Study of radially excited $D_s(2^1S_0)$ and $D^+_{sJ}(2632)$

Yu Tian; Ze Zhao; Ailin Zhang

arXiv:1701.05678·hep-ph·September 13, 2017

Study of radially excited $D_s(2^1S_0)$ and $D^+_{sJ}(2632)$

Yu Tian, Ze Zhao, Ailin Zhang

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TL;DR

This paper investigates the properties and decay channels of the radially excited $D_s(2^1S_0)$ and the $D^+_{sJ}(2632)$, suggesting the latter is unlikely to be a conventional charmed strange meson based on decay analysis.

Contribution

The study provides theoretical predictions for the masses, decay widths, and dominant decay channels of $D_s(2^1S_0)$ and $D^+_{sJ}(2632)$ using the $^3P_0$ model, offering insights into their nature.

Findings

01

$D_s(2^1S_0)$ expected at 2650 MeV with specific decay channels.

02

$D^+_{sJ}(2632)$ unlikely to be a conventional $c\bar s$ meson.

03

Identified dominant decay modes for these states.

Abstract

The $J^{P} = 0^{-}$ radial excitation $D_{s} (2^{1} S_{0})$ is anticipated to have mass $2650$ MeV (denoted with $D_{s} (2650)$ ) though it has not been observed. $D_{s} (2650)$ is anticipated to be observed in inclusive $e^{+} e^{-}$ and $pp$ collisions in $D^{*} K$ channel. $D_{s} (2650)$ can be produced from hadronic decays of higher excited resonances. In a $^{3} P_{0}$ model, hadronic production of $D_{s} (2650)$ has been studied, relevant decay widths have been estimated. The hadronic decays of $D_{s} (3 P)$ have been explored in detail, and the dominant decay channels have been pointed out. Two hadronic decay channels of $D_{s J}^{+} (2632)$ were observed by SELEX, but have not been observed by any other experiment. Hadronic decays of $D_{s J}^{+} (2632)$ in different assignments have been explored in the $^{3} P_{0}$ model. In view of the hadronic decay manner of $D_{s J}^{+} (2632)$ , we conclude that $D_{s J}^{+} (2632)$ cannot be a…

Tables8

Table 1. Table 1: Tentative assignment of 2 S 2 𝑆 2S and 1 D 1 𝐷 1D excited D 𝐷 D and D s subscript 𝐷 𝑠 D_{s} resonances.

	$∖ n^{2 S + 1} L_{J}$	$2^{1} S_{0}$	$2^{3} S_{1}$	$1^{1, 3} D_{2}$	$1^{3} D_{1}$	$1^{3} D_{3}$
	$D$	$D (2550)$	$D^{*} (2600)$	$D (2750)$	$D_{1}^{*} (2760)$	$D_{3}^{*} (2760)$
	$D_{s}$	$-, -$	$D_{s 1}^{*} (2700)$	$-, -$	$D_{s 1}^{*} (2860)$	$D_{s 3}^{*} (2860)$

Table 2. Table 2: Masses and β 𝛽 \beta values of mesons (MeV).

States	Mass	$β$	States	Mass	$β$
$(1^{1} S_{0}) K^{\pm}$	$493.677$	$400$	$(1 P_{1}) K_{1} {(1270)}^{0 (\pm)}$	$1272$	$400$
$(1^{1} S_{0}) K^{0}$	$497.614$	$400$	$(1 P_{1}^{'}) K_{1} {(1400)}^{0 (\pm)}$	$1403$	$400$
$(1^{3} S_{1}) K^{* \pm}$	$891.66$	$400$	$(1^{3} P_{2}) K_{2}^{*} {(1430)}^{0}$	$1432.4$	$400$
$(1^{3} S_{1}) K^{* 0}$	$895.81$	$400$	$(1^{3} P_{2}) K_{2}^{*} {(1430)}^{\pm}$	$1425.6$	$400$
$(1^{1} S_{0}) η$	$547.862$	$400$	$(1^{3} P_{0}) K_{0}^{*} {(1430)}^{0 (\pm)}$	$1425$	$400$
$(1^{3} S_{1}) ϕ$	$1019.461$	$400$	$(1 P_{1}) D_{1} {(2430)}^{0 (\pm)}$	$2427$	$475$
$(1^{1} S_{0}) D^{\pm}$	$1869.61$	$601$	$(1 P_{1}^{'}) D_{1} {(2420)}^{0}$	$2421.4$	$475$
$(1^{1} S_{0}) D^{0}$	$1864.84$	$601$	$(1 P_{1}^{'}) D_{1} {(2420)}^{\pm}$	$2423.2$	$475$
$(1^{3} S_{1}) D^{* \pm}$	$2010.26$	$516$	$(1^{3} P_{0}) D_{0} {(2400)}^{0}$	$2318$	$516$
$(1^{3} S_{1}) D^{* 0}$	$2006.96$	$516$	$(1^{3} P_{0}) D_{0} {(2400)}^{\pm}$	$2403$	$516$
$(1^{1} S_{0}) D_{s}$	$1968.30$	$651$	$(1^{3} P_{2}) D_{2} {(2460)}^{\pm}$	$2464.3$	$437$
$(1^{3} S_{1}) D_{S}^{*}$	$2112.1$	$562$	$(1^{3} P_{2}) D_{2} {(2460)}^{0}$	$2462.6$	$437$
$(2^{1} S_{0}) K {(1460)}^{0 (\pm)}$	$1460$	$400$	$(2^{3} S_{1}) D^{*} {(2600)}^{0}$	$2608.7$	$434$
$(2^{3} S_{1}) K^{*} {(1410)}^{0 (\pm)}$	$1414$	$400$	$(2^{3} S_{1}) D^{*} {(2600)}^{+}$	$2621.3$	$434$
$(2^{1} S_{0}) D {(2550)}^{0 (\pm)}$	$2539.4$	$450$	$(1^{3} D_{1}) D_{1}^{*} {(2760)}^{0}$	$2763.3$	$456$
$(1^{3} P_{0}) D_{s 0} (2317)$	$2317.7$	$542$	$(1^{3} D_{1}) D_{1}^{*} {(2760)}^{+}$	$2769.7$	$456$
$(1 P_{1}) D_{s 1} (2460)$	$2459.5$	$498$	$(1^{3} D_{3}) D_{3}^{*} {(2760)}^{0}$	$2763.3$	$407$
$(1 P_{1}^{'}) D_{s 1} (2536)$	$2535.10$	$498$	$(1^{3} D_{3}) D_{3}^{*} {(2760)}^{+}$	$2769.7$	$407$
$(1^{3} P_{2}) D_{s 2} (2573)$	$2571.9$	$464$	$(1^{1} D_{2}) D {(2750)}^{0}$	$2752.4$	$428$
$(2^{3} S_{1}) D_{s 1}^{*} (2700)$	$2708.3$	$458$	$(1^{1} D_{2}) D {(2750)}^{+}$	$2752.4$	$428$
$(1^{3} D_{1}) D_{s 1}^{*} (2860)$	$2859$	$469$	$(1^{3} D_{2}) D {(2750)}^{0}$	$2752.4$	$428$
$(1^{3} D_{3}) D_{s 3}^{*} (2860)$	$2860.5$	$426$	$(1^{3} D_{2}) D {(2750)}^{+}$	$2752.4$	$428$

Table 3. Table 3: Hadronic partial decay widths of D s ( 2650 ) subscript 𝐷 𝑠 2650 D_{s}(2650) from highly excited resonances (MeV)

Mode	Width	Mode	Width
$D (4^{3} S_{1}) \to D_{s} (2650) K$	$0.1$	$D_{s} (4^{3} S_{1}) \to D_{s} (2650) η$	$0.0$
$D (5^{3} S_{1}) \to D_{s} (2650) K$	$0.0$	$D_{s} (5^{3} S_{1}) \to D_{s} (2650) η$	$0.0$
$D (3^{3} P_{0}) \to D_{s} (2650) K$	$1.8$	$D_{s} (3^{3} P_{0}) \to D_{s} (2650) η$	$0.1$
$D (3^{3} P_{2}) \to D_{s} (2650) K$	$2.8$	$D_{s} (3^{3} P_{2}) \to D_{s} (2650) η$	$2.4$
$D (4^{3} P_{0}) \to D_{s} (2650) K$	$0.2$	$D_{s} (4^{3} P_{0}) \to D_{s} (2650) η$	$0.0$
$D (4^{3} P_{2}) \to D_{s} (2650) K$	$0.0$	$D_{s} (4^{3} P_{2}) \to D_{s} (2650) η$	$0.2$
$D (2^{3} D_{1}) \to D_{s} (2650) K$	$6.1$	$D_{s} (2^{3} D_{1}) \to D_{s} (2650) η$	$4.7$
$D (2^{3} D_{3}) \to D_{s} (2650) K$	$0.0$	$D_{s} (2^{3} D_{3}) \to D_{s} (2650) η$	$0.1$
$D (3^{3} D_{1}) \to D_{s} (2650) K$	$0.0$	$D_{s} (3^{3} D_{1}) \to D_{s} (2650) η$	$0.5$
$D (3^{3} D_{3}) \to D_{s} (2650) K$	$1.4$	$D_{s} (3^{3} D_{3}) \to D_{s} (2650) η$	$1.0$
$D (1^{3} F_{2}) \to D_{s} (2650) K$	$-, -$	$D_{s} (1^{3} F_{2}) \to D_{s} (2650) η$	$0.0$
$D (2^{3} F_{2}) \to D_{s} (2650) K$	$4.5$	$D_{s} (2^{3} F_{2}) \to D_{s} (2650) η$	$3.0$
$D (2^{3} F_{4}) \to D_{s} (2650) K$	$0.6$	$D_{s} (2^{3} F_{4}) \to D_{s} (2650) η$	$0.8$
$D (1^{3} G_{3}) \to D_{s} (2650) K$	$0.6$	$D_{s} (1^{3} G_{3}) \to D_{s} (2650) η$	$0.6$
$D (1^{3} G_{5}) \to D_{s} (2650) K$	$0.0$	$D_{s} (1^{3} G_{5}) \to D_{s} (2650) η$	$0.0$
$D (2^{3} G_{3}) \to D_{s} (2650) K$	$1.6$	$D_{s} (2^{3} G_{3}) \to D_{s} (2650) η$	$1.0$
$D (2^{3} G_{5}) \to D_{s} (2650) K$	$0.7$	$D_{s} (2^{3} G_{5}) \to D_{s} (2650) η$	$1.0$

Table 4. Table 4: Hadronic decay widths of D s ( 3 3 P 0 ) subscript 𝐷 𝑠 superscript 3 3 subscript 𝑃 0 D_{s}(3^{3}P_{0}) (MeV).

Channels	Width	Channels	Width
$D^{0} K^{+}$	$10.5$	$D_{1} {(2420)}^{+} K^{0}$	$0.1$
$D^{+} K^{0}$	$10.3$	$D_{s 1} (2536) η$	$0.3$
$D_{s} η$	$0.2$	$D_{1} {(2430)}^{0} K^{+}$	$7.8$
$D {(2550)}^{0} K^{+}$	$9.8$	$D_{1} {(2430)}^{+} K^{0}$	$7.6$
$D {(2550)}^{+} K^{0}$	$9.7$	$D_{s 1} (2460) η$	$0.2$
$D^{0} K {(1460)}^{+}$	$5.5$	$D_{1} {(2420)}^{0} K^{* +}$	$8.1$
$D^{+} K {(1460)}^{0}$	$6.9$	$D_{1} {(2420)}^{+} K^{* 0}$	$8.2$
$D^{* 0} K^{* +}$	$3.2$	$D_{1} {(2430)}^{0} K^{* +}$	$5.7$
$D^{* +} K^{* 0}$	$2.8$	$D_{1} {(2430)}^{+} K^{* 0}$	$5.7$
$D_{s}^{*} ϕ$	$4.0$	$D^{* 0} K_{1} {(1270)}^{+}$	$1.1$
$D_{0}^{} {(2400)}^{0} K^{ +}$	$0.6$	$D^{* +} K_{1} {(1270)}^{0}$	$1.1$
$D_{0}^{} {(2400)}^{+} K^{ 0}$	$2.2$	$D^{* 0} K_{1} {(1400)}^{+}$	$0.3$
$D_{s 0}^{*} (2317) ϕ$	$2.0$	$D_{2}^{} {(2460)}^{0} K^{ +}$	$3.0$
$D^{0} K_{1} {(1270)}^{+}$	$0.8$	$D_{2}^{} {(2460)}^{+} K^{ 0}$	$2.5$
$D^{+} K_{1} {(1270)}^{0}$	$0.6$	$D {(2750)}^{0} K^{+}$	$19.3$
$D^{0} K_{1} {(1400)}^{+}$	$1.5$	$D {(2750)}^{+} K^{0}$	$18.8$
$D^{+} K_{1} {(1400)}^{0}$	$1.7$	$(2^{1} S_{0}) D_{s} (2650) η$	$0.1$
$D_{1} {(2420)}^{0} K^{+}$	$0.1$	$Γ_{t o t a l}$	$162.3$

Table 5. Table 5: Hadronic decay widths of D s ( 3 1 P 1 ) subscript 𝐷 𝑠 superscript 3 1 subscript 𝑃 1 D_{s}(3^{1}P_{1}) (MeV).

Channels	Width	Channels	Width
$D^{0} K^{* +}$	$3.9$	$D_{s 1} (2460) η$	$0.0$
$D^{+} K^{* 0}$	$3.7$	$D_{1} {(2420)}^{0} K^{* +}$	$7.9$
$D_{s} ϕ$	$0.1$	$D_{1} {(2420)}^{+} K^{* 0}$	$8.1$
$D^{0} K^{*} {(1410)}^{+}$	$13.0$	$D_{1} {(2430)}^{0} K^{* +}$	$9.3$
$D^{+} K^{*} {(1410)}^{0}$	$12.4$	$D_{1} {(2430)}^{+} K^{* 0}$	$9.3$
$D^{* 0} K^{+}$	$7.7$	$D^{* 0} K_{1} {(1270)}^{+}$	$8.1$
$D^{* +} K^{0}$	$7.6$	$D^{* +} K_{1} {(1270)}^{0}$	$8.4$
$D_{s}^{*} η$	$0.2$	$D^{* 0} K_{1} {(1400)}^{+}$	$2.6$
$D^{* 0} K^{* +}$	$2.7$	$D^{* +} K_{1} {(1400)}^{0}$	$1.8$
$D^{* +} K^{* 0}$	$2.4$	$D^{0} K_{2}^{*} {(1430)}^{+}$	$3.1$
$D_{s}^{*} ϕ$	$2.0$	$D^{+} K_{2}^{*} {(1430)}^{0}$	$3.2$
$D^{* 0} K^{*} {(1410)}^{+}$	$15.9$	$D_{2}^{*} {(2460)}^{0} K^{+}$	$14.6$
$D^{* +} K^{*} {(1410)}^{0}$	$4.5$	$D_{2}^{*} {(2460)}^{+} K^{0}$	$14.4$
$D_{0}^{*} {(2400)}^{0} K^{+}$	$2.5$	$D_{s 2}^{*} (2573) η$	$3.9$
$D_{0}^{*} {(2400)}^{+} K^{0}$	$1.3$	$D_{2}^{} {(2460)}^{0} K^{ +}$	$13.0$
$D_{s 0}^{*} (2317) η$	$0.2$	$D_{2}^{} {(2460)}^{+} K^{ 0}$	$12.4$
$D^{0} K_{0}^{*} {(1430)}^{+}$	$1.4$	$D^{*} {(2600)}^{0} K^{+}$	$11.9$
$D^{+} K_{0}^{*} {(1430)}^{0}$	$1.6$	$D^{*} {(2600)}^{+} K^{0}$	$12.0$
$D_{0}^{} {(2400)}^{0} K^{ +}$	$0.0$	$D {(2750)}^{0} K^{+}$	$0.0$
$D_{0}^{} {(2400)}^{+} K^{ 0}$	$0.0$	$D {(2750)}^{+} K^{0}$	$0.0$
$D_{s 0}^{*} (2317) ϕ$	$0.0$	$D_{1}^{*} {(2760)}^{0} K^{+}$	$0.7$
$D^{0} K_{1} {(1270)}^{+}$	$0.0$	$D_{1}^{*} {(2760)}^{+} K^{0}$	$0.7$
$D^{+} K_{1} {(1270)}^{0}$	$0.0$	$D_{3}^{*} {(2760)}^{0} K^{+}$	$14.4$
$D^{0} K_{1} {(1400)}^{+}$	$0.0$	$D_{3}^{*} {(2760)}^{+} K^{0}$	$13.1$
$D^{+} K_{1} {(1400)}^{0}$	$0.0$	$D_{s 1}^{*} (2860) η$	$0.5$
$D_{1} {(2420)}^{0} K^{+}$	$0.0$	$D_{s 3}^{*} (2860) η$	$0.0$
$D_{1} {(2420)}^{+} K^{0}$	$0.0$	$D_{s 1}^{*} (2700) η$	$3.5$
$D_{s 1} (2536) η$	$0.0$	$Γ_{t o t a l}$	$260.4$
$D_{1} {(2430)}^{0} K^{+}$	$0.0$
$D_{1} {(2430)}^{+} K^{0}$	$0.4$

Table 6. Table 6: Hadronic decay widths of D s ( 3 3 P 1 ) subscript 𝐷 𝑠 superscript 3 3 subscript 𝑃 1 D_{s}(3^{3}P_{1}) (MeV).

Channels	Width	Channels	Width
$D^{0} K^{* +}$	$4.1$	$D_{s 1} (2460) η$	$0.0$
$D^{+} K^{* 0}$	$3.8$	$D_{1} {(2420)}^{0} K^{* +}$	$4.5$
$D_{s} ϕ$	$0.2$	$D_{1} {(2420)}^{+} K^{* 0}$	$4.6$
$D^{0} K^{*} {(1410)}^{+}$	$7.2$	$D_{1} {(2430)}^{0} K^{* +}$	$11.9$
$D^{+} K^{*} {(1410)}^{0}$	$6.6$	$D_{1} {(2430)}^{+} K^{* 0}$	$12.0$
$D^{* 0} K^{+}$	$8.5$	$D^{* 0} K_{1} {(1270)}^{+}$	$7.6$
$D^{* +} K^{0}$	$8.3$	$D^{* +} K_{1} {(1270)}^{0}$	$8.0$
$D_{s}^{*} η$	$0.1$	$D^{* 0} K_{1} {(1400)}^{+}$	$2.3$
$D^{* 0} K^{* +}$	$3.7$	$D^{* +} K_{1} {(1400)}^{0}$	$1.7$
$D^{* +} K^{* 0}$	$3.3$	$D^{0} K_{2}^{*} {(1430)}^{+}$	$2.5$
$D_{s}^{*} ϕ$	$2.0$	$D^{+} K_{2}^{*} {(1430)}^{0}$	$2.8$
$D^{* 0} K^{*} {(1410)}^{+}$	$0.0$	$D_{2}^{*} {(2460)}^{0} K^{+}$	$9.1$
$D^{* +} K^{*} {(1410)}^{0}$	$0.0$	$D_{2}^{*} {(2460)}^{+} K^{0}$	$9.0$
$D_{0}^{*} {(2400)}^{0} K^{+}$	$0.0$	$D_{s 2}^{*} (2573) η$	$2.6$
$D_{0}^{*} {(2400)}^{+} K^{0}$	$0.0$	$D_{2}^{} {(2460)}^{0} K^{ +}$	$3.6$
$D_{s 0}^{*} (2317) η$	$0.0$	$D_{2}^{} {(2460)}^{+} K^{ 0}$	$3.2$
$D^{0} K_{0}^{*} {(1430)}^{+}$	$0.0$	$D^{*} {(2600)}^{0} K^{+}$	$9.8$
$D^{+} K_{0}^{*} {(1430)}^{0}$	$0.0$	$D^{*} {(2600)}^{+} K^{0}$	$9.3$
$D_{0}^{} {(2400)}^{0} K^{ +}$	$0.3$	$D {(2750)}^{0} K^{+}$	$6.7$
$D_{0}^{} {(2400)}^{+} K^{ 0}$	$1.7$	$D {(2750)}^{+} K^{0}$	$6.5$
$D_{s 0}^{*} (2317) ϕ$	$1.8$	$D_{1}^{*} {(2760)}^{0} K^{+}$	$0.1$
$D^{0} K_{1} {(1270)}^{+}$	$0.0$	$D_{1}^{*} {(2760)}^{+} K^{0}$	$0.1$
$D^{+} K_{1} {(1270)}^{0}$	$0.0$	$D_{3}^{*} {(2760)}^{0} K^{+}$	$12.0$
$D^{0} K_{1} {(1400)}^{+}$	$1.2$	$D_{3}^{*} {(2760)}^{+} K^{0}$	$11.0$
$D^{+} K_{1} {(1400)}^{0}$	$1.4$	$D_{s 1}^{*} (2860) η$	$0.3$
$D_{1} {(2420)}^{0} K^{+}$	$1.7$	$D_{s 3}^{*} (2860) η$	$0.0$
$D_{1} {(2420)}^{+} K^{0}$	$1.6$	$D_{s 1}^{*} (2700) η$	$2.3$
$D_{s 1} (2536) η$	$0.2$	$Γ_{t o t a l}$	$204.8$
$D_{1} {(2430)}^{0} K^{+}$	$1.8$
$D_{1} {(2430)}^{+} K^{0}$	$1.8$

Table 7. Table 7: Hadronic decay widths of D s ( 3 3 P 2 ) subscript 𝐷 𝑠 superscript 3 3 subscript 𝑃 2 D_{s}(3^{3}P_{2}) (MeV).

Channels	Width	Channels	Width
$D^{0} K^{+}$	$1.0$	$D^{* 0} K_{0}^{*} {(1430)}^{+}$	$0.3$
$D^{+} K^{0}$	$1.1$	$D^{* +} K_{0}^{*} {(1430)}^{0}$	$0.2$
$D_{s} η$	$0.3$	$D_{1} {(2420)}^{0} K^{* +}$	$4.1$
$D {(2550)}^{0} K^{+}$	$2.6$	$D_{1} {(2420)}^{+} K^{* 0}$	$4.3$
$D {(2550)}^{+} K^{0}$	$2.7$	$D_{1} {(2430)}^{0} K^{* +}$	$3.1$
$D^{0} K {(1460)}^{+}$	$5.3$	$D_{1} {(2430)}^{+} K^{* 0}$	$3.0$
$D^{+} K {(1460)}^{0}$	$5.0$	$D^{* 0} K_{1} {(1270)}^{+}$	$3.1$
$D^{0} K^{* +}$	$2.3$	$D^{* +} K_{1} {(1270)}^{0}$	$3.2$
$D^{+} K^{* 0}$	$2.2$	$D^{* 0} K_{1} {(1400)}^{+}$	$1.8$
$D_{s} ϕ$	$0.0$	$D^{* +} K_{1} {(1400)}^{0}$	$1.6$
$D^{0} K^{*} {(1410)}^{+}$	$11.1$	$D^{0} K_{2}^{*} {(1430)}^{+}$	$1.1$
$D^{+} K^{*} {(1410)}^{0}$	$10.9$	$D^{+} K_{2}^{*} {(1430)}^{0}$	$1.0$
$D {(2550)}^{0} K^{* +}$	$0.0$	$D_{2}^{*} {(2460)}^{0} K^{+}$	$6.5$
$D {(2550)}^{+} K^{* 0}$	$0.0$	$D_{2}^{*} {(2460)}^{+} K^{0}$	$6.5$
$D^{* 0} K^{+}$	$3.8$	$D_{s 2}^{*} (2573) η$	$1.8$
$D^{* +} K^{0}$	$3.8$	$D^{* 0} K_{2}^{*} {(1430)}^{+}$	$1.6$
$D_{s}^{*} η$	$0.3$	$D_{2}^{} {(2460)}^{0} K^{ +}$	$25.0$
$D^{* 0} K^{* +}$	$4.8$	$D_{2}^{} {(2460)}^{+} K^{ 0}$	$24.9$
$D^{* +} K^{* 0}$	$4.4$	$D^{*} {(2600)}^{0} K^{+}$	$7.1$
$D_{s}^{*} ϕ$	$1.7$	$D^{*} {(2600)}^{+} K^{0}$	$7.6$
$D^{* 0} K^{*} {(1410)}^{+}$	$45.2$	$D {(2750)}^{0} K^{+}$	$2.0$
$D^{* +} K^{*} {(1410)}^{0}$	$44.5$	$D {(2750)}^{+} K^{0}$	$2.0$
$D_{0}^{} {(2400)}^{0} K^{ +}$	$0.1$	$D {(2750)}^{0} K^{+}$	$0.4$
$D_{0}^{} {(2400)}^{+} K^{ 0}$	$1.1$	$D {(2750)}^{+} K^{0}$	$0.4$
$D_{s 0}^{*} (2317) ϕ$	$1.3$	$D_{1}^{*} {(2760)}^{0} K^{+}$	$0.2$
$D^{0} K_{1} {(1270)}^{+}$	$2.9$	$D_{1}^{*} {(2760)}^{+} K^{0}$	$0.2$
$D^{+} K_{1} {(1270)}^{0}$	$2.7$	$D_{3}^{*} {(2760)}^{0} K^{+}$	$4.7$
$D^{0} K_{1} {(1400)}^{+}$	$1.6$	$D_{3}^{*} {(2760)}^{+} K^{0}$	$4.3$
$D^{+} K_{1} {(1400)}^{0}$	$1.5$	$D_{s} (2650) η$	$2.4$
$D_{1} {(2420)}^{0} K^{+}$	$1.9$	$D_{s 1}^{*} (2860) η$	$0.0$
$D_{1} {(2420)}^{+} K^{0}$	$1.8$	$D_{s 3}^{*} (2860) η$	$0.0$
$D_{s 1} (2536) η$	$0.1$	$D_{s 1}^{*} (2700) η$	$3.0$
$D_{1} {(2430)}^{0} K^{+}$	$4.9$	$Γ_{t o t a l}$	$307.9$
$D_{1} {(2430)}^{+} K^{0}$	$4.9$
$D_{s 1} (2460) η$	$2.7$

Table 8. Table 8: Partial and total widths of D s J + ( 2632 ) subscript superscript 𝐷 𝑠 𝐽 2632 D^{+}_{sJ}(2632) (MeV) and Γ ( D 0 K + ) / Γ ( D s + η ) Γ superscript 𝐷 0 superscript 𝐾 Γ subscript superscript 𝐷 𝑠 𝜂 \Gamma(D^{0}K^{+})/\Gamma(D^{+}_{s}\eta) in different assignments.

	$A s s i g n m e n t$	$D^{0} K^{+}$	$D^{+} K^{0}$	$D_{s} η$	$D^{* 0} K^{+}$	$D^{* +} K^{0}$	$Γ_{t o t a l}$	$\frac{Γ (D^{0} K^{+})}{Γ (D_{s}^{+} η)}$
	$D_{s J}^{+} (2632) (2^{3} S_{1})$	$17.8$	$17.5$	$3.5$	$22.9$	$21.6$	$83.2$	$5.02$
	$D_{s J}^{+} (2632) (1^{3} D_{1})$	$34.2$	$33.4$	$4.3$	$8.8$	$8.3$	$88.9$	$7.90$

Equations43

Γ = π^{2} \frac{∣ p ∣}{M _{A}^{2}} J L \sum ∣ M^{J L} ∣^{2}

Γ = π^{2} \frac{∣ p ∣}{M _{A}^{2}} J L \sum ∣ M^{J L} ∣^{2}

∣ p ∣ = \frac{[ m _{A}^{2} - ( m _{B} - m _{C} ) ^{2} ] [ m _{A}^{2} - ( m _{B} + m _{C} ) ^{2} ]}{2 m _{A}}

∣ p ∣ = \frac{[ m _{A}^{2} - ( m _{B} - m _{C} ) ^{2} ] [ m _{A}^{2} - ( m _{B} + m _{C} ) ^{2} ]}{2 m _{A}}

M^{J L} (A \to B C)

M^{J L} (A \to B C)

\times M_{J_{B}}, M_{J_{C}} \sum ⟨ L 0 J M_{J_{A}} ∣ J_{A} M_{J_{A}} ⟩

\times ⟨ J_{B} M_{J_{B}} J_{C} M_{J_{C}} ∣ J, J M_{J_{A}} ⟩

\times M^{M_{J_{A}} M_{J_{B}} M_{J_{C}}} (p)

δ^{3} (p_{B} + p_{C}) M^{M_{J_{A}} M_{J_{B}} M_{J_{C}}}

δ^{3} (p_{B} + p_{C}) M^{M_{J_{A}} M_{J_{B}} M_{J_{C}}}

\displaystyle=\sqrt{8E_{A}E_{B}E_{C}}\gamma\sum_{\mbox{\tiny$\begin{array}[]{c}M_{L_{A}},M_{S_{A}},\\ M_{L_{B}},M_{S_{B}},\\ M_{L_{C}},M_{S_{C}},m\end{array}$}}\langle{L_{A}M_{L_{A}}S_{A}M_{S_{A}}}|{J_{A}M_{J_{A}}}\rangle

\times ⟨ L_{B} M_{L_{B}} S_{B} M_{S_{B}} ∣ J_{B} M_{J_{B}} ⟩ ⟨ L_{C} M_{L_{C}} S_{C} M_{S_{C}} ∣ J_{C} M_{J_{C}} ⟩

\times ⟨ 1 m; 1 - m ∣ 00 ⟩ ⟨ χ_{S_{B} M_{S_{B}}}^{13} χ_{S_{C} M_{S_{C}}}^{24} ∣ χ_{S_{A} M_{S_{A}}}^{12} χ_{1 - m}^{34} ⟩

\times ⟨ φ_{B}^{13} φ_{C}^{24} ∣ φ_{A}^{12} φ_{0}^{34} ⟩ I_{M_{L_{B}}, M_{L_{C}}}^{M_{L_{A}}, m} (p)

I_{M_{L_{B}}, M_{L_{C}}}^{M_{L_{A}}, m} (p)

I_{M_{L_{B}}, M_{L_{C}}}^{M_{L_{A}}, m} (p)

\times δ^{3} (k_{1} + k_{2} - p_{A}) δ^{3} (k_{3} + k_{4})

\times δ^{3} (p_{B} - k_{1} - k_{3}) δ^{3} (p_{C} - k_{2} - k_{4})

\times Ψ_{n_{B} L_{B} M_{L_{B}}}^{*} (k_{13}) Ψ_{n_{c} L_{C} M_{L_{c}}}^{*} (k_{24})

\times Ψ_{n_{A} L_{A} M_{L A}} (k_{12}) Y_{1 m} (k_{34})

⟨ φ_{B}^{13} φ_{C}^{24} ∣ φ_{A}^{12} φ_{0}^{34} ⟩

⟨ φ_{B}^{13} φ_{C}^{24} ∣ φ_{A}^{12} φ_{0}^{34} ⟩

\times [(2 I_{B} + 1) (2 I_{C} + 1) (2 I_{A} + 1)]^{1/2}

\times ⎩ ⎨ ⎧ I_{1} I_{2} I_{A} I_{3} I_{4} 0 I_{B} I_{C} I_{A} ⎭ ⎬ ⎫

⟨ χ_{S_{B} M_{S_{B}}}^{13} χ_{S_{C} M_{S_{C}}}^{24} ∣ χ_{S_{A} M_{S_{A}}}^{12} χ_{1 - m}^{34} ⟩

⟨ χ_{S_{B} M_{S_{B}}}^{13} χ_{S_{C} M_{S_{C}}}^{24} ∣ χ_{S_{A} M_{S_{A}}}^{12} χ_{1 - m}^{34} ⟩

= (- 1)^{S_{C} + 1} [3 (2 S_{B} + 1) (2 S_{C} + 1) (2 S_{A} + 1)]^{1/2}

\times S, M_{S} \sum ⟨ S_{B} M_{S_{B}} S_{C} M_{S_{C}} ∣ S M_{S} ⟩ ⟨ S M_{S} ∣ S_{A} M_{S_{A}}; 1, - m ⟩

\times ⎩ ⎨ ⎧ 1/2 1/2 S_{A} 1/2 1/2 1 S_{B} S_{C} S ⎭ ⎬ ⎫ .

Γ (D_{s} (2^{1} S_{0}) \to D^{*} K) = 78 MeV,

Γ (D_{s} (2^{1} S_{0}) \to D^{*} K) = 78 MeV,

Γ (D_{s} (2^{1} S_{0}) \to D_{s}^{*} η) = 0.

Γ (D_{s} (2^{1} S_{0}) \to D^{*} K) = 94 MeV,

Γ (D_{s} (2^{1} S_{0}) \to D^{*} K) = 94 MeV,

Γ (D_{s} (2^{1} S_{0}) \to D_{s}^{*} η) = 0.6 MeV .

Γ (D_{s} (2^{1} S_{0}) \to D^{*} K) = 73.6 MeV,

Γ (D_{s} (2^{1} S_{0}) \to D^{*} K) = 73.6 MeV,

Γ (D_{s} (2^{1} S_{0}) \to D_{s}^{*} η) = 0.

\frac{Γ ( D K )}{Γ ( D ^{*} K )} = 0.79; \frac{Γ ( D _{s}^{+} η )}{Γ ( D ^{*} K )} = 0.08.

\frac{Γ ( D K )}{Γ ( D ^{*} K )} = 0.79; \frac{Γ ( D _{s}^{+} η )}{Γ ( D ^{*} K )} = 0.08.

\frac{Γ ( D K )}{Γ ( D ^{*} K )} = 3.95; \frac{Γ ( D _{s}^{+} η )}{Γ ( D ^{*} K )} = 0.25.

\frac{Γ ( D K )}{Γ ( D ^{*} K )} = 3.95; \frac{Γ ( D _{s}^{+} η )}{Γ ( D ^{*} K )} = 0.25.

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\CJKtilde

Study of radially excited $D_{s}(2^{1}S_{0})$ and $D^{+}_{sJ}(2632)$

Yu Tian, Ze Zhao and Ailin Zhang111Corresponding author: [email protected]

Department of Physics, Shanghai University, Shanghai 200444, China

Abstract

The $J^{P}=0^{-}$ radial excitation $D_{s}(2^{1}S_{0})$ is anticipated to have mass $2650$ MeV (denoted with $D_{s}(2650)$ ) though it has not been observed. $D_{s}(2650)$ is anticipated to be observed in inclusive $e^{+}e^{-}$ and $pp$ collisions in $D^{*}K$ channel. $D_{s}(2650)$ can be produced from hadronic decays of higher excited resonances. In a ${}^{3}P_{0}$ model, hadronic production of $D_{s}(2650)$ has been studied, relevant decay widths have been estimated. The hadronic decays of $D_{s}(3P)$ have been explored in detail, and the dominant decay channels have been pointed out. Two hadronic decay channels of $D^{+}_{sJ}(2632)$ were observed by SELEX, but have not been observed by any other experiment. Hadronic decays of $D^{+}_{sJ}(2632)$ in different assignments have been explored in the ${}^{3}P_{0}$ model. In view of the hadronic decay manner of $D^{+}_{sJ}(2632)$ , we conclude that $D^{+}_{sJ}(2632)$ cannot be a conventional charmed strange $c\bar{s}$ meson.

pacs:

13.25.Ft; 12.39.-x

Keywords: ${}^{3}P_{0}$ model, Hadronic decay, $D_{s}(2650)$ , $D^{+}_{sJ}(2632)$

I Introduction

The lowest lying $1S$ and $1P$ excited $D$ and $D_{s}$ states are believed established pdg ; gi . In recent years, more and more higher excited $D$ and $D_{s}$ resonances have been observed though some of them have not been identified. Study of their spectroscopy, production and decay will be helpful for their identification and quark dynamics.

For $D$ resonances, $D(2550)$ , $D^{*}(2600)$ , $D(2750)$ and $D^{*}(2760)$ were observed in inclusive $e^{+}e^{-}$ and $pp$ collisions by BaBar amo and LHCb lhcb1 collaborations, respectively. $D^{*}(2760)$ was subsequently found consisting of $D^{*}_{1}(2760)$ and $D^{*}_{3}(2760)$ lhcb2 ; lhcb3 .

For $D_{s}$ resonances, $D^{*}_{s1}(2700)$ and $D^{*}_{sJ}(2860)$ were observed by Belle belle and BaBar babar collaborations. $D^{*}_{sJ}(2860)$ was also found consisting of $D^{*}_{s1}(2860)$ and $D^{*}_{s3}(2860)$ lhcb4 ; lhcb5 . According to Refs. colangelo ; gm ; liu3 ; zhong ; wang ; chen2 ; chen ; zhang3 ; zhang4 ; zhu , we have a tentative assignment of these $D$ and $D_{s}$ resonances given in Table 1

When these highly excited $D$ and $D_{s}$ resonances are examined, it is interesting to note that the masses of $D^{*}_{s1}(2700)$ , $D^{*}_{s1}(2860)$ and $D^{*}_{s3}(2860)$ are about $100$ MeV higher than those of $D^{*}(2600)$ , $D^{*}_{1}(2760)$ and $D^{*}_{3}(2760)$ , respectively. Therefore, two missing $D_{s}$ with masses $2650$ MeV and $2850$ MeV are expected to correspond to the observed $D(2550)$ and $D(2750)$ . In Ref. chen2 , $D_{sJ}(2850)^{\pm}$ has been predicted with $\Gamma_{total}=125.1$ MeV and $\Gamma_{total}=51.6$ MeV corresponding to $(J^{P}=2^{-},j_{q}={3\over 2})$ and $(J^{P}=2^{-},j_{q}={5\over 2})$ , respectively.

$D(2550)$ was suggested as the $J^{P}=0^{-}$ radial excitation $D(2^{1}S_{0})$ wang ; chen2 ; colangelo ; gm , and $D(2750)$ was suggested as the $J^{P}=2^{-}$ admixture of $D(1^{1}D_{2})$ and $D(1^{3}D_{2})$ chen2 ; gm ; zhong . Therefore, the two missing $D_{s}$ are expected to have $J^{P}=0^{-}$ and $J^{P}=2^{-}$ . In this paper, the $D_{s}(2^{1}S_{0})$ with mass around $2650$ MeV will be denoted as $D_{s}(2650)$ and be studied. The $J^{P}=2^{-}$ $D_{s}$ resonance will not be addressed for an unclear mixing.

In $2600\sim 2700$ MeV energy region, a surprisingly narrow charmed strange state, $D^{+}_{sJ}(2632)$ , was first reported by SELEX SELEX in $D^{+}_{s}\eta$ and $D^{0}K^{+}$ decay channels: $D^{+}_{sJ}(2632)\rightarrow D^{0}K^{+},D^{+}_{s}\eta$ . $D_{sJ}(2632)$ was observed with mass $M=2632.5\pm 1.7$ MeV and $\Gamma<17$ MeV. In particular, it has an exotic relative branching ratio $\Gamma(D^{0}K^{+})/\Gamma(D^{+}_{s}\eta)=0.14\pm 0.06$ . However, subsequent search for $D^{+}_{sJ}(2632)$ by BaBar babar2 , FOCUS or BELLE has not found any evidence.

Since the report of $D^{+}_{sJ}(2632)$ , there appear many interpretations of this state. In Refs. chao ; ted ; rupp , $D^{+}_{sJ}(2632)$ was interpreted as the $J^{P}=1^{-}$ first radial excitation of $D^{\ast}_{s}(2112)$ . The special decay manner was attributed to the node structure of the wave functions in Ref. chao or to the couple channels effect in Ref. rupp . However, the assignment of $D^{+}_{sJ}(2632)$ with $2^{+}$ $D_{s}(^{3}P_{2})$ or $1^{-}$ $D_{s}(2^{3}S_{1})$ was pointed out impossible in Refs. zhang1 ; chang , and the assignment of $D^{+}_{sJ}(2632)$ with $D_{s}(1^{3}D_{1})$ was advocated in Refs. chang ; zhang2 . In the frame of heavy quark effective theory (HQET) sum rule huang , it was pointed out that $D^{+}_{sJ}(2632)$ was unlikely to be a conventional orbitally higher excited $c\bar{s}$ resonance.

$D^{+}_{sJ}(2632)$ was alternatively interpreted as a four-quark state in Refs. Maiani ; Y.-R ; Yu-Qi . The narrow width and exotic decay manner were interpreted in this assignment. However, the predicted partners and more decay channels of $D^{+}_{sJ}(2632)$ have not been observed. In Refs. ted ; swanson , the authors concluded that $D^{+}_{sJ}(2632)$ was an experimental artefact.

The ${}^{3}P_{0}$ model is a phenomenological method first put forth by Micu micu1969 and subsequently developed by Yaouanc et al. yaouanc1 ; yaouanc2 ; yaouanc3 . It has been employed successfully to explore Okubo-Zweig-Iizuka-allowed (OZI-allowed) hadronic decays of hadrons. Hadronic decays of the $J^{P}=0^{-}$ radial excitation $D_{s}$ with $M=2673$ MeV and $M=2650$ MeV have been explored in the ${}^{3}P_{0}$ model gm2 ; gm , but the hadronic production of this missing $D_{s}(2650)$ from higher excited resonances has not been explored. As that in Refs. zhang3 ; zhang4 , it is interesting to explore the hadronic production of $D_{s}(2650)$ from higher excited resonances.

As for $D^{+}_{sJ}(2632)$ , its hadronic decay has not yet been systematically studied in the ${}^{3}P_{0}$ model. In order to identify $D^{+}_{sJ}(2632)$ , it is required to explore the hadronic decay features of $D^{+}_{sJ}(2632)$ no matter whether $D^{+}_{sJ}(2632)$ is an experimental artefact or not.

The paper is organized as follows. In the second section, the ${}^{3}P_{0}$ model and the parameters are briefly introduced. In Sec. III, the numerical results relevant to hadronic decays of $D_{s}(2650)$ , $D_{s}(3P)$ and $D^{+}_{sJ}(2632)$ are presented. The summaries and discussions are presented in Sec. IV.

II ${}^{3}P_{0}$ MODEL

Among models for hadronic decays, the ${}^{3}P_{0}$ model is popularly known as a quark-pair creation (QPC) model. Since the propose of ${}^{3}P_{0}$ model micu1969 ; yaouanc1 ; yaouanc2 ; yaouanc3 , it has been extensively applied to mesons and baryons with success. To proceed with a practical computation, the formula in Refs. zhang4 ; zhang5 are presented. In the ${}^{3}P_{0}$ model, the hadronic decay width of $A\to BC$ is

[TABLE]

where $\vec{p}$ is the momentum of the final mesons $B$ and $C$ in the initial meson $A$ 's center-of-mass frame

[TABLE]

and $M^{JL}$ is the partial wave amplitude of $A\rightarrow BC$ . The partial wave amplitude $M^{JL}$ is obtained from the helicity amplitude $\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}$ in terms of the Jacob-Wick formula as follows

[TABLE]

where $\vec{J}=\vec{J_{B}}+\vec{J_{C}}$ , $\vec{J_{A}}=\vec{J_{B}}+\vec{J_{C}}+\vec{L}$ and $M_{J_{A}}=M_{J_{B}}+M_{J_{C}}$ . The helicity amplitude reads

[TABLE]

where $\vec{p}_{B}$ and $\vec{p}_{C}$ ( $\vec{p}_{B}$ =- $\vec{p}_{C}$ = $\vec{p}$ ) are the momenta of final mesons $B$ and $C$ in $A$ 's center-of-mass frame, respectively. $\gamma$ is a phenomenological parameter, which indicates the strength of the quark-pair creation from the vacuum. $I_{M_{L_{B}},M_{L_{C}}}^{M_{L_{A}},m}(\vec{p})$ is a momentum integral

[TABLE]

with a relative momentum of quark $i$ and quark $j$ : $\vec{k}_{ij}={{m_{j}\vec{k}_{i}-m_{i}\vec{k}_{j}}\over{m_{i}+m_{j}}}$ ( $i,j=1,2,3,4$ ). $\Psi_{nLM_{L}}(\vec{k})$ are simple harmonic oscillator (SHO) wave functions for the mesons, and $y_{1m}(\vec{k})=|\vec{k}|Y_{1m}(\Omega)$ is the solid harmonic polynomial of the created P-wave quark pair.

In terms of $9j$ symbols yaouanc3 , the flavor matrix element reads

[TABLE]

where $I_{i}~{}(i=1,2,3,4)$ is the isospin of the four $u$ , $d$ , $s$ or $c$ quark. $I_{A}$ , $I_{B}$ and $I_{C}$ are the isospins of the mesons $A$ , $B$ and $C$ , respectively. $I^{3}_{A}$ , $I^{3}_{B}$ and $I^{3}_{C}$ are the third isospin components of the mesons, and $\langle{I_{C}I_{C}^{3};I_{B}I^{3}_{B}}|{I_{A},I_{A}^{3}}\rangle$ is the isospin matrix element.

The spin matrix element reads

[TABLE]

With these formula in hand, we proceed with our calculation. In the practical calculation, the parameters have been employed as follows. The masses of the constituent quarks are taken to be, $m_{c}=1628$ MeV, $m_{u}=m_{d}=220$ MeV, and $m_{s}=419$ MeV gm . The masses and the effective scale parameters $\beta$ of relevant mesons are presented in Table 2 pdg ; gm , where a common value $\beta=0.4$ GeV is employed for all light flavor mesons. For resonances included in PDG, their masses in PDG are employed (unlike those in Ref. gm ), while for resonances not included in PDG or not observed by experiment, their masses and $\beta$ are taken from Ref. gm . $\gamma=6.95$ is employed in this paper, where $\gamma$ is $\sqrt{96\pi}$ times as the $\gamma=0.4$ adopted in Ref. gm for a different definition. The meson flavor follows the convention gm : $D^{0}=c\bar{u}$ , $D^{+}=-c\bar{d}$ , $D_{s}^{+}=-c\bar{s}$ , $K^{+}=-u\bar{s}$ , $K^{-}=s\bar{u}$ , $\phi=-s\bar{s}$ , and $\eta=(u\bar{u}-d\bar{d})/2-s\bar{s}/\sqrt{2}$ .

III Hadronic decays

III.1 $D_{s}(2650)$

Two possible hadronic decay channels and partial decay widths of the $J^{P}=0^{-}$ radial excitation $D_{s}$ with $M=2650$ MeV ( $D_{s}(2650)$ ) were calculated in the ${}^{3}P_{0}$ model gm2 :

[TABLE]

Once the $J^{P}=0^{-}$ radial excitation $D_{s}$ has a mass $M=2673$ MeV, the partial decay widths become gm2 :

[TABLE]

In Ref. gm , the results were improved as follows

[TABLE]

$D^{*}K$ decay channel is the dominant hadronic decay of $D_{s}(2650)$ , which may be used as a typical evidence to identify the $J^{P}=0^{-}$ radial excitation $D_{s}$ . $D(2550)$ was observed in inclusive $e^{+}e^{-}$ and $pp$ collisions, $D_{s}(2650)$ is naturally anticipated to be observed in these inclusive $e^{+}e^{-}$ and $pp$ collisions in $D^{*}K$ channel.

In other ways, $D_{s}(2650)$ can be produced from the hadronic decays of higher excited $D$ and $D_{s}$ mesons through $D_{s}(2650)K$ or $D_{s}(2650)\eta$ channels, respectively. Relevant hadronic partial decay widths have been estimated and presented in Table 3.

III.2 Hadronic decay of $D_{s}(3P)$ resonances

$D_{s}(2650)$ can be produced from hadronic decays of higher excited $D$ and $D_{s}$ mesons. Hadronic decays of $D(2D)$ and $D_{s}(2D)$ have been explored in the ${}^{3}P_{0}$ model in Refs. zhang3 ; zhang4 . Hadronic decays of $D_{s}(3P)$ have been studied in Ref. gm , where some decay channels were ignored. In this subsection, hadronic decays of $D_{s}(3P)$ resonances as an example are explored in detail.

In the $D_{s}(3P)$ multiplets, $D_{s}(3^{3}P_{1})$ and $D_{s}(3^{1}P_{1})$ may mix with each other. For lack of information, the detail of the mixing of $D_{s}(3^{3}P_{1})$ with $D_{s}(3^{1}P_{1})$ is not clear. Therefore, the OZI-allowed hadronic decay channels of $D_{s}(3^{3}P_{0})$ , $D_{s}(3^{1}P_{1})$ , $D_{s}(3^{3}P_{1})$ and $D_{s}(3^{3}P_{2})$ are studied separately.

The masses of $D_{s}(3P)$ are employed as follows gm : $M(D_{s}(3^{3}P_{0}))=3412$ MeV, $M(D_{s}(3^{1}P_{1}))=M(D_{s}(3^{3}P_{1}))=3425$ MeV, $M(D_{s}(3^{3}P_{2}))=3439$ MeV. The masses of $D_{s}(3^{1}P_{1})$ and $D_{s}(3^{3}P_{1})$ are assumed equal to the mean mass value of $D_{s}(3P_{1})$ and $D_{s}(3P^{\prime}_{1})$ in Ref. gm . The final decay widths are given in Tables 4-7. All the four $D_{s}(3P)$ resonances have large total decay widths, which are larger than those in Ref. gm .

From Table 4, $\Gamma_{total}(D_{s}(3^{3}P_{0}))=162.3$ MeV and the dominant decay channels of $D_{s}(3^{3}P_{0})$ are: $(1^{1}D_{2})D(2750)K$ , $DK$ , $D(2550)K$ and $D_{1}(2420)K^{*}$ .

From Table 5, $\Gamma_{total}(D_{s}(3^{1}P_{1}))=260.4$ MeV and the dominant decay channels of $D_{s}(3^{1}P_{1})$ are: $D^{*}_{2}(2460)K$ , $D_{3}^{*}(2760)K$ , $DK^{*}(1410)$ and $D^{*}_{2}(2460)K^{*}$ .

From Table 6, $\Gamma_{total}(D_{s}(3^{3}P_{1}))=204.8$ MeV and the dominant decay channels of $D_{s}(3^{3}P_{1})$ are: $D_{1}(2430)K^{*}$ , $D_{3}^{*}(2760)K$ , $D(2600)K$ and $D^{*}_{2}(2460)K$ .

From Table 7, $\Gamma_{total}(D_{s}(3^{3}P_{2}))=307.9$ MeV and the dominant decay channels of $D_{s}(3^{3}P_{2})$ are: $D^{*}K^{*}(1410)$ , $D^{*}_{2}(2460)K^{*}$ , $DK^{*}(1410)$ and $D(2600)K$ .

III.3 $D^{+}_{sJ}(2632)$

In Table I, $D^{*}_{s1}(2700)$ and $D^{*}_{s3}(2860)$ are tentatively identified with $D_{s}(2^{3}S_{1})$ and $D_{s}(1^{3}D_{1})$ , respectively. However, as introduced in the first section, $D^{+}_{sJ}(2632)$ was once regarded as a $D_{s}(2^{3}S_{1})$ , $D_{s}(1^{3}D_{1})$ or their admixture if its lower mass is ignored. In order to have a look at these possibilities, let's explore the hadronic decay of $D^{+}_{sJ}(2632)$ . The mixing of $D_{s}(2^{3}S_{1})$ and $D_{s}(1^{3}D_{1})$ will not be taken into account for its unclear mixing feature.

In $D_{s}(2^{3}S_{1})$ or $D_{s}(1^{3}D_{1})$ assignment, $D^{+}_{sJ}(2632)$ may decay into $D^{0}K^{+}$ , $D^{+}K^{0}$ , $D_{s}\eta$ , $D^{*0}K^{+}$ and $D^{*+}K^{0}$ final states. The partial and total decay widths are given in Table 8.

In the case of a $D_{s}(2^{3}S_{1})$ , we have the following ratios:

[TABLE]

In the case of a $D_{s}(1^{3}D_{1})$ , we have the following ratios:

[TABLE]

Obviously, the decay manner of $D^{+}_{sJ}(2632)$ is different in different assignments. In fact, this decay manner can be employed to distinguish $D_{s}(2^{3}S_{1})$ from $D_{s}(1^{3}D_{1})$ .

In particular, the estimated ratios ${\Gamma(D^{0}K^{+})\over\Gamma(D^{+}_{s}\eta)}$ , $5.02$ and $7.90$ , are much larger than the observed $0.14\pm 0.06$ . Obviously, $D^{+}_{sJ}(2632)$ cannot be a conventional charmed strange $c\bar{s}$ meson.

IV CONCLUSIONS AND DISCUSSIONS

The missing $J^{P}=0^{-}$ radial excitation $D_{s}(2650)$ is anticipated to have mass $2650$ MeV. $D^{*}K$ channel is the dominant hadronic decay channel of $D_{s}(2650)$ , and $D_{s}(2650)$ is anticipated to be observed in inclusive $e^{+}e^{-}$ and $pp$ collisions in $D^{*}K$ channel. $D_{s}(2650)$ can be produced from higher excited $D$ through $D_{s}(2650)K$ channel or from $D_{s}$ meson through $D_{s}(2650)\eta$ channel.

Hadronic decays of $D_{s}(3P)$ have been studied. All the hadronic decay channels are given, and the decay widths have been estimated. The four $D_{s}(3P)$ resonances have large total decay widths. The dominant decay channels of $D_{s}(3^{3}P_{0})$ are: $(1^{1}D_{2})D(2750)K$ , $DK$ , $D(2550)K$ and $D_{1}(2420)K^{*}$ . The dominant decay channels of $D_{s}(3^{1}P_{1})$ are: $D^{*}_{2}(2460)K$ , $D_{3}^{*}(2760)K$ , $DK^{*}(1410)$ and $D^{*}_{2}(2460)K^{*}$ . The dominant decay channels of $D_{s}(3^{3}P_{1})$ are: $D_{1}(2430)K^{*}$ , $D_{3}^{*}(2760)K$ , $D(2600)K$ and $D^{*}_{2}(2460)K$ . The dominant decay channels of $D_{s}(3^{3}P_{2})$ are: $D^{*}K^{*}(1410)$ , $D^{*}_{2}(2460)K^{*}$ , $DK^{*}(1410)$ and $D(2600)K$ .

$D^{+}_{sJ}(2632)$ has a mass comparable to those of predicted $D_{s}(2^{1}S_{0})$ , $D_{s}(2^{3}S_{1})$ and $D_{s}(1^{3}D_{1})$ . However, $D^{+}_{sJ}(2632)$ has an exotic ratio ${\Gamma(D^{0}K^{+})\over\Gamma(D^{+}_{s}\eta)}=0.14\pm 0.06$ , which is much smaller than the predicted ones in these possibilities. In short, $D^{+}_{sJ}(2632)$ cannot be a conventional charmed strange $c\bar{s}$ meson, which keeps rooms for $D_{s}(2650)$ , $D^{*}_{s1}(2700)$ and $D^{*}_{s1}(2860)$ .

Acknowledgements.

This work is supported by National Natural Science Foundation of China under the grants: 11475111 and 11075102. It is also supported by the Innovation Program of Shanghai Municipal Education Commission under the grant No. 13ZZ066.

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TL;DR

Contribution

Findings

Abstract

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Study of radially excited Ds(21S0)D_{s}(2^{1}S_{0})Ds​(21S0​) and DsJ+(2632)D^{+}_{sJ}(2632)DsJ+​(2632)

Abstract

pacs:

I Introduction

II 3P0{}^{3}P_{0}3P0​ MODEL

III Hadronic decays

III.1 Ds(2650)D_{s}(2650)Ds​(2650)

III.2 Hadronic decay of Ds(3P)D_{s}(3P)Ds​(3P) resonances

III.3 DsJ+(2632)D^{+}_{sJ}(2632)DsJ+​(2632)

IV CONCLUSIONS AND DISCUSSIONS

Acknowledgements.

Study of radially excited $D_{s}(2^{1}S_{0})$ and $D^{+}_{sJ}(2632)$

II ${}^{3}P_{0}$ MODEL

III.1 $D_{s}(2650)$

III.2 Hadronic decay of $D_{s}(3P)$ resonances

III.3 $D^{+}_{sJ}(2632)$