Sum rule for the partial decay rates of bottom hadrons based on the dynamical supersymmetry of the $\bar s$ quark and the $ud$ diquark

We investigate the weak decays of $\bar B_{s}^{0}$ and $\Lambda_{b}$ to charm hadrons based on the dynamical supersymmetry between the $\bar s$ quark and the $ud$ diquark. We derive a new sum rule relating the decay rates of the processes $\bar B_{s}^{0} \to D_{s}^{+} P^{-}$, $\bar B_{s}^{0} \to D_{s}^{*+} P^{-}$ and $\Lambda_{b} \to \Lambda_{c} P^{-}$, where $P^{-}$ is a negatively charged meson, such as $\pi^{-}$ and $K^{-}$. It is found that the observed decay rates satisfy the sum rule very well. This implies that the supersymmetry between the $\bar s$ quark and the $ud$ diquark is also seen in the wavefunctions of the heavy hadrons and suggests that the $ud$ diquark can be regarded as a valid effective constituent for heavy hadrons.

Finding fundamental correlations is a clue to understand the structure of strongly interacting systems.In electron systems the Cooper pair is a key ingredient and its condensation leads to superconductivity [1].In nuclear physics, the nucleon pair correlation is an important object to describe the nuclear structure in the interacting boson model [2,3].Also the two-neutron correlation can be a hint to understand the structure of unstable light nuclei [4,5].In hadron physics, the two-quark correlation called diquark has been already mentioned in Ref. [6] when quarks were proposed, and it can be used as an effective constituent in many-body systems.The importance of the diquark correlation in hadronic systems was discussed phenomenologically in [7,8].It is also known that diquark condensation induces color superconductivity at high density quark systems [9,10].
The diquark is a colored object that cannot be observed at low energies as an isolated particle due to color confinement.Its existence, however, is expected as a constituent inside hadrons similar to the constituent quark, which is a quasi-particle of the fundamental particles and is regarded as an effective building block of hadrons.The role of the diquark in the baryon structure has been extensively investigated by diquark pictures, in which baryons are composed of a diquark and a constituent quark [11][12][13][14][15][16][17][18][19][20][21].Light scalar mesons may be described by a configuration of diquark and antidiquark [22][23][24][25] and their decay properties are reproduced reasonably well [24,25].Lattice QCD calculations also have suggested attractive diquark correlations [26,27].
Recently a dynamical supersymmetry between the ud scalar diquark and the s constituent quark has been proposed in Ref. [28].Both objects have the same color charge 3 and same electric charge.Phenomenologically they are known to have a similar mass around 500 MeV.This is a supersymmetry between a boson and a fermion, but not a symmetry for fundamental particles, rather a dynamical symmetry for quasi particles which are regarded as effective elements of the dynamics like the constituent quarks.If this supersymmetry is realized univer-sally in hadronic systems, one may conclude the existence of the diquark inside hadrons as seen for the constituent quarks that were established from the symmetry arguments of the light hadrons.Historically such a dynamical supersymmetry was introduced first by Miyazawa for mesons and baryons [29] and later applied to the light hadron spectra [30].One can also use holographic QCD to motivate a supersymmetry connecting baryons and mesons [31][32][33].
The supersymmetry among the scalar ud diquark and the s quark works rather well for the hadron spectra [28].For instance, combining a bottom quark b with the ud diquark and the s quark, we have three hadrons ( B0 s , B * 0 s , Λ b ), which are a spin 0 pseudoscalar meson, a spin 1 vector meson and a spin 1/2 baryon.The observed masses are found as (5367, 5415, 5620) in units of MeV, respectively.Similarly for the charm quark c, we have (D + s , D * + s , Λ c ) and these masses are (1968,2112,2286) in units of MeV.The symmetry breaking on these hadron masses is about 300 MeV, which is as good as the flavor symmetry breaking stemming from the mass difference among the light constituent quarks.
The symmetry among these hadrons may be based on a similar mass for the ud diquark and the s constituent quark.Color electric interactions play the main role for confinement and are mainly determined by the masses and color of the interacting particles.Because ud diquark and s quark have same color and a similar mass, the interactions of the heavy quark with the ud diquark and the s quark must be very similar.Possible sources of symmetry breaking are the mass difference between the ud diquark and the s quark and spin dependent forces such as the spin-spin interaction between quarks.The former is responsible for the mass difference of the mesons and the baryon, while the latter induces the mass difference between pseudoscalar and vector mesons.
The purpose of this article is to investigate whether this supersymmetry is realized also in the wavefunctions in heavy hadrons.The symmetry of the wavefunctions can be seen in the decay of the heavy hadrons, where the decay rates are expressed by the matrix elements of the parent and daughter particles with the wavefunctions of the initial and final states.For this purpose, we compare the weak decays of B0 s into D + s and D * + s with those of Λ b into Λ c .
From now on, let us call the s quark and the ud diquark collectively as ψ and consider a spin doublet s and a scalar ud to form a triplet ψ of the V(3) supersymmetry introduced by Miyazawa [29].We denote hadrons composed of the triplet ψ and a heavy quark h collectively as ψh.This yields ψb = ( B0 s , B * 0 s , Λ b ) for the bottom hadrons and ψc = (D + s , D * + s , Λ c ) for the charm hadrons.Hadrons ψh form a sextet of V(3) ⊗ SU (2) where SU(2) denotes the spin symmetry of the heavy quark.
Let us first consider pionic decay ψb → π − ψc.This decay is induced by transition b → cW − and then either the weak boson W − turns into a pion π − or W − is absorbed into ψ.In the former process, the s quark or the ud diquark is a spectator in the weak decay, and thus the weak transition of the b quark commonly contributes to the decays of B0 s and Λ b and the wavefunctions of the s quark in B0 s and of the ud diquark in Λ b are responsible for the difference of their decay rates.The latter process involves two particles in the initial state.Because such a two-body process is known to be strongly suppressed compared to one-body processes [35], we can safely neglect it.Therefore, the decay process ψb → π − ψc is good to investigate the supersymmetry in the s and ud wavefunctions.This situation is also true for kaonic, ρ mesonic and semileptonic decays.
2 0 1 Y i v T y f e + 9 7 + W L X 5 R T I p X n / d 7 a r t X v 3 X + w 8 7 D x 6 P G T p 7 t 7 + 8 / O J C 8 E w q e I U y 4 u I i g x J Q y f K q I o v s g F h l l E 8 X l 0 8 9 7 y 5 0 M s J O H s s x r n + C q D A 0 Y S g q A y 0 P X + 9 p 8 w w g P r X 2 v f a z / q x / W g T u p 3 C 9 P N j e W b t 9 b a q c t / g u 4 n c A = = < / l a t e x i t > More systematically, we show the relevant diagrams of the weak decays ψb → P ψc in Fig. 1 by making use of the topological classification of Ref. [36].Based on the supersymmetry we extend this classification from mesons to baryons and use it for both.In the "external Wemission" diagram (a), the weak decay is induced by the transition of the b quark to the c quark with emitting a meson P directly from the W boson.The "horizontal W -loop" diagram (b) contains charm quark pair creation and contributes to the D, D s and D * s mesonic decays.(The D mesonic decay is doubly Cabibbo suppressed.)Also in this diagram, ψ is a spectator.Diagrams (c) and (d) contribute differently to the B0 s and Λ b .These two diagrams, however, contain two-body processes.There are two more diagrams in the classification of Ref. [36]: the internal W -emission and the W -annihilation diagrams.These diagrams are irrelevant for the present calculation, because the former diagram does not contain D s , D * s nor Λ c in the final state and the latter is only relevant for a charged meson decay.In order to explore the diquark ansatz, we did not consider in Fig. 1 the processes in which the ud diquark falls apart during the weak decay.
The decays of ψb can be calculated from diagrams (a) and (b), in which ψ can be regarded as a spectator of the decay process.The effective Hamiltonian that we consider here for the transition b to c reads where P µ is the weak current for each weak process, such as P µ = ∂ µ π † for the pionic decay, P µ = ρ µ † for the ρ mesonic decay, and P µ = ¯ γ µ (1 − γ 5 )ν for the leptonic decay.The effective coupling strengths A and B in the current J µ h depend on the weak process specified by P µ , but do not depend on whether the spectator is a ud diquark or an s quark.Here the supersymmetry enters.
The decay rate of a bottom hadron ψb (with mass M ψb ) to a charm hadron ψc (mass M ψc ) by emitting particle(s) P is calculated as with the phase space element of the final states Spin average of the initial state and spin summation of the final states are taken.The matrix elements M µ h and M µ P are defined by The latter matrix element M µ P describes the particle emission during the transition and is common for the process ψb → P ψc, irrespective of the choice of the triplet member from ψ and irrespective of the spin orientations of the heavy quarks.For two-body decays in the rest frame, the decay rate ( 2) is written as with the center of mass momentum of the final states p where m denotes the mass of particle P .Thanks to the symmetry of the masses in the same multiplet, the mass of the decaying hadron and the phase space of the final states are also the same in each decay mode specified by P .
Because the s quark and the ud diquark can be regarded as spectators in the transition, the hadronic matrix element M µ h can be evaluated in terms of the heavy quark states.Specifying the quark spins, we write the matrix element of the hadronic current for the bottom and charm quarks with spin α and β as Under the assumption that the wavefunctions are the same due to the supersymmetry, the matrix elements M µ αβ appear commonly in the calculations of each decay mode.
The spin of the heavy baryon Λ h and the heavy quark coincide thanks to the spinless diquark.Thus the spin wavefunctions of the heavy baryon spin doublet Λ (1) h and Λ (2) h are given by (ud)h (1) and (ud)h (2) , respectively.For the decay rate of an unpolarized Λ b to Λ c , we take a spin average of the initial Λ b and sum up all of the spin states of the final Λ c : ). ( 6) The spin configuration of a pseudoscalar meson composed of a heavy quark h and an s quark is given by 1 √ 2 (s (1) h (1) + s(2) h (2) ).For the weak decay of the pseudoscalar B0 s , the spin of the s quark does not change in the decay as it is a spectator.The hadronic part of the decay amplitude of B0 s to the pseudoscalar D + s is calculated as where we have used the orthogonality of the states having different spin for the s quark.This implies that the weak decay of B0 s to D + s has only the spin non-flip amplitude A. The square of the amplitude is given by The spin configurations for the vector mesons D * + s with s z = +1, 0, −1 are given by s(2) h (1) , 1 √ 2 (s (1) h (1) − s(2) h (2) ) and s(1) h (2) , respectively.In analogy to Eq. ( 7), we calculate the decay amplitudes of the pseudoscalar B0 s to the vector D * + s .Summing up the spin of D * + s in the final state, we obtain The heavy hadrons ( B0 s , Λ b ) and (D + s , D * + s , Λ c ) are in the same multiplets, respectively, and the V(3) supersymmetry demands the kinematical factors of these decays to be the same.In addition, if the wavefunctions of the heavy hadrons are the same in each multiplet, the hadronic matrix elements can be calculated commonly using the amplitude (5).Under these conditions, we find that the sum of Eqs. ( 8) and ( 9) coincides with Eq. ( 6).This implies that we have a sum rule for the decay probabilities of B0 s and Λ b as With this sum rule, we can check the symmetry of the wavefunctions for the heavy hadrons ψh.We will examine whether the sum rule (10) agrees with experimental observations and we will derive predictions for partial decay rates that have not been measured yet.The experimental data collected by the Particle Data Group (PDG) [34] are summarized in Table I, where the partial decay rates are evaluated in units of 10 9 /s using the central values of the mean life of the decaying particle and the branching fraction of the corresponding decay mode.For B0 s , we use the average of the mean lives of the heavy and light CP eigenstates [37].Although the branching fractions for B 0 s are provided by the PDG, we use them for B0 s since CP violation is very small.First of all, it is very interesting to note that for each decay mode the partial decay rates of the B0 s meson and the Λ b baryon have the same order of magnitude.This can be interpreted already as a consequence of the supersymmetry between the s quark and the ud diquark.
For the decays ψb → π − ψc, the sum of decay rates of .Therefore we consider the kaonic decay fractions for B0 s as upper limits.The sum of the decay rates of B0 s to K ∓ D ± s and K ∓ D * ± s is found to be (2.37 ± 0.26) × 10 8 /s, while the decay rate of Λ b to K − Λ c is observed as (2.44 ± 0.020) × 10 8 /s.The sum rule may work well.
For the ρ, D − and D s mesonic decays, one of the branching fractions has not been measured yet.Assuming the sum rule (10), we can predict the partial decay rates of these missing decay modes.The predicted values are shown as values in square brackets in Table I.It will be very interesting to see if future measurements of the branching rates of presently missing decays will confirm the validity of our sum rule (10).The decay branching fraction of B0 s → D − s D * + s + D * − s D + s has been observed as (1.39 ± 0.17) × 10 −2 , which corresponds to (9.17 s Λ c as (14.1 ± 1.9) × 10 9 /s.For the semileptonic decays, exclusive measurements exist only for the baryon case.For the B0 s decays they have not been performed yet.But the three inclusive decay modes collected in Table I have a similar magnitude to the baryon decay rate.This may be a consequence of the supersymmetry between s and ud.In order to confirm the sum rule for the semileptonic decays, exclusive observations are strongly desired.
It is interesting to estimate the magnitude of symmetry breaking of the sum rule (10) coming from the kinematical factor K of Eq. ( 4).This factor is a function of M ψb , M ψc and m.The observed heavy hadron masses deviate from the symmetric mass M h .The latter is given by a spin average M b = (M B0 s + 3M B * 0 s + 2M Λ b )/6 and similar for the charm sector.Numerically one obtains M b = 5475 MeV and M c = 2146 MeV.The deviation of the kinematical factor K from the symmetry limit can be evaluated as where δm b and δm c are the deviations of the bottom and charm hadron masses from their symmetric mass, respectively.Evaluating Eq. ( 11) using the observed masses, we find that the deviation of the kinematical factor from the symmetric limit is 5 % at most for these decay modes.Therefore, the fact that the sum rule works very well for the observed weak decay processes implies that the wavefunctions for the heavy hadrons ψh have also good symmetry stemming from the supersymmetry between the s constituent quark and the ud diquark.
In conclusion, based on the supersymmetry between the s quark and the ud scalar diquark, we have derived a sum rule for the weak transition rates of the bottom B0 s meson and Λ b baryon to charm hadrons.The sum rule is well satisfied by the observed weak decays for pionic and kaonic decay modes.This implies that the ud scalar diquark behaves as a quasi-particles inside of the Λ b and s and Λ b and the corresponding branching fractions and rates.The partial decay rates Γi are shown in units of 10 9 /s and are evaluated using the central values of the observed mean life and branching fraction.The values of the partial decay rates in square brackets are predictions based on the sum rule (10).The value of the observed mean life of B0 s is (1.515±0.004)×10−12 s, which is the average mean life of the heavy and light CP eigenstates, and that of Λ b is (1.471 ± 0.009) × 10 −12 s.The charge of the kaonic decay of B0 s cannot be discriminated due to the B0 s -B 0 s mixing.The data are taken from [34].The original experiments are found in Refs.[38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56].Λ c baryons like the s quark in heavy mesons and can be a clue for the nature of the diquark.We have also predicted from the sum rule several weak decay rates of B0 s and Λ b that have not been observed yet.If these missing decay modes are observed in future experiments, they can give us further support for the importance of the diquark correlation.
This work was partially supported by the Grant-in-Aid FIG.1.Relevant diagrams for the weak decay of ψb based on the topological classification given in Ref.[36].
B0 s → π − D + s and → π − D * + s yields (3.3 ± 0.4) × 10 9 /s, while the decay rate Λ b → π − Λ c is (3.3 ± 0.3) × 10 9 /s.Thus, the sum rule (10) is satisfied extremely well.Next, we discuss the sum rule (10) for ψb → K − ψc.Unfortunately, present experiments provide only the branching fraction of the B 0 s → K ± D ( * )∓ s decay, i.e. one cannot discriminate between B 0 s → K + D ( * )− s and B 0 s → K − D ( * )+ s ± 1.12) × 10 9 /s for the partial decay rate.Using the partial decay rate of B0 s → D − s D * + s obtained from the sum rule, we estimate the partial decay rate of B0 s → D − s D * + s as (4.6 ± 0.8) × 10 9 /s.Using the sum rule again, we can predict the partial decay rate of Λ b → D −

TABLE I .
Weak decay modes of B0