Branching Fractions of $B$ Meson Decays in Mesogenesis

Under the standard paradigm of inflationary cosmology the Universe is born with equal parts matter and anti-matter. This necessitates a dynamical mechanism of baryogenesis to generate the asymmetry necessary to seed the complex structures observed today. The required primordial baryon asymmetry is inferred to be $Y_{\mathcal{B}}^{\rm meas}\equiv(n_{B}-n_{\bar{B}})/s=\left(8.718\pm 0.004\right)\times 10^{-11}$ from measurements of the Cosmic Microwave Background (CMB) Planck:2015fie ; Planck:2018vyg and light element abundances after Big Bang Nucleosynthesis (BBN) Cyburt:2015mya . Another outstanding mystery is the nature and origin of dark matter; the gravitationally inferred, but yet to be experimentally detected, component of matter which makes up roughly 26% of the energy of the Universe today Planck:2015fie ; Planck:2018vyg . Recently proposed mechanisms of Mesogenesis Elor:2018twp ; Elor:2020tkc ; Elahi:2021jia use Standard Model (SM) meson systems to generate both the baryon asymmetry and the dark matter abundance of the Universe.

Many mechanisms have been proposed since 1967 when Sakharov first introduced the conditions sakharov necessary for baryogenesis; C and CP Violation (CPV), baryon number violation, and departure from thermal equilibrium. Traditional mechanisms of baryogenesis often involve high scales and massive particles leading to dismal prospects for experimental verification¹¹1See Elor:2022hpa for a summary of some interesting new ideas for testing traditional mechanisms.. In contrast, mechanisms of Mesogenesis generate the baryon asymmetry of the Universe by leveraging the CPV within SM meson systems. Excitingly, Mesogenesis is directly testable at hadron colliders and $B$-factories Alonso-Alvarez:2021qfd ; Borsato:2021aum , with additional indirect signals at Kaon and Hyperon factories Alonso-Alvarez:2021oaj ; Goudzovski:2022vbt and even upcoming neutrino experiments such as DUNE and Hyper-Kamiokande MesoDUNE (see also Elor:2022hpa ; Asadi:2022njl ; Barrow:2022gsu ; Fox:2022tzz for summaries). A search for operator through which neutral $B$-Mesogenesis can proceed has already been conducted by the Belle-II collaboration Belle:2021gmc , and a search has also been proposed at LHCb Rodriguez:2021urv .

Production of the baryon asymmetry in the Neutral $B$-Mesogenesis mechanism Elor:2018twp is directly related to the branching fraction of seemingly baryon number violating decays of neutral $B_{s,d}^{0}$ mesons into a light SM baryon and missing energy. Successful baryogenesis in this framework predicts this observable should be $\text{Br}(B_{s,d}^{0}\rightarrow\mathcal{B}^{0}_{\rm SM}+\text{MET})\gtrsim\mathcal{O}(10^{-5})$ to generate the observed baryon asymmetry Elor:2018twp ; Alonso-Alvarez:2021qfd . Charged $B$ Mesogenesis Elahi:2021jia , links the baryon asymmetry to the decays of the $B^{\pm}$ mesons and requires $\text{Br}(B^{+}\rightarrow\mathcal{B}^{+}_{\rm SM}+\text{MET})\gtrsim\mathcal{O}(10^{-6})$. Furthermore, a measurement of the branching fraction as low as $10^{-7}$ could be an indication that a Mesogenesis mechanism is at play.

The nature of the final state SM baryon $\mathcal{B}_{\rm SM}$ in these decays depends on the flavor structure of the operator through which $B$-Mesogenesis proceeds (see Table 1 for a list of all the possible contributing decays). Experimental searches target $B$ meson decay modes to specific final states, for instance the Belle-II collaboration set a limit on the branching fraction for $B_{d}^{0}$ decaying into a neutral $\Lambda$ baryon and missing energy Belle:2021gmc . Therefore, the computation of the exclusive branching fractions of the various possible decay modes is critical for testing $B$-Mesogenesis. Searching for such $B$ meson decays down to the $\text{Br}\sim 10^{-7}$ level will fully test the landscape of different Mesogenesis possibilities.

In this work we leverage the powerful machinery of QCD Light Cone Sum Rules (LCSRs), first introduced in BALITSKY1989509 ; Braun:1988qv ; Chernyak:1990ag , reviewed in Colangelo:2000dp , to compute the branching fractions of the various channels that could generate the baryon asymmetry. The hard scattering amplitude of the $B$ meson to baryon transition is convoluted with the Distribution Amplitude (DA) of the baryon; these are fundamental non perturbative functions that can be interpreted as light-cone wave functions integrated over transverse quark momenta Lepage:1980fj ; RQCD:2019hps . Using our results, we comment on implications for current and future experimental searches in light of existing collider and flavor constraints on the model.

This paper is organized as follows: in Sec. 2 we first review the mechanism of $B$-Mesogenesis. We specifically emphasize the details most useful for an experimentalist eager to conduct such searches. Next, in Sec. 3, we detail the results of the LCSRs derivation of the form factors and branching fraction of the relevant $B$ meson decays. Our main results are presented in Figure 2 and 3 and in Table 1, we further discuss the results and implications for Mesogenesis in Sec. 4. We conclude in Sec. 5 with an outlook on future directions. Appendices  A.1, A.2, and A.3 contain supplementary details on the formalism and derivation.

Here we provide a brief review of the neutral $B$-Mesogenesis mechanism, with particular emphasis on the those ingredients most relevant for experimental searches.

The baryon asymmetry and dark matter abundance are produced through $B$-Mesogenesis as follows: we first assume the late time out of equilibrium production of equal numbers of $B_{s,d}^{0}$, $\bar{B}_{s,d}^{0}$ and $B^{\pm}$ mesons²²2This can be achieved if a scalar field, perhaps related to inflation, comes to dominate the energy density of the Universe at late times. when the temperature of the Universe was $5\,\text{MeV}\lesssim T\lesssim 100\,\text{MeV}$ i.e. after the QCD phase transition but before BBN. The produced $B$ mesons will then undergo SM CP violating processes before decaying into a dark sector state charged under SM baryon number and a SM baryon. The end result is an equal and opposite baryon asymmetry in the dark and the visible sector which is directly proportional to the branching fraction of the exotic decay of the $B$ mesons. In the case of Neutral $B$-Mesogenesis Elor:2018twp , the generated asymmetry will also depend on the charge asymmetry of the $\bar{B}^{0}-B^{0}$ oscillations, and indirectly on a host of other observables (see Alonso-Alvarez:2021qfd for an overview). Achieving the observed baryon asymmetry Alonso-Alvarez:2021qfd , in light of current world averages limiting the CPV in the meson systems, requires an inclusive branching fraction $\text{Br}(B^{0}_{s,d}\rightarrow\psi_{\mathcal{B}}+\mathcal{B}_{\rm SM})\gtrsim 10^{-5}$. Where $\psi_{\mathcal{B}}$ is a dark fermion carrying baryon number $-1$, and $\mathcal{B}_{\rm SM}$ is a neutral SM baryon. Charged $B$ Mesogenesis Elahi:2021jia , leverages CPV in SM meson decays and requires $\text{Br}(B^{+}\rightarrow\mathcal{B}^{+}_{\rm SM}+\psi_{\mathcal{B}})\gtrsim\mathcal{O}(10^{-6})$ to generate the observed baryon asymmetry³³3Note that fully verifying this flavor of Mesogenesis requires future generation experiments, and as such is less testable on shorter time scales than its neutral counterpart.. In both cases the exact nature of the SM baryon depends on the flavor structure of the UV model as will be discussed below. Indeed, various different decay modes can contribute to the generation of the baryon asymmetry.

The seemingly baryon number violating decay of the charged or neutral $B$ meson can generically proceed as follows: a colored scalar mediator⁴⁴4 See Alonso-Alvarez:2019fym for a consistent UV realization of this theory where the heavy colored mediator is identified with a squark of a supersymmetric theory. $Y$ can mediate the decay of a $b$-quark within the $B$ meson to a SM baryon $\mathcal{B}_{\rm SM}$ and dark sector baryon $\psi_{\mathcal{B}}$ (which appears as missing energy in the detector). The Lagrangian for the model, allowed by all the symmetries, is

where all SM quarks are right handed, and “c” indicates a charge conjugation. $i,j,k$ are color indices, and $\alpha,\beta,\gamma$ are flavor indices. Here $Y$ has the SM charge assignment (3,1,-1/3) ⁵⁵5Note that other charge assignments for the mediator are also possible e.g. $(3,1,2/3)$ or $(3,2,-1/6)$ which lead to a variety of different flavor and collider constraints Alonso-Alvarez:2021qfd ; Alonso-Alvarez:2021oaj ), and furthermore could allow the couplings of $\psi_{\mathcal{B}}$ to left handed quarks. For simplicity, we consider only the hypercharge $-1/3$ mediator case in the present work.. The colored mediator $Y$ can be produced at the LHC and thus must have a mass at the TeV scale to be consistent with collider constraints Alonso-Alvarez:2021qfd . Integrating out $Y$ leads to the following effective operator at the low energies:

There are four different flavor combinations that correspond to operators that can mediate the decay of the $b$ quark within the $B^{0}$ meson and thus contribute to the generated baryon asymmetry in the $B$-Mesogenesis framework. Each flavor combination can give rise to two possible effective operators:

where again the quarks are right handed and $u_{\alpha}=u\,,c$ and $d_{\beta}=d\,,s$; giving eight operators in total.

In Table 1 we list the four possible flavorful variations of the operator in Eq. (3) along with the corresponding $B$ meson decay modes, through which the baryon asymmetry can be generated can proceed. Note that the model also allows the decay of $B^{+}$ mesons which, while not directly responsible for the generated baryon asymmetry, serves as an indirect probe of the mechanism. In Figure 1 we show the decays through the $\mathcal{O}_{b,us}$/$\mathcal{O}_{s,ub}$ operator for the $B_{s,d}^{0}$ meson as well as the $B^{+}$ meson. Note that the coefficients $y_{\psi d_{\gamma}}y_{u_{\alpha}d_{\beta}}/M_{Y}^{2}$ associated with the above operators are constrained by a combination of LHC searches and flavor observables (see Alonso-Alvarez:2021qfd for a detailed analysis). For simplicity, where obvious, we will omit the flavor indices on the Wilson coefficient and use refer to it as $y^{2}/M_{Y}^{2}$.

Paramount to confirming Mesogenesis as the mechanism by which Nature chose to generate the baryon asymmetry is the measurement of the various decay modes in Table 1. This is the main result of the present work.

Some final comments on the parameter space are now in order. The mass of the dark baryon $\psi_{\mathcal{B}}$, and thus the corresponding missing energy, is required to be within a window of roughly 1-4 GeV. The reason is as follows: First it must be the case that $m_{\rm\psi_{\mathcal{B}}}<m_{B}-m_{p}\simeq 4.34\,\text{GeV}$, such that the requisite decay of the $B$ meson is kinematically allowed. Second, since the operator Eq. (2) can lead to proton decay, we simply require that any dark sector state charged under baryon number be sufficiently heavy to kinematically forbid such decays and therefore stabilize matter⁶⁶6Neutron star bounds may place slightly more stringent constraints on the mass of dark baryons (see McKeen:2018xwc ) but these are model dependent so we can allow ourselves to be more relaxed.: $m_{\psi_{\mathcal{B}}}>m_{p}-m_{e}\simeq 937.8\text{MeV}$. In summary, the dark baryon should have a mass in the range

corresponding to a range about 1-4 GeV.

Finally, note that $\psi_{\mathcal{B}}$ cannot be the dark matter as it is unstable. Indeed, the operators of Eq. (2) would allow the decay $\psi_{\mathcal{B}}\rightarrow p+e+\bar{\nu}_{e}$ which is kinematically allowed, and would washout the generated asymmetry. To evade washout and stabilize the dark matter additional dark sector states must be introduced. The dark sector dynamics has no implications on the current work and we simply refer the interested reader to Elor:2018twp ; Elahi:2021jia for more details.

Historically, the LCSR technique was developed to explain the mass difference between the $\Sigma$–hyperion and the proton in the weak decay $\Sigma\to p+\gamma$ BALITSKY1989509 . Technically, LCSR combines the QCD sum rules technique with the theory of hard exclusive processes. Intuitively, the short distance expansion is replaced by the expansion in the transverse distance between partons in light-cone coordinates  Braun:1997kw . This allows one to incorporate additional information about QCD correlators related to the approximate conformal symmetry of the theory. The order of the expansion is regulated by the twist of the operators OHRNDORF198226 ; Braun:1989iv . Twist is defined as the difference between the dimension and the spin of an operator; increasing the twist accuracy takes into account higher transverse momentum and higher Fock state contributions Ball:1998fj .

To compute the branching fractions of the two body meson to baryon decays listed in Table 1 via the QCD LCSR technique, one introduces a baryon to vacuum correlator. This object depends on the specific meson interpolating current and on the effective three-quark operator coupled to $\psi_{\mathcal{B}}$. In the recent paper Khodjamirian:2022vta ⁷⁷7Khodjamirian:2022vta appeared while the present work was being prepared. the two body decay $B^{+}\to p+\psi_{\mathcal{B}}$ was obtained using the LCSR method. Our approach reproduces the result of Khodjamirian:2022vta , and we further generalize it to all the remaining decays that could contribute to the production of the baryon asymmetry. Unless explicitly stated otherwise, we will mostly refer to the $B$–meson triplet, $(B_{d},B_{s},B^{+})$, without specifying the quark content, simply as $B$. A brief summary of the formalism is given in Appendix A.1, and we extended it to heavy baryons in Appendix A.2.

The two body decay amplitudes of interest all have the form:

where $u^{c}_{\psi_{\mathcal{B}}}$ is the charge conjugated bispinor of the $\psi_{\mathcal{B}}$ field. Here $P+q$ and $P$ are the momenta of the $B$-meson and $\mathcal{B}_{SM}$ respectively while $y_{\psi d_{\gamma}}y_{u_{\alpha}d_{\beta}}/M_{Y}^{2}$ is the Wilson coefficient associated with the specific operator $\mathcal{O}$ defined in Eq. (2). Lorentz covariance dictates that the hadronic matrix element in Eq. (5) can be parameterized in terms of four independent form factors as follows:

where $\bar{u}_{\mathcal{B}_{R,L}}$ is the chiral spinor associated with the baryon. The calculation of the various $F_{I}$’s appearing in Eq. (6) is analogous to the ones discussed in Khodjamirian:2022vta , and we refer the interested reader to the details in Appendix A.1. Of these four possible structures only two are found to be non-zero for the specific choice of transition operators we consider, namely, $F_{R}$ and $\tilde{F}_{L}$ — a consequence of taking only $R-$handed fields in Eq. (3). The different flavor structures produce three distinct forms of $F_{R}$ and a single form $\tilde{F}_{L}$. All the transitions in Table 1 are calculated using the leading twist-3 DAs:

Where in Eqs. (7–10) $s(\alpha)=[m_{b}^{2}-\alpha q^{2}+\alpha(1-\alpha)m_{\mathcal{B}_{\rm SM}}^{2}]/(1-\alpha)$, and

Here $s_{0}^{B}$ and $M^{2}$ are the effective threshold and the Borel parameter respectively. They correspond to “internal” parameters of the LCSR description that are to be fixed. The two functions appearing in the definitions of the sum rules above are the, once integrated, leading twist-3 DAs of for the specific baryon, given by RQCD:2019hps :

The functions $V_{1}$, $A_{1}$ and $T_{1}$ are defined in Eqs. (1-9) of RQCD:2019hps or in Eqs. (6-9) of Anikin:2013aka . The definitions of the sum rules for charmed operators are not modified however subtleties are involved in the DAs of heavy–baryons; a dedicated discussion is given in A.2. From Eqs. (5–6) one can recover a convenient expression for the two-body branching ratio mediated by a single operator:

where $\lambda$ is the Källen function. The appropriate $R_{I}$ and $\widetilde{L}_{I}$ can be read off the corresponding entry in Table 1.

In this work we have used LCSRs to compute the exclusive branching fractions for each decay mode in Table 1 that could generate the baryon asymmetry in the Mesogenesis framework. Our results for the maximum possible branching fraction (in light of LHC constraints) are presented in Figure 2 and Figure 3 for $B$ meson decays to light and heavy (charmed) baryons respectively. In what follows we will discuss some interesting features of these results, before moving on to comment on implications for testing $B$-Mesogenesis.

In this work we have presented the results of a LCSR calculation of the form factors and branching fractions for the seemingly baryon number violating decays of $B_{s,d}^{0}$ and $B^{+}$ mesons into a dark sector baryon $\psi_{\mathcal{B}}$ and various SM baryon final states. Figures 2 and 3 show the results for the maximum possible branching fractions (in light of LHC constraints on the Wilson coefficients) and associated uncertainties for each of the four flavorful variation on the operators that could produce the baryon asymmetry. These results will serve to guide experimental searches for these exclusive decay modes. Several interesting future directions exist, which we now comment briefly upon.

In the present work we considered operators involving only right handed fermions as would arise in the model defined by the Lagrangian in Eq. (1) i.e. where the colored mediator is a scalar and has SM hyper-charge assignment $-1/3$. However, other UV completions do exist corresponding to different charge assignments for the mediator (see the models in Alonso-Alvarez:2021qfd ). For instance, the mediator could be a doublet under $SU(2)_{L}$ which would give rise to four fermion operators containing left handed quarks. The branching fractions arising from these operators would lead to non-vanishing contributions from the $F_{L}^{\mathcal{O}}(q^{2})$ and $\widetilde{F}_{R}^{\mathcal{O}}(q^{2})$ form factors. The computation of these form factors is a trivial extension and we leave it to future work.

The model of Eq. (1) would also give rise to the decay of a baryon into light mesons and missing energy e.g. $\Lambda_{b}\rightarrow\bar{\psi}_{\mathcal{B}}+\pi^{0}$ through the $\mathcal{O}_{b,ud}$ and $\mathcal{O}_{d,ub}$ (see Elor:2018twp for other decay modes). While the branching fraction of such decays do not directly feed into the Boltzmann equations that track the generated baryon asymmetry, they do serve as an indirect probe of $B$-Mesogenesis. Therefore, it would be interesting for experimental searches to target these decays as well. Indeed, the proposed search at LHCb Rodriguez:2021urv would be capable of measuring these $b$-flavored baryon decays. As such it is also a worthy peruse to also compute the predicted branching fractions for these decays. To do so one can once again use the LCSR machinery. The starting point would be a three point correlator and we leave the details of this interesting technical pursuit to future work.

Another application of the LCSR technique would to calculate the branching fractions of the $B$ meson decay’s in Table 1 where multiple final state light mesons are radiated e.g. $\pi^{0}$.