):
which above the neutrino decoupling temperatures simplifies to Venumadhav et al. (2016):
We can therefore use Equation (20) in place of Equation (10). For the number of relativistic species contributing to entropy and energy and , we use the values obtained in Laine and Schroder (2006). Finally, since the DM particles generically have masses greater then a GeV we can safely neglect the inverse scatterings in the DM Boltzmann equations i.e. the term. To make the integration numerically straightforward we change variables and solve the equations for and , such that . Note, that we also convert to the convenient yield variables .
The parameter space of our model includes the particle masses, the Inflation decay width, the dark Yukawa coupling, the branching ratio of mesons to DM and a hadrons, the leptonic asymmetry, and the dark sector annihilation cross sections. Table 2 summarizes the parameters and the range of over which they are allowed to vary taking into account all constraints.
The upper limit on the mass is imposed because above , the scalar could potentially have a small branching fraction to quarks (see e.g. Djouadi et al. (1998)).
DM masses are constrained by kinematics, and neutron star stability – Equations (6) and (7). We take the Yukawa coupling in the dark sector to be since this value enables an efficient depletion of the heavier DM state to the lower one, thus simplifying the phenomenology. For sufficiently lower values of this coupling we may require interactions of both the and states with additional particles. The current bounds Tanabashi et al. (2018) on the leptonic asymmetry read and for the and systems respectively. Note that these values allow for additional new physics contributions beyond those expected from the SM alone Artuso et al. (2016); Lenz and Nierste (2011): and . While there is no direct search for the branching ratio , we can constrain the range of experimentally viable values. For instance, in the example of Figure 2 where the produced baryon is a , we can, based on the decay to , set the bound at 95% CL Tanabashi et al. (2018).
iii.3 Results and Discussion
The recent Planck CMB observations imply a comoving baryon asymmetry of Aghanim et al. (2018). In our scenario, even without fully solving the system of Boltzmann equations, we can see from integrating Equation (III.1) that the baryon asymmetry directly depends upon the product of leptonic asymmetry times branching fraction:
Meanwhile, the DM relic abundance is measured to be Aghanim et al. (2018) and reads (where is the current entropy density and is the critical density). In Figure 3 we display the results (the comoving number density of the various components) of numerically solving the Boltzmann equations for two sample benchmark points that reproduce the observed DM abundance and baryon asymmetry.
Consider the plot on the right panel of Figure 3, which corresponds to the case where DM is comprised of and particles. We can understand the behavior of the particle yields as follows: particles starts to decay at , thereby increasing the abundance of the dark particles and until at which point decay completes (as it must, so that the predictions of BBN are preserved). The dip in the dark particle yields at lower temperatures is the necessary effect of the additional annihilations – which reduce the yield to reproduce to the observed DM abundance. Meanwhile, the asymmetric component is only generated for , as it is only then that the CPV oscillations are active in the early Universe. The decrease in the asymmetric component at is due to the negative contribution of the decays, since in this case the leptonic asymmetry is chosen to be negative. Note that for the case in which the DM is mostly comprised of and particles the observed baryon asymmetry and DM abundance imply an asymmetry of
The plot on the left panel of Figure 3 corresponds to the case where DM is mostly comprised of particles. In this case the evolution of the dark particles is rather similar. Here we have chosen , so that the asymmetric component gets two positive contributions at from both and CPV oscillations. While at the change in behavior of the yield curve corresponds to the contribution from the oscillations – given that the oscillation time scale is 20 times smaller than the one, the contribution it is active at higher temperatures.
The Baryon Asymmetry
In order to make quantitative statements, beyond the benchmark examples discussed above, we have explored the parameter space outlined in Table 2 and mapped out the regions that reproduce the observed baryon asymmetry of the Universe. From Equation (III.1), we see the baryon asymmetry depends on the product of the leptonic asymmetry times branching fraction (with contributions from both and mesons), as well as the Inflaton mass and width. The result of this interplay is displayed in Figure 3, where the contours correspond to the value the product of needed to reproduce the asymmetry for a given point in space. For simplicity, the left and right panels show the effect of considering either the or the contributions but generically both will contribute.
While the entire parameter space in Figure 4 is allowed by the range of uncertainty in the experimentally measured values of , our range of prediction is further constrained. In particular, the blue region in Figure 4 is excluded by a combination of constraints on the leptonic asymmetry and the branching ratio Tanabashi et al. (2018) (see Section IV), while the lower bound comes from requiring that the Inflaton not spoil the measured effective number of neutrino species from CMB and the measured primordial nuclei abundances de Salas et al. (2015). Therefore, to reproduce the expected asymmetry coming from, for instance, only , we find (depending upon the Inflaton width and mass).
Interestingly, the baryon asymmetry can be generated with only the SM leptonic asymmetry , provided that and that (which is compatible with current data) – see the green region in the right panel of Figure 4. Additionally, if new physics enhances up to the current limit , Baryogenesis could take place with a branching fraction as low as . Figure 5 shows that even with a negative , as expected in the SM, the baryonic asymmetry can be generated with provided that the . We reiterate that both the leptonic asymmetry and the decay of a meson to a baryon and missing energy are measurable quantities at B-factories and hadron colliders (see Section IV).
The Dark Matter Abundance
As previously argued, in the absence of additional interactions, our set-up generically tends to overproduce the DM since the leptonic asymmetry is . By examining the DM yield curve in Figure 3 we see that annihilations (the dip in the curve) deplete the DM abundance that would otherwise be overproduced from the Inflaton decay.
Recall that for a stable particle species annihilating into particles in the early Universe when neglecting inverse-annihilations: . For WIMPs produced through thermal freeze-out , while, in our scenario . Therefore, an annihilation cross section roughly one order of magnitude higher than that of the usual WIMP is required to obtain the right DM abundance. We have analyzed the extrema of the parameter space and found that we require the dark cross section to be
where , and the spread of values correspond to varying the DM mass over the range specified in Table 2 (with only a very slight sensitivity to other parameters). In particular .
Primordial Antimatter with a low Reheat Temperature
Finally, note that since we are considering rather low reheat temperatures, there could be a significant change to the primordial antimatter abundance. In the case of a high reheat temperature scenario, the primordial antinucleon abundance is tiny: Kolb and Turner (1990). In our scenario, we can track the antinucleon abundance from the following Boltzmann equation:
where is the produced number of antinucleons per decay Cirelli et al. (2011). By solving this Boltzmann equation we find that for the primordial antinucleon abundance is (and usually way smaller) and too small to have any phenomenological impact at the CMB or during BBN.
Iv Searches and Constraints
Developing a testable mechanism of Baryogenesis has always been challenging. Likewise, should a DM detection occur, nailing down the specific model in set-ups where a rich hidden dark sector is invoked, is generally daunting. The scenario described in the present work is therefore unique in that it is potentially testable by future searches at current and upcoming experiments, while being relatively unconstrained at the moment.
iv.1 Searches at LHCb and Belle-II
As discussed above, a positive leptonic asymmetry in meson oscillations, and the existence of the new decay mode of mesons into visible hadrons and missing energy, would both indicate that our mechanism may describe reality. Both these observables are testable at current and upcoming experiments.
Semileptonic Asymmetry in decays
As shown in Section III.3, the model we present requires a positive and relatively large leptonic asymmetry: . The current measurements of the semileptonic asymmetry Tanabashi et al. (2018) (recall that in our setup the semileptonic and leptonic asymmetries may be used interchangeably) are:
These are extracted from a combination of various analysis of LHCb and -factories. Future and current experiments will improve upon this measurement. In particular, the future reach of LHCb with for the measurement of the leptonic asymmetry is estimated to be Artuso et al. (2016), and a similar sensitivity should be expected for . To our knowledge, the reach of Belle-II on the semileptonic asymmetries has not been addressed Kou et al. (2018). However, a naive rescaling of the Belle 2005 result Nakano et al. (2006) could optimistically lead to sensitivities at the level of for . In addition, Belle-II is planning on collecting of data at the resonance Kou et al. (2018) which could potentially result on a sensitivity comparable to that of LHCb for .
meson decays into a Baryon and missing energy
For our mechanism to produce the observed Baryon asymmetry in the Universe, from Figure 5, we notice that moderately large are required. To our knowledge there are no searches available to measure this branching fraction, and no published data on the inclusive branching fraction for mesons either. We expect that existing data from Babar, Belle and LHC can already be used to place a meaningful limit. The search for this channel should in principle be similar to other meson missing energy final states like or with current bounds at the level of Tanabashi et al. (2018). Given this, the reach of Belle-II Kou et al. (2018) could be of . Thus, potentially our mechanism is fully testable.
Exotic -flavored Baryon decays
Our Baryogenesis and DM production mechanism requires the presence of the new exotic meson decays. However, once these decays are kinematically allowed, the -flavored baryons will also decay in an apparently baryon violating way to mesons and DM in the final state. For instance, given the interaction (3) the baryon could decay into provided which will always be the case since in this case. In addition, the rate of this process should be very similar to that of mesons. To our knowledge there is no current search for this decay channel, but in principle LHCb could search for it. In particular, Ref. Stone and Zhang (2014) pointed out that it is possible to identify the initial energy of a if it comes from the decay of by measuring the kinematic distribution of the process. The LHCb collaboration has very recently observed Aaij et al. (2018) candidates produced via this process, thus making the measurement of potentially viable at hadron colliders. We refer to Appendix .4 where the lightest states of the possible decay processes are outlined for the four different flavor operators.
Considerations from Flavor Observables
Recall that, the semileptonic asymmetry may be computed theoretically from the off-diagonal elements of the Hamiltonian describing the system:
In the SM, oscillations arise from quark- box diagrams, whose vertices contain CPV phases from the CKM matrix. In particular, the SM predictions for read and Lenz and Nierste (2011); Artuso et al. (2016). Since these are substantially smaller then the current measurements (IV.1) there is room to accommodate new physics.
The large positive leptonic asymmetry required in our set-up could differ considerably from the SM values, depending on the value of . There are many BSM models that allow for a substantial enlargement of the semileptonic asymmetries of both the and systems (see e.g. Artuso et al. (2016); Botella et al. (2015) and references therein). In addition, it is worth mentioning, that the flavorful models invoked to explain the recent -anomalies also induce sizable mixing in the system (see e.g. Altmannshofer et al. (2014); Celis et al. (2015); Gripaios et al. (2015); Bečirević, Damir and Fajfer, Svjetlana and Košnik, Nejc and Sumensari, Olcyr (2016)).
Note, that while the elements of the evolution Hamiltonian (25) are not directly probed in experiment, they can be related to additional experimental observables as:
where, for instance , the meosn oscillation length, is related to the mass eigenstates. Therefore, any new physics that modifies away from the SM value will also modify and , and must not be in conflict with current bounds on these observables. For an overview of the allowed BSM modifications to and see Artuso et al. (2016).
Here we comment on collider, cosmological, and DM direct detection constraints.
Within our set-up, the heavy colored scalar is responsible for inducing the meson decays into the dark sector via Equation (2). The colored scalar may be produced at colliders and its decay products searched for, thus resulting on an indirect constraint on the model. This branching ratio was calculated in Aitken et al. (2017) and we quote the result here;