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Invisible and semi-invisible decays of bottom baryons

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Yong Zheng, Jian-Nan Ding, Dong-Hao Li, Lei-Yi Li, Cai-Dian Lü and Fu-Sheng Yu. Invisible and Semi-invisible Decays of Bottom Baryons[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad4afa
Yong Zheng, Jian-Nan Ding, Dong-Hao Li, Lei-Yi Li, Cai-Dian Lü and Fu-Sheng Yu. Invisible and Semi-invisible Decays of Bottom Baryons[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad4afa shu
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Invisible and semi-invisible decays of bottom baryons

    Corresponding author: Jian-Nan Ding, dingjn23@pku.edu.cn (Corresponding author)
    Corresponding author: Dong-Hao Li, lidonghao@ihep.ac.cn (Corresponding author)
    Corresponding author: Lei-Yi Li, lileiyi@ihep.ac.cn (Corresponding author)
  • 1. MOE Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China
  • 2. School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China
  • 3. Center for High Energy Physics, Peking University, Beijing 100871, China
  • 4. Institute of High Energy Physics, CAS, Beijing 100049, China
  • 5. School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 101408, China

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.

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    I.   INTRODUCTION
    • 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 and B0s¯B0s oscillations, which naturally provide CP violation. Finally, the bottom mesons decay into an ordinary baryon with the baryon number B=+1 and a dark fermion ψ that is charged under baryon number with B=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 [822]. 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 B0Λ0ψ and B+pψ [1921]. However, it is also possible for the bottom baryons to decay into a meson and a dark baryon, such as Λb(Ξb)Pˉψ, with P denoting a light pseudo-scalar π,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 (Λ0bξˉϕ) and semi-invisible radiative decay (Λ0bγˉψ) of Λ0b baryons. In particular, the fully invisible decay of Λ0b 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 Λ0bξˉϕ, the fully invisible decays are also very interesting for neutron and hyperon decays, which share the same type of interaction as uadbdcψ, where a,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 of Λ0 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 as BR(Λ0invisible)<7.4×105 [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 [2932]. For example, this kind of form factor also emerges in the baryon and lepton number violation decays of ΛbP 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 in Λ0bγˉψ can also be calculated within the same QCD factorization method, while those of Λ0bξˉϕ 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 and Ξb 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.

    II.   BRIEF INTRODUCTION TO B-MESOGENESIS
    • As the relic densities of DM and net baryons are similar, they may arise from the same UV completion. In practice, B-Mesogenesis [7, 1012], 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:

      LintyRuadbϵijkYiˉujRadk,CRbyLuadbϵijkYiˉujLadk,CLbyψdcYiˉψdi,CRcydˉψϕξ+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 charge QY=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 number Bψ=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 of Yiˉψdi,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 with Bϕ=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 Λ0bπ0ˉψ, (c) for Λ0bK0ˉψ, (d) for Ξbπˉψ, (e) for ΞbKˉψ, (f,g,h) for the semi-invisible radiative decay of Λ0bγˉψ, and (i) for the fully invisible decay of Λ0bξˉϕ. The dark sector particles of ˉψ and ˉϕ are charged under the baryon number of Bˉψ,ˉϕ=+1, so that the total baryon number are conserved in these processes.

      Field SQB
      Y01323
      ψ120−1
      ξ1200
      ϕ00−1

      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 by

      LEFT=CL,Ruadb,dcOL,Ruadb,dc+CL,Ruadb,dcˉOL,Ruadb,dc,

      (2)

      where CL,Ruadb,dc=yL,Ruadbyψdc/M2Y are the Wilson coefficients and the effective operators OL,Ruadb,dc and ˉOL,Ruadb,dc are given by

      OLuadb,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,dcˉψCOL,Ruadb,dc+h.c.,

      (4)

      where the operators OL,Ruadb,dc are expressed as

      OLuadb,dc=ϵijk dkRc(ˉui,CLadjLb),ORuadb,dc=ϵijk dkRc(ˉui,CRadjRb).

      (5)
    III.   INVISIBLE AND SEMI-INVISIBLE DECAYS OF BOTTOM BARYONS
    • 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

      Λ0bP0ˉψ,ΞbPˉψ,Λ0bγˉψ,Λ0bξˉϕ.

      (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 operators OL,Ruadb,dc can be represented as a convolution of the perturbative part with the leading twist heavy baryon and light meson distribution amplitudes.

    • A.   Hadronic semi-invisible decays of Λb/ΞbPˉψ

    • In this subsection, we construct the factorization formula and explicitly calculate the hadronic matrix elements of Λ0b/ΞbP0,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 operator OL,Ruadb,dc as

      Figure 2.  (color online) The Feynman diagrams of the Λ0b/ΞbPˉψ decays, where the bold line represents the b quark and the circle cross denotes the currents in the OL,R operators.

      iM=CL,Ruadb,dc(ˉvCψ(q)P(p)|OL,Ruadb,dc|B(p)),

      (7)

      where vψ is the spinor of dark baryon ˉψ with momentum q=pp, and B represents Λ0b or Ξb with momentum p=mBv. The hadronic matrix elements can be parameterized in terms of two form factors:

      P(p)|OL,Ruadb,dc|B(p)=PR(ζBP(L,R)1(q2)+qmBζBP(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+m2ψm2P2mB,Ep=m2Bm2ψ+m2P2mB,|q|=|p|=(m2B(mψ+mP)2)(m2B(mψmP)2)2mB,

      (10)

      with

      pμ=(np)ˉnμ2+(ˉnp)nμ2+pμ(np,ˉnp,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 λmψmBmPmBΛQCDmB, 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 ORuadb,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 matrices PR=1+γ52 within the operator ORuadb,db. For example, in the heavy quark limit, the Dirac structure of the Λ0bπ0ˉψ transition amplitude shown in Fig. 2 (a) is given by

      iMvTψCPR(pγ5)γμ(ˉnγ5C)(γμ)TxpTPL,RuΛb,

      (13)

      where (pγ5) and (ˉnγ5C) are the light-cone projectors corresponding to the leading twist distribution amplitudes of Λ0b and π0, respectively [31, 35]. The subscripts of PL,Rcorrespond to the insertion of operators OLuadbdc or ORuadbdc. Because there is an odd number of Dirac matrices between PR and PL,R, the contribution of right-handed operators ORuadbdc 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 and OLus,b owing to the parity conservation. Taking Λ0bπ0ˉψ as an example, the partonic amplitude 1 can be factorized as a product of two transition matrix elements in the spin space, iMˉψ(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 that γ5 in the operator ˉuCLdL cannot contribute to the trace because there are only two independent Lorentz vectors in this process to contract with the 4-dimensional antisymmetric tensor ϵμνρσ. Actually, the JP quantum number of quark pair (ud) in the Λ0b baryon is 0+ 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 OLub,d and OLub,s. Strange number conversed or violated processes are induced by the corresponding operators, which is summarized as

      Λ0bπ0ˉψandΞbKˉψ,inducedbyOLub,d,Λ0bK0ˉψandΞbπˉψ,inducedbyOLub,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 order mbΛQCD, 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 as

      iM=ηΔsCLuadb,dcζBP(ˉvCψ(q)PRn2uB(p)),

      (15)

      where the sign factor ηΔs=1 for strange flavor conserved processes Λ0bπ0 and ΞbK, while ηΔs=+1 for strange flavor violated processes Λ0bK0 and Ξbπ. The convention for the sign factor arises from the inconsistency between the quark flavor order in the operator O and the definition of the light-cone distribution amplitudes.

      The factorization formula of the form factor ζBP is given by

      ζBP=fPf(2)B0dωω10du10dxJBP×(x,u,ω,μ)ϕP(x)ψ2(u,ω),

      (16)

      where fP(B) is the decay constant of particle P(B), JBP(x,u,ω,μ) is the corresponding hard-scattering kernel, and ϕP(x) and ψ2(u,ω) are the leading twist light-cone distribution amplitudes of the P meson and B baryon [30, 3840], respectively,

      P(p)|[ˉq(tˉn)]A[tˉn,0][q(0)]B|0=ifP4ˉnp[n2γ5]BA10dxeixtˉnpϕP(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]BA0dωω×10dueiω(t1u+t2ˉu)ψ2(u,ω),0|[di(t1ˉn)]A[0,t1ˉn][sj(t2ˉn)]B[0,t2ˉn][bk(0)]C|Ξb(p)=16ϵijk4f(2)Ξb(μ)[uΞb(p)]C[ˉn2γ5CT]BA0dωω×10dueiω(t1u+t2ˉu)ψ2(u,ω),

      (18)

      where ωi=(nki) represent the momentum components of the spectator quark. We use the convention ω1=uω, ω2=ˉuω with ω=ω1+ω2, where u and ˉu represent the momentum fraction of the spectator quark in the light quark pair. In the heavy-quark limit, the coupling constants fB for Λ0b and Ξb are respectively defined by [39]

      ϵijk0|uiTα(0)djβ(0)bkγ(0)|Λ0b(p)14{f(1)Λb[γ5CT]βα+f(2)Λb[vγ5CT]βα}[uΛb(p)]γ,ϵijk0|diTα(0)sjβ(0)bkγ(0)|Ξb(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[igt0dxˉnA(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(1x)[1+n=1aPn(μ)C(3/2)n(2x1)],

      (21)

      and the leading-twist wave function for the Λ0b baryon is given by [38, 39, 42],

      ψ2(u,ω)=u(1u)ω21ω40eω/ω0.

      (22)

      In the limit of the SU(3) flavor symmetry, the wave function of Ξb is the same as that of Λ0b. Table 2 summarizes the numerical input parameters.

      mΛb=5.6196GeV [43] τΛb=1.471ps [43]
      mΞb=5.7970GeV [43] τΞb=1.572ps [43]
      ω0=0.280+0.0470.038GeV [42] f(2)Λb(μ0)=0.030±0.005GeV3 [42]
      fπ=0.1304±0.0002GeV [40] fK=0.1562±0.0007GeV [40]
      aπ1(μ0)=0 [40] aπ2(μ0)=0.29±0.08 [40]
      aK1(μ0)=0.07±0.04 [40] aK2(μ0)=0.24±0.08 [40]

      Table 2.  Input parameters with renormalization scale μ0=1GeV.

      The corresponding hard-scattering functions are calculated as

      JΛ0bπ0(x,u,ω,μ)=12(Ju(x,u,ω,μ)+Jˉu(x,u,ω,μ)),JΛ0bK0(x,u,ω,μ)=JΞbK(x,u,ω,μ)=Jˉu(x,u,ω,μ),JΞbπ(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=12(|uˉu|dˉd). The asymptotic behaviors of the leading twist light-cone distribution amplitudes in the endpoint region are

      ϕP(x)x(1x),ψ2(u,ω)u(1u)ω2,

      (25)

      which will cancel the divergent behavior in the hard-scattering functions Ju(x,u,ω,μ) and Jˉu(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 of αs(2GeV)0.3 and use the other inputs displayed in Table 2. The numerical results for the form factors are

      ζΛ0bπ0=(1.43+0.460.54)×102[GeV2],ζΛ0bK0=(1.23+0.400.47)×102[GeV2],ζΞbK=(1.23+0.400.47)×102[GeV2],ζΞbπ=(1.01+0.330.38)×102[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 and ζ2 resulting from the match between Eq. (15) and Eqs. (7) and (8):

      ηΔsζBP=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., Bπˉν [36] or ΛbΛˉ [42]), the relevant form factors can be determined only at one or two values of q2 using an assumed q2-dependence to obtain the numerical results at an arbitrary q2. 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 factors ζ1,2(q2=0) toward ζ1,2(q2=m2ψ) [18, 4446]:

      ζ1,2(q2)=11q2/m2poleζ1,2(q2=0).

      (28)

      Here, ζ1,2(q2=0) are obtained through the values in Eq. (26) and the relationship between ζBP and ζ1,2 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 of Λ0b(12+) for the Λ0bπ0/ΞbK transition, as Λ0b(12+) is the lowest state of operator OLub,d, while mpole is the mass of Ξ0b(12+) for the Λ0bK0/Ξbπ transition.

      Starting from Eqs. (7) and (8), the branching fractions of BPˉψ are expressed as

      BR(BPˉψ)=|qψ|16πΓB|CLub,d|2[ζ21(q2)+ζ22(q2)m2ψm2B+(ζ21(q2)+ζ22(q2)m2ψm2B)m2ψm2Pm2B+4ζ1(q2)ζ2(q2)m2ψm2B],

      (29)

      where (uadb,dc) represents (ub,d) or (ub,s) for effective operators OLub,d or OLub,s, |qψ| is the momentum of dark baryon ˉψ,

      |qψ|=(m2B(mψ+mP)2)(m2B(mψmP)2)2mB,

      (30)

      and ΓB (mB) is the decay width (mass) of baryon B.

    • B.   Radiative semi-invisible decay of Λ0bγˉψ

    • For the radiative decay processes Λ0bγˉψ showed in Fig. 3, the decay amplitudes can be expressed as,

      Figure 3.  (color online) The Feynman diagram for Λ0bγˉψ with the bold line representing the b quark.

      iM=CL,Ruadb,dc(ˉvCψ(q)γ(p)|OL,Ruadb,dc|Λ0b(p)).

      (31)

      The transition amplitudes associated with the right-handed operators ORub,d and ORud,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)|OLuadb,dc|Λ0b(p)=PR(ζΛbγ1(q2)pm2Λbiσμν+ζΛbγ2(q2)qm2Λbiσμν)uΛb(p)ϵμpν.

      (32)

      In the heavy quark limit, the transition amplitude for operator OLud,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, iMˉψ(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 vector ϵμ of the photon. Considering momentum conservation (k+r=p) and the on-shell condition (p2=0), there are two independent Lorentz-invariant amplitudes proportional to pϵ and rϵ, respectively. In the heavy quark limit, we neglect the momentum k ΛQCD of the spectator quarks. Therefore, the second transition matrix element must be proportional to pϵ=rϵ, which is zero after implementing the Ward Identity: pϵ=0.

      The non-vanishing amplitude Λ0bγˉψ defined by OLub,d is given by

      iM=CLub,dˉvCψ(q)PR(ζΛ0bγuˉn2ϵn2+ζΛ0bγdn2ϵˉn2)uΛb(p),

      (33)

      The form factors can be factorized as

      ζΛ0bγu,d=f(2)Λb0dωω10duJΛ0bγu,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, JΛ0bγu,d(u,ω,μ) is the hard-scattering kernel for the Λbγ transition,

      JΛ0bγu(u,ω,μ)=14(Qu(μ)uω),JΛ0bγd(u,ω,μ)=14(Qd(μ)ˉuω),

      (35)

      where the electric charges of the light quarks u and d are Qu(μ)=23e(μ) and Qd(μ)=13e(μ), respectively. Combining Eqs. (31)−(33), we have

      ζΛ0bγ1=ζΛ0bγd,ζΛ0bγ2=ζΛ0bγu+ζΛ0bγd.

      (36)

      The numerical results for the form factors of the spectator processes Λ0bγˉψ are

      ζΛ0bγu=(5.49+1.181.30)×103[GeV2],ζΛ0bγd=(2.74+0.590.65)×103[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 the q2-dependence of form factor ζ1,2(q2),

      ζ1,2(q2)=11q2/m2Λbζ1,2(q2=0),

      (38)

      where mΛb is the mass of Λ0b(12+). Then, the branching ratio for Λ0bγˉψ with operator OLub,d is estimated as

      BR(Λ0bγˉψ)=|qψ|8πm2ΛbΓΛb12spin|M|2=|qψ|8πΓΛb|CLub,d|2[ζ21(q2)+(ζ21(q2)+ζ22(q2))×m2ψm2Λb+2ζ1(q2)ζ2(q2)m2ψm2Λb],

      (39)

      with q2=m2ψ.

    • C.   Fully invisible decay for Λ0b

    • Under the B-Mesogenesis scenario, the Λ0bξˉϕ channel is governed by both effective operators OL,Rud,b and OL,Rub,d. The Feynman diagram is shown in Fig. 4. The decay amplitudes can be respectively expressed as

      Figure 4.  (color online) The Feynman diagram for Λ0bξˉϕ process, with the bold line denoting the b quark.

      iML,Rud,b=ydCL,Rud,b(ˉuCξ(q)p+mψp2m2ψ0|OL,Rud,b|Λ0b(p)),iML,Rub,d=ydCL,Rub,d(ˉuCξ(q)p+mψp2m2ψ0|OL,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|OL,Rud,b|Λ0b(p)=λL,Rud,b PRuΛb(p),0|OL,Rub,d|Λ0b(p)=λL,Rub,d PRuΛb(p).

      (41)

      We found that the couplings λL,Rud,b and λL,Rub,d could be related to the coupling f(1,2)Λb defined in Eq. (19) by multiplying the corresponding matrix element. For instance,

      0|OL,Rud,b|Λ0b(p)=ϵijk0|[uiTCPL,Rdj]bkR|Λ0b(p)=ρ(CPL,R)αβ(PR)ργ{ϵijk0|uiTαdjβbkγ|Λ0b(p)}=14{f(1)ΛbTr[CPL,Rγ5CT]+f(2)ΛbTr[CPL,Rvγ5CT]}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 in

      λLub,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)Λbf(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 Λbξˉϕ are given by

      BR(Λbξˉϕ)L,Ruadb,dc=|qξ|8πmΛbΓΛb(|yd|2|CL,Ruadb,dc|2|λL,Ruadb,dc|2)×Eξ(m2Λb+m2ψ)+2mΛbmψmξ(m2Λbm2ψ)2,

      (46)

      where (uadb,dc) represents (ud,b) or (ub,d) for different effective operators OL,Rud,b and OL,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 and Eξ=m2Λb+m2ξm2ϕ2mΛb are the momentum and kinetic energy of ξ in the heavy baryon Λ0b rest frame, respectively.

    IV.   PROBING B-MESOGENESIS WITH INVISIBLE AND SEMI-INVISIBLE DECAYS OF BOTTOM BARYONS
    • 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 for Λ0b/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 of Λ0b/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 of Λ+cΛ0e+νe at the Λ+cˉΛc threshold, which determines Λ+c by reconstructing the charmed anti-baryon via its hadronic decays, such as ˉΛcˉpK+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 of Λ+c decays by reconstructing the recoiling D()ˉpπ+ system in the event of e+eD()ˉpπ+Λ+c at the colliding energy of approximately 10 GeV [51]. Therefore, the invisible and semi-invisible decays of Λ0b/Ξb can be measured using the double tag method at e+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=nSnS+nB=nS=NB×Br(BX)=2,

      (47)

      where nS and nB 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 fractions Br(BX) with X=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 is NB=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>mpmemνe937.8MeV,mξ+mˉϕ>mpmemνe937.8MeV,

      (48)

      where the former prohibits pˉψe+νe decay and the latter prevents pˉϕξe+νe decay through an off-shell ˉψ propagation. In addition, the stability of dark matter requires that the mass difference between the dark scalar ˉϕ and dark Majorana fermion ξ must obey [10]

      |mξmˉϕ|<mp+me+mνe938.8 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.

    • A.   Probing B-Mesogenesis in the semi-invisible decays of bottom baryons

    • 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| versus mψ 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 be NΛb,Ξb=108. The constraints on the right-handed operators ORuda,db are omitted because they could not project out the leading-twist wave function of theΛb and Ξb 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 mass mψ close to the mass of Λ0b baryon, the soft photon radiation in Λ0bγˉψ channel will induce a serious infrared divergence, which invalid the leading order estimation in this region. Hence, we omit the constraints in Λ0bγˉψ process when mψ>5 GeV (blue dot dashed line).

      For the Λb(Ξb)Pˉψ processes, we found that the semi-invisible decay Λ0bπ0ˉψ is most sensitive to operator OLub,d, while the ΛbK0ˉψ process is most sensitive to operator OLub,s. When assuming the production number of bottom baryons in the double tag method is NΛb,Ξb=108, the constraints on the Wilson coefficients can reach O(108) 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 Λ0bγˉψ channel, the constraints on the Wilson coefficients can reach at least O(107) 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 mass mψ is similar to the mass of initial baryons mΛb, the divergence in the evolution of the form factors invalidates the estimation of the branching ratio. Therefore, we omit the constraints in the Λ0bγˉψ channel when mψ>5GeV.

      Finally, because the final state hadrons in the Ξb 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.

    • B.   Probing B-Mesogenesis in the invisible decay of bottom baryons

    • As shown in Refs. [7, 13, 1518], 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(ψϕξ)= 100%. Therefore, the coupling of yd and the interaction of ˉψϕξ are very challenging to explore in the semi-invisible decays in which ψ is always on-shell. The width of ψ (depending on yd) 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 of Λ0bξˉϕ, as seen in Fig. 1 (i) or Fig. 4. Then the stable dark matter particle ξ or ˉϕ can be explored, and the interaction of ydˉψϕξ can be measured, as shown in the branching fraction of Λ0bξˉϕ in Eq. (46). Therefore, we argue that the fully invisible decay of Λ0bξˉϕ 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 is NΛb=108, the constraints on the parameters |ydCL,Ruda,db| can be taken straight from Eq. (46), depending on the different masses of ˉψ, ˉϕ, and ξ. The constraints at the 95% C.L. are displayed in Fig. 6 and Fig. 7 for the variations of mψ and mξ+mϕ, respectively.

      Figure 6.  (color online) The constraints on the parameters |ydCL,Rud,b| and |ydCL,Rub,d| versus mψ from the invisible decay of Λ0b 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 operators OL,Rud,b and OL,Rub,d, respectively. The dark matter (DM) stability in Eq. (49) prohibits a large mass difference between dark fermion ξ and dark scalar ˉϕ, which imposes a truncation at large mass region if mϕmξ. For the three cases where mξ+mϕ=0.5mψ (dashed line), there is another truncation at small mψ region to forbid the decay of proton into ξˉϕ through an off-shell ˉψ particle.

      Figure 7.  (color online) The constraints on the parameters |ydCL,Rud,b| and |ydCL,Rub,d| versus total mass mξϕ=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 and ˉϕ carry most of the mass of ˉψ, the decay of protons into ξˉϕ through off-shell ˉψ particles is forbidden for mψ1 GeV. In addition, to directly illustrate the dependence of the constraints on the mass distribution of dark particles, three other cases of mξ and mϕ 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 into ξˉϕ. 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 reach O(107) 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 of 2 in the constraint on |ydCL,Rub,d| arises from the half suppression of the decay constants λL,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 of Λb baryons is insensitive to the mass distribution of dark particles. The drop in the constraints at mψvalues close to mΛb arises from a nearly on-shell transition of Λ0bˉψ, 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 of mξϕ 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 of mξ and mϕ. 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 of Λ0b 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.

    V.   CONCLUSION
    • 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 of Λ0bπ0(K0)ˉψ and Ξbπ(K)ˉψ, the radiative semi-invisible decay of Λ0bγˉψ, and the fully invisible decay of Λ0bξˉϕ. 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 was NΛb,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 reach O(108) GeV-2, while the constraints on |ydCL,Rud,b| and |ydCL,Rub,d| can reach O(107) GeV-2 from the fully invisible decay of Λ0b. Although the invisible decay of Λ0b 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.

    ACKNOWLEDGMENTS
    • We are grateful to Sheng-Qi Zhang and Man-Qi Ruan for useful discussions.

Reference (51)

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