Semi-inclusive decays of B meson into a dark anti-baryon and baryons

Using the recently developed B-Mesogenesis scenario, we studied the semi-inclusive decays of B meson into a dark anti-baryon ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} plus any possible states X containing u/c and d/s quarks with unit baryon number. The two types of effective Lagrangians proposed by the scenario are both considered in the study. The semi-inclusive decay branching fractions of B→Xψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\rightarrow X \psi $$\end{document} are calculated by the method of heavy quark expansion, where the non-perturbative contributions from the matrix elements of dimension-5 operators are included. We obtained the branching fractions as functions of the dark anti-baryon mass. Using the experimental upper limits of the branching fractions, we presented the constraints of the coupling constants in the B-Mesogenesis scenario.


I. INTRODUCTION
The Standard Model of particle physics and the standard cosmological model are two highly successful frameworks for describing the most microscopic and macroscopic physics respectively.However, these two models are not consistent with each other, which leaves many unanswered questions including the existence of dark matter and the asymmetry of matter and anti-matter.To answer these questions, many mechanisms have been proposed since Sakharov firstly introduced the conditions necessary for baryogenesis [1].The traditional mechanisms generally include high scales and extremely massive particles which makes them difficult to be tested by experiments.Recently, a new B-Mesogenesis scenario is proposed by Refs.[2][3][4], which can simultaneously explain the relic dark matter abundance and the baryon asymmetry and in our Universe.The main advantage of this scenario is that it is not only directly testable at hadron colliders and B-factories [3,5], but also indirectly testable at Kaon and Hyperon factories [6,7].Nowadays, the search for B meson decays into baryon with missing energy through B-Mesogenesis has been independently started by the Belle-II collaboration [8] and the LHCb collaboration [9].
In the B-Mesogenesis scenario, a new mechanism for Baryogenesis and DM production is proposed.The b, b quarks are produced by decays of some heavy scalar field Φ during a late era in the history of the early universe.The produced b, b quarks hadronize to charged and neutral Bmesons.The neutral ones B 0 , B0 quickly undergo CP violating oscillations, and then decay into a dark sector baryon with baryon number −1 as well as visible hadron states with baryon number +1.
As a result, the asymmetry of baryon and anti-baryon is produced in the B-Mesogenesis without violating the baryon number.The exclusive decay B → pψ in the framework of B-Mesogenesis was firstly studied by Ref. [10] using light-cone sum rules (LCSR).After that, with the use of LCSR, a more complete study of B meson decays into an octet baryon or charmed anti-triplet baryon and ψ was given by Ref. [11].In addition, similar exclusive decays of B meson into a baryon plus missing energy are studied by Ref. [12] for probing the lightest neutralino.
Recently, there are no strict theoretical studies on inclusive B meson decays in the B-Mesogenesis.Compared with the exclusive decays, inclusive decay branching fractions are more likely to be measured in the experiments.On the other hand, from the theoretical point of view, another advantage of inclusive decays is that the summation over various of hadronic final states eliminates bound-state effects of individual hadrons, which is due to the hypothesis of quark-hadron duality [13].In Ref. [3], using the data of bottom hadron decays with missing energy from the ALEPH experiment [14][15][16], the authors obtained the upper limits on the inclusive decay branching fractions of B → X u/c,d/s ψ, where X u/c,d/s denotes any possible hadron states containing u/c and d/s quarks with unit baryon number.Therefore, compared with the experimental upper limits, a strict theoretical calculation on the B → Xψ branching fraction enables us to determine the upper limits on the coupling constants in the B-Mesogenesis.Nowadays the heavy quark expansion (HQE) [17][18][19][20] has been successfully applied for the studies of inclusive decays as well as lifetime calculations of heavy hadron decays [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].In this work, we will use HQE to calculate the inclusive decay branching fractions of B → X u/c,d/s ψ, where the bound-state effects related to the initial state can be can systematically accounted for by introducing matrix elements of high dimension operators.This article is organized as follows: Section II is a brief introduction to the B-Mesogenesis scenario proposed by Refs.[2][3][4].Section III present a detailed HQE calculation for the B → X u/c,d/s ψ decays.Section IV gives the numerical results for decay branching fractions and constraints on the coupling constants in the B-Mesogenesis.

II. B-MESOGENESIS SCENARIO
The B-Mesogenesis scenario firstly proposed by Refs.[2][3][4] aims to simultaneously explain the baryon asymmetry and the existence of dark matter in our Universe.This B-Mesogenesis model offers a mechanism where an anti-b quark can decays into u/c, d/s quarks and a dark anti-baryon ψ.Although the baryon number is conserved, ψ is invisible so that only the baryons composed of u, d/s quarks can be detected by the experiments.In Refs.[2,3], such baryon number violating decays are realized by exchanging a charged color triplet scalar Y i .There two types of effective Lagrangians in the B-Mesogenesis model with the charge of Y i being Q Y = −1/3: where all the quark fields are taken as right handed and the superscript c indicates charge conjugate.
Y is assumed to be heavy with its mass denoted as M Y .The y s are unknown coupling constants.
In the Type-I model the b quark couples with u, c quarks, while in the Type-II model the b quark couples with the dark anti-baryon ψ.It should be mentioned that in Ref. [3] there is a third type of effective Lagrangian with Q Y = 2/3, which reads as In this work, for simplicity we will only consider the case of Q Y = −1/3, which is consistent with the exclusive decay studies in Refs.[10,11].Integrating out the heavy boson Y in Eq. ( 1), one arrives at the effective Hamiltonian for the two types of models as: Here for simplicity, q = s, d and u denotes u or c quark simultanously.We have defined threequark operatora ŌI (q) = −iϵ ijk ( bi R u c,j R )q k R and ŌII (q) = −iϵ ijk (q i R u c,j R ) bk R , which transform an anti-b quark into two light quarks u, q.In this work, we will calculate the semi-inclusive decay width of B → X uq ψ induced by H I,uq eff and H II,uq eff respectively, with X uq being the summation of any states containing u, q quarks.
In the rest frame of B meson, denoting the momentum and energy of the outgoing dark antibaryon ψ as q and E, we can express the differential decay width of B → X uq ψ as where the spin of ψ and any possible X uq states with momentum p X are summed.The integration of E is equivalent to averaging over a range of final-state hadronic masses.Since ψ has no strong interaction with quarks, the matrix element in Eq. ( 4) can be factoraized as with a being a spinor index.For simplicity we have omitted the superscripts I, II here.Now we introduce a rank-two tensor W with two spinor indexes: which can be generally parameterized as Note that the appearance of P L on the right hand side is due to the identity Ō(uq) P R = 0. Now the differential decay width can be expressed in terms of W or A 1,2 as It is difficult to calculate the W tensor directly due to the infinite summation on the X uq states.
Actually, the W tensor can be extracted from the imaginary part of a correlation function: with The correlation function defined in Eq. ( 10) can be calculated by HQE, where it is expanded according to the power of 1/m b .Each term in the expansion is factorized into perturbative part and non-perturbative part.The former one can be calculated perturbatively, while the later one are parameterized by matrix elements of B meson.We will perform an explicit calculation of T ba by HQE in the next section.

B. Heavy quark expansion in the Type-I model
We firstly consider the type-I model.The T ba is calculated by HQE with the expansion for the power of 1/m b .Using the explicit form of ŌI The diagram of T ba .The initial and final B mesons are replaced byt free b quarks with momentum The two crossed dots denote Ō † (q) (x) and ŌI (uq) (0) respectively.and free quark propagators, one can obtain To extract the perturbative part of the matrix element above, one can temporary replace the initial where b(p b ) denotes the b quark spinor.In the last step the external states are transformed back to B meson.Now the diagram of the correlation function T is shown by Fig. 1, where the two crossed dots denote Ō † (q) (x) and Ō(q) (0) respectively.The W ba can be calculated by extracting the discontinuity part of T ba using cutting rules, namely all the internal quark lines in Fig. 1 Then we arrive at where Q = m b v − q.The A 2bd , B 2bd are the two scalar functions of the rank-2 two-body phase space integration, which is generally defined as: The explicit expression of A 2bd and B 2bd are given in the Appendix A.
The 1/m b expansion is equivalent to the expansion in terms of the small momentum k.At O(k 0 ) all the k s in Eq. ( 14) vanishes and the b quark field are replaced by the effective one b v .
The axial-vector matrix element in Eq. ( 14) vanishes due to the parity.The vector matrix element can be calculated straightforwardly as In terms of the k expansion, the lowest order of W ba reads as where The O(k 1 ) contribution to W ba comes from the terms linear in k in Eq. ( 14).The procedure to extract the perturbative part by temporarily changing the external B mesons to free b quark is almost the same as that at O(k 0 ).However, now the non-perturbative matrix element becomes ferred to coordinate space.Note that the γ 5 term vanishes again due to the reason of parity conservation.Changing the b quark field into the heavy quark field in HQET, one arrives at where is the O(1/m b ) interaction term of the HQET Lagrangian.The matrix element of the first term in Eq. ( 18) vanishes because of the equation of motion, while the second term in Eq. ( 18) can be parameterized as [17]: with where The matrix element of the last two terms in Eq. ( 18) can be parameterized as where Y = (2/3)λ 1 , Z = −4λ 2 [17].Using the non-perturbative matrix elements defined in Eq. (20) and Eq. ( 22), we obtain the O(k 1 ) contribution to W ba as Similarly, the O(k 2 ) contribution to W ba comes from: where 2bd .Each k in the matrix element above is replaced by iD − mv when transferred to coordinate space.Transforming the b into b v and using the results given in Ref. [17], we have The explicit expressions of A 1,2 at O(k 1 ) and O(k 2 ) are given in the Appendix B.
Up to now we have only considered the case of free quark propagation when calculating the T ba as shown in Fig. 1.When considering the interaction of the internal quarks and the background gluon fields, one has to calculated the one gluon emission diagrams as shown in Fig. 2.Here the external B mesons are also replaced by free b states.We have set the incoming and outgoing b quark momentums as p 1 = m b v + k/2 and p 2 = m b v − k/2 respectively.Here we take the u quark emission as an example, the corresponding T ba tensor is: The emitted gluon has momentum k and note that now the O(k 1 ) term in the denominator vanishes.
The O(k 1 ) contribution to T ug ba is where we have used the trick 1/(l The corresponding W ba can still be extracted by cutting rules, and thus we obtain The combination of k and ϵ a * (k) can be replaced by the gluon tensor field when transferred to coordinate space.Explicitly, we can do the replacement k α ϵ a * µ t a ij → (−i/2)G a αµ t a ij and b → b v , and also note that ⟨B(p B )| bi v (0)γ ν gG αµ b j v (0)|B(p B )⟩ = 0 and then we obtain the W ba for u and q emission as The corresponding explicit expressions of their contribution to A 1,2 are given in the Appendix B.

C. Heavy quark expansion in the Type-II model
In this section we will consider the type-II model.Now the HQE calculation of T ba is almost the same as that in the type-I model.Using the explicit form of ŌII and extracting the imaginary part of T ba as shown in Fig. 1, we can obtain the corresponding W ba through W = −(1/π)ImT : where C 2bd = A 2bd + 4B 2bd .Note that now the spinor structure of the matrix element above is different from that of Eq. ( 14), and it seems not straightforward to read out the A 1,2 defined in Eq. (7).However, instead one can use the following trick: On the other hand, it can be found that in the Type-II model the O(k 1 ) contribution from the one gluon emission diagrams as shown in Fig. 2 vanishes.The explicit expression of A 1,2 at O(k 0 ), O(k 1 ) and O(k 2 ) are given in the Appendix C, which are proportional to m 2 q .Therefore, in the Type-II model the decay width of B → X ud /X cd ψ vanishes in the chiral limit m u,d = 0, and the decay width of B → X us /X cs ψ is suppressed compared with that in the Type-I model.

IV. NUMERICAL RESULTS
In this section, we will present the numerical results on the various of B → X uq ψ branching fractions as functions of ψ mass.The mass parameters are m B = 5.28 GeV, m s = 87 MeV, m c = 1.0 GeV, m b = 4.47 ± 0.03 GeV [36], where the quark masses are chosen at µ = 3 GeV as that used in Ref. [10].The non-perturbative parameters λ 1,2 are related with the kinetic term µ 2 π and the chromo-magnetic term µ 2 G of B meson as: λ 1 = −µ 2 π = −0.414± 0.078 GeV 2 and λ 2 = µ 2 G /3 = 0.117 ± 0.023 GeV 2 respectively [37,38].The errors of m b and λ 1,2 will be used for estimating the uncertainty of the numerical results.
Before calculating the decay width by Eq. ( 8), one has to determine the integration range of E.
Obviously, the lower bound of E must be m ψ .On the other hand, the upper bound of E seems to be Buq .The contribution of the spectator quark in B meson to X uq is omitted since the energy of X uq is mostly given by the heavy b quark.
m Buq (GeV) [36] 1.0 1.115 2.286 2.468 flowing into the loop bubble as shown in Fig. 1 and Fig. 2 becomes Q 2 = (m q +m u ) 2 .However, it can be found that the terms proportional to λ 1,2 in the results of A 1,2 contain end point singularities at 2bd and B (1,2) 2bd , with n being 1/2 or 3/2.Note that although A 2bd and B 2bd also have such pole structures, they are actually finite in the limit Q 2 → (m q + m u ) 2 .The reason why this end point singularity emerges is due to the fact that HQE breaks down at this region, where single states or resonances dominate.
Before the expansion of k, the W ba contain the terms like It should be noted that the final states X uq observed in the experiment are baryons, not the quarks.Practically, one has to sum the inclusive states X uq from the lowest baryon state B uq .For example, in terms of the B 0 → X ud ψ decay, B uq is a proton or neutron.Accordingly, the lower bounds of Q 2 should be set as the mass square of the corresponding lowest baryon state, namely Buq or equivalently uq × 10 8 .It should be mentioned that in the type-II model, the A 1,2 are proportional to m q , and thus vanishes in the case of B 0 → X ud ψ and B 0 → X cd ψ.In Fig. 4 only the branching fractions of B 0 → X us ψ and B 0 → X cs ψ are presented.The uncertainty mainly comes from λ 1,2 , 1 The error is estimated by the formula: , where δλ1,2 and δm b are the errors of λ1,2 and m b respectively.
uq × 10 10 .The band width shows the uncertainty coming from the uncertainties of λ 1,2 and m b .The maximum m ψ is reached when m ψ = E upper .
which is tiny and can be ignored.Since the masses and lifetimes of B ± and B s are similar to those of B 0 , in this work we only present the branching fractions of B 0 decays and the decay branching fractions of B ± and B s are assumed to be the same.
In addition, instead of the complicated analytical expression given in the Appendix B and C, for practice we can parameterize the branching fraction curves shown in Fig. 3 and Fig. 4 by a simpler formula.Here we use the following polynomial form to fit the curves: where n = 10, 8 for the case of type-I, II. a i have unit GeV 4 and are listed in Table .II.It should be mentioned that the number of the polynomial terms given above is chosen arbitrarily, which is enough for parameterizing the curves perfectly.
In the Ref. [3], 95% CL constraints on the inclusive B meson decays into baryons and missing energy are estimated according to the ALEPH analysis [14], which is shown by the red curves in Fig. 5 and Fig. 6.The corresponding error bands come from the 20% QCD corrections, which is a inferred percentage ratio according to the estimation on the exclusive decay branching fraction given in the Eq.(39) of Ref. [3].On the other hand, it can be found that the branching fractions FIG. 4: The type-II model branching fractions of B 0 → X us ψ and B 0 → X cs ψ as functions of m ψ in the unit of G 2 uq × 10 8 .The maximum m ψ is reached when m ψ = E upper .The branching fractions of B 0 → X ud ψ and B 0 → X cd ψ vanish due to m u = m d = 0.The uncertainty mainly comes from λ 1,2 , which is tiny and can be ignored.

TABLE II:
The coefficients a i defined in the Eq. ( 35) (in unit GeV 4 ).Due to the tiny error bar of the type-II branching fraction, here we ignore the errors of a i in the type-II case.decrease with the increasing of m ψ as shown in Fig. 3 and Fig. 4. Therefore, using the minimal possible value of m ψ , one can in principle obtain the upper limits of the coupling constants G uq .The restriction on m ψ given by [2] reads as: 1.5GeV < m ψ < 4.2GeV.Setting m ψ at its minimum value: m ψ = 1.5 GeV for the red bands in Fig. 5, we can obtain the upper limits and the corresponding errors for the branching fractions as: At the point m ψ = 1.5 GeV, comparing the center values of the branching fractions given in Fig. 3 and Fig. 4 with the constraints given above we can obtain the upper limits of G uq as Type I : FIG. 5: The red curves are 95% CL constraints on the inclusive B meson decays into baryons and missing energy are estimated according to the ALEPH analysis given in the Ref. [3].The corresponding error bands come from the 20% QCD corrections, which is a inferred percentage ratio according to the estimation on the exclusive decay branching fraction given in Eq. (39) of Ref. [3].The blue curves are the branching fractions calculated in the type-I model, where the G uq are taken as their upper limit values given in Eq. ( 37 FIG. 6: The red curves are the same as those in Fig. 5.The blue curves are the branching fractions calculated in the type-II model, where the G uq are taken as their upper limit values given in Eq. ( 37): Using the maximum value of G 2 uq given above, we also present the branching fraction curves in Fig. 5 and Fig. 6.It can be found that with this setting of G uq the branching fraction curves are safely below the upper limit curves in the region m ψ > 1.5 GeV.Since the upper limit curves (red band) have much larger error than those of the branching fraction curves (blue band).To obtain the error for the constraints of G uq in Eq. ( 37), we just compare the center value of the branching fraction with the upper and lower bounds of the red band at m ψ = 1.5 GeV.
Note that the branching fractions of B 0 → X ud ψ and B 0 → X cd ψ vanish in the type-II model due to the chiral limit, thus they cannot be used to constrain G 2 ud and G 2 cd .The branching fractions of B 0 → X us ψ and B 0 → X cs ψ are suppressed by m 2 s , so they produce larger upper limits for G 2 uq .

FIG. 2 :
FIG. 2: One gluon emission diagrams of T ba .The incoming and outgoing B mesons are replaced by free b states, with momentums being p 1 = m b v + k/2 and p 2 = m b v − k/2 respectively.
2m b , and thus the end point singularity is avoided.Here we have omitted the contribution of the spectator quark in B meson to X uq , because the energy of X uq is mostly given by the heavy b quark.The lower bounds of Q 2 corresponding to various of b → uqψ transitions are listed in Table.I.Now, integrating E in the region m ψ < E < E upper , and using the lifetime of B 0 : τ B 0 = 1.519 × 10 −12 fs, we can obtain the branching fractions of B 0 → X uq ψ as functions of m ψ .The branching fractions calculated in the type-I model are shown in Fig. 3 in the unit of G 2 uq × 10 10 .The band width shows the uncertainty coming from the uncertainties of λ 1,2 and m b . 1 In fact, the results are insensitive to the values of λ 1,2 , and most of the uncertainties come from the b quark mass since in the type-I model A 1,2 are proportional to m n b .The maximum m ψ is reached when m ψ = E upper .The branching fractions calculated in the type-II model are shown in Fig. 4 in the unit of G 2 G 2 ud < (1.8 ± 0.35) × 10 −14 GeV −4 , G 2 us < (3.75 ± 0.74) × 10 −14 GeV −4 , G 2 cd < (1.06 ± 0.21) × 10 −12 GeV −4 , G 2 cs < (1.63 ± 0.33) × 10 −12 GeV −4 ; ): G 2 ud = 1.8 × 10 −14 GeV −4 , G 2 us = 3.75 × 10 −14 GeV −4 , G 2 cd = 1.06 × 10 −12 GeV −4 and G 2 cs = 1.63 × 10 −12 GeV −4 .
. CONCLUSION In this work, using the recently developed B-Mesogenesis scenario, we have studied the semiinclusive decays of B meson into a dark anti-baryon ψ plus any possible states X containing u/c and d/s quarks with unit baryon number.The two types of effective Lagrangians proposed by the scenario are both considered in this work.The semi-inclusive decay branching fractions of B → Xψ are calculated by the method of heavy quark expansion, where the non-perturbative contributions from the matrix elements of dimension-5 operators are included.We obtained the branching fractions as functions of the dark anti-baryon mass.Using the experimental upper limits of the branching fractions, we provided the upper limits on the coupling constants in the B-Mesogenesis scenario.In the Type-I model, the upper limits on G 2 ud and G 2 us are around 10 −14 GeV −4 , while the upper limits on G 2 cd and G 2 cs are around 10 −12 GeV −4 .The upper limits on G 2 us and G 2 cs in the Type-II model are around 10 −11 GeV −4 and 10 −10 GeV −4 respectively.+(9 − 7ϵ)s 3 + (ϵ + 3)s 2 u + 2q s (s − u) 2 −2s 3ϵ 2 + 2ϵ(u − 3) − 3u + 3 +u −18ϵ 2 + 2ϵ(u + 18) − 3(u + 6) + (2ϵ − 3)s 2 +(ϵ − 3)q 5