1 Introduction

Standard model (SM) is a well-established theory describing all elementary particles and their interactions. From the beginning, the Gauge and CPT symmetries as well as the Lorentz invariance have been the foundation of this model. The Lorentz symmetry refers to an important feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space while the CPT theorem predicts the equality of some quantities such as life-time, mass, gyromagnetic ratio and charge-to-mass ratio, for particle and anti-particle. Despite of all successes, the SM does not represent a theory of everything because, in one hand, it does not include the gravity and, on the other hand, does not have convincing answers for the candidates of dark matter, matter–antimatter asymmetry, hierarchy problem, etc. To solve these difficulties, new models and extended theories have been developed during past decades. Among all, a fundamental theory that unifies the SM and the gravity would emerge at energies approaching Planck scale (\(\approx 10^{19}\) GeV). Basically, tiny violations of the Lorentz and CPT symmetries could emerge in models unifying gravity with quantum physics [1, 2]. In this context, the Lorentz violations are allowed in string theory, supersymmetry and Horava-Lifshitz gravity [3]. To study Lorentz violations in the context of quantum field theory a new theory has been constructed by Colladay and Kostelecky [4, 5] which is called the SM Extension (SME). It contains the SM, General Relativity and all possible operators that break Lorentz symmetry. Generally, effective field theories (EFTs) such as the SME, introduce Lorentz and CPT violations through spontaneous symmetry breaking caused by hypothetical background fields. The SME is classified into two parts: (i) the minimal version including the operators with dimensions \(d\le 4\) which preserves gauge invariance, conventional quantization, hermiticity, power counting renormalizability and positivity of energy and (ii) the nonminimal prescription which also includes operators of higher dimensions. The structure of SME is one way to study the Lorentz violation (LV) so that an alternative procedure, which is adopted in our work to study the QED interaction processes, is to modify just the SME interactions part via nonminimal couplings. It leads to the improved photon-fermion vertices in QED. In fact, through this approach a nonminimal coupling term is added to the covariant derivative which may be CPT-odd or CPT-even. Heretofore, using this prescription the possibility of Lorentz covariance breakdown in the context of quantum field theory has been extensively studied in the literatures. For example, in Ref. [6] this possibility was employed to determine the LV bounds on Bhabha scattering process and in Ref. [7] the same procedure has been used for studying the spectrum of hydrogen atom to specify the magnitude of bounds on the LV coefficients. Simultaneously with the theoretical developments, many experimental tests on LV corrections have also been carried out and several constraints on LV parameters have been established, e.g. hyperfine structure of muonium ground state [8], clock-comparison experiments [9], hyperfine spectroscopy of hydrogen and anti-hydrogen [10], etc., see also Refs. [11,12,13,14]. The clock anisotropy, which is a spectroscopic experiment, is one of the most precise experiments [15] where the LV parameters are introduced as in the SME. Note that, the SME includes a number of possible terms which violate local Lorentz invariance by coupling to particle spin [9]. Then, in Ref. [15] authors have performed a search for neutron spin coupling to a Lorentz- and CPT-violating background field using a magnetometer with overlapping ensembles of K and \(^3H\) atoms. Therefore, they determined bounds of \(10^{-33}\) GeV on the LV parameters.

In a realistic theory, if the spontaneous CPT and partial Lorentz violation are extended to the four-dimensional spaces, detectable effects might occur in interferometric experiments with neutral kaons [1, 2], neutral \(B_d\) or \(B_s\) mesons [16, 17], or neutral D mesons [16,17,18]. For instance, the quantities parameterizing indirect CPT violation in these systems could be nonzero. However, for scattering processes there are few studies about possible effects of LV on cross sections in order to determine upper bounds on the breaking parameters, see for example the study on Bhabha scattering in Refs. [6, 19] where authors have determined the effects induced by the nonminimal CPT-odd coupling. Collision experiments in high energy scales provide a suitable environment where Lorentz symmetry breaking can be tested. This motivates us to study the effects of LV on inclusive production process of heavy flavored hadrons in pair annihilation. To be specific, we concentrate on the bottom-flavored mesons in the process \(e^+e^-\rightarrow B+jets\) and estimate the magnitude of breaking parameters. To this aim, we first calculate the partonic cross section \(e^+e^-\rightarrow q\bar{q}\) to leading order in QED. To impose the LV corrections, we take a modified nonminimal coupling for the photon-fermions vertex like the terms \(\epsilon _{\mu \nu \alpha \beta }v^\nu F^{\alpha \beta }\) and \(\epsilon _{\mu \nu \alpha \beta }\gamma _5 b^\nu F^{\alpha \beta }\) (vectorial and axial nonminimal couplings) which are added to the covariant derivative [4]. Here, the quantities \(v^\nu \) and \(b^\nu \) must be real as consequences of their origins in spontaneous symmetry breaking and of the presumed hermiticity of the underlying theory. Our analytical results show that the breaking of Lorentz symmetry leads to an unusual dependence of cross section on the orientation of the scattering plane in the center-of-mass (COM) frame. In the following, using the fragmentation function of \(q/\bar{q}\rightarrow B\) we compute the cross section of B-hadron production in pair annihilation. Considering the data at the collision energy \(\sqrt{s}=10.52\) GeV (scale above the \(B\bar{B}\) threshold) we determine the magnitude of LV coefficients in both cases. Since, there are few works on scattering processes in the framework of QED including the mentioned nonminimal couplings, then our work could be a new channel to study the LV effects. Our analytical results for the partonic cross section, which are presented for the first time, could be applied to explore the LV effect on the production process of various types of hadrons at the present and future electron-positron colliders.

This paper is organized as follows. In Sect. 2, the differential cross section for pair annihilation in the presence of vectorial nonminimal coupling is computed. In Sect. 3, the axial-like nonminimal coupling is considered and our computation is iterated. In both sections, the hadron production process through pair annihilation is analyzed and the magnitudes of LV parameters are estimated. Section 4 is devoted to the summary and conclusion.

Fig. 1
figure 1

Feynman diagram for the process \(e^+e^-\rightarrow q\bar{q}\) including the QED (cross symbol) and LV (solid circle) vertices

Full size image

2 Pair annihilation: vectorial nonminimal coupling

In lowest-order, the differential cross section for the pair annihilation process:

$$\begin{aligned} e^+(p_1)+e^-(p_2)\rightarrow \bar{q}(p_1^\prime )+q(p_2^\prime ) \end{aligned}$$
(1)

is given by

$$\begin{aligned} \frac{d\sigma ^{LO}}{d\Omega }=\frac{\overline{|\mathcal {M}|^2}}{64\pi ^2 s}\frac{|\vec {p}_1{^\prime }|}{|\vec {p}_1|}, \end{aligned}$$
(2)

where, \(s=(p_1+p_2)^2=(2E)^2=E_{com}^2\) is the collision energy in the center-of-mass frame and \(|\vec {p_1^\prime }|=|\vec {p}_1|\sqrt{1-4m_q^2/s}\). If we assume \(s\ll m_z^2\), then the contribution of Z-exchange is not taken into account so there will be just one Feynman diagram at LO with a virtual photon intermediated (Fig. 1). As was mentioned in the Introduction, an interesting way to introduce the Lorentz violation in QED is to modify the electron-photon vertex directly. This LV scenario has been proposed in Ref. [20] in the context of topological phases and represents a very simple gauge invariant nonminimal coupling possibility (note that, this coupling does not exist in the SME [4, 5]). In this way, one obtains an extended version of QED characterized by the following nonminimal covariant derivative:

$$\begin{aligned} D_\mu =\partial _\mu +ieA_\mu +igv^\nu F_{\mu \nu }^\star \end{aligned}$$
(3)

where \(F_{\mu \nu }^\star \) is related to the usual field-strength tensor as \(F_{\mu \nu }^\star =\frac{1}{2}\epsilon _{\mu \nu \alpha \beta }F^{\alpha \beta }\) with \(\epsilon ^{0123}=1\) (four-dimensional antisymmetric Levi-Civita tensor). Moreover, e, g and \(v^\mu \) are the electron charge, a coupling constant and a constant four-vector, respectively. In this situation, the additional term sets a nonminimal coupling of the fermion sector to a fixed background \(v^\mu \), however it is clearly gauge invariant. The LV background \(v^\mu \) is responsible for the breaking of Lorentz symmetry at the particle frame, as it may select a preferred direction in space-time. With this modification, the generalized QED Lagrangian (including Lorentz violating CPT-odd term) in the Feynman gauge is written as:

$$\begin{aligned} L= & {} \bar{\psi }(i\gamma ^\mu \partial _\mu -m)\psi -\frac{1}{4} F^{\mu \nu }F_{\mu \nu }-\frac{1}{2}(\partial _\mu A^\mu )^2\nonumber \\{} & {} -e\bar{\psi } \not \!\!A \psi -gv^\nu \bar{\psi }\gamma ^\mu \psi (\partial ^\alpha A^\beta )\epsilon _{\mu \nu \alpha \beta }. \end{aligned}$$
(4)

The additional vertex in the last term is gauge invariant but explicitly violates Lorentz symmetry. Meanwhile, it is not perturbatively renormalizable since the LV coupling constant has mass dimension \([gv^\nu ]=-1\). Considering the above Lagrangian, the Feynman rules could be extracted as in the standard QED. Therefore, the coupling factor in the photon-electron vertex (\(e(p_{in})- e(p_{out})-\gamma (k)\)) is modified as:

$$\begin{aligned} ie\gamma _\alpha \rightarrow ie\gamma _\alpha +g v^\nu k^\beta \gamma ^\mu \epsilon _{\mu \nu \beta \alpha }, \end{aligned}$$
(5)

and the coupling in the photon-quark vertex, i.e., \(q(p_{in})-q(p_{out})-\gamma (k)\), is corrected as:

$$\begin{aligned} iee_q\gamma ^\alpha \rightarrow iee_q\gamma ^\alpha +g v^\sigma k^\eta \gamma ^\kappa \epsilon _{\sigma \eta \kappa \lambda }g^{\lambda \alpha }, \end{aligned}$$
(6)

where, \(e_q\) stands for the fractional charge of quarks and \(k^\eta \) represents the four-momentum carried by the photon line. For the annihilation process (1), at lowest-order perturbative QED the corresponding Feynman amplitude reads:

$$\begin{aligned} \mathcal {M}= & {} -\frac{1}{s}[\bar{u}_q(p_2^\prime )(ie\gamma _{\alpha }+gv^\nu k^\beta \gamma ^\mu \epsilon _{\mu \nu \beta \alpha }) v_q(p_1^\prime )] \nonumber \\{} & {} \times {[}\bar{v}_e(p_1)(iee_q\gamma ^\alpha +gv^\sigma k^\eta \gamma ^\kappa g^{\lambda \alpha }\epsilon _{\kappa \sigma \eta \lambda }) u_e(p_2)]\nonumber \\= & {} \mathcal {M}_0+\mathcal {M}_1+\mathcal {M}_2, \end{aligned}$$
(7)

where \(k=p_1+p_2\). Here, \(\mathcal {M}_0\) is the matrix element in conventional QED (Fig. 1A):

$$\begin{aligned} \mathcal {M}_0=\frac{e_qe^2}{s}[\bar{u}_q(p_2^\prime )\gamma _{\alpha } v_q(p_1^\prime )][\bar{v}_e(p_1)\gamma ^\alpha u_e(p_2)]. \end{aligned}$$
(8)

The term \(M_1\), linear in \(gv^\mu \), is formed by a usual vertex and another with the Lorentz-violating term (Fig. 1B, C):

$$\begin{aligned} \mathcal {M}_1= & {} -i\frac{e(1+e_q)}{s}gv^\nu (p_1+p_2)^\beta \epsilon _{\mu \nu \beta \alpha }\nonumber \\{} & {} \times {[}\bar{u}_q(p_2^\prime )\gamma ^\mu v_q(p_1^\prime )][\bar{v}_e(p_1)\gamma ^\alpha u_e(p_2)]. \end{aligned}$$
(9)

The matrix element \(M_2\) results purely from the Lorentz-violating vertices then it is quadratic in \(gv^\mu \) as (Fig. 1D):

$$\begin{aligned} \mathcal {M}_2= & {} -\frac{g^2}{s}(\epsilon _{\mu \nu \beta \alpha }\epsilon _{\kappa \sigma \eta \lambda } g^{\lambda \alpha }v^\nu v^\sigma (p_1+p_2)^\beta (p_1+p_2)^\eta )\nonumber \\{} & {} \times {[}\bar{u}_q(p_2^\prime )\gamma ^\mu v_q(p_1^\prime )][\bar{v}_e(p_1)\gamma ^\kappa u_e(p_2)]. \end{aligned}$$
(10)

To evaluate the unpolarized cross section, we need the mean squared amplitude which is computed taking an average over the spin of incoming particles and summing over the outgoing particles. This could be accomplished using the completeness relations which lead to traces of Dirac matrices products. By setting \(p_1^2=p_2^2=m_e^2\approx 0\), one has:

$$\begin{aligned} \overline{|\mathcal {M}_0|^2}= & {} \frac{1}{1+2s_e}\frac{1}{1+2s_e}\sum _{spin,color}\nonumber \\ |\mathcal {M}_0|^2= & {} \frac{N_c}{4}\sum _{spin}|\mathcal {M}_0|^2\nonumber \\= & {} \frac{N_ce^4e_q^2}{4s^2}A_{\alpha \beta }A^{\alpha \beta }, \end{aligned}$$
(11)

where \(N_c=3\) is the color number of quarks and \(A_{\alpha \beta }=Tr[(\displaystyle \not \!\!p_2^\prime +m_q)\gamma _{\alpha } (\displaystyle \not \!\!p_1^\prime -m_q)\gamma _\beta ]\) and \(B^{\alpha \beta }=Tr[\displaystyle \not \!\!p_1\gamma ^\alpha \displaystyle \not \!\!p_2\gamma ^\beta ]\). Using the FeynCalc package [21], it reads:

$$\begin{aligned} \overline{|\mathcal {M}_0|^2}= & {} \frac{8N_ce^4e_q^2}{s^2}\bigg [(p_1\cdot p_1^\prime )(p_2\cdot p_2^\prime )\nonumber \\{} & {} +(p_1\cdot p_2^\prime )(p_2\cdot p_1^\prime )+m_q^2(p_1\cdot p_2)\bigg ]. \end{aligned}$$
(12)

Considering the kinematics of process (1) in the COM frame, the four-vector momenta take the form:

$$\begin{aligned}{} & {} p_1=E(1, 0, 0, 1),\nonumber \\{} & {} p_1^\prime =(E, p^\prime sin\theta \cos \phi , p^\prime sin\theta \sin \phi , p^\prime \cos \theta )\nonumber \\{} & {} p_2=E(1, 0, 0, -1),\nonumber \\{} & {} p_2^\prime =(E, -p^\prime sin\theta \cos \phi , -p^\prime sin\theta \sin \phi , -p^\prime \cos \theta ) \end{aligned}$$
(13)

where we assumed the \(\hat{Z}\)-axis as the collision axis and labeled \(|\vec {p_1^\prime }|=|\vec {p_2^\prime }|=p^\prime \) in which \(p^\prime =\sqrt{s/4-m_q^2}\). Then, one has:

$$\begin{aligned}{} & {} \overline{|\mathcal {M}_0(e^+e^-\rightarrow q\bar{q})|^2}\nonumber \\{} & {} \quad = (4\pi \alpha )^2N_ce_q^2 \bigg [1+\cos ^2\theta +\frac{4m_q^2}{s}(1-\cos ^2\theta )\bigg ], \end{aligned}$$
(14)

where \(\alpha \) is the tiny coupling constant. The above result is the mean squared amplitude in QED which is independent of the azimuthal angle \(\phi \). Using the same technique, we arrive at:

$$\begin{aligned} \overline{|\mathcal {M}_1|^2}= & {} \frac{N_ce^2g^2(1+e_q)^2}{4s^2}v^\nu v^\kappa (p_1+p_2)^\beta (p_1+p_2)^\gamma \nonumber \\{} & {} \times \epsilon _{\mu \nu \beta \alpha }\epsilon _{\eta \kappa \gamma \theta } Tr[(\displaystyle \not \!\!p_2^\prime +m_q) \gamma ^\mu (\displaystyle \not \!\!p_1^\prime -m_q)\gamma ^\eta ] \nonumber \\{} & {} \times Tr[\displaystyle \not \!\!p_1 \gamma ^\alpha \displaystyle \not \!\!p_2 \gamma ^\theta ]. \end{aligned}$$
(15)

In computing this contribution, we assume an arbitrary direction for the space-like component of the four-vector \(v^\mu \) and define it as \(v^\mu =(v_0, v\sin \theta _v\cos \phi _v, v\sin \theta _v\sin \phi _v, v\cos \theta _v)\). The polar (\(\theta _v\)) and azimuthal (\(\phi _v\)) angles are fixed relative to the experimental setup at a given time, so for the dot-products of four-momenta we have:

$$\begin{aligned}{} & {} p_1\cdot v=E(v_0-v\cos \theta _v), \nonumber \\{} & {} p_1^\prime \cdot v=Ev_0-vp^\prime \sin \theta \sin \theta _v\cos (\phi -\phi _v)\nonumber \\ {}{} & {} \quad -vp^\prime \cos \theta \cos \theta _v\nonumber \\{} & {} p_2\cdot v=E(v_0+v\cos \theta _v), \nonumber \\{} & {} p_2^\prime \cdot v=Ev_0+vp^\prime \sin \theta \sin \theta _v\cos (\phi -\phi _v)\nonumber \\ {}{} & {} \quad +vp^\prime \cos \theta \cos \theta _v \end{aligned}$$
(16)

Therefore, the mean squared amplitude reads:

$$\begin{aligned} \overline{|\mathcal {M}_1|^2}= & {} 4\pi \alpha N_cg^2(1+e_q)^2sv^2\bigg \{\bigg [1-\frac{4m_q^2}{s}\bigg ] \nonumber \\{} & {} \times \bigg (\frac{1}{2} \sin 2\theta \sin 2\theta _v\cos (\phi -\phi _v)\nonumber \\{} & {} +2\cos ^2\theta \cos ^2\theta _v +\sin ^2\theta _v+\sin ^2\theta \bigg ) \nonumber \\ {}{} & {} -\sin ^2\theta _v +\frac{8m_q^2}{s}\bigg \}. \end{aligned}$$
(17)

Three important conclusions are: (i) the LV result is of first-order in the fine structure constant, as opposed to QED which is of second order in \(\alpha \) (14), (ii) if we take \(m_q=0\) the squared amplitude \(\overline{|\mathcal {M}_0|^2}\) is independent of the collision energy (s) while the term \(\overline{|\mathcal {M}_1|^2}\) depends on the collision energy in any case, and (iii) the amplitude \(\overline{|\mathcal {M}_1|^2}\) depends on the azimuthal angle \(\phi \) so that this nonstandard feature acts as LV signature to be searched in experiments. This means that the Lorentz symmetry breaking leads to an unusual dependence of cross section on the orientation of the scattering plane in the COM reference frame. Moreover, the amplitude \(\overline{|\mathcal {M}_2|^2}\) (which is proportional with \(g^4v^4\)) reads:

$$\begin{aligned} \overline{|\mathcal {M}_2|^2}= & {} N_cg^4v^4s^2\bigg \{\bigg [1-\frac{4m_q^2}{s}\bigg ] \nonumber \\ {}{} & {} \times \bigg (\sin ^2\theta \sin ^2\theta _v\cos ^2(\phi -\phi _v)(1+\cos ^2\theta _v)\nonumber \\{} & {} +\sin 2\theta \sin \theta _v\cos ^3\theta _v\cos (\phi -\phi _v)\nonumber \\{} & {} +\frac{1}{8}\cos ^2\theta (7+\cos 4\theta _v)\bigg )+\cos ^2\theta _v+\frac{4m_q^2}{s}\bigg \}.\nonumber \\ \end{aligned}$$
(18)

The differential cross section is now expressed as

$$\begin{aligned} \frac{d\sigma _0^{LV}}{d\Omega }=\frac{\sqrt{1-\frac{4m_q^2}{s}}}{64\pi ^2 s}\overline{|\mathcal {M}|^2}, \end{aligned}$$
(19)

where \(\overline{|\mathcal {M}|^2}=\overline{|\mathcal {M}_0|^2}+\overline{|\mathcal {M}_1|^2}+\overline{|\mathcal {M}_2|^2}+ {\textbf {Interference}} {\textbf {terms}}\).

The total mean squared amplitude considering all contributions is expressed as:

$$\begin{aligned}{} & {} \overline{|\mathcal {M}|^2}\nonumber \\ {}{} & {} \quad =(4\pi \alpha )^2N_ce_q^2\bigg [1+\cos ^2\theta +\frac{4m_q^2}{s}(1-\cos ^2\theta )\bigg ]\nonumber \\{} & {} \quad +4\pi \alpha sN_c(gv)^2\bigg (1-\frac{4m_q^2}{s}\bigg )\bigg [-2e_q\sin ^2\theta \sin ^2\theta _v\cos ^2\nonumber \\{} & {} \quad \times (\phi -\phi _v)- 4e_q\cos ^2\theta \cos ^2\theta _v-e_q(3+\cos 2\theta _v)\frac{1+\frac{4m_q^2}{s}}{1-\frac{4m_q^2}{s}}\nonumber \\{} & {} \quad +\frac{1}{2}(1-3e_q+e_q^2)\sin 2\theta \sin 2\theta _v\cos (\phi -\phi _v)\nonumber \\ {}{} & {} \quad +(1+e_q^2)\cos ^2\theta \cos 2\theta _v\nonumber \\{} & {} \quad +(1+e_q^2)\bigg (1+\frac{4m_q^2}{s}\cos ^2\theta _v\bigg )\bigg (1-\frac{4m_q^2}{s}\bigg )^{-1}\bigg ]\nonumber \\{} & {} \quad +\frac{N_cs^2(gv)^4}{8}\bigg [8+8\cos ^2\theta _v+\bigg (1-\frac{4m_q^2}{s}\bigg )\nonumber \\{} & {} \quad \times \bigg (\cos ^2\theta (7+\cos 4\theta _v) + 8\sin 2\theta \sin \theta _v\cos ^3\theta _v\cos \nonumber \\{} & {} \quad (\phi -\phi _v)-8+4\sin ^2\theta \sin ^2 \theta _v\nonumber \\ {}{} & {} \quad \cos ^2(\phi -\phi _v)(3+\cos 2\theta _v)\bigg )\bigg ]. \end{aligned}$$
(20)

By integrating over the polar (\(0\le \theta \le \pi \)) and azimuthal (\(0\le \phi \le 2\pi \)) angles the total cross section is obtained as:

$$\begin{aligned} \sigma _0^{LV}= & {} N_c\frac{\sqrt{1-\frac{4m_q^2}{s}}}{48\pi } \bigg (1+\frac{2m_q^2}{s}\bigg )\Big \{\frac{64\pi ^2\alpha ^2 e_q^2}{s}\nonumber \\ {}{} & {} +[4\pi \alpha (gv)^2(1+e_q^2-4e_q) +s(gv)^4]\nonumber \\ {}{} & {} [3+\cos 2\theta _v]\Big \}. \end{aligned}$$
(21)

In Ref. [6], it is shown that for the fixed background \(\vec {v}\) perpendicular to the beam collision (\(\theta _v=\pi /2\)), the LV effect is maximal and it is characterized by a set of periodic sharp peaks, see Fig. 2 in Ref. [6]. A similar result was reported in Ref. [22] for the Compton scattering with the LV term in the kinetic sector. We also adopt this case in our computations. To conclude this section, we consider the inclusive pair annihilation process to produce B-hadron:

$$\begin{aligned} e^{+}e^{-}\rightarrow B+X, \end{aligned}$$
(22)

where, X shows unobserved jets. To study the hadron production process in the perturbative QED, the factorization theorem of improved parton model plays the key role. Based on this theorem [23], the cross section of hadron production can be expressed as a sum of convolutions of hard scattering result \(d\sigma _q(y, \mu _R, \mu _F)/dy\) with the nonperturbative fragmentation functions (FFs) of B-meson from the initial parton \(q(=u,\bar{u}, \ldots , b,\bar{b})\), i.e., \(D^B_q (x, \mu _F)\). It reads:

$$\begin{aligned} \frac{d\sigma _B}{dz}= \sum _q\int \limits _z^1\frac{dy}{y} \frac{d\sigma _q}{dy}\left( y, \mu _R, \mu _F\right) D_q^B\bigg (\frac{z}{y}, \mu _F\bigg ). \end{aligned}$$
(23)

We label the four-momenta of B-meson and the intermediate gauge photon by \(p_B\) and k, respectively, so that \(s=k^2\). Then, the scaling variable z is defined as \(z=2(p_B.k)/k^2\) which is simplified to \(z=2E_B/\sqrt{s}\) in the COM frame. It now refers to the B-meson energy scaled to the beam energy. In the COM frame, the variable y is also defined as \(y=2E_q/\sqrt{s}\). The \(D_q^B(x,\mu _F)\)-FF (with \(x=z/y\)) indicates the probability density of B-meson production from the initial quark q which carries the fraction \(x(=E_B/E_q)\) of the energy of parent quark. In Eq. (23), the scales \(\mu _R\) and \(\mu _F\) show the renormalization and factorization scales, respectively, so we conventionally adopt \(\mu _R^2=\mu _F^2=s\).

Having the cross section (21), the LO differential cross section at the parton level is given by:

$$\begin{aligned} \frac{d}{dy}\sigma _0^{LV}(e^+e^-\rightarrow q\bar{q})=\sigma ^{LV}_0\delta (1-y). \end{aligned}$$
(24)

By substituting the above result into Eq. (23) one gets:

$$\begin{aligned} \frac{d\sigma _B}{dz}(e^+e^-\rightarrow B+jets)=2 \sigma ^{LV}_0 \sum _q D_q^B(z, \mu _F), \end{aligned}$$
(25)

where we assumed \(D_q^B(z, \mu )=D_{\bar{q}}^{\bar{B}}(z, \mu )\). The primary products of \(b/\bar{b}\) hadronization are the bottom-flavored mesons \(B^+, B^0, B^0_s\) and their charge conjugates. Since, the most probability density for production of a specific meson comes from the fragmentation of a quark which is itself a constituent quark of desired meson then the splitting contribution of \(b\rightarrow B\) is governed. Then, the relation (25) for the inclusive process \(e^+e^-\rightarrow BX\) is estimated as:

$$\begin{aligned} \frac{d\sigma _B}{dz}(e^+e^-\rightarrow B+X)\approx 2 \sigma ^{LV}_0 D_b^B(z, \mu _F). \end{aligned}$$
(26)

Then, the total cross section reads:

$$\begin{aligned} \sigma _B(e^+e^-\rightarrow BX)\approx 2 \sigma ^{LV}_0 Br(b\rightarrow B), \end{aligned}$$
(27)

where \(Br(b\rightarrow B)=\int _0^1 dz D_b^B(z, \mu _F)\) represents the branching fraction for the transition \(b\rightarrow B\).

In Ref. [24], the nonperturbative fragmentation function of \(b\rightarrow B\) at next-to-next-to-leading order (NNLO) is obtained through a global fit to \(e^+e^-\rightarrow B+Jets\) annihilation data from the ALEPH, DELPHI, and OPAL collaborations at CERN LEP1 and the SLD collaboration at SLAC SLC. Specifically, the power ansatz \(D_b^B(z,\mu _0=m_b)=Nz^\alpha (1-z)^\beta \) was used as the initial condition for the \(b\rightarrow B\) FF at \(\mu _0=4.5\) GeV, while the gluon and light-quark FFs were generated via the DGLAP evolution [25]. The fit yielded \(N =1805.896, \alpha =14.168\), and \(\beta =2.341\) with \(\chi ^2=1.104\). Having the partonic cross section (Eq. (21)) and \(D_b^B(z, \mu _0)\)-FF, the theoretical cross section of B-hadron production could be computed. Note that, the \(D_b^B\)-FF given in Ref. [24] has been extracted at the initial scale \(\mu _{0F}=m_b\) which should be evaluated to the desired energy by solving the DGLAP evaluation equations [25]. Considering the experimental data for the inclusive production cross section of B-mesons through pair annihilation, reported by the Belle and OPAL Collaborations [26, 27], one can approximate the products of parameters gv which are free parameters in Eq. (21). To determine the value of gv we consider the size of cross section at the energy scale \(E_{com}=10.52\) GeV which is just above threshold for B-meson production. Since the data has been reported for \(\sqrt{s}\ge 2m_b\) then the b-quark mass effect seems to be important so we preserve it in our calculation. For the COM energy \(E_{cm}=10.52\) GeV, the cross section has been reported as \(\sigma =3.27\) nb. Considering the theoretical cross section (27) and the unit conversion 1 GeV\(^{-2}=0.389379\times 10^6\) nb we obtain \(gv\approx 7.5\times 10^{-13}\) eV\(^{-1}\) which is comparable with the value \(gv\le 10^{-12}\) eV\(^{-1}\) reported in Ref. [6] and \(gv\le 10^{-3}\) GeV\(^{-1}\) in Ref. [28]. In Ref. [6], authors have studied the effect of Lorentz violating nonminimal coupling on the Bhabha scattering and in Ref. [28] the Compton and the Bhabha scatterings as well as electron-positron annihilation have been studied in QED. We also checked our result with the Data Tables for Lorentz and CPT Violation [29] and found good agreement with the one presented in Table D22-part 3 (Nonminimal QED couplings, \(d \ge 5\)), i.e., \(gv \equiv \xi <10^{-3}\) GeV\(^{-1}\) (the LV background \(\xi \) is a nonminimal coupling with canonical dimensions of inverse mass, as in our work.).

In above calculation, we provided a way to obtain the value of LV parameter from the analyses of B-meson production in the theory of QED. In the SM, the total partonic cross section in higher-order approximations is expressed as [30]:

$$\begin{aligned} \frac{\sigma _{tot}}{\sigma _0}(e^+e^-\rightarrow q\bar{q})\approx 1 +\frac{\alpha _s}{\pi }+1.411 \alpha _s^2+\cdots , \end{aligned}$$
(28)

where, \(\alpha _s\) is the QCD coupling constant and \(\sigma _0=4\pi \alpha ^2/(3s)\) is the cross section of \(e^+e^-\rightarrow \mu ^-\mu ^+\) at LO when one sets \(m_e=m_\mu =0\). The inclusion of QCD effects would improve the value of gv and should allow a better comparison with the results encountered for atomic clocks or torsion balances.

3 Pair annihilation: axial-like nonminimal coupling

Another possible way for coupling the Lorentz-violating background to the fermion field is a torsion-like nonminimal coupling which has a chiral character. In this way, the nonminimal covariant derivative reads [7, 31]:

$$\begin{aligned} D_\mu =\partial _\mu +ieA_\mu +ig_5 \gamma ^5 b^\nu F_{\mu \nu }^\star \end{aligned}$$
(29)

This leads to the modified photon-electron and photon-quark vertices as:

$$\begin{aligned}{} & {} ie\gamma _\beta \rightarrow ie\gamma _\beta +g_5\gamma ^5 b^\nu k^\alpha \gamma ^\mu \epsilon _{\mu \nu \alpha \beta }, \nonumber \\{} & {} iee_q\gamma ^\alpha \rightarrow iee_q\gamma ^\alpha +g_5\gamma ^5 b^\sigma k^\eta \gamma ^\kappa \epsilon _{\kappa \sigma \eta \lambda }g^{\lambda \alpha }, \end{aligned}$$
(30)

where, \(b^\mu =(b_0, \vec {b})\) defines a privileged direction in the space-time. With this new coupling, the LO amplitude for the annihilation process (1) reads:

$$\begin{aligned} \mathcal {M}= & {} -\frac{1}{s}[\bar{u}_q(p_2^\prime )(ie\gamma _{\alpha }+g_5b^\nu k^\beta \gamma ^5\gamma ^\mu \epsilon _{\mu \nu \beta \alpha }) v_q(p_1^\prime )]\nonumber \\ {}{} & {} \times [\bar{v}_e(p_1)(iee_q\gamma ^\alpha +g_5b^\sigma k^\eta \gamma ^5\gamma ^\kappa g^{\lambda \alpha }\epsilon _{\kappa \sigma \eta \lambda }) u_e(p_2)],\nonumber \\ \end{aligned}$$
(31)

where as before, \(k=p_1+p_2\) is the four-momentum of intermediate photon. Then, the corresponding mean squared amplitude reads:

$$\begin{aligned}{} & {} \overline{|\mathcal {M}|^2}=N_ce^4e_q^2\bigg (1+\cos ^2\theta +\frac{4m_q^2}{s}(1-\cos ^2\theta )\bigg )\nonumber \\{} & {} \quad +N_c\sqrt{1-\frac{4m_q^2}{s}}\Bigg \{ (bg_5)e^3e_q\sqrt{s}\bigg (2\sin \theta \sin \theta _v\nonumber \\{} & {} \quad \times \cos (\phi -\phi _v)\bigg [\cos \theta \sqrt{1-\frac{4m_q^2}{s}}-e_q\bigg ]\nonumber \\{} & {} \quad +\cos \theta _v\bigg [\frac{3+\cos 2\theta +\frac{8m_q^2}{s} \sin ^2\theta }{\sqrt{1-\frac{4m_q^2}{s}}}-4e_q\cos \theta \bigg ]\bigg )\nonumber \\{} & {} \quad +se^2(bg_5)^2\bigg (-e_q\cos \theta (5+3\cos 2\theta _v)+\bigg (e_q^2\nonumber \\{} & {} \quad +(1+e_q^2)\cos ^2\theta \cos 2\theta _v\bigg )\sqrt{1-\frac{4m_q^2}{s}}\nonumber \\{} & {} \quad +\bigg (1+\frac{4m_q^2}{s}\cos ^2\theta _v\bigg )\bigg (1 -\frac{4m_q^2}{s}\bigg )^{-\frac{1}{2}}\nonumber \\{} & {} \quad +\sin \theta \sin 2\theta _v\cos (\phi -\phi _v)\nonumber \\ {}{} & {} \quad \times \bigg [-3e_q+(1+e_q^2)\cos \theta \sqrt{1-\frac{4m_q^2}{s}}\bigg ]\bigg )\nonumber \\{} & {} \quad +e(bg_5)^3s^{\frac{3}{2}}\bigg (-\frac{1}{2} \cos \theta (7\cos \theta _v+\cos 3\theta _v)\nonumber \\{} & {} \quad - \sin \theta \sin \theta _v(3+\cos 2\theta _v)\cos (\phi -\phi _v)+e_q\sqrt{1 -\frac{4m_q^2}{s}}\nonumber \\{} & {} \quad \times \bigg [2\cos \theta _v(1+\cos ^2\theta \cos ^2\theta _v)+2\sin 2 \theta \sin \theta _v\cos ^2\theta _v\nonumber \\{} & {} \quad \times \cos (\phi -\phi _v)+\sin ^2\theta \sin \theta _v\sin 2\theta _v\cos ^2(\phi -\phi _v)\bigg ]\bigg )\nonumber \\{} & {} \quad +\frac{s^2(bg_5)^4}{16}\sqrt{1-\frac{4m_q^2}{s}}\nonumber \\{} & {} \quad \times \bigg [15+8\cos 2\theta _v +2\cos ^2\theta \cos 4\theta _v+7\cos 2\theta \nonumber \\{} & {} \quad + 8\sin ^2\theta \sin ^2\theta _v(3+\cos 2\theta _v)\cos ^2(\phi -\phi _v)\nonumber \\{} & {} \quad +16\sin 2\theta \sin \theta _v\cos ^3\theta _v\cos (\phi -\phi _v)\bigg ]\Bigg \}, \end{aligned}$$
(32)

where the presence of odd-order corrections in \(bg_5\) is to be noticed. In the previous result, Eq. (20), we notice that the interference terms of odd-order cancel each other and just even-order corrections in gv are appeared. Putting the mean squared amplitude in Eq. (19) and integrating over the polar and azimuthal angles the total cross section is now obtained as:

$$\begin{aligned} \sigma _0^{LV}= & {} \frac{N_c}{48\pi }\sqrt{1-\frac{4m_q^2}{s}} \Bigg \{\frac{4\pi \alpha }{s}\bigg [1+\frac{2m_q^2}{s}\bigg ]\nonumber \\{} & {} \times \bigg (16e_q(bg_5)\sqrt{\pi \alpha s}\cos \theta _v\nonumber \\ {}{} & {} +s(bg_5)^ 2(3+\cos 2\theta _v)+16\pi \alpha e_q^2\bigg )\nonumber \\{} & {} +\bigg [1-\frac{4m_q^2}{s}\bigg ]\bigg (\big (s(bg_5)^4+4\pi \alpha e_q^2(bg_5)^2\big )\nonumber \\{} & {} (3+\cos 2\theta _v)+16e_q\sqrt{\pi \alpha s}(bg_5)^3\cos \theta _v\bigg )\Bigg \}. \end{aligned}$$
(33)

Now, an numerical analysis similar to the one described in previous section allows us to determine the breaking parameter as \(bg_5=6.53\times 10^{-15}\) eV\(^{-1}\) which is comparable with the result reported in Ref. [6], i.e., \(bg_5\le 10^{-14}\) eV\(^{-1}\).

4 Summary and conclusion

Since, collision experiments in high energy physics provide a suitable environment where Lorentz symmetry breaking can be tested then in this work we, for the first time, studied the implications of this symmetry breaking on B-meson production through pair annihilation. For this aim, we first discussed the modifications on the QED process \(e^+e^-\rightarrow q\bar{q}\) due to the inclusion of a nonminimal coupling between the fermion and gauge fields. We presented our analytical results for the corresponding differential cross sections in the presence of vector and axial couplings by introducing the vector backgrounds \(v^\nu \) and \(b^\nu \). Our results show that, in this case the Lorentz symmetry breaking leads to an nontrivial dependence of cross section on the orientation of the scattering plane in the COM frame.

In the following, considering the nonperturbative \(D_b^B(z, \mu _F)\)-FF we computed the LO cross section of \(\sigma (e^+e^-\rightarrow B+jets)\) and compared with the experimental data taken from the OPAL and Belle Collaborations. From this comparison, we obtained the magnitude of corresponding LV coefficients which are in good consistencies with other analysis resulted from different processes. Since, there is no mathematical formalism or physical principle which can constrain the magnitude order of the LV coefficients, thus we hope that our results may be useful as a guide in the investigation of Lorentz violation phenomena in high energy scattering processes. Note that, in the present work we used the \(D_b^B(z, \mu )\)-FF to investigate the B-meson production in pair annihilation but it would also be possible to study other hadron production processes considering their corresponding FFs. For example, in Refs. [32,33,34,35,36], the FFs of D- and \(\eta \)-mesons as well as \(\Lambda _c\)- and \(\Omega _c\)-baryons have been calculated up to NNLO in the theory of perturbative QCD. These processes have enough potential to be tested in the present and future \(e^+e^-\) colliders such as the international linear collider (ILC) and circular electron-positron collider (CEPC). The CEPC is initially planned to have a collision energy of 500 GeV with the possibility for a later upgrade to 1 TeV. Up to now, most studies on the production of doubly heavy meson \(B_c\) have concentrated on the hadronic production due to the numerous production rate at high energy colliders such as CERN LHC or Tevatron [37,38,39]. The CEPC machine can be a potentially good platform for investigating the \(B_c\) meson. In comparison with hadron high energy colliders there will be less backgrounds at the \(e^+e^-\) collider. Thus, it would be suitable for studying this doubly heavy meson, precisely. Furthermore, the production mechanism of \(B_c\) at the \(e^+e^-\) collider is much simpler than the one at hadron colliders. Therefore, our analytical results (Eqs. (21) and (33)) as well as our proposed channel could be applicable in the present and future generation of high energy colliders.