This is the accepted manuscript made available via CHORUS. The article has been published as: Baryogenesis from oscillations of charmed or beautiful baryons Kyle Aitken, David McKeen, Thomas Neder, and Ann E. Nelson Phys. Rev. D 96, 075009 — Published 9 October 2017 DOI: 10.1103/PhysRevD.96.075009 http://dx.doi.org/10.1103/PhysRevD.96.075009 PITT-PACC-1709 Baryogenesis from Oscillations of Charmed or Beautiful Baryons Kyle Aitken,1, ∗ David McKeen,2, † Thomas Neder,3, ‡ and Ann E. Nelson1, § 1Department of Physics, University of Washington, Seattle, WA 98195, USA 2Pittsburgh Particle Physics, Astrophysics, and Cosmology Center, Department of Physics and Astronomy, University of Pittsburgh, PA 15260, USA 3AHEP Group, Instituto de Física Corpuscular — C.S.I.C./Universitat de València, Parc Científic de Paterna, C/ Catedrático José Beltrán 2 E-46980 Paterna (Valencia), Spain (Dated: September 7, 2017) We propose a model for CP violating oscillations of neutral, heavy-flavored baryons into an- tibaryons at rates which are within a few orders of magnitude of their lifetimes. The flavor structure of the baryon violation suppresses neutron oscillations and baryon number violating nuclear decays to experimentally allowed rates. We also propose a scenario for producing such baryons in the early Universe via the out-of-equilibrium decays of a neutral particle, after hadronization but before nucleosynthesis. We find parameters where CP violating baryon oscillations at a temperature of a few MeV could result in the observed asymmetry between baryons and antibaryons. Furthermore, part of the relevant parameter space for baryogenesis is potentially testable at Belle II via decays of heavy flavor baryons into an exotic neutral fermion. The model introduces four new particles: three light Majorana fermions and a colored scalar. The lightest of these fermions is typically long lived on collider timescales and may be produced in decays of bottom and possibly charmed hadrons. PACS numbers: 11.30.Fs,14.20.-c,14.80.Ly,14.80.Nb I. INTRODUCTION The puzzle of baryogenesis, how the Universe came to be composed primarily of matter rather than equal amounts of matter and antimatter, has led to numerous theories about physics beyond the standard model (SM), beginning with the pioneering work of Sakharov [1]. Three ingredients are present in one form or another in any baryogenesis theory: baryon number violation, C and CP violation, and departure from thermal equilibrium. Because baryon number violation is required, initially baryogenesis was thought to involve new baryon number violating processes which are only important at very high energies, although it was later realized that anomalous electroweak processes could do the job at temperatures as low as the weak phase transition [2]. Most baryogenesis models require the Universe to re- heat after inflation to a high temperature, typically well above the weak scale. However, many theories of physics beyond the SM are inconsistent with a high inflation scale or are inconsistent with a high postinflation reheat scale. Axion dark matter, if the axion is present during infla- tion, requires a low inflation scale in order to avoid ex- cessive isocurvature perturbations [3]. Supersymmetry requires a low reheat scale in order to avoid overproduc- tion of the gravitino [4]. The relaxion solution to the hierarchy problem requires a low inflation scale so that the Hubble temperature during inflation does not sup- press instantons [5]. In addition, avoiding the need for ∗ kaitken17@gmail.com † dmckeen@pitt.edu ‡ neder@ific.uv.es § aenelson@uw.edu a high reheat temperature or production of heavy parti- cles during reheating means a low baryogenesis scale is consistent with a wider variety of inflationary models [6]. Lower reheat temperatures are possible provided the inflaton decays produce heavy particles which decay out of thermal equilibrium in a baryon and CP violating manner [7]. In Ref. [7] baryogenesis occurs due to the baryon number violating decays of TeV mass squarks in an R-parity violating supersymmetric model, in which the reheat temperature could be as low as an MeV, pro- vided that the heavy squarks can be produced out of equilibrium at the end of inflation. Such squark medi- ated baryon number violation is consistent with the ob- served lifetime of the proton, due to the conservation of lepton number, and, depending on the flavor structure of the baryon number violating operators, can be consistent with the stability of heavy nuclei as well. A similar model involving the decays of Majorana fermions was consid- ered in Ref. [8]. In Ref. [9] it was pointed out that heavy flavor baryon number violation could lead to oscillations of the Ξ0b baryon at a rate comparable to its lifetime, while being consistent with the lifetime of heavy nuclei. Enhanced baryon number violation involving heavy fla- vors was also studied in Ref. [10]. Here we present a baryogenesis model which is consis- tent with a reheat temperature as low as a few MeV, and which requires no postinflationary production of any par- ticle heavier than about 6—10 GeV. The required baryon number violation is conceivably observable via the oscil- lations of heavy-flavor neutral baryons, and the required CP violation is potentially of O(1) in such oscillations. The processes that produce the baryon asymmetry in the early Universe involve particles and phenomena which can be directly studied in the laboratory – a unique fea- ture of our theory. Our proposal is that certain neutral mailto:kaitken17@gmail.com mailto:dmckeen@pitt.edu mailto:neder@ific.uv.es mailto:aenelson@uw.edu 2 heavy flavor baryons undergo CP and baryon number violating oscillations and decays, and are produced in the early Universe via the out of equilibrium decays of a weakly coupled neutral particle whose lifetime is of or- der 0.1 s, a time when the temperature is of order a few MeV. The basic scenario was outlined in Ref. [11], and the model we study has the same field content and cou- plings as Ref. [12]. The basic formalism for analyzing CP violation in fermion antifermion oscillations was worked out in Ref. [13]. The outline of the paper is as follows. In section II the model is introduced, and the effective operator responsi- ble for baryon oscillations is constructed. In section III, general ∆B = 2, six-quark effective operators are ana- lyzed for their contribution to dinucleon decay, that is, the decay of two nucleons into mesons. Currently, dinu- cleon decay places similar or stronger constraints on all such operators than does neutron oscillations. For oper- ators that cannot contribute to dinucleon decay at tree level, electroweak corrections to the six-quark operators are examined. In section IV, the general formalism for CP-violating oscillations of fermions is reviewed, and the oscillation parameters are calculated for the model in- troduced in section II. In sections V and VI, direct con- straints on the masses and couplings of the new φ and χ particles from collider searches, and indirect constraints from rare decays of mesons and baryons are derived, re- spectively. Section VII contains our analysis of how in this model the baryon asymmetry of the Universe (BAU) is produced. Finally, in section VIII we conclude and point at possible directions for future work. II. MODEL We wish to find a theory which allows for sufficiently large baryon number and CP violation to explain baryo- genesis at relatively low energy. In order to ensure suf- ficient stability of the proton, we assume lepton number is not violated, other than perhaps via the tiny ∆L = 2 terms that could account for Majorana neutrino masses. The lowest dimension terms which violate baryon num- ber and not lepton number are dimension 9, six-quark ∆B = 2 operators. Such operators can lead to neutral baryon oscillations and conceivably CP violation [11], and can arise as an effective field theory description of physics at some higher energy scale. A minimal renor- malizable model for generating such terms involves a new charge −1/3 color triplet scalar and two Majorana fermions, as described in Ref. [12]. A third Majorana fermion, which decays out of thermal equilibrium, allows for the fulfillment of the out-of-equilibrium Sakharov con- dition. We note that this model for baryon number violation can easily be embedded in an R-parity violating super- symmetric (RPV SUSY) theory. In such theories, the neutralinos would play the role of the Majorana fermions and a down-type SU(2) singlet squark can be the colored scalar. For simplicity, we do not explore this embedding in a SUSY framework in this paper and we stick to the minimal version of the model. Our model thus adds four new particles: three Ma- jorana fermions, χ1,2,3, and a single color triplet scalar, φ. The interactions involving the new particles and weak SU(2) singlet SM quarks are given by Lint ⊃−g∗udφ∗ūRdcR −yidφχ̄idcR + h.c., (1) along with terms involving other generations, d → s,b and u → c,t. By convention we take all two component fields to transform in the left-handed representation un- der Lorentz transformations. dcR stands for the charge conjugate of the right-handed down quark field, which is in the left-handed Lorentz representation. In this ex- pression and throughout the paper, we work in the mass basis, which is unambiguous for SU(2) singlet quarks. The required new particles and their interactions are motivated as follows. A natural way to construct the ∆B = 2 six-quark operator we require for baryon oscilla- tions is from two ∆B = 1 four-fermion interactions con- nected by an exotic neutral Majorana fermion. Thus we introduce an exotic, electrically-neutral, colorless, Majo- rana fermion, χ1, which couples to other fermions via a four-fermion interaction of the form uRdRd′Rχ1 (using u and d here to represent any up- or down-type quark).1 Since such a four-fermion interaction is itself nonrenor- malizable, we also introduce a complex, color triplet, scalar particle (diquark) φ to mediate the ∆B = 1 in- teractions. Note that if χ1 is heavier than the difference in mass between the proton and electron, mp − me = 937.76 MeV, this interaction does not give rise to proton decay.2 In the presence of only χ1, there is no physical CP violation, as there is enough reparameterization free- dom to remove the phases in the couplings. We introduce a second fermion, χ2 (with mχ2 > mχ1), in order to give rise to CP violation. Finally, for baryogenesis, the oscil- lating baryons must be produced out of thermal equilib- rium. As described in Sec. VII, this is most simply ac- complished by introducing a third Majorana fermion, χ3, which decays out of equilibrium to produce the baryons whose oscillations result in baryogenesis. Note that we only consider operators constructed out of right-handed quarks, for two reasons. Our phenomeno- logical reason is that, as we will show in Sec. III, right handed quark operators are less constrained by dinucleon decay due to the requirement of light quark mass inser- tions in flavor-changing loops. Our top down theoretical reason is that, as mentioned, the interactions in Eq. (1) occur in RPV SUSY models, suggesting a possible em- bedding of our model into a more complete theory. 1 Models of baryogenesis that generate four-fermion interactions of this form and identify χi with right-handed neutrinos can be found in Ref. [14]. 2 The stability of 9Be leads to a marginally stronger lower bound of mχ1 > 937.9 MeV [11]. 3 uR dR dR ūR d̄R d̄R φφ χ mχ FIG. 1. The basic six-quark ∆B = 2 operator generated by φ and χ exchange. u and d here represent any of the up- or down-type quark flavors. The quarks involved are all weak SU(2) singlets, as emphasized by the R subscripts. Figure 1 shows the ∆B = 2 six-quark operators that are generated by the interactions in Eq. (1). Such oper- ators, which can mediate the transition of a baryon B to an antibaryon B̄′, can be written as OBB′ = �abc�def × [ (qR) i a (qR)i,b (qR) j c (q ′ R)j,d (q ′ R) k e (q ′ R)k,f + . . . ] , (2) where a,. . . ,f are color indices, i,j,k spinor indices, and q = u,d,s,c,b any of the quark flavors (because of its short lifetime, the top quark does not hadronize and is not important in the low-energy effective theory) which are all right-chiral. The ellipsis represents other possible permutations of color or spinor indices. Here, B denotes an arbitrary standard model baryon with the quark con- tent qqq while B′ contains q′q′q′. [We will use both the baryon name B or the quark content (qqq) to label the operators in question throughout this paper.] The pre- cise index structure of OBB′ is not important for the pur- poses of this paper. Therefore, in what follows, we will suppress the indices on OBB′ and generically denote the operators we are interested in that appear in the effective Lagrangian via the shorthand Leff ⊃ CBB′ (qqq) (q′q′q′) ≡ CBB′OBB′, (3) keeping in mind that the leading operators that are gen- erated involve only right-chiral quarks. Matching the interactions generated by Eq. (1) to the effective theory at tree-level gives the coefficient of the operator that generates oscillations between a neutral baryon and its antiparticle, B ↔B̄, CBB ∼ ∑ i mχi m2B −mχ2i ( g∗udyid′ + g ∗ ud′yid m2φ )2 , (4) with u, d, and d′ labeling the quarks comprising B. For example, the operator (ddc)2 would allow the processes Σ̄c ↔ Σc. Given this operator, we will find it useful to relate the coefficient to the (dispersive) transition ampli- tude, defined by δBB ≡〈B̄|CBBOBB|B〉, with δBB = κ 2CBB, (5) where κ ∼ 10−2 GeV3 [15]. In analogy with meson oscil- lations, when the two-state system in question is unam- biguous, δBB can also be referred to as M12. Operators which involve different baryons of the form OBB′ would allow for a common decay product between B and B̄ and could also give rise to oscillations. For example, (uss) (uds) would allow for Ξ0 and Ξ̄0 to have a common decay product, through Ξ0 → Λ0 and Ξ̄0 → Λ0 (ignoring any neutral meson products). However, we will find such processes are suppressed relative to their direct oscillation cousins, so we will ignore them in our analysis. There are also baryon-number–preserving operators that contribute to the masses and mixings of SM baryons. These are greatly suppressed relative to those that occur in the SM and we do not consider them further either. Therefore, in what follows, we focus on operators OBB with coefficients of the form of Eq. (4). III. DINUCLEON DECAY CONSTRAINTS As described in the preceding section, we would like our six-quark operators to allow for the oscillation of heavy baryons in order to produce the Universe’s ob- served baryon asymmetry. In Sec. VII, we will show that the ideal width for such an oscillation, which is depen- dent upon the value of CBB in Eq. (4), is a few orders of magnitude smaller than B’s decay width. However, mod- els with B violation by two units are certainly not a new idea, and so significant experimental effort has been put fourth into constraining ∆B = 2 processes. The most immediate constraint on our six-quark operators is the lack of observed dinucleon decay, which we quantify in this section. The analysis we perform here applies to six-quark operators in general and is independent of the origin of the new physics introduced in Sec. II. Dinucleon constraints come from underground detec- tors whose primary purpose is the detection of proton decay and neutrino oscillations. For example, in a nu- cleus, a n → n̄ transition will be shortly followed by the annihilation of the n̄ with one of the other nucleons, leading to the decay of the nucleus of mass number A to a nucleus with A′ = A − 2 plus mesons. The lack of observation of such decays can therefore bound the transition amplitude δnn [16] which is related to the co- efficient of the (udd)2 operator, Cnn [cf. Eqs. (4) and (5)]. Currently, the lower bound on the 16O lifetime (in decays to pions) of 1.9 × 1032 years from the Super- Kamiokande collaboration [17] places the strongest limit, δnn < 1.9 × 10−33 GeV. Operators that also violate strangeness do not directly induce n → n̄ transitions in a nucleus. However, they can also lead to dinucleon decays, A → (A−2) + mesons, through the reaction NN → kaons + X where N is a nucleon. For example, the diagram on the left of Fig. 2 shows how the operator (uds)2 can lead to dinucleon de- cay to a pair of kaons. The Super-Kamiokande collabo- ration [18] has searched for such decays and has placed 4 s̄ s̄d u d u/d u/d u p/n p/n K+/K0 K+/K0 s̄ s̄ s̄ u d u d u/d u/d u s W ± p/n p/n K+/K0 K+/K0 K0 FIG. 2. Left: Dinucleon decay via the ∆B = ∆S = 2 (uds)2 operator that mediates Λ0 ↔ Λ̄0 oscillations. Right: Din- ucleon decay mediated by the ∆B = 2, ∆S = 4 (uss)2 operator that becomes ∆S = 3 (uds)(uss) operator in the presence of flavor-changing weak interactions. Because the short-distance ∆B = 2 operators we consider involve weak isosinglets, this operator requires light quark chirality flips, indicated by crosses. See text for discussion of the match- ing of the short distance theory onto the (chiral symmetry violating) long distance theory. an upper bound on the pp → K+K+ decay rate by lim- iting the lifetime for 16O → 14C K+K+ to more than 1.7 × 1032 years. To make use of this limit, we start with the effective operator OBB. The dinucleon decay rate through direct nucleon annihilation can then be roughly approximated by considering the decay rate to a meson pair [19], ΓNN→X ∼ 9 32π |CBB|2 m2N |〈2 mesons|OBB|NN〉|2 ρN (6) where mN is the nucleon mass, ρN ' 0.25 fm−3 is the nucleon density, and we have ignored the masses of the final state particles. In the case of operators that can con- tribute at tree level, the matrix element can be estimated as roughly 〈2 mesons|OBB|NN〉∼ Λ5QCD ' (200 MeV) 5. Using this and Eq. (5), the limit on the rate for NN → KK from Super Kamiokande translates to a limit on the transition amplitude of δ(uds)2 . 10 −30 GeV. (7) In what follows, we also take operators that change strangeness by one or three units to have roughly the same bound as this. Kinematic constraints protect certain operators from contributing to dinucleon decay at leading order. Op- erators such as (uss)2 that change strangeness by four units (i.e. ∆S = 4) are kinematically forbidden from con- tributing to dinucleon decay at tree level since 2mN < 4mK. Similarly, those that involve charm3 or bottom quarks also do not lead to dinucleon decay at leading or- der. However, when combined with flavor-violating weak 3 Depending on the nucleon binding energy, nn → Dγ through a ∆C = 1 operator is kinematically allowed for some nuclei, interactions, these operators involving heavy quarks can lead to dinucleon decay. An illustration of this is shown on the right of Fig. 2. To properly estimate the rate for dinucleon decay from ∆B = 2 operators (involving heavy flavors), we must match the UV theory involving quarks to a low energy effective theory involving baryons valid at momentum transfers below 4πfπ ∼ 1 GeV where fπ = 93 MeV. This consists of writing down an operator in the UV theory and treating the coefficient of this operator as a spurion that transforms in a particular way under the global chiral quark flavor symmetry SU(3)L×SU(3)R so as to make the operator invariant. This operator is then matched onto an operator in the effective theory that transforms in the same way under the chiral symmetry with the same spurion coefficient. In the UV, the light quarks qL,R transform as triplets under SU(3)L,R. In the low energy theory, the meson octet, Π, is described by a field Σ = exp (2iΠ/fπ) which transforms under the chi- ral symmetry as Σ → LΣR† where L, R are SU(3)L,R transformations, respectively. Incorporating the baryon octet (see, e.g., Ref. [20]) can be done by defining a field ξ = exp (iΠ/fπ) which transforms as ξ → LξU†, ξ → UξR† under SU(3)L,R. U is an SU(3) matrix that depends nonlinearly on the meson fields. The baryon octet B is defined to transform as B → UBU†. Oper- ators in the effective theory are then constructed out of Σ, B, and ξ along with spurions from the UV theory to be invariant under the flavor symmetry. Since the chiral symmetry is dynamically broken by the strong coupling of QCD around 4πfπ, one can use naive dimensional anal- ysis to properly account for factors of 4π (that come from the strong coupling) and the cutoff, 4πfπ, that appear in this matching procedure, as described in, e.g., Ref. [21]. We will first illustrate this matching procedure in our theory with interactions given by Eq. (1), assuming for now that only the light quarks u, d, and s are involved. We will deal with heavy quarks c and b below. Since distinction between chiralities is necessary, we will tem- porarily denote them explicitly. After integrating out the scalar, φ, and the Majorana fermions, χi, we are left with a ∆B = 2 operator involving only (light) right-chiral quarks, CBB(qRqRqR)2. This operator must be matched onto an operator valid at long distances involving baryons at the scale of chiral symmetry breaking. The coeffi- cient CBB can be treated as a spurion that transforms under SU(3)R in a representation that appears in the tensor decomposition of 6 triplets. For definiteness, take it to transform as an SU(3)R octet. Then the object C̃BB ≡ ξCBBξ† transforms as C̃BB → UC̃BBU† and the but due to the dependence of the amplitude on the photon mo- mentum and coupling and phase space suppression the rate is proportional to (α/4π) (kγ/mN ) 3 ∼ 10−9, where kγ is the pho- ton energy, suppressing the rate below other decays with less constrained phase space. 5 operator matching is CBB(qRqRqR) 2 → ( 4πf3π )2 trBC̃BBB + . . . , (8) where the ellipsis represents other possible orderings of the baryon octets and the spurion. Note that this gives an understanding of the size of κ ∼ 4πf3π ' 10−2 GeV3 in Eq. (5). Adjusting this analysis if CBB transforms under a different representation of SU(3)R is straightforward; one inserts the required numbers of ξ and ξ† into the definition of C̃BB so that it transforms in such a way as to leave trBC̃BBB invariant. For example, if CBB is a singlet then one simply takes C̃BB ≡ CBB. Four-quark weak operators involving light quarks can be matched onto the low energy effective the- ory in much the same way. The coefficient of the operator ūLγµq j Lq̄LiγµuL can be viewed as a spurion that transforms as an octet under SU(3)L and the strangeness changing (∆S = 1) coefficient takes a value ∝ GFVusV ∗udh with hij = δi2δ3j . Then ξ†hξ → Uξ†hξU† and the matching is GF√ 2 VusV ∗ udūγ µ ( 1 −γ5 ) sd̄γµ ( 1 −γ5 ) u → GF√ 2 VusV ∗ ud ( 4πf3π ) trB̄ξ†hξB + . . . , (9) where again the ellipsis represents other possible order- ings of B, B̄, and ξ†hξ. Now we can combine a ∆B = 2 operator that also changes strangeness by n units with the weak ∆S = 1 operator to form a ∆B = 2, ∆S = n−1 operator that is given by GF√ 2 VusV ∗ udf 2 π ( 4πf3π )2 trB̄C̃BBξ †hξB + . . . . (10) In other words, if the leading ∆B = 2 operator has ∆S = n, the ∆B = 2, ∆S = n − 1 operator that is generated due to weak interactions is suppressed relative to it by the factor GF√ 2 VusV ∗ udf 2 π ∼ 10−8. (11) Thus, for example, the bound on a leading ∆S = 4 op- erator (uRsRsR)2 from the lack of dinucleon decay is around eight orders of magnitude weaker than that on the ∆S = 3 operator (uds)(uss) [which we take to be comparable to that on the ∆S = 2 operator (uds)2], δ(uss)2 . 10 −22 GeV. (12) Now, we consider the case where the leading ∆B = 2 operators contain heavy quarks. Consider, for example, if after integrating out the heavy scalar φ and Majo- rana fermions χi, that the leading operator we generate is C(udb)2 (uRdRbR)2. Before matching onto the theory valid after chiral symmetry breaking we must first inte- grate out the b quarks. In the presence of weak interac- tions, as shown in Fig. 3, doing so will lead to a ten-quark ⟨ūLuR⟩ ⟨ūLuR⟩dL dL uL dRdR uL b b FIG. 3. The ten-quark ∆B = 2 operator that results from the leading (uRdRbR)2 operator after integrating out the b quarks. The crosses represent chirality-flipping b quark mass insertions. We use 〈ūLuR〉 to indicate the pairs of light quark fields that can be replaced by the chiral condensate when matching onto the long distance theory relevant for dinucleon decay. operator, C(udb)2 (uRdRbR) 2 → ( GF√ 2 VubV ∗ ud 1 mb )2 ×C(udb)2 (uRdRdLūLuL)2. (13) After chiral symmetry breaking, ūLuR can be replaced by the quark condensate which is roughly 4πf3π. This means that the induced ∆B = 2 operator (uLdRdL)2 is suppressed relative to (uRdRbR)2 by the factor( GF√ 2 VubV ∗ ud 4πf3π mb )2 ∼ 10−20. (14) Additionally in the case of a leading operator containing a b and c quark, e.g. (dRcRbR)2, there are perturbative loops that generate dimension-nine operators involving light quarks above the chiral symmetry breaking scale. In the case of the operator (dRcRbR)2, two such loops can be used to generate the operator (uLdLdR)2 with a coefficient suppressed relative to the leading one by( GF√ 2 VubV ∗ cd mbmc 4π2 log m2W m2b )2 ∼ 10−16. (15) In Table I, we list operators that can mediate B ↔ B̄ transitions along with the number of loops required for each operator to mediate (∆S = 0, 1, 2, 3) dinucleon decay. We show the resulting limits on the transition amplitudes δBB = |M12| = κ2CBB of each operator from the lack of observation of dinucleon decay, accounting for the appropriate suppression factors. In general we find that only operators which require 2 or more weak interactions to contribute to dinucleon decay can give baryon oscillations at a rate which is large enough to be relevant for either experimental searches or baryogenesis. The last column of the table gives the limit on the size of 6 the operator that can be produced in our specific model when collider constraints on new particles are considered, which will be discussed in Sec. V. IV. CP VIOLATION IN HEAVY BARYON OSCILLATIONS The evolution of the (B, B̄) system in vacuum, assum- ing CPT conservation, can be described [11] by a 2 × 2 Hamiltonian, H = M − i 2 Γ = ( MB − i2 ΓB M12 − i 2 Γ12 M∗12 − i2 Γ ∗ 12 MB − i2 ΓB ) . (16) M and Γ are both Hermitian matrices that describe the dispersive and absorptive parts of the B, B̄ → B, B̄ am- plitude, respectively. This system is entirely analogous to the very well known case of neutral mesons and an- timesons. Because of the off-diagonal terms in H, the mass eigenstates |BL,H〉 with masses mL,H are linear com- binations of the flavor eigenstates |B〉 and |B̄〉, |BL,H〉 = p|B〉±q|B̄〉. (17) The mass difference is ∆m = mH−mL > 0 and the width difference between the states is ∆Γ = ΓH−ΓL and can be of either sign. The flavor admixtures can be determined by ( q p )2 = M∗12 − (i/2)Γ∗12 M12 − (i/2)Γ12 . (18) A state that begins at t = 0 as a |B〉 or |B̄〉 is at time t |B(t)〉 = g+(t)|B〉− q p g−(t)|B̄〉, |B̄(t)〉 = g+(t)|B̄〉− p q g−(t)|B〉 (19) with g±(t) = 1 2 ( e−imHt− 1 2 ΓHt ±e−imLt−12 ΓLt ) . (20) A particularly useful quantity that measures the level of CP and baryon number violation is the quantity, AB = PB→B −PB→B̄ + PB̄→B −PB̄→B̄ PB→B + PB→B̄ + PB̄→B + PB̄→B̄ , (21) where, e.g., PB→B̄ is the time integrated probability for an intitial B state to oscillate into a B̄ and the other terms are defined analogously. In terms of the elements of H, this can be concisely expressed, AB = 2Im (M∗12Γ12) Γ2B + 4 |M12| 2 . (22) This expresses the familiar fact that CP violation requires a phase difference between the absorptive and dispersive parts of the transition amplitudes. The dispersive part of the transition amplitude, M12, is dominantly given by off-shell χi exchange in our model, as seen in Fig. 1. We have already written down what we need to estimate this in Eqs. (4) and (5), resulting in M12 ∼ κ2 ∑ i mχi m2B −mχ2i ( g∗udyid′ m2φ )2 . (23) Here, u, d, and d′ refer to the flavors that comprise B and we have assumed, if d 6= d′, that g∗udyid′ � g∗ud′yid. If we concentrate on the contribution due to a particular χi and express it in terms of its mass difference from the baryon, ∆mBi = mB −mχi, we have |M12|i ∼ κ2 2∆mBi ∣∣∣∣∣g ∗ udyid′ + g ∗ ud′yid m2φ ∣∣∣∣∣ 2 ' 8 × 10−16 GeV ( 500 MeV ∆mBi ) × ( 600 GeV mφ/ √ |g∗udyid′ + g∗ud′yid| )4 . (24) The absorptive part of the transition amplitude re- quires an on-shell state into which both B and B̄ can decay. This requires at least χ1 to be light enough for either baryon or antibaryon to decay into it. CP viola- tion will be largest when the mass splitting between χ1 and B is not too large. In this case the most important states for Γ12 are decays of B to χ1 plus a meson. The contribution from χ1π0, for instance, can be estimated using the effective Lagrangian, Leff ⊃−yiBπ0B̄iγ5χi + h.c., (25) where yiB ∼ 4πκ mB g∗udyid′ m2φ . (26) The factor of 4π in this expression accounts for the non- perturbative nature of the interaction, which is similar to the pion-nucleon vertex. This interaction gives a con- tribution to Γ12 of Γ12 ∼ ∑ i y2iBmχi 32π (1 + rχi −rπ0 ) λ1/2 (1,rχi,rπ0 ) , (27) where rχi,π0 = m 2 χi,π0 /m2B and λ (a,b,c) = a 2 + b2 + c2 − 2ab − 2ac − 2bc. The magnitude of the contribution to Γ12 from a particular χi is roughly |Γ12|i ∼ y2iB 8π ∆mBi ∼ 2πκ2 m2B ∣∣∣∣∣g ∗ udyid′ m2φ ∣∣∣∣∣ 2 ∆mBi ' 1 × 10−16 GeV ( ∆mBi 500 MeV ) × ( 5 GeV mB )2 ( 600 GeV mφ/ √ |g∗udyid′| )4 . (28) 7 Operator B Weak Insertions Measured Limits on δBB = M12 (GeV) Required Γ (GeV) [22] Dinucleon decay Collider (udd)2 n None (7.477 ± 0.009) × 10−28 10−33 10−17 (uds)2 Λ None (2.501 ± 0.019) × 10−15 10−30 10−17 (uds)2 Σ0 None (8.9 ± 0.8) × 10−6 10−30 10−17 (uss)2 Ξ0 One (2.27 ± 0.07) × 10−15 10−22 10−17 (ddc)2 Σ0c Two ( 1.83+0.11−0.19 ) × 10−3 10−17 10−16 (dsc)2 Ξ0c Two ( 5.87+0.58−0.61 ) × 10−12 10−16 10−15 (ssc)2 Ω0c Two (9.5 ± 1.2) × 10−12 10−14 10−15 (udb)2 Λ0b Two (4.490 ± 0.031) × 10 −13 10−13 10−17 (udb)2 Σ0b ∗ Two ∼ 10−3∗ 10−13 10−17 (usb)2 Ξ0b Two (4.496 ± 0.095) × 10 −13 10−10 10−17 (dcb)2 Ξ0cb † Two ∼ 10−12† 10−17 10−15 (scb)2 Ω0cb † Two ∼ 10−12† 10−14 10−15 (ubb)2 Ξ0bb ‡ Four ∼ 10−13‡ >1 10−17 (cbb)2 Ω0cbb † Four ∼ 10−12† >1 10−15 TABLE I. Operators that mediate B ↔B̄ oscillations and the number of weak interaction insertions required for each of these to contribute to dinucleon decay. The resulting limit from dinucleon decay on the transition amplitude, defined in Eq. (5), for each operator is shown. An ∗ indicates a baryon that has not yet been observed and which has a strong decay channel open. A † (‡) indicates an unobserved baryon which primarily decays through a weak interaction of a c (b) quark. We now see the reason we need at least two Majorana fermions. If there were only a single Majorana fermion, χ1, that contributed to M12 and Γ12, they would have the same phase and AB in Eq. (22) would vanish. Thus, we use contributions from χ1 and χ2 exchange to obtain a physical, CP-violating phase difference between M12 and Γ12. The ratio of the single meson contribution to Γ12 from χ1 to its contribution to M12 is∣∣∣∣ Γ12M12 ∣∣∣∣ 1 ∼ 4π ( ∆mB1 mB )2 ' 0.1 ( ∆mB1 500 MeV )2 ( 5 GeV mB )2 . (29) The CP-violating quantity AB in Eq. (22) linearly de- pends on |Γ12|. Without finely tuning the contributions due to χ1 and χ2 against each other, this value of the ratio due to χ1 alone is roughly as large as the total ratio |Γ12/M12| can get. V. COLLIDER CONSTRAINTS To obtain a large amount of CP violation in heavy baryon oscillation, it will be clear that the lightest two Majorana fermions must have masses on the order of a few GeV along with couplings to quarks that are not too small. In this discussion, we consider the two lightest Ma- jorana fermions. The third, χ3 must be weakly coupled in the minimal version of the model, due to cosmological considerations as we will see in Sec. VII. The constraints that we will discuss in this section re- quire that φ have a mass of at least a few hundred GeV. In this case, χi decays can be analyzed by integrating out the scalar in Eq. (1) resulting in four-fermion inter- actions, − gudyid ′ m2φ χ̄iūRd c Rd ′c R + h.c. (30) (For d 6= d′, we have again assumed that gudyid′ � gud′yid.) Both the interactions responsible for the de- cay of the Majorana fermions and those that source CP-violating baryon oscillations are of the same form. The quarks involved in the decay operator must be lighter than those responsible for baryon oscillations and the couplings responsible for decay must be relatively smaller, to avoid stronger dinucleon decay limits from ∆B = 2 quarks involving light quarks. The interaction in Eq. (30) allows for the decay χi → udd′, where u, d, and d′ are up- and down-type quarks light enough for this to be kinematically allowed. It is reasonable to assume that one mode dominates their al- lowed branchings and in this case, their lifetimes are τχi ∼ 2 (8π) 3 m5χ1,2 ∣∣∣∣∣ m 2 φ gud yid′ ∣∣∣∣∣ 2 ' 10−6 s ( 5 GeV mχi )5 ( mφ/ √ gud yid′ 20 TeV )4 . (31) For mχi = 5 GeV, with couplings gud yid′ . (mφ/20 TeV) 2 the ∆B = 2 transition amplitude in the udd′ system is less than 10−22 GeV, avoiding con- flict with constraints from dinucleon decay (see Table I). 8 Furthermore, if mχi = 5 GeV, as long as gud yid′ & (mφ/350 TeV) 2, τχi . 0.1 s which is a short enough lifetime to avoid spoiling successful BBN (see, e.g., [23]). We might ask whether instead of ensuring that χ1,2 decay fast enough to avoid spoiling BBN, the lightest fermion χ1 could instead be long enough lived to serve as dark matter. First we note that the range allowed for kinematic stability of both proton and χ1 is extremely fine tuned, with the χ1 mass between mp − me and mp + me. If we assume all the χ’s participate in a vi- able heavy flavor baryogenesis mechanism, we will see that we must require the χ1,2 masses to be around 3– 5 GeV, and we also need sufficiently large four-fermion interactions involving χ1,2 and heavy flavor quarks, sup- pressed by a scale Λheavy ≡ mφ/ √ gy ∼ 600 GeV. Here g and y here label couplings with the relevant flavor struc- ture. Based on our discussion in Sec. III, at the one loop level we must generate four-fermion operators involving light quarks (into which χ1,2 can decay) are generated with a scale Λlight & 104Λheavy ∼ 106 GeV. This pro- vides a lower bound on the strength of the light quark four-fermion operator which gives an upper bound on the χ1,2 lifetime of τχ1,2 . 1000 s in the absence of fine- tuning against some other source of this operator. There is also an unavoidable decay channel that comes from the mixing of χ1,2 with the heavy flavor baryon, B, whose oscillations are responsible for the BAU, with a mixing angle θ ∼ |M12|/∆m where ∆m is the mass splitting be- tween the Majorana fermions and B. This mixing leads to the decay of χ1,2 into B’s decay channels with a partial width proportional to θ2ΓB, which is much shorter than the lifetime of the universe. We see that the Majorana fermions are generically un- stable but long-lived on the scale of collider experiments and appear as missing energy. Decay lengths on the or- der of 102 to 107 m are expected, potentially relevant for the recently proposed MATHUSLA detector [24] which is optimized to search for long-lived particles. In what follows, to analyze collider constraints on the new scalar φ we will assume that any χi produced at a collider is invisible and defer discussion of the displaced decay sig- natures at, e.g., MATHUSLA. Now that we know that the χi’s are invisible at col- liders, we can understand how the scalars appear when produced in hadron collisions. Because φ is a color fun- damental, if it is kinematically accessible, φφ∗ pairs are easily produced in proton-(anti)proton collisions, and the signatures are essentially those of squarks in RPV SUSY. In addition to QCD production, (single) scalars can be resonantly produced in the presence of some nonzero gud. Once produced, the scalar decays through one of the in- teractions in Eq. (1), either to quark pairs with a rate Γφ→ūd̄ ' ∑ i,j ∣∣guidj∣∣2 16π mφ, (32) or to χi plus a quark, Γφ→χd ' ∑ i,j ∣∣yidj∣∣2 16π mφ, (33) where we have assumed that mφ is much larger than the mass of any decay product. Therefore, these scalars can appear in searches for dijet resonances (either singly or pair produced) and (mono)jets and missing energy. Which search is most sensitive depends on mφ and the branching fractions for φ → ūd̄ and φ → χd. Taken together, LHC searches for pair produced dijet resonances, both with [25] and without [26] heavy flavor in the final states, as well as standard SUSY searches for (b-tagged [27, 28] or not [28, 29]) jets plus missing en- ergy rule out φ masses below about 400 GeV. Above this mass, limits from pair produced dijet resonances are no longer constraining while resonant production of a sin- gle φ with a rate proportional to |gud|2 for some u, d is important [30]. We use the limits from resonant di- jet production from [30] and recast searches for jets and missing energy [27–29] as well as monojets [31] to find limits on the couplings gud and yid′ as functions of mφ. We find this limit for every flavor u, d, and d′, assuming that only gud and yid′ are relevant. Given these limits, the maximum value of the product of couplings gudyid′ at each mφ can be found, and taking a value of the mass splitting between χi and the udd′ baryon, which can be turned into an upper limit on the transition amplitude δudd′ = M12 in the udd′ system.4 We show the upper limit on M12 as a function of mφ for each pattern of flavors u, d, and d′, assuming the dominance of one par- ticular pair of couplings gud, yid′ and a mass splitting between mχi and the udd ′ baryon of 200 MeV in Fig. 4. We also show the largest value of M12 allowed from col- lider searches in each neutral baryon system in Table I. Lastly, we note that the six-quark ∆B = 2 operators themselves can lead to interesting signatures at the LHC. These were studied in Ref. [32]. VI. HADRON PHENOMENOLOGY A. Hadron decays After integrating out the heavy colored scalars, four- fermion interactions between the Majorana fermions and quarks are generated as in Eq. (30). These can lead to new decays of hadrons to final states that differ in baryon 4 Note that we perform this scan for a single Majorana fermion. Including a second, as we must to obtain CP violation, does not change the allowed values by more than an O(1) factor which, given our level of precision, is unimportant. The third, χ3 must be more weakly coupled than χ1,2 and can be even more safely neglected here. 9 ��� ��� ��� ���� ���� ���� ��-�� ��-�� ��-�� ��-�� �ϕ (���) |� � � | (� �� ) ������ ������ ������ ������ ������ ������ ��-�� ��-�� ��-�� ��-�� |� � � | (� �� ) �������� �������� �������� �������� �������� �������� FIG. 4. Upper limits on M12 as functions of mφ that re- sult from collider searches for diject resonances and jets plus missing energy, assuming the dominance of the product of couplings gudyid′ indicated, where u and d (′) label generic up- and down-type quarks, respectively. We have taken ∆mB = mB −mχ = 200 MeV. Top: The limits when yid or yis are dominant. Bottom: The limits when yib is dominant. Solid curves show the limits in the case where the charge 2/3 quark involved is u while dashed lines show the limit in the case of the c quark. number by one unit along with any kinematically acces- sible χi, e.g., meson → baryon + χi [+ meson(s)], baryon → meson(s) + χi. (34) As we showed in Sec. V, on the scale of particle physics experiments, χi appear as missing energy. For definiteness, let us focus now on four-fermion in- teractions that involve the b quark and the lightest Ma- jorana fermion. This is potentially relevant to the case where baryons containing b quarks undergo CP-violating oscillations in the early Universe, producing the BAU; operators involving heavy quarks are less constrained by dinucleon decay and are therefore more promising candi- dates, cf. Table I. Similar considerations apply for oper- ators involving lighter quarks. Consider, as a definite example, b decays through the operator − guby1d m2φ χ1uRdRbR, (35) where u and d here are the actual up and down quarks. (We have omitted the contribution to this operator from gudy1b which is more constrained by collider searches.) The rate for the b quark to decay through such an inter- action is Γb→χ1ūd̄ ∼ mb∆m 4 60 (2π) 3 ( guby1d m2φ )2 + O ( ∆m5 m5b ) ' 2 × 10−15 GeV ( ∆m 2 GeV )4 ( 1.2 TeV mφ/ √ guby1d )4 . (36) In this expression ∆m is the mass splitting between χ1 and the bottom quark (we have ignored masses in the final state besides mχ1). We have chosen to nor- malize this expression on values of the mass splitting and mφ/ √ guby1d that result in a transition amplitude of |M12| ∼ 10−17 GeV in the Λ0b = (udb) baryon system, which is the rough collider limit. Given this mass split- ting, this can lead to decays of B+ mesons to a nucleon and χ1 with a branching ratio of BrB±→Nχ1+X ∼ 6 × 10 −3 ( ∆m 2 GeV )4 × ( 1.2 TeV mφ/ √ guby1d )4 , (37) where X represents possible additional pions. This is not a small branching fraction, although final states of this form have not yet been searched for in B meson decays. However, the requirement that the final state hadrons carry baryon number means that this decay is kinemati- cally forbidden if mχ1 > mB± −mp = 4.34 GeV. Decays of bottom baryons would be allowed to proceed for split- tings down to mπ, and one could expect branching ratios on the order of 10−3 for the parameters in Eq. (37).5 We also expect “wrong sign” decays of heavy baryons in this model, following a B →B̄ oscillation, with a branch- ing fraction that is roughly 1 2 |M12|2 Γ2B . (38) Consider, e.g., the Ω0c. Given the constraints that appear in Table I, this branching could potentially be as large as 10−7. The Belle II experiment hopes to collect ∼ 50 ab−1 of e+e− data at √ s = 10.56 GeV collecting about 50×109 B meson pairs. If Ω0c baryons are produced in 2% of B 5 The calculation of the baryon decay rate to χ1 and a single meson is essentially the same as that of Γ12 in Sec. IV, modulo a factor of mb/mχ1 ∼O (1). 10 meson decays (comparable to the measured production of Λc baryons), then there would be a sample of about 109 Ω0c’s and Ω̄ 0 c’s. Thus, there could be a few hundred “wrong sign” decays in the data sample. While this would be a challenging measurement, it is interesting that it is in principle observable at the next generation B-factory given current experimental limits. We mention here that baryon-number–violating decays of baryons along these lines have been searched for by the CLAS Collaboration [33]. The branching fraction for Λ → K0S + inv. is limited to less than 2 × 10−5 while that for Λ → p̄π+ must be less than 9 × 10−7 which are sensitive to the operator (uds)2. While interesting, these limit δ(uds)2 = |M12| to less than about 10−18 GeV, which is less strong than the limit on this operator from null searches for dinucleon decay. In light of the less stringent limits from dinucleon decay on operators involving heavy flavor, it would be highly desirable for searches for ∆B = 2 decays of baryons with heavy quarks to be performed. B. Meson oscillations In addition to the decays described above, the new interactions could lead to flavor-changing oscillations of neutral mesons. The limits from these processes on this model were considered in Ref. [12]; we refer the reader to [12] and references therein for further details. Avoiding these constraints requires a suppression of particular combinations of flavor-violating couplings. For example, considering Kaon oscillations, given mφ & 400 GeV, ys1 and ys2 could be O(1) provided yd1,yd2 . 10−2. Similar considerations apply for flavor-violating combinations of the couplings gus and gud. The con- straints on charm and bottom couplings from D and B oscillations are less severe. From the model building point of view, flavor-changing meson oscillations can be naturally avoided, e.g. charging the scalar under a symmetry so that F1−F2 is conserved, where F1,2 label flavor quantum numbers. VII. COSMOLOGICAL PRODUCTION OF THE BARYON ASYMMETRY We now answer in detail the question of how the baryon asymmetry of the Universe is produced in this model. In addition to the CP and baryon number violation de- scribed above, a nonzero asymmetry requires a departure from thermal equilibrium. The simplest possibility for this is to assume that χ3 is very weakly coupled. It is therefore long-lived and decays out of equilibrium, pro- ducing the baryons that undergo CP- and B-violating oscillations. At temperatures below mχ3, the equations that deter- mine the radiation and χ3 energy densities are dρrad dt + 4Hρrad = Γχ3ρχ3, (39) dρχ3 dt + 3Hρχ3 = −Γχ3ρχ3. (40) H is the Hubble parameter which is related to the total energy density, H = √ 8π 3 ρ M2Pl ' √ 8π 3 ρrad + ρχ3 M2Pl , (41) where MPl = 1.22 × 1019 GeV is the Planck mass. In the absence of χ3 decays, ρrad and ρχ3 simply redshift like radiation and matter energy densities, respectively. The right-hand sides of these equations describe how χ3 decays cause the energy density in matter to decrease while dumping energy into the plasma. In addition to depositing energy in the plasma, some of the χ3 decays produce baryons and antibaryons, B and B̄, that can oscillate and decay, violating CP and B. For this to occur, the temperature of the Universe needs to be below the QCD confinement temperature, TQCD ' 200 MeV. On the timescale of the expansion of the Universe, H−1, the (anti-)baryons produced this way rapidly oscillate and decay, producing a net B asym- metry. However, because of the presence of the plasma, with which they can interact, as well as their large anni- hilation cross section, B ↔ B̄ can decohere in this envi- ronment, suppressing the asymmetry that is generated. Properly accounting for this requires a density matrix treatment, which has been used in a cosmological con- text for neutrino oscillations and oscillating asymmetric dark matter [34, 35]. Following Ref. [35] (see [36] for a similar analysis in the context of baryogenesis-related os- cillations), we can write the Boltzmann equations that govern the evolution of the number density of the B-B̄ system, dn dt + 3Hn = −i ( Hn−nH† ) − Γ± 2 [O±, [O±,n]] −〈σv〉± ( 1 2 {n,O±n̄O±}−n2eq ) + 1 2 Γχ3ρχ3 mχ3 Brχ3→BO+, (42) where the last term describes B and B̄ production through χ3 decay. Brχ3→B is the branching ratio for χ3 to decay to B or B̄. In this equation n and n̄ are density matrices, n = ( nBB nBB̄ nB̄B nB̄B̄ ) , n̄ = ( nB̄B̄ nBB̄ nB̄B nBB ) , (43) and neq is the equilibrium density of baryons plus an- tibaryons. H is the Hamiltonian seen in Eq. (16). 〈σv〉± 11 and Γ± are thermally-averaged annihilation cross sec- tions and scattering rates on the plasma, respectively. O± is a matrix O± = ( 1 0 0 ±1 ) . (44) The subscript of 〈σv〉± and Γ±, i.e. whether they appear with O+ or O− in Eq. (42), is determined by the be- havior of the effective Lagrangian that gives rise to these interactions under charge conjugation of only the heavy baryons, B ↔ B̄, Leff ↔ ±Leff. Interactions that do not change sign are said to be flavor-blind while those that do are flavor-sensitive. For example, B and B̄ can scat- ter on light charged particles in the plasma through their magnetic moment, µ, which corresponds to a term in the effective Lagrangian of iµ 4 B̄ [γν,γρ]BFνρ. (45) Under B ↔ B̄ this term changes sign, so the rate for scattering via the magnetic moment appears with O− in the Botzmann equation. It is useful to work in terms of the quantities Σ ≡ nBB + nB̄B̄, ∆ ≡ nBB −nB̄B̄, Ξ ≡ nBB̄ −nB̄B, Π ≡ nBB̄ + nB̄B. (46) In this basis the Boltzmann equations are( d dt + 3H ) Σ = Γχ3ρχ3 mχ3 Brχ3→B − ΓBΣ − (Re Γ12) Π + i (Im Γ12) Ξ − 1 2 [( 〈σv〉+ + 〈σv〉− )( Σ2 − ∆2 − 4n2eq ) + ( 〈σv〉+ −〈σv〉− )( Π2 − Ξ2 )] ,( d dt + 3H ) ∆ = −ΓB∆ + 2i (Re M12) Ξ + 2 (Im M12) Π,( d dt + 3H ) Ξ = − ( ΓB + 2Γ− + 〈σv〉+Σ ) Ξ + 2i (Re M12) ∆ − i (Im Γ12) Σ,( d dt + 3H ) Π = − ( ΓB + 2Γ− + 〈σv〉+Σ ) Π − 2 (Im M12) ∆ − (Re Γ12) Σ. (47) Coherent oscillations from a flavor-symmetric state to an asymmetric state proceed through Σ → Ξ, Π → ∆. Flavor-sensitive scattering and flavor-blind annihilation suppress Ξ and Π and therefore lead to decoherence. When they decay, B and B̄ create states that carry baryon number. The flavor-asymmetric configuration contributes to the difference between the baryon and an- tibaryon number densities,( d dt + 3H ) (nB −nB̄) = ΓB∆. (48) The dominant interaction of B and B̄ with the plasma is scattering on charged particles (mostly electrons at T . 100 MeV) via the magnetic moment term in Eq. (45). The cross section for this at temperatures well below mB is dσsc dΩ = α2µ2 ( 1 + sin2 θ/2 sin2 θ/2 ) (49) which diverges at small scattering angle, θ → 0. This divergence is cut off at finite temperature by the inverse photon screening length, mγ. Using this, the total cross section can be estimated as σsc ∼ 4πα2µ2 log ( 4E2 m2γ ) (50) where E is the electron energy. Taking E ∼ T , mγ ∼ eT/3 [37] and µ ∼ 1/(2mB), this gives σsc ∼ πα2 m2B log ( 9 πα ) . (51) The (flavor-sensitive) scattering rate is therefore Γ− = Γsc ∼ σsc (ne− + ne+ ) ∼ πα 2 m2B log ( 9 πα ) × 3ζ(3) π2 T 3 ∼ 10−11 GeV ( 5 GeV mB )2 ( T 10 MeV )3 . (52) At temperatures above a few MeV, as is needed for BBN, this rate is larger than a typical heavy baryon width and therefore strongly affects the B-B̄ oscillations. When solving the Boltzmann equations, we take an annihilation cross section that is similar to that for pp̄ annihilation at low energies, 〈σv〉+ + 〈σv〉− = 400 mb. (53) We will find that only the total annihilation cross section and not whether it is flavor-blind or -sensitive is impor- tant, since Σ � ∆, Ξ, Π. Furthermore the annihilation rate is always much smaller than the scattering rate at temperatures we are interested in, so its effect on the final asymmetry is subdominant and can generally be ignored. A. Sudden Decay Approximation Having removed the heavy baryons from the problem due to the short timescales in their system, the evolution equations are Eqs. (39), (40), and (59). These involve only the radiation energy density, χ3 density, and the baryon asymmetry. They can be simply studied using a sudden decay approximation to gain a rough estimate of the baryon asymmetry. We outline this estimate below. 12 At some high temperature above mχ3, we assume that χ3 was in thermal equilibrium with the plasma, fixing its number density for T . mχ3 to roughly nχ3 ' 3 4 ζ(3) π2 T 3. (54) As the Universe cools the energy density in χ3 and radi- ation are equal. This occurs at the temperature Teq = 45ζ(3) 2π4g∗(T0) mχ3. (55) g∗ is the effective number of relativistic degrees of free- dom and here it is evaluated at T0 & mχ3. This corre- sponds to the time teq = √ 45 16π3g∗(Teq) MPl T 2eq = 1√ 5πg∗(Teq) π7g∗(T0) 2 135ζ(3)2 MPl m2χ3 . (56) After this the Universe is matter dominated and the en- ergy density in radiation and χ3 redshift as ρrad = 1 2 ρeq ( teq t )8/3 , ρχ3 = 1 2 ρeq ( teq t )2 . (57) We then assume that all of the χ3’s decay at the time tdec = 1/Γχ3. The ratio of the energy densities just before decay is ξ ≡ ρχ3 (t − dec) ρrad(t − dec) = (teqΓχ3 ) −2/3 = 15 [ g∗(Teq) g∗(T0) ]1/3 [ 50 g∗(T0) ] × ( mχ3 10 GeV )4/3 (10−22 GeV Γχ3 )2/3 . (58) We use t−dec here to indicate the time infinitesimally be- fore decay. The dominance of the scattering rate over other scales in the problem allows us to make some simplifications of the evolution equations that are useful here. In this limit we can ignore the Hubble rate as well as annihi- lation and the equations governing B and B̄ in (47) can be integrated. This results in the evolution equation for the difference between baryon and antibaryon densities, Eq. (48), becoming( d dt + 3H ) (nB −nB̄) = Γχ3ρχ3 mχ3 × 2Im (M ∗ 12Γ12) Brχ3→B ΓB (ΓB + 2Γ−) + 4 |M12|2 ' Γχ3ρχ3 mχ3 ΓB 2Γ− �, (59) which is valid for the cases we consider with |M12| � ΓB � Γ−. We have defined � ≡ 2Im (M ∗ 12Γ12) Γ2B Brχ3→B ' ABBrχ3→B, (60) with AB from Eq. (22). Using �, we can then relate the baryon asymmetry to the χ3 number density at decay, ηB = nB −nB̄ s(t+dec) = nχ3 (t − dec) s(t−dec) [ T(t−dec) T(t+dec) ]3 ΓB 2Γ− � = 3 4 T(t−dec) mχ3 ξ [ T(t−dec) T(t+dec) ]3 ΓB 2Γ− �. (61) Here, t+dec is the time just after decay. The ratio of the temperatures just before and after decay is determined by ρrad(t + dec) = (1 + ξ)ρrad(t − dec) so that T(t−dec) T(t+dec) = (1 + ξ)−1/4 ' ξ−1/4, (62) and ηB ' 3 4 ξ1/4T(t−dec) mχ3 ΓB 2Γ− �. (63) The temperature just before decay can be arrived at by evolving the radiation energy density, resulting in ηB ' 3 8 √ 3 π [ 5 2πg∗(Tdec) ]1/4 √ MPlΓχ3 mχ3 ΓB 2Γ− �. (64) Using the expression for the scattering rate in Eq. (52) evaluated at T(t+dec), ηB ' π3 3ζ(3) √ πg∗(Tdec) 10 ΓB� σscmχ3 Γχ3MPl ≈ 9 × 10−11 [ g∗(Tdec) 50 ]1/2 ( mB 5 GeV )2 ( ΓB 10−13 GeV ) × ( 8 GeV mχ3 )( 10−22 GeV Γχ3 )( � 10−5 ) . (65) Therefore we see that a baryon asymmetry of the required size is possible for a heavy baryon system with � ∼ 10−5, which requires |M12|/ΓB ∼ 10−2 with |M12|/ |Γ12| not small. B. Full Solution of the Boltzmann Equations To get a more precise estimate of the baryon asymme- try, we numerically solve the system in Eqs. (39), (40), and (47). As mentioned above, we need |M12|/ΓB to not be much smaller than around 10−2. Looking at Ta- ble I, one potential candidate is the Ω0cb where the dom- inant coupling involves the operator (dcb)2. In Fig. 5 13 ����� ��-�� ��-�� ��-�� ��-�� ��-� ��-� ��-� ��-� ��-� ���� ��� � (���) � (�) η�η� ���� (�ℬ+�ℬ _ )/� ��-� (�χ�/�) FIG. 5. ηB = (nB −nB̄)/s (solid, black) as a function of the temperature or time from a numerical solution of Eqs. (39), (40), and (47) for parameters relevant to the Ω0cb-Ω̄ 0 cb system: mB = 7 GeV, ΓB = 3 × 10−12 GeV, |M12| = 3 × 10−15 GeV, |Γ12/M12| = 0.3, arg(M∗12Γ12) = π/2, mχ3 = 7.5 GeV, Γχ3 = 3 × 10−23 GeV, and Brχ3→B = 0.35. We have taken the rate for heavy baryon scattering on the plasma from Eq. (52) and the annihilation cross section to be 400 mb. This can be compared against the value of ηB (solid, gray) from a solution of Eqs. (39), (40), and (59) as well as using the sudden decay approximation (dashed, gray) in Eq. (65). Also shown are the ratio of the number density of χ3 to the entropy density (multiplied by 10−4, solid, orange) and the ratio of the B plus B̄ number densities to the entropy density (solid, purple). The dashed purple line shows the equilibrium B and B̄ density (in units of the entropy density). The measured value of ηB = 8.8 × 10−11 is given by the solid red line. we show the value of ηB as a function of temperature in the case of the asymmetry being sourced by the Ω0cb-Ω̄ 0 cb system, taking mB = 7 GeV, ΓB = 3 × 10−12 GeV [38], |M12| = 3×10−15 GeV, |Γ12/M12| = 0.3, arg(M∗12Γ12) = π/2, mχ3 = 7.5 GeV, Γχ3 = 3 × 10−23 GeV, and Brχ3→B = 0.35. We have used an annihilation cross sec- tion of 400 mb (the results do not depend on whether it is flavor-blind or -sensitive) and the scattering rate given in Eq. (52). In addition, the temperature dependence of the scat- tering and annihilation rates is compared to the expan- sion rate of the Universe as well as to the rates governing the baryon-antibaryon system in Fig. 6. As mentioned before, the (decohering) scattering is the dominant pro- cess above temperatures of about 1 MeV and, in partic- ular, is always much larger than the annihilation rate. At high temperatures, the heavy baryon density tracks its equilibrium value and it begins to deviate from its equilibrium value when χ3’s begin to decay. Although not directly evident from the plots (except through the change in the temperature vs. time), the out-of- equilibrium χ3 particles actually come to dominate the energy density of the Universe prior to their decay. Af- ter the χ3 decays, which we assume happens in less than ����� ��-�� ��-�� ��-�� ��-�� ��-� ��-� ��-� ��-� ���� ��� � (���) � �� � (�) Γ����� 〈σ�〉(� ℬ+� ℬ _ ) � Γℬ ��� Γ�� FIG. 6. The temperature dependence of the rates involved in the numerical solution of Eqs. (39), (40), and (47). The parameters are the same as in Fig. 5. In orange, from top to bottom are the scattering, annihilation, and Hubble rates. The purple lines indicate the rates relevant to the B-B̄ system itself, ΓB, |M12|, and |Γ12|, from top to bottom, respectively. ∼ 0.1s, the Universe undergoes a transition from being matter-dominated to radiation-dominated, reheating to a temperature above a few MeV. We have numerically confirmed the rough accuracy of the sudden decay approximation prediction for ηB over much of the parameter space. Maximal CP violation, and thus more baryon asymmetry per oscillation, oc- curs for arg (M∗12Γ12) = π/2 and larger values of |M12| and |Γ12|. A larger branching ratio, Brχ3→B, would pro- duce more oscillating baryons per Majorana decay. The value of ηB that is generated is maximized if χ3 decays when the Universe’s temperature is about 10 MeV, i.e. τχ3 = 1/Γχ3 ∼ 10−2 s. If it decays earlier than this, heavy baryon scattering on the plasma leads to decoher- ence, suppressing the asymmetry. If if decays later, the Universe does not have a sufficient baryon asymmetry at the time the neutrinos begin to decouple, when the Universe is around 3 MeV. Given the constraints on the transition amplitudes in Table I, the most promising baryon that could allow for a large enough transition amplitude to source the BAU is the as yet unobserved Ω0cb. A relatively large value for |M12| is needed in this case, not far from the collider limit, unless Brχ3→Ω0cb were rather large. It should be noted that the collider limits discussed in Sec. V which appear in Table I depend on the specific model that we considered. It is conceivable that the model could be ex- tended in a way that makes the standard collider searches that we considered less constraining. For example, one could imagine making the φ decay to a large number of relatively soft jets by coupling to a heavy vector-like quark and a singlet which decay to a large number of col- ored objects. Relaxing these limits could allow for other heavy flavor baryons to source the BAU, potentially even observed baryons like the Ω0c, Λ 0 b, and Ξ 0 b. On the other 14 hand, since they involve low-energy effective operators, the dinucleon decay constraints are less model depen- dent. Weakening them would require significant tuning of tree-level operators against those induced by weak in- teractions. VIII. SUMMARY AND OUTLOOK We have presented a model for producing the observed baryon asymmetry of the Universe which avoids high re- heat temperatures. The asymmetry is generated through CP and B-violating oscillations of baryons occurring late in the hadronization era. Our model minimally intro- duces three neutral Majorana fermions and a single col- ored scalar, and could potentially be embedded into RPV SUSY. The Ωcb ∼ (scb) baryon emerges as our most promising candidate when constraints due to collider data and din- ucleon decay are taken into account. Note that the con- straints from colliders are more model-dependent than those from the absence of dinucleon decay. Consider- ing only the constraints from dinucleon decay, additional baryons, e.g., Ω0c ∼ (ssc), Λ0b ∼ (udb), and Ξ0b ∼ (usb), become viable candidates for baryogenesis via their os- cillation. An interesting avenue for future work would be constructing models that are less constrained by collider experiments while preserving a large baryon oscillation rate. Interesting signatures of this scenario could be present in the large dataset of the upcoming Belle II experi- ment. If the lightest Majorana fermion is sufficiently light, one possible signature would be decays of heavy flavor hadrons that violate baryon number and involve missing energy. Additionally there could be heavy fla- vor baryons that oscillate into their antiparticles at po- tentially measurable rates. Exploring the experimental prospects of this model at high luminosity, lower energy colliders in more detail will be left for future work. Constraints from the LHC and the lack of dinucleon decay observation are quite important, suggesting the possibility of the detection of a signal in one or both areas. Dinucleon decays are a more model-independent consequence of this scenario, and because of the require- ment of baryon number violation involving heavy flavors, it is likely to assume that dinucleon decay to kaons would be dominant. In the case of the LHC, a particular com- bination of signals in dijet resonances (singly and pair produced) along with an excess in jets plus missing en- ergy should be expected. We should mention in this case that a long-lived neutral particle, χ1, that decays hadron- ically is a generic prediction of this model. The typical χ1 decay length is in the range of 102−7 m, which could be well probed by the MATHUSLA detector that was re- cently proposed. The signal of a long-lived but unstable particle at this experiment could help disentangle this scenario from others that lead to excesses in jets plus missing energy. ACKNOWLEDGMENTS We would like to thank Brian Batell, Kristian Hahn, and Ahmed Ismail for useful conversations. The work of DM is supported by PITT PACC through the Samuel P. Langley Fellowship. The work of TN is supported by Spanish grants FPA2014-58183-P and SEV-2014-0398 (MINECO), and PROMETEOII/2014/084 (Generalitat Valenciana). 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