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Cosmological observations, such as the velocity dispersions of galaxies [1], Big Bang nucleosynthesis (BBN) [2, 3] and cosmic microwave background (CMB) [4], provide strong evidence for the existence of dark matter (DM) and baryon asymmetry in the universe. Cosmological measurements [5] indicate that the relic densities of dark matter (
ΩDM ) and baryons (ΩB ) are of the same order, i.e.,ΩDM=(5.36±0.06) ΩB , which implies that the dark matter and baryon asymmetry may stem from the same ultraviolet (UV) model [6].Recently, a model of B-Mesogenesis was proposed to simultaneously explain the origins of DM and baryon asymmetry by assuming that dark matter is charged under baryon number [7]. The Sakharov conditions are satisfied in the following ways. A long-lived particle produces bottom mesons and their anti-mesons out of the thermal equilibrium in the early universe. Then the neutral bottom mesons undergo
B0−¯B0 andB0s−¯B0s oscillations, which naturally provide CP violation. Finally, the bottom mesons decay into an ordinary baryon with the baryon numberB=+1 and a dark fermion ψ that is charged under baryon number withB=−1 ; thus, the visible baryon number is violated but the total baryon number is conserved. To preserve the DM and baryon asymmetry, the dark fermion ψ should decay into a dark Majorana fermion ξ and a dark scalar baryon ϕ. In this way, both the DM relic abundance and baryon asymmetry in the universe can be simultaneously explained by this model. Additionally, most of the new particles in this scenario exist at the GeV scale, which can be precisely examined at the current B-factories and hadron colliders or by experiments in the near future.This model has garnered significant interest from both theorists and experimentalists during the past a few years [8–22]. To explore the B-Mesogenesis scenario, most theoretical studies concentrate on the semi-invisible decays of B mesons. The BABAR and Belle experiments have measured the decay of bottom mesons into a baryon and a dark sector anti-baryon ψ, such as
B0→Λ0ψ andB+→pψ [19–21]. However, it is also possible for the bottom baryons to decay into a meson and a dark baryon, such asΛb(Ξb)→Pˉψ , with P denoting a light pseudo-scalarπ,K meson, which was mentioned in the B-Mesogenesis proposal [7] and precisely estimated in a recent study [18]. However, for bottom baryons, there are two processes that have not ever been studied in the context of B-Mesogenesis, which are the fully invisible decay (Λ0b→ξˉϕ ) and semi-invisible radiative decay (Λ0b→γˉψ ) ofΛ0b baryons. In particular, the fully invisible decay ofΛ0b yields a distinctive signal with which the stable dark particles in B-Mesogenesis can be explored. All these processes can be explored at future colliders with high accuracy. For instance, the upgrade of Belle-II [23] is expected to produce the bottom baryons pair at the threshold, while the LHCb can measure the signals with its vertices detector [24]. In addition, as the Circular Electron Positron Collider (CEPC) [25] and Future Circular Collider (FCC-ee) [26] will produce substantial bottom baryons, these rare decays of bottom baryons are expected to be explored at future lepton colliders. Therefore, to precisely probe B-Mesogenesis, systematic analyses of the invisible and semi-invisible decays of bottom baryons are crucial and may shed light on the study of dark matter and baryon asymmetry in the universe.Except for
Λ0b→ξˉϕ , the fully invisible decays are also very interesting for neutron and hyperon decays, which share the same type of interaction asuadbdcψ , wherea,b,c represent the flavors of up- and down-type quarks. To explain the neutron lifetime puzzle (i.e., the discrepancy between the bottle method and the beam method), the idea of allowing neutrons to decay into dark matter was proposed [27]. The invisible decays ofΛ0 hyperons were also allowed for in the new physics models in a fashion similar to the B-Mesogenesis model and the ones for the neutron lifetime anomaly [12]. The BESIII collaboration has measured it asBR(Λ0→invisible)<7.4×10−5 [28]. The invisible decays of bottom baryons rely on the interaction with a bottom quark, which is beneficial for exploring B-Mesogenesis.When searching for new physics in the decays of hadrons, hadronic matrix elements are crucial for determining the observables, except for the interaction of UV models. Therefore, it is necessary to calculate the corresponding matrix elements to examine the new physics or constrain the parameters of a specific model of hadron decays. Because the matrix elements relate to the strong interaction, we need to utilize a perturbative or non-perturbative method to determine the hadronic matrix elements. In practice, the decays of
Λb(Ξb)→Pˉψ involve bottom baryon to light-meson transition form factors, which have been investigated recently in the light-cone sum rules (LCSR) method [18]. Because the baryon-to-meson form factors are a new kind of physical quantity, they require more detailed discussions in theoretical studies. Under the heavy quark limit, these matrix elements can also be estimated within the QCD factorization approach [29–32]. For example, this kind of form factor also emerges in the baryon and lepton number violation decays ofΛb→Pℓ from a Leptoquark model, and have been studied recently within the QCD factorization [33]. Although the higher dimensional operators of the B-Mesogenesis model are different to those of the Leptoquark model, the bottom baryon to light meson transition form factors can be similarly determined within the QCD factorization. Our results suggest that the transition form factors can be factorized into a convolution of the perturbatively calculable hard-scattering kernel and the non-perturbative distribution amplitudes without endpoint divergence [33, 34]. The matrix elements involved inΛ0b→γˉψ can also be calculated within the same QCD factorization method, while those ofΛ0b→ξˉϕ can be estimated by directly relying on the matrix elements in the SM. With these matrix elements, we can calculate the branching ratios of invisible and semi-invisible decays of bottom baryons and predict the potential of exploring B-Mesogenesis using future measurements.This manuscript is organized as follows. In Sec. II, we briefly introduce the B-Mesogenesis scenario. In Sec. III, the hadronic matrix elements in the invisible and semi-invisible decays of
Λb andΞb baryons are calculated within the QCD factorization approach. In Sec. IV, the sensitivities of the parameters of B-Mesogenesis are discussed for the invisible and semi-invisible decays of bottom baryons at future colliders. Finally, we conclude the study in Sec. V. -
As the relic densities of DM and net baryons are similar, they may arise from the same UV completion. In practice, B-Mesogenesis [7, 10–12], which assumes dark matter is charged under the baryon number, attempts to simultaneously explain the existence of DM and baryon asymmetry in the Universe. In particular, this scenario induces the decays of bottom baryons into dark baryons, which significantly violate the baryon number conservation in the visible regime. The corresponding Lagrangian is given by:
Lint⊃−yRuadbϵijkY∗iˉujRadk,CRb−yLuadbϵijkY∗iˉujLadk,CLb−yψdcYiˉψdi,CRc−ydˉψϕξ+h.c..
(1) The u (d) represents the up-type (down-type) quarks, the indices
a,b,c (i,j,k ) denote the quark flavors (colors), L(R) represents the left-handed (right-handed) component of quarks,C represents a charge conjugation, Y is a color-triplet scalar with electric chargeQY=−1/3 (which couples to ordinary quarks in the first two terms and to a right-handed quark and a dark fermion in the third term; it is a TeV-scale particle which will be integrated out in the bottom hadron decays), and ψ is a dark fermion with the baryon numberBψ=−1 , which is a singlet under the standard model gauge group that can only couple to the right-handed down-type quarks under that color conservation. The total baryon number is conserved in the interaction ofYiˉψdi,CRc , while the visible baryon number is violated. This is illustrated in Fig. 1, in which ψ completely decays into stable dark matter, a dark scalar ϕ withBϕ=−1 , and a dark Majorana fermion ξ. Their quantum numbers are summarized in Table 1.Figure 1. (color online) The diagrammatic representations for the invisible and semi-invisible decays of bottom baryons in the B-Mesogenesis. (a) and (b) are the diagrams for the semi-invisible hadronic decay of
Λ0b→π0ˉψ , (c) forΛ0b→K0ˉψ , (d) forΞ−b→π−ˉψ , (e) forΞ−b→K−ˉψ , (f,g,h) for the semi-invisible radiative decay ofΛ0b→γˉψ , and (i) for the fully invisible decay ofΛ0b→ξˉϕ . The dark sector particles ofˉψ andˉϕ are charged under the baryon number ofBˉψ,ˉϕ=+1 , so that the total baryon number are conserved in these processes.Field S Q B Y 0 −13 −23 ψ 12 0 −1 ξ 12 0 0 ϕ 0 0 −1 Table 1. Quantum numbers for the new particles in B-Mesogenesis, including particle spin S, electric charge Q, and baryon number B.
If the mass of color-triplet scalar
MY is much heavier than that of bottom baryons, we can integrate out such particles and obtain an effective Lagrangian related to the (semi-)invisible decay, which is given byLEFT=CL,Ruadb,dcOL,Ruadb,dc+CL,R∗uadb,dcˉOL,Ruadb,dc,
(2) where
CL,Ruadb,dc=yL,R∗uadby∗ψdc/M2Y are the Wilson coefficients and the effective operatorsOL,Ruadb,dc andˉOL,Ruadb,dc are given byOLuadb,dc=ϵijk(ˉui,CLadjLb)(ˉψCdkRc),ˉOLuadb,dc=ϵijk(ˉuiLadj,CLb)(ˉψdk,CRc),ORuadb,dc=ϵijk(ˉui,CRadjRb)(ˉψCdkRc),ˉORuadb,dc=ϵijk(ˉuiRadj,CRb)(ˉψdk,CRc).
(3) For convenience, we factorize out the external field ψ from the effective operators and rewrite the Lagrangian as
LEFT=CL,Ruadb,dcˉψCO′L,Ruadb,dc+h.c.,
(4) where the operators
O′L,Ruadb,dc are expressed asO′Luadb,dc=ϵijk dkRc(ˉui,CLadjLb),O′Ruadb,dc=ϵijk dkRc(ˉui,CRadjRb).
(5) -
With the effective Lagrangian for B-Mesogenesis shown in Eq. (4), we can systematically study the invisible and semi-invisible decays of bottom baryons. To explore B-Mesogenesis, we primarily focus on the exclusive decay modes of
Λ0b→P0ˉψ,Ξ−b→P−ˉψ,Λ0b→γˉψ,Λ0b→ξˉϕ.
(6) The diagrammatic representation of these processes are shown in Fig. 1. In the heavy quark limit, we calculate the relevant diagrams and apply the QCD factorization method for these heavy baryon to meson transition form factors at the leading order of
αs . The non-perturbative inputs are the heavy baryon and light meson distribution amplitudes, and we restrict our calculation to the leading twist (twist-2) distribution amplitudes. Finally, the transition matrix elements defined by the effective operatorsO′L,Ruadb,dc can be represented as a convolution of the perturbative part with the leading twist heavy baryon and light meson distribution amplitudes. -
In this subsection, we construct the factorization formula and explicitly calculate the hadronic matrix elements of
Λ0b/Ξ−b→P0,−ˉψ decays in the heavy quark limit. The relevant diagrams are shown in Fig. 2. Using the effective Lagrangian given in Eq. (4), one can express the decay amplitude for the operatorO′L,Ruadb,dc asFigure 2. (color online) The Feynman diagrams of the
Λ0b/Ξ−b→Pˉψ decays, where the bold line represents the b quark and the circle cross denotes the currents in theOL,R operators.iM=CL,Ruadb,dc(ˉvCψ(q)⟨P(p)|O′L,Ruadb,dc|B(p′)⟩),
(7) where
vψ is the spinor of dark baryonˉψ with momentumq=p′−p , andB representsΛ0b orΞ−b with momentump′=mBv . The hadronic matrix elements can be parameterized in terms of two form factors:⟨P(p)|O′L,Ruadb,dc|B(p′)⟩=PR(ζB→P(L,R)1(q2)+⧸qmBζB→P(L,R)2(q2))uB(p′).
(8) We will calculate the form factors working in the rest frame of the heavy baryon and choose two light-cone vectors along the fast-moving final state particles with the conventions of
nμ=(1,0,0,1),ˉnμ=(1,0,0,−1),vμ=(1,0,0,0).
(9) Then the four-momentum of the final state particles can be written as
q=(Eq+|→q|,Eq−|→q|,0),p=(Ep−|→p|,Ep+|→p|,0),Eq=m2B+m2ψ−m2P2mB,Ep=m2B−m2ψ+m2P2mB,|→q|=|→p|=√(m2B−(mψ+mP)2)(m2B−(mψ−mP)2)2mB,
(10) with
pμ=(n⋅p)ˉnμ2+(ˉn⋅p)nμ2+pμ⊥≡(n⋅p,ˉn⋅p,p⊥).
(11) When computing the Feynman diagrams in the heavy quark limit, we list the momenta involved as
p′=(mB,mB,0),q=(mB,mBλ2,0),p=(mBλ2,mB,0),k=(ΛQCD,ΛQCD,ΛQCD),
(12) where k corresponds to the momentum of the light spectator quark in the baryon. The parameter
λ∼mψmB∼mPmB∼ΛQCDmB , which vanishes in the heavy quark limit.When considering only the leading twist distribution amplitudes and the tree-level QCD calculations, the contribution of operator
O′Ruadb,db in Eq. (4) to the transition matrix element always vanishes because there is an odd number of Dirac matrices between the two chiral projection matricesPR=1+γ52 within the operatorO′Ruadb,db . For example, in the heavy quark limit, the Dirac structure of theΛ0b→π0ˉψ transition amplitude shown in Fig. 2 (a) is given byiM∝vTψCPR(⧸pγ5)γμ(⧸ˉnγ5C)(γμ)Tx⧸pTPL,RuΛb,
(13) where
(⧸pγ5) and(⧸ˉnγ5C) are the light-cone projectors corresponding to the leading twist distribution amplitudes ofΛ0b andπ0 , respectively [31, 35]. The subscripts ofPL,R correspond to the insertion of operatorsOLuadbdc orORuadbdc . Because there is an odd number of Dirac matrices betweenPR andPL,R , the contribution of right-handed operatorsORuadbdc will vanish. The contributions of the right-handed operators in other semi-invisible decays of bottom baryons will vanish in the same way.In heavy quark limit, the transition amplitude in Eq. (7) also vanishes for operators
OLud,b andOLus,b owing to the parity conservation. TakingΛ0b→π0ˉψ as an example, the partonic amplitude1 can be factorized as a product of two transition matrix elements in the spin space,iM∝⟨ˉψ(q)|ˉψCbR|b(mbv)⟩⟨ˉq(xp)q(ˉxp)|ˉuCLdL|u(k1)d(k2)⟩. We will find that there is a trace in the second matrix element in our calculation, which is always zero because there is an odd number of γ matrices. We find thatγ5 in the operatorˉuCLdL cannot contribute to the trace because there are only two independent Lorentz vectors in this process to contract with the 4-dimensional antisymmetric tensorϵμνρσ . Actually, theJP quantum number of quark pair(ud) in theΛ0b baryon is0+ in the leading twist approximation, and⟨q(xp)ˉq(ˉxp)|ˉuCLdL|u(k1)d(k2)⟩ describes a scalar decay into a pseudoscalar induced by a scalar operator, which violates parity conservation. The hadronic effects will not affect this conclusion cause the perturbative results in the final factorization formula are independent of the external state [37].In the following calculation, we only consider the contribution of the operators
O′Lub,d andO′Lub,s . Strange number conversed or violated processes are induced by the corresponding operators, which is summarized asΛ0b→π0ˉψandΞ−b→K−ˉψ,inducedbyO′Lub,d,Λ0b→K0ˉψandΞ−b→π−ˉψ,inducedbyO′Lub,s.
(14) In the framework of the QCD factorization approach, the longitudinal momentum of the quarks in the energetic final state meson is at the scale of
mb and the exchanged gluons have virtuality of ordermbΛQCD , which can be treated perturbatively. Next, we calculate the hard-scattering contributions, such as the ones shown in Fig. 2. The decay amplitudes are expressed asiM=ηΔsCLuadb,dcζB→P(ˉvCψ(q)PR⧸n2uB(p′)),
(15) where the sign factor
ηΔs=−1 for strange flavor conserved processesΛ0b→π0 andΞ−b→K− , whileηΔs=+1 for strange flavor violated processesΛ0b→K0 andΞ−b→π− . The convention for the sign factor arises from the inconsistency between the quark flavor order in the operatorO and the definition of the light-cone distribution amplitudes.The factorization formula of the form factor
ζB→P is given byζB→P=fPf(2)B∫∞0dωω∫10du∫10dxJB→P×(x,u,ω,μ)ϕP(x)ψ2(u,ω),
(16) where
fP(B) is the decay constant of particleP(B) ,JB→P(x,u,ω,μ) is the corresponding hard-scattering kernel, andϕP(x) andψ2(u,ω) are the leading twist light-cone distribution amplitudes of the P meson andB baryon [30, 38–40], respectively,⟨P(p)|[ˉq(tˉn)]A[tˉn,0][q(0)]B|0⟩=ifP4ˉn⋅p[⧸n2γ5]BA∫10dxeixtˉn⋅pϕP(x,μ),
(17) with the momentum fraction x of anti-quark
ˉq in the P meson, and⟨0|[ui(t1ˉn)]A[0,t1ˉn][dj(t2ˉn)]B[0,t2ˉn][bk(0)]C|Λ0b(p′)⟩=16ϵijk4f(2)Λb(μ)[uΛb(p′)]C[⧸ˉn2γ5CT]BA∫∞0dωω×∫10due−iω(t1u+t2ˉu)ψ2(u,ω),⟨0|[di(t1ˉn)]A[0,t1ˉn][sj(t2ˉn)]B[0,t2ˉn][bk(0)]C|Ξ−b(p′)⟩=16ϵijk4f(2)Ξb(μ)[uΞ−b(p′)]C[⧸ˉn2γ5CT]BA∫∞0dωω×∫10due−iω(t1u+t2ˉu)ψ2(u,ω),
(18) where
ωi=(n⋅ki) represent the momentum components of the spectator quark. We use the conventionω1=uω ,ω2=ˉuω withω=ω1+ω2 , where u andˉu represent the momentum fraction of the spectator quark in the light quark pair. In the heavy-quark limit, the coupling constantsfB forΛ0b andΞ−b are respectively defined by [39]ϵijk⟨0|uiTα(0)djβ(0)bkγ(0)|Λ0b(p′)⟩≡14{f(1)Λb[γ5CT]βα+f(2)Λb[⧸vγ5CT]βα}[uΛb(p′)]γ,ϵijk⟨0|diTα(0)sjβ(0)bkγ(0)|Ξ−b(p′)⟩≡14{f(1)Ξb[γ5CT]βα+f(2)Ξb[⧸vγ5CT]βα}[uΞb(p′)]γ.
(19) The gauge links in Eq. (18) are used to preserve the gauge invariance,
[tˉn,0]=Pexp[ig∫t0dxˉn⋅A(xˉn)],
(20) which does not affect our calculations at the leading order of QCD. The leading-twist LCDAs of pseudoscalar mesons can be expanded using Gegenbauer polynomials [41],
ϕP(x,μ)=6x(1−x)[1+∞∑n=1aPn(μ)C(3/2)n(2x−1)],
(21) and the leading-twist wave function for the
Λ0b baryon is given by [38, 39, 42],ψ2(u,ω)=u(1−u)ω21ω40e−ω/ω0.
(22) In the limit of the SU(3) flavor symmetry, the wave function of
Ξ−b is the same as that ofΛ0b . Table 2 summarizes the numerical input parameters.Table 2. Input parameters with renormalization scale
μ0=1GeV .The corresponding hard-scattering functions are calculated as
JΛ0b→π0(x,u,ω,μ)=1√2(Ju(x,u,ω,μ)+Jˉu(x,u,ω,μ)),JΛ0b→K0(x,u,ω,μ)=JΞ−b→K−(x,u,ω,μ)=Jˉu(x,u,ω,μ),JΞ−b→π−(x,u,ω,μ)=Ju(x,u,ω,μ),
(23) with
Ju(x,u,ω,μ)=19παs(μ)uω2x,Jˉu(x,u,ω,μ)=19παs(μ)ˉuω2x,
(24) where the
1/√2 is the isospin factor from|π0⟩=1√2(|uˉu⟩−|dˉd⟩) . The asymptotic behaviors of the leading twist light-cone distribution amplitudes in the endpoint region areϕP(x)∼x(1−x),ψ2(u,ω)∼u(1−u)ω2,
(25) which will cancel the divergent behavior in the hard-scattering functions
Ju(x,u,ω,μ) andJˉu(x,u,ω,μ) . Then the convolution integral in Eq. (16) converges, as noted already in [33, 34]. Next, we assign the QCD coupling constants a value ofαs(2GeV)≈0.3 and use the other inputs displayed in Table 2. The numerical results for the form factors areζΛ0b→π0=(1.43+0.46−0.54)×10−2[GeV2],ζΛ0b→K0=(1.23+0.40−0.47)×10−2[GeV2],ζΞ−b→K−=(1.23+0.40−0.47)×10−2[GeV2],ζΞ−b→π−=(1.01+0.33−0.38)×10−2[GeV2].
(26) The mass dimension of the form factors is
[ζ]=2 , as the dimension of the baryon coupling constant is[fB]=3 .The form factors in Eq. (26) are related to the form factors
ζ1 andζ2 resulting from the match between Eq. (15) and Eqs. (7) and (8):ηΔsζB→P=ζ1=−ζ2,
(27) We use the heavy quark limit during the calculation of the heavy baryon to light meson transition form factors, and the reliability of our results is best when the momentum transfer
q2=0 . In most of the calculations of heavy to light decays, (e.g.,B→πℓˉνℓ [36] orΛb→Λℓˉℓ [42]), the relevant form factors can be determined only at one or two values ofq2 using an assumedq2 -dependence to obtain the numerical results at an arbitraryq2 . The parameterization and extrapolation of the transition form factors have been extensively studied; thus, we introduce the simplest single-pole model to extrapolate the form factorsζ1,2(q2=0) towardζ1,2(q2=m2ψ) [18, 44–46]:ζ1,2(q2)=11−q2/m2poleζ1,2(q2=0).
(28) Here,
ζ1,2(q2=0) are obtained through the values in Eq. (26) and the relationship betweenζB→P andζ1,2 in Eq. (27). The denominator is introduced to describe the threshold behaviour below the onset of the continuum. The form factor has a pole at the mass of the lowest state that relates to the operators in Eq. (14). For instance,mpole is the mass ofΛ0b(12+) for theΛ0b→π0/Ξ−b→K− transition, asΛ0b(12+) is the lowest state of operatorO′Lub,d , whilempole is the mass ofΞ0b(12+) for theΛ0b→K0/Ξ−b→π− transition.Starting from Eqs. (7) and (8), the branching fractions of
B→Pˉψ are expressed asBR(B→Pˉψ)=|→qψ|16πΓB|CLub,d|2[ζ21(q2)+ζ22(q2)m2ψm2B+(ζ21(q2)+ζ22(q2)m2ψm2B)m2ψ−m2Pm2B+4ζ1(q2)ζ2(q2)m2ψm2B],
(29) where
(uadb,dc) represents(ub,d) or(ub,s) for effective operatorsOLub,d orOLub,s ,|→qψ| is the momentum of dark baryonˉψ, |→qψ|=√(m2B−(mψ+mP)2)(m2B−(mψ−mP)2)2mB,
(30) and
ΓB (mB ) is the decay width (mass) of baryonB . -
For the radiative decay processes
Λ0b→γˉψ showed in Fig. 3, the decay amplitudes can be expressed as,Figure 3. (color online) The Feynman diagram for
Λ0b→γˉψ with the bold line representing the b quark.iM=CL,Ruadb,dc(ˉvCψ(q)⟨γ(p)|O′L,Ruadb,dc|Λ0b(p′)⟩).
(31) The transition amplitudes associated with the right-handed operators
O′Rub,d andO′Rud,b always vanish for the same reason as in the hadronic semi-invisible decays, which we have mentioned previously. The hadronic matrix elements in Eq. (31) can be parameterized in terms of two form factors:⟨γ(p)|O′Luadb,dc|Λ0b(p′)⟩=PR(ζΛb→γ1(q2)⧸p′m2Λbiσμν+ζΛb→γ2(q2)⧸qm2Λbiσμν)uΛb(p′)ϵ∗μp′ν.
(32) In the heavy quark limit, the transition amplitude for operator
O′Lud,b vanishes in our calculation. We can also arrive at this conclusion by considering the amplitude at the partonic level, which can be factorized as a product of two transition matrix elements in the spin space,iM∝⟨ˉψ(q)|ˉψCbR|b(mbv)⟩⟨γ(p)|ˉuCLdL|u(k1)d(k2)⟩ . The second matrix element describes the transition of a quark pair into a photon, with an initial momentum k, transfer momentum r, and final state momentum p. Its amplitude is proportional to the polarization vectorϵμ of the photon. Considering momentum conservation (k+r=p ) and the on-shell condition (p2=0 ), there are two independent Lorentz-invariant amplitudes proportional top⋅ϵ andr⋅ϵ , respectively. In the heavy quark limit, we neglect the momentumk∼ ΛQCD of the spectator quarks. Therefore, the second transition matrix element must be proportional top⋅ϵ=r⋅ϵ , which is zero after implementing the Ward Identity:p⋅ϵ=0 .The non-vanishing amplitude
Λ0b→γˉψ defined byO′Lub,d is given byiM=CLub,dˉvCψ(q)PR(ζΛ0b→γu⧸ˉn2⧸ϵ⧸n2+ζΛ0b→γd⧸n2⧸ϵ⧸ˉn2)uΛb(p′),
(33) The form factors can be factorized as
ζΛ0b→γu,d=f(2)Λb∫∞0dωω∫10duJΛ0b→γu,d(u,ω,μ)ψ2(u,ω),
(34) where the index u (d) denotes the contributions of photons emitted from a u (d) quark. The photon radiation from a b quark is suppressed by
mb in a heavy quark propagator. Furthermore,JΛ0b→γu,d(u,ω,μ) is the hard-scattering kernel for theΛb→γ transition,JΛ0b→γu(u,ω,μ)=−14(Qu(μ)uω),JΛ0b→γd(u,ω,μ)=−14(Qd(μ)ˉuω),
(35) where the electric charges of the light quarks u and d are
Qu(μ)=23e(μ) andQd(μ)=−13e(μ) , respectively. Combining Eqs. (31)−(33), we haveζΛ0b→γ1=−ζΛ0b→γd,ζΛ0b→γ2=−ζΛ0b→γu+ζΛ0b→γd.
(36) The numerical results for the form factors of the spectator processes
Λ0b→γˉψ areζΛ0b→γu=−(5.49+1.18−1.30)×10−3[GeV2],ζΛ0b→γd=(2.74+0.59−0.65)×10−3[GeV2],
(37) the input of which is shown in Table 2, with
αem(2GeV)≈1/133 [47]. The single-pole model is also used to describe theq2 -dependence of form factorζ1,2(q2) ,ζ1,2(q2)=11−q2/m2Λbζ1,2(q2=0),
(38) where
mΛb is the mass ofΛ0b(12+) . Then, the branching ratio forΛ0b→γˉψ with operatorO′Lub,d is estimated asBR(Λ0b→γˉψ)=|→qψ|8πm2ΛbΓΛb12∑spin|M|2=|→qψ|8πΓΛb|CLub,d|2[ζ21(q2)+(ζ21(q2)+ζ22(q2))×m2ψm2Λb+2ζ1(q2)ζ2(q2)m2ψm2Λb],
(39) with
q2=m2ψ . -
Under the B-Mesogenesis scenario, the
Λ0b→ξˉϕ channel is governed by both effective operatorsO′L,Rud,b andO′L,Rub,d . The Feynman diagram is shown in Fig. 4. The decay amplitudes can be respectively expressed asFigure 4. (color online) The Feynman diagram for
Λ0b→ξˉϕ process, with the bold line denoting the b quark.iML,Rud,b=ydCL,Rud,b(ˉuCξ(q)⧸p′+mψp′2−m2ψ⟨0|O′L,Rud,b|Λ0b(p′)⟩),iML,Rub,d=ydCL,Rub,d(ˉuCξ(q)⧸p′+mψp′2−m2ψ⟨0|O′L,Rub,d|Λ0b(p′)⟩),
(40) where
uξ(q) is the spinor for ξ with four-momentum q. To estimate the branching ratios of the invisible baryon decay, we first calculate the corresponding hadronic matrix elements in Eq. (40), which can be parameterized in terms of the coupling constants defined by⟨0|O′L,Rud,b|Λ0b(p′)⟩=λL,Rud,b PRuΛb(p′),⟨0|O′L,Rub,d|Λ0b(p′)⟩=λL,Rub,d PRuΛb(p′).
(41) We found that the couplings
λL,Rud,b andλL,Rub,d could be related to the couplingf(1,2)Λb defined in Eq. (19) by multiplying the corresponding matrix element. For instance,⟨0|O′L,Rud,b|Λ0b(p′)⟩=ϵijk⟨0|[uiTCPL,Rdj]bkR|Λ0b(p′)⟩=∑ρ(CPL,R)αβ(PR)ργ{ϵijk⟨0|uiTαdjβbkγ|Λ0b(p′)⟩}=14{f(1)ΛbTr[CPL,Rγ5CT]+f(2)ΛbTr[CPL,R⧸vγ5CT]}PRuΛb(p′).
(42) The last line is the result of using Eq. (19). After the trace, we can obtain the relations,
λLud,b=−12f(1)Λb,λRud,b=+12f(1)Λb.
(43) The same operation can be used for
λL,Rub,d , which results inλLub,d=−14f(2)Λb,λRub,d=+14f(1)Λb.
(44) For the numerical value of the couplings, we quote the result of the NLO QCD sum rules analysis in Ref. [48]:
f(2)Λb≈f(1)Λb=0.030±0.005[GeV3].
(45) Then, we can estimate the branching ratios of the invisible baryon decay. Assuming the new physics individually exists in either the left-handed or right-handed component, the branching ratios of
Λb→ξˉϕ are given byBR(Λb→ξˉϕ)L,Ruadb,dc=|→qξ|8πmΛbΓΛb(|yd|2|CL,Ruadb,dc|2|λL,Ruadb,dc|2)×Eξ(m2Λb+m2ψ)+2mΛbmψmξ(m2Λb−m2ψ)2,
(46) where
(uadb,dc) represents(ud,b) or(ub,d) for different effective operatorsOL,Rud,b andOL,Rub,d according to the convention in Eq. (3), respectively;mi(i=Λ0b,ˉψ,ˉϕ,ξ) is the mass of particle i;ΓΛb is the total width of heavy baryons; and→qξ andEξ=m2Λb+m2ξ−m2ϕ2mΛb are the momentum and kinetic energy of ξ in the heavy baryonΛ0b rest frame, respectively. -
Using the hadronic matrix elements obtained in the section above, we can explore B-Mesogenesis via the invisible and semi-invisible decays of bottom baryons detectable at future lepton colliders. The reconstructions of the invisible and semi-invisible decays of bottom baryons are always difficult in experiments. However, these difficulties can be overcome by the double tag method [49]. In the
e+e− collisions, the initial energy is well known in each event. Assuming the detectors have a sufficient resolution, we reconstruct all the recoiling particles except forΛ0b/Ξ0,−b in the colliding event, which helps to determine the invariant mass of the bottom-baryon candidate using energy-momentum conservation. The production of a bottom baryon will be identified if the invariant mass is consistent with the mass ofΛ0b/Ξ0,−b and if the conservation laws for the electric charge, baryon number, bottom number, and strange number are satisfied. Then we are able to measure the invisible and semi-invisible decays of bottom baryons. The double tag method has been widely used by BESIII, BABAR and Belle. For example, BESIII measured the absolute branching fraction ofΛ+c→Λ0e+νe at theΛ+cˉΛ−c threshold, which determinesΛ+c by reconstructing the charmed anti-baryon via its hadronic decays, such asˉΛ−c→ˉpK+π− [50]. It is also possible to use the double tag method at energies much higher than the baryon-antibaryon threshold. The Belle collaboration measured the absolute branching fractions ofΛ+c decays by reconstructing the recoilingD(∗)−ˉpπ+ system in the event ofe+e−→D(∗)−ˉpπ+Λ+c at the colliding energy of approximately 10 GeV [51]. Therefore, the invisible and semi-invisible decays ofΛ0b/Ξ−b can be measured using the double tag method ate+e− colliders, such as BelleII at the bottom baryon-antibaryon threshold, or CEPC and FCC-ee at a higher energy as well.In practice, to estimate the sensitivity of the parameters of B-Mesogenesis, we define the significance s of the signal at the 95% C.L. as
s=nS√nS+nB=√nS=√NB×Br(B→X)=2,
(47) where
nS andnB denote the numbers of signal and background events at future lepton colliders, respectively. The background events are negligible using the double tag method because the three kinds of processes studied in this work are all forbidden in the Standard Model by the visible baryon number violation. The branching fractionsBr(B→X) withX=πˉψ,Kˉψ,γˉψ,ξˉϕ are given in Eqs. (29), (39), and (46), whereNB represents the production number of bottom baryons detected by the double tag method, which is different from the total production of bottom baryons in the collisions. We assume an efficiency of reconstruction of 100% for pions, kaons, and photons in the semi-invisible decays. We also assume that the production number of bottom baryons in the double tag method isNB=NΛb,Ξb=108 as a benchmark for future lepton colliders. The corresponding constraints can directly scale up to the results for the actual production number of bottom baryons at future lepton colliders, which are still indistinct in the current theoretical study.There are some constraints on the masses of dark particles under the B-Mesogenesis scenario. To prevent the decay of protons into dark particles, the masses of dark matter particles must satisfy
mˉψ>mp−me−mνe≃937.8MeV,mξ+mˉϕ>mp−me−mνe≃937.8MeV,
(48) where the former prohibits
p→ˉψe+νe decay and the latter preventsp→ˉϕξe+νe decay through an off-shellˉψ propagation. In addition, the stability of dark matter requires that the mass difference between the dark scalarˉϕ and dark Majorana fermion ξ must obey [10]|mξ−mˉϕ|<mp+me+mνe≃938.8 MeV.
(49) Otherwise, the dark particles could decay into each other by emitting a proton, an electron, and a neutrino, which will diminish the baryon asymmetry generated in the Universe.
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First, we estimate the sensitivity required for B-Mesogenesis by searching for the semi-invisible decay of bottom baryons at future lepton colliders. Their branching ratios are only sensitive to the mass
mψ and Wilson coefficients|CLub,d| or|CLub,s| of the scenario. The constraints on the Wilson coefficients|CLub,d| and|CLub,s| versusmψ at the 95% C.L. are illustrated in Fig. 5, for which the production number of bottom baryons in the double tag method is assumed to beNΛb,Ξb=108 . The constraints on the right-handed operatorsORuda,db are omitted because they could not project out the leading-twist wave function of theΛb andΞb baryons.Figure 5. (color online) The constraints on the Wilson coefficients at 95% C.L. versus
mψ in the semi-invisible decays of bottom baryons. The left panel illustrates the constraints on|CLub,d| , while the right panel display the constraints on|CLub,s| . The declines of the constraints at the tail result from the suppression of phase space. When dark fermion massmψ close to the mass ofΛ0b baryon, the soft photon radiation inΛ0b→γˉψ channel will induce a serious infrared divergence, which invalid the leading order estimation in this region. Hence, we omit the constraints inΛ0b→γˉψ process whenmψ>5 GeV (blue dot dashed line).For the
Λb(Ξb)→Pˉψ processes, we found that the semi-invisible decayΛ0b→π0ˉψ is most sensitive to operatorOLub,d , while theΛb→K0ˉψ process is most sensitive to operatorOLub,s . When assuming the production number of bottom baryons in the double tag method isNΛb,Ξb=108 , the constraints on the Wilson coefficients can reachO(10−8) GeV-2. The decline in the Wilson coefficients at the tail result from the suppression of phase space. The form factors and hadron mass in the semi-invisible decay are similar, and the constraints on the Wilson coefficients are similar in these processes.For the
Λ0b→γˉψ channel, the constraints on the Wilson coefficients can reach at leastO(10−7) GeV-2. This constraint is looser than that in the hadronic semi-invisible decay in the small-mass region owing to the fact that the form factors of photon radiation are smaller than those of meson final states. However, in the large-mass region, the constraints will be tighter because of the evolution of the form factors. Moreover, when the dark fermion massmψ is similar to the mass of initial baryonsmΛb , the divergence in the evolution of the form factors invalidates the estimation of the branching ratio. Therefore, we omit the constraints in theΛ0b→γˉψ channel whenmψ>5GeV .Finally, because the final state hadrons in the
Ξ−b semi-invisible decay are all electrically charged, these decay modes should be probed at future high-energy lepton colliders (i.e., CEPC or FCC-ee) using a displaced vertex method. Therefore, we expect a thorough exploration of these semi-invisible decays of bottom baryons to be conducted at future lepton colliders. -
As shown in Refs. [7, 13, 15–18], previous studies primarily focused on searching for the dark fermion ψ in the semi-invisible decay of bottom baryons or mesons. To keep the dark matter stable, B-Mesogenesis must involve a dark Majorana fermion ξ and a dark scalar baryon ϕ, and requires all the dark fermions ψ to completely decay into these two invisible particles, with
BR(ψ→ϕξ)= 100%. Therefore, the coupling ofyd and the interaction ofˉψϕξ are very challenging to explore in the semi-invisible decays in which ψ is always on-shell. The width of ψ (depending onyd ) cannot be measured in experiments because the final states ϕ and ξ are not measurable. On the contrary,ˉψ is off-shell in the fully invisible decay ofΛ0b→ξˉϕ , as seen in Fig. 1 (i) or Fig. 4. Then the stable dark matter particle ξ orˉϕ can be explored, and the interaction ofydˉψϕξ can be measured, as shown in the branching fraction ofΛ0b→ξˉϕ in Eq. (46). Therefore, we argue that the fully invisible decay ofΛ0b→ξˉϕ is crucial for probing B-Mesogeneisis, as it provides a distinctive signal that can be used to examine the stable dark particles in this scenario.Assuming the production number of
Λ0b baryons using the double-tag method isNΛb=108 , the constraints on the parameters|ydCL,Ruda,db| can be taken straight from Eq. (46), depending on the different masses ofˉψ ,ˉϕ , and ξ. The constraints at the 95% C.L. are displayed in Fig. 6 and Fig. 7 for the variations ofmψ andmξ+mϕ , respectively.Figure 6. (color online) The constraints on the parameters
|ydCL,Rud,b| and|ydCL,Rub,d| versusmψ from the invisible decay ofΛ0b with 95% C.L. We estimate the constraints in six different mass distribution of dark matters. The left and right panels illustrate the constraints related to operatorsOL,Rud,b andOL,Rub,d , respectively. The dark matter (DM) stability in Eq. (49) prohibits a large mass difference between dark fermion ξ and dark scalarˉϕ , which imposes a truncation at large mass region ifmϕ≠mξ . For the three cases wheremξ+mϕ=0.5mψ (dashed line), there is another truncation at smallmψ region to forbid the decay of proton intoξˉϕ through an off-shellˉψ particle.Figure 7. (color online) The constraints on the parameters
|ydCL,Rud,b| and|ydCL,Rub,d| versus total massmξϕ=mξ+mϕ with 95% C.L.. As the DM stability prohibits a large mass splitting of dark fermion and dark baryons, we estimate the constraints in three different mass distribution of dark matters.In Fig. 6, the constraints of the parameters depending on
mψ are given for|ydCL,Rud,b| in the left panel, and for|ydCL,Rub,d| in the right panel. We estimate the constraints in six different mass distributions of dark matter particles. Three cases of the masses of ξ and ϕ are shown as the blue, green, and red curves, with (i)mξ=mϕ=0.49mψ , (ii)mξ=0.90mψ ,mϕ=0.08mψ , and (iii)mξ=0.08mψ ,mϕ=0.90mψ , respectively. Because the stable dark particles ξ andˉϕ carry most of the mass ofˉψ , the decay of protons intoξˉϕ through off-shellˉψ particles is forbidden formψ≃1 GeV. In addition, to directly illustrate the dependence of the constraints on the mass distribution of dark particles, three other cases ofmξ andmϕ are shown as the orange, cyan, and magenta curves, with (i)mξ=mϕ=0.25mψ , (ii)mξ=0.4mψ,mϕ=0.1mψ , and (iii)mξ=0.1mψ,mϕ=0.4mψ , respectively. For these cases, a truncation at the small-mass region was imposed to prevent the decay of protons intoξˉϕ . Because the DM stability in Eq. (49) prohibits a large mass difference between dark fermions ξ and dark scalar baryons ϕ, it also imposes a truncation at the large-mass region with a specific mass distribution. We found that the constraints on the parameters of B-Mesogenesis can reachO(10−7) GeV-2. Because the values of the form factors from the left- and right-handed operators are the same except for their signs, the constraints on the operators with different chirality are equal to each other. For the operators with different flavor structures, a factor of2 in the constraint on|ydCL,Rub,d| arises from the half suppression of the decay constantsλL,Rub,d in the comparison between Eqs. (43) and (44). In the allowed mass region, we found that the differences between the constraints for various mass distributions are very small, indicating that the fully invisible decay ofΛb baryons is insensitive to the mass distribution of dark particles. The drop in the constraints atmψ values close tomΛb arises from a nearly on-shell transition ofΛ0b→ˉψ , which significantly enlarges the invisible decay rate.Similarly, the constraints on the parameters of the mass of
mξϕ=mξ+mϕ are shown in Fig. 7; the left and right panels correspond to|ydCL,Rud,b| and|ydCL,Rub,d| , respectively. The three cases of (i)mϕ=mξ , (ii)mξ=mϕ+0.9GeV , and (iii)mϕ=mξ+0.9GeV are displayed as the blue, green, and red curves, respectively, with the mass of ψ fixed at 4.0 GeV. The curves are roughly flat in the region ofmξϕ between 1 GeV and 5 GeV. Furthermore, the differences between the three curves in each panel are also very small. These phenomena indicate that the constraints on the parameters are insensitive to the mass distribution ofmξ andmϕ . This is beneficial to the search for the stable particles in B-Mesogenesis. If there is a deviation in the semi-invisible decays of bottom baryons or mesons, the fully invisible decay ofΛ0b can further indicate the stable components that are inherent in B-Mesogenesis. The declines in the constraints within the large-mass region result from the suppression of phase space. -
The similar densities of baryons and dark matter in the universe imply that they may arise from the same UV complete model. B-Mesogensis, which assumes dark matter can be charged under the baryon number, attempts to simultaneously explain the origins of dark matter and baryon asymmetry. Since this model was proposed, most studies have primarily concentrated on exploring the semi-invisible decays of B mesons. However, both invisible and semi-invisible decays of bottom baryons are crucial for exploring B-Mesogenesis at the GeV scale. In particular, as B-Mesogenesis must involve a dark Majorana fermion and a dark scalar baryon to keep the DM and baryon asymmetry stable, the fully invisible decay of
Λ0b yields a distinctive signal that can be used to directly probe the stable particles of this scenario. In addition, when exploring B-Mesogenesis in terms of the decays of bottom baryons, the hadronic matrix elements are vital for determining the physical observables. Accordingly, we systematically examined the hadronic semi-invisible decays ofΛ0b→π0(K0)ˉψ andΞ−b→π−(K−)ˉψ , the radiative semi-invisible decay ofΛ0b→γˉψ , and the fully invisible decay ofΛ0b→ξˉϕ . The relevant hadronic matrix elements were calculated using the QCD factorization method in the heavy quark limit. Furthermore, we analyzed the sensitivities of the parameters of B-Mesogenesis by searching for the invisible and semi-invisible decays of bottom baryons at future lepton colliders. Assuming the production of bottom baryons in the double-tag method wasNΛb,Ξb=108 , we found that the constraints on the Wilson coefficients|CLub,d| and|CLub,s| in the semi-invisible decay of bottom baryons can reachO(10−8) GeV-2, while the constraints on|ydCL,Rud,b| and|ydCL,Rub,d| can reachO(10−7) GeV-2 from the fully invisible decay ofΛ0b . Although the invisible decay ofΛ0b baryons was sensitive to new physics parameters, this decay mode provides a distinctive signal with which the stable dark particles of B-Mesogenesis can be directly explored. -
We are grateful to Sheng-Qi Zhang and Man-Qi Ruan for useful discussions.
Invisible and semi-invisible decays of bottom baryons
- Received Date: 2024-04-07
- Available Online: 2024-08-15
Abstract: The similar densities of dark matter and baryons in the universe imply that they may arise from the same ultraviolet model. B-Mesogenesis, which assumes dark matter is charged under the baryon number, attempts to simultaneously explain the origin of baryon asymmetry and dark matter in the universe. In particular, B-Mesogenesis may induce bottom-baryon decays into invisible or semi-invisible final states, which provide a distinctive signal for probing this scenario. In this work, we systematically study the invisible decays of bottom baryons into dark matter and the semi-invisible decays of bottom baryons into a meson or a photon together with a dark matter particle. In particular, the fully invisible decay can reveal the stable particles in B-Mesogenesis. Some QCD-based frameworks are used to calculate the hadronic matrix elements under the B-Mesogenesis model. We estimate the constraints on the Wilson coefficients or the product of some new physics couplings with the Wilson coefficients according to the semi-invisible and invisible decays of bottom baryons detectable at future colliders.