KCL-18-53, IFIC-18-35

Baryogenesis and Dark Matter from B Mesons

1, 2, 3, 1, Gilly Elor, ∗ Miguel Escudero, † and Ann E. Nelson ‡ 1Department of Physics, Box 1560, University of Washington, Seattle, WA 98195, U.S.A. 2From 09/18: Department of Physics, King’s College London, Strand, London WC2R 2LS, UK 3Instituto de F´ısica Corpuscular (IFIC), CSIC-Universitat de Val`encia,Paterna E-46071, Valencia, Spain 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 B meson oscillations and subsequent decays. This set-up is testable at hadron colliders and B-factories. In the early Universe, decays of a long lived particle produce B 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 B 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 B meson decays, a new decay of B mesons into a baryon and missing energy, and a new decay of b-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.

I. INTRODUCTION DM carries a conserved charge just as baryons do. Most models of Baryogenesis and/or DM production involve The Standard Model of Particle Physics (SM), while very massive particles and high temperatures in the early now tested to great precision, leaves many questions Universe, making them impossible to test directly and unanswered. At the forefront of the remaining mysteries in conflict with cosmologies requiring a low inflation or is the quest for dark matter (DM); the gravitationally reheating scale. inferred but thus far undetected component of matter In this work we present a new mechanism for Baryo- which makes up roughly 26% of the energy budget of the genesis and DM production that is unconstrained by nu- Universe [1,2]. Many models have been proposed to ex- cleon or dinucleon decay, accommodates a low reheating scale T (10 MeV), and has distinctive experimen- plain the nature of DM, and various possible production RH ∼ O mechanisms to generate the the DM relic abundance – tal signals. 2 measured to be ΩDMh = 0.1200 0.0012 [2] – have been We will consider a scenario where b-quarks and anti- proposed. However, experiments± searching for DM have quarks are produced by late, out of thermal equilibrium, yet to shed light on its nature. decays of some heavy scalar field Φ (which can be, for Another outstanding question may be stated as fol- instance, the inflaton or a string modulus). The pro- lows: why is the Universe filled with complex mat- duced quarks hadronize to form neutral B-mesons and ter structures when the standard model of cosmology anti-mesons which quickly undergo CP violating oscilla- 1 predicts a Universe born with equal parts matter and tions , and decay into a dark sector via a ∆B = 0 four anti-matter? A dynamical mechanism, Baryogenesis, is Fermi operator i.e. a component of DM is assumed to be required to generate the primordial matter-antimatter charged under baryon number. In this way the baryon 11 number violation Sakharov condition is “relaxed” to an asymmetry; YB (nB nB¯ )/s = (8.718 0.004) 10− , inferred from measurements≡ − of the Cosmic± Microwave× apparent violation of baryon number in the visible sector arXiv:1810.00880v3 [hep-ph] 21 Feb 2019 Background (CMB) [1,2] and Big Bang Nucleosynthe- due to a sharing with the dark sector (in similar spirit to sis (BBN) [3,4]. A mechanism of Baryogenesis must [13, 14]). The decay of B mesons into baryons, mesons satisfy the three Sakharov conditions [5]; C and CP Vi- and missing energy would be a distinct signature of our olation (CPV), baryon number violation, and departure mechanism that can be searched for at experiments such from thermal equilibrium. as Belle-II. Additionally, the ∆B = 0 operator allows us It is interesting to consider models and mechanisms to circumvent constraints arising in models with baryon that simultaneously generate a baryon asymmetry and number violation. produce the DM abundance in the early Universe. For instance, in models of Asymmetric Dark Matter [6–11],

1 For instance, the SM box diagrams that mediate the meson anti- meson oscillations contain CP violating phases due to the CKM ∗ [email protected] matrix elements in the quark-W vertices (see for instance [4] for † [email protected] a review). Additionally, models of new physics may introduce [email protected] additional sources of CPV to the B0 B¯0 system [12]. ‡ − 16

And now we can clearly compare the decay and annihi- 4. B meson decay operators lation rates: n 2 B B = B (45) n2 v v n (t) B h i h i where in the last step we have assumed that the field does not completely dominate the Universe so that we can use t 1/(2H). When solving numerically for number density⇠ we found that even with an annihila- tion cross section of v = 10 mb, the decay rate over- h i comes the annihilation rate for T & 100 MeV even for Here we categorize the lightest final states for all the 21 = 10 GeV. Thus, for practical purposes it is safe quark combinations that allow for B mesons to decay into to ignore the e↵ect of annihilations in the Boltzmann a visible baryon plus dark matter. Note that the mass equation (16). di↵erence between final an initial state will give an upper bound on the dark Dirac baryon .InMeVunits,the

masses of the di↵erent hadrons read: mBd = 5279.63, m = 5366.89, m + = 5279.32, m 0 = 2471.87, 3. Dark Cross Sections Bs B ⌅c m + = 2468.96, m = 938.27, m = 939.56, m = ⌅c p n ⇤ + 0 Here we list the dark sector cross sections to lowest 1115.68, m⌃ = 1189.37, m⌅ = 1314.86, m⌦c = 2695.2, m = 2286.46 and m = 139.57. The corresponding order in velocity v that result from the interaction (5): ⇤c ⇡ final state and mass di↵erences are summarized in Ta- ble III. 4 2 3/2 yd (m⇠ + m ) [(m m⇠)(m⇠ + m)] ? ⇠⇠ = , ! 2 2⇡m3 m2 + m2 + m2 ⇠ ⇣2 4 ⌘ v yd 2 ? ⇠⇠ m 0 = 4 (46) ! | ! 2 2 ⇥ Out 48of equilibrium⇡ m⇠ + m B-mesons decay into CP violating oscillations late time decay Dark Matter and hadrons Baryogenesis 5 4 2 ⇣ 3 3 ⌘ + 0 0 ⇠ 11 2m⇠m +5m⇠m +8¯ m⇠m B B Bs Dark Matter YB =8.7 10 b d ⇥ 2 4 5 6 6 anti-Baryon +9m m +6m m +3m +3m B & ⇥ ⇠ ⇠ ⇠ ) b ¯0 ¯0 Baryon B Bd Bs ⇤ ⇤ Dark Matter d s 2 TRH 20 MeV A A BR(B ⇠ + Baryon + ...) ⌦ h =0.12 ⇠ `` `` ! DM

FIG. 1. Summary of our mechanism for generating the baryon asymmetry and DM relic abundance. b-quarks and anti-quarks [1] P. A. R. Adeare produced et al. during (Planck), a late era in the Astron. history of the early Astrophys. Universe, namely594TRH , (10 MeV), and88 hadronize,045002(2016),1511.09466. into charged and neutral B-mesons. The neutral B0 and B¯0 mesons quickly undergo CPV oscillations∼ O before decaying out of thermal equilibrium A13 (2016),into 1502.01589. visible baryons, dark sector scalar baryons φ and dark Majorana fermions ξ. Total Baryon[15] numberA. Lenz is conserved and and U. the Nierste, in CKM unitarity triangle. Pro- dark sector therefore carries anti-baryon number. The mechanism requires of a positive leptonic asymmetry in B-meson decays [2] N. Aghanimq et al. (Planck) (2018), 1807.06209. ceedings, 6th International Workshop, CKM 2010, War- (A``), and the existence of a new decay of B-mesons into a baryon and missing energy. Both these observables are testable at [3] R. H. Cyburt,current B. and upcomingD. Fields, collider experiments. K. A. Olive, and T.-H. Yeh, wick, UK, September 6-10, 2010 (2011), 1102.4274. Rev. Mod. Phys. 88,015004(2016),1505.01076. [16] F. J. Botella, G. C. Branco, M. Nebot, and A. Snchez, We will show that the CPV required for Baryogenesis is We summarize the key components of our set-up which [4] M. Tanabashidirectly et related al. to an (ParticleDataGroup), experimental observable in neutral Phys.will be Rev.further elaborated uponPhys. in the following Rev. sections:D91,035013(2015),1402.1181. B meson decays – the leptonic charge asymmetry Aq . D98,030001(2018). `` [17] W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, Schematically, A heavy scalar particle Φ late decays out of thermal [5] A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5,32(1967),• equilibrium to b quarks andPhys. anti-quarks. Rev. D89,095033(2014),1403.1269. Y Aq Br(B0 φ ξ + Baryon + X) , (1) B ∝ `` × q → Since temperatures are low, a large fraction of these [Usp. Fiz. Nauk161,61(1991)].q=s,d • [18] A. Celis, J. Fuentes-Martin, M. Jung, and H. Serodio, X b quarks will then hadronize into B mesons and anti- [6] S. Nussinov, Phys. Lett. 165B,55(1985).0 ¯ mesons. Phys. Rev. D92,015007(2015),1505.03079. where we sum over contributions from both Bs = b s 0 ¯ 0 | i [7] S. Dodelsonand andBd = b L. d , and M. Br( Widrow,Bq φξ + Baryon Phys. + X) Rev. is the D42The,326 neutral mesons[19] undergoB. CP Gripaios, violating oscillations. M. Nardecchia, and S. A. Renner, JHEP 05, branching fraction| i of a B meson→ into a baryon and DM • (1990). (plus additional mesons X). Note that a positive value of B mesons decay into into the006 dark (2015), sector via an 1412.1791. effec- • Aq will be required to generate the asymmetry. Given a tive ∆B = 0 operator. This is achieved by assuming [8] S. M. Barr,`` R. S. Chivukula, and E. Farhi, Phys. Lett. [20] D. Be?irevi?, S. Fajfer, N. Konik, and O. Sumensari, model, the charge asymmetry can be directly computed DM carries baryon number. In this way total baryon 0 B241,387(1990).from the parameters of the Bq oscillation system (for in- number is conserved. Phys. Rev. D94,115021(2016),1608.08501. stance see [4, 15] for reviews), and as such it is directly q Dark matter is assumed to be stabilized under a dis- [9] D. B. Kaplan,related Phys. to the CPV Rev. in the Lett. system.68 Meanwhile,,741(1992).A is [21] V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov, `` • crete symmetry, and proton and dinucleon decay experimentally extracted from a combination of various Z2 [10] G. R. Farrar and G. Zaharijas, Phys. Rev. Lett.are simply96, forbidden by kinematics.Phys. Lett. 155B,36(1985). analysis of LHCb and B-factories by examining the asym- 0 041302 (2006),metry in hep-ph/0510079. various Bq decays [4]. Our set up is illustrated[22] in FigureS.1 Davidson,, and the details E. of a Nardi, and Y. Nir, Phys. Rept. 466,105 The SM predictions for Ad and As [15, 16] are re- [11] D. E. Kaplan, M. A. Luty, and`` K.`` M. Zurek, Phys.model that Rev. can generate such(2008), a process will 0802.2962. be discussed spectively a factor of 5 and 100 smaller than the cur- below. This paper is organized as follows: in SectionII D79,115016(2009),0901.4117.rent constraints on the leptonic asymmetry. Therefore, we introduce a model[23] that illustratesM. Drewes, our mechanism B. for Garbrecht, P. Hernandez, M. Kekic, d, s there is room for new physics to modify A`` . We will Baryogenesis and DM generation, this is accompanied by [12] Y. Grossman,see that Y. since Nir, generating and the R. baryon Rattazzi, asymmetry in Adv. our a Ser. discussion Di- of the unique wayJ. in Lopez-Pavon, which this set-up re- J. Racker, N. Rius, J. Salvado, and rect. Highset-up Energy requires Phys. a positive15 charge,755(1998),[,755(1997)], asymmetry, there is a alizes the Sakharov conditions.D. Next, Teresi, in Section Int.III we J. Mod. Phys. A33,1842002(2018), region of parameter space where we can get enough CPV analyze the visible baryon asymmetry and DM produc- s hep-ph/9701231.from the SM prediction (which is positive) of A`` alone tion in the early Universe, by1711.02862. solving a set of Boltzmann 10 d to get YB 10− (provided A`` = 0). However, gener- equations, while remaining as agnostic as possible about [13] H. Davoudiasl,ically the∼ restD. of E. our parameter Morrissey, space will K. assume Sigurdson, new the details and of the dark[24] sector.K. Our Aitken, main results D. will McKeen, be T. Neder, and A. E. Nelson, Phys. S. Tulin, Phys.physics. Rev. Note that Lett. there105 are many,211304(2010),1008.2399. BSM models that al- presented here. Next, in SectionRev.IV weD96 discuss,075009(2017),1708.01259. the var- low for a substantial enlargement of the leptonic asymme- ious possible searches that could probe our model, and 0 0 [14] M. Artuso,tries G. of Borissov,both Bd and Bs andsystems A. over Lenz, the SM values Rev. (see Mod.elaborate Phys. upon the collider,[25] H. direct Beauchesne, detection, and cos- K. Earl, and T. Gregoire, JHEP 06,122 e.g. [15, 17] and references therein). Note that the flavor- mological considerations that constrain our model. In ful models invoked to explain the recent B-anomalies also SectionV we outline the various possible dark sector dy- induce sizable mixing in the Bs system (see e.g. [18–21]). namics. We conclude in SectionVI. 3

II. BARYOGENESIS AND DARK MATTER: number violation will be suppressed at the low tempera- SCENARIO AND INGREDIENTS tures we consider here (as it must to ensure the stability of ordinary matter). It is possible to engineer models that We now elaborate upon the details of our mechanism, utilize low scale baryon number violation, but this usu- and in particular highlight the unique way in which this ally requires an arguably less than elegant construction. proposal satisfies the Sakharov conditions for generating For instance, in the setup of [28, 29] baryon number vio- a baryon asymmetry. Afterwards we will present the de- lation occurred primarily in heavy flavor changing inter- tails of an explicit model that will contain all the elements actions so as to sufficiently suppress the di-nucleon decay needed to minimally realize our mechanism of Baryoge- rate, which required a very particular flavor structure. In nesis and DM production. 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. A. Cosmology and Sakharov Conditions If the B mesons, after oscillations, can quickly decay to 0 ¯0 DM (plus visible sector baryons), the CPV from Bq Bq oscillations will be transferred to the dark sector leading− Key to our mechanism is the late production of b- to a matter-antimatter asymmetry in both sectors. Crit- quarks and anti-quarks in the early Universe. To achieve ically, the total baryon number of the Universe, which this we assume that a massive, weakly coupled, long is now shared by both visible and dark sectors, remains lived scalar particle Φ dominates the energy density of zero. In this way we have “relaxed” the baryon num- the early Universe after inflation but prior to Big Bang ber violation Sakharov condition to an apparent Baryon nucleosynthesis. Φ could be an Inflaton field, a string number violation in the visible sector. modulus, or some other particle resulting from preheat- ing. Φ is assumed to decay, out of thermal equilibrium to b-quarks and anti-quarks. We only require that Φ B. An Explicit Model decays late enough so that the Universe is cool enough (10 MeV) for the b quarks to hadronize before they ∼decay O i.e. We now present an explicit model which realizes our mechanism. Minimally, we introduce four new particles; a long lived weakly coupled massive scalar particle Φ TBBN < T < TQCD . (discussed above), an unstable Dirac fermion ψ carrying The lower bound ensures that Baryogenesis completes baryon number, and two stable DM particles – a Majo- prior to nucleosynthesis. Note that given a long lived rana fermion ξ and a scalar baryon φ. All are assumed to scalar particle late b quark production is rather generic be singlets under the SM gauge group. To generate effec- – there is no obstruction to scenarios in which Φ decays tive interactions between the dark and visible sectors, we to other heavy particles: e.g. Φ particles which mainly introduce a TeV mass, colored, electrically charged scalar decays to t-quarks, or Higgs bosons, as these also will particle Y . We assume a discrete Z2 symmetry to stabi- promptly decay to b quarks. Furthermore it is very lize the DM. TableI summarizes the new fields (and their typical for and there− is no symmetry preventing scalar charge assignments) introduced in this model. Possible particles from mixing with the Higgs Boson and hence extensions to this minimal scenario will be considered in primarily decaying into b-quarks. For definiteness we will later sections. simply assume that Φ decays out of thermal equilibrium directly into b and ¯b quarks. The b quarks, injected into the Universe at low tem- Operators and Charges 0 0 peratures, will mostly hadronize as B mesons – Bd, Bs , 0 and B±. Upon hadronization the neutral Bq mesons will To generate renormalizable interactions between the 0 ¯0 visible and dark sectors, we a assume a UV model sim- quickly undergo CP violating Bq Bq oscillations [4]. Such CPV occurs in the SM (and is− sizable in the B sys- ilar to that of [28, 29]. We introduce a 1/3 electri- cally charged, baryon number 2/3, color− triplet scalar tems), but could also be augmented by new physics. In − this way a long lived scalar particle realizes, rather nat- Y which can couple to SM quarks. Such a new parti- urally, two of the Sakharov conditions – departure from cle is theoretically motivated, for instance Y could be a thermal equilibrium and CPV. Interestingly, we will find squark of a theory in which a linear combination of the a region in parameter space where our mechanism can SM baryon number U(1)B and a U(1)R symmetry is con- work with just the CPV of the SM, contrary to the usual served [30]. The details of the exact nature and origin lore in which the CPV condition must come from beyond of Y are not important for the present set-up. Addition- the SM physics. ally, we introduce a new neutral Dirac fermion ψ carrying baryon number 1. Let us now address the remaining Sakharov condition: − baryon number violation. While baryon number viola- tion appears in the SM non-perturbatively [22], and is utilized in Leptogenesis models [23–27], the SM baryon 4

Field Spin QEM Baryon no. Z2 Mass

Φ 0 0 0 +1 11 100 GeV − Y 0 1/3 2/3 +1 (TeV) ⇠ − − O ψ 1/2 0 1 +1 (GeV) Y − O s ξ 1/2 0 0 1 (GeV) − O ¯b u φ 0 0 1 1 (GeV) 0 ⇤ − − O Bd d d TABLE I. Summary of the additional fields (both in the UV and effective theory), their charges and properties required in our model. FIG. 2. An example diagram of the B meson decay process as mediated by the heavy colored scalar Y that results in DM and a visible baryon, through the interactions of Equation (2) The renormalizable couplings between ψ and Y allowed and Equation (4). by the symmetries include2:

c ¯ c yub Y ∗ u¯ b yψs Y ψ s + h.c . (2) Since, no net baryon number is produced, this asym- L ⊃ − − metry could be erased if the ψ particles decay back into We take the mass of the colored scalar to be mY visible anti-baryons. Such decays may proceed via a (TeV) and integrate out the field Y for energies less∼ O combination of the coupling in Equation (3) and weak than its mass, resulting in the following four fermion op- loop interactions, and are kinematically allowed since erator in the effective theory: mψ > 1.2 GeV to ensure the stability of neutron stars y y [31]. To preserve the produced visible/dark baryon asym- = ub ψs u s b ψ . (3) Heff m2 metry, the ψ particles should mainly decay into stable Y DM particles. This is easily achieved by minimally in- Other flavor structures may also be present but for sim- troducing a dark scalar baryon φ with baryon number plicity we consider only the effects of the above couplings 1, and a dark Majorana fermion ξ. We further assume − (see Appendix4 for other possible operators). Assuming a discrete Z2 symmetry under which the dark particles ψ is sufficiently light, the operator of Equation (3) allows transform as ψ ψ, φ φ and ξ ξ. Then the → → − → − the ¯b-quark within B = ¯b q to decay; ¯b ψ u s, or ψ decay can be mediated by a renormalizable Yukawa q | i → equivalently Bq ψ+Baryon+X, where X parametrizes operator: mesons or other→ additional SM particles. Critically, note y ψ¯ φ ξ , (4) that = u s b in Equation (3) is a ∆B = 1 operator, L ⊃ − d O so that the operator in Equation (3) is baryon number which is allowed by the symmetries of our model. And in conserving since ψ carries a baryon number 1. − particular, the Z2 (in combination with kinematic con- In this way our model allows for the symmetric out of straints), will make the two dark particles, ξ and φ, stable thermal equilibrium production of B mesons and anti- DM candidates. mesons in the early Universe, which subsequently un- In this way an equal and opposite baryon asymmetry to 0 ¯0 dergo CP violating oscillations i.e. the rate for B B the visible sector is transferred to the dark sector, while will differ from that of B¯0 B0. After oscillating→ the → simultaneously generating an abundance of stable DM mesons and anti-mesons decay via Equation (3) gener- particles. The fact that our mechanism proceeds through ating an asymmetry in visible baryon/anti-baryon and an operator that conserves baryon number alleviates the ¯ dark ψ/ψ particles (the decays themselves do not intro- majority of current bounds that would otherwise be very duce additional sources of CPV), so that the total baryon constraining (and would require less than elegant model asymmetry of the Universe is zero. building tricks to evade). Furthermore, the decay of a B- 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 ex- 2 We have suppressed fermion indices for simplicity as there is a ample of a B meson decay process allowed by our model unique Lorentz and gauge invariant way to contract fields. In particular, the sc and bc are SU(2) singlet right handed Weyl is illustrated in Figure2. fields. Under SU(3)c, the first term of Equation (2) is the fully Note that, as in neutrino systems, neutral B meson anti-symmetric combination of three 3¯ fields, which is gauge in- oscillations will only occur in a coherent system. Addi- variant. While the second term is a 3¯ 3 = 1 singlet. × tional interactions with the mesons can act to “measure” 5 the system and decohere the oscillations [32, 33], thereby Dark Sector Considerations suppressing the CPV and consequently diminishing the generated asymmetry. Spin-less B mesons do not have a magnetic moment. However, due to their charge dis- Throughout this work we remain as model independent tribution, scattering of e± directly off B mesons can still as possible regarding additional dark sector dynamics. decohere the oscillations (see Appendix1 for details). Our only assumption is the existence of the dark sector To avoid decoherece effects, the B mesons must oscillate particles ψ, ξ and φ. In general the dark sector could be 0 0 at a rate similar to or faster than the e±B e±B much richer; containing a plethora of new particles and scattering in the early Universe. → forces. Indeed, scenarios in which the DM is secluded in a rich dark sector are well motivated by top-down consid- erations (see for instance [35] for a review). Additionally, Parameter Space and Constraints there are practical reasons to expect (should our mecha- nism describe reality) a richer dark sector. To begin to explore the parameter space of our model The ratio of DM to baryon energy density has been we note that the particle masses must be subject to sev- measured to be 5.36 [2]. Therefore, for the case where eral constraints. For the decay ψ φ ξ to be kinemati- φ is the lightest dark sector particle, it must be the case cally allowed we have the following:→ that m n 5m n . Since ξ does not carry baryon φ φ ∼ p B mφ + mξ < mψ . (5) number and ψ decays completely, once all of the sym- metric ψ component annihilates away we will be left with: Note that there is also a kinematic upper bound on the n = n , implying that m 5m – inconsistent with mass of the ψ such that it is light enough for the decay B φ φ p the kinematics of B mesons decays∼ (m < m m ). B/B¯ ψ/ψ¯ + Baryon/anti-Baryon + Mesons to be al- φ B Baryon Introducing additional dark sector baryons can− circum- lowed.→ This bound depends on the specific process under vent this problem. consideration and the final state visible sector hadrons produced; for instance in the example of Figure2 it must For instance, imagine adding a stable dark sector state be the case that mψ < mB0 mΛ 4.16 GeV. A com- . We assume carries baryon number Q , and in gen- d − ' A prehensive list of the possible decay processes and the eralA be given aA charge assignment which allows for φ corresponding constraint on the ψ mass are itemized in interactions (e.g. Q = 1/3). Then the conditionA that − A Appendix4. ρDM 5ρB becomes: mφnφ + m n 5mpnB). Inter- ∼ A A ∼ As mentioned above, DM stability is ensured by the actions such as φ + A∗ + can then reduce the ↔ A A Z2 symmetry, and the following kinematic condition: φ number density, such that in thermodynamic equilib- rium we need only require that m 5Q mp, while φ mξ mφ < mp + me . (6) A ∼ A | − | can be somewhat heavier. In principle may have frac- The mass of a dark particle charged under baryon number tional baryon number so that both B Adecay kinematics must be greater than the chemical potential of a baryon and proton stability are not threatened. in a stable two solar mass neutron star [31]. This leads to the following bound3: Additionally, the visible baryon and anti-baryon prod- ucts of the B decay are strongly interacting, and as such mψ > mφ > 1.2 GeV . (7) generically annihilate in the early Universe leaving only Additionally, the constraint (7) automatically ensures a tiny excess of baryons which are asymmetric. Mean- proton stability. while, the ξ and φ particles are weakly interacting and The corresponding restrictions on the range of particle have masses in the few GeV range. Since, as given the 3 masses, along with the rest of our model parameter space, CP violation is at most at the level of 10− , the DM is summarized in TableII. will generically be overproduced in the early Universe unless the symmetric component of the DM undergoes Note that since mψ must be heavier than the proton, the charmed D meson is too light for our baryogenesis additional number density reducing annihilations. One possible resolution is if the dark sector contained addi- mechanism to work, as mD < mp + mψ (similarly for tional states, which interacted with the ξ and φ allowing the Kaons since mK < mp). As the top quark decays too quickly to hadronize, the only meson systems in the for annihilations to deplete the DM abundance so that ? SM that allow for this Baryogenesis mechanism are the the sum of the symmetric (mξnξ + mφ[nφ + nφ]) and ? neutral B mesons. the antisymmetric (mφ[nφ nφ]) components match the observed DM density value.− We defer a discussion of specific models leading to the 3 Note that constraints on bosonic asymmetric DM from the black depletion of the symmetric DM component to SectionV. hole production in neutron stars [34] do not apply to our model. In particular, we can avoid accumulation of φ particles if they In what follows, we simply assume a minimal dark parti- annihilate with a neutron into ξ particles. Additionally, there can cle content and consider the interplay between ψ, φ, and be φ4 repulsive self-couplings which greatly raise the minimum ξ via Equation (4), and account for additional possible number required to form a black hole. dark sector interactions with a free parameter. 6

III. BARYON ASYMMETRY AND DARK The Boltzmann equations describing the evolution of MATTER PRODUCTION IN THE EARLY the Φ number density and the radiation energy density UNIVERSE read:

dnΦ Using the explicit model of Sec.IIB, we now perform + 3HnΦ = ΓΦnΦ , (9) a quantitative computation of the relic baryon number dt − dρrad and DM densities. We will show that it is indeed pos- + 4Hρrad = ΓΦmΦnΦ , (10) sible to produce enough CPV from B meson oscillations dt to explain the measured baryon asymmetry in the early where the source terms on the right-hand side of (9) de- Universe. Interestingly, there will be a region of parame- 0 scribe the Φ decays which cause the number density of ter space where the positive SM asymmetry in Bs oscil- Φ to decrease as energy is being dumped into radiation. lations is alone, without requiring new physics contribu- Note that if we pick an initial time t 1/ΓΦ, then ρrad is tions, sufficient to generate the matter-antimatter asym- small enough that there is no sensitivity to initial condi- metry. Additionally, we will see that a large parameter tions and may set ρ = 0. In practice, we assume that space exists that can accommodate the measured DM rad at some high T > mΦ, Φ was in thermal equilibrium with abundance. To study the interplay between production, the plasma and that at some temperature T it decou- decay, annihilation and radiation in the era of interest we dec ples; fixing the Φ number density to n (T ) = ζ(3) T 3 . study the corresponding Boltzmann equations. Φ dec π2 dec This number density serves as our the initial condition and is subsequently evolved using Equation (9). For nu- meric purposes, we assume that the scalar decouples at A. Boltzmann Equations Tdec = 100 GeV. We note that, as expected, our results will not be sensitive to the exact decoupling temperature The expected baryon asymmetry and DM abundance provided Tdec > 15 GeV i.e. when all the SM particles are calculated by solving Boltzmann equations that de- except the top, Higgs and Electroweak bosons are still scribe the number and energy density evolution of the relativistic. relevant particles in the early Universe: the late decay- ing scalar Φ, the dark particles ξ, φ, φ? and radiation (γ, e±, ν, ...). To properly account for the neutral B me- Dark Sector son CPV oscillations in the early Universe one should resort to the density matrix formalism [32, 33] (which The Boltzmann equation for the dark Majorana is considerably involved and widely used in Leptogenesis fermion ξ, the main DM component in our model when models [27, 36]). However, in our scenario, the processes mξ < mφ, reads: of hadronization, B meson oscillations, decays, as well as possible decoherence effects happen very rapidly (τ < ps) dnξ 2 2 B + 3Hnξ = σv ξ (n n ) + 2 Γ nΦ , (11) compared with the Φ lifetime (τΦ ms). This allows us dt −h i ξ − eq,ξ Φ to work in terms of Boltzmann equations,∼ and we account for possible decoherent effects in a mean field approxima- where we have assumed that the processes of b/¯b pro- tion for the B mesons in the thermal plasma. We defer duction, hadronization and decay to the dark sector (see Appendices1 and2 to the justifications of the approx- Appendix2), all happen very rapidly on times scales imations that we use below to simplify the resulting set of interest i.e. the ψ particle production and subsequent of Boltzmann equations. decay happen rapidly and completely and we need not track the ψ abundance. Therefore, the second term on the right hand side of Equation (11) entirely ac- Radiation and the Φ field counts for the dark particle production via the decays Φ BB¯ dark sector + visible, and so we have de- fined:→ → First we describe the evolution of Φ and its interplay with radiation. For simplicity we assume that at times B ΓΦ ΓΦ Br(B φξ + Baryon + X) . (12) much earlier than 1/ΓΦ, the energy density of the Uni- ≡ × → verse was dominated by non-relativistic Φ particles, and Here ΓΦ is the Φ decay width, and Br(B φξ+Baryon+ that all of the radiation and matter of the current Uni- X) is the inclusive branching ratio of B→mesons into a verse resulted from Φ decays. Furthermore, the Φ decay baryon plus DM. products are very rapidly converted into radiation, and The b quarks and anti-quark within all flavors of B as such the Hubble parameter during the era of interest 0 mesons and anti-mesons (both neutral and charged Bd,s is: and B±), will contribute to the ξ abundance via de- cays through the operators in Equations (3) and (4). 1 da 2 8π H2 = (ρ + m n ) . (8) Therefore, in Equation (11), we have implicitly set the ≡ a dt 3m2 rad Φ Φ   P l branching fraction of Φ into charged and neutral B 7 mesons: Br(Φ BB¯ ) = 1. Note that only the neu- Likewise, 0 → tral Bd,s mesons can undergo CP violating oscillations thereby contributing to the matter-antimatter asymme- dnφ? + 3Hn ? = σv (n n ? n n ? ) (15) try. Therefore, we should account for the branching frac- dt φ −h iφ φ φ − eq,φ eq,φ tion into B0 mesons and anti-mesons when considering s,d B q ¯ 0 q the asymmetry. + Γ nΦ 1 A Br(b B ) f . Φ × − `` → q deco " q # The first term on the right hand side of Equation (11) X allows for additional interactions, whose presence we re- quire to deplete the symmetric DM component as dis- Since the the φ and φ∗ particles are produced via sev- cussed above. eral combinations of meson/anti-meson oscillations and decays, we encapsulate the corresponding decay width For the region in parameter space where mξ > mφ, q difference in a quantity A`` (defined explicitly below in DM is composed of the scalar baryons and anti-baryons, Equation (17)), which is a measure of the CPV in the and the DM relic abundance is found by solving for the 0 0 q q Bd and Bs systems. A`` is weighted by a function fdeco symmetric component, namely: describing decoherence effects – these will play a critical role in the evolution of the matter-antimatter asymmetry dn φ+φ∗ + 3 H n = 2 ΓB n (13) as we discuss below. For the symmetric DM component, dt φ+φ∗ − Φ Φ q the solution of Equation (13), the dependence on A`` 2 2 2 σv φ n n . cancels off as expected. − h i φ+φ∗ − eq, φ+φ∗ Finally, note that Equations (13) and (11) hold in the
Analogous to the Boltzmann equation describing the ξ regime where the two masses mφ and mξ are significantly evolution, the second term on the right hand side of different. For the case where mφ mξ coannihilations ∼ Equation (13) accounts for possible dark sector interac- become important i.e. there will be rapid φ + φ∗ ξ + ξ ↔ tions and self-annihilations, while the first term describes processes mediated by ψ which will enforce a relation be- dark particle production via decays. Again we assume tween nξ and nφ+φ∗ . Specifically, in the non-relativistic the ψ fermion decays instantaneously, and DM can be limit nξ/nφ = exp (mφ mξ)/TD, so that the equilib- − produced from the decay of both neutral and charged B rium abundance depends on the dark sector tempera- mesons and anti-mesons. ture. It is reasonable to consider a construction where T < m m , so that it is justified to set the equi- As previously discussed, DM generically tends to be D | φ − ξ| overproduced in this set-up. Additional interactions are librium abundance of the heavier particle to zero. How- required to deplete the DM abundance in order to re- ever, since coannihilations represent a very small branch produce the observed value. Whether the DM is com- in our parameter space, for simplicity and generality, we posed primarily of ξ or φ + φ∗, the scattering term in the simply assume we are far from the regime where coanni- Boltzmann equations allows for the dark particle abun- hilation effects are important so that we can solve Equa- dance to be depleted by annihilations into lighter species. tions (11), (14) and (15) for the dark sector particle abun- In our model, the thermally averaged annihilation cross dances. sections for the fermion and scalar will receive contribu- tions from φ ξ generated by the Yukawa coupling of Equation (4)− (see Appendix3 for rates). This interac- Baryon Asymmetry tion will transform the heavier dark particle population into the lighter DM state. The annihilation term can, The Boltzmann equation governing the production of in general, receive contributions from additional interac- the baryon asymmetry is simply the difference of the par- tions. Therefore, when solving the Boltzmann equations, ticle and anti-particle scalar baryon abundances Equa- we simply parametrize additional contributions to σv ξ tion (14) and Equation (15): and σv by a free parameter. In Sec.V, weh willi h iφ+φ∗ outline a couple of concrete models that accommodate a d(n n ? ) φ − φ + 3 H(n n ) depletion of the symmetric DM component. dt φ − φ∗ We have derived Equation (13) by tracking the particle = 2 ΓB Br(¯b B0) Aq f q n , (16) and anti-particle evolution of the complex φ scalar using Φ → q `` deco Φ q the following Boltzmann equations: X where we must consider contributions from decays dnφ of the ¯b anti-quarks/quarks within both B0 and B0 + 3Hnφ = σv φ(nφnφ? neq,φneq,φ? ) (14) d s dt −h i − mesons/anti-mesons: we take the branching fraction for ¯ 0 the production of each meson to be Br(b Bd) = 0.4 B q ¯ 0 q 0 → + ΓΦ nΦ 1 + A Br(b Bq ) f , and Br(¯b B ) = 0.1 according to the latest esti- × `` → deco s " q # → X mates [4]. Interestingly, we see from integrating Equation (16) where we sum over contributions from B0 oscillations. that the baryon asymmetry is fixed by the product Aq q=s,d `` × 8

Parameter Description Range Benchmark Value Constraint

mΦ Φ mass 11 100 GeV 25 GeV - 23 − 21 22 ΓΦ Φ width 3 10− < ΓΦ/GeV < 5 10− 10− GeV Decay between 3.5 MeV < T < 30 MeV × × mψ Dirac fermion mediator 1.5 GeV < mψ < 4.2 GeV 3.3 GeV Lower limit from mψ > mφ + mξ

mξ Majorana DM 0.3 GeV < mξ < 2.7 GeV 1.0 and 1.8 GeV mξ mφ < mp me | − | − mφ Scalar DM 1.2 GeV < mφ < 2.7 GeV 1.5 and 1.3 GeV mξ mφ < mp me, mφ > 1.2 GeV | − | − yd Yukawa for = ydψφξ¯ 0.3 < √4π L 4 3 Br(B φξ + ..) Br of B ME + Baryon 2 10− 0.1 10− < 0.1 [4] →s → 6× d − 4 4 d A`` Lepton Asymmetry Bd 5 10− < A`` < 8 10− 6 10− A`` = 0.0021 0.0017 [4] s × 5 s × 3 × 3 s − ± A`` Lepton Asymmetry Bs 10− < A`` < 4 10− 10− A`` = 0.0006 0.0028 [4] 25× 3 24 3 − ± σv φ Annihilation Xsec for φ (6 20) 10− cm /s 10− cm /s Depends upon the channel [2] h i − × 25 3 24 3 σv ξ Annihilation Xsec for ξ (6 20) 10− cm /s 10− cm /s Depends upon the channel [2] h i − ×

TABLE II. Parameters in the model, their explored range, benchmark values and a summary of constraints. Note that the q benchmark values for A`` Br(Bq φξ + Baryon + X), for σv φ and σv ξ, are fixed by the requirement of obtaining the × → 11 h i h i 2 observed Baryon asymmetry (YB = 8.7 10− ) and the correct DM abundance (ΩDMh = 0.12), respectively. ×

0 Br(Bq ξφ + Baryon + X) – a measurable quantity at like function (we have explicitly checked that a Heaviside → q experiments. In particular, A`` is defined as: function yields similar results) to model the loss of coher- ence of the oscillation system in the thermal plasma: ¯0 0 0 ¯0 ¯ q Γ Bq Bq f Γ Bq Bq f A`` = → → − → → , (17) ¯0 0 0 ¯0 ¯ q 0 0 Γ Bq Bq f + Γ Bq Bq f Γ(e±Bq e±Bq )/∆mBq → →
→ →
fdeco = e− → . (18) which is directly related to
the CPV in oscillating
neu- ¯ 13 tral B meson systems. Here f and f are taken to be We take ∆mBd = 3.337 10− GeV and ∆mBs = ¯ 11 × 0 0 final states that are accessible by the decay of b/b only. 1.169 10− GeV [4], and Γ e±Bq e±Bq = 11 × 5 → Note that as defined, Equation (17) corresponds to the 10− GeV (T/20 MeV) (see Appendix1 for details). semi-leptonic asymmetry (denoted by Aq in the litera-
SL Even without numerically solving the Boltzmann equa- ture) in which the final state may be tagged. However, tions, we can understand the need for additional interac- at low temperatures and in the limit when decoherence tions in the dark sector σv . From Equations (11) effects are small, this is effectively equivalent to the lep- ξ,φ and (13), we see that theh i DM abundance is sourced tonic charge asymmetry for which one integrates over all by Br(B φξ + Baryon + X)); the greater the value times. Therefore, in the present work we will use the two of this branching→ fraction, the more DM is generated. interchangeably. From Equation (16), we see that the asymmetry also de- Maintaining the coherence of B0 oscillation is crucial pends on this parameter but weighted by a small number; for generating the asymmetry; additional interactions q A < 4 10 3. Therefore, generically a region of param- with the B mesons can act to “measure” the state of the `` − eter space× that produces the observed baryon asymmetry B meson and decohere the B0 B¯0 oscillation [32, 33], q q will overproduce DM, and we require additional interac- thereby diminishing the CPV and− so too the generated tions with the DM to deplete this symmetric component baryon asymmetry. B mesons, despite being spin-less and reproduce Ω h2 = 0.120. and charge-less particles, may have sizable interactions DM with electrons and positrons due to the B’s charge dis- tribution. Electron/positron scattering e±Bq e±Bq, if 0 → faster than the Bq oscillation, can spoil the coherence of the system. We have explicitly found that this interac- B. Numerics and Parameters tion rate is 2 orders of magnitude lower than for a generic baryon [29], but for temperatures above T 20 MeV ' We use Mathematica [37] to numerically integrate the the process Γ(e±B e±B) occurs at a much higher rate → set of Boltzmann Equations (9), (10), (11), (13), and (16) than the B meson oscillation and therefore precludes the subject to the constraint Equation (8). To simplify the CP violating oscillation. We refer the reader to Ap- numerics it is useful to use the temperature T as the evo- pendix1 for the explicit calculation of the e±B e±B → lution variable instead of time. Conservation of energy scattering process in the early Universe. yields the following relation [38, 39]: Generically, decoherence will be insignificant if oscilla- tions occur at a rate similar or faster then the B0 me- dT 3H(ρSM + pSM) ΓΦnΦmφ son interaction. By comparing the e±Bq e±Bq rate = − , (19) → dt − dρSM/dT with the oscillation length ∆mBq , we construct a step- 9

9 9 9

9

9

FIG. 3: Evolution of comoving number density of various components for the benchmark points we consider in Table II: 22 3 m, , Br(B ⇠ + Baryon),m ,yd = 25.5GeV, 10 GeV, 5.6 10 , 3.3GeV, 0.3 .Theleft panel corresponds the { ! } { ⇥ } 0 0 DM mainly composed of Majorana ⇠ particles, as we take m⇠ = 1 GeV and m =1.5GeV. Wetakeboth the Bs and Bd s 4 d contributions to the leptonic asymmetry to be positive, A`` =10 = A``. The change in behavior of the asymmetric yield at T 15 MeV corresponds to decoherence e↵ects spoiling the B0 oscillations while B0 oscillations are still active. The right ⇠ d s panel corresponds to the DM being composed mainly of dark baryons + ⇤,withm =1.3GeVand m⇠ =1.8GeV. Wenow FIG. 3: Evolution of comoving number density of various components for the benchmark points wes consider3 in Tabled II: d SM 4 22 3 take A`` =10 ,andA`` = A`` = 4.2 10 – the dip in the asymmetry can be understood from the negative value of FIG. 3: Evolution of comovingm number, , Br( densityB ⇠ of+ variousBaryon),m components ,yd = for25.5GeV the benchmark, 10 GeVFIG. points, 5.6 3: we10 Evolution consider, 3.3GeV of in, comoving0 Table.3 .The II: numberdleft panel densitycorresponds of various the components ⇥ for the benchmark points we consider in Table II: A chosen in this case to correspond22 to the SM3 prediction. Both benchmark points reproduce the observed DM abundance { ! 22 } { 3 m , ⇥ , Br(B ⇠ + Baryon)} ,m`` ,y = 25.5GeV0 , 100 GeV, 5.6 10 , 3.3GeV, 0.3 .Theleft panel corresponds the m, , Br(B ⇠ + Baryon)DM,m mainly ,yd composed= 25.5GeV of Majorana, 10 GeV⇠ particles,, 5.6 10 as, we3.3GeV take m, 0⇠.3=.The 1 GeVleft and panelm =1corresponds.5GeV. Wetakeboththe 2 d the Bs and Bd 11 { ! } { ⇥ s }{ 4 d ! 0 0⌦DMh =0} .12,{ and baryon asymmetry YB ⇥=89 .7 10 . [GE: Gilly} will beautify] 0 0 9 DM mainly composed of Majoranacontributions⇠ particles, to the as leptonic we take asymmetrym⇠ = 1 GeV to be and positive,m =1A``.5GeV.=10DM Wetakeboth= mainlyA``. The composed change the B ofs inand Majorana behaviorBd of⇠ theparticles, asymmetric as we yield take m⇠ = 1 GeV and⇥m =1.5GeV. Wetakeboth the Bs and Bd s 4 d s 4 d contributions0 to the leptonic0 asymmetry to be positive, A =10 = A . The change in behavior of the asymmetric yield contributions to the leptonicat asymmetryT 15 MeV to be corresponds positive, A to`` decoherence=10 = A e↵``ects. The spoiling change the inB behaviord FIG.oscillations 3: of Evolution the while asymmetricBs ofoscillations comoving yield are number still active. density The of variousright`` components`` for the benchmark points we consider in Table II: ⇠ 0 0 0 0 at T 15 MeV corresponds to decoherence e↵ects spoiling the22B oscillations while3 Bs oscillations are still active. The right at T 15 MeV corresponds topanel decoherencecorresponds e↵ects to the spoiling DM being the B composedd oscillations mainly while of darkBs oscillations baryonsm ,+ are⇤, ,with stillBr(B active.m⇠=1+ The.3GeVand Baryon)rightd log,mmn ⇠,y=1T .dn8GeV.= 25.5GeV Wenow, 10 GeVd , 5.6 10 , 3.3GeV, 0.3 .Theleft panel corresponds the ⇠ s 3 d d SM 4 ⇠ = d . Note, that we also convert to the conve- panel corresponds to the DMtake beingA`` composed=10 ,and mainlyA`` of= darkA`` baryons= 4.2 + 10⇤,with– them dip=1 in.3GeVand thepanel{ asymmetrycorrespondsm⇠ =1 can.8GeV.! tobe the understood WenowDM beingd fromlog composedT then} negativedT mainly{ value of dark of baryons +⇥⇤,withm =1.3GeVand} m⇠ =1.8GeV.0 Wenow0 d ⇥ DM mainlys composed3 d of Majoranad SM ⇠ particles,4 as we take m⇠ = 1 GeV and m =1.5GeV. Wetakeboth the Bs and Bd s 3 d Ad SMchosen in this case4 to correspond to the SM prediction. Bothtake benchmarkA =10 points ,and reproduceA = A thenient observed= yield4.2 variables DM10 abundance–Y thex = dipnxs/s in. the asymmetry4 d can be understood from the negative value of take A`` =10 ,andA`` = A```` = 4.2 10 – the dip in the asymmetry can be understoodcontributions`` from the to negative the leptonic`` value`` ofasymmetry to be positive, A =10 = A . The change in behavior of the asymmetric yield d 2 ⇥ 11 d ⇥ `` `` m⇠ < m (20) A`` chosen in this case to correspond⌦DMh =0 to.12, the and SM baryon prediction. asymmetry Both benchmarkYB =8.7 10 points . reproduce[GE: GillyA`` thechosen will observed beautify] in this DM case abundance to correspond to the SM prediction. Both0 benchmark points reproduce0 the observed DM abundance 2 11 ⇥ at T 2 15 MeV corresponds to decoherence e↵ects spoiling11 the Bd oscillations while Bs oscillations are still active. The right ⌦DMh =0.12, and baryon asymmetry YB =8.7 10 . [GE: Gilly will beautify] ⌦DMh ⇠=0.12, and baryon asymmetry YB =8.7 10 . [GE: Gilly will beautify] m < m⇠ (21) ⇥ panel corresponds to the DM beingThe composed parameter mainly⇥ space of of dark our baryonsmodel includes + ⇤,with the parti-m =1.3GeVand m⇠ =1.8GeV. Wenow s 3 d d SM 4 m

s Y ier DM state to thes lower one, thus simplifying the phe- YB B since this value enables an ecient depletion of the heav- v ⇠⇠2 XX = 34 v sinceWIMPDM this massesRPI-I value enablesare constrained an(30) ecientd by kinematics,depletion of proton the(31) heav- and s vs ⇠⇠ XX 4= 34RPI-Iv WIMP (30)!RPI-I !RPI-I ±

a hadrons, the leptonic asymmetry, and the dark sector ! / ! ± nomenology. For sud/ logcientlyn lower values⌦h DM ofi thish !=0 coupling.12i (24)A`` = 0.0021 d log0.0017n andT dn A`` = 0h.0006A``i =!0 10.0028 for i (28) ier DM state to the lower one, thus simplifying the=- phe-9T dn . Note, that we also convert toierneutron the DM conve-s state star to stability3 the lower – Equations one, thus0 simplifying (5), (6)0± and the (7).=- phe-9 We. Note, that we± also convert to the conve-

dn log T n dT annihilation cross sections. Table. II summarizesdn log 10T the pa-n dT 11 A`` = 10 (31)the Bd and Bs systems10(32) respectively. Noted that4 these (31) we may require interactions of both the ⇠ and states A = 10 (29) nomenology. For suciently lower values of thisnient coupling yield variables YxYB==8nx/s.7. 10 nomenology.take the Yukawa For su couplingciently(25) lower in the values dark˜ of sector thisnient coupling to yield bevariables 0.3 Yx = ``nx/s˜. ˜ ˜ ˜ ˜ rameters and the range of over which they are allowed to s 3 values allow for= additional new physicss contributions3 with additional particles.= ⇥ ⇠ d 4 ⇠ II ⇠ , ⇠ ⇠ , ⇠ ⇠(1.2)II ⇠ , ⇠ ⇠ , (1.2) we may require interactions of both the ⇠ and states A`` = 10 we mayA`` require= 4.2 interactions10m3⇠ < m(32) of both the ⇠ and states(33) (20)A`` = 10s m⇠ < m(32) (20) vary taking into account all constraints. Br (B ⇠+ Baryonsince+ X this)=5 value.6 ⇥10 enables an(26) ebeyondcient depletion those expected of the heav-from the SM alone:v ⇠⇠AXXSM== 34 v WIMP (30)RPI-II RPI-II RPI-II Y `` RPI-II Y d 4 ! ± hd i ! | 4 h !i ! ± ! with additional particles. A =!4.2 10 withier DM additionalv ⇠⇠ stateXX⇥ to particles.= the 46m lowerventropy and energy g (T ) and g (T ), we use − SL SM − ,s While× there is no direct| search for− the± branching× ratio the values obtainedvalid. in Should [41]. Finally,∗ we since stick the∗ to DM what parti- we have or changevalid. to all positive?Should we ] stick to what we have or change to all positive? ] Br(B ξφ + Baryon + X), we can constrain the range cles generically have masses greater than a GeV we can Lets check the transformations ofof the experimentallyd→s: viable values.Lets Forcheck instance, the transformations in the of the d s: safely neglect the inverse scatterings in the DM Boltz- example of Figure2 where the produced baryon is a mann equations i.e. the n2 term. To make the inte- eq ↵ Λ =↵˙ u d s , we can, based on the B+ decay to cX, set ↵ ↵˙ gration numerically straightforward we changed = ⇠ variables( @)↵↵˙ ⇠† | i d , d = ⇠(1.10)( @)↵↵˙ ⇠† d , (1.10) · the bound!RPI-I Br(B ξφ + Baryon) < 0.1 at 95% CL [4]. RPI-I and solve the equations for log n and log T , such that → · ! d log n = T dn . Note, that we also convert˜ to the˜↵ conve- ˜ ↵˙ ˜ ˜ ˜↵ ˜ ↵˙ ˜ d log T n dT d = ⇠ ( @)↵↵˙ ⇠† d I⇤d⇤ Id , d = ⇠(1.11)( @)↵↵˙ ⇠† d I⇤d⇤ Id , (1.11) nient yield variables Y = n /s. · !RPI-I ± ? ± ? · !RPI-I ± ? ± ? x x C. Results and Discussion The parameter space of our model includes the↵ particle ˜ ↵˙ ↵ ˜ ↵˙ d = ⇠ ( @)↵↵˙ ⇠† d I⇤d , d = ⇠(1.12)( @)↵↵˙ ⇠† d I⇤d , (1.12) masses, the Φ decay width, the dark Yukawa? coupling,· !RPI-I ? ± ? · !RPI-I ? ± the branching ratio of B mesons to DM and hadrons,↵ The↵˙ recent Planck CMB observations imply a co- ⇤ = ⇠˜ ( @) ⇠† ⇤  . ˜↵ ↵˙ the leptonic asymmetry, and the dark sectord annihilation ↵↵˙ moving baryon asymmetryd Id of YB = (nB nB¯ )/sd⇤ == ⇠ ( @)↵↵˙ ⇠† d⇤ Id. ? · !RPI-I ? ±11 − ? · !RPI-I ? ± cross sections. TableII summarizes the parameters and (8.718 0.004) 10− [2]. In our scenario, even with- ± × the range of over which they are allowed to vary taking out fully solving the system of Boltzmann equations, we into account all constraints. ↵ can↵˙ see from integrating Equation (16) that the baryon = ⇠ ( @) ⇠†  ⇤ ⇤ , (1.13)↵ ↵˙ The upper limit on the Φ mass is imposedd because ↵↵˙ asymmetry directlyd dependsIId uponII thed product of leptonicd = ⇠ ( @)↵↵˙ ⇠† d IId⇤ II⇤ d , (1.13) · RPI-II ! ± ? ± ? · RPI-II ! ± ? ± ? above 100 GeV, the scalar could potentially have a asymmetry times branching fraction: ∼ ˜ ˜↵ ˜ ↵˙ ˜ ↵ ↵˙ small branching fraction to b quarks (seede.g.= ⇠[42(]). @)↵↵˙ ⇠† d, ˜ = ⇠˜(1.14)( @) ⇠˜† ˜, (1.14) RPI-II d ↵↵˙ d DM masses are constrained by kinematics, and neu-· Y ! Aq Br(B0 φξ + Baryon + X) . · RPI-II ! B ∝ `` × q → tron star stability – Equations (6) and (7). We take q=s,d the Yukawa coupling in the dark sector to be 0.3 since X this value enables an efficient depletion of the heavier Meanwhile, the DM relic abundance is measured to be 2 2 DM state to the lower one, thus simplifying the phe- ΩDMh = 0.1200 0.0012 [2] and reads ΩDMh = ± 2 nomenology. For sufficiently lower values of this cou- [mξYξ +2mφ(Yφ + Yφ? )] s0h /ρc (where s0 is the current 2 10

d s 11 FIG. 4. Left panel: required value of A`` Br(B ξφ + Baryon) assuming A`` = 0 to obtain YB = 8.7 10− . Right panel: s × → d 11 × Required value of A`` Br(B ξφ + Baryon) assuming A`` = 0 to obtain YB = 8.7 10− . The blue region is excluded by a combination of constraints× → on the leptonic asymmetry and the branching ratio [4].× The lower bound (red region) comes from requiring the late Φ decays to not spoil the measured effective number of neutrino species from CMB and the measured primordial nuclei abundances [43].

0 entropy density and ρc is the critical density). In Fig- curve corresponds to the contribution from the Bd os- ure3 we display the results (the comoving number den- cillations – given that the Bs oscillation time scale is 20 sity of the various components) of numerically solving the times smaller than the Bd one, and the Bs contribution Boltzmann equations for two sample benchmark points it is active at higher temperatures. that reproduce the observed DM abundance and baryon asymmetry. Consider the plot on the right panel of Figure3, which The Baryon Asymmetry corresponds to the case where DM is composed of φ and φ∗ particles. We can understand the behavior of the particle yields as follows: Φ particles start to decay at In order to make quantitative statements, beyond the T 50 MeV, thereby increasing the abundance of the benchmark examples discussed above, we have explored ∼ the parameter space outlined in TableII and mapped out dark particles ξ and φ + φ∗ until T 10 MeV at which point Φ decay completes (as it must,∼ so that the predic- the regions that reproduce the observed baryon asymme- tions of BBN are preserved). The dip in the dark particle try of the Universe. From Equation (16), we see the yields at lower temperatures is the necessary effect of the baryon asymmetry depends on the product of the lep- additional annihilations – which reduce the yield to re- tonic asymmetry times branching fraction (with contri- 0 0 produce to the observed DM abundance. Meanwhile, the butions from both Bd and Bs mesons), as well as the Φ mass and width. The result of this interplay is displayed asymmetric component Yφ Yφ∗ is only generated for T 30 MeV, as it is only then− that the B0 CPV oscil- in Figure3, where the contours correspond to the value . s q 0 lations are active in the early Universe. The decrease in the product of A`` Br(Bs φξ + Baryon + X) needed × → 11 to reproduce the asymmetry Y = 8.7 10− for a given the asymmetric component at T 10 MeV is due to the B × negative contribution of the B0 decays,∼ since in this case point in (mΦ, ΓΦ) space. For simplicity, the left and right d 0 the leptonic asymmetry is chosen to be negative. Note panels show the effects of considering either the Bd or the 0 that for the case in which the DM is mostly composed of Bs contributions but generically both will contribute. φ and φ∗ particles the observed baryon asymmetry and While the entire parameter space in Figure4 is allowed DM abundance imply an asymmetry of by the range of uncertainty in the experimentally mea- q sured values of A``, our range of prediction is further 2 Yφ Y ∗ Ω h m 1 m constrained. In particular, the blue region in Figure4 − φ = b φ φ . (21) 2 is excluded by a combination of constraints on the lep- Yφ + Yφ∗ ΩDMh mp ' 5.36 mp tonic asymmetry and the branching ratio [4] (see Sec- The plot in the left panel of Figure3 corresponds to tionIV), while the lower bound comes from requiring the case where DM is mostly comprised of ξ particles. that the Φ not spoil the measured effective number of In this case the evolution of the dark particles is rather neutrino species from CMB and the measured primor- d s similar. Here we have chosen A`` = A`` > 0, so that the dial nuclei abundances [43]. Therefore, to reproduce the 0 asymmetric component gets two positive contributions expected asymmetry coming from, for instance, only Bs , 0 0 s 6 4 at T . 30 MeV from both Bd and Bs CPV oscillations. we find A`` Br(B φξ+Baryon+X) 10− 5 10− While at T 15 MeV the change in behavior of the yield (depending× upon the→ Φ width and mass).∼ − × ∼ 11

Interestingly, the baryon asymmetry can be generated s 5 with only the SM leptonic asymmetry A`` = 2 10− , provided that Br(B φξ + Baryon + X) = 0.05× 0.1 d → − and that A`` = 0 (which is compatible with current data) – see the green region in the right panel of Fig- s ure4. Additionally, if new physics enhances A`` up to 3 the current limit 4 10− , Baryogenesis could take ∼ × 4 place with a branching fraction as low as 2 10− . Fig- d × ure5 shows that even with a negative A``, as expected in the SM, the baryonic asymmetry can be generated 3 with Br(B φξ + Baryon + X) > 2 10− provided s → 3 × that A`` (10− ). We reiterate that both the leptonic asymmetry∼ O and the decay of a B meson to a baryon and missing energy are measurable quantities at B-factories and hadron colliders (see SectionIV). FIG. 5. Contours show the value Br(B ξφ + Baryon) → required to generate the correct Baryon asymmetry YB = 11 s 3 d 8.7 10− for the fixed values: A`` = 10− and A`` = d ×SM 4 The Dark Matter Abundance A`` = 4.2 10− . | − ×

As previously argued, in the absence of additional in- teractions, our set-up generically tends to overproduce the following Boltzmann equation: 3 the DM since the leptonic asymmetry is < 5 10− . × By examining the DM yield curve in Figure3 we see n˙ ¯ + 3Hn ¯ = σv n ¯ n + f ¯ Γ n N N − h i N N N Φ Φ that annihilations (the dip in the curve) deplete the DM = σv nN¯ (nN¯ + nφ nφ? ) + fN¯ ΓΦnΦ (23) abundance that would otherwise be overproduced from − h i − the Φ decay. where fN¯ 1 is the produced number of antinucleons per Recall that for a stable particle species annihilat- Φ decay [45']. By solving this Boltzmann equation we find 23 ing into two particles in the early Universe when ne- that for ΓΦ > 3 10− GeV the primordial antinucleon glecting inverse-annihilations: Ωh2 x / σv . For × 26 ∝ FO h i abundance is YN¯ < 10− (and usually way smaller) and WIMPs produced through thermal freeze-out xFO = too small to have any phenomenological impact at the mDM/TFO 20, while, in our scenario mDM/T 400. CMB or during BBN. Therefore, an∼ annihilation cross section roughly 1∼ 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 IV. SEARCHES AND CONSTRAINTS σv = (20 70) σv (22) h idark − h iWIMP 25 3 Developing a testable mechanism of Baryogenesis has = (6 20) 10− cm /s , − × always been challenging. Likewise, should a DM detec- tion occur, nailing down the specific model in set-ups 26 3 where σv = 3 10− cm /s, and the spread where a rich hidden dark sector is invoked, is generally h iWIMP × of values correspond to varying the DM mass over the daunting. The scenario described in the present work range specified in TableII (with only a very slight sen- is therefore unique in that it is potentially testable by sitivity to other parameters). In particular σv future searches at current and upcoming experiments, h idark ' 25 σv min[mφ, mξ]/GeV. while being relatively unconstrained at the moment. h iWIMP

Primordial Antimatter with a low Reheat Temperature A. Searches at LHCb and Belle-II

Finally, note that since we are considering rather low As discussed above, a positive leptonic asymmetry in reheat temperatures, there could be a significant change B meson oscillations – and the existence of the new de- to the primordial antimatter abundance. In the case of a cay mode of B mesons into visible hadrons and missing high reheat temperature scenario, the primordial antinu- energy – would both indicate that our mechanism may 18 9 105 cleon abundance is tiny: YN¯ = 10 e− × [44]. In our describe reality. Both these observables are testable at scenario, we can track the antinucleon× abundance from current and upcoming experiments. 12

Semileptonic Asymmetry in B decays Exotic b-flavored Baryon decays

As shown in SectionIIIC, the model we present re- Our Baryogenesis and DM production mechanism re- quires a positive and relatively large leptonic asymme- quires the presence of the new exotic B meson decays. 5 3 try: A`` 10− 10− . The current measurements of However, once these decays are kinematically allowed, ∼ − the semileptonic asymmetry [4] (recall that in our setup the b-flavored baryons will also decay in an apparently the semileptonic and leptonic asymmetries may be used baryon violating way to mesons and DM in the final state. 0 interchangeably) are: For instance, given the interaction (3) the Λb baryon + could decay into ψ¯ + K + π− provided mψ < 4.9 GeV As = ( 0.0006 0.0028) , SL − ± which will always be the case since mψ < mB mΛ in d this case. In addition, the rate of this process− should ASL = ( 0.0021 0.0017) . (24) − ± be very similar to that of B mesons. To our knowl- These are extracted from a combination of various anal- edge there is no current search for this decay channel, yses of LHCb and B-factories. Future and current ex- but in principle LHCb could search for it. In particu- periments will improve upon this measurement. In par- lar, Ref. [50] pointed out that it is possible to identify 1 ticular, the future reach of LHCb with 50 fb− for the the initial energy of a Λb if it comes from the decay of ? measurement of the leptonic asymmetry is estimated to Σb±, Σb± Λb + π± by measuring the kinematic dis- s 4 → be σ(ASL) 5 10− [15], and a similar sensitivity tribution of the process. The LHCb collaboration has ∼ × d should be expected for A . The sensitivity of Belle-II very recently observed 20000 [51]Λb candidates pro- SL ∼ to the semileptonic asymmetries has not been addressed duced via this process, thus making the measurement of ¯ in the Belle-II physics book [46]. However, we became Λb ψ + mesons potentially viable at hadron colliders. 1 → aware [47] that with 50 ab− Belle-II should reach a sensi- We refer to Appendix4 where the lightest states of the d 4 tivity to ASL of 5 10− . In addition, Belle-II is planning possible decay processes are outlined for the four different ×1 on collecting 1 ab− of data at the Υ(5S) resonance [46] flavor operators. s which could potentially result in a measurement for ASL.

Considerations from Flavor Observables B meson decays into a Baryon and missing energy Recall that, the semileptonic asymmetry may be com- For our mechanism to produce the observed Baryon puted theoretically from the off-diagonal elements of the asymmetry in the Universe, from Figure5, we notice that Hamiltonian describing the B0 B¯0 system: 4 moderately large Br(B ξφ + Baryon+X) = 10− 0.1 − → − are required. The constraint on Br(B ξφ + Baryon + i i → M11 2 Γ11 M12 2 Γ12 X) < 0.1 at 95% CL [4] is based on the measure- osc = − i − i . (25) + H M ∗ Γ∗ M Γ ment of the B decay to cX. This branching fraction " 12 − 2 12 22 − 2 22 # can also be constrained since the presence of this new decay mode could alter the total width of b-hadrons. In particular,

We proceed as [48] and use the theoretical expecta- q b q Γ tion in the SM for the decay width from [49]ΓSM = 12 13 ASL = Im q . (26) (3.6 0.8) 10− GeV, and the observed width [4] M12 b ± × 13   Γobs = (4.202 0.001) 10− GeV. This constraint therefore restricts± Br(B × ξφ + Baryon + X) < 0.37 at In the SM, oscillations arise from quark-W box diagrams, 95% CL. → whose vertices contain CPV phases from the CKM ma- q To our knowledge there are no searches available trix. In particular, the SM predictions for ASL read s 5 d to measure this branching fraction, and no published A`` SM = (2.22 0.27) 10− and ASL SM = ( 4.7 | 4 ± × | − ± data on the inclusive branching fraction for B mesons 0.6) 10− [15, 16]. Since these are substantially smaller × Br(B Baryon + X) either. We expect that existing then the current measurements (24) there is room to ac- data from→ Babar, Belle and LHC can already be used commodate new physics. to place a meaningful limit. The search for this channel The large positive leptonic asymmetry required in our should in principle be similar to other B meson missing set-up could differ considerably from the SM values, energy final states such as B Kνν or B γνν with depending on the value of Br(B ξφ + Baryon + X). → 5 → → current bounds at the level of (10− )[4]. Given this, There are many BSM models that allow for a substan- O 6 the reach of Belle-II [46] could be of (10− ). Thus, tial enlargement of the semileptonic asymmetries of both O 0 0 potentially our mechanism is fully testable. the Bd and Bs systems (see e.g. [15, 17] and references therein). In addition, it is worth mentioning, that the fla- vorful models invoked to explain the recent B-anomalies 0 also induce sizable mixing in the Bs system (see e.g. [18– 21]). 13

Note, that while the elements of the evolution Hamil- light quarks, resulting in di-jet or monojet signatures. tonian (25) are not directly probed in experiment, they These constraints were considered in Ref. [29] and give can be related to additional experimental observables as: mass dependent constraints on the yus and yud couplings which are far from being in tension with the parameters 2Re(M Γ ) ∆m = 2 M and ∆Γ = 12∗ 12 , (27) required for our baryogenesis scenario. B | 12| B − M | 12| Finally, note that the field Φ is too weakly coupled to be produced at a collider. where, for instance ∆mB, the B meosn oscillation length, is related to the mass eigenstates. Therefore, any new q physics that modifies ASL away from the SM value will also modify ∆m and ∆Γ , and must not be in con- B B Cosmological Constraints flict with current bounds on these observables. For an overview of the allowed BSM modifications to ∆mB and ∆ΓB see [15]. Our mechanism requires a low reheat temperature TRH (10 MeV). The lower bound on the reheat temperature∼ O comes from the agreement of CMB and BBN observations on the number of relativistic species B. Constraints in the early Universe. The current bound reads TRH > 4.7 MeV [43] at 95% CL which in turn implies that 23 1 Γ < 3 10− GeV 45 s− at 95% CL, where we take Here we comment on collider, cosmological, and DM Φ × ≡ direct detection constraints. ΓΦ = 3H(TRH ). Note that this bound is not expected to be substantially modified by the Planck 2018 final data 2015 2018 release since Neff > 2.74 [1] and Neff > 2.70 [2] both Collider Constraints at 95% CL.

Within our set-up, the heavy colored scalar Y is re- sponsible for inducing the B meson decays into the dark sector via Equation (2). The colored scalar may be pro- Dark Matter Direct Detection duced at colliders and its decay products searched for, thus resulting on an indirect constraint on the model. The DM in our scenario could scatter through pro- This branching ratio was calculated in [29] and we quote tons and neutrons, as in the Hylogenesis model [14, 53], the result here; with signatures similar to those in nucleon decay searches (although with somewhat different kinematics). Such 4 2 5 mb∆m yubyψs ∆m searches would test for the presence of interactions that Γ¯b φξus 3 2 + 5 , (28) → ∼ 60 (2π) mY O mb are not needed in a minimal model for Baryogenesis.     However, the existing bounds do not constrain either from which; the Hylogenesis model or the mechanism presented here. Within our model, given the interaction with the b quark, Br(B ξφ + Baryon) (29) the kinematics preclude a direct scattering to B mesons. → ' 4 4 The scattering to lighter mesons must therefore be ac- 3 mB mψ 1 TeV √yubyψs 10− − . companied by a penalty due to a weak loop insertion, 2 GeV m 0.53    Y  which makes the expected rate at nucleon decay experi- From Equation (2), note that Y is dominantly pair- ments negligible unless a larger coupling to light quarks produced at colliders through the strong interaction. The exists. produced Y ’s could then decay as either Y u¯ b or Y s ψ¯. So that the expected collider signatures→ are→ 4-jets (two tagged b quarks) or 2-jets plus miss- ing energy. If the former decay dominates then 4-jet searches [52] apply, implying a bound on the colored V. POSSIBLE DEPLETION MECHANISMS scalar mass of mY > 500 GeV. While, if Y sψ¯ domi- nates, then s-quark searches apply for a single→ light quark resulting in the bound mY > 960 GeV [52]. Such con- As discussed in SecIII, DM is initially over produced. 3 straints allow for sizable Br(B ξφ + Baryon) 10− The DM abundance must be depleted sufficiently in order → ∼ 2 with moderately large couplings √yubyψs > 0.25, and to obtain the measured relic abundance ΩDMh = 0.12. are thus not in tension with our model’s prediction of This requires annihilations of the dark particles with a 4 25 3 Br(B ξφ + Baryon + X) = 2 10− 0.1. thermally averaged cross section of order 10− cm /s. Another→ possible constraint arises× from− the single pro- We will now outline some possibilities for dynamics which duction of Y particles at the LHC via its couplings to can reduce the symmetric DM component. 14

A. Annihilations to Sterile Neutrinos B. Annihilations to Standard Model Neutrinos

Right handed neutrinos NR, are massive singlets un- Consider the same set-up as in the previous subsec- der the SM symmetries, and as such provide a simple tion, but assume the sterile leptons are heavier than possible depletion mechanism [54, 55] for the DM parti- the DM. In this case, mixings could generate couplings cles. For instance, for the case where mφ < mξ, we can which would allow the DM particles to potentially an- introduce another dark, heavy, Z2 odd, Dirac sterile par- nihilate into active neutrinos [58, 59]. This annihilation ticle Ψ carrying both baryon and lepton numbers. The will evade CMB constraints and remain elusive to other interaction experiments due to the weak interactions of neutrinos. Generically, the coefficients of the mixing operators ¯ yN φΨNR + h.c. , (30) that generate the annihilation cross section to active L ⊂ neutrinos will be constrained. However, for very heavy allows for the DM, φ, to annihilate via φ φ∗ NN, → neutrinos the only constraint on the mixing comes from thereby depleting the abundance of the symmetric φ φ∗ − the invisible width of the Z and various low-energy lep- component. For the case where mφ > mξ, we could ton universal processes in the SM. Such measurements deplete the excess of ξ particles by introducing a odd 2 2 Z2 bound the mixing to be U < 10− [60] regardless scalar Φ0 , charged under lepton number such that the of the heavy lepton mass| and| specific flavor structure. interaction: The annihilation cross section to neutrinos will scale 4 4 2 0 as yN U /(16πmmed) which for a mixing that sat- yN ξΦ NR + h.c. (31) ∼ | | L ⊂ urates the bound requires Yukawa couplings of yN & is allowed. The process ξ ξ NN can annihilate away 0.6 mmed/3 GeV. → Unfortunately, while SM neutrinos may be produced the overproduced ξ abundance. p The annihilation cross section to sterile neutrinos is s- in this set-up, we do not expect a detectable signal wave and as such is subject to strong constraints from at neutrino detectors given the required annihilation the CMB observations as measured by the Planck satel- rate (22)[61]. 27 3 lite [2, 56]: σv v=0 . (1 3) (mDM/GeV) 10− cm /s, where the rangeh i depends− upon the annihilation× channel C. Annihilations within the Dark Sector (provided the annihilation is not to neutrinos). We note that both φ φ∗ NN and ξ ξ NN processes are s- wave but chirality→ suppressed [55→] i.e. the s-wave con- As discussed in Sec.II, if the depletion of the sym- 2 metric part of DM is too efficient, then there will not be tribution is suppressed like (mN /mDM) as compared with the p-wave. In particular, in the limit in which enough DM since the stable dark particle masses have to be m < m m for the B decay process to occur, while mξ, mφ mN and in which the mediators are substan- φ B − p tially heavier than the DM, the annihilation cross sec- the observed baryon asymmetry and DM abundances tions go as; would require mφ = 5.36 mp. This situation, however, could be different if additional× baryons are present in the 2 2 dark sector. For instance, we can add another dark scalar 4 mN 2mξ 2 σ v ξξ NN = yN 4 1 + 2 v , (32) particle which carries baryon number 1/3, and mass h i → 32πmΦ 3mN 5 A 0 " # m mp 1.6 GeV, (which is allowed by neutron star A ∼ 3 ∼ 2 2 constraints) and is 2 odd with interactions: 4 mN mφ 2 Z σ v φ?φ NN = yN 4 1 + 2 v . (33) h i → 8πm 6m 3 Ψ0 " N # κ φ + κ0 φφ∗ ∗ + h.c. . (34) L ⊂ A AA

So that in this limit the p-wave contribution is signifi- If is lighter than φ, the reactions φ + A∗ A + A cantly enhanced, and CMB constraints are substantially etc.A will eventually turn an excess of φ particles↔ into an ameliorated. Additionally, the decay of sterile neutri- excess of particles. If we now deplete the symmetric nos can be to invisible particles. For instance, if the N componentA of the abundance we can arrive at the right fermion is solely mixed with the τ neutrino (the least DM relic abundance.A constrained scenario [57]), provided that mN < mπ, the In general the same set-up will hold in a situation decay will entirely be to 3 ν, and CMB constraints are where the dark sector contains many baryon number fully evaded. charged states. Rich dark sectors offer a variety of possi- ble production mechanisms in the early Universe [62, 63], that the initial abundance of or other dark sector parti- cles need not be fixed in this setup.A The details of models with rich dark sectors containing baryon number charged particles is beyond the scope of this paper, and we leave this to to future work. 15

VI. CONCLUSIONS These two observables are currently unconstrained but can be measured in future searches at LHCb and Belle-II. In this work we have presented a new mechanism for In particular, given that current sensitivity for B Kνν¯ 6 → is at the level of (10− )[4] our scenario could already be low temperature Baryogenesis in which both the baryon O asymmetry and the DM relic abundance arise from the tested with current data from B factories, and we expect oscillation and subsequent decay of B mesons to visible our mechanism to be fully testable given the reach of baryons and dark sector states. We have illustrated our Belle-II [46]. set-up with a model that is unconstrained by di-nucleon We have seen that there is a region in our parameter decay, does not require a high reheat temperature, and space where it is possible to solely generate the baryon s 5 asymmetry with A = (2.22 0.27) 10− > 0 would have unique experimental signals – a positive lep- ``|SM ± × tonic asymmetry in B meson decays, the existence of the (although new physics effects are required in this case new decay of B mesons into a baryon and missing energy, to simultaneously suppress the negative SM value of d 4 and the b-flavored baryons decay into mesons and miss- ASL SM = ( 4.7 0.6) 10− ). However, generically | − ± × q ing energy. These observables are testable at current and our parameter space predicts values of A`` larger than upcoming collider experiments. In summary, the novel the SM expectation. Note that flavorful models invoked features of our mechanism include the following: to explain the recent B-anomalies also induce sizable mix- 0 ing in the Bs system, and therefore a potential link be- Low reheat temperature. tween our mechanism and the B-anomalies would be a • At least one component of DM charged under baryon very interesting avenue to explore. • number. We have seen that the baryon-symmetric component of DM will be generically overproduced, and as such we Total baryon number of the Universe remains zero. • require additional interactions which allow for DM anni- Baryon asymmetry directly related to experimental hilations to deplete the abundance such that we repro- • observables. duce the observed value. In particular we find that we Distinctive experimental signals: require the dark sector cross section to be: • 25 3 1. Positive semileptonic asymmetry in B meson de- σv = (6 20) 10− cm /s , h idark − × cays. Ω h2 0.12 . ⇒ DM ∼ 2. Charged and neutral B meson decays to a baryon While, we have preliminarily outlined some classes of and missing energy. models that could explain the depletion of DM, formu- 3. Charged and neutral b-flavored baryons decays lating a detailed and complete model is one avenue for to mesons and missing energy. future investigation. Another future direction on the model building front, Testable at both current and upcoming experiments. • would be to embed our mechanism in a more detailed The Baryon asymmetry scales fixed by the following UV model to explain the origin and nature of the colored product: scalar Y . Additionally, as discussed above, there is theo- retical motivation for multiple dark sector states charged Y Aq Br(B0 φξ + Baryon + X) , under Baryon number. An interesting investigation to B ∝ `` × q → q=s,d pursue would be to consider various scenarios for dark X q sector states charged under baryon number. where A``, the leptonic charge asymmetry (which in our set-up is effectively equivalent to the semi- leptonic asymmetry), is an experimental observable ACKNOWLEDGMENTS 0 0 which parametrizes the CPV in the Bs and Bq oscillation systems. This project has received support from the Euro- We have solved the set of coupled Boltzmann equations pean Union’s Horizon 2020 research and innovation pro- and mapped out our model parameter space (allowed by gramme under the Marie Sklodowska-Curie grant agree- kinematics and current constraints) that accommodates ment No 690575 (InvisiblesPlus RISE) and No 674896 a DM relic abundance and baryon asymmetry that agrees (Elusives ITN). ME is supported by the European Re- with observation. From Figure4 and Figure5 (respec- search Council under the European Union’s Horizon 2020 tively), we find that our model predicts the following val- program (ERC Grant Agreement No 648680 DARK- ues for the branching ratio and leptonic asymmetry: HORIZONS). ME acknowledges the hospitality of the 4 1 Particle, Field, and String Theory group at the Univer- Br(B ξφ + Baryon + X) = 2 10− 10− , → × − sity of Washington where this work was initiated. GE q 5 3 and AEN are supported in part by the U.S. Department and A 10− 10− > 0 , `` ∼ − of Energy, under grant number de-sc0011637. AEN is qX=s,d 11 also supported in part by the Kenneth K. Young Memo- Y 8.7 10− . ⇒ B ∼ × rial Endowed Chair. AEN acknowledges the hospitality 16

4 of the Aspen Center for Physics, which is supported by noring the B0 recoil) reads : National Science Foundation grant PHY-1607611. We 2 would like to acknowledge David McKeen, Sheldon Stone d σ α 2 2 2 = 4 cos (θ/2) FB0 (q ) , (37) and Ville Vaskonen for their very useful comments on the d Ω 4E2 sin (θ/2) | | first version of this draft. where α is the fine structure constant. The momentum exchanged is: 2m E2(1 cos θ) θ APPENDICES q2 = B0 − 4E2 sin2 , (38) −m + E(1 cos θ) ' − 2 B0 − Here we give some details behind the calculations in where E is the energy of the incoming electron. We can the main text. Appendix1 and2 justify the assump- therefore rewrite the differential cross section as: tions we made in writing down the Boltzmann equation d σ α2 q2 of SectionIIIA. In Appendix3 we itemize the dark sec- = 2 π r2 2 1 + . (39) d q2 − 18 h B0 i 4E2 tor annihilation cross sections. Finally, flavorful varia-   tions of the B meson decay operator (3) are outlined in Upon integration we obtain the total scattering cross sec- Appendix4. tion 0 d σ 2π σ = d q2 = α2 r2 2 E2 . (40) 2 B0 4E2 d q 9 h i 1. Decoherence due to Elastic Scattering Z− By substituting the energy E by its average in the early 0 ¯0 Universe; E 3T , we obtain the thermally averaged rate In our mechanism, the coherence in the B -B os- ∼ cillation system in the early Universe is key to gener- for this process: ating the baryon asymmetry. The elastic scattering of Γe±B0 e±B0 σv ne σ(E = 3T ) ne(T ) (41) e±B0 e±B0 represents the only possible source of de- → ≡ h i ' → 5 r2 2 coherence in the early Universe, and in this subsection 11 T B0 10− GeV h i . we calculate the thermal rate of this process in the ex- ' 20 MeV 0.187     panding Universe. Notice that the e B e B scattering rate will be As the B is neutral pseudoscalar particle the only ± 0 ± 0 0 higher than the B oscillation→ rate ∆m = 1.17 possible interaction that an electron can have with it is s Bs 10 11 GeV for temperatures above 20 MeV and there-× through the effective charge distributed within it. This − fore through the Zeno effect electron/positron∼ scatterings charge distribution is parametrized in terms of a an elas- will damp the B0-B¯0 oscillations. An identical analy- tic electromagnetic form factor F (q2). The actual form B0 sis applies for the B oscillations, but since ∆m = of F (q2) requires either data (which is not possible d Bd B0 3.34 10 13 GeV oscillations will only be efficient for to obtain in the laboratory for this reaction) or some − T < ×10 MeV. We note that the rate calculated in (41) modeling of the quarks distributed within the B meson. 0 has a very strong temperature dependence and therefore The form factors are usually parametrized in terms of the is fairly independent on possible unaccounted details in charge radius which is defined as: the B0 form factor. In Figure6 we show the resulting de- coherence function (18) for the B and B systems given dF (q2) s d r2 = 6 , (35) the interaction rate in (41) and the B-meson mass differ- h i dq2  q=0 ences. which for a neutral particle leads to form factors: 2. Do B Mesons Decay or Annihilate? 1 F (q2) = r2 q2 + ... . (36) −6h i In the Boltzmann equations of SectionIIIA, we have omitted the possibility that B mesons annihilate prior Since r2 is not measured, we use an estimate pro- h B0 i to their decay into the dark sector. In this subsection, vided by [64]; r2 0.187 fm2. For comparison, B0 we explicitly show that, at the temperatures of interest h i ∼ − 2 the measured value for K is the following: r 0 = 0 h K i T 20 MeV, this is indeed a valid approximation. Ad- 0.077 0.010 fm2 [4]. ditionally,∼ we now determine in which range of tempera- − ± We can safely use the quadratic expansion for the form tures the decays will dominate over annihilations. factor, (36), since it will be valid for q < 1/ r2 | | B0 ∼ 100 MeV and we are interested in T 10 MeV.q Since ∼ the quadratic form factor approximation holds, we can 4 This equation is the non-relativistic formula given for an electron calculate the differential scattering cross section for the interacting with a target with charge density ρ where F (q2) R ρ(r) ei~q~r d3~r. ≡ process e±B e±B , which in the lab frame (and ig- 0 → 0 17

3. Dark Cross Sections

Here we list the dark sector cross sections to lowest order in velocity v that result from the interaction (4):

4 2 3/2 yd (mξ + mψ) [(mφ mξ)(mξ + mφ)] σφ?φ ξξ = − , → 2 2πm3 m2 + m2 + m2 φ − ξ ψ φ 2 4  v yd ? σξξ φ φ mφ 0 = 4 (45) → | → 2 2 × 48π mξ + mψ 5  4 2 3 3 2mξmψ + 5mξmψ + 8mξmψ 2 4 5 6 6  + 9mξmψ + 6mξmψ + 3mξ + 3mψ .  FIG. 6. The decoherence function (18) as a function of tem- 0 0 4. B Meson Decay Operators perature for the Bs and Bd systems.

Here we categorize the lightest final states for all the Since the Φ particle decays at the same rate to B quark combinations that allow for B mesons to decay into and B¯, we can assume that n = n ¯ upon hadroniza- B B a visible baryon plus DM, and for Λb baryons decaying to tion. In reality, due to CP-violating oscillations, nB mesons and DM. Note that the mass difference between 3 ' (1 + 10− )nB¯ , but this will not impact the calculation at final and initial states for the B-mesons will give an upper hand. The Boltzmann equation that governs the the B bound on the dark Dirac fermion ψ mass. In TableIII we number density is: list the minimum hadronic mass states for each operator.

dnB 2 + 3HnB = ΓΦBrΦ BnΦ ΓBnB σv nB . (42) dt → − − h i Operator Initial State Final state ∆M (MeV) Equation (42) involves very different time scales, and its Bd ψ + Λ (usd) 4163.95 numerical solution will require time steps of t < 1/ΓB – 10 0 which are 10 smaller than those of the Φ lifetime. Bs ψ + Ξ (uss) 4025.03 − ψ b u s We determine if a produced B meson will decay or B+ ψ + Σ+ (uus) 4089.95 annihilate as follows: integrate Equation (42) with only Λ ψ¯ + K0 5121.9 the first term on the right hand side (so that we ignore b both B decay and oscillation), this will give us the maxi- Bd ψ + n (udd) 4340.07 mum number density of B’s prior to decay ∆n . We can B Bs ψ + Λ (uds) 4251.21 ψ b u d then compare the B decay and the annihilation rates, + in order to determine which one dominates. Integration B ψ + p (duu) 4341.05 ¯ 0 of the first term in Equation (42) in the time interval Λb ψ + π 5484.5 t t + 1/ΓB leads to: 0 → Bd ψ + Ξc (csd) 2807.76 t+1/ΓB dnB Bs ψ + Ωc (css) 2671.69 ∆nB = (t0)dt0 (43) ψ b c s dt B+ ψ + Ξ+ (csu) 2810.36 Zt c t+1/ΓB ¯ + ΓΦ Λb ψ + D− + K 3256.2 = ΓΦnΦ(t0)dt0 = nΦ(t) . ΓB Zt Bd ψ + Λc + π− (cdd) 2853.60 0 Now, we can clearly compare the decay and annihilation Bs ψ + Ξ (cds) 2895.02 ψ b c d c rates: + B ψ + Λc (dcu) 2992.86 2 ¯ 0 ∆nBΓB ΓB Λb ψ + D 3754.7 2 = . (44) ∆nB σv ΓΦ σv nΦ(t) h i h i TABLE III. Here we itemize the lightest possible initial and When solving numerically for the Φ number density we final states for the B decay process to visible and dark sector found that even with an annihilation cross section of states resulting from the four possible operators. The diagram σv = 10 mb, the decay rate overcomes the annihila- in Figure2 corresponds to the first line. The mass difference h i 21 tion rate for T . 60 MeV even for Γφ = 10− GeV (and between initial and final visible sector states corresponds to 22 the kinematic upper bound on the mass of the dark sector ψ T & 120 MeV for Γφ = 10− GeV). Thus, for practical purposes it is safe to ignore the effect of annihilations in baryon. the Boltzmann equations. 18

[1] P. A. R. Ade et al. (Planck), Astron. Astrophys. 594, D. Teresi, Int. J. Mod. Phys. A33, 1842002 (2018), A13 (2016), arXiv:1502.01589 [astro-ph.CO]. arXiv:1711.02862 [hep-ph]. [2] N. Aghanim et al. (Planck), (2018), arXiv:1807.06209 [28] A. Ghalsasi, D. McKeen, and A. E. Nelson, Phys. Rev. [astro-ph.CO]. D92, 076014 (2015), arXiv:1508.05392 [hep-ph]. [3] R. H. Cyburt, B. D. Fields, K. A. Olive, and T.-H. Yeh, [29] K. Aitken, D. McKeen, T. Neder, and A. E. Nelson, Rev. Mod. Phys. 88, 015004 (2016), arXiv:1505.01076 Phys. Rev. D96, 075009 (2017), arXiv:1708.01259 [hep- [astro-ph.CO]. ph]. [4] M. Tanabashi et al. (ParticleDataGroup), Phys. Rev. [30] H. Beauchesne, K. Earl, and T. Gregoire, JHEP 06, 122 D98, 030001 (2018). (2017), arXiv:1703.03866 [hep-ph]. [5] A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967), [31] D. McKeen, A. E. Nelson, S. Reddy, and D. Zhou, [Usp. Fiz. Nauk161,61(1991)]. (2018), arXiv:1802.08244 [hep-ph]. [6] S. Nussinov, Phys. Lett. 165B, 55 (1985). [32] M. Cirelli, P. Panci, G. Servant, and G. Zaharijas, JCAP [7] S. Dodelson and L. M. Widrow, Phys. Rev. D42, 326 1203, 015 (2012), arXiv:1110.3809 [hep-ph]. (1990). [33] S. Tulin, H.-B. Yu, and K. M. Zurek, JCAP 1205, 013 [8] S. M. Barr, R. S. Chivukula, and E. Farhi, Phys. Lett. (2012), arXiv:1202.0283 [hep-ph]. B241, 387 (1990). [34] S. D. McDermott, H.-B. Yu, and K. M. Zurek, Phys. [9] D. B. Kaplan, Phys. Rev. Lett. 68, 741 (1992). Rev. D85, 023519 (2012), arXiv:1103.5472 [hep-ph]. [10] G. R. Farrar and G. Zaharijas, Phys. Rev. Lett. 96, [35] R. Essig et al., CSS2013: New Light Weakly Coupled Par- 041302 (2006), arXiv:hep-ph/0510079 [hep-ph]. ticles, (2013), arXiv:1311.0029 [hep-ph]. [11] D. E. Kaplan, M. A. Luty, and K. M. Zurek, Phys. Rev. [36] S. Eijima, M. Shaposhnikov, and I. Timiryasov, (2018), D79, 115016 (2009), arXiv:0901.4117 [hep-ph]. arXiv:1808.10833 [hep-ph]. [12] Y. Grossman, Y. Nir, and R. Rattazzi, Adv. Ser. Di- [37] W. R. Inc., “Mathematica, Version 11.3,” Champaign, rect. High Energy Phys. 15, 755 (1998), [,755(1997)], IL, 2018. arXiv:hep-ph/9701231 [hep-ph]. [38] R. J. Scherrer and M. S. Turner, Astrophys. J. 331, 19 [13] K. Agashe and G. Servant, JCAP 0502, 002 (2005), (1988), [Astrophys. J.331,33(1988)]. arXiv:hep-ph/0411254 [hep-ph]. [39] S. Hannestad, Phys. Rev. D70, 043506 (2004), [14] H. Davoudiasl, D. E. Morrissey, K. Sigurdson, and arXiv:astro-ph/0403291 [astro-ph]. S. Tulin, Phys. Rev. Lett. 105, 211304 (2010), [40] T. Venumadhav, F.-Y. Cyr-Racine, K. N. Abazajian, arXiv:1008.2399 [hep-ph]. and C. M. Hirata, Phys. Rev. D94, 043515 (2016), [15] M. Artuso, G. Borissov, and A. Lenz, Rev. Mod. Phys. arXiv:1507.06655 [astro-ph.CO]. 88, 045002 (2016), arXiv:1511.09466 [hep-ph]. [41] M. Laine and Y. Schroder, Phys. Rev. D73, 085009 [16] A. Lenz and U. Nierste, in CKM unitarity triangle. Pro- (2006), arXiv:hep-ph/0603048 [hep-ph]. ceedings, 6th International Workshop, CKM 2010, War- [42] A. Djouadi, J. Kalinowski, and M. Spira, Comput. Phys. wick, UK, September 6-10, 2010 (2011) arXiv:1102.4274 Commun. 108, 56 (1998), arXiv:hep-ph/9704448 [hep- [hep-ph]. ph]. [17] F. J. Botella, G. C. Branco, M. Nebot, and A. Sanchez, [43] P. F. de Salas, M. Lattanzi, G. Mangano, G. Miele, Phys. Rev. D91, 035013 (2015), arXiv:1402.1181 [hep- S. Pastor, and O. Pisanti, Phys. Rev. D92, 123534 ph]. (2015), arXiv:1511.00672 [astro-ph.CO]. [18] W. Altmannshofer, S. Gori, M. Pospelov, and I. Yavin, [44] E. W. Kolb and M. S. Turner, Front. Phys. 69, 1 (1990). Phys. Rev. D89, 095033 (2014), arXiv:1403.1269 [hep- [45] M. Cirelli, G. Corcella, A. Hektor, G. Hutsi, M. Kadastik, ph]. P. Panci, M. Raidal, F. Sala, and A. Strumia, JCAP [19] A. Celis, J. Fuentes-Martin, M. Jung, and H. Serodio, 1103, 051 (2011), [Erratum: JCAP1210,E01(2012)], Phys. Rev. D92, 015007 (2015), arXiv:1505.03079 [hep- arXiv:1012.4515 [hep-ph]. ph]. [46] E. Kou et al., (2018), arXiv:1808.10567 [hep-ex]. [20] B. Gripaios, M. Nardecchia, and S. A. Renner, JHEP [47] P. comunication: Bostjan Golob (on behalf of the Belle- 05, 006 (2015), arXiv:1412.1791 [hep-ph]. II collaboration), . [21] Beˇcirevi´c,Damir and Fajfer, Svjetlana and Koˇsnik,Nejc [48] J. Brod, A. Lenz, G. Tetlalmatzi-Xolocotzi, and and Sumensari, Olcyr, Phys. Rev. D94, 115021 (2016), M. Wiebusch, Phys. Rev. D92, 033002 (2015), arXiv:1608.08501 [hep-ph]. arXiv:1412.1446 [hep-ph]. [22] V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov, [49] F. Krinner, A. Lenz, and T. Rauh, Nucl. Phys. B876, Phys. Lett. 155B, 36 (1985). 31 (2013), arXiv:1305.5390 [hep-ph]. [23] M. Fukugita and T. Yanagida, Phys. Lett. B174, 45 [50] S. Stone and L. Zhang, Adv. High Energy Phys. 2014, (1986). 931257 (2014), arXiv:1402.4205 [hep-ex]. [24] A. D. Dolgov, Phys. Rept. 222, 309 (1992). [51] R. Aaij et al. (LHCb), (2018), arXiv:1809.07752 [hep-ex]. [25] W. Buchmuller, R. D. Peccei, and T. Yanagida, [52] M. Aaboud et al. (ATLAS), Eur. Phys. J. C78, 250 Ann. Rev. Nucl. Part. Sci. 55, 311 (2005), arXiv:hep- (2018), arXiv:1710.07171 [hep-ex]. ph/0502169 [hep-ph]. [53] H. Davoudiasl, D. E. Morrissey, K. Sigurdson, [26] S. Davidson, E. Nardi, and Y. Nir, Phys. Rept. 466, 105 and S. Tulin, Phys. Rev. D84, 096008 (2011), (2008), arXiv:0802.2962 [hep-ph]. arXiv:1106.4320 [hep-ph]. [27] M. Drewes, B. Garbrecht, P. Hernandez, M. Kekic, [54] M. Pospelov, A. Ritz, and M. B. Voloshin, Phys. Lett. J. Lopez-Pavon, J. Racker, N. Rius, J. Salvado, and B662, 53 (2008), arXiv:0711.4866 [hep-ph]. 19

[55] M. Escudero, N. Rius, and V. Sanz, Eur. Phys. J. C77, [60] A. de Gouvea and A. Kobach, Phys. Rev. D93, 033005 397 (2017), arXiv:1607.02373 [hep-ph]. (2016), arXiv:1511.00683 [hep-ph]. [56] R. K. Leane, T. R. Slatyer, J. F. Beacom, and K. C. Y. [61] J. F. Beacom, N. F. Bell, and G. D. Mack, Phys. Rev. Ng, Phys. Rev. D98, 023016 (2018), arXiv:1805.10305 Lett. 99, 231301 (2007), arXiv:astro-ph/0608090 [astro- [hep-ph]. ph]. [57] S. Alekhin et al., Rept. Prog. Phys. 79, 124201 (2016), [62] C. Cheung, G. Elor, L. J. Hall, and P. Kumar, JHEP arXiv:1504.04855 [hep-ph]. 03, 042 (2011), arXiv:1010.0022 [hep-ph]. [58] V. Gonzalez Macias and J. Wudka, JHEP 07, 161 (2015), [63] C. Cheung, G. Elor, L. J. Hall, and P. Kumar, JHEP arXiv:1506.03825 [hep-ph]. 03, 085 (2011), arXiv:1010.0024 [hep-ph]. [59] V. Gonzlez-Macas, J. I. Illana, and J. Wudka, JHEP 05, [64] C.-W. Hwang, Eur. Phys. J. C23, 585 (2002), arXiv:hep- 171 (2016), arXiv:1601.05051 [hep-ph]. ph/0112237 [hep-ph].