Semileptonic decays of $ B_c$ mesons into charmonium states

TL;DR

This paper calculates the form factors for semileptonic $B_c$ meson decays into various charmonium states using a covariant quark model, providing predictions for branching ratio ratios to be tested at LHC.

Contribution

It presents a comprehensive calculation of $B_c$ decay form factors across the full kinematic range within a covariant quark model, including ratios of branching fractions.

Findings

Calculated form factors for multiple $B_c$ decay channels.

Predicted ratios of branching fractions ($R_{\eta_c}$, $R_{J/\psi}$, etc.).

Results are consistent with other theoretical models.

Abstract

In this work we study the semileptonic decays of $B_c$ meson. We evaluated $B_{c}\rightarrow D(D^{\ast})$, $B_{c}\rightarrow D_s(D_s^{\ast})$ and $B_{c}\rightarrow \eta_{c}(J/\psi)$ transitions form factors in the full kinematical region within the covariant quark model. The calculated form factors are used to evaluate the semileptonic decays of $B_c$ meson and it was defined ratios ($R_{\eta_{c}}$, $R_{J/\psi}$, $R_{ D}$ ,$R_{ D^{\ast}}$) of the branching ratios, which will be hopefully tested on LHC experiments.We compare the obtained results with the results from other theoretical approaches.

Tables

Table 1. Table 1: Leptonic decay constants f H subscript 𝑓 𝐻 f_{H} (MeV).

	This work	Other	Ref.
$f_{B_c}$	489	$489\pm 4\pm 3$	LAT Chiu:2007km
		$395\pm 15$	Kiselev:2003uk
$f_D$	206	$222.6\pm {16.7}_{-3.4}^{+2.8}$	CLEO Artuso:2005ym
		$201\pm 3\pm 17$	MILC LAT Aubin:2005ar
		$235\pm 8\pm 14$	LAT Chiu:2005ue
		$210\pm {10}_{-16}^{+17}$	UKQCD LAT Lellouch:2000tw
		$211\pm {14}_{-12}^{+2}$	LAT Becirevic:1998ua
		204.6$\pm$5.0	PDG Agashe:2014kda
$f_{D^{\ast }}$	244	$245\pm {20}_{-2}^{+3}$	LAT Becirevic:1998ua
		$278\pm 13\pm 10$	LAT Becirevic:2012ti
		$252.2\pm 22.3\pm 4$	QCD SR Lucha:2014xla
$f_{D_s}$	257	257.5$\pm$4.6	PDG Agashe:2014kda
		$249\pm 3\pm 16$	MILC LAT Aubin:2005ar
		$266\pm 10\pm 18$	LAT Chiu:2005ue
		$290\pm 20\pm 29\pm 29\pm 6$	LAT Wingate:2003gm
		$236\pm 8_{-14}^{+17}$	UKQCD LAT Lellouch:2000tw
		$231\pm {12}_{-1}^{+8}$	LAT Becirevic:1998ua
$f_{D_s^{\ast }}$	272	311$\pm$9	LAT Becirevic:2012ti
		$272{\left(16\right)}_{-20}^{+3}$	LAT Becirevic:1998ua
		$305.5\pm 26.8\pm 5$	QCD SR Lucha:2014xla
$\frac{f_{D_s}}{f_D}$	1.25	1.258$\pm$0.038	PDG Agashe:2014kda
		$1.24\pm 0.01\pm 0.07$	MILC LAT Aubin:2005ar
		$1.13\pm 0.03\pm 0.05$	LAT Chiu:2005ue
		$1.13\pm {0.02}_{-0.02}^{+0.04}$	UKQCD LAT Lellouch:2000tw
		$1.10\pm 0.02$	LAT Becirevic:1998ua
$f_{{\eta }_c}$	628	$420\pm 52$	Hwang:1997ie
		$337.7\pm 18.2$	pQCDSun:2017lla
$f_{J/\psi }$	415	$405\pm 14$	pQCDRui:2014tpa
		$416.2\pm 7.4$	pQCDSun:2017lla

Table 2. Table 2: Form factors for B c → D ​ ( D ∗ ) → subscript 𝐵 𝑐 𝐷 superscript 𝐷 ∗ B_{c}\rightarrow D(D^{\ast}) , B c → D s ​ ( D s ∗ ) → subscript 𝐵 𝑐 subscript 𝐷 𝑠 superscript subscript 𝐷 𝑠 ∗ B_{c}\rightarrow D_{s}(D_{s}^{\ast}) and B c → η c ​ ( J / ψ ) → subscript 𝐵 𝑐 subscript 𝜂 𝑐 𝐽 𝜓 B_{c}\rightarrow\eta_{c}(J/\psi) transitions. Form factors are approximated by the form F ​ ( q 2 ) = F ​ ( 0 ) / ( 1 − a ​ s ^ + b ​ s ^ 2 ) 𝐹 superscript 𝑞 2 𝐹 0 1 𝑎 ^ 𝑠 𝑏 superscript ^ 𝑠 2 F(q^{2})=F(0)/(1-a\,\hat{s}+b\,\hat{s}^{2}) with s ^ = q 2 / m B c 2 ^ 𝑠 superscript 𝑞 2 superscript subscript 𝑚 subscript 𝐵 𝑐 2 \hat{s}=q^{2}/m_{B_{c}}^{2} .

	$B_c\to D\left(D^{\ast }\right)$	$B_c\to D_s\left(D_s^{\ast }\right)$	$B_c\to {\eta }_c\left(J/\psi \right)$
$F_{+}\left(0\right)$	0.186	0.254	0.74
$F_{-}\left(0\right)$	-0.160	-0.202	-0.39
$A_0\left(0\right)$	0.276	0.365	1.65
$A_{+}\left(0\right)$	0.151	0.190	0.55
$A_{-}\left(0\right)$	-0.236	-0.293	-0.87
$V\left(0\right)$	0.230	0.282	0.78

Table 3. Table 3: Branching ratios (in % percent \% ) of semileptonic B c subscript 𝐵 𝑐 B_{c} decays into ground state charmonium states.

Mode	This work	Ivanov:2006ni	Ivanov:2000aj	KKL ; exBc	Chang:1992pt	narod	Wang:2012
$B_c^{-}\to {\eta }_c\ell \nu$	0.95	0.81	0.98	0.75	0.97	0.59	0.44
$B_c^{-}\to {\eta }_c\tau \nu$	0.24	0.22	0.27	0.23		0.20	0.14
$B_c^{-}\to J/\psi \ell \nu$	1.67	2.07	2.30	1.9	2.35	1.20	1.01
$B_c^{-}\to J/\psi \tau \nu$	0.40	0.49	0.59	0.48		0.34	0.29
$B_c^{-}\to {\overline{D}}^{-}\ell \nu$	0.0033	0.0035	0.018		0.004	0.006	0.0032
$B_c^{-}\to {\overline{D}}^{-}\tau \nu$	0.0021	0.0021	0.0094	0.002			0.0022
$B_c^{-}\to {\overline{D}}^{\ast -}\ell \nu$	0.006	0.0038	0.034		0.018	0.018	0.011
$B_c^{-}\to {\overline{D}}^{\ast -}\tau \nu$	0.0034	0.0022	0.019	0.008			0.006

Table 4. Table 4: Ratios of semileptonic decays of the B c subscript 𝐵 𝑐 B_{c} meson

Decay rate	This work	Ivanov:2006ni	Wang:2012
$R_{{\eta }_c}=\frac{B_c^{-}\to {\eta }_c\ell \nu }{B_c^{-}\to {\eta }_c\tau \nu }$	3.96	3.68	3.2
$R_{J/\psi }=\frac{B_c^{-}\to J/\psi \ell \nu }{B_c^{-}\to J/\psi \tau \nu }$	4.18	4.22	3.4
$R_D=\frac{B_c^{-}\to D^{-}\ell \nu }{B_c^{-}\to D^{-}\tau \nu }$	1.57	1.67	1.42
$R_{D^{\ast }}=\frac{B_c^{-}\to D^{\ast -}\ell \nu }{B_c^{-}\to D^{\ast -}\tau \nu }$	1.76	1.72	1.66

Equations

$${\mathcal{L}}_{\mathrm{int}}(x)$$

$$J_M(x)$$

$$F_M(x,x_1,x_2)=\delta (x-w_1{x}_1-w_2{x}_2){\Phi }_M((x_1-x_2{)}^2),$$

$${\bar{\Phi }}_M(-k^2)=\exp \left(k^2/{\Lambda }_M^2\right)$$

$$S_i(k)=\frac{1}{m_{q_i}-/k}$$

$$Z_M=1-\frac{3g_M^2}{4{\pi }^2}{\bar{\Pi }}_M^{\prime }(m_M^2)=0,$$

$${\Pi }_P(x-y)$$

$${\Pi }_V^{\mu \nu }(x-y)$$

$$\begin{array}[]{ccccc|c}m_{u/d}&m_{s}&m_{c}&m_{b}&\lambda\\ \hline\cr\ \ 0.241&\ \ 0.428&\ \ 1.67&\ \ 5.05&\ \ 0.181&\ {\rm GeV}\end{array}$$

$$\begin{array}[]{cccccccccc|c}\Lambda_{B_{c}}&\Lambda_{\eta_{c}}&\Lambda_{J/\psi}&\Lambda_{D}&\Lambda_{D^{\ast}}&\Lambda_{D_{s}}&\Lambda_{D^{\ast}_{s}}&\Lambda_{B}&\Lambda_{B^{\ast}}&\Lambda_{B_{s}}&\\ \hline\cr\ \ 2.73&\ \ 3.97&\ \ 1.74&\ \ 1.6&\ \ 1.53&\ \ 1.75&\ \ 1.56&\ \ 1.96&\ \ 1.8&\ \ 2.05&\ {\rm GeV}\\ \end{array}$$

$$M(H_{12}\to \bar{l}\nu )=\frac{G_F}{\sqrt{2}}{V}_{q_1{q}_2}{\mathcal{M}}_H^{\mu }(p){\bar{u}}_l(k_l)O^{\mu }{u}_{\nu }(k_{\nu }),$$

$${\mathcal{M}}_H^{\mu }(p)=-3g_{12}\int \frac{d^{4}k}{(2\pi {)}^{4}i}{\tilde{\Phi }}_{12}\left(-k^2\right)\mathrm{tr}\left[{\Gamma }_H{\tilde{S}}_2(k-c_{12}^{2}p)O^{\mu }{\tilde{S}}_1(k+c_{12}^{1}p)\right],$$

$${\Gamma }_P=i{\gamma }^5,{\Gamma }_V={\epsilon }_V\cdot \gamma ,$$

$${\mathcal{M}}_P^{\mu }(p)=-i{f}_P{p}^{\mu },{\mathcal{M}}_V^{\mu }(p)=f_V{m}_V{\epsilon }_V^{\mu }.$$

$$M(H_{13}\to H_{23}+\bar{l}\nu )$$

$${\mathcal{M}}_{12}^{\mu }$$

$${\mathcal{M}}_{S=0}^{\mu }$$

$${\mathcal{M}}_{S=1}^{\mu }$$

$$P$$

$$H_t$$

$$H_{\pm }$$

$$H_0$$

$$H_t$$

$$H_{\pm }$$

$$H_0$$

$$\Gamma (B_c^{-}\to M_{\bar{c}c}l\bar{\nu })$$

$$\Gamma (B_c^{-}\to {\overline{D}}^{0}l\bar{\nu })$$

$$\begin{array}[]{ccccccc|c|c}m_{B_{c}}&m_{\eta_{c}}&m_{J/\psi}&m_{D}&m_{D^{\ast}}&m_{D_{s}}&m_{D^{\ast}_{s}}&&\tau_{B_{c}}\\ \hline\cr\ \ 6.275&\ \ 2.984&\ \ 3.097&\ \ 1.869&\ \ 2.010&\ \ 1.968&\ \ 2.112&\ {\rm GeV}&\ \ 0.507ps\end{array}$$

$$\begin{array}[]{ccccccc}|V_{ud}|&|V_{us}|&|V_{cd}|&|V_{cs}|&|V_{cb}|&|V_{ub}|\\ \hline\cr\ \ 0.974&\ \ 0.225&\ \ 0.220&\ \ 0.995&\ \ 0.0405&\ \ 0.00409\\ \end{array}$$

Full text

1 Model

The covariant quark model was developed by G.V.Efimov and M.A.Ivanov Efimov:1988yd ; Faessler:2002ut ; Branz:2009cd .

The effective Lagrangian describing the transition of a meson $M(q_{1}\bar{q}_{2})$ to its constituent quarks $q_{1}$ and $\bar{q}_{2}$ in model looks like

[TABLE]

with $\Gamma_{M}$ a Dirac matrix which projects onto the spin quantum number of the meson field $M(x)$. The vertex function $F_{M}$ characterizes the finite size of the meson. Translational invariance requires the function $F_{M}$ to fulfill the identity $F_{M}(x+a,x_{1}+a,x_{2}+a)=F_{M}(x,x_{1},x_{2})$ for any four-vector $a$. A specific form for the vertex function is adopted

[TABLE]

where $\Phi_{M}$ is the correlation function of the two constituent quarks with masses $m_{q_{1}}$, $m_{q_{2}}$ and the mass ratios $w_{i}=m_{q_{i}}/(m_{q_{1}}+m_{q_{2}})$.

A simple Gaussian form of the vertex function $\bar{\Phi}_{M}(-\,k^{2})$ is selected

[TABLE]

with the parameter $\Lambda_{M}$ linked to the size of the meson. The minus sign in the argument is chosen to indicate that we are working in the Minkowski space. Since $k^{2}$ turns into $-\,k_{E}^{2}$ in the Euclidean space, the form (3) has the appropriate fall-off behavior in the Euclidean region. Any choice for $\Phi_{M}$ is appropriate as long as it falls off sufficiently fast in the ultraviolet region of the Euclidean space to render the corresponding Feynman diagrams ultraviolet finite. We choose a Gaussian form for calculational convenience.

The fermion propagators for the quarks are given by

[TABLE]

with an effective constituent quark mass $m_{q_{i}}$.

The so-called compositeness condition Weinberg:1962hj ; Salam:1962ap is used to determine the value of the coupling constants $g_{M}$. It means that the renormalization constant $Z_{M}$ of the elementary meson field $M(x)$ is to be set to zero, i.e.,

[TABLE]

where $\bar{\Pi}^{\prime}_{M}$ is the derivative of the meson mass operator. Its physical meaning in Eq. (5) becomes clear when interpreted as the matrix element between the physical and the corresponding bare state: $Z_{M}=0$ implies that the physical state does not contain the bare state and is appropriately described as a bound state. The interaction makes the physical particle dressed, i.e. its mass and wave function have to be renormalized. The condition $Z_{M}=0$ also effectively excludes the constituent degrees of freedom from the space of physical states. It thereby guarantees the absence of double counting for the physical observable under consideration, the constituents exist only in virtual states. The tree-level diagram together with the diagrams containing self-energy insertions into the external legs (i.e. the tree-level diagram times $Z_{M}-1$) give a common factor $Z_{M}$ which is equal to zero.

The mass functions for the pseudoscalar meson (spin $S=0$) and vector meson (spin $S=1$) are defined as

[TABLE]

Herein we use the updated values of the model parameters from Dubnicka:2016nyy which are shown in Eq. (8,9).

[TABLE]

[TABLE]

2 Semileptonic decays

We give the necessary definitions of the leptonic decay constants, invariant form factors and helicity amplitudes. The leptonic decay constants are defined by

[TABLE]

The semileptonic decays of the $B_{c}$-meson may be induced by a b-quark transition.

[TABLE]

where $q_{1}\equiv b$ and $q_{3}\equiv c$ whereas $q_{2}$ denotes either of $c,u,d,s$.

The invariant form factors for the semileptonic $B_{c}$-decay into the hadron with spin $S=0,1$ are defined by

[TABLE]

It is convenient to express all physical observables through the helicity form factors $H_{m}$. The helicity form factors $H_{m}$ can be expressed in terms of the invariant form factors in the following way Ivanov:2000aj :

(a) Spin $S=0$:

[TABLE]

(b) Spin $S=1$:

[TABLE]

where $|{\bf p_{2}}|=\lambda^{1/2}(m_{1}^{2},m_{2}^{2},q^{2})/(2\,m_{1})$ is the momentum of the outgoing particles in the $B_{c}$ rest frame. The semileptonic $B_{c}$-decay widths are given by

[TABLE]

where $q^{2}=(m_{1}-m_{2})^{2}$, $m_{1}\equiv m_{B_{c}}$, and $m_{2}\equiv m_{f}$. Note that $M_{\bar{c}c}$ and $\overline{D}^{0}$ denote both the pseudoscalar and vector cases.

3 Numerical results

We take the following values (16) of the meson masses and the $B_{c}$-meson’s lifetime from the PDG Olive:2016xmw .

[TABLE]

The calculation of the semileptonic decay widths is straightforward. For the CKM-matrix elements we use

[TABLE]

The value of the decay constant $f_{\eta_{c}}$ was calculated from the branching ratio for the $\eta_{c}$ meson decay into two photons using the last data Olive:2016xmw . The quality of the fit may be assessed from the entries in Table 1.

The form factors are calculated in the full kinematical region of momentum transfer squared and are shown in Table 2. The curves are depicted in Fig. 1 and Fig. 2.

The results of our evaluation of the branching ratios of the semileptonic $B_{c}$ decays appear in Table 3, which contains our predictions for the semileptonic $B_{c}$ decays into ground state charmonium states and charm meson states. We compare our ratios of semileptonic decays of the $B_{c}$ meson with those of other models in Table 4.

Acknowledgment

Authors A. Issadykov, M.A. Ivanov and G.S. Nurbakova acknowledge the partial support by the Ministry of Education and Science of the Republic of Kazakhstan, grant 3092/GF4, state registration No. 0115RK01040.

Author A. Issadykov is grateful for the support by the JINR, grant number 17-302-03.