We calculate cross sections of a pair $B_c$ meson production on the basis of two-photon mechanism from electron-positron annihilation. We investigate the production cross sections in nonrelativistic approximation and with the account of relativistic corrections. Relativistic production amplitudes of S-wave pair pseudoscalar, vector and pseudoscalar+vector $B_c$-mesons are constructed on the basis of relativistic quark model. Numerical values of the production cross sections are obtained at different center-of-mass energies. The comparison of one-photon and two-photon annihilation contributions is presented.
I review recent progress in the theory of gamma gamma annihilation into meson or baryon pairs at large energy, and of the process gamma* gamma* -> pi0 at large photon virtuality.
Software package for Monte-Carlo simulation of e+e- exclusive annihilation channels written in the C++ language for Linux/Solaris platforms has been developed. It incorporates matrix elements for several mechanisms of multipion production in a model of consequent two and three-body resonance decays. Possible charge states of intermediate and final particles are accounted automatically under the assumption of isospin conservation. Interference effects can be taken into acccount. Package structure allows adding new matrix elements written in a gauge-invariant form.
Exclusive proton-antiproton annihilation into two photons at large s (~10 GeV^2) and |t|,|u| ~ s can be described by a generalized parton picture analogous to the 'soft mechanism' in wide-angle real Compton scattering. The two photons are emitted in the annihilation of a single fast quark and antiquark. The matrix element describing the transition of the p-pbar system to a q-qbar pair can be related to the timelike proton elastic form factors as well as to the quark/antiquark distributions measured in inclusive deep-inelastic scattering. The reaction could be studied with the proposed 1.5-15 G…
We analyze the exclusive charmonium $J/ψ+η_c$ pair production in $e^+e^-$ annihilation using the nonfactorized perturbative QCD and the light-front quark model(LFQM) that goes beyond the peaking approximation. We effectively include all orders of higher twist terms in the leading order of QCD coupling constant and compare our nonfactorized analysis with the usual factorized analysis in the calculation of the cross section. We also calculate the quark distribution amplitudes, the Gegenbauer moments, and the decay constants for $J/ψ$ and $η_c$ mesons using our LFQM. Our nonfactorized result enha…
A completely relativistic quark model in the Bethe-Salpter framework is employed to calculate the exclusive production ratio of the neutral over charged Kaon pair in $e^+e^-$ annihilation continuum region for center of mass energies smaller than the $J/Ψ$ mass. The valence quark charge plays the key rôle. The cancellation of the diagrams for the same charge case (in $K_S + K_L$) and the non-cancellation of the diagrams for the different charge case (in $K^-+K^+$) lead to the ratio as $(m_s-m_d)^2/M_{Kaon}^2 \sim 1/10$.
Order alpha^2 contribution to the orthopositronium decay rate due to one-photon virtual annihilation is found to be delta Gamma = (alpha/pi)^2 (pi^2 ln(alpha) - 0.8622(9))Gamma_LO.
We study the exclusive double-photon annihilation processes, $e^+ e^- \to γγ^\ast\to γV^0$ and $e^+ e^- \to γ^\ast γ^\ast \to V^0_a V^0_b,$ where the $V^0_i$ is a neutral vector meson produced in the forward kinematical region: $s \gg -t$ and $-t \gg Λ_{\rm QCD}^2$. We show how the differential cross sections $\frac{dσ}{dt}$, as predicted by QCD, have additional falloff in the momentum transfer squared $t$ due to the QCD compositeness of the hadrons, consistent with the leading-twist fixed-$θ_{\rm CM} $ scaling laws. However, even though they are exclusive channels and not associated with the …
We report on a recent work proposing measurements of the deeply virtual Compton amplitude (DVCS) $γ^* \to h \bar h γ$ in the timelike kinematic domain which is accessible at electron-positron colliders via the radiative annihilation process $e^+ e^- \to h \bar h γ$.
We calculate the exclusive cross sections for e^+ e^- annihilation into two charmonium states through a virtual photon. Purely electromagnetic contributions are surprisingly large, changing the cross sections by as much as 21%. The predicted cross section for J/psi + eta_c is about an order of magnitude smaller than a recent measurement by the BELLE Collaboration, although part of the discrepancy can be attributed to large relativistic corrections. The cross sections for S-wave + P-wave, P-wave + P-wave, and S-wave + D-wave charmonium states are also calculated. It may be possible to discover …
The exclusive production of pairs of vector mesons with J^{PC} = 1^{--} in e^+ e^- collisions can proceed through e^+ e^- annihilation into two virtual photons. At energies much greater than the meson masses, the cross section is dominated by the independent fragmentation of the virtual photons into the vector mesons. The fragmentation approximation is used to calculate the cross sections and angular distributions for pairs of vector mesons that can be produced at the B factories. The predicted cross sections for rho^0 + rho^0 and rho^0 + phi production agree with recent measurements by the Ba…
We compute the cross section for exclusive production of a photon associated with a heavy quarkonium H of charge-conjugation parity C=+1 from e+ e- annihilation into a virtual photon at the center-of-momentum energy root-s=10.58 GeV. The NRQCD factorization formulas for the differential and total cross sections are obtained at leading order in the strong coupling and in the relative velocity of the heavy quark in the quarkonium rest frame. The predicted cross sections for the S-wave spin-singlet cases are about 80 fb and 50 fb for H = eta_c and eta_c(2S), respectively. Among P-wave spin-triple…
Exclusive vector meson production and deeply virtual Compton scattering are ideally suited reactions for studying the structure of the proton and the transition from soft to hard processes. The main experimental data obtained at HERA are summarised and presented in the light of QCD approaches.
We study the factorization of the dijet cross section in $e^+ e^-$ annihilation using the generalized exclusive jet algorithm which includes the cone-type, the JADE, the $k_T$, the anti-$k_T$ and the Cambridge/Aachen jet algorithms as special cases. In order to probe the characteristics of the jet algorithms in a unified way, we consider the generalized $k_T$ jet algorithm with an arbitrary weight of the energies, in which various types of the $k_T$-type algorithms are included for specific values of the parameter. We show that the jet algorithm respects the factorization property for the para…
In reference to the recently observed high Q^2, large x events in deep-inelastic positron-proton scattering at HERA, various leptoquark and supersymmetric scenarios are discussed. We study the impact of virtual leptoquark or R-parity breaking squark exchange as well as generic contact interaction on the production of quark-antiquark pairs in $e^+e^-$ annihilation, in particular at LEP2.
In the talk we present the Coherent Exclusive Exponentiation (CEEX) which is implemented in the KK MC event generator for the process e+e- to f bar f +n gamma, f=mu,tau,d,u,s,c,b for center of mass energies from tau lepton threshold to 1TeV, that is for LEP1, LEP2, SLC, future Linear Colliders, b,c,tau-factories etc. We will attempt a short discussion of the theoretical concepts necessary in our approach, in particular the relations between the rigorous calculation of spin amplitudes (perturbation expansion), phase space parametrisation and exponentiation. In CEEX effects due to photon emissio…
We present comparisons of the fully differential exact virtual correction to the important single hard bremsstrahlung process in e+e- annihilation at high energies, which is essential for precision studies of the Standard Model from 1 GeV to 1 TeV, as calculated by two completely independent methods and groups. We show that the two sets of results are in excellent agreement. Phenomenological implications are discussed.
Neutrino scattering on nucleons in the regime of deeply virtual kinematics is studied both in the charged and the neutral electroweak sectors using a formalism developed by Blumlein, Robaschik, Geyer and Collaborators for the analysis of the Virtual Compton amplitude in the generalized Bjorken region. We discuss the structure of the leading twist amplitudes of the process.
A model based on a dipole Pomeron framework for vector meson exclusive photoproduction by real and virtual photons shows very good agreement with the experimental data. The model does not violate unitarity constrains and describes in a universal manner all the available data for $ρ$, $ω$, $ϕ$, $J/ψ$ and $Υ$ vector meson photoproduction in the region of energies $1.7\le W \le 250 GeV$ and photon virtualities $0 \le Q^2 \le 35 GeV^2$.
We have compared the virtual corrections to electron-positron annihilation to fermion pairs with single hard bremsstrahlung as calculated by S. Jadach, M. Melles, B.F.L. Ward and S.A. Yost to several other expressions. The most recent of these comparisons is to the leptonic tensor calculated by J.H. Kuhn and G. Rodrigo for radiative return. Agreement is found to within $10^{-5}$ or better, as a fraction of the Born cross section.
Recently dispersion relations have been applied to hard exclusive processes such as deeply virtual Compton scattering, and a holographic principle was proposed that maps out the generalized parton distributions entering the soft matrix elements for the processes from their values on a given kinematical ridge. We examine possible pitfalls associated with the implicit, direct identification in this approach of the physical hadronic states with colored partons, and suggest an improved treatment of this assumption.
We calculate the contributions from QED processes involving two virtual photons to the cross sections for e^+ e^- annihilation into two charmonium states with the same C-parity. Generically, the cross sections are three orders of magnitude smaller than those for charmonia with opposite C-parity because they are suppressed by a factor of alpha^2/alpha_s^2. However, if both charmonia have quantum numbers J^{PC} = 1^{--}, then there is a contribution to the cross section that involves the fragmentation of each photon into a charmonium. The fragmentation contribution is enhanced by powers of E_{be…
We outline the proofs of the factorization theorems for exclusive two-body charmonium production in B-meson decay and e^+e^- annihilation to all orders in perturbation theory in quantum chromodynamics. We find that factorized expressions hold up to corrections of order m_c/m_b in B-meson decay and corrections of order m_c^2/s in e^+e^- annihilation, where m_c is the charm-quark mass, m_b is the bottom-quark mass, and root-s is the e^+e^- center-of-momentum energy.
We investigate the possibility to constrain the pion distribution amplitude from the gamma* gamma* -> pi transition. For a surprisingly large range in the two photon virtualities we find that the transition form factor is essentially independent of the distribution amplitude. This in turn entails a parameter-free prediction of QCD. The gamma* gamma* -> eta, eta' form factors are also briefly discussed. We estimate that experimental studies might be feasible at the existing e+ e- experiments BaBar, Belle, and CLEO.
Decays into two charged pions of the f0(1370) are the main source of an isolated structure localized between 1.2 and 1.5 GeV in the two charged pions mass spectrum measured in pp Central Exclusive Production (CEP) at 200 GeV at very low four momentum transfer ltl by the STAR experiment. These data confirm in the two charged pions decay channel the existence of the f0(1370) as an isolated well identified structure previously observed in K+K-, KsKs, 4 charged pions, two charged and two neutral pions and 4 neutral pions decays measured in antiproton annihilations at rest. The ensemble of these da…
We develop a framework to establish benchmarks for machine learning and deep neural networks analyses of exclusive scattering cross sections (FemtoNet). Within this framework we present an extraction of Compton form factors for deeply virtual Compton scattering from an unpolarized proton target. Critical to this effort is a study of the effects of physics constraint built into machine learning (ML) algorithms. We use the Bethe-Heitler process, which is the QED radiative background to deeply virtual Compton scattering, to test our ML models and, in particular, their ability to generalize inform…
We perform a systematic analysis of exclusive hadronic channels in e+e- collisions at centre-of-mass energies between 2.1 and 2.6 GeV within the statistical hadronization model. Because of the low multiplicities involved, calculations have been carried out in the full microcanonical ensemble, including conservation of energy-momentum, angular momentum, parity, isospin, and all relevant charges. We show that the data is in an overall good agreement with the model for an energy density of about 0.5 GeV/fm^3 and an extra strangeness suppression parameter gamma_S ~ 0.7, essentially the same values…
A Pomeron model applicable to both "`soft" and "hard" processes is suggested and tested against the high-energy data from virtual photon-induced reactions. The Pomeron is universal, containing two terms, a "soft" and a "hard" one, whose relative weight varies with $\widetilde {Q^2}=Q^2+M_V^2$, where $Q^2$ is the virtuality of the incoming photon and $M_V$ is the mass of the produced vector particle. With a small number of adjustable parameters, the model fits all available data on vector meson production and deeply virtual Compton scattering from HERA. Furthermore, we attempt to apply the mode…
A Reggeometric (Regge+Geometry) model, based on the observed proportionality between the forward slope of the differential cross section and the interaction radius, the latter depending on virtuality Q2 of the incoming virtual photon and on the mass M 2 of the produced particle, is constructed. The objective of this study is the dependence of the Regge-pole amplitude on the virtuality Q2 and masses of the external particles, which remains an open problem for the theory. The present analysis is based on the HERA data on Deeply Virtual Compton Scattering (DVCS) and exclusive diffractive Vector M…
We compare fully differential exact results for the virtual photon correction to single hard photon bremsstrahlung obtained using independent calculations, both for electron-positron annihilation at high-energy colliders and for radiative return applications. The results are compared using Monte Carlo evaluations of the matrix elements as well as by direct analytical evaluation of certain critical limits. Special attention is given to the issues of numerical stability and the treatment of finite-mass corrections. It is found that agreement on the order of 10^{-5} or better is obtained over mos…
The soft physics approach to Compton scattering at moderately large momentum transfer is reviewed. It will be argued that in that approach the Compton cross section as well as other exclusive observables exhibit approximate scaling in a limited range of momentum transfer. Invited talk held at the workshop on exclusive and semi-exclusive processes at high momentum transfer, Jefferson Lab, May (1999).
We have compared the virtual corrections to single hard bremsstrahlung as calculated by S. Jadach, M. Melles, B.F.L. Ward and S.A. Yost to several other expressions. The most recent of these comparisons is to the leptonic tensor calculated by J.H. Kuhn and G. Rodrigo for radiative return. Agreement is found to within a part in 10^5 or better, as a fraction of the Born cross section, for most of the range of photon energies. The massless limits have been shown to agree analytically to NLL order.
The existence and properties of the f0(1370) scalar meson are rather well established from data of antiproton annihilations at rest. However conflicting results from Central Exclusive Production (CEP) experiments of the last millennium and ignorance of data from antiproton annihilations at rest in H2 and D2 bubble chambers have generated doubts on the very existence of the f0(1370). Properties of π+π- pairs produced in central exclusive production (CEP) reactions observed in old data together with data collected in the current decade at high energy colliders permit to show that π+π- decays of …
By including the interference effect between the QCD and the QED diagrams, we carry out a complete analysis on the exclusive productions of $e^+e^- \to J/ψ+χ_{cJ}$ ($J=0,1,2$) at the $B$ factories with $\sqrt{s}=10.6$ GeV at the next-to-leading-order (NLO) level in $α_s$, within the nonrelativistic QCD framework. It is found that the $\mathcal O (α^3α_s)$-order terms that represent the tree-level interference are comparable with the usual NLO QCD corrections, especially for the $χ_{c1}$ and $χ_{c2}$ cases. To explore the effect of the higher-order terms, namely $\mathcal O (α^3α_s^2)$, we perf…
In this work, we investigate the exclusive production of particles in scattering processes in the so-called saturation region. Within this scheme the phenomenon of geometric scaling takes place: cross sections are functions only of a dimensionless combination of the relevant kinematic variables, which happens both in inclusive and diffractive cases, as in the production of vector mesons. In particular, the scaling variable is given in general by $τ= Q^2/Q_s^2$, where $Q^2$ is the photon virtuality and $Q_s$ represents the saturation scale, which drives the energy dependence and the correspondi…
We propose measurements of the deeply virtual Compton amplitude (DVCS), gamma* to H H-bar gamma, in the timelike t = (p_{H} + p_{H-bar})^2 > 0 kinematic domain which is accessible at electron-positron colliders via the radiative annihilation process e+ e- to H H-bar gamma. These processes allow the measurement of timelike deeply virtual Compton scattering for a variety of H H-bar hadron pairs such as pi+ pi-, K+ K-, and D D-bar as well as p p-bar. As in the conventional spacelike DVCS, there are interfering coherent amplitudes contributing to the timelike processes involving C= - form factors.…
We propose to investigate a largely unexplored sector that is unique to the formulation of hard exclusive processes in terms of GPDs, namely the electroproduction of strange and charmed mesons in the kinematical ranges of Jefferson Lab's 12 GeV upgrade, and of the proposed Electron Ion Collider (EIC). In this contribution we focus on charmed meson production that is unique to the EIC. Exclusive strange and charmed meson production provides new insights in the connection of the quark/gluon degrees of freedom with the meson-baryon description, both in the unpolarized and polarized sectors. Howev…
Depending on their masses, Supersymmetric particles can affect various measurements in Z decay. Among these are the total width (or consequent extracted value of $α_s$), enhancement or suppression of various flavors, and left-right and forward-backward asymmetries. The latter depend on squark mass splittings and are, therefore, a possible test of the Supergravity related predictions. We calculate leading order corrections for these quantities considering in particular the case of light photino and gluino where the SUSY effects are enhanced. In this limit the effect on $α_s$ is appreciable, the…
We review the current theoretical situation with regard to the anomalously large cross section for exclusive J/psi + eta_c production in e^+ e^- annihilation, as measured by BELLE Collaboration.
Recently we have shown that exclusive QCD photon-induced reactions at low Mandelstam-t are best described by Regge exchanges in the entire scaling region, and not only for low values of Bjorken-x. In this paper we explore this crucial Regge behavior in Deeply Virtual Compton Scattering from the point of view of collinear factorization, with the proton tensor written in terms of Generalized Parton Distributions, and we reproduce this feature. Thus it appears that in the Bjorken limit, a large class of hard, low-t exclusive processes are more sensitive to the meson cloud of the proton than to it…
We calculate the contribution of the weak annihilation to the $B\rightarrow ργ$ decay amplitude by means of QCD sum rules using the photon light-cone wave function. We find that this long-distance contribution amounts to about 10\% of the leading short-distance effect in $B^-\rightarrow ρ^-γ$. On the other hand, weak annihilation is the dominant source of the corresponding $D$ meson decays and according to our estimates, yields branching ratios of $O(10^{-4})$ for $D^0\rightarrow \bar{K}^{*0}γ$, $O(10^{-5})$ for $D_s\rightarrow ρ^+γ$, and $O(10^{-6})$ for $D^-\rightarrow ρ^-γ$ and for $D^0\rig…
We study the production of $J/ψ+c\bar{c}$ in $e^+e^-$ annihilation through two virtual photons. The cross section is estimated to be 23 fb at $\sqrt{s}=10.6$ GeV, which is smaller by a factor of six than the calculated cross section for the same process but through one virtual photon. As a result, while the annihilation into two photons may be important for certain exclusive production processes, the big gap between the inclusive production cross section $σ(e^+e^- \to J/ψ+ c\bar{c})\simeq$ 0.9pb observed by Belle and the current nonrelativistic QCD prediction of $\simeq$0.15pb still remains ve…
We study the scaling regime of hadron-(anti)-hadron annihilation into a deeply virtual photon and a real photon, H anti-H -> gamma^* gamma, and deep backward virtual Compton scattering, gamma^* H -> H gamma. We advocate that there is a kinematical region where the scattering amplitude factorizes into a short-distance matrix element and a long-distance dominated object: a transition distribution amplitude which describes the hadron to photon transition.
Events with tagged photons in the process of electron-positron annihilation into hadrons are considered. The initial state radiation is suggested to scan the hadronic cross section with the energy. QED radiative corrections are taken into account. The results for the total and exclusive cross sections are given in an analytic form. Some numerical estimates are presented.
In inclusive decays of the $Ψ$ (3097), electromagnetic and gluonic annihilation amplitudes add incoherently, namely they are 90$^\circ$ out of phase. We argue that this incoherence must persist in each exclusive decay channel. For inclusive $Ψ'$ (3686) decays, we suggest the absence of a significant direct annihilation amplitude into three gluons and propose a new amplitude via QCD anomalies and the $h_c$ (3526) off shell. Phenomenological implications for exclusive decay channels are pointed out.
Within the framework of the non-relativistic QCD, we present a detailed discussion on the heavy quarkonium production at the leading order in $α_s$ at a $e^+ e^-$ collider with the collision energy around the $Z^0$ peak. Quarkonia are produced through the semi-exclusive channels $e^{+}e^{-} \rightarrow |H_{Q\bar{Q}}\rangle +X$ with $X=Q\bar{Q}$ or $gg$, where $Q$ indicates a heavy quark (respectively $b$ or $c$). It is noted that in addition to the color-singlet 1S-level quarkonium states, the 2S and 1P color-singlet states and the color-octet $|(Q\bar{Q})[1^3S_1^{({\bf 8})}]g\rangle$ state al…
We calculate the cross section for the exclusive production of J^{PC}=0^{++} glueballs G_0 in association with the J/psi in e^+e^- annihilation using the pQCD factorization formalism. The required long-distance matrix element for the glueball is bounded by CUSB data from a search for resonances in radiative Upsilon decay. The cross section for e^+e^- -> J/psi+ G_0 at sqrt{s}=10.6 GeV is similar to exclusive charmonium-pair production e^+e^- -> J/psi+h for h=eta_c and chi_{c0}, and is larger by a factor 2 than that for h=eta_{c}(2S). As the subprocesses gamma^* -> (c c-bar) (c c-bar) and gamma^…
We compute analytically the tree-level annihilation rates of a collection of non-relativistic neutralino and chargino two-particle states in the general MSSM, including the previously unknown off-diagonal rates. The results are prerequisites to the calculation of the Sommerfeld enhancement in the MSSM, which will be presented in subsequent work. They can also be used to obtain concise analytic expressions for MSSM dark matter pair annihilation in the present Universe for a large number of exclusive two-particle final states.
I survey a number of interesting tests of quantum chromodynamics at the amplitude level which can be carried out in electron-positron annihilation and in photon-photon collisions at low energy, including measurements of fundamental processes such as timelike form factors and transition amplitudes, timelike Compton scattering, timelike photon to meson transition amplitudes, and two-photon exclusive processes. Many of these reactions test basic principles of QCD such as hadronization at the amplitude level, factorization, and hadron helicity conservation, tools also used in the analysis of exclu…
We make a comparative and comprehensive study on the charmonium exclusive productions at the $e^+e^-$ collider with the collision energy either round the $Z^0$-boson mass for a super $Z$ factory or equals to 10.6 GeV for the $B$ factories as Belle and BABAR. We study the total cross sections for the charmonium production via the exclusive processes $e^+e^- \to γ^*/Z^0 \to H_{1}+H_{2}$ and $e^+e^- \to γ^*/Z^0 \to H_{1} +γ$, where $H_{1}$ and $H_{2}$ represent the dominant color-singlet $S$-wave and $P$-wave charmonium states respectively. Total cross sections versus the $e^+e^-$ collision energ…
Recent measurements of the Belle Collaboration of the exclusive production of two charmonia in the $e^+e^-$ annihilation differ substantially from theoretical predictions. Recently it was suggested that a significant part of this discrepancy can be explained by the process $e^2e^- --> 2J/psi$ and value for its cross-section was published. In this paper it is shown, that these results are incorrect and the values for the cross-sections of the production of different vector charmonia are presented.
The polarization of virtual photons produced in relativistic nucleus-nucleus collisions provides information on the conditions in the emitting medium. In a hydrodynamic framework, the resulting angular anisotropy of the dilepton final state depends on the flow as well as on the transverse momentum and invariant mass of the photon. We illustrate these effects in dilepton production from quark-antiquark annihilation in the QGP phase and $π^+π^-$ annihilation in the hadronic phase for a static medium in global equilibrium and for a longitudinally expanding system.
We present factorization theorems for two exclusive heavy-quarkonium production processes: production of two quarkonia in e^+e^- annihilation and production of a quarkonium and a light meson in B-meson decays. We describe the general proofs of factorization and supplement them with explicit one-loop analyses, which illustrate some of the features of the soft-gluon cancellations. We find that violations of factorization are generally suppressed relative to the factorized contributions by a factor v^2m_c/Q for each S-wave charmonium and a factor m_c/Q for each L-wave charmonium with L>0. Here, v…
We present complete results for the $O(α_s)$ corrections to the production of a heavy quark pair plus a hard photon, in $e^+ e^-$ annihilation, including both photon and $Z$ intermediate states. Virtual corrections are calculated analytically in $4 - 2 ε$ dimensions, together with the soft approximation to the real emission diagrams. After cancellation of infrared divergences, real and virtual contributions are combined numerically to obtain the cross section for the production of two heavy quark jets plus a hard photon in next-to-leading order. The resulting fully differential cross section c…
Using the renormalon technique, we estimate higher-twist contributions in deeply virtual Compton scattering and in hard exclusive $π^0$ leptoproduction.
We present theoretical analysis of paired $B_c$ mesons production in $e^+e^-$ annihilation at different energy scales taking into account full next-to-leading order QCD corrections. Both possible electroweak channels are considered: production via virtual photon and via virtual $Z$-boson. We study in detail the role of radiative QCD corrections, which were found to be significant especially at low energies. It is shown that the contribution from $Z$-boson is significant at high energies ($\sqrt{s} > M_Z/2$) especially in the case of paired production of pseudo-scalar and vector ($B_c + B_c^*$)…
