The Rate for Bb̄ Production Accompanied by a Single Pion

We study the rate for the production of BBπ, where the sign of the charged pion tags the flavor content of the neutral B meson. We estimate this branching ratio, employing the heavy meson chiral effective theory. We find that at center of mass energy of approximately 12 GeV, a B meson pair should be produced as often with and without an accompanying charged pion. We also calculate two pion production at this center of mass energy, and find that it is negligible, as is the rate for rho production. We consider the implications for CP violation studies. MIT-CTP 2214 May 1993 We study an alternative method for tagging neutral B mesons for the purpose of CP violation studies which can be used at a symmetric collider. This talk is based on ref. [1]We consider Yamamoto’s suggestion [2] to study B mesons produced in conjunction with a single charged pion at or near an Υ resonance, so that the sign of the charged pion tags the single neutral B meson as B or B̄. Neutral B mesons can also be used, since they decay immediately via photon emission into pseudoscalars, before weak mixing or decays have occurred. This method should provide a simple and efficient tag of the neutral B meson: in addition to the decay products of the neutral B, one only needs to detect the additional soft pion. The charged B is then tagged by the invariant mass of the missing four momentum, so it need not be reconstructed. Unlike many conventional proposals for the study of CP violation in the neutral B system, essentially all the events are tagged. The utility of this method depends on the event rate. We calculate this rate, employing the heavy meson chiral effective theory, in order to evaluate the potential of Yamamoto’s proposal. We find that Yamamoto’s proposal could prove a competitive method for tagging neutral B’s for the study of CP violation at a symmetric collider at slightly higher center of mass energy, about 12GeV, where nevertheless multipion final states still have a negligible branching fraction. We first construct the effective lagrangian for BB and BBπ production. Since the kinetic energies of the B and B̄ are on the order of the QCD scale, both the B and B̄ mesons have essentially timelike four velocities in the center of mass frame: v ≈ v ≈ (1, 0, 0, 0). We couple a source S to the b-quark current as bv′S γμbv (i.e., with the heavy quark spins coupled to the spin of the source). When we match onto the heavy meson theory, we can couple the source to the mesons as B(v)SγμB(v). This gives the same matrix elements as if we had coupled a source to heavy quarks, and then evaluated the heavy quark matrix elements. The lagrangian applicable to low-energy production of B and B̄ meson is Leff =− itr{Ba(v)v∂μBa(v)} − itr{Ba(v)v∂μBa(v)} + gtr{Ba(v)Bb(v)Abaγνγ5}+ gtr{Ba(v)Bb(v)Abaγνγ5} + LS , LS = −iλ 2 Str{γμBa(v) ↔ D abγνBb(v)} + λgStr{γμBa(v)Aabγνγ5Bb(v)}. (1) Here D = ∂ + V is the chiral covariant derivative incorporating the pion fields. As usual, a factor of √ MB has been absorbed into the heavy meson fields along with the positiondependent phase corresponding to the momentum of the heavy quark (so that a derivative acting on these fields only gives a factor of the residual momentum), in order to suppress the appearance of the heavy quark mass and emphasize the heavy quark symmetry. Because this is the low energy theory, no large momentum transfers are permitted. At higher energies, the appropriate lagrangian would be the heavy meson lagrangian with velocities v 6= v. The result for two meson production matches smoothly, as the difference in residual momenta in the amplitude gets replaced by the difference in heavy meson velocities. The kinetic and axial coupling terms for theB mesons have been discussed previously and result from the straightforward application of heavy quark and chiral effective field theories [3]. LS is the new term and follows from the assumptions described above. Note that with the trace the heavy quark spin labels are coupled to Sγμ. We now summarize the result of calculating the process of interest. From the lagrangian (1) we see that two types of diagrams contribute to BBπ production. The pion can be produced “indirectly” by being emitted from a virtual B meson through the heavy-meson axial coupling. Or, the pion can be produced “directly”, together with the B mesons at a single vertex. This diagram comes from the contact term in the lagrangian in which the source couples directly to the B meson and axial fields. The direct contribution is much the smaller of the two, because it is higher order in the heavy quark mass expansion [1], so the discussion concentrates on the indirect contribution. In order to compare BBπ with BB̄ as sources of neutral B mesons we normalize the BBπ cross sections by dividing them by the cross section for ee → neutral B mesons, σ0. The density of states for a final state of B mesons and a pion simplifies for nonrelativistic B mesons to D = 1 64π3 MB|∆k| |k| dEπ d(cos θ) = 1 64π3 M 3/2 B pπ(r − Eπ)dEπ d(cos θ). (2) where r is the center of mass energy that remains after supplying the rest mass energies of the heavy mesons. As discussed in the last section, we treat the B mesons as nonrelativistic in the laboratory frame, working to lowest nonvanishing order in the heavy-meson threemomenta. Implicit in our approximations is the realization that the kinetic energy is fairly evenly shared between the pion and the heavy mesons.Therefore, when we work at energies where the heavy mesons are nonrelativistic, there is a hierarchy of energy scales, TB , Eπ, pπ (∝ M B) ≪ |kB| (∝ M 1/2 B ) ≪ MB, (3) where TB is the kinetic energy of the B mesons. Since the dimensionful quantities that compensate the different powers of MB here are of order r, we are essentially working to lowest order in r/MB. Taking sums and differences of the B meson momenta, this hierarchy can be reexpressed as k 0 ,∆k, |k| (∝ M B) ≪ |∆k| (∝ M 1/2 B ) ≪ MB. (4) In calculating a given amplitude, we retain only the leading term according to this hierarchy. Specifically, we drop k 0 and ∆k compared to MB, incurring errors of order r/MB . We also drop |k| compared to |∆k| and |∆k| compared to MB. This would seem to mean dropping terms of order √ r/MB . However, once an amplitude is squared, averaged/summed over Table 1. The ratios σ(ee → BBπ)/σ(ee → BB̄) expressed as percentages. √ s(MeV) R PP R i PV R i VV R i R PV R d VV R d R 10865 0.12 0.12 0.02 0.26 0.03 0.01 0.04 0.3