Placing the newly observed state $B_{J}(5840)$ in bottom spectra along with states $B_{1}(5721)$, $B_{2}^{*}(5747)$, $B_{s1}(5830)$, $B_{2s}^{*}(5840)$ and $B_{J}(5970)$

TL;DR

This paper classifies and analyzes the strong decay properties of excited bottom mesons, including recently observed states, using HQET to assign quantum numbers and predict decay widths for unobserved states, aiding future experiments.

Contribution

It provides a detailed classification of the $B_{J}(5840)$ state among possible quantum numbers and predicts decay widths for missing bottom mesons using HQET-based coupling constants.

Findings

$B_{J}(5840)$ is most likely assigned as a specific $J^{P}$ state based on decay analysis.

Predicted decay widths and branching ratios for several unobserved bottom mesons.

Identification of quantum numbers for $B_{J}(5970)$, $B_{J}(5721)$, and $B_{2}^{*}(5747)$.

Abstract

In this article, we study the two body strong decays with the emission of light pseudo-scalar mesons $(\pi, \eta, K)$ for higher excited bottom states with in the framework of HQET. Inspired from the recent observation of bottom meson $B_{J}(5840)$ by LHCb collaboration \cite{9}, we classify the six possible $J^{P}$'s for this state on the basis of the theoretically available masses. By analyzing the strong decay widths and the branching ratios for all these six cases of $B_{J}(5840)$, we justify one of them to be the most favorable assignment for it. We also examined the recently observed bottom state $B_{J}(5970)$ as 2S$1^{-}$ and states $B_{J}(5721)$ and $B_{2}^{*}(5747)$ with their strange partners $B_{s1}(5830)$ and $B_{2s}^{*}(5840)$ for their $J^{P}$'s as $1P_{3/2}1^{+}$ and $1P_{3/2}2^{+}$ respectively. The predicted coupling constants $g_{XH}$, $\widetilde{g}_{HH}$ and $g_{TH}$ helps in redeeming the strong decay width of experimentally missing bottom states $B(2 ^{1}S_{0})$, $B_{s}(2 ^{3}S_{1})$, $B_{s}(2 ^{1}S_{0})$, $B(1 ^{1}D_{2})$, $B_{s}(1 ^{3}D_{1})$ and $B_{s}(1 ^{1}D_{2})$. These predictions provide a crucial information for upcoming experimental studies.

Tables

Table 1. Table 1: Values of the masses and the decay widths of bottom mesons observed by various collaborations.

State	$J^P$	Mass/MeV	Width/MeV	Experiments	Observed Decay Modes
$B_J\left(5721\right)$	$1^{+}$	$5727.7\pm 0.7$	$30.1\pm 1.5$	LHCb 9	$B^{\ast }\pi$
		$5720.6\pm 2.4$	-	D010	$B^{\ast }\pi$
		$5725.3\pm 1.6$	-	CDF11	$B^{\ast }\pi$
$B_2^{\ast }\left(5747\right)$	$2^{+}$	$5739.44\pm 0.37$	$24.5\pm 1.0$	LHCb 9	$B^{\ast }\pi$,$B\pi$
		$5746.8\pm 2.4$	-	D010	$B^{\ast }\pi$,$B\pi$
		$5740.2\pm 1.7$	$22.7\pm 3.2$	CDF11	$B^{\ast }\pi$,$B\pi$
$B_J\left(5840\right)$	$-$	$5862.9\pm 5.0$	$127.4\pm 16.7$	LHCb 9	$B\pi$
$B_J\left(5960\right)$	$-$	$5978\pm 5$	-	CDF11	$B\pi$
		$5969.2\pm 2.9$	$82.3\pm 7.7$	LHCb 9	$B\pi$
$B_{s1}\left(5830\right)$	$1^{+}$	$5828.40\pm 0.04$	-	LHCb37	$B^{\ast }K$
		$5828.3\pm 0.1$	$0.5\pm 0.3$	CDF38	$B^{\ast }K$
		$5829.4\pm 0.7$	-	CDF7	$B^{\ast }K$
$B_{s2}^{\ast }\left(5840\right)$	$2^{+}$	$5839.6\pm 1.1$	-	D039	$B^{\ast }K$,BK
		$5839.70\pm 0.7$	-	CDF7	$B^{\ast }K$,BK
		$5839.70\pm 0.1$	$1.40\pm 0.4$	CDF38	$B^{\ast }K$,BK
		$5839.99\pm 0.05$	$1.56\pm 0.13$	Lhcb37	$B^{\ast }K$,BK

Table 2. Table 2: Numerical value of the meson masses used in this work 8 .

States	$B^0$	$B^{\pm }$	$B^{\ast }$	$B_s$	$B_s^{\ast }$
Masses/MeV	5279.58	5279.25	5325.20	5366.77	5415.40
States	${\pi }^{\pm }$	${\pi }^0$	$\eta$	$K^{+}$	$K^0$
Masses/MeV	139.57	134.97	547.85	493.67	497.61

Table 3. Table 3: Strong decay channels for all the six possible spin-parity J P superscript 𝐽 𝑃 J^{P} values for B J ​ ( 5840 ) subscript 𝐵 𝐽 5840 B_{J}(5840) state.

Decay Mode	2S$0^{-}$	2S$1^{-}$	1D$1^{-}$	1D$2_{3/2}^{-}$	1D$2_{5/2}^{-}$	1D$3^{-}$
$B^0{\pi }^0$	-	220.64${\tilde{g}}_{HH}^2$	182.58$g_{XH}^2$	-	-	12.84$g_{YH}^2$
$B^{+}{\pi }^{-}$	-	439.40${\tilde{g}}_{HH}^2$	364.14$g_{XH}^2$	-	-	25.44$g_{YH}^2$
$B^0\eta$	-	12.47${\tilde{g}}_{HH}^2$	11.60$g_{XH}^2$	-	-	0.01$g_{YH}^2$
$B_{s}K$	-	-	-	-	-	-
$B^{\ast }{\pi }^0$	523.71${\tilde{g}}_{HH}^2$	347.46${\tilde{g}}_{HH}^2$	61.63$g_{XH}^2$	184.91$g_{XH}^2$	16.98$g_{YH}^2$	9.70$g_{YH}^2$
$B^{\ast }{\pi }^{+}$	1040.10${\tilde{g}}_{HH}^2$	690.08${\tilde{g}}_{HH}^2$	122.46$g_{XH}^2$	367.39$g_{XH}^2$	33.41$g_{YH}^2$	19.09$g_{YH}^2$
$B^{\ast }\eta$	-	-	-	-	-	-
$B_s^{\ast }K$	-	-	-	-	-	-
Total	1563.82${\tilde{g}}_{HH}^2$	1710.09${\tilde{g}}_{HH}^2$	742.43$g_{XH}^2$	552.30$g_{XH}^2$	50.40$g_{YH}^2$	67.09$g_{YH}^2$
Ratio $R_1$	0	0.63	2.96	0	0	1.32

Table 4. Table 4: Strong decay width of newly observed bottom mesons B J ∗ ​ ( 5840 ) superscript subscript 𝐵 𝐽 5840 B_{J}^{*}(5840) and its spin and strange partners B ​ ( 1 1 ​ D 2 ) 𝐵 superscript 1 1 subscript 𝐷 2 B(1^{1}D_{2}) , B s ​ ( 1 1 ​ D 2 ) subscript 𝐵 𝑠 superscript 1 1 subscript 𝐷 2 B_{s}(1^{1}D_{2}) and B s ∗ ​ ( 1 3 ​ D 1 ) superscript subscript 𝐵 𝑠 superscript 1 3 subscript 𝐷 1 B_{s}^{*}(1^{3}D_{1}) . Ratio in 5th column represents the 𝚪 ^ = Γ Γ ​ ( B J ( ∗ ) → B ∗ + ​ π − ) ^ 𝚪 Γ Γ → superscript subscript 𝐵 𝐽 superscript 𝐵 absent superscript 𝜋 \widehat{{\bf\Gamma}}=\frac{\Gamma}{\Gamma(B_{J}^{(*)}\rightarrow B^{*+}\pi^{-})} for the non-strange mesons and 𝚪 ^ = Γ Γ ​ ( B s ​ J ∗ → B ∗ 0 ​ K + ) ^ 𝚪 Γ Γ → superscript subscript 𝐵 𝑠 𝐽 superscript 𝐵 absent 0 superscript 𝐾 \widehat{{\bf\Gamma}}=\frac{\Gamma}{\Gamma(B_{sJ}^{*}\rightarrow B^{*0}K^{+})} for the strange mesons. Branching fraction (B.R) gives the percentage of the partial decay width with respect to the total decay width.

State	$nL{s}_l{J}^P$	Decay channel	Decay Width/MeV	Ratio	B.F $\%$	Experimental/Theoretical value(MeV)
$B_J\left(5840\right)$	1$D_{3/2}{1}^{-}$	$B^{\ast }{\pi }^{+}$	122.46$g_{XH}^2$	1	16.49
		$B^{\ast }{\pi }^0$	61.63$g_{XH}^2$	0.50	8.30
		$B^{\ast }\eta$	-	-	-
		$B_s^{\ast }K$	-	-	-
		$B^0{\pi }^0$	182.58$g_{XH}^2$	1.49	24.59
		$B^{+}{\pi }^{-}$	364.14$g_{XH}^2$	2.97	49.04
		$B^0\eta$	11.60	0.09	1.56
		$B_{s}K$	-	-	-
		Total	742.43$g_{XH}^2$			127.40 9 , 127 41b
$B_J\left(5967.20\right)$	1$D_{3/2}{2}^{-}$	$B^{\ast }{\pi }^{+}$	866.42$g_{XH}^2$	1	60.34
		$B^{\ast }{\pi }^0$	435.16$g_{XH}^2$	0.50	30.31
		$B^{\ast }\eta$	94.61$g_{XH}^2$	0.10	6.58
		$B_s^{\ast }K$	39.42$g_{XH}^2$	0.04	2.74
		Total	1435.69$g_{XH}^2$			250.6920 , 98 41b
$B_{sJ}\left(6083.00\right)$	1$D_{s\left(3/2\right)}{1}^{-}$	$B_s^{\ast }{\pi }^0$	175.31$g_{XH}^2$	0.57	5.64
		$B_s^{\ast }\eta$	12.39$g_{XH}^2$	0.04	0.39
		$B^{\ast 0}{K}^0$	300.48$g_{XH}^2$	0.98	9.67
		$B^{\ast -}{K}^{+}$	305.62$g_{XH}^2$	1	9.84
		$B_s^0{\pi }^0$	488.80$g_{XH}^2$	1.59	15.73
		$B_s\eta$	49.01$g_{XH}^2$	0.16	1.57
		$B^{+}{K}^{-}$	894.37$g_{XH}^2$	2.92	28.79
		$B^0{K}^0$	879.81$g_{XH}^2$	2.87	28.32
		Total	3105.83$g_{XH}^2$			213.3820 , 137 41b
$B_{sJ}\left(6057.50\right)$	1$D_{s\left(3/2\right)}{2}^{-}$	$B_s^{\ast }{\pi }^0$	436.79$g_{XH}^2$	0.60	23.04
		$B_s^{\ast }\eta$	23.62$g_{XH}^2$	0.03	1.24
		$B^{\ast -}{K}^{+}$	724.42$g_{XH}^2$	1	38.21
		$B^{\ast 0}{K}^0$	710.60$g_{XH}^2$	0.98	37.48
		Total	1895.45$g_{XH}^2$			198.6420 , 89 41b

Table 5. Table 5: Strong decay width of newly observed bottom mesons B 1 ​ ( 5721 ) subscript 𝐵 1 5721 B_{1}(5721) and B 2 ∗ ​ ( 5747 ) superscript subscript 𝐵 2 5747 B_{2}^{*}(5747) and their strange partners B 1 ​ s ​ ( 5830 ) subscript 𝐵 1 𝑠 5830 B_{1s}(5830) and B 2 ​ s ∗ ​ ( 5840 ) superscript subscript 𝐵 2 𝑠 5840 B_{2s}^{*}(5840) . Ratio in 5th column represents the 𝚪 ^ = Γ Γ ​ ( B J ( ∗ ) → B ∗ + ​ π − ) ^ 𝚪 Γ Γ → superscript subscript 𝐵 𝐽 superscript 𝐵 absent superscript 𝜋 \widehat{{\bf\Gamma}}=\frac{\Gamma}{\Gamma(B_{J}^{(*)}\rightarrow B^{*+}\pi^{-})} for the non-strange mesons and 𝚪 ^ = Γ Γ ​ ( B s ​ J ∗ → B ∗ 0 ​ K + ) ^ 𝚪 Γ Γ → superscript subscript 𝐵 𝑠 𝐽 superscript 𝐵 absent 0 superscript 𝐾 \widehat{{\bf\Gamma}}=\frac{\Gamma}{\Gamma(B_{sJ}^{*}\rightarrow B^{*0}K^{+})} for the strange mesons. Branching fraction (B.F) gives the percentage of the partial decay width with respect to the total decay width.

State	$nL{s}_l{J}^P$	Decay channel	Decay Width(MeV)	Ratio	B.F $\%$	Experimental/Theoretical value(MeV)
$B_1\left(5721\right)$	1$P_{3/2}{1}^{+}$	$B^{\ast }{\pi }^{+}$	74.83$g_{TH}^2$	1	66.16
		$B^{\ast }{\pi }^0$	38.25$g_{TH}^2$	0.51	33.82
		Total	113.09$g_{TH}^2$			30.1 9 , 55 41b
$B_2^{\ast }\left(5747\right)$	1$P_{3/2}{2}^{+}$	$B^{\ast }{\pi }^{+}$	52.47$g_{TH}^2$	1	30.65
		$B^{\ast }{\pi }^0$	26.78$g_{TH}^2$	0.51	15.64
		$B^0{\pi }^0$	30.91$g_{TH}^2$	0.58	18.05
		$B^{+}{\pi }^{-}$	61.02$g_{TH}^2$	1.16	35.64
		Total	171.18$g_{TH}^2$			24.59 , 7 41b
$B_{1s}\left(5830\right)$	1$P_{s3/2}{1}^{+}$	$B_s^{\ast }{\pi }^0$	44.45$g_{TH}^2$	-	100
		$B^{\ast +}{K}^0$	$\sim 𝒪\left({10}^{-10}\right)$$g_{TH}^2$	-	-
		$B^{\ast -}{K}^{+}$	$\sim 𝒪\left({10}^{-10}\right)$$g_{TH}^2$	-	-
		Total	44.45$g_{TH}^2$			0.5 38 , 30 41b
$B_{2s}^{\ast }\left(5840\right)$	1$P_{s3/2}{2}^{+}$	$B_s^{\ast }{\pi }^0$	30.41$g_{TH}^2$	87.38	40.27
		$B^{\ast +}{K}^0$	$0.20g_{TH}^2$	0.59	0.27
		$B^{\ast -}{K}^{+}$	$0.34g_{TH}^2$	1	0.46
		$B^{+}{K}^{-}$	3.97$g_{TH}^2$	11.40	5.25
		$B^0{K}^0$	4.64$g_{TH}^2$	13.33	6.14
		$B_s{\pi }^0$	35.92$g_{TH}^2$	103.21	47.53
		Total	75.57$g_{TH}^2$			1.4038 , 1 41b

Table 6. Table 6: Strong decay width of bottom meson B J ​ ( 5970 ) subscript 𝐵 𝐽 5970 B_{J}(5970) with its spin and strange partners B ​ ( 2 1 ​ S 0 ) 𝐵 superscript 2 1 subscript 𝑆 0 B(2^{1}S_{0}) , B s ​ ( 2 1 ​ S 0 ) subscript 𝐵 𝑠 superscript 2 1 subscript 𝑆 0 B_{s}(2^{1}S_{0}) and B s ​ ( 2 3 ​ S 1 ) subscript 𝐵 𝑠 superscript 2 3 subscript 𝑆 1 B_{s}(2^{3}S_{1}) . Ratio in 5th column represents the 𝚪 ^ = Γ Γ ​ ( B J ( ∗ ) → B ∗ + ​ π − ) ^ 𝚪 Γ Γ → superscript subscript 𝐵 𝐽 superscript 𝐵 absent superscript 𝜋 \widehat{{\bf\Gamma}}=\frac{\Gamma}{\Gamma(B_{J}^{(*)}\rightarrow B^{*+}\pi^{-})} for the non-strange mesons and 𝚪 ^ = Γ Γ ​ ( B s ​ J ∗ → B ∗ 0 ​ K + ) ^ 𝚪 Γ Γ → superscript subscript 𝐵 𝑠 𝐽 superscript 𝐵 absent 0 superscript 𝐾 \widehat{{\bf\Gamma}}=\frac{\Gamma}{\Gamma(B_{sJ}^{*}\rightarrow B^{*0}K^{+})} for the strange mesons. Branching fraction (B.F) gives the percentage of the partial decay width with respect to the total decay width.

State	$nL{s}_l{J}^P$	Decay channel	Decay Width(MeV)	Ratio	B.F $\%$	Experimental/Theoretical value(MeV)
$B_0\left(5881\right)$	2${}_{}{}^{1}S_0^{}{0}^{-}$	$B^{\ast }{\pi }^{+}$	1148.08${\tilde{g}}_{HH}^2$	1	66.48
		$B^{\ast }{\pi }^0$	577.80${\tilde{g}}_{HH}^2$	0.50	33.27
		$B^{\ast }\eta$	1.00${\tilde{g}}_{HH}^2$	0.00	0.05
		$B_s^{\ast }K$	-	-	-
		Total	1726.89${\tilde{g}}_{HH}^2$			91 41b
$B_J\left(5970\right)$	2${}_{}{}^{3}S_1^{}{1}^{-}$	$B^{\ast }{\pi }^{+}$	1178.23${\tilde{g}}_{HH}^2$	1	36.30
		$B^{\ast }{\pi }^0$	591.95${\tilde{g}}_{HH}^2$	0.50	18.23
		$B^{\ast }\eta$	122.22${\tilde{g}}_{HH}^2$	0.10	3.76
		$B_s^{\ast }K$	69.94${\tilde{g}}_{HH}^2$	0.05	2.15
		$B^0{\pi }^0$	359.11${\tilde{g}}_{HH}^2$	0.30	11.06
		$B^{+}{\pi }^{-}$	716.21${\tilde{g}}_{HH}^2$	0.60	22.06
		$B^0\eta$	113.37${\tilde{g}}_{HH}^2$	0.09	3.49
		$B_{s}K$	94.43${\tilde{g}}_{HH}^2$	0.08	2.90
		Total	3245.49${\tilde{g}}_{HH}^2$			82.30 9 , 107 41b
$B_{s0}\left(5976.0\right)$	$\left(2^1{S}_0\right)0^{-}$	$B^{\ast 0}{K}^0$	521.96${\tilde{g}}_{HH}^2$	0.96	31.38
		$B^{\ast +}{K}^{-}$	539.41${\tilde{g}}_{HH}^2$	1	32.43
		$B_s^{\ast }{\pi }^0$	593.72${\tilde{g}}_{HH}^2$	1.10	35.70
		$B_s^{\ast }\eta$	7.85${\tilde{g}}_{HH}^2$	0.01	0.47
		Total	1662.97${\tilde{g}}_{HH}^2$			75.8023 , 106 41b
$B_s^{\ast }$(6007.8)	$2^3{S}_1{1}^{-}$	$B^0{K}^0$	342.71${\tilde{g}}_{HH}^2$	0.97	15.87
		$B^{+}{K}^{-}$	350.66${\tilde{g}}_{HH}^2$	1	16.24
		$B_s{\pi }^0$	292.67${\tilde{g}}_{HH}^2$	0.83	13.55
		$B_s\eta$	58.29${\tilde{g}}_{HH}^2$	0.16	2.70
		$B^{\ast 0}{K}^0$	474.96${\tilde{g}}_{HH}^2$	1.35	22.00
		$B^{\ast +}{K}^{-}$	486.47${\tilde{g}}_{HH}^2$	1.38	22.53
		$B_s^{\ast }{\pi }^0$	466.53${\tilde{g}}_{HH}^2$	1.33	21.61
		$B_s^{\ast }\eta$	36.98${\tilde{g}}_{HH}^2$	0.10	1.71
		Total	2158.65${\tilde{g}}_{HH}^2$			114.023 , 127 41b

Equations

$$H_a=\frac{1+/v}{2}\{P_{a\mu }^{\ast }{\gamma }^{\mu }-P_a{\gamma }_5\}.$$

$$T_a^{\mu }=\frac{1+/v}{2}\{P_{2a}^{\ast \mu \nu }{\gamma }_{\nu }-P_{1a\nu }\sqrt{\frac{3}{2}}{\gamma }_5[g^{\mu \nu }-\frac{{\gamma }^{\nu }({\gamma }^{\mu }-{\upsilon }^{\mu })}{3}]\}.$$

$$X_a^{\mu }=\frac{1+/v}{2}\{P_{2a}^{\mu \nu }{\gamma }_5{\gamma }_{\nu }-P_{1a\nu }^{\ast }\sqrt{\frac{3}{2}}[g^{\mu \nu }$$

$$-\frac{{\gamma }_{\nu }({\gamma }^{\mu }+v^{\mu })}{3}]\}.$$

$$Y_a^{\mu \nu }=\frac{1+/v}{2}\{P_{3a}^{\ast \mu \nu \sigma }{\gamma }_{\sigma }-P_{2a}^{\alpha \beta }\sqrt{\frac{5}{3}}{\gamma }_5[g_{\alpha }^{\mu }{g}_{\beta }^{\nu }$$

$$-\frac{g_{\beta }^{\nu }{\gamma }_{\alpha }({\gamma }^{\mu }-v^{\mu })}{5}-\frac{g_{\alpha }^{\mu }{\gamma }_{\beta }({\gamma }^{\nu }-v^{\nu })}{5}]\}.$$

$$\mathcal{M}=\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}{\pi }^0+\frac{1}{\sqrt{6}}\eta & {\pi }^{+}& K^{+}\\ {\pi }^{-}& -\frac{1}{\sqrt{2}}{\pi }^0+\frac{1}{\sqrt{6}}\eta & K^0\\ K^{-}& {\overline{K}}^0& -\sqrt{\frac{2}{3}}\eta \end{array}\right).$$

$$L_{HH}=g_{HH}Tr\{{\overline{H}}_a{H}_b{\gamma }_{\mu }{\gamma }_5{A}_{ba}^{\mu }\}.$$

$$L_{TH}=\frac{g_{TH}}{\Lambda }Tr\{{\overline{H}}_a{T}_b^{\mu }(i{D}_{\mu }/A+i/D{A}_{\mu }{)}_{ba}{\gamma }_5\}+h.c..$$

$$L_{XH}=\frac{g_{XH}}{\Lambda }Tr\{{\overline{H}}_a{X}_b^{\mu }(i{D}_{\mu }/A+i/D{A}_{\mu }{)}_{ba}{\gamma }_5\}+h.c..$$

$$L_{YH}=\frac{1}{{\Lambda }^2}Tr\{{\overline{H}}_a{Y}_b^{\mu \nu }[k_1^Y\{D_{\mu },D_{\nu }\}{A}_{\lambda }+k_2^Y(D_{\mu }{D}_{\lambda }{A}_{\nu }$$

$$+D_{\nu }{D}_{\lambda }{A}_{\mu }){]}_{ba}{\gamma }^{\lambda }{\gamma }_5\}+h.c..$$

$$D_{\mu }={\partial }_{\mu }+V_{\mu },\&\{D_{\mu },D_{\nu }\}=D_{\mu }{D}_{\nu }+D_{\nu }{D}_{\mu },$$

$$\{D_{\mu },D_{\nu }{D}_{\rho }\}=D_{\mu }{D}_{\nu }{D}_{\rho }+D_{\mu }{D}_{\rho }{D}_{\nu }+D_{\nu }{D}_{\mu }{D}_{\rho }$$

$$+D_{\nu }{D}_{\rho }{D}_{\mu }+D_{\rho }{D}_{\mu }{D}_{\nu }+D_{\rho }{D}_{\nu }{D}_{\mu }$$

$$\Gamma =\frac{1}{(2J+1)}\sum \frac{p_M}{8\pi M_i^2}|A{|}^2$$

$$\Gamma (1^{-}\to 1^{-})=C_M\frac{g_{HH}^2{M}_f{p}_M^3}{3\pi f_{\pi }^2{M}_i}$$

$$\Gamma (1^{-}\to 0^{-})=C_M\frac{g_{HH}^2{M}_f{p}_M^3}{6\pi f_{\pi }^2{M}_i}$$

$$\Gamma (0^{-}\to 1^{-})=C_M\frac{g_{HH}^2{M}_f{p}_M^3}{2\pi f_{\pi }^2{M}_i}$$

$$\Gamma (2^{+}\to 1^{-})=C_M\frac{2g_{TH}^2{M}_f{p}_M^5}{5\pi f_{\pi }^2{\Lambda }^2{M}_i}$$

$$\Gamma (2^{+}\to 0^{-})=C_M\frac{4g_{TH}^2{M}_f{p}_M^5}{15\pi f_{\pi }^2{\Lambda }^2{M}_i}$$

$$\Gamma (1^{+}\to 1^{-})=C_M\frac{2g_{TH}^2{M}_f{p}_M^5}{3\pi f_{\pi }^2{\Lambda }^2{M}_i}$$

$$\Gamma (1^{-}\to 0^{-})=C_M\frac{4g_{XH}^2}{9\pi f_{\pi }^2{\Lambda }^2}\frac{M_f}{M_i}[p_M^3(m_M^2+p_M^2)]$$

$$\Gamma (1^{-}\to 1^{-})=C_M\frac{2g_{XH}^2}{9\pi f_{\pi }^2{\Lambda }^2}\frac{M_f}{M_i}[p_M^3(m_M^2+p_M^2)]$$

$$\Gamma (2^{-}\to 1^{-})=C_M\frac{2g_{XH}^2}{3\pi f_{\pi }^2{\Lambda }^2}\frac{M_f}{M_i}[p_M^3(m_M^2+p_M^2)]$$

$$\Gamma (2^{-}\to 1^{-})=C_M\frac{4g_{YH}^2}{15\pi f_{\pi }^2{\Lambda }^4}\frac{M_f}{M_i}[p_M^7]$$

$$\Gamma (3^{-}\to 0^{-})=C_M\frac{4g_{YH}^2}{35\pi f_{\pi }^2{\Lambda }^4}\frac{M_f}{M_i}[p_M^7]$$

$$\Gamma (3^{-}\to 1^{-})=C_M\frac{16g_{YH}^2}{105\pi f_{\pi }^2{\Lambda }^4}\frac{M_f}{M_i}[p_M^7]$$

$$BR=R_1=\frac{\Gamma (B_J(5840)\to B\pi )}{\Gamma (B_J(5840)\to B^{\ast }\pi )}$$

$$g_{XH}=0.41\pm \mathrm{0.02.}$$

$$\Gamma (B_{s1}^{\ast })=522.09MeV.$$

$$\Gamma (B_{s2})=318.62MeV.$$

$$(B_1(5721),B_2^{\ast }(5747))=(1^{+},2^{+}{)}_{3/2}\text{with }n=1,L=1.$$

$$(B_{1s}(5830),B_{2s}^{\ast }(5840))=(1^{+},2^{+}{)}_{3/2}\text{with }n=1,L=1.$$

$$R_2=\frac{\Gamma (B_1(5721))}{\Gamma (B_1(5721))+\Gamma (B_2^{\ast }(5747))}=0.60$$

$$R_3=\frac{\Gamma (B_2^{\ast }\to B^{\ast }\pi )}{(\Gamma (B_2^{\ast }\to B^{\ast }\pi )+\Gamma (B_2^{\ast }\to B\pi ))}=0.46$$

$$R_4=B_2^{\ast }(5747)\to \frac{B^{\ast +}{\pi }^{-}}{B^{+}{\pi }^{-}}=0.85$$

$${\tilde{g}}_{HH}=0.15\pm \mathrm{0.01.}$$

$$\Gamma (B_0)=38.85MeV$$

$$\Gamma (B_{s1}^{\ast })=37.41MeV$$

Full text

Heavy quark effective theory,Decays of bottom mesons, Bottom mesons $(|B|>0)$

I Introduction

In the recent decades, a significant experimental progress have been achieved in studying the heavy-light meson spectroscopy. Heavy-light mesons composed of one heavy quark Q and a light quark $\overline{q}$ are useful in understanding the strong interactions in the non perturbative regime. Recently, many new charm states like $D^{*}_{2}(3000)$, $D_{J}(3000)$, $D_{J}^{*}(3000)$, $D^{*}_{3}(2760)$, $D_{1}^{*}(2680)$, $D^{*}_{2}(2460)$, $D^{*}_{J}(2650)^{0}$, $D^{*}_{J}(2760)^{0}$ etc announced by LHCb 1 ; 2 and BABAR 3 have successfully stimulated the charm meson spectroscopy. Whereas in bottom sector, only ground state $B^{0}(5279)$, $B^{\pm}(5279)$, $B^{*}(5324)$, $B_{s}(5366)$, $B^{*}_{s}(5415)$ and few of the low lying excited bottom mesons $B_{1}(5721)$, $B^{*}_{2}(5747)$ are experimentally well known 4 ; 5 ; 6 ; 7 ; 10 ; 11 which are listed in PDG 8 . But the information for other excited bottom mesons is rather limited as compared to the charm mesons. However, the recent measurement of newly observed bottom mesons by LHCb have opened the gate to extend our understanding for these higher excited bottom states.

Recently in 2015, LHCb has reported the observation of $B_{J}(5721)^{0,+}$ and $B_{2}^{*}(5747)^{0,+}$ states, along with the observation of two new resonances $B_{J}(5840)^{0,+}$ and $B_{J}(5960)^{0,+}$ in the pp collision data, at center-of-mass energies of 7 and 8 TeV 9 . Also in 2013, the CDF collaboration analyzed a new state $B_{J}(5960)$ both in the $B^{0}\pi^{+}$ and $B^{+}\pi^{-}$ mass distribution from the p$\overline{p}$ collision data at $\sqrt{s}$ = 1.96 TeV 12 .

And in the strange sector of the bottom mesons, $B_{1s}(5830)$ and $B_{2s}^{*}(5840)$ states are well observed by CDF 38 ; 7 , D039 and LHCb 37 collaborations and are assigned the $J^{P}$ states $1P_{3/2}1^{+}$ and $1P_{3/2}2^{+}$ respectively. The masses and the widths of the recently measured experimental bottom states $B_{J}(5721)$, $B_{2}^{*}(5747)$, $B_{J}(5840)$, $B_{J}(5960)$, $B_{sJ}(5830)$ and $B_{2s}^{*}(5840)$ are listed in Table 1. Assigning a place in the mass spectra for such newly observed experimental states is very important, as the $J^{P}$’s helps in redeeming many crucial strong interaction properties of the states. To assign a $J^{P}$, many theoretical models are available such as quark model 13 ; 14 ; 15 ; 41b , Heavy Quark Effective Theory (HQET)17 , ${}^{3}P_{0}$ model 16 ; 41b and many more 18 . Many theoretical predictions have been made for assigning a particular $J^{P}$’s to these newly observed states. Different theoretical approaches uses different theoretical parameters and therefore the predictions are not completely consistent with each other, hence a particular $J^{P}$ is not confirmed for these experimentally observed bottom states.

The first two bottom states in Table 1 $B_{J}(5721)$ and $B_{2}^{*}(5747)$ are being analyzed theoretically by various models 19 ; 20 ; 21 ; 22 ; 41b and their analysis have interpreted the $B_{2}^{*}(5747)$ state to belong to $J^{P}$ $2^{+}$. For the $B_{J}(5721)$ state, some of the theoretical works 21 ; 17 favors it to be the spin partner of the $B_{2}^{*}(5747)$ state and hence $J^{P}$ as $1^{+}$ for $j_{l}=3/2$ P-wave bottom meson. And other papers 19 ; 20 ; 22 suggests $B_{J}(5721)$ to be the mixture of the $1P_{1/2}$ and $1P_{3/2}$ state. The other two bottom states $B_{1s}(5830)$ and $B_{2s}^{*}(5840)$, being the strange partners of $B_{J}(5721)$ and $B_{2}^{*}(5747)$ states, too belongs to $1P_{s3/2}1^{+}$ and $1P_{s3/2}2^{+}$ $J^{P}$’s respectively.

For the bottom state $B_{J}(5960)$, authors in Ref. 23 claimed, that the properties of the $B(5970)$ seen by CDF collaboration 12 are consistent with the properties of the $B_{J}(5960)$ measured by LHCb 9 , so they may be the same state. The theoretical analysis made by studying decay widths for $B_{J}(5960)$ by QPC model 22 and HQET 22a , favors it to belong to 2S$1^{-}$ state. This prediction is also supported by work in 24 , where the authors have used the relativistic quark model. In our previous work 22b , where masses were predicted using the QCD and $1/m_{Q}$ corrections to the flavor independent parameters $\Delta_{F}$ and $\lambda_{F}$, $B_{J}(5960)$ is again favored to be the 2S$1^{-}$ state. But, Qi-Fang Lu et. al. in Ref.20 studied masses and strong decays of $B_{J}(5960)$ states with different spin parity hypothesis and identified that the $B_{J}(5960)$ belongs to 1D$3^{-}$ state. A review of the open charm and open bottom systems by Hua-Xing Chen in Ref.25 undergoes various theoretical analysis with conclusion that $B_{J}(5960)$ belongs to 2S$1^{-}$ state.

As most of the analysis favored the $2S1^{-}$ spin parity, thus $B_{J}(5960)$ is considered to be the radial excited $2S1^{-}$ state. And lastly for the $B_{J}(5840)$ bottom state, two spin parity proposals has been put forward. First one is given by authors in 20 , where they suggested it to belong to 2S$0^{-}$ state. This interpretation matches with the LHCb collaboration analysis 9 . Second possible $J^{P}$ is given in 24 , where authors suggested the $B_{J}(5840)$ state to be the member of 1P$1^{+}$ doublet with $j_{l}=1/2$. As $B_{J}(5840)$ is studied only in few papers, so in this scenario, $B_{J}(5840)$ needs to be properly placed in the bottom meson spectra. In Ref. 20 , the $J^{P}$ for bottom state $B_{J}(5840)$ has been analysed by predicting the masses and decay widths using non-relativistic quark model and ${}^{3}P_{0}$ model respectively. Whereas the $J^{P}$ in Ref. 24 has been decided just on the basis of theoretically predicted bottom meson masses. In both the references, the models have some unknown parameters, which are fitted by using experimental data’s like decay width of bottom state $B_{2}^{*}(5747)$. So the accuracy of these predictions cannot be completely justified.

We apply the Heavy Quark Effective Theory (HQET) to discuss the assignments of the quantum numbers $J^{P}$ of the open bottom states recently reported by LHCb 9 . In the past decades, HQET have successfully explained the properties of heavy-light hadrons. The effectiveness of this theory lies on the fact that, heavy quark is treated as a dynamically degree of freedom. As a result, number of unknown parameters is greatly reduced by using the heavy quark spin and flavor symmetry. Another peculiar property of HQET is that each effective quark field is written in terms of four vector $v_{\mu}$ of heavy quark, which remains conserved in the strong interactions in the infinitely heavy quark mass limit. In our paper, we predicted $J^{P}$ using the branching ratio $\frac{B\pi}{B^{*}\pi}$ which is free from any theoretical parameter, hence the prediction made by HQET are supposed to be more accurate and logical. The HQET was originally proposed to study the interactions among heavy-light charm and bottom mesons through the emission of light pseudo-scalar mesons ($\pi,\eta,K$)25a ; 25b ; 25c ; 25d ; 25e ; 25f ; 25g ; 25h ; 25i ; 33 . The paper is arranged as follows: section 2 gives the description about the model ”Heavy Quark Effective Theory”. Section 3 represents the numerical analysis where we investigated the $B_{J}(5840)$ state by considering all the possible quark model assignments based on its mass and spin parity and analyze the branching ratio $\frac{B\pi}{B^{*}\pi}$ for all these possible $J^{P}$ states and confirm one of them to be the suitable $J^{P}$ for this state. Along with this, we also analyze the bottom states $B_{1}(5721)$, $B_{2}^{*}(5747)$, $B_{1s}(5830)$, $B_{2s}^{*}(5840)$ and $B_{J}(5960)$ for their respective $J^{P}$’s. In addition to this, we also study the strong decays for the experimentally unobserved but theoretically predicted states $B(2^{1}S_{0})$, $B_{s}(2^{3}S_{1})$, $B_{s}(2^{1}S_{0})$, $B(1^{1}D_{2})$, $B_{s}(1^{3}D_{1})$ and $B_{s}(1^{1}D_{2})$ and discuss their strong coupling constants involved. Section 4 presents the summary of our work.

II Framework

We use heavy quark effective theory for the study of heavy light bottom mesons. As in our analysis we use the decay width expressions calculated in Ref. 17 by means of HQET, it is useful to remind the theoretical framework of this theory. In the heavy quark limit $m_{Q}\rightarrow\infty$,(Q=c,b), spin of the heavy quark $s_{Q}$ decouples from the light degree of freedom which includes light anti-quark and the gluons. Thus the spin of the heavy quark $s_{Q}$ and the total angular momentum of light degree of freedom $s_{l}$ are separately conserved. The total angular momentum of light degree of freedom $s_{l}$ is given by ($s_{l}=s_{\overline{q}}+l$) where, $s_{\overline{q}}$ is the spin of the light quark and l is the orbital angular momentum of the light degree of freedom, therefore the resultant angular momentum J for each heavy-light meson is $J=s_{l}+s_{Q}$. Thus for each state, there is a degenerate doublet of meson state with $J^{P}=s_{l}^{P}\pm 1/2$ which for S-wave $(\textit{l}=0)$, gives the doublet which is represented by $(P,P^{*})$ with $J^{P}_{s_{l}}=(0^{-},1^{-})_{\frac{1}{2}}$. For the P-wave $(\textit{l}=1)$, we get two doublets which are represented by $(P^{*}_{0},P^{{}^{\prime}}_{1})$ and $(P_{1},P^{*}_{2})$ with $J^{P}_{s_{l}}=(0^{+},1^{+})_{\frac{1}{2}}$ and $(1^{+},2^{+})_{\frac{3}{2}}$ respectively. Similarly, two doublets for the D-wave ($\textit{l}=2$) are represented by $(P^{*}_{1},P_{2})$ and $(P_{2}^{{}^{\prime}},P^{*}_{3})$ belonging to $J^{P}_{s_{l}}=(1^{-},2^{-})_{\frac{3}{2}}$ and $(2^{-},3^{-})_{\frac{5}{2}}$ respectively. The above mentioned doublets for each wave are expressed by the effective super-field $H_{a},T_{a},X_{a}$ and $Y_{a}$ 28 ; 33 .

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the field $H_{a}$ describe the doublet of S-wave and field $T_{a}$ represents the P-wave doublet $(1^{+},2^{+})_{\frac{3}{2}}$. D-wave doublets are represented by $X_{a}$ and $Y_{a}$ fields for the $(1^{-},2^{-})_{\frac{3}{2}}$ and $(2^{-},3^{-})_{\frac{5}{2}}$ $J^{P^{\prime}}s$ respectively. Here indices a or b in the subsequent fields are $SU(3)$ flavor index (u, d or s). The heavy meson field $P^{(*)}$ contain a factor $\sqrt{m_{Q}}$ with mass dimension of $\frac{1}{2}$. For the radially excited states for radial quantum number n=2, these states are replaced by notation with $\sim$ on their heads e.g. $\widetilde{P},\widetilde{P}^{*}$ and so on. The strong interaction for these heavy-light mesons involves their decay with the emission of light pseudoscalar mesons($\pi,\eta,K$), which can be studied with the help of chiral perturbation theory.

The light pseudoscalar mesons are described by the fields $\xi=exp^{\frac{i\mathcal{M}}{f_{\pi}}}$, where $\mathcal{M}$ is defined as

[TABLE]

The pion octet is introduced by the vector and axial vector combinations $V^{\mu}=\frac{1}{2}(\xi\partial^{\mu}\xi^{{\dagger}}+\xi^{{\dagger}}\partial^{\mu}\xi)$ and $A^{\mu}=\frac{1}{2}(\xi\partial^{\mu}\xi^{{\dagger}}-\xi^{{\dagger}}\partial^{\mu}\xi)$. We choose $f_{\pi}=130MeV$. The Dirac structure of the chiral Lagrangian is given by the velocity vector v/c. The interaction terms between the ground state doublet ($H_{a}$) and the excited states ($T_{a},X_{a},Y_{a}$) through light pseudoscalar mesons are written as :

[TABLE]

[TABLE]

In these equations

[TABLE]

Here $g_{HH}$, $g_{TH}$, $g_{XH}$ and $g_{YH}=k^{Y}_{1}+k^{Y}_{2}$ are the strong coupling constants, $\Lambda$ is the chiral symmetry breaking scale which is taken as 1 GeV. Using the lagrangian $L_{HH},L_{TH},L_{XH}$ and $L_{YH}$ the two body strong decay widths of $Q\bar{q}$ heavy-light bottom mesons are calculated in Ref. 17 as:

[TABLE]

where $A$ is the scattering amplitude, $p_{M}$ and $m_{M}$ are the final momentum and mass of the light pseudo-scalar meson with $p_{M}$ = $\sqrt{\lambda(M_{i}^{2},m_{M}^{2},M_{f}^{2})}/2M_{i}$, where $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$ is the K llen function. $M_{i}$ and $M_{f}$ stand for initial and final heavy-meson mass.

$(0^{-},1^{-})\rightarrow(0^{-},1^{-})+M$

[TABLE]

$(1^{+},2^{+})\rightarrow(0^{-},1^{-})+M$

[TABLE]

$(1^{-},2^{-})\rightarrow(0^{-},1^{-})+M$

[TABLE]

$(2^{-},3^{-})\rightarrow(0^{-},1^{-})+M$

[TABLE]

In these equations, the coefficients $C_{\pi^{\pm}},C_{K^{\pm}},C_{K^{0}},C_{\overline{K}^{0}}=1$, $C_{\pi^{0}}=\frac{1}{2}$ and $C_{\eta}=\frac{2}{3}$ or $\frac{1}{6}$ as from Ref.17 . Different values of $C_{\eta}$ corresponds to the initial state being $b\overline{u},b\overline{d}$ or $b\overline{s}$ respectively. For the decay within n=1, the hadronic coupling constants are notated as $g_{HH}$, $g_{TH}$ etc, and for the decay from n=2 to n=1 these couplings are notated as $\widetilde{g}_{HH}$, $\widetilde{g}_{TH}$ respectively. Higher order corrections for spin and flavor violation of order $\frac{1}{m_{Q}}$ are excluded to avoid new unknown coupling constants. The coupling constants involved in these widths, can either be theoretically predicted or can be determined indirectly from the known experimental values of the decay widths. The numerical masses of various mesons used in the calculation are listed in Table 2

III Numerical Analysis

To assign a particular $J^{P}$ to the experimental available states is very important, as the $J^{P}$’s helps in redeeming many crucial strong interaction properties of the states like their decay widths, masses, branching ratios, hadronic coupling constants etc. The recently observed state $B_{J}(5840)$ has gone through various theoretical analysis 20 ; 24 for its strong decay, but an unique $J^{P}$ is not yet confirmed for it.

In this paper, we confirm a particular $J^{P}$ to the bottom state $B_{J}(5840)$ recently observed by LHCb. On the basis of the theoretically predicted masses Ref.20 ; 24 ; 34 ; 35 ; 36 , $B_{J}(5840)$ can be a member of the doublets for radially excited S-wave 2S$(0^{-},1^{-})$ , or for orbitally excited D-wave doublets 1D$(1^{-},2^{-})$ or 1D$(2^{-},3^{-})$. These six possible $J^{P}$ states are tabulated in Table 3 with their allowed strong decays to the ground state bottom mesons 1S$(0^{-},1^{-})$.

To choose the best possible $J^{P}$ among these, we study the branching ratio

[TABLE]

for all these suggested $J^{P}$’s and their masses. This ratio $R_{1}$ is effective in distinguishing these six possible assignments, as this ratio $R_{1}$ gives result independent of the coupling constants $\widetilde{g}_{HH}$, $g_{XH}$ and $g_{YH}$, thus making the predictions model independent. This ratio gives different values for all these six states, thus allowing us to notate the proper $J^{P}$ for the bottom state $B_{J}(5840)$.

We have also plotted the graphs for the $R_{1}$ with the masses for these $J^{P}$ states which are shown in Figure 1. It is worth noticing that the Fig. 1(a) shows, the $R_{1}$ remains 0 for the entire mass range which depicts that $B\pi$ decay mode is either suppressed or not allowed for $J^{P}$’s $2S0^{-}$, $1D_{3/2}2^{-}$ and $1D_{5/2}2^{-}$. The graph 1(b), 1(c) and 1(d) shows the variation of $R_{1}$ with the masses and give the values of $R_{1}$ as 0.63, 2.96 and 1.32 for the $J^{P}$ states $2S1^{-}$, $1D1^{-}$ and $1D3^{-}$ respectively, corresponding to the M(5840) =5862.90 MeV. The values 2.96 for $1D1^{-}$ and 1.32 for $1D3^{-}$ point towards the dominancy of $B\pi$ mode, whereas the value 0.63 for the $2S1^{-}$ directs the dominancy of $B^{*}\pi$ decay mode. The calculation of the total decay widths for all these six classifications of $B_{J}(5840)$ requires the value of the coupling constants $\widetilde{g}_{HH}$, ${g}_{XH}$ and ${g}_{YH}$ which are experimentally unknown. Nevertheless, on the basis of the theoretically available values of these couplings, following results can be seen.

• •

If $B_{J}(5840)$ is classified as the one of the member of the doublet 2S$(0^{-},1^{-})$, then the total decay width for these states comes out to be 150.28 MeV and 165.13 MeV respectively for $2S0^{-}$ and $2S1^{-}$. This prediction is made using the theoretical data $\widetilde{g}_{HH}$ = 0.31 41 . Both these decay widths matches very well with the experimentally observed broad decay width of 127 MeV for $B_{J}(5840)$. Since the $2S1^{-}$ state is known to be filled by the experimentally seen bottom state $B_{J}(5970)$, and the experimentally observed decay mode $B^{0}\pi^{+}$ is not possible for $2S0^{-}$. So the possibility of these both $J^{P}$’s $2S0^{-}$ and $2S1^{-}$ are excluded.

• •

If $B_{J}(5840)$ is the member of the doublet 1D$(1^{-},2^{-})$ with $s_{l}^{P}=3/2^{-}$ , then the total strong decay width comes out to be 42.76 MeV and 31.81 MeV for $J^{P}$ $1^{-}$ and $2^{-}$ respectively. For this, ${g}_{XH}$ is taken as 0.24 which is derived using the charm state $D_{1}^{*}(2760)$ information observed by LHCb in 2016 42 . The 0 $R_{1}$ value and the narrow decay width for state 1D$2^{-}$ also rules out this option for the $B_{J}(5840)$.

• •

The last possibility for $B_{J}(5840)$ can be the member of the doublets 1D$(2^{-},3^{-})$. Using the available data for coupling constant ${g}_{YH}=0.61$ 41 , the total decay widths for $J^{P}$ states $2^{-}$ and $3^{-}$ comes out to be 18.75 MeV and 24.96 MeV respectively. Even for such high value of ${g}_{YH}$, the decay widths are very narrow. So, the classification of $B_{J}(5840)$ as member of 1D$(2^{-},3^{-})$ is completely ruled out.

Thus the left out possibility of spin parity for $B_{J}(5840)$ is $1D(1^{-})_{3/2}$. It is interesting to notice that the ratio $R_{1}$ of $B_{J}(5840)$ for $J^{P}$ state $1D(1^{-})_{3/2}$ also comes out to be maximum with value of 2.96, thus favoring $1D(1^{-})_{3/2}$ as the most favorable $J^{P}$ for $B_{J}(5840)$. However, if we consider the fact that photon from the $B^{*}\rightarrow B\gamma$ is too low in energy to be detected and $B^{*}$ mesons are partially reconstructed as B mesons. Then the $J^{P}$ option for $B_{J}(5840)$ belonging to $2S0^{-}$ cannot be fully ignored because of its large decay width. The identification of $B_{J}(5840)$ as $2S0^{-}$ is also supported by the work in 9 ; 20 . So, in future one may expect that, experimental information about the decay modes for $B_{J}(5840)$ broaden up, to clearly identify the exact $J^{P}$ for this state. In this paper, due to the only $B^{0}\pi^{+}$ experimentally observed decay mode, we expect $B_{J}(5840)$ to belong to $1D(1^{-})_{3/2}$ J value.

III.1 Prediction of Spin and Strange partners for $B_{J}(5840)$

On the basis of the spin parity assignment of $B_{J}(5840)$, it is interesting to look for some features of its spin and strange partners. As discussed, $B_{J}(5840)$ is assigned as the orbitally excited D-wave state with $J^{P}$ as 1D$1^{-}$. The complete picture of the partial decay widths for $B(1^{1}D_{2})$, $B_{s}(1^{3}D_{1})$ and $B_{s}(1^{1}D_{2})$ being the spin and strange partners of $B_{J}(5840)$ state is listed in Table 4. Along with the partial decay widths, Table also shows the branching ratios $\widehat{{\bf\Gamma}}=\frac{\Gamma}{\Gamma(B_{J}(5840)\rightarrow B^{*+}\pi^{-})}$, $\widehat{{\bf\Gamma}}=\frac{\Gamma}{\Gamma(B_{sJ}^{*}\rightarrow B^{*0}K^{+})}$ and branching fractions for all the mentioned decay modes. Apart from the decay channels mentioned in this Table, $B_{J}(5840)$ being $1D(1^{-})$ also decays to $1P(1^{+})$, $1P^{{}^{\prime}}(1^{+})$ and $1P(2^{+})$ states along with pseudoscalar mesons $(\pi,\eta,K)$. Since these decays occur via D-wave, so their contribution is relatively suppressed. Here, we mentioned only the dominant decay modes with which total decay width of $B_{J}(5840)$ comes out to be 742.43$g^{2}_{XH}$.

The information in the Table 4 reveals that, for $B_{J}(5840)$ state $B^{+}\pi^{-}$ and $B^{0}\pi^{0}$ are the main decay modes as compared to the $B^{*+}\pi^{-}$ and $B^{*0}\pi^{0}$ mode. The decay width obtained in this work is finally compared with the experimental data, and hence the coupling constant $g_{XH}$ is obtained to be

[TABLE]

This information can be a beneficial in finding total and the partial decay widths of unobserved highly excited bottom meson states. Theoretically, this coupling values are also obtained as 0.45 41a , 0.5343 and 0.1921 from the charm states $D_{sJ}(2860),D_{J}(2600)$ and bottom state $B_{J}(5960)$ assuming them to be in $1D1^{-}$ state. As the $D_{J}(2600)$ and $B_{J}(5960)$ belongs to $2S1^{-}$, so the last two values of coupling $g_{XH}$ = 0.53 and 0.19 predicted from $D_{J}(2600)$ and $B_{J}(5960)$ are not useful for our study.

Theoretically, mass of the spin partner of $B_{J}(5840)$ i.e. $B(1^{1}D_{2})$ is predicted to be $5967.20\pm 30$ MeV in Ref.20 ; 24 ; 34 ; 35 ; 36 . Column 5 of the Table gives the ratio of the partial decay widths for $B(1^{1}D_{2})$ with respect to its partial decay width $B^{*-}\pi^{+}$. Apart from the decay channels listed in this Table, $B(1^{1}D_{2})$ also decays to P-wave bottom meson states $1P(0^{+})$, $1P(1^{+})$, $1P^{{}^{\prime}}(1^{+})$ and $1P(2^{+})$ which occurs via D-wave, and thus due to the small phase space, these decay modes are suppressed when compared with decays to ground state S-wave mesons and hence are not shown in Table 4. From the listed decay channels, $B^{*-}\pi^{+}$ comes out to be the dominant decay mode for $B(1^{1}D_{2})$ with branching fraction $60.34\%$. Decay width calculated in Ref. 41b , also shows $B^{*}\pi$ to be the dominant decay mode. Hence, the decay mode $B^{*-}\pi^{+}$ can be a motivation for the experimental search for the missing bottom state $B(1^{1}D_{2})$ in future. Using the value of the coupling constant $g_{XH}$ obtained from equation 25, the total decay width of the bottom state $B(1^{1}D_{2})$ is obtained as 241.33 MeV. This decay width value is in same range as given in Ref. 20 with 3.87 % deviation.

Masses for the strange partners of these bottom states are taken as $6083.06$ MeV and $6057.50$ MeV from the theoretical work 20 ; 24 ; 34 ; 35 ; 36 . Referring to the branching fractions in the Table 4, $B^{+}K^{0}$ and $B^{*-}K^{+}$ seems to be the dominant decay modes with contribution $28.79\%$ and $38.21\%$ for the bottom strange states $B_{s1}^{*}$ and $B_{s2}$ respectively, which are comparable with the dominant modes seen in Ref. 41b . These strange states also allow decays to P-wave bottom meson states, but are relatively suppressed. Hence the total decay width for these strange state comes out to be

[TABLE]

The results conclude $B_{s1}^{*}$ to be a broader state as compared to its spin partner $B_{s2}$. Moreover, if we use the coupling $g_{XH}=0.45$ obtained in Ref.41a , the decay widths for states $B(1^{1}D_{2}),B_{s}(1^{3}D_{1})$ and $B_{s}(1^{1}D_{2})$ deviate from our results by $16\%$.

III.2 Analysis for bottom states $B_{1}(5721)$, $B_{2}^{*}(5747)$, $B_{1s}(5830)$ and

$B_{2s}^{*}(5840)$

We have also analyzed the bottom states $B_{1}(5721)$, $B_{2}^{*}(5747)$, $B_{1s}(5830)$ and $B_{2s}^{*}(5840)$ for their $J^{P}$’s. On the basis of their available theoretical and experimental information, the states $B_{1}(5721)$, $B_{2}^{*}(5747)$, $B_{1s}(5830)$ and $B_{2s}^{*}(5840)$ are identified as the P- wave bottom mesons with $j_{l}$ = 3/2.

[TABLE]

We study their strong decay widths using the equations 14-16 and calculate the various branching ratios involved. The numerical value of the partial decay widths for the bottom states $B_{1}(5721)$, $B_{2}^{*}(5747)$, $B_{1s}(5830)$ and $B_{2s}^{*}(5840)$ are given in Table 5. The obtained decay widths are then compared with the experimental data to obtain the strong coupling constant $g_{TH}$. Since the strange states $B_{1s}(5830)$ and $B_{2s}^{*}(5840)$ are very narrow, we exclude them to calculate the coupling constant $g_{TH}$. $g_{TH}$ comes out to be $0.50\pm 0.01$ and $0.37\pm 0.01$ for bottom states $B_{1}(5721)$ and $B_{2}^{*}(5747)$ respectively. This is consistent with the theoretical values of $g_{TH}$ in Ref.41 , 21 , 40 obtained from charm mesons. Here, the consistency in the hadronic coupling constant $g_{TH}$ beautifully describes the heavy quark symmetry between the charm and the bottom mesons.

We also obtained the ratios $R_{2}$, $R_{3}$ and $R_{4}$ as

[TABLE]

which are consistent with their experimental value $R_{2}=0.47\pm 0.06$ and $R_{3}=0.47\pm 0.09$ observed by D0 collaboration 10 and $R_{4}=0.71\pm 0.14$ measured by LHCb 9 . Table 5 also shows the decay widths of the strange bottom states $B_{1s}(5830)$ and $B_{2s}^{*}(5840)$. The negligible values of the decay widths (of order of $10^{-10}$ MeV) for $B_{1s}(5830)$ state decaying to $B^{*+}K^{-}$ and $B^{*-}K^{0}$ are consistent with its very small decay width 0.5 MeV measured by CDF collaboration 38 in 2014. Table 5 reveals that $B^{*+}\pi^{-}$ and $B^{+}\pi^{-}$ are main decay modes for $B_{1}(5721)$ and $B_{2}^{*}(5747)$ with branching fraction $66.16\%$ and $35.64\%$ respectively. Similarly $B^{*}_{s}\pi^{0}$ and $B_{s}\pi^{0}$ are observed to be the dominating decay modes for their strange partners $B_{1s}(5830)$ and $B_{2s}^{*}(5840)$ respectively.

III.3 Prediction of Spin and Strange partners for $B_{J}(5970)$

Now, we will proceed in the similar manner to study the spin and strange partners for bottom state $B_{J}(5970)$. As we have discussed, $B_{J}(5970)$ is fitted to be the radially excited state with $J^{P}$ $1^{-}$. Table 6 shows the partial decay widths for $B_{J}(5970)$ along with its spin and strange partners $B(2^{1}S_{0})$, $B_{s}(2^{3}S_{1})$ and $B_{s}(2^{1}S_{0})$. Along with the partial decay widths, Table also shows the branching ratio $\widehat{{\bf\Gamma}}=\frac{\Gamma}{\Gamma(B_{J}^{(*)}\rightarrow B^{*+}\pi^{-})}$ and $\widehat{{\bf\Gamma}}=\frac{\Gamma}{\Gamma(B_{sJ}^{*}\rightarrow B^{*0}K^{+})}$ for the non-strange and strange states $B(2^{1}S_{0})$,$B(2^{3}S_{1})$, $B_{s}(2^{3}S_{1})$ and $B_{s}(2^{1}S_{0})$ respectively.

From the experimental decay widths of $B_{J}(5970)$, we obtain the strong coupling constant $\widetilde{g}_{HH}$ as

[TABLE]

The error in the value of coupling comes from the statistical error in experimental mass and decay width values of these bottom states. Using HQET, this coupling constant $\widetilde{g}_{HH}$ is also predicted as 0.1421 , 0.3141 , 0.2817 and 0.4043 . The first value is obtained from bottom state $B_{J}(5960)$ and other three values are obtained from the charm state sector by assuming the charm states to be in $2S0^{-}$ state.

From the listed decay channels mentioned in Table 6, $B^{*-}\pi^{+}$ comes out to be the dominant decay mode for $B_{J}(5970)$ and its spin partner $B(2^{1}S_{0})$ with branching fraction $36.30\%$ and $66.48\%$ respectively.

Apart from the decay channels listed in Table 6, we also find its partial decays to $1P(0^{+})$, $1P(2^{+})$, $1D(1^{-})$ and $1D(3^{-})$ states, but due to the small phase space, these decay modes are suppressed and are not considered in this work. And for their strange partners $B_{s0}$ and $B_{s1}^{*}$, we observe, $B^{*+}K^{-}$ and $B^{*}_{s}\pi^{0}$ as the dominant decay modes for $B_{s1}^{*}$ and $B_{s0}$ bottom states respectively. Thus these decay modes are suitable for the experimental search for these missing radially excited strange bottom mesons $B_{s1}^{*}$ and $B_{s0}$. Using the result in equation 33, their total decay widths corresponding to the mass $M(B_{0})$ = 5881.00 MeV, $M(B_{s1}^{*})$ = 6007.80 MeV and $M(B_{s0})$ = 5976.00 MeV 20 ; 24 ; 34 ; 35 ; 36 are obtained as

[TABLE]

This shows that the strange partners follow the same pattern as the non-strange bottom states i.e. $B_{s1}^{*}$ state is seen to be broader as compared to its spin partner $B_{s0}$. From the other theoretical available coupling values of $\widetilde{g}_{HH}$, the upper most theoretical value predicted from charm states is 0.4043 . The results for decay widths obtained using this higher value are very large $(\simeq 254\pm 35)$ MeV from the values obtained in our result. However, if we use the coupling value $\widetilde{g}_{HH}$ = 0.14 21 obtained from bottom sector, it give decay widths as 33.84 MeV, 32.59 MeV, 42.30 MeV for states $B(2^{1}S_{0}),B_{s}(2^{1}S_{0}),B_{s}(2^{3}S_{1})$, which deviates by $11\%$ from our results. The results for dominating decay modes for all these four states are same as observed in Ref. 41b . If we look at the leading order terms of coupling constants, it will remain same for both charm and bottom sectors. It may vary if we go for corrections upto $1/m_{Q}$ order. So using the coupling constant values obtained from experimental charm states to theoretically predict the information for bottom states may change the actual results. Moreover, one can also extend the work by studying the decays decaying to ground state through vector mesons with $J^{P}=1^{-}$ 42 .

IV Conclusion

In the present article, we have used the Heavy Quark Effective Theory of Ref.17 to investigate the recently observed bottom mesons, $B_{J}(5840)$, $B_{2}^{*}(5747)$,$B_{J}(5840)$, $B_{J}^{*}(5960)$, $B_{s1}^{*}(5830)$ and $B_{s2}^{*}(5840)$ by calculating the $B/B^{*}$ -light pseudoscalar meson decay widths. We also calculate the strong decay widths for the experimentally unobserved but theoretically predicted states $B(2^{1}S_{0})$, $B_{s}(2^{3}S_{1})$, $B_{s}(2^{1}S_{0})$, $B(1^{1}D_{2})$, $B_{s}(1^{3}D_{1})$ and $B_{s}(1^{1}D_{2})$. In particular, we have identified the six possible spin parity assignments for $B_{J}(5840)$ state, observed by the LHCb in 2015 9 . We have analyzed the total decay widths and branching ratio ($R_{1}$) $\frac{B\pi}{B^{*}\pi}$ for all these six assignments in Table 3 and concluded that the only favorable J value for $B_{J}(5840)$ state is $1D1^{-}$. This ratio has very different values for $B_{J}(5840)$ belonging to these two $J^{P}$’s, so experimental measurement of such a branching ratio in the future will be very helpful in clearly identifying one of them to be the most favorable $J^{P}$ for $B_{J}(5840)$.

We have also obtained coupling constant $g_{XH}$, $\widetilde{g}_{HH}$ and $g_{TH}$ governing the strong decays of bottom states to the light pseudo-scalar mesons. These obtained couplings allowed us to compute the strong decay widths of the above mentioned experimentally missing bottom states. Along with this, we examine the recently observed bottom state $B_{J}(5721)$ and $B_{2}^{*}(5747)$ and their strange partners $B_{sJ}(5830)$ and $B_{2s}^{*}(5840)$ for their $J^{P}$’s as $1P_{3/2}1^{+}$ and $1P_{3/2}2^{+}$ respectively. Thus, these predictions have opened a window to investigate the higher excitations of bottom mesons at the LHCb, D0, CDF.