Exclusive double $B_c$ meson production from $e^+e^-$ annihilation into two virtual photons

TL;DR

This paper calculates the production cross sections of $B_c$ meson pairs via two-photon mechanisms in electron-positron annihilation, considering relativistic effects and comparing with one-photon processes.

Contribution

It introduces a relativistic quark model approach to compute $B_c$ meson pair production amplitudes and cross sections in electron-positron collisions.

Findings

Relativistic corrections significantly affect cross section values.

Two-photon production mechanism is quantitatively analyzed.

Comparison shows differing contributions from one-photon and two-photon processes.

Abstract

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.

Tables

Table 1. Table 1: Numerical values of relativistic parameters ( 23 ) in pair B c subscript 𝐵 𝑐 B_{c} meson production cross section ( 20 ).

$B_c$	$n^{2S+1}{L}_J$	$M_{B_{\overline{b}c}}$, GeV	${\Psi }_{B_{\overline{b}c}}^0\left(0\right)$, GeV3/2	${\omega }_{10}$	${\omega }_{01}$	${\omega }_{\frac{1}{2}\frac{1}{2}}$
meson
$B_{\overline{b}c}$	$1^1{S}_0$	6.276	0.250	0.0489	0.0060	0.0171
$B_{\overline{b}c}^{\ast }$	$1^3{S}_1$	6.317	0.211	0.0540	0.0066	0.0188

Table 2. Table 2: Nonrelativistic and relativistic production cross sections of B c subscript 𝐵 𝑐 B_{c} mesons.

Final state	Center-of-mass	Nonrelativistic cross	Relativistic cross
$B_{\overline{b}c}+B_{b\overline{c}}$	energy s	section ${\sigma }_{nr}$	section ${\sigma }_r$
$B_{\overline{b}c}^{+}+B_{b\overline{c}}^{-}$	22.0 GeV	$0.03\times {10}^{-3}$ fb	$0.02\times {10}^{-3}$ fb
$B_{\overline{b}c}^{\ast +}+B_{b\overline{c}}^{-}$	22.0 GeV	$0.18\times {10}^{-3}$ fb	$0.11\times {10}^{-3}$ fb
$B_{\overline{b}c}^{\ast +}+B_{b\overline{c}}^{\ast -}$	22.0 GeV	$1.43\times {10}^{-3}$ fb	$0.46\times {10}^{-3}$ fb

Equations

$$p_1={\eta }_{1}P+p,p_2={\eta }_{2}P-p,(p\cdot P)=0,{\eta }_{1,2}=\frac{M_{B_{\bar{b}c}}^2\pm m_1^2\mp m_2^2}{2M_{B_{\bar{b}c}}^2},$$

$$q_1={\rho }_{1}Q+q,q_2={\rho }_{2}Q-q,(q\cdot Q)=0,{\rho }_{1,2}=\frac{M_{B_{b\bar{c}}}^2\pm m_1^2\mp m_2^2}{2M_{B_{b\bar{c}}}^2},$$

$$k_1^2=(p_1+q_1{)}^2=({\eta }_{1}P+{\rho }_{1}Q+p+q{)}^2,k_2^2=(p_2+q_2{)}^2=({\eta }_{2}P+{\rho }_{2}Q-p-q{)}^2,$$

$${\mathcal{T}}_1(p,q,P,Q)=\frac{16{\pi }^2{\alpha }^2{Q}_c{Q}_b}{k_1^2{k}_2^2}\bar{v}(p_{+})\frac{{\gamma }^{\alpha }({\hat{p}}_{-}-{\hat{k}}_1+m_e){\gamma }^{\beta }}{(p_{-}-k_1{)}^2-m_e^2}u(p_{-}){\bar{u}}_1^i(p_1){\gamma }^{\beta }{v}_1^i(q_1){\bar{u}}_2^j(q_2){\gamma }^{\alpha }{v}_2^j(p_2),$$

$${\mathcal{M}}_1(p_{-},p_{+},P,Q)=\int \frac{d\mathbf{p}}{(2\pi {)}^3}{\bar{\Psi }}_{\mathcal{P}}(p,P)\int \frac{d\mathbf{q}}{(2\pi {)}^3}{\bar{\Psi }}_{\mathcal{V}}(q,Q){\mathcal{T}}_1(p,q,P,Q).$$

$${\Psi }_P^{\rho \omega }(\mathbf{p})=D_1^{1/2,\rho \alpha }(R_{L_P}^W)D_2^{1/2,\omega \beta }(R_{L_P}^W){\Psi }_0^{\alpha \beta }(\mathbf{p}),$$

$${\bar{\Psi }}_P^{\lambda \sigma }(\mathbf{p})={\bar{\Psi }}_0^{\epsilon \tau }(\mathbf{p})D_1^{+1/2,\epsilon \lambda }(R_{L_P}^W)D_2^{+1/2,\tau \sigma }(R_{L_P}^W),$$

$$\left(\genfrac{}{}{0ex}{}{10}{01}\right)D_{1,2}^{1/2}(R_{L_P}^W)=S^{-1}({\mathbf{p}}_{1,2})S(\mathbf{P})S(\mathbf{p}),$$

$$S(\mathbf{p})=\sqrt{\frac{ϵ(p)+m}{2m}}\left(1+\frac{(\mathbf{\alpha }\mathbf{p})}{ϵ(p)+m}\right).$$

$$S_{\alpha \beta }(\Lambda )u_{\beta }^{\lambda }(p)=\sum _{\sigma =\pm 1/2}{u}_{\alpha }^{\sigma }(\Lambda p)D_{\sigma \lambda }^{1/2}(R_{\Lambda p}^W),$$

$${\bar{u}}_{\beta }^{\lambda }(p)S_{\beta \alpha }^{-1}(\Lambda )=\sum _{\sigma =\pm 1/2}{D}_{\lambda \sigma }^{+1/2}(R_{\Lambda p}^W){\bar{u}}_{\alpha }^{\sigma }(\Lambda p).$$

$${\bar{u}}_1(\mathbf{p})={\bar{u}}_1(0)\frac{({\hat{p}}_1^{\prime }+m_1)}{\sqrt{2{ϵ}_1(\mathbf{p})({ϵ}_1(\mathbf{p})+m_1)}},p_1^{\prime }=({ϵ}_1,\mathbf{p}),$$

$$v_2(-\mathbf{p})=\frac{({\hat{p}}_2^{\prime }-m_2)}{\sqrt{2{ϵ}_2(\mathbf{p})({ϵ}_2(\mathbf{p})+m_2)}}{v}_2(0),p_2^{\prime }=({ϵ}_2,-\mathbf{p}),$$

$${\hat{\Pi }}^{\mathcal{P},\mathcal{V}}=[v_2(0){\bar{u}}_1(0){]}_{S=0,1}={\gamma }_5({\widehat{ϵ}}^{\ast })\frac{1+{\gamma }^0}{2\sqrt{2}}.$$

$${\cal M}(p_{-},p_{+},P,Q)=\int\frac{d{\bf p}}{(2\pi)^{3}}\int\frac{d{\bf q}}{(2\pi)^{3}}\frac{4\pi^{2}\alpha^{2}Q_{c}Q_{b}}{k_{1}^{2}k_{2}^{2}}\sqrt{M_{B_{\bar{b}c}}M_{B_{b\bar{c}}}}\bar{v}(p_{+})\Bigl{[}\frac{\gamma^{\alpha}(\hat{p}_{-}-\hat{k}_{1}+m_{e})\gamma^{\beta}}{(p_{-}-k_{1})^{2}-m_{e}^{2}}+$$

$$\frac{\gamma^{\beta}(\hat{p}_{-}-\hat{k}_{2}+m_{e})\gamma^{\alpha}}{(p_{-}-k_{2})^{2}-m_{e}^{2}}\Bigr{]}u(p_{-})Sp\left\{\Psi^{\cal P}_{B_{\bar{b}c}}(p,P)\gamma^{\beta}\Psi^{\cal V}_{B_{b\bar{c}}}(q,Q)\gamma^{\alpha}\right\},$$

$${\Psi }_{B_{\bar{b}c}}^{\mathcal{P}}(p,P)$$

$${\Psi }_{B_{b\bar{c}}^{\ast }}^{\mathcal{V}}(q,Q)$$

$$\frac{1}{k_{1,2}^2}=\frac{1}{{\eta }_{1,2}^2{s}^2}\left[1\mp \frac{2(pQ+qP)}{{\eta }_{1,2}{s}^2}-\frac{(p+q{)}^2}{{\eta }_{1,2}^2{s}^2}+\cdots \right],$$

$$\frac{1}{(k_{1,2}-p_{-}{)}^2-m_e^2}=-\frac{1}{{\eta }_1{\eta }_2{s}^2}\left[1\pm \frac{2(pQ+qP)}{{\eta }_{2,1}{s}^2}\mp \frac{2p_{-}(p+q)}{{\eta }_1{\eta }_2{s}^2}+\frac{(p+q{)}^2}{{\eta }_1{\eta }_2{s}^2}+\cdots \right].$$

$$\frac{d\sigma_{B_{\bar{b}c}B_{b\bar{c}}}}{dz}=\frac{8\pi^{3}\alpha^{4}q_{c}^{2}q_{b}^{2}M_{B_{\bar{b}c}}M_{B_{b\bar{c}}}|{\bf P}|M^{4}}{r_{1}^{6}r_{2}^{6}s^{15}}|\Psi^{0}_{B_{\bar{b}c}}(0)|^{2}|\Psi_{B_{b\bar{c}}}(0)|^{2}\Bigl{[}f^{LO}_{B_{\bar{b}c}B_{b\bar{c}}}(z,s)+$$

$$\frac{(B_{B_{\bar{b}c}}+B_{B_{b\bar{c}}})}{2M}f^{bind}_{B_{\bar{b}c}B_{b\bar{c}}}(z,s)+\omega_{\frac{1}{2}\frac{1}{2}}f^{1,rel}_{B_{\bar{b}c}B_{b\bar{c}}}(z,s)+\omega_{01}f^{2,rel}_{B_{\bar{b}c}B_{b\bar{c}}}(z,s)+\omega_{10}f^{3,rel}_{B_{\bar{b}c}B_{b\bar{c}}}(z,s)\Bigr{]},$$

$${\Psi }_{B_{\bar{b}c}}^0(0)=\int \sqrt{\frac{({ϵ}_1(p)+m_1)({ϵ}_2(p)+m_2)}{2{ϵ}_1(p)\cdot 2{ϵ}_2(p)}}{\Psi }_{B_{\bar{b}c}}^0(\mathbf{p})\frac{d\mathbf{p}}{(2\pi {)}^3}.$$

$$I_{B_{\bar{b}c},B_{b\bar{c}}}^{nk}={\int }_0^{\infty }{q}^2{R}_{B_{\bar{b}c},B_{b\bar{c}}}(q)\sqrt{\frac{({ϵ}_1(q)+m_1)({ϵ}_2(q)+m_2)}{2{ϵ}_1(q)\cdot 2{ϵ}_2(q)}}{\left(\frac{{ϵ}_1(q)-m_1}{{ϵ}_1(q)+m_1}\right)}^n{\left(\frac{{ϵ}_2(q)-m_2}{{ϵ}_2(q)+m_2}\right)}^{k}dq,$$

$${\omega }_{10}^{B_{\bar{b}c},B_{b\bar{c}}}=\frac{I_{B_{\bar{b}c},B_{b\bar{c}}}^{10}}{I_{B_{\bar{b}c},B_{b\bar{c}}}^{00}},{\omega }_{01}^{B_{\bar{b}c},B_{b\bar{c}}}=\frac{I_{B_{\bar{b}c},B_{b\bar{c}}}^{01}}{I_{B_{\bar{b}c},B_{b\bar{c}}}^{00}},{\omega }_{\frac{1}{2}\frac{1}{2}}^{B_{\bar{b}c},B_{b\bar{c}}}=\frac{I_{B_{\bar{b}c},B_{b\bar{c}}}^{\frac{1}{2}\frac{1}{2}}}{I_{B_{\bar{b}c},B_{b\bar{c}}}^{00}},$$

$$|\mathbf{p}|=2m_1\left[\sqrt{\frac{{ϵ}_1-m_1}{{ϵ}_1+m_1}}+{\left(\frac{{ϵ}_1-m_1}{{ϵ}_1+m_1}\right)}^{3/2}+{\left(\frac{{ϵ}_1-m_1}{{ϵ}_1+m_1}\right)}^{5/2}+...\right]=$$

$$2m_2\left[\sqrt{\frac{{ϵ}_2-m_2}{{ϵ}_2+m_2}}+{\left(\frac{{ϵ}_2-m_2}{{ϵ}_2+m_2}\right)}^{3/2}+{\left(\frac{{ϵ}_2-m_2}{{ϵ}_2+m_2}\right)}^{5/2}+...\right],$$

$$H=H_0+\Delta U_1+\Delta U_2,H_0=\sqrt{{\mathbf{p}}^2+m_1^2}+\sqrt{{\mathbf{p}}^2+m_2^2}-\frac{4{\tilde{\alpha }}_s}{3r}+(Ar+B),$$

$$\Delta U_1(r)=-\frac{{\alpha }_s^2}{3\pi r}\left[2{\beta }_0\ln (\mu r)+a_1+2{\gamma }_E{\beta }_0\right],a_1=\frac{31}{3}-\frac{10}{9}{n}_f,{\beta }_0=11-\frac{2}{3}{n}_f,$$

$$\Delta U_2(r)=-\frac{2{\alpha }_s}{3m_1{m}_{2}r}\left[{\mathbf{p}}^2+\frac{\mathbf{r}(\mathbf{r}\mathbf{p})\mathbf{p}}{r^2}\right]+\frac{2\pi {\alpha }_s}{3}\left(\frac{1}{m_1^2}+\frac{1}{m_2^2}\right)\delta (\mathbf{r})+\frac{4{\alpha }_s}{3r^3}\left(\frac{1}{2m_1^2}+\frac{1}{m_1{m}_2}\right)({\mathbf{S}}_1\mathbf{L})+$$

$$+\frac{4{\alpha }_s}{3r^3}\left(\frac{1}{2m_2^2}+\frac{1}{m_1{m}_2}\right)({\mathbf{S}}_2\mathbf{L})+\frac{32\pi {\alpha }_s}{9m_1{m}_2}({\mathbf{S}}_1{\mathbf{S}}_2)\delta (\mathbf{r})+\frac{4{\alpha }_s}{m_1{m}_2{r}^3}\left[\frac{({\mathbf{S}}_1\mathbf{r})({\mathbf{S}}_2\mathbf{r})}{r^2}-\frac{1}{3}({\mathbf{S}}_1{\mathbf{S}}_2)\right]-$$

Full text

Hadron production in $e^{+}e^{-}$ interactions, Relativistic quark model

I Introduction

The production of bound states of heavy quarks (double-heavy mesons and baryons) in such fundamental processes as electron-positron annihilation and proton-proton interaction is of great importance for testing the perturbative and non-perturbative aspects of quantum chromodynamics and the theory of bound states. Among these reactions stands out the process of exclusive pair production of heavy quarkonia and double heavy baryons, since a pair of bound states of heavy quarks is produced in it and the binding effects of quarks are significantly enhanced. The activity in the theoretical study of the pair charmonium production in electron-positron annihilation was largely connected with the experimental results obtained by the Belle and BaBar collaborations belle ; babar ; pahlova ; brambilla2011 , which differed significantly from the predictions of non-relativistic quantum chromodynamics bl1 ; qiao ; chao1 . During a short period of research, it has been shown that the theoretical results for the pair production of charmonium can be reconciled with experimental data, taking into account perturbative corrections of order $O(\alpha_{s})$ to the production amplitudes and relativistic corrections, due to the relative motion of heavy quarks bodwin1 ; bodwin2 ; bodwin3 ; chao2 ; chao3 ; bll1 ; bll2 ; em2006 ; ji ; jia ; gong ; akl1 ; efgm2009 . Along with the mechanism of one-photon annihilation, the production process of a pair of charmoniums in two-photon annihilation was also studied at this time bbl1 . Despite the fact that such contributions contain an additional small factor $\alpha^{2}/\alpha_{s}^{2}$, nevertheless, the fragmentation contribution has a structure that compensates for the suppression of the production amplitude by coupling constants.

In our work apm2016 , we used the developed methods of studying the reactions of pair exclusive quarkonia production in the case of $B_{c}$-mesons. Our results apm2016 show that, as in the case of $(c\bar{b})$ mesons, relativistic effects substantially change the magnitude of the cross sections for the production of a pair of $B_{c}$-mesons. In akl2 , the calculation was performed for one-loop corrections in the pair-production of $B_{c}$-mesons in a one-photon annihilation. An approximation was used in which the relative motion of heavy quarks was not taken into account. In this paper, we explore the contribution of two-photon annihilation process in reactions of the production of a pair of $B_{c}$-mesons. We calculate the cross sections for the production of a pair of $B_{c}$-mesons in the nonrelativistic approximation and with the account of relativistic corrections. A comparison of numerical contributions of one-photon and two-photon mechanisms in total cross section is performed.

While extensive literature is devoted to the issues of the production of single $B_{c}$ mesons gklt , the problem of the production of a pair of $B_{c}$ mesons or heavy diquarks $(bc)$ in different reactions has been discussed to a much lesser extent apm2016 ; mt ; mt1 ; mt2 . This is mainly due to the lack of a clear experimental perspective to observe a sufficient number of such events in existing experiments. This work continues our research of exclusive double heavy meson production in $e^{+}e^{-}$ annihilation. Our approach to calculating the observed cross sections for the production of a pair of mesons is based on methods of relativistic quark model (RQM) and perturbative quantum chromodynamics em2006 ; efgm2009 ; apm2016 ; em2010 ; apm2005 ; apm2007 . In this approach we can take into account relativistic effects in the construction of relativistic production amplitudes, relativistic production cross sections, and in the description of bound states of heavy quarks through the use of the corresponding quark interaction operator. We can say that in this approach we have microscopic picture of the photon, quark and gluon interaction at different stages of meson production. The approach based on relativistic quark model allows you to perform a self-consistent calculation of various theoretical parameters, which ultimately determine the total numerical values of the production cross sections.

II General formalism

Two-photon annihilation amplitudes leading to the production of a pair of $B_{c}$ mesons, at leading order are presented in Fig. 1. The production process can be divided into two stages. At the first stage, a pair of virtual photons is formed. At the second stage, each virtual photon produces a pair of quark-antiquark, which then with some probability combine into $B_{c}$ mesons with definite spin. To properly take into account the quark binding effects and relativistic corrections, we express quark four-momenta in terms of the total and relative four-momenta in the form:

[TABLE]

[TABLE]

where $M_{B_{\bar{b}c}}$ is the mass of pseudoscalar or vector $B_{c}^{+}$ ($B_{c}^{\ast+}$) meson consisting of $\bar{b}$-antiquark and $c$-quark. $P(Q)$ are the total four-momenta of mesons $B_{c}^{+}$ and $B_{c}^{\ast-}$, relative quark four-momenta $p=L_{P}(0,{\bf p})$ and $q=L_{P}(0,{\bf q})$ are obtained from the rest frame four-momenta $(0,{\bf p})$ and $(0,{\bf q})$ by the Lorentz transformation to the system moving with the momenta $P$ and $Q$. It can be noted that a good choice of coefficients in the formulas (1) leads to the orthogonality of the total and relative four-momenta. In turn, the virtual photon momenta $k_{1,2}$ can also be expressed through P, Q, p, q in the form:

[TABLE]

and the virtuality of each photon is large. We note that in our approach, quarks are not in an intermediate state on the mass shell, since after their production there is always an interaction between them, and there is a symmetric escape of particles beyond the mass shell: $p_{1,2}^{2}=\eta_{1,2}^{2}P^{2}-{\bf p}^{2}=\eta_{1,2}^{2}M_{B_{\bar{b}c}}^{2}-{\bf p}^{2}\not=m_{1,2}^{2}$, $p_{1}^{2}-m_{1}^{2}=p_{2}^{2}-m_{2}^{2}$. Since we are discussing the creation of a pair of S-wave states, in the case of a pair of pseudoscalar or pair of vector mesons $\eta_{1,2}=\rho_{1,2}$, and at the production of pseudoscalar and vector mesons $\eta_{1,2}\approx\rho_{1,2}$ with good accuracy. Indeed, the numerical values of these coefficients can be obtained by choosing the masses of the mesons $M_{\cal P}=6.2749$ GeV pdg , $M_{\cal V}=6.332$ GeV efg2003 : $\eta_{1}=0.228$, $\rho_{1}=0.233$, $\eta_{2}=0.772$, $\rho_{2}=0.767$ (hereinafter, the indices denote pseudoscalar ${\cal P}$ and vector ${\cal V}$ states.).

The construction of the relativistic amplitude of the two-photon pair production includes a number of intermediate steps. In the color singlet model the production amplitude of four quarks and anti-quarks presented in first Feynman diagram takes the form:

[TABLE]

where $p_{-}$, $p_{+}$ are four momenta of electron and positron, $u^{i}_{1,2}$, $v^{j}_{1,2}$ are wave functions of quarks and anti-quarks playing the role of projection operators on positive energy states. Accounting for color part of the meson wave function ($\delta_{ij}/\sqrt{3}$) we obtain total color factor in the amplitude (3) equal to 1. In quasipotential approach we can express the amplitude of the reaction $e^{+}+e^{-}\to B^{+}_{\bar{b}c}(B^{\ast+}_{\bar{b}c})+B^{-}_{\bar{c}b}(B^{\ast-}_{\bar{b}c})$ as a convolution of ${\cal T}_{1}(p,q,P,Q)$ and quasipotential wave functions of produced quark bound states:

[TABLE]

In this matrix element we have the wave functions of mesons moving with four-momenta P and Q. The transformation law of the bound state wave function from the rest frame $\Psi_{0}({\bf p})$ to the moving one with four-momentum P was derived in the Bethe-Salpeter approach in brodsky and in quasipotential method in faustov . We use the quasipotential method, so the wave function transformation looks as follows:

[TABLE]

[TABLE]

where $R^{W}$ is the Wigner rotation, $L_{P}$ is the Lorentz boost from the meson rest frame to a moving one, and the rotation matrix $D^{1/2}(R)$ is defined by

[TABLE]

where the explicit form for the Lorentz transformation matrix of the four-spinor is

[TABLE]

Further transformations of the amplitude (4) can be carried out by means of the following relations:

[TABLE]

[TABLE]

Using also the transformation property of the Dirac bispinors to the rest frame

[TABLE]

we can introduce the projection operators $\hat{\Pi}^{{\cal P},{\cal V}}$ onto the states $(c\bar{b})$, $(b\bar{c})$ with total spin 0 and 1 as follows:

[TABLE]

As a result of such transformations the total amplitude of pair $B_{c}$ meson production can be written in the form:

[TABLE]

[TABLE]

where $s$ is the center-of-mass energy, a superscript ${\cal P}$ indicates a pseudoscalar $B_{c}$ meson, a superscript ${\cal V}$ indicates a vector $B_{c}$ meson, $\alpha$ is the fine structure constant. The transition wave functions $\Psi^{\cal V}_{B_{b\bar{c}}}(q,Q)$ and $\Psi^{\cal P}_{B_{\bar{b}c}}(p,P)$ (form factors) have the following form:

[TABLE]

[TABLE]

where the symbol hat denotes convolution of four-vector with the Dirac gamma matrices, $v_{1}=P/M_{B_{\bar{b}c}}$, $v_{2}=Q/M_{B_{b\bar{c}}}$; $\varepsilon_{\cal V}(Q,S_{z})$ is the polarization vector of the $B^{\ast-}_{c}(1^{-})$ meson, relativistic quark energies $\epsilon_{1,2}(p)=\sqrt{p^{2}+m_{1,2}^{2}}$ and $m_{1,2}$ are the masses of $c$ and $b$ quarks. In (13) we have complicated factor including the bound state wave function in the rest frame. Therefore instead of the substitutions $M_{B_{\bar{b}c}}=\epsilon_{1}({\bf p})+\epsilon_{2}({\bf p})$ and $M_{B^{\ast}_{b\bar{c}}}=\epsilon_{1}({\bf q})+\epsilon_{2}({\bf q})$ in the production amplitude we carry out the integration over the quark relative momenta ${\bf p}$ and ${\bf q}$. Relativistic wave functions in (15) and (17) are equal to the product of wave functions in the rest frame $\Psi^{0}_{B_{\bar{b}c}}({\bf p})$ and spin projection operators that are accurate at all orders in $|{\bf p}|/m$. An expression of spin projector in different form for $(c\bar{c})$ system was obtained in bodwin2002 ; bodwin2004 where spin projectors are written in terms of heavy quark momenta $p_{1,2}$ lying on the mass shell. We can consider (15)-(17) as a transition form factors for heavy quark-antiquark pair from free state to bound state.

Relative momenta of heavy quarks $p$ and $q$ enter in the propagators of photons and electron in intermediate state as well as in transition wave functions (15) and (17). The ratios of relative momenta to s or quark masses are small so we can use an expansion of all quantities depending on p and q. In the decomposition of the corresponding factors, we take into account the terms of the second order in p and q in the form:

[TABLE]

[TABLE]

The coefficients $\eta_{1,2}$ include also bound state effects which we express in terms of bound state energies setting $M_{\cal P}=m_{1}+m_{2}+B_{\cal P}$, $M_{\cal V}=m_{1}+m_{2}+B_{\cal V}$. Despite expansions (18)-(19) with respect to relative momenta, the integrals in (13) remain convergent when taking corrections of second-order in p or q and can be calculated if the $\Psi^{0}_{B^{\ast}_{b\bar{c}}}({\bf p})$ functions are known. It is convenient to perform further transformations using the Form package form immediately when calculating the cross section for the production of a pair.

When calculating the differential cross section, we introduce the angle $\theta$ between the electron momentum ${\bf p}_{e}$ and momentum ${\bf P}$ of $B_{c}$ meson. After all the simplifications and the explicit separation of relativistic corrections of the second order and the bound state effects, we obtain the differential cross section $d\sigma/d\cos\theta$ ($z=\cos\theta$) as a function of center-of-mass energy $s$ with a number of parameters containing quark masses, binding energies, and relativistic corrections. The differential cross sections for the production of pair pseudoscalar mesons, pair vector mesons and pair pseudoscalar plus vector mesons can be written in the following form:

[TABLE]

[TABLE]

where $B_{B_{\bar{b}c}}$ is the binding energy, $|{\bf P}|=\sqrt{[s^{2}-(M_{B_{\bar{b}c}}+M_{B_{b\bar{c}}})^{2}][s^{2}-(M_{B_{\bar{b}c}}-M_{B_{b\bar{c}}})^{2}]/4s^{2}}$ is the meson three momentum in center-of-mass frame, $r_{1.2}=m_{1,2}/(m_{1}+m_{2})=m_{1,2}/M$. The value of bound state wave function at the origin is equal

[TABLE]

In the integral function (21) , we have identified the relativistic factor that usually arises in the relativistic quark model. Explicit analytical expressions for the functions $f^{1,rel}_{B_{\bar{b}c}B_{b\bar{c}}}(z,s)$, $f^{LO}_{B_{\bar{b}c}B_{b\bar{c}}}(z,s)$, $f^{bind}_{B_{\bar{b}c}B_{b\bar{c}}}(z,s)$ are presented in Appendix A. At the first stage of transformations of the production cross sections in the Forms package, we decompose the integrand function in powers of relativistic factors $C_{nk}=[(m_{1}-\epsilon_{1}(p))/(m_{1}+\epsilon_{1}(p))]^{n}[(m_{2}-\epsilon_{2}(q))/(m_{2}+\epsilon_{2}(q))]^{k}$, where $n$, $k$ are integers and half-integers with $n+k\leq 1$. To preserve the symmetry of the expression on the quark masses we make following substitution in some expansion terms: ${\bf p}^{2}/4m_{1}m_{2}\approx\sqrt{(\epsilon_{1}-m_{1})(\epsilon_{2}-m_{2})/(\epsilon_{1}+m_{1})(\epsilon_{2}+m_{2})}[1+(\epsilon_{1}-m_{1})/(\epsilon_{1}+m_{1})+(\epsilon_{2}-m_{2})/(\epsilon_{2}+m_{2})+...]$. On the second stage after simplifications based on the symmetry properties of integral functions when replacing ${\bf p}\to{\bf q}$ we extract in the cross sections specific relativistic parameters $\omega_{nk}$. These parameters can be expressed in terms of momentum integrals $I^{nk}$ and calculated in the quark model:

[TABLE]

[TABLE]

where $R_{B_{\bar{b}c},B_{b\bar{c}}}(q)$ is the radial wave function of the mesons $B_{\bar{b}c}$,$B_{b\bar{c}}$ in momentum space. The expansion in (20) can extend and take into account terms of higher order in p and q. In the process of obtaining the necessary integrand functions we use different substitutions for ${\bf p}^{2}$ and ${\bf q}^{2}$. All of them can be obtained using the following expansion

[TABLE]

[TABLE]

which allows you to save, if necessary, the symmetry of the particles.

There are two groups of relativistic corrections to the production cross sections (20) connected with relative quark momenta p and q. First group is connected with different relativistic factors in the production amplitude (13) containing relative momenta of heavy quarks ${\bf p}$ and ${\bf q}$. They are presented in cross section (20) in terms of parameters $\omega_{nk}$. It is important to emphasize that all these parameters can be calculated numerically within the quark model itself. In this sense, the proposed approach is self-consistent. Despite the convergence of the integrals (22) determining the relativistic parameters, when calculating them, we introduce an additional cutoff for relativistic momenta near the mass of c-quark $m_{c}$. The reason for this is that in the field of such relativistic momenta, the wave function is already very small and is not defined accurately enough, since its calculation uses the non-relativistic Shrödinger equation. Relativistic corrections of second group are determined by bound state wave functions of pseudoscalar and vector $B_{c}$ mesons $\Psi^{0}_{B_{\bar{b}c}}({\bf q})$. For their calculation with the account of relativistic corrections we use corresponding QCD generalization of the Breit Hamiltonian in the center-of-mass reference frame repko1 ; pot1 ; pot11 ; pot3 ; capstick ; isgur :

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where ${\bf L}=[{\bf r}\times{\bf p}]$, ${\bf S}_{1}$, ${\bf S}_{2}$ are spins of heavy quarks, $n_{f}$ is the number of flavors, $\gamma_{E}\approx 0.577216$ is the Euler constant. To improve an agreement of theoretical hyperfine splittings in $(\bar{b}c)$ mesons with experimental data and other calculations in quark models we add to the Breit potential (28) the spin confining potential obtained in repko1 ; repko2 ; repko22 ; repko222 :

[TABLE]

where we take the parameter $f_{V}=0.9$. For the dependence of the QCD coupling constant $\tilde{\alpha}_{s}(\mu^{2})$ on the renormalization point $\mu^{2}$ in the pure Coulomb term in (26) we use the three-loop result kniehl1997

[TABLE]

In other terms of the Hamiltonians (27) and (28) we use the leading order approximation for $\alpha_{s}$. The typical momentum transfer scale in a quarkonium is of order of double reduced mass, so we set the renormalization scale $\mu=2m_{1}m_{2}/(m_{1}+m_{2})$ and $\Lambda=0.168$ GeV, which gives $\alpha_{s}=0.265$ for $(\bar{b}c)$ meson. The coefficients $b_{i}$ are written explicitly in kniehl1997 . The parameters of the linear confinement potential $A=0.18$ GeV2 and $B=-0.16$ GeV were obtained in quark models and lattice calculations rqm1 ; rqm11 ; rqm111 ; godfrey .

To calculate relativistic corrections of second group to the pseudoscalar and vector $B_{c}$-meson wave functions $\Psi^{0}_{B_{\bar{b}c}}({\bf p})$ we take the Breit potential (26) and construct the effective potential model as in em2010 ; lucha by means of the rationalization of kinetic energy operator. Numerical values of relativistic parameters entering the cross section (20) are calculated on the basis of (22) and by means of numerical solution of the Schrödinger equation ls ; sch and presented in Table 1. On the basis of this model we calculate a number of observed quantities such as the masses of charmonium, bottomonium and $B_{c}$ mesons and compare the results with existing experimental data and other theoretical predictions. The obtained results are in good agreement with them (the accuracy amounts to about one percent). For example, in the case of low lying $(b\bar{c})$ mesons we obtain $M(1^{1}S_{0})=6.276$ GeV and $M(1^{3}S_{1})=6.317$ GeV (compare with experimental value $M(1^{1}S_{0})=6.2749$ GeV pdg and $M(1^{3}S_{1})=6.332$ GeV efg2003 ). The similar situation occurs for charmonium states em2010 . Our estimates of the charmonium massess agree with experimental data with more than a per cent accuracy em2010 ; pdg . Our nonrelativistic values of $B_{c}$ meson wave functions at the origin differs on 20 per cent from the values presented in rqm1 ; godfrey ; gklt . Taking the values of parameters of the $B_{c}$ mesons, we calculate the production cross sections as functions of center-of-mass energy $s$. The plots of total production cross sections of pair $B_{c}$ mesons are presented in Fig. 2. In Table 2 we give numerical values of total production cross sections at $s=22~{}GeV$ and compare them with nonrelativistic result in our quark model. The effect of relativistic corrections to the bound state wave functions (the Breit potential) plays a key role in total decreasing of the production cross sections as compared with nonrelativistic results. The decreasing of nonrelativistic values of cross sections in our model with regard to relativistic corrections ranges from 40 to 70 percent.

III Numerical results and conclusion

The studies in many previous papers bodwin1 ; bodwin2 ; bodwin3 ; chao2 ; chao3 ; bll1 ; bll2 ; em2006 ; ji ; jia ; gong ; akl1 ; efgm2009 have convincingly shown that in such reactions as the production of bound states of heavy quarks, relativistic corrections should be taken into account in order to obtain more accurate values of the observed production cross sections. Therefore, in this work we investigate not only another mechanism for the production of a pair of $B_{c}$-mesons, but also carried out an account of relativistic effects in the production cross section. We develop our formalism, which was used in previous work apm2016 , in case of two-photon processes. In this case, the general structure of the relativistic amplitudes of the pair $B_{c}$ meson production remains the same, but the vertex functions change. The important role of relativistic effects in exclusive processes of $B_{c}$ pair production from two-photon electron-positron annihilation is confirmed in this case. It is useful to say once again that when constructing the relativistic production amplitude (4) we keep two types of relativistic corrections which act in different directions. The first type corrections can be qualified as relativistic corrections to the amplitude connected with the relative quark momenta ${\bf p}$ and ${\bf q}$. The corrections of second type appear from the perturbative and nonperturbative treatment of the quark-antiquark interaction operator which leads to the modification of the quark bound state wave functions $\Psi^{0}_{B_{\bar{b}c}}({\bf p})$ as compared with nonrelativistic case. We also systematically account for the bound state corrections working with masses of $B_{c}$ mesons. The calculated masses of $B_{c}$ mesons agree well with previous theoretical results and experimental data pdg ; rqm1 ; godfrey ; gklt . Note that the quark model, which we have used in the calculations is based on quantum chromodynamics and has certain characteristics of universality.

Total cross sections for the exclusive pair production of pseudoscalar and vector $B_{c}$ mesons in $e^{+}e^{-}$ annihilation can be obtained from (20) after angular integration in the form:

[TABLE]

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[TABLE]

[TABLE]

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[TABLE]

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[TABLE]

where $\tilde{s}=s/M=s/(m_{1}+m_{2})$.

The plots of total cross sections for the production of two pseudoscalar, pseudoscalar and vector and two vector $B_{c}$ mesons as functions of center-of-mass energy $s$ are presented in Fig. 2. A comparison of numerical results of the two-photon mechanism of electron-positron annihilation with the one-photon mechanism shows that the two-photon contribution to the total cross sections for the production of a pair is suppressed by a small factor $10^{-3}$, which is associated with the ratio of interaction constants $\alpha^{2}/\alpha_{s}^{2}$ and qualitatively corresponds to the results for $D^{+}D^{-}$ chao2010 . There are no factors that could lead to an increase in the cross sections for two-photon annihilation compared with one-photon annihilation. At the production of $B_{c}$ mesons, there is only a recombination mechanism shown in Fig. 1, while the fragmentation mechanism of the production is absent in contrast to the production of a pair of charmonia. The developed technique will be applied to the $e^{+}e^{-}\to B_{c}B_{c}e^{+}e^{-}$ process, which is not suppressed by the propagators, and for which it can be expected that its contribution will be comparable to the contribution of one-photon annihilation, as shown for pair production $D^{+}D^{-}$ in berezhnoy . Note also that we are discussing a comparison of the relativistic and nonrelativistic cross sections in the tail section of the plots in Fig. 2, in which the $M^{2}/s^{2}$ corrections of higher order are strongly suppressed. Accounting for the interference effects from the amplitudes of one-photon and two-photon annihilation gives terms of order $\alpha^{2}\alpha_{s}^{2}\frac{\alpha}{\alpha_{s}}$ in the differential cross section, which essentially has the same order of smallness as the radiative corrections of order $O(\alpha_{s})$ to the amplitudes of one-photon annihilation. But since such terms are proportional to odd powers of $z=\cos\theta$ in the differential cross section, their contribution to the total cross section is zero.

From the results presented in Fig. 2, it follows that an account of relativistic and bound state corrections decreases the values of nonrelativistic production cross sections. We would like to emphasize that the term nonrelativistic limit means that $M_{B_{\bar{b}c}}=m_{1}+m_{2}$, and the wave function of the bound state of quarks was determined by solving the Schrödinger equation with a purely nonrelativistic Hamiltonian. In the resulting expressions for relativistic cross sections, there are various relativistic factors that affect the change in the original non-relativistic section in different ways. The greatest reduction in cross sections is given by the value of bound state wave function at the origin. Relativistic effects in the production amplitude, which are determined by the parameters $\omega_{10}$, $\omega_{01}$, $\omega_{\frac{1}{2}\frac{1}{2}}$ give an increase in the cross sections by several tens of percent. It is useful to recall that effects of order $O(\alpha_{s})$ also increase the values of the cross sections.

As is well known brambilla2011 ; bbl , calculations similar to those given in this paper contain a number of theoretical uncertainties. Our calculation of production cross sections of a $B_{c}$ meson pair via two-photon annihilation mechanism is based on relativistic quark model, which can be considered as a microscopic theory of quark-gluon and photon interactions. The quark model allows you to perform the calculation of observables and different parameters describing the formation of bound states of heavy quarks can be found within its framework. This property of the quark model represents its obvious advantage. In this work we take into account corrections of the second order in relative momenta p and q. The used calculation method can be extended to include fourth-order corrections in accordance with p and q. The arising new parameters can also be estimated within the framework of the quark model itself. But in this calculation, these corrections are included in the theoretical calculation error, which we define at 30 percent. Another important theoretical calculation uncertainty is associated with the determination of the wave functions of the bound states of quarks in the region of relativistic momenta $m_{c}$. We estimate the total error of this type at $5\%$. Then the corresponding error in determining the cross sections (20) for the production of a pair should not exceed $20\%$. In our opinion, this rather approximate estimate is consistent with calculations of the mass spectrum of heavy quarkonia, in which a large error in determining the wave functions in the field of relativistic momenta will give a discrepancy with the observed masses of more than one percent. An important part of the total theoretical error is connected with radiative corrections of order $\alpha_{s}$, which are not considered in this work. It can be said that the use of the Breit Hamiltonian in the calculation of the wave functions leads only to a partial account of corrections of this type. We assume that such radiative corrections may give a 20 percent change in the production cross sections. The total maximum theoretical error can be estimated in $40\%$. To get it, we add the above estimates in quadrature.

Acknowledgements.

The work is supported by Russian Foundation for Basic Research (grant No. 18-32-00023) (F.A.M. and O.S.S.). A.V. Berezhnoy acknowledges the support from ”Basis” Foundation (grant No. 17-12-244-1).

Appendix A The coefficient functions $f^{i,rel}$, $f^{bind}$ and $f^{LO}$ entering in

the $B_{c}$ meson production cross section (16)

$e^{+}+e^{-}\to B_{\bar{b}c}^{+}+B_{b\bar{c}}^{-}$.

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$e^{+}+e^{-}\to B^{\ast+}_{\bar{b}c}+B^{-}_{b\bar{c}}$.

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$e^{+}+e^{-}\to B^{\ast+}_{\bar{b}c}+B^{\ast-}_{b\bar{c}}$.

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