Searching $B_{c}^{\ast}$ via conservation laws

The $B_{c}$ meson is unique in the Standard Model (SM) as its members are composed of heavy quarks with two different flavors, beauty ($b$) and charm ($c$). The $B_{c}$ mesons lie intermediate between ($c\bar{c}$) and ($b\bar{b}$) states both in mass and size, while the different quark flavors leads to much richer dynamics. On the other hand, the ground state of $B_{c}$ mesons, unlike the charmonium and bottomonium, cannot annihilate into gluons or photons, providing an idea place to examine the heavy quarks. Study on the $B_{c}$ mesons can deepen our understanding of both the strong and the weak interactions, revealing the underlying physics of the heavy quark dynamics. Last but not least , it provides a unique hunting ground for searching new physics beyond the SM.

The ground state of $B_{c}$ meson was first observed by the CDF Collaboration at Fermilab Abe:1998fb in 1998, and there have been continuous measurements on both the mass CDF:2007umr ; LHCb:2012ihf ; LHCb:2020ayi and the lifetime CDF:2006kbk ; D0:2008thm via the exclusive decay $B_{c}^{+}\to J/\psi\pi^{+}$ and the semileptonic decay $B_{c}^{+}\to J/\psi l^{+}\nu_{l}$. In 2014, the ATLAS Collaboration reported a structure with the mass of $(6842\pm 9)$ MeV Aad:2014laa , which is consistent with the value predicted for $B_{c}(2S)$. In 2019, the excited $B_{c}(2^{1}S_{0})$ was confirmed and $B_{c}^{\ast}(2^{3}S_{1})$ states have been observed in the $B_{c}^{+}\pi^{+}\pi^{-}$ invariant mass spectrum by the CMS and LHCb Collaborations, with their masses determined to be $(6872.1\pm 2.2)$ and $(6841.2\pm 1.5)$ MeV Sirunyan:2019osb ; Aaij:2019ldo , respectively. The $B_{c}(2^{1}S_{0})^{+}$ decays to $B_{c}^{+}(1^{1}S_{0})\pi^{+}\pi^{-}$ directly, and the $B_{c}^{\ast}(2^{3}S_{1})^{+}$ state decay to $B_{c}^{\ast+}(1^{3}S_{1})\pi^{+}\pi^{-}$ followed by $B_{c}^{\ast+}(1^{3}S_{1})\to B_{c}^{+}(1^{1}S_{0})\gamma$. Since the soft photon in the intermediate decay $B_{c}^{\ast+}(1^{3}S_{1})\to B_{c}^{+}(1^{1}S_{0})\gamma$ was not reconstructed, the mass of $B_{c}^{\ast}(2^{3}S_{1})$ meson appears lower than that of $B_{c}(2^{1}S_{0})$. This peculiar behaviors of the mass hierarchy makes $B_{c}^{\ast}(1^{3}S_{1})$ uniquely important in studying the $B_{c}$ meson family.

In the following, we will abbreviate $B_{c}^{\ast}(1^{3}S_{1})$ as $B_{c}^{\ast}$ so long as it does not cause confusion. Study on the $B_{c}^{\ast}$ can complete the precise measurements of the spectrum of the $B_{c}$ family, and the confirmation of its existence is of great importance for the understanding of strong interaction dynamics at low energy. On the mass of $B_{c}^{\ast}$, the theoretical predictions range discrepantly from $6326$ to $6346$ MeV Ding:2021dwh ; Eichten:1994gt ; Godfrey:2004ya ; Mathur:2018epb ; Li:2019tbn ; Asghar:2019qjl ; Eichten:2019gig , and an experimental measurement is still lacking. The dominant decay mode $B_{c}^{\ast}\to B_{c}\gamma$ has not yet been observed, partly due to the noisy soft photon background of the hadron collider. To identify $B_{c}^{\ast}$ in the experiments, one of the important tasks is to distinguish them from $B_{c}$. In this study, we propose two methods based on the conservation laws:

• •

  From the angular momentum conservation, the $J/\psi$ can only possess a zero helicity from $B_{c}^{+}\to J/\psi\pi^{+}$ as $B_{c}^{+}$ is spin-0. In contrast, the $J/\psi$ of $B_{c}^{\ast+}\to J/\psi\pi^{+}$ can have either positive, zero, or negative helicities (see Fig. 1).

• •

  As $B_{c}^{+}\to B^{+}\phi$ is kinematically forbidden, $B_{c}^{\ast+}\to B^{+}\phi$ provides a clean channel.

Their responsible quark diagrams at the tree level are given in Fig. 2 , where the hadronizations take place in the blue regions. As the W boson is color blind, the decays are color allowed and color suppressed, respectively.

The rest of the paper is organized as follows. The primary formulas in our calculation are presented in Sec. II. We give the numerical analysis and results in Sec. III. We conclude the study in Sec. IV.

To extract the helicity information of $J/\psi$ as well as calculate the branching fractions, we give the helicity formalism of the decays in this section. The helicity information of $J/\psi$ can be obtained from $B_{c}^{(*)+}\to J/\psi(\to l^{-}l^{+})\pi^{+}$ with $l=e,\mu$ . The advantage of the helicity analysis is that it can easily cooperate with the sequential decays and has a clear view of physical meaning Gutsche:2013oea .

Taking the initial $B_{c}^{\ast+}$ as unpolarized, the angular distributions of $B_{c}^{(\ast)+}\to J/\psi(\to l^{-}l^{+})\pi^{+}$ are given as

$$\frac{\partial\Gamma^{(\ast)}}{\partial\cos\theta}\propto\sum_{\lambda=\pm,0\,,l=\pm}\left|H_{\lambda}^{(\ast)}d^{1}(\theta)^{\lambda}\,_{l}\right|^{2}\propto 1-P_{2}+\frac{3}{2}\alpha^{(\ast)}P_{2}\,,$$

(1)

where $H_{\lambda}^{(\ast)}$ are the helicity amplitudes with the subscripts denoting the helicity of $J/\psi$, $d^{1}(\theta)$ the Wigner $d$-matrix for $J=1$, $\theta$ defined in the helicity frame of $J/\psi$ (see Fig. 1), and

	$\displaystyle P_{2}=\frac{1}{2}\left(3\cos^{2}\theta-1\right)\,,$
	$\displaystyle\alpha^{(\ast)}=\frac{|H_{+}^{(\ast)}|^{2}+|H_{-}^{(\ast)}|^{2}}{|H_{+}^{(\ast)}|^{2}+|H_{-}^{(\ast)}|^{2}+|H_{0}^{(\ast)}|^{2}}\,.$		(2)

Here, $\alpha$ has the physical meaning of the nonzero-polarized fraction of $J/\psi$. Notice that $H_{\pm}$ are forbidden by the angular momentum conservation, resulting in

$$\alpha=0\,.$$

(3)

To further extract the helicity information, we define

$${\cal A}^{(\ast)}=\frac{1}{\Gamma}\left(\int_{|\cos\theta|<x_{0}}\frac{\partial\Gamma^{(\ast)}}{\partial\cos\theta}d\cos\theta-\int_{|\cos\theta|>x_{0}}\frac{\partial\Gamma^{(\ast)}}{\partial\cos\theta}d\cos\theta\right)=\left(3x_{0}-\frac{3}{2}\right)\alpha^{(\ast)}\,,$$

(4)

where $x_{0}$ is chosen to satisfy

$$x_{0}^{3}-3x_{0}+1=0\,,$$

(5)

which is found to be $x_{0}\approx 0.3473$ .

The experiments of $B_{c}^{\ast+}$ are polluted by the off-shell contributions from $B_{c}^{+}$ at LHC. Thus, we define the event-average $\overline{{\cal A}}$ as

$$\overline{{\cal A}}=r{\cal A}+r^{*}{\cal A}^{\ast}=r^{*}{\cal A}^{\ast}\,,$$

(6)

as well as the event-average nonzero-polarized fraction as

$$r\frac{\partial\Gamma}{\partial\cos\theta}+r^{*}\frac{\partial\Gamma^{\ast}}{\partial\cos\theta}\propto 1-P_{2}+\frac{3}{2}\overline{\alpha}P_{2}\,,$$

(7)

with

$$r^{(\ast)}=\frac{N_{B_{c}^{(\ast)}}}{N_{B_{c}}+N_{B_{c}^{\ast}}}\,,$$

(8)

where $N_{B_{c}^{(\ast)}}$ is the number of the observed events in $B_{c}^{(\ast)+}\to J/\psi(\to l^{+}l^{-})\pi^{+}$. The second equality in Eq. (6) is attributed to Eq. (3) .

To get an estimation on the experiments, we calculate the amplitudes within the factorization framework. The helicity amplitudes of $B_{c}^{(\ast)+}\to J/\psi\pi^{+}$ are given as

	$\displaystyle(2\pi)^{4}\delta^{4}(p_{B_{c}}-p_{J/\psi}-p_{\pi})H_{\lambda}=$
	$\displaystyle~{}~{}~{}~{}i\frac{G_{F}}{\sqrt{2}}V_{cb}^{\ast}V_{ud}f_{\pi}p_{\pi}^{\mu}a_{1}\langle J/\psi;p\hat{z},J_{z}=\lambda|\overline{b}\gamma_{\mu}(1-\gamma_{5})c|B_{c}^{(\ast)+};J_{z}=\lambda\rangle\,,$		(9)

where $p$ is for the 4-momentum of the hadron in the subscript, $G_{F}$ and $f_{\pi}$ the Fermi and the pion decay constants, $a_{1}$ the effective Wilson coefficient for the color-allowed decays, $J_{z}$ the angular momentum at the $z$ direction, and $p\hat{z}$ indicates $\vec{p}_{J/\psi}\mathrel{\mkern-5.0mu\leavevmode\leavevmode\hbox{ \leavevmode\hbox to7.5pt{\vbox to5.77pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{{}{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{5.77084pt}\pgfsys@lineto{7.5pt}{5.77084pt}\pgfsys@lineto{7.5pt}{0.0pt}\pgfsys@closepath\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{ }{{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} {{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{ }\hbox{{\leavevmode\hbox{\hskip 0.0pt\raise-1.22916pt\hbox{\set@color{${\sslash}$}}}}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}} {}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\!}\hat{z}$.

On the other hand, the helicity amplitudes of $B_{c}^{\ast+}\to B^{+}\phi$ are given as

	$\displaystyle(2\pi)^{4}\delta^{4}(p_{B_{c}}-p_{B^{+}}-p_{\phi})H_{\lambda}=$
	$\displaystyle~{}~{}~{}~{}-\frac{G_{F}}{\sqrt{2}}V_{cs}^{\ast}V_{su}f_{\phi}\epsilon^{\mu\ast}_{\lambda}a_{2}\langle B^{+};p\hat{z}|\overline{u}\gamma_{\mu}(1-\gamma_{5})c|B_{c}^{(\ast)+};J_{z}=-\lambda\rangle\,,$		(10)

where $a_{2}$ is the effective Wilson coefficient for the color-suppressed decays, $f_{\phi}$ the $\phi$ decay constant, and $\epsilon^{\mu\ast}_{\lambda}$ the polarization 4-vector of $\phi$ with $\lambda$ its helicity.

Finally, the decay width for $B_{c}^{+}$ is given as

$$\Gamma=\frac{|\vec{p}_{\text{cm}}|}{8\pi M_{B_{c}}}\left|H_{0}\right|^{2}\,,$$

(11)

whereas the decay widths of $B_{c}^{*+}$ with the daughter vector meson having $\lambda$ helicity are given as

$$\Gamma_{\lambda}=\frac{|\vec{p}_{\text{cm}}|}{24\pi M_{B_{c}^{\ast}}}\left|H_{\lambda}^{\ast}\right|^{2}\,.$$

(12)

The total decays widths of $B_{c}^{\ast}\to J/\psi\pi^{+}$ and $B_{c}^{\ast}\to B^{+}\phi$ can be easily obtained by adding up the contributions from $\lambda=0,\pm$ .

The meson transition matrix elements require the knowledge of the hadron wave functions. In this work, we employ the ones from the homogeneous bag model, in which the center motions of the hadrons in the original bag model are removed Geng:2020ofy . The bag radius $(R)$ and the quark masses can be extracted from the mass spectra, which are found to be Zhang:2021yul

$$R=(2.81\pm 0.30)~{}\text{GeV}^{-1}\,,~{}~{}~{}M_{u,d}=0\,,~{}~{}~{}M_{c}=1.641~{}\text{GeV}\,,~{}~{}~{}M_{b}=5.093~{}\text{GeV}\,.$$

(13)

The details of the calculation can be found in the Appendix. In this study, $f_{\pi}$ and $f_{\phi}$ are taken from the experiments and the Lattice QCD Chen:2020qma ; pdg

$$f_{\pi}=131~{}\text{MeV}\,,~{}~{}~{}f_{\phi}=(241\pm 9)~{}\text{MeV}\,,$$

(14)

and the effective Wilson coefficients are taken to be

$$|a_{1}|=1.0\pm 0.1\,,~{}~{}~{}~{}|a_{2}|=0.27\pm 0.07\,.$$

(15)

The results are given in Table I , where we also include $\Gamma(B_{c}^{\ast+}\to B_{c}^{+}\gamma)$, which can be safely approximated as $1/\tau$ with $\tau$ the lifetime of $B_{c}^{\ast+}$. The calculated lifetime is consistent with most of the literature Li:2019tbn ; consistent , but significantly smaller than the one from the nonrelativistic potential model Eichten:1994gt , and twice larger than the one from the relativistic independent quark model twice . Nonetheless, a large part of the uncertainties that arises from the hadron wave functions is canceled in the branching ratios of $B_{c}^{\ast+}$, as the lifetime is calculated under the same framework.

Our ${\cal B}(B_{c}^{+}\to J/\psi\pi^{+})$ is consistent with the relativistic quark model BcRQ , but two times smaller compared to most of the literature Jpsipip  , which can be partly attributed to that we use a smaller $|a_{1}|$. As our estimation is a more conservative one, the angular analysis is promising to be carried out in the experiments for there are more data points to reconstruct the distribution than we expect.

The decay of $B_{c}^{\ast+}\to B^{+}\phi$ is color suppressed and suffers large uncertainties from $a_{2}$ as well as $M_{B_{c}^{\ast}}$. In particular, as $M_{B_{c}^{\ast}}$ is close to the mass threshold of $B^{+}\phi$, the decay width can range from $0$ to $10^{-6}$ eV, depending on $M_{B_{c}^{\ast+}}$. The dependency on $M_{B_{c}^{*+}}$ as well as the uncertainties caused by $a_{2}$ are plotted in Fig. 3. Taking $M_{B_{c}^{\ast+}}=6331$ MeV, the calculated decay width is given in Table 1, which is consistent with Ref. Search  , within the range of the error.

From Table 1, for $B_{c}^{\ast+}\to J/\psi\pi^{+}$ we obtain

$$\alpha^{\ast}=0.82\pm 0.01\,,~{}~{}~{}{\cal A}^{\ast}=0.38\pm 0.01\,,$$

(16)

in which the theoretical uncertainty is canceled for the correlations between $H_{\lambda}^{\ast}$. The cross section of $B_{c}^{\ast}$ meson at the LHC is expected to be $\sigma\left({B_{c}^{\ast}}\right)=29\ {\rm nb}$ Chang:1992jb . At an integrated luminosity of $150\ \text{fb}^{-1}$ during LHC Run-2, $300\ \text{fb}^{-1}$ during LHC Run-3, and $3000\ \text{fb}^{-1}$ after High Luminosity upgrade (HL-LHC) Apollinari:2017lan , the numbers of $B_{c}^{\ast}$ events are $8.7\times 10^{9}$, $1.74\times 10^{10}$ and $1.74\times 10^{11}$, resulting in $270$, $540$, and $5400$ events of $B_{c}^{\ast+}\to J/\psi\pi^{+}$, respectively. Taking the branching ratios ${\cal B}(J/\psi\to l^{+}l^{-})\approx 12\%$ pdg , there are expected to be $33$, $65$, and $650$ events of $B_{c}^{\ast+}\to\pi^{+}J/\psi(\to l^{+}l^{-})$ being able to be reconstructed at LHC Run-2, LHC Run-3 and HL-LHC, respectively.

By choosing $6400~{}\text{MeV}>M(J/\psi\pi^{+})>6325~{}\text{MeV}$ with $M(J/\psi\pi^{+})$ the invariant mass of $J/\psi\pi^{+}$, most of the off-shell contribution from $B_{c}^{+}$ would be filtered, in which $N_{B_{c}}$ is expected to be less than 20 at running LHC from the Fig. 1 of Ref. LHCb:2016vni ¹¹1In fact, at the bottom right figure, there appears to have a little bump around $6340$ MeV.. Thus, $N_{B_{c}}$ and $N_{B_{c}^{*}}$ can be safely taken as equal in the simulation.

We generate the pseudodata based on the experimental conditions at LHC Run-2, LHC Run-3, and HL-LHC. The off-shell contributions from $B_{c}^{+}$ are also included with $N_{B_{c}}=N_{B_{c}^{\ast}}$ as discussed in the previous paragraph. The numbers of the events are plotted against $\cos\theta$ in Fig. 4 , and the numerical results of $\overline{\alpha}$ and $\overline{{\cal A}}$ are given in Table 2 . Our analysis show that there would be a $1.5\sigma$ signal of nonzero $\overline{{\cal A}}$ at LHC Run-2 , and a $5\sigma$ signal at HL-LHC, which would be a solid evidence of $B_{c}^{*}$.

On the other hand, at the forthcoming experiments at FCC-hh FCC:2018vvp , the number of $B_{c}^{\ast}$ events are expected to be $10^{12}$. Hence, there would be about $2000$ and $10^{5}$ $B_{c}^{\ast+}\to B^{+}\phi$ events at HL-LHC and FCC-hh, respectively, which would be sufficient for the experiments to determine the mass.

Utilizing the conservation laws, we propose two novel methods for distinguishing $B_{c}^{*}$ and $B_{c}$ in the experiments. The calculated branching fractions of $B_{c}^{+}\to J/\psi\pi^{+}$ and $B_{c}^{*+}\to B^{+}\phi$ are compatible with the literature, indicating that our analysis is reliable. The nonzero polarized fraction of $J/\psi$ from $B_{c}^{(*)+}\to J/\psi\pi^{+}$ has been found to be $\alpha=0~{}(0.82)$. Furthermore, the branching fractions of $B_{c}^{*+}\to J/\psi\pi^{+}$ and $B_{c}^{*+}\to\phi\pi^{+}$ have been obtained as $(2.4\pm 0.5)\times 10^{-8}$ and $(7.0\pm 3.0)\times 10^{-9}$, respectively. To calculate the lifetime of $B_{c}^{*}$, we have found that $\Gamma(B_{c}^{*+}\to B_{c}^{+}\gamma)=(53\pm 3)$ eV with the homogeneous bag model, consistent with most of the literature.

We have shown that $B_{c}^{*+}\to B^{+}\phi$ would be promising to be measured at HL-LHC as well as FCC-hh. To examine the feasibilities of the measurements, we have conducted simulations based on the experimental conditions. Remarkably, we have shown that the helicity analysis on $B_{c}^{(*)+}\to J/\psi\pi^{+}$ is ready to be performed at LHC. Thus, we urge the experimentalists to probe the angular distributions of $B_{c}^{(*)+}\to J/\psi(\to l^{+}l^{-})\pi^{+}$ in the region of $M(J/\psi\pi^{+})>6325$ MeV, which can be served as an evidence of $B_{c}^{(*)+}$.

Acknowledgments

The authors would like to acknowledge the helpful discussion with Chao-Qiang Geng, Ying-Rui Hou and Cong-Feng Qiao.