$B_c$ excitations at LHC experiments

TL;DR

This paper discusses the current observations and future prospects of excited $B_c$ meson states at LHC experiments, including potential detection of various higher wave states.

Contribution

It provides an analysis of observed $2S$ $B_c$ states and estimates the observability of multiple higher excited states at LHC.

Findings

Observation of $2S$ $B_c$ states by ATLAS, CMS, LHCb

Estimated detection prospects for $B_c^*$, $2P$, $3P$, and $D$ wave states

Discussion of experimental challenges and opportunities

Abstract

The $2S$ $B_c$ states observed by ATLAS, CMS and LHCb Collaborations are discussed. The observation perspectives of $B_c^*$, $2P$ wave, $3P$ wave and $D$ wave states of $B_c$ at LHC experiments are estimated.

Figures

Tables

Table 1. Table 1: Mass values and row relative yields of B c ​ ( 2 ​ S ) subscript 𝐵 𝑐 2 𝑆 B_{c}(2S) states estimated from the experimental data.

	experiment	ATLAS Aad et al. (2014)	CMS Sirunyan et al. (2019)	LHCb Aaij et al. (2019)
	luminosity (energy)	24.1 fb-1 (7, 8 TeV)	140 fb-1 (13 TeV)	8.7 fb-1 (7, 8, 13 TeV)
mass, MeV	$2^3{S}_1$ , shifted	$6842\pm 6$	$6842\pm 2$	$6841\pm 1$
	$2^1{S}_0$		$6871.0\pm 1.6$	$6872.1\pm 1.6$
row relative yield	$2^3{S}_1$		$0.0088\pm 0.0014$	$0.0136\pm 0.0027$
	$2^1{S}_0$		$0.0068\pm 0.0014$	$0.0063\pm 0.0024$
	total	$0.18\pm 0.05$	$0.0156\pm 0.0019$	$0.0198\pm 0.0036$
$N\left(2^3{S}_1\right)/N\left(2^1{S}_0\right)$			$1.31\pm 0.32$	$2.1\pm 0.9$

Table 2. Table 2: Masses of D 𝐷 D -wave B c subscript 𝐵 𝑐 B_{c} meson states in MeV.

State	EQ Eichten and Quigg (1994)	GKLT Gershtein et al. (1995)	ZVR Zeng et al. (1995)	FUI Fulcher (1999)	EFG Ebert et al. (2003)	GI Godfrey (2004)	MBV Monteiro et al. (2017)	SJSCP Soni et al. (2018)	LLLGZ Li et al. (2019)
$3^3{D}_1$	7012	7008	7010	7024	7072	7028	6973	6998	7020
$3D_2^{\prime }$	…	7016	…	…	7079	7036	7003	…	7032
$3D_2$	…	7001	…	…	7077	7041	6974	…	7024
$3^1{D}_2$	7009	…	7020	7023	…	…	…	6994	…
$3^3{D}_2$	7012	…	7030	7025	…	…	…	6997	…
$3^3{D}_3$	7005	7007	7040	7022	7081	7045	7004	6990	7030

Table 3. Table 3: Radiative decays of B c subscript 𝐵 𝑐 B_{c} meson P 𝑃 P -wave excitations (see  Godfrey ( 2004 ); Gupta and Johnson ( 1996 ); Kiselev et al. ( 1995 ) )

initial state	final state	Br, %	$\Delta M$, MeV
$2^3{P}_0$	$1^3{S}_1+\gamma$	100	363-366
$2P{1}^{+}$	$1^3{S}_1+\gamma$	87	393-400
	$1^1{S}_0+\gamma$	13	393-400
$2P{1}^{\prime +}$	$1^1{S}_0+\gamma$	94	472-476
	$1^3{S}_1+\gamma$	6	472-476
$2^3{P}_2$	$1^3{S}_1+\gamma$	100	410-426
$3^3{P}_0$	$1^3{S}_1+\gamma$	2	741
$3P{1}^{+}$	$1^3{S}_1+\gamma$	8.5	761
	$1^1{S}_0+\gamma$	3.3	820
$3P{1}^{\prime +}$	$1^1{S}_0+\gamma$	22.6	825
	$1^3{S}_1+\gamma$	0.7	769
$3^3{P}_2$	$1^3{S}_1+\gamma$	18	778

Table 4. Table 4: Conversion factors for B c ∗ , B c ​ ( 1 ​ P 1 ) , B c ​ ( 1 ​ P 1 ) ′ → B c ​ γ → superscript subscript 𝐵 𝑐 subscript 𝐵 𝑐 1 subscript 𝑃 1 subscript 𝐵 𝑐 superscript 1 subscript 𝑃 1 ′ subscript 𝐵 𝑐 𝛾 B_{c}^{*},B_{c}(1P_{1}),B_{c}(1P_{1})^{\prime}\to B_{c}\gamma decays

	GI Godfrey and Isgur (1985)	EFG Ebert et al. (2003)	FUII Fulcher (1999)	GKLT Kiselev et al. (1995)	EQ Eichten and Quigg (1994)	GJ Gupta and Johnson (1996)	ZVR Zeng et al. (1995)
mass of $B_c^{\ast }$, MeV	$6.338$	$6.332$	$6.341$	$6.317$	$6.337$	$6.308$	$6.34$
${10}^3\cdot I^{ee}$	$6.105$	$5.616$	$5.431$	$5.665$	$5.869$	$5.591$	$6.011$
mass of $B_c\left(1P_1\right)$, MeV	$6.741$	$6.734$	$6.737$	$6.717$	$6.73$	$6.738$	$6.73$
${10}^3\cdot I^{ee}$	$8.811$	$8.733$	$8.689$	$8.733$	$8.74$	$8.821$	$8.753$
${10}^3\cdot I^{\mu \mu }$	$0.7192$	$0.6538$	$0.6176$	$0.6538$	$0.6593$	$0.7272$	$0.6703$
mass of $B_c\left(1P_1^{\prime }\right)$,MeV	$6.75$	$6.749$	$6.76$	$6.729$	$6.736$	$6.757$	$6.74$
${10}^3\cdot I^{ee}$	$8.84$	$8.782$	$8.766$	$8.773$	$8.76$	$8.88$	$8.786$
${10}^3\cdot I^{\mu \mu }$	$0.7432$	$0.6949$	$0.6813$	$0.6867$	$0.6758$	$0.7774$	$0.6976$

Equations

$$M_{B_c}=6274.9\pm 0.8\text{mbox}MeV,$$

$${\tau }_{B_c}=0.507\pm 0.009\text{mbox}ps.$$

$$2^1{S}_0(B_c)$$

$$2^3{S}_1(B_c)$$

$$\sigma (B_c(2S))$$

$$\sigma (2^3{S}_1)/\sigma (2^1{S}_0)\sim \mathrm{2.6.}$$

$$\Delta {\tilde{M}}_{2S}<2\frac{\Delta M^{\ast }\sqrt{\Delta M^2-4m_{\pi }^2}}{M}\approx 10\text{leavevmode}\text{MeV},$$

$$\Delta {\tilde{M}}_{2S}\sim 2\frac{\Delta M^{\ast }⟨\sqrt{\Delta M^2-m_{\pi \pi }^2}⟩}{M}.$$

$$M(B_c^{\ast })-M(B_c)\approx 65\text{leavevmode}\text{mbox}MeV.$$

$$\omega_{T}^{max}\approx\Bigl{(}\sqrt{M_{B_{c}^{*}}^{2}+p_{T}^{2}}+p_{T}\Bigr{)}\frac{\Delta M^{*}}{M_{B_{c}^{*}}}\approx 0.01\Bigl{(}\sqrt{M_{B_{c}^{*}}^{2}+p_{T}^{2}}+p_{T}\Bigr{)}$$

$$2P{1}^{+}(B_c)$$

$$2P{1}^{{}^{\prime }+}(B_c)$$

$$2P(B_c)\stackrel{{\gamma }^{\text{hard}}}{\to }{1}^3{S}_1(B_c^{\ast })$$

$$\Delta \tilde{M}={\tilde{M}}_{max}-{\tilde{M}}_{min}\approx 2\frac{\Delta M^{\ast }\Delta M}{M}$$

$$\Delta {\tilde{M}}_{2P}\approx 10\text{leavevmode}\text{MeV}.$$

$$\Delta {\tilde{M}}_{3P}\approx 20\text{leavevmode}\text{MeV}.$$

$$\frac{d{\mathrm{Br}}_{\ell \ell }}{d{q}^2}$$

$$\lambda (M;m_1,m_2)$$

$$M_{B_c^{\ast }}={\omega }^{\prime }+\sqrt{{\omega }^{\prime 2}+M_{B_c}^2}$$

$${\omega }^{\prime }=\Delta M^{\ast }\left(1-\frac{\Delta M^{\ast }}{2M_{B_c^{\ast }}}\right),$$

$${\omega }^{max}=\frac{(1+v){\omega }^{\prime }}{\sqrt{1-v^2}}=\frac{{\omega }^{\prime }}{M_{B_c^{\ast }}}\left(\sqrt{p_{B_c^{\ast }}+M_{B_c^{\ast }}^2}+p_{B_c^{\ast }}\right),$$

$$\omega^{max}_{T}(p_{B_{c}^{*}}^{T},p_{B_{c}^{*}})=\omega^{max}\frac{p_{B_{c}^{*}}^{T}}{p_{B_{c}^{*}}}=\frac{\omega^{\prime}}{M_{B_{c}^{*}}}\left(\sqrt{p_{B_{c}^{*}}^{T}+M_{B_{c}^{*}}^{2}\Bigl{(}\frac{p_{B_{c}^{*}}^{T}}{p_{B_{c}^{*}}}\Bigr{)}^{2}}+p_{B_{c}^{*}}^{T}.\right)$$

$$\omega_{T}^{max}=\left(1-\frac{\Delta M^{*}}{2M_{B_{c}^{*}}}\right)\Bigl{(}\sqrt{M_{B_{c}^{*}}^{2}+{p_{B_{c}^{*}}^{T}}^{2}}+p_{B_{c}^{*}}^{T}\Bigr{)}\frac{\Delta M^{*}}{M_{B_{c}^{*}}}.$$

$$\omega_{T}^{max}\approx 0.01\Bigl{(}\sqrt{M_{B_{c}^{*}}^{2}+{p_{B_{c}^{*}}^{T}}^{2}}+p_{T}\Bigr{)}$$

$$B_c(2^3{S}_1):M+\Delta M^{\ast }+\Delta M$$

$$B_c(1^3{S}_1):M+\Delta M^{\ast }$$

$$B_c(1^1{S}_0):M$$

$${\left(M+\Delta M^{\ast }+\Delta M\right)}^2={\left(M+\epsilon +\omega \right)}^2-{\left(k+p\right)}^2,$$

$${\left(M+\Delta M^{\ast }\right)}^2={\left(M+\omega \right)}^2-k^2.$$

$$\omega =\Delta M^{\ast }\left(1+\frac{\Delta M^{\ast }}{2M}\right)$$

$${\epsilon }_{max}$$

$${\epsilon }_{min}$$

$${\epsilon }_{min}^{max}\approx \Delta M\left(1-\frac{\Delta M^{\ast }}{M}\right)\pm \Delta M^{\ast }\sqrt{{\left(\frac{\Delta M}{M}\right)}^2-{\left(\frac{m_{\pi \pi }}{M}\right)}^2}$$

Full text

I Introduction

$B_{c}$-meson is a nonrelativistic two heavy quark system, and like $(b\bar{b})$ and $(c\bar{c})$ states can be considered in the potential model framework that predicts 19 bounded states $(b\bar{c})$ below the decay threshold to $B\bar{D}$ (see Gershtein et al. (1995, 1997); Gouz et al. (2004); Godfrey (2004)). Unlike $(b\bar{b})$- or $(c\bar{c})$-quarkonium, there are no annihilation decay modes for these states, which makes the $(b\bar{c})$ system similar to usual $b\bar{q}$ or $c\bar{q}$-meson. The first observation of the ground state $B_{c}$-meson was done at Tevatron collider (CDF and D0 experiments) in two decay modes: $B_{c}\to J\psi l\nu$ ($l=e,\mu$) and $B_{c}\to J\psi\pi$ Aaltonen et al. (2008); Abazov et al. (2009, 2008); Abe et al. (1998). Now this discovery is repeatedly confirmed by the LHC Collaborations LHCb, CMS, ATLAS in numeral decay modes: $B_{c}\to J\psi\pi$ Aaij et al. (2012a); Adam et al. (2012); Aad et al. (2012), $B_{c}\to J\psi\pi\pi\pi$ Aaij et al. (2012b); Adam et al. (2012), $B_{c}\to J/\psi l\nu$ Aaij et al. (2018), $B_{c}\to J/\psi K$ Aaij et al. (2013a), $B_{c}\to\psi(2S)\pi$ Aaij et al. (2013b), $B_{c}\to J/\psi\pi KK$ Aaij et al. (2013c), $B_{c}\to J\psi D^{(*)}K^{(*)}$ Aaij et al. (2017a, b), $B_{c}\to J/\psi D^{(*)}_{s}$ Aaij et al. (2013d); Aad et al. (2016), $B_{c}\to J/\psi\pi p\bar{p}$ Aaij et al. (2014a), $B_{c}\to J/\psi 3\pi^{+}2\pi^{-}$ Aaij et al. (2014b) and $B_{c}\to B_{s}\pi^{+}$ Aaij et al. (2013e). It is worth to note that most of these decays occur due to transformation $b\to cW$. The only observed mode with $c$ quark decay is $B_{c}\to B_{s}\pi^{+}$. The annihilation channel is not observed yet. It is interesting to note that according to theoretical predictions the dominant decay channel is $c\to sW$, the contribution of $b\to cW$ is smaller, and the smallest contribution is due to weak annihilation channel $bc\to W$.

Also the mass and life time are known with excellent accuracy Aaij et al. (2017a, 2015); Tanabashi et al. (2018):

[TABLE]

[TABLE]

There are some differences in the production of ordinary quarkonium with hidden flavor and the production of $B_{c}$-meson. In case of $B_{c}$-meson it is necessary to produce two pairs of heavy quarks, $b\bar{b}$ and $c\bar{c}$, which leads to the strong suppression of the $B_{c}$ production cross section. Unlike production of quarkonium with hidden flavor, the production of $P$-wave states is suppressed comparing to $S$-wave states. The production mechanism of $B_{c}$-meson and its excitations is well studied theoretically in Berezhnoy et al. (1995); Chang et al. (1995); Berezhnoy et al. (1996a); Kolodziej et al. (1995); Berezhnoy et al. (1996b, 1997a); Baranov (1997a, b); Berezhnoy et al. (1997b); Chang et al. (2004); Berezhnoy (2005); Chang et al. (2005, 2006); Berezhnoy et al. (2011); Gao et al. (2010). The ratio $R_{B_{c}}=\sigma(B_{c}^{*})/\sigma(B_{c})$ is predicted to be about $\sim 2.6$ under assumption that the values of wave functions at origin for $B_{c}^{*}$ and $B_{c}$ are the same. It is worth to note that according this study the $B_{c}$-meson production at fixed gluon interaction energy resolves itself to just a $b$-quark fragmentation (like fragmentation $b\to B$) only at high transverse momenta, where the cross section ratio between the production of vector state $B_{c}^{*}$ and pseudoscalar state is predicted to be $R_{B_{c}}\sim 1.4$.111It is worth to mention the study Zheng et al. (2019), where the fragmentation functions $b\to B_{c}$ and $b\to B_{c}^{*}$ are estimated within one loop approximation and it is shown that value of $R_{B_{c}}$ does not change dramatically. However the transverse momentum value where the fragmentation becomes dominant depends on the gluon energy: the higher gluon energy, the higher this transverse momentum value. This is why the non-fragmentation mechanisms contribute to all transverse momenta.

Unfortunately the observation of $B_{c}^{*}$ is extremely difficult due to the low energy of the photon in the decay $B_{c}^{*}\to B_{c}\gamma$. This is why in study Berezhnoy and Likhoded (2013) we discussed the other $B_{c}$ excitations, namely, $2S$-wave states and $P$-wave states, and these states are more preferable for observation. The recent observation of $2S$ states in ATLAS Aad et al. (2014), CMS Sirunyan et al. (2019) and LHCb Aaij et al. (2019) made us return to this discussion.

Although the current state and prospects of research of $\bar{b}c$ quarkonia are discussed in an excellent review Eichten and Quigg (2019), we nevertheless would like to pay attention to some details.

The rest of the paper is organized as follows. In the next section decays of $B_{c}(2S)$ states with the emission of two $\pi$ mesons are discussed. Section III is devoted to the description of $D$-wave states’ spectroscopy and production. In sections IV and V we give theoretical predictions for radiative decays of the excited states and lepton pair production in these processes. The results of our paper are summarized in the Conclusion.

II $B_{c}(2S)\to B_{c}(1S)+\pi\pi$ decays.

The excitations of $B_{c}$ meson were first observed by ATLAS Aad et al. (2014) and CMS Sirunyan et al. (2019) collaborations. Recently the results of CMS are confirmed by LHCb Collaboration in Aaij et al. (2019). The experimental results are shown in Table 1.

Before discussing these measurements, let us remind what was theoretically predicted. According to Berezhnoy and Likhoded (2013) about one half of such excitations decay to $B_{c}$ ($B_{c}^{*}$) and $\pi^{+}\pi^{-}$ pair:

[TABLE]

Therefore, the predicted relative yield of $2S$-excitations with $B_{c}(2S)\to B_{c}(B_{c}^{*})+\pi^{+}\pi^{-}$ decay is up to 10% of total $B_{c}$-meson yield. Under assumption that the radial wave functions at origin for $2^{3}S_{1}$ and for $2^{1}S_{0}$ states are approximately equal to each other, we predicted that for the total phase space

[TABLE]

However, some models Martynenko (2019); Galkin (2019); Ebert et al. (2011) predict that the wave function value is essentially larger for pseudoscalar $2S$ state, than for vector one. According  Martynenko (2019) $|R(B_{c}^{*}(2S))(0)|/|R(B_{c}(2S))(0)|=0.87$ that leads to decreasing of $\sigma(2^{3}S_{1})/\sigma(2^{1}S_{0})$ from 2.6 to 2. Within the approach Galkin (2019); Ebert et al. (2011) the $\sigma(2^{3}S_{1})/\sigma(2^{1}S_{0})$ decreases even more dramatically: $|R(B_{c}^{*}(2S))(0)|/|R(B_{c}(2S))(0)|=0.567$ and therefore the ratio $\sigma(2^{3}S_{1})/\sigma(2^{1}S_{0})$ becomes close to 0.9.

It was also predicted in Berezhnoy and Likhoded (2013) that the loss of ‘‘soft’’ photon from $B_{c}^{*}$ shifts the vector $2S$-state approximately by 65 MeV and insignificantly broadens the peak:

[TABLE]

where $M$ is a ground state mass, $\Delta M^{*}=M(B_{c}^{*})-M(B_{c})$ is a difference between masses of lowest vector and pseudoscalar states and $\Delta M=M(B_{c}(2^{3}S_{1}))-M(B_{c}^{*})$ is a difference between the masses of $2^{3}S_{1}$-wave excitation and lowest vector state. As a result, the mass peak on $B_{c}+\pi\pi$ spectrum for more massive vector state $2^{3}S_{1}$ will appear about 30 MeV lower, than for less massive pseudoscalar state $2^{1}S_{0}$. It is worth noting that in our previous study Berezhnoy and Likhoded (2013) we have skipped the fact that the $\Delta\tilde{M}_{2S}$ is not a value of additional width itself, but an upper value of additional width. The real value of additional width depends on shape of $\pi\pi$ distribution: the smaller the average value of $\pi\pi$ invariant mass $m_{\pi\pi}$, the narrower the peak. The width can be estimated as follows:

[TABLE]

Now let us return to the experimental results (see Table 1). At first sight the measured $B_{c}(2S)$ yields are essentially smaller than the predicted one, which is about 10%. However the registration efficiencies in these experimental studies were not taken into account and therefore at the moment this theoretical prediction does not contradict CMS and LHCb results.

Also it is worth to note that the ATLAS measurement Aad et al. (2014) is significantly out of the range of others, and we think that such a huge difference can not be explained by the difference in the registration efficiencies. However the mass value measured by ATLAS is in consistence with shifted value of the vector $B_{c}(2S)$ excitation. The resolution in ATLAS is not enough to separate peaks and it makes awkward the detailed comparison with the results of other experiments, where the two peaks are clearly seen.

It is very interesting to compare the ratio of yields of $B_{c}^{*}(2S)$ and $B_{c}(2S)$ in CMS and LHCb experiments. If LHCb measured the central value which is close to the predicted one in Berezhnoy and Likhoded (2013): $\sim 2.1$, the CMS Collaboration measured the central value $\sim 1.3$. However, the errors are quite large, and within uncertainties these results do not contradict each other.

We hope that new data will allow to compare the ratio of yields for different experiments. The situation would be most interesting if the ratio essentially depended on kinematic region, which would indicate a crucial change of production mechanism. It is worth noting that the sharp change of this value is not expected within the conventional production model.

It will be very interesting to measure the shape of the spectrum of the invariant mass of $\pi\pi$-pair in this decay and compare it with similar spectrum shape for $\psi^{\prime}\to\psi+\pi^{+}\pi^{-}$ decay. A theoretical distribution of the $\pi\pi$-pair mass in this decay has been investigated since 1970s. In Brown and Cahn (1975); Novikov and Shifman (1981); Voloshin (1975); Voloshin and Zakharov (1980) from the considerations based on chiral theory it was concluded that for small invariant masses of $\pi\pi$-pair the amplitude of this process is approximately proportional to $m_{\pi\pi}^{2}-4m_{\pi}^{2}$. This approximation was extended to the whole phase space of the decay (see Figure 2). However, the data from the BESII experiment Ablikim et al. (2007) shows, that it is very likely that resonant contribution to this process should be also considered. The resonant contribution is made by $f_{0}(600)$- or $\sigma$-resonance with $J^{PC}=0^{++}$ and the mass $(400\div 550)-i(200\div 350)$ MeV (Figure 2).

It should be noted that there is no consensus about the nature of the $\sigma$-meson. Currently, there are generally two approaches about it. The first approach Albaladejo and Oller (2012) is based on unitary chiral perturbation theory, considering $\sigma$ as a dynamically generated resonance in $\pi\pi$ interactions. The $\sigma$-meson apprears to be an $S$-wave $\pi\pi$ bound state that can be also four-quark state. Another approach Black et al. (2001); Harada et al. (2012) originates from linear sigma model. Within this approach, $\sigma$ is a mixed state of two-quark and four-quark states. As seen from $\psi(2S)$ decay, the contribution from $\sigma$ dominates over raw $\pi\pi$-pair production. It is likely that $B_{c}(2S)$ meson will hadronically decay via the same or similar mechanisms, so the role of $\sigma$-meson in this case will be also dominant. The role of $\sigma$-meson in $D_{1}(2430)\to D\pi\pi$ decay was studied in paper Harada et al. (2012).

III $D$-wave excitations at LHC

The production of $D$-wave states of $B_{c}$ quarkonium is not broadly discussed due toupposedly small relative yield $\sim 1$% (see for example Cheung and Yuan (1996), where the production of $D$-waves in $e^{+}e^{-}$ annihilation was studied), as well as due to technical difficulties in the cross section estimation. However, at the moment, when $2S$ states are already experimentally observed, the theoretical study of hadronic production of $D$-wave states looks quite reasonable. Indeed, despite that the dominant decay mode for $D$-wave states is electromagnetic Gouz et al. (2004); Godfrey (2004); Ebert et al. (2003); Li et al. (2019), it is shown in Eichten and Quigg (1994) that about 20% of such states decay to $1S$ state radiating two $\pi$ mesons, as well as $2S$ excitations. Thus it provides a chance to extract the $D$-wave states in the $B_{c}\pi^{+}\pi^{-}$ mass spectrum with large statistics.

Similar to registered $B_{c}(2S)\xrightarrow{\pi\pi}B_{c}(1S)$ decay there should be several peaks: corresponding to direct decay to $B_{c}$ ground state and corresponding to decay with intermediate $B_{c}^{*}$, i.e. lowest vector state. The predicted $B_{c}$ spectroscopy comprises four $D$-wave states (we indicate their masses gained by different groups in table (2)). If $B_{c}(3D)\xrightarrow{\pi\pi}B_{c}(1S)$ decay goes with conservation of spin (as supposed in Eichten and Quigg (1994)), than we should obtain one peak for $3^{1}D_{2}$ state and three peaks for $3^{3}D_{1},3^{3}D_{2},3^{3}D_{3}$ states shifted by the value close to $\Delta M^{*}=M_{B_{c}^{*}}-M_{B_{c}}$. Most likely the latter three ones will overlap each other and will appear earlier than $3^{1}D_{2}$ peak at the $B_{c}\pi^{+}\pi^{-}$ invariant mass scale. Therefore we could expect, that one narrow peak from $D$ wave states can be observed at $\sim$ 7000 MeV and one broad peak can be observed at $\sim 6930$ MeV.

IV Radiative $B_{c}$ meson decays

According to predictions of the potential model, the mass difference between lowest vector and pseudoscalar states of $\bar{b}c$-quarkonium ($B_{c}^{*}$ and $B_{c}$) is fairly small:

[TABLE]

Since LHCb is unable to detect photon with transverse momentum about $65$ MeV it is necessary for decaying $B_{c}^{*}$ to have fairly large transverse momentum. It is known that production cross section is greatly reduced with increasing of the transverse momentum, so that it leads to significant decreasing of amount of events, where such photon could be detected.

The maximum photon transverse energy in the laboratory system can be calculated using the expression

[TABLE]

This leads to, for example, that photons with transverse energy $\omega_{T}>0.5$ GeV may be produced only if transverse momentum of $B_{c}^{*}$-meson $p_{T}(B_{c}^{*})>24$ GeV. Such a cut on transverse momentum decreases the yield of $B_{c}^{*}$-mesons approximately by two orders (see Berezhnoy and Likhoded (2013)). It is not so for radiation transitions of $2P$-wave states. In the latter case the transverse energy could be sufficiently large, even if initial state of $B_{c}(2P)$-meson would have small momentum. Therefore, despite the fact that the yield of $2P$-excitations is about 6-20% of total yield of $B_{c}$-meson Berezhnoy et al. (1996b); Chang et al. (2006), it is much easier to register decays of $2P$-excitations. As it was estimated in Berezhnoy and Likhoded (2013), the yield of $2P$-excitations emitting photon with $\omega_{T}>0.5$ GeV is $25\div 50$ times more, than yield of vector $B_{c}^{*}$, emitting photon with the same transverse energy.

It should be stressed that only 20% of all $2P$-excitations emit only one photon, immediately transforming to lowest pseudoscalar state:

[TABLE]

In all other cases $2P$-states first decay to $B_{c}^{*}$ while emitting ‘‘hard’’ photon and then decay to $B_{c}$ emitting ‘‘soft’’ photon:

[TABLE]

The second photon (‘‘soft’’) will be lost almost always. However, as shown below, it doesn’t prevent the experimental observation of $2P$-wave states of $B_{c}$-meson.

Really, it is easy to show that the loss of the ‘‘soft’’ photon in the cascade decay of $2P$-wave states leads to broadening of the peak by the value

[TABLE]

and to the left shifting it by $\Delta M^{*}$. Considering $\Delta M^{*}\approx 65$ MeV and $\Delta M\approx 400$ MeV we can get that the value of broadening for $2P$-wave states is

[TABLE]

It is clear that this width is smaller than hardware width of the resonance and doesn’t affect its detection quality at all. It should be noted that the value of broadening from ‘‘soft’’ photon loss for $3P$-wave states is also fairly small:

[TABLE]

Predicted distributions over invariant mass of the final pseudoscalar $B_{c}$-meson and the ‘‘hard’’ photon in cascade decays of $2P$-wave states of the $B_{c}$-meson are shown in Figure 3. Unshifted peaks from $2P\leavevmode\nobreak\ 1^{+}$ and $2P\leavevmode\nobreak\ 1^{{}^{\prime}+}$ states are depicted with blue lines, red lines depict shifted and broadened peaks from $2^{3}P_{0}$, $2P\leavevmode\nobreak\ 1^{+}$, $2P\leavevmode\nobreak\ 1^{\prime+}$ and $2^{3}P_{2}$ states. At last black line shows sum of the peaks. Similar distributions for $3P$-wave states are shown in Figure 4. The pictures are plotted by the data from the table 3 and $B_{c}$-meson mass spectrum from the work Gouz et al. (2004). At all figures peaks are plotted with respect to rough model of detector resolution: broadening the source histograms with Gaussian function with dispersion of $6$ MeV (pictures (a)) and $12$ MeV (pictures (b)). It should be stressed that such simulation is just an estimation and doesn’t reflect the actual properties of detector. Nevertheless, such estimations allow us to understand what shapes would have peaks at finding them out in the experiment (see also Eichten and Quigg (2019)).

It is important to note that in spite of the fact that the yields of $2P$- and $3P$-states in the proton-proton interaction are nearly the same, $3P$-excitations are harder to detect in the mass spectrum of $B_{c}+\gamma$. $2P$-excitations always decay via electromagnetic transitions, while $3P$-excitations — only in 20% of the cases. Moreover, it is clear that the shapes of peaks broadened due to the loss of photon will repeat the shapes of distributions over the cosine of the angular between the directions of motion of ‘‘soft’’ and ‘‘hard’’ photons in rest frame of the decaying $B_{c}$ excitation.

Is is shown that the more minimal transverse energy of photon, the less probability it is radiated by $B_{c}^{*}$-meson.

V Lepton pair production in radiative $B_{c}$ meson decays

The branching fraction of lepton pair production in radiative decays of the excited $B_{c}$ meson $B_{1}\to B_{2}\ell\ell$ can be written in the form Faessler et al. (2000); Luchinsky (2017)

[TABLE]

where $\mathrm{Br}_{\gamma}$ is the branching fraction of the original radiative decay, $q^{2}$ is the squared invariant mass of the lepton pair, $\alpha=e^{2}/4\pi$ is the fine structure coupling constant, $M_{1,2}$ are the masses of initial and final mesons respectively, and

[TABLE]

is the velocity of the final particles in $M\to m_{1}m_{2}$ decay.

It should be noted that the relation (5) is universal and does not depend on the physics of the process. The only assumption was that we neglect the $q^{2}$ dependence of the $B_{1}\to B_{2}\gamma^{*}$ decay vertex. This assumption looks quit reasonable since typical energy deposit in the radiative decays of the doubly heavy mesons is small in comparison with quarks’ masses. As a result, the conversion factor $I^{\ell\ell}$ depends only on the masses of the initial and final particles. Masses of the leptons and ground state $B_{c}$ meson can be found easily Tanabashi et al. (2018), while for the initial excited particles some theoretical models are required.

In the following we will present numerical values of the $\gamma^{*}\to\ell\ell$ conversion factors for all the initial particles.

The spectroscopy of $B_{c}$ mesons was studied already in details. In our work we will use results presented in papers Godfrey and Isgur (1985); Ebert et al. (2003); Fulcher (1999); Kiselev et al. (1995); Eichten and Quigg (1994); Gupta and Johnson (1996); Zeng et al. (1995) (see also Godfrey (2004)). The mass of $B_{c}^{*}$ meson is not high enough for muon pair production, so only the $ee$ channel is opened. The corresponding results are shown in Table 4. As for $P$ wave excitations, both electronic and muonic decays are allowed. One can see in Table 4, that in all cases electron pair emission leads to suppression of the branching fraction by a factor $\sim 10^{-2}$, while in the case of $\mu\mu$ channel the suppression is about an order of magnitude stronger. However, unlike soft photon, the lepton-antilepton pair can be easily detected by the modern detectors, we think that the excited $B_{c}$ mesons could be observed in the discussed modes.

VI Conclusion

In this work we briefly discuss the excellent results of LHC Collaborations CMS, ATLAS and LHCb, which recently opened a new era in the heavy quark spectroscopy by observing $2S$ excitations of $B_{c}$ meson. We show that the study of $\pi^{+}\pi^{-}$ distribution could provide a new information about the nature of $\sigma$-meson and about the chiral symmetry breaking. We emphasize that the ratio dependence between yields of $2^{3}S_{1}$ and $2^{1}S_{0}$ states of $B_{c}$ meson on the kinematical conditions is the very important source of information about the $B_{c}$ meson production mechanisms. Also we estimate the perspectives of observation of $B_{c}$ states, such as $B_{c}^{*}$, $P$ wave and $D$ wave excitations. We think, that at least for $P$ excitations such prespectives are fairly optimistic. In addition we suggest to study $P$ wave states in their radiative decays to the lepton pair.

Authors thank V. Galkin and A. Martynenko for help and fruitful discussion.

A. Berezhnoy and I. Belov acknowledge the support from ‘‘Basis’’ Foundation (grants 17-12-244-1 and 17-12-244-41).

Appendix A Maximum $p_{T}$ of the photon emitted by the excited $B_{c}$

We consider single-photon transition $B_{c}^{*}\xrightarrow{\gamma}{B_{c}}$. In the $B_{c}^{*}$ rest frame photon energy $\omega^{\prime}$ can be determined from the equation:

[TABLE]

[TABLE]

where $\Delta M^{*}$ is a mass difference between $B_{c}$ states: $\Delta M^{*}=M_{B_{c}^{*}}-M_{B_{c}}$.

The energy $\omega$ of the photon in the laboratory system achieves maximum, if the direction of the photon motion coincides with the direction of the $B_{c}^{*}$ motion in this system:

[TABLE]

where $v=\frac{p_{B_{c}^{*}}}{E_{B_{c}^{*}}}$.

The maximal transverse energy for the given values of $p_{B_{c}^{*}}$ and $p_{B_{c}^{*}}^{T}$ can be estimated as

[TABLE]

For the maximal transverse energy for the given value of $p_{B_{c}^{*}}^{T}$ and arbitrary $p_{B_{c}^{*}}$ value we get in terms of $p_{B_{c}^{*}}^{T}$:

[TABLE]

Expression (9) is valid for any electromagnetic transition. Particularly for the considered case (lowest vector state decaying into the ground state):

[TABLE]

Appendix B The additional decay width due to the loss of soft photon in the decay $B_{c}(2^{3}S_{1})\xrightarrow{\pi\pi}{B_{c}(1^{3}S_{1})}\xrightarrow{\gamma}{B_{c}(1^{1}S_{0})}.$

We denote masses of $B_{c}$ states as:

[TABLE]

Invariant mass squared for the initial and final states:

[TABLE]

where $(\varepsilon,p)$ and $(\omega,k)$ are four-momenta of $\pi\pi$ and $\gamma$ correspondingly in the $B_{c}$ ground state rest frame.

Invariant mass squared for the decay $B_{c}(1^{3}S_{1})\rightarrow B_{c}(1^{1}S_{0})+\gamma:$

[TABLE]

[TABLE]

Solving (10) by $\varepsilon$ we get:

[TABLE]

where $F=\frac{1}{2}\left(\Delta M^{2}+2\Delta M\left(\Delta M^{*}+M\right)-m_{\pi\pi}^{2}\right)$

Further we write approximately with $\frac{m_{\pi\pi}}{M},\frac{\Delta M^{*}}{M},\frac{\Delta M^{*}}{M}$ treated as very small.

[TABLE]

Invariant mass of reсonstructed $B_{c}(1^{1}S_{0})+\pi\pi$ system:

[TABLE]

Taking $\tilde{M}_{2S}$ with $\varepsilon_{min},\varepsilon_{max}$ values we can write it in the following form:

[TABLE]

where we do not specify the terms of higher order contributing to $\tilde{M}_{2S}$ shift from the exact value $M_{2S}=M+\Delta M+\Delta M^{*}$. Thus we obtain the extra broadening

[TABLE]

Appendix C The additional decay width due to the loss of soft photon in the decay $B_{c}(2^{3}P_{1})\xrightarrow{\gamma^{\textrm{hard}}}{B_{c}(1^{3}S_{1})}\xrightarrow{\gamma^{\textrm{soft}}}{B_{c}(1^{1}S_{0})}.$

We denote masses of $B_{c}$ states as:

[TABLE]

Invariant mass squared for the initial and final states:

[TABLE]

where $(\omega,k)$ and $(\omega^{*},k^{*})$ – four momenta of $\gamma^{\textrm{hard}}$ and $\gamma^{\textrm{soft}}$ in the $B_{c}$ ground state rest frame and $\cos\Theta$ is an angle between $k$ and $k^{*}$.

Invariant mass squared for the decay $B_{c}(1^{3}S_{1})\rightarrow B_{c}(1^{1}S_{0})+\gamma^{\textrm{soft}}:$

[TABLE]

From (14) and (15) we get:

[TABLE]

Approximately:

[TABLE]

where $\frac{\Delta M}{M}$ and $\frac{\Delta M^{*}}{M}$ are very small.

We consider that $\gamma^{\textrm{soft}}$ is being lost in the reconstructed invariant mass:

[TABLE]

We obtain that

[TABLE]

and $\tilde{M}_{2P}$ peak gains extra broadening with varying $\Theta$:

[TABLE]