Amplitude analysis of $D^{0} \rightarrow K^{-} \pi^{+} \pi^{+} \pi^{-}$

TL;DR

This paper performs a detailed amplitude analysis of the decay D0 to K- pi+ pi+ pi- using BESIII data, identifying key intermediate processes and their contributions.

Contribution

It introduces a comprehensive amplitude model for the decay, including multiple intermediate states, and quantifies their relative fractions and phases.

Findings

The dominant process is D0 to K- a1+(1260), with a 54.6% fit fraction.

The analysis reveals the substructure and relative contributions of various intermediate states.

Nearly background-free sample of 16,000 events enables precise amplitude modeling.

Abstract

We present an amplitude analysis of the decay $D^{0} \rightarrow K^{-} \pi^{+} \pi^{+} \pi^{-}$ based on a data sample of 2.93 ${\mbox{\,fb}^{-1}}$ acquired by the BESIII detector at the $\psi(3770)$ resonance. With a nearly background free sample of about 16000 events, we investigate the substructure of the decay and determine the relative fractions and the phases among the different intermediate processes. Our amplitude model includes the two-body decays $D^{0} \rightarrow \bar{K}^{*0}\rho^{0}$, $D^{0} \rightarrow K^{-}a_{1}^{+}(1260)$ and $D^{0} \rightarrow K_{1}^{-}(1270)\pi^{+}$, the three-body decays $D^{0} \rightarrow \bar{K}^{*0}\pi^{+}\pi^{-}$ and $D^{0} \rightarrow K^{-}\pi^{+}\rho^{0}$, as well as the four-body decay $D^{0} \rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$. The dominant intermediate process is $D^{0} \rightarrow K^{-}a_{1}^{+}(1260)$, accounting for a fit fraction of $54.6\%$.

Tables

Table 1. Table 1: Spin factors S ​ ( p ) 𝑆 𝑝 S(p) for different decay modes.

Decay mode	$S\left(p\right)$
$D\left[S\right]\to {\text{ V}}_1{\text{ V}}_2$, ${\text{ V}}_1\to {\text{ P}}_1{\text{ P}}_2$, ${\text{ V}}_2\to {\text{ P}}_3{\text{ P}}_4$	${\tilde{t}}^{\left(1\right)\mu }\left({\text{ V}}_1\right){\tilde{t}}_{\mu }^{\left(1\right)}\left({\text{ V}}_2\right)$
$D\left[P\right]\to {\text{ V}}_1{\text{ V}}_2$, ${\text{ V}}_1\to {\text{ P}}_1{\text{ P}}_2$, ${\text{ V}}_2\to {\text{ P}}_3{\text{ P}}_4$	${ϵ}_{\mu \nu \lambda \sigma }{p}^{\mu }\left(D\right){\tilde{T}}^{\left(1\right)\nu }\left(D\right){\tilde{t}}^{\left(1\right)\lambda }\left({\text{ V}}_1\right){\tilde{t}}^{\left(1\right)\sigma }\left({\text{ V}}_2\right)$
$D\left[D\right]\to {\text{ V}}_1{\text{ V}}_2$, ${\text{ V}}_1\to {\text{ P}}_1{\text{ P}}_2$, ${\text{ V}}_2\to {\text{ P}}_3{\text{ P}}_4$	${\tilde{T}}^{\left(2\right)\mu \nu }\left(D\right){\tilde{t}}_{\mu }^{\left(1\right)}\left({\text{ V}}_1\right){\tilde{t}}_{\nu }^{\left(1\right)}\left({\text{ V}}_2\right)$
$D\to {\text{ AP}}_1,\text{ A}\left[S\right]\to {\text{ VP}}_2$, $\text{ V}\to {\text{ P}}_3{\text{ P}}_4$	${\tilde{T}}_1^{\mu }\left(D\right)P_{\mu \nu }^{\left(1\right)}\left(\text{ A}\right){\tilde{t}}^{\left(1\right)\nu }\left(\text{ V}\right)$
$D\to {\text{ AP}}_1,\text{ A}\left[D\right]\to {\text{ VP}}_2$, $\text{ V}\to {\text{ P}}_3{\text{ P}}_4$	${\tilde{T}}^{\left(1\right)\mu }\left(D\right){\tilde{t}}_{\mu \nu }^{\left(2\right)}\left(\text{ A}\right){\tilde{t}}^{\left(1\right)\nu }\left(\text{ V}\right)$
$D\to {\text{ AP}}_1,\text{ A}\to {\text{ SP}}_2$, $\text{ S}\to {\text{ P}}_3{\text{ P}}_4$	${\tilde{T}}^{\left(1\right)\mu }\left(D\right){\tilde{t}}_{\mu }^{\left(1\right)}\left(\text{ A}\right)$
$D\to \text{ VS}$, $\text{ V}\to {\text{ P}}_1{\text{ P}}_2$, $\text{ S}\to {\text{ P}}_3{\text{ P}}_4$	${\tilde{T}}^{\left(1\right)\mu }\left(D\right){\tilde{t}}_{\mu }^{\left(1\right)}\left(\text{ V}\right)$
$D\to {\text{ V}}_1{\text{ P}}_1,{\text{ V}}_1\to {\text{ V}}_2{\text{ P}}_2$, ${\text{ V}}_2\to {\text{ P}}_3{\text{ P}}_4$	${ϵ}_{\mu \nu \lambda \sigma }{p}_{{\text{ V}}_1}^{\mu }{q}_{{\text{ V}}_1}^{\nu }{p}_{{\text{ P}}_1}^{\lambda }{q}_{{\text{ V}}_2}^{\sigma }$
$D\to {\text{ PP}}_1,\text{ P}\to {\text{ VP}}_2$, $\text{ V}\to {\text{ P}}_3{\text{ P}}_4$	$p^{\mu }\left({\text{ P}}_2\right){\tilde{t}}_{\mu }^{\left(1\right)}\left(\text{ V}\right)$
$D\to \text{ TS}$, $\text{ T}\to {\text{ P}}_1{\text{ P}}_2$, $\text{ S}\to {\text{ P}}_3{\text{ P}}_4$	${\tilde{T}}^{\left(2\right)\mu \nu }\left(D\right){\tilde{t}}_{\mu \nu }^{\left(2\right)}\left(\text{ T}\right)$

Table 2. Table 2: K ​ π 𝐾 𝜋 K\pi S 𝑆 S -wave parameters, obtained from the fit to the D 0 → K S 0 ​ π + ​ π − → superscript 𝐷 0 superscript subscript 𝐾 𝑆 0 superscript 𝜋 superscript 𝜋 D^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-} Dalitz plot from B ​ A ​ B ​ A ​ R 𝐵 𝐴 𝐵 𝐴 𝑅 BABAR   KPiS .

$M$(GeV/$c^2$)	$1.463\pm 0.002$
$\Gamma$(GeV/$c^2$)	$0.233\pm 0.005$
$F$	$0.80\pm 0.09$
${\varphi }_F$	$2.33\pm 0.13$
$R$	$1$(fixed)
${\varphi }_R$	$-5.31\pm 0.04$
$a$	$1.07\pm 0.11$
$r$	$-1.8\pm 0.3$

Table 3. Table 3: Masses and widths of intermediate resonances K ¯ ∗ 0 superscript ¯ 𝐾 absent 0 \bar{K}^{*0} and ρ 0 superscript 𝜌 0 \rho^{0} , the first and second uncertainties are statistical and systematic, respectively.

Resonances	Mass (MeV/$c^2$)	Width (MeV/$c^2$)
${\overline{K}}^{\ast 0}$	$894.78\pm 0.75\pm 1.66$	$44.18\pm 1.57\pm 1.39$
${\rho }^0$	$779.14\pm 1.68\pm 3.98$	$148.42\pm 2.87\pm 3.36$

Table 4. Table 4: Statistical significances for different amplitudes.

Component	Amplitude	Significance ($\sigma$)
$D^0\to {\overline{K}}^{\ast 0}{\rho }^0$	$D^0\left[S\right]\to {\overline{K}}^{\ast 0}{\rho }^0$	$>10.0$
	$D^0\left[P\right]\to {\overline{K}}^{\ast 0}{\rho }^0$	$>10.0$
	$D^0\left[D\right]\to {\overline{K}}^{\ast 0}{\rho }^0$	$>10.0$
$D^0\to K^{-}{a}_1^{+}\left(1260\right)$, $a_1^{+}\left(1260\right)\to {\rho }^0{\pi }^{+}$	$D^0\to K^{-}{a}_1^{+}\left(1260\right)$, $a_1^{+}\left(1260\right)\left[S\right]\to {\rho }^0{\pi }^{+}$	$>10.0$
	$D^0\to K^{-}{a}_1^{+}\left(1260\right)$, $a_1^{+}\left(1260\right)\left[D\right]\to {\rho }^0{\pi }^{+}$	$7.4$
$D^0\to K_1^{-}\left(1270\right){\pi }^{+}$, $K_1^{-}\left(1270\right)\to {\overline{K}}^{\ast 0}{\pi }^{-}$	$D^0\to K_1^{-}\left(1270\right){\pi }^{+}$, $K_1^{-}\left(1270\right)\left[S\right]\to {\overline{K}}^{\ast 0}{\pi }^{-}$	$4.3$
	$D^0\to K_1^{-}\left(1270\right){\pi }^{+}$, $K_1^{-}\left(1270\right)\left[D\right]\to {\overline{K}}^{\ast 0}{\pi }^{-}$	$9.6$
$D^0\to K_1^{-}\left(1270\right){\pi }^{+}$, $K_1^{-}\left(1270\right)\to K^{-}{\rho }^0$	$D^0\to K_1^{-}\left(1270\right){\pi }^{+}$, $K_1^{-}\left(1270\right)\left[S\right]\to K^{-}{\rho }^0$	$>10.0$
$D^0\to K^{-}{\pi }^{+}{\rho }^0$	$D^0\to {\left({\rho }^0{K}^{-}\right)}_A{\pi }^{+}$, ${\left({\rho }^0{K}^{-}\right)}_A\left[D\right]\to K^{-}{\rho }^0$	$9.6$
	$D^0\to {\left(K^{-}{\rho }^0\right)}_P{\pi }^{+}$	$7.0$
	$D^0\to {\left(K^{-}{\pi }^{+}\right)}_{S-\mathrm{wave}}{\rho }^0$	$5.1$
	$D^0\to {\left(K^{-}{\rho }^0\right)}_V{\pi }^{+}$	$6.8$
$D^0\to {\overline{K}}^{\ast 0}{\pi }^{+}{\pi }^{-}$	$D^0\to {\left({\overline{K}}^{\ast 0}{\pi }^{-}\right)}_P{\pi }^{+}$	$8.5$
	$D^0\to {\overline{K}}^{\ast 0}{\left({\pi }^{+}{\pi }^{-}\right)}_S$	$8.9$
	$D^0\to {\left({\overline{K}}^{\ast 0}{\pi }^{-}\right)}_V{\pi }^{+}$	$9.7$
$D\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$	$D^0\to {\left({\left(K^{-}{\pi }^{+}\right)}_{S-\mathrm{wave}}{\pi }^{-}\right)}_A{\pi }^{+}$	$>10.0$
	$D^0\to K^{-}{\left({\left({\pi }^{+}{\pi }^{-}\right)}_S{\pi }^{+}\right)}_A$	$>10.0$
	$D^0\to {\left(K^{-}{\pi }^{+}\right)}_{S-\mathrm{wave}}{\left({\pi }^{+}{\pi }^{-}\right)}_S$	$>10.0$
	$D^0\left[S\right]\to {\left(K^{-}{\pi }^{+}\right)}_V{\left({\pi }^{+}{\pi }^{-}\right)}_V$	$8.8$
	$D^0\to {\left(K^{-}{\pi }^{+}\right)}_{S-\mathrm{wave}}{\left({\pi }^{+}{\pi }^{-}\right)}_V$	$5.8$
	$D^0\to {\left(K^{-}{\pi }^{+}\right)}_V{\left({\pi }^{+}{\pi }^{-}\right)}_S$	$>10.0$
	$D^0\to {\left(K^{-}{\pi }^{+}\right)}_T{\left({\pi }^{+}{\pi }^{-}\right)}_S$	$6.8$
	$D^0\to {\left(K^{-}{\pi }^{+}\right)}_{S-\mathrm{wave}}{\left({\pi }^{+}{\pi }^{-}\right)}_T$	$9.7$

Table 5. Table 5: Fit fractions for different components. The first and second uncertainties are statistical and systematic, respectively.

Component	Fit fraction (%)	Mark III’s result	E691’s result
$D^0\to {\overline{K}}^{\ast 0}{\rho }^0$	$12.3\pm 0.4\pm 0.5$	$14.2\pm 1.6\pm 5$	$13\pm 2\pm 2$
$D^0\to K^{-}{a}_1^{+}\left(1260\right)\left({\rho }^0{\pi }^{+}\right)$	$54.6\pm 2.8\pm 3.7$	$49.2\pm 2.4\pm 8$	$47\pm 5\pm 10$
$D^0\to K_1^{-}\left(1270\right)\left({\overline{K}}^{\ast 0}{\pi }^{-}\right){\pi }^{+}$	$0.8\pm 0.2\pm 0.2$	$6.6\pm 1.9\pm 3$	-
$D^0\to K_1^{-}\left(1270\right)\left(K^{-}{\rho }^0\right){\pi }^{+}$	$3.4\pm 0.3\pm 0.5$
$D^0\to K^{-}{\pi }^{+}{\rho }^0$	$8.4\pm 1.1\pm 2.5$	$8.4\pm 2.2\pm 4$	$5\pm 3\pm 2$
$D^0\to {\overline{K}}^{\ast 0}{\pi }^{+}{\pi }^{-}$	$7.0\pm 0.4\pm 0.5$	$14.0\pm 1.8\pm 4$	$11\pm 2\pm 3$
$D^0\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$	$21.9\pm 0.6\pm 0.6$	$24.2\pm 2.5\pm 6$	$23\pm 2\pm 3$

Table 6. Table 6: Phases and fit fractions for different amplitudes. The first and second uncertainties are statistical and systematic, respectively.

Amplitude	${\varphi }_i$	Fit fraction (%)
$D^0\left[S\right]\to {\overline{K}}^{\ast }{\rho }^0$	$2.35\pm 0.06\pm 0.18$	$6.5\pm 0.5\pm 0.8$
$D^0\left[P\right]\to {\overline{K}}^{\ast }{\rho }^0$	$-2.25\pm 0.08\pm 0.15$	$2.3\pm 0.2\pm 0.1$
$D^0\left[D\right]\to {\overline{K}}^{\ast }{\rho }^0$	$2.49\pm 0.06\pm 0.11$	$7.9\pm 0.4\pm 0.7$
$D^0\to K^{-}{a}_1^{+}\left(1260\right)$, $a_1^{+}\left(1260\right)\left[S\right]\to {\rho }^0{\pi }^{+}$	$0$(fixed)	$53.2\pm 2.8\pm 4.0$
$D^0\to K^{-}{a}_1^{+}\left(1260\right)$, $a_1^{+}\left(1260\right)\left[D\right]\to {\rho }^0{\pi }^{+}$	$-2.11\pm 0.15\pm 0.21$	$0.3\pm 0.1\pm 0.1$
$D^0\to K_1^{-}\left(1270\right){\pi }^{+}$, $K_1^{-}\left(1270\right)\left[S\right]\to {\overline{K}}^{\ast 0}{\pi }^{-}$	$1.48\pm 0.21\pm 0.24$	$0.1\pm 0.1\pm 0.1$
$D^0\to K_1^{-}\left(1270\right){\pi }^{+}$, $K_1^{-}\left(1270\right)\left[D\right]\to {\overline{K}}^{\ast 0}{\pi }^{-}$	$3.00\pm 0.09\pm 0.15$	$0.7\pm 0.2\pm 0.2$
$D^0\to K_1^{-}\left(1270\right){\pi }^{+}$, $K_1^{-}\left(1270\right)\to K^{-}{\rho }^0$	$-2.46\pm 0.06\pm 0.21$	$3.4\pm 0.3\pm 0.5$
$D^0\to {\left({\rho }^0{K}^{-}\right)}_A{\pi }^{+}$, ${\left({\rho }^0{K}^{-}\right)}_A\left[D\right]\to K^{-}{\rho }^0$	$-0.43\pm 0.09\pm 0.12$	$1.1\pm 0.2\pm 0.3$
$D^0\to {\left(K^{-}{\rho }^0\right)}_P{\pi }^{+}$	$-0.14\pm 0.11\pm 0.10$	$7.4\pm 1.6\pm 5.7$
$D^0\to {\left(K^{-}{\pi }^{+}\right)}_{S-\mathrm{wave}}{\rho }^0$	$-2.45\pm 0.19\pm 0.47$	$2.0\pm 0.7\pm 1.9$
$D^0\to {\left(K^{-}{\rho }^0\right)}_V{\pi }^{+}$	$-1.34\pm 0.12\pm 0.09$	$0.4\pm 0.1\pm 0.1$
$D^0\to {\left({\overline{K}}^{\ast 0}{\pi }^{-}\right)}_P{\pi }^{+}$	$-2.09\pm 0.12\pm 0.22$	$2.4\pm 0.5\pm 0.5$
$D^0\to {\overline{K}}^{\ast 0}{\left({\pi }^{+}{\pi }^{-}\right)}_S$	$-0.17\pm 0.11\pm 0.12$	$2.6\pm 0.6\pm 0.6$
$D^0\to {\left({\overline{K}}^{\ast 0}{\pi }^{-}\right)}_V{\pi }^{+}$	$-2.13\pm 0.10\pm 0.11$	$0.8\pm 0.1\pm 0.1$
$D^0\to {\left({\left(K^{-}{\pi }^{+}\right)}_{S-\mathrm{wave}}{\pi }^{-}\right)}_A{\pi }^{+}$	$-1.36\pm 0.08\pm 0.37$	$5.6\pm 0.9\pm 2.7$
$D^0\to K^{-}{\left({\left({\pi }^{+}{\pi }^{-}\right)}_S{\pi }^{+}\right)}_A$	$-2.23\pm 0.08\pm 0.22$	$13.1\pm 1.9\pm 2.2$
$D^0\to {\left(K^{-}{\pi }^{+}\right)}_{S-\mathrm{wave}}{\left({\pi }^{+}{\pi }^{-}\right)}_S$	$-1.40\pm 0.04\pm 0.22$	$16.3\pm 0.5\pm 0.6$
$D^0\left[S\right]\to {\left(K^{-}{\pi }^{+}\right)}_V{\left({\pi }^{+}{\pi }^{-}\right)}_V$	$1.59\pm 0.13\pm 0.41$	$5.4\pm 1.2\pm 1.9$
$D^0\to {\left(K^{-}{\pi }^{+}\right)}_{S-\mathrm{wave}}{\left({\pi }^{+}{\pi }^{-}\right)}_V$	$-0.16\pm 0.17\pm 0.43$	$1.9\pm 0.6\pm 1.2$
$D^0\to {\left(K^{-}{\pi }^{+}\right)}_V{\left({\pi }^{+}{\pi }^{-}\right)}_S$	$2.58\pm 0.08\pm 0.25$	$2.9\pm 0.5\pm 1.7$
$D^0\to {\left(K^{-}{\pi }^{+}\right)}_T{\left({\pi }^{+}{\pi }^{-}\right)}_S$	$-2.92\pm 0.14\pm 0.12$	$0.3\pm 0.1\pm 0.1$
$D^0\to {\left(K^{-}{\pi }^{+}\right)}_{S-\mathrm{wave}}{\left({\pi }^{+}{\pi }^{-}\right)}_T$	$2.45\pm 0.12\pm 0.37$	$0.5\pm 0.1\pm 0.1$

Table 7. Table 7: Systematic uncertainties on masses and widths of intermediate resonances K ¯ ∗ 0 superscript ¯ 𝐾 absent 0 \bar{K}^{*0} and ρ 0 superscript 𝜌 0 \rho^{0} .

Parameter	Source (${\sigma }_{stat}$)				total (${\sigma }_{stat}$)
	I	II	III	IV
$m_{{\overline{K}}^{\ast 0}}$	2.21	0.04	0.13	0.10	2.22
${\Gamma }_{{\overline{K}}^{\ast 0}}$	0.87	0.05	0.17	0.07	0.89
$m_{{\rho }^0}$	2.37	0.08	0.12	0.08	2.37
${\Gamma }_{{\rho }^0}$	1.16	0.04	0.11	0.12	1.17

Table 8. Table 8: Systematic uncertainties on fit fractions for different components.

Fit fraction	Source (${\sigma }_{stat}$)				total (${\sigma }_{stat}$)
	I	II	III	IV
$D^0\to {\overline{K}}^{\ast 0}{\rho }^0$	1.12	0.06	0.11	0.08	1.13
$D^0\to K^{-}{a}_1^{+}\left(1260\right)$	1.32	0.09	0.12	0.06	1.33
$D^0\to K_1^{-}\left(1270\right)\left({\overline{K}}^{\ast 0}{\pi }^{-}\right){\pi }^{+}$	1.41	0.02	0.12	0.10	1.42
$D^0\to K_1^{-}\left(1270\right)\left(K^{-}{\rho }^0\right){\pi }^{+}$	1.58	0.04	0.23	0.06	1.60
$D^0\to K^{-}{\pi }^{+}{\rho }^0$	2.22	0.10	0.12	0.15	2.23
$D^0\to {\overline{K}}^{\ast 0}{\pi }^{+}{\pi }^{-}$	1.32	0.08	0.13	0.10	1.34
$D^0\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$	0.94	0.10	0.09	0.12	1.00

Table 9. Table 10: Pull mean and pull width of the pull distributions for the fitted masses and widths of intermediate resonances K ¯ ∗ 0 superscript ¯ 𝐾 absent 0 \bar{K}^{*0} and ρ 0 superscript 𝜌 0 \rho^{0} from simulated data using either the generated or fitted four-momenta.

Parameter	Generated $p_i$		Fitted $p_i$
	pull mean	pull width	pull mean	pull width
$m_{{\overline{K}}^{\ast 0}}$	$0.07\pm 0.07$	$1.05\pm 0.05$	$0.06\pm 0.07$	$1.04\pm 0.05$
${\Gamma }_{{\overline{K}}^{\ast 0}}$	$-0.03\pm 0.06$	$0.97\pm 0.04$	$-0.17\pm 0.06$	$0.97\pm 0.04$
$m_{{\rho }^0}$	$0.03\pm 0.07$	$1.06\pm 0.05$	$-0.02\pm 0.07$	$1.06\pm 0.05$
${\Gamma }_{{\rho }^0}$	$0.10\pm 0.07$	$1.08\pm 0.05$	$0.06\pm 0.07$	$1.07\pm 0.05$

Table 10. Table 13: Absolute branching fractions of the seven components and the corresponding values in the PDG. Here, we denote K ¯ ∗ 0 → K − ​ π + → superscript ¯ 𝐾 absent 0 superscript 𝐾 superscript 𝜋 \bar{K}^{*0}\rightarrow K^{-}\pi^{+} and ρ 0 → π + ​ π − → superscript 𝜌 0 superscript 𝜋 superscript 𝜋 \rho^{0}\rightarrow\pi^{+}\pi^{-} . The first two uncertainties are statistical and systematic, respectively. The third uncertainties are propagated from the uncertainty of ℬ ​ ( D 0 → K − ​ π + ​ π + ​ π − ) ℬ → superscript 𝐷 0 superscript 𝐾 superscript 𝜋 superscript 𝜋 superscript 𝜋 \mathcal{B}(D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}) .

Component	Branching fraction (%)	PDG value (%)
$D^0\to {\overline{K}}^{\ast 0}{\rho }^0$	$0.99\pm 0.04\pm 0.04\pm 0.03$	$1.05\pm 0.23$
$D^0\to K^{-}{a}_1^{+}\left(1260\right)\left({\rho }^0{\pi }^{+}\right)$	$4.41\pm 0.22\pm 0.30\pm 0.13$	$3.6\pm 0.6$
$D^0\to K_1^{-}\left(1270\right)\left({\overline{K}}^{\ast 0}{\pi }^{-}\right){\pi }^{+}$	$0.07\pm 0.01\pm 0.02\pm 0.00$	$0.29\pm 0.03$
$D^0\to K_1^{-}\left(1270\right)\left(K^{-}{\rho }^0\right){\pi }^{+}$	$0.27\pm 0.02\pm 0.04\pm 0.01$
$D^0\to K^{-}{\pi }^{+}{\rho }^0$	$0.68\pm 0.09\pm 0.20\pm 0.02$	$0.51\pm 0.23$
$D^0\to {\overline{K}}^{\ast 0}{\pi }^{+}{\pi }^{-}$	$0.57\pm 0.03\pm 0.04\pm 0.02$	$0.99\pm 0.23$
$D^0\to K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}$	$1.77\pm 0.05\pm 0.04\pm 0.05$	$1.88\pm 0.26$

Equations

$$\begin{array}{r}{M}_{\mathrm{BC}}\equiv \sqrt{E_{\mathrm{beam}}^2-{\vec{p}}_D^2},\end{array}$$

$$\begin{array}{r}\Delta E\equiv E_D-E_{\mathrm{beam}},\end{array}$$

$$\begin{array}{rl}& N(K_S^0{K}^{-}{\pi }^{+}|K^{+}{\pi }^{-})=\\ & \frac{Y(K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}|K^{+}{\pi }^{-})}{ϵ(K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-}|K^{+}{\pi }^{-})}\times \frac{\mathcal{B}(K_S^0{K}^{-}{\pi }^{+})}{\mathcal{B}(K^{-}{\pi }^{+}{\pi }^{+}{\pi }^{-})},\end{array}$$

$$\begin{array}{r}{f}_S(p_j)=\frac{ϵ(p_j)|M(p_j){|}^2{R}_4(p_j)}{\int ϵ(p_j)|M(p_j){|}^2{R}_4(p_j)d{p}_j},\end{array}$$

$$\begin{array}{r}{R}_4(p_j)d{p}_j={\delta }^4\left(p_{D^0}-\sum _{j=1}^4{p}_j\right)\prod _{j=1}^4\frac{d^3{\mathbf{p}}_j}{(2\pi {)}^{3}2E_j}.\end{array}$$

$$\begin{array}{r}M(p_j)=\sum _n{c}_n{A}_n(p_j),\end{array}$$

$$\begin{array}{r}{A}_n(p_j)=P_n^1(m_1)P_n^2(m_2)S_n(p_j)F_n^1(p_j)F_n^2(p_j)F_n^D(p_j),\end{array}$$

$$\begin{array}{r}\ln L=\sum _{k=1}^{N_{\mathrm{data}}}\ln f_S(p_j^k)-\sum _{k^{\prime }=1}^{N_{\mathrm{bkg}}}{w}_{k^{\prime }}^{\mathrm{bkg}}\ln f_S(p_j^{k^{\prime }}),\end{array}$$

$$\begin{array}{r}{\gamma }_{ϵ}(p_j)=\prod _j\frac{{ϵ}_{j,\mathrm{data}}(p_j)}{{ϵ}_{j,\mathrm{MC}}(p_j)},\end{array}$$

$$\begin{array}{rl}& \int ϵ(p_j)|M(p_j){|}^2{R}_4(p_j)d{p}_j\\ & =\frac{1}{N_{\mathrm{MC}}}\sum _{k_{\mathrm{MC}}}^{N_{\mathrm{MC}}}\frac{|M(p_j^{k_{\mathrm{MC}}})){|}^2{\gamma }_{ϵ}(p_j^{k_{\mathrm{MC}}})}{|M^{\mathrm{gen}}(p_j^{k_{\mathrm{MC}}}){|}^2},\end{array}$$

$$\begin{array}{r}{P}_{\mu \nu }^{(1)}=-g_{\mu \nu }+\frac{p_{a\mu }{p}_{a\nu }}{p_a^2}\end{array}$$

$$\begin{array}{rl}& P_{{\mu }_1{\mu }_2{\nu }_1{\nu }_2}^{(2)}=\\ & \frac{1}{2}(P_{{\mu }_1{\nu }_1}^{(1)}{P}_{{\mu }_2{\nu }_2}^{(1)}+P_{{\mu }_1{\nu }_2}^{(1)}{P}_{{\mu }_2{\nu }_1}^{(1)})-\frac{1}{3}{P}_{{\mu }_1{\mu }_2}^{(1)}{P}_{{\nu }_1{\nu }_2}^{(1)}\end{array}$$

$$\begin{array}{r}{\tilde{t}}_{{\mu }_1\cdot \cdot \cdot {\mu }_L}^L=(-1{)}^L{P}_{{\mu }_1\cdot \cdot \cdot {\mu }_L{\nu }_1\cdot \cdot \cdot {\nu }_L}^{(L)}{r}^{{\nu }_1}\cdot \cdot \cdot r^{{\nu }_L},\end{array}$$

$$\begin{array}{r}q=\sqrt{\frac{(s_a+s_b-s_c{)}^2}{4s_a}-s_b}\end{array}$$

$$\begin{array}{r}{F}_L(q)=z^L{X}_L(q),\end{array}$$

$$\begin{array}{r}{X}_{L=0}(q)=1,\end{array}$$

$$\begin{array}{r}{X}_{L=1}(q)=\sqrt{\frac{2}{z^2+1}},\end{array}$$

$$\begin{array}{r}{X}_{L=2}(q)=\sqrt{\frac{13}{z^4+3z^2+9}}.\end{array}$$

$$\begin{array}{r}P(m)=\frac{1}{(m_0^2-s_a)-i{m}_0\Gamma (m)},\end{array}$$

$$\begin{array}{r}\Gamma (m)={\Gamma }_0{\left(\frac{q}{q_0}\right)}^{2L+1}\left(\frac{m_0}{m}\right){\left(\frac{X_L(q)}{X_L(q_0)}\right)}^2,\end{array}$$

$$\begin{array}{r}{P}_{\text{GS}}(m)=\frac{1+d\frac{{\Gamma }_0}{m_0}}{(m_0^2-m^2)+f(m)-i{m}_0\Gamma (m)},\end{array}$$

$$\displaystyle\begin{aligned} f(m)=\Gamma_{0}\frac{m_{0}^{2}}{q_{0}^{3}}\Big{[}q^{2}(h(m)-h(m_{0}))\\ +(m_{0}^{2}-m^{2})q_{0}^{2}\frac{dh}{d(m^{2})}\Big{|}_{m^{2}=m^{2}_{0}}\Big{]},&\end{aligned}$$

$$\begin{array}{r}h(m)=\frac{2}{\pi }\frac{q}{m}\ln \left(\frac{m+2q}{2m_{\pi }}\right),\end{array}$$

$$\displaystyle\begin{aligned} \frac{dh}{d(m^{2})}\Big{|}_{m^{2}=m^{2}_{0}}=h(m_{0})[(8q_{0}^{2})^{-1}-(2m_{0}^{2})^{-1}]+(2\pi m_{0}^{2})^{-1},\end{aligned}$$

$$\begin{array}{r}d=\frac{3}{\pi }\frac{m_{\pi }^2}{q_0^2}\ln \left(\frac{m_0+2q_0}{2m_{\pi }}\right)+\frac{m_0}{2\pi q_0}-\frac{m_{\pi }^2{m}_0}{\pi q_0^3}.\end{array}$$

$$\begin{array}{r}A(m_{K\pi })=F\sin {\delta }_F{e}^{i{\delta }_F}+R\sin {\delta }_R{e}^{i{\delta }_R}{e}^{i2{\delta }_F},\end{array}$$

$${\delta }_F$$

$${\delta }_R$$

$$\begin{array}{r}FF(n)=\frac{{\sum }_{k=1}^{N_{\mathrm{gen}}}|{\tilde{A}}_{\mathbf{n}}(p_j^k){|}^2}{{\sum }_{k=1}^{N_{\mathrm{gen}}}|M(p_j^k){|}^2},\end{array}$$

$$\begin{array}{rl}{m}_{a_1^{+}(1260)}& =1362\pm 13\text{mbox}MeV/c^2,\\ {\Gamma }_{a_1^{+}(1260)}& =542\pm 29\text{mbox}MeV/c^2,\end{array}$$

Full text

I Introduction

The decay $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ is one of the three golden decay modes of the neutral $D$ meson (the other two are $D^{0}\rightarrow K^{-}\pi^{+}$ and $D^{0}\rightarrow K^{-}\pi^{+}\pi^{0}$). Due to a large branching fraction and low background it is well suited to use as a reference channel for other decays of the $D^{0}$ meson PDG . An accurate knowledge of its resonant substructure and the relative amplitudes and phases are important to reduce systematic uncertainties in analyses that use this channel for reference. In particular, the lack of knowledge of the substructure leads to one of the largest systematic uncertainties in the measurement of the absolute branching fractions of the $D$ hadronic decays CLEODdecay . The knowledge of the decay substructure in combination with a precise measurement of strong phases can also help to improve the measurement of the CKM angle $\gamma$ (the phase of $V_{cb}$ relative to $V_{ub}$) ADS . In the measurement of $\gamma$, the parametrization model is an important input information in a model dependent method and also can be used to generate Monte Carlo (MC) simulations to check the sensitivity in a model independent method K3Pigamma . Furthermore, the branching fractions of intermediate processes can be used to understand the $D^{0}-\bar{D}^{0}$ mixing in theory AFFalk ; HYCheng .

The decay $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ was studied by Mark III MarKIII and E691 E691 more than twenty years ago. Both measurements are affected by low statistics. Using about $1300$ signal events, Mark III obtained the branching fractions for $D^{0}\rightarrow K^{-}a^{+}_{1}(1260)$, $D^{0}\rightarrow\bar{K}^{*0}\rho^{0}$, $D^{0}\rightarrow K_{1}^{-}(1270)\pi^{+}$, as well as for the three- and four-body nonresonant decays. Based on $1745$ signal events and $800$ background events, E691 obtained a similar result but without considering the $D^{0}\rightarrow K^{-}_{1}(1270)\pi^{+}$ decay mode. The results from Mark III and E691 have large uncertainties. Therefore, further experimental study of $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ decay is of great importance for improving the precision of future measurements.

In this paper, a data sample of about 2.93 ${\mbox{\,fb}^{-1}}$ datasample ; datasample2 collected at the $\psi(3770)$ resonance with the BESIII detector in 2010 and 2011 is used. We perform an amplitude analysis of the decay $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ (the inclusion of charge conjugate reactions is implied) to study the resonant substructure in this decay. The $\psi(3770)$ decays into a $D^{0}\bar{D}^{0}$ pair without any additional hadrons. We employ a double-tag method to measure the branching fraction. In order to suppress the backgrounds from other charmed meson decays and continuum (QED and $q\bar{q}$) processes, only the decay mode $\bar{D}^{0}\rightarrow K^{+}\pi^{-}$ is used to tag the $D^{0}\bar{D}^{0}$ pair. A detailed discussion of background can be found in Sec. III. The amplitude model is constructed using the covariant tensor formalism Zou .

II Detection and Data Sets

The BESIII detector is described in detail in Ref. detector . The geometrical acceptance of the BESIII detector is 93% of the full solid angle. Starting from the interaction point (IP), it consists of a main drift chamber (MDC), a time-of-flight (TOF) system, a CsI(Tl) electromagnetic calorimeter (EMC) and a muon system (MUC) with layers of resistive plate chambers (RPC) in the iron return yoke of a 1.0 T superconducting solenoid. The momentum resolution for charged tracks in the MDC is 0.5% at a transverse momentum of 1 GeV$/c$.

Monte Carlo (MC) simulations are based on GEANT4 sim . The production of $\psi(3770)$ is simulated with the KKMC KKMC package, taking into account the beam energy spread and initial-state radiation (ISR). The PHOTOS FSR package is used to simulate the final-state radiation (FSR) of charged tracks. The MC samples, which consist of $\psi(3770)$ decays to $D\bar{D}$, non-$D\bar{D}$, ISR production of low mass charmonium states and continuum processes, are referred to as “generic MC” samples. The EvtGen EvtGen package is used to simulate the known decay modes with branching fractions taken from the Particle Data Group (PDG) PDG , and the remaining unknown decays are generated with the LundCharm model LundCharm . The effective luminosities of the generic MC samples correspond to at least 5 times the data sample luminosity. They are used to investigate possible backgrounds. The decay $D^{0}\rightarrow K_{S}^{0}(\pi^{+}\pi^{-})K^{-}\pi^{+}$ has the same final state as signal and is investigated using a dedicated MC sample with the decay chain of $\psi(3770)\rightarrow D^{0}\bar{D}^{0}$ with $D^{0}\rightarrow K_{S}^{0}K^{-}\pi^{+}$ and $\bar{D}^{0}\rightarrow K^{+}\pi^{-}$, referred to as the “$K_{S}^{0}K\pi$ MC”. The decay model of $D^{0}\rightarrow K_{S}^{0}K^{-}\pi^{+}$ is generated according to CLEO’s results KsKPi . In amplitude analysis, two sets of signal MC samples using different decay models are generated. One sample is generated with an uniform distribution in phase space for the $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ decay, which is used to calculate the MC integrations and called the “PHSP MC” sample. The other sample is generated according to the results obtained in this analysis for the $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ decay. It is used to check the fit performance, calculate the goodness of fit and estimate the detector efficiency, and is called the “SIGNAL MC” sample.

III Event Selection

Good charged tracks are required to have a point of closest approach to the interaction point (IP) within $10$ cm along the beam axis and within $1$ cm in the plane perpendicular to beam. The polar angle $\theta$ between the track and the $e^{+}$ beam direction is required to satisfy $|\cos\theta|<0.93$. Charged particle identification (PID) is implemented by combining the energy loss ($dE/dx$) in the MDC and the time-of-fight information from the TOF. Probabilities $P(K)$ and $P(\pi)$ with the hypotheses of $K$ or $\pi$ are then calculated. Tracks without PID information are rejected. Charged kaon candidates are required to have $P(K)>P(\pi)$, while the $\pi$ candidates are required to have $P(\pi)>P(K)$. The average efficiencies for the kaon and pions in $K^{-}\pi^{+}\pi^{+}\pi^{-}$ are $\sim 98$% and $\sim 99$% respectively. The $D^{0}\bar{D^{0}}$ pair with $\bar{D}^{0}\rightarrow K^{+}\pi^{-}$ and $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ is reconstructed with the requirement that the two $D^{0}$ mesons have opposite charm and do not have any tracks in common. Since the tracks in $K^{-}\pi^{+}\pi^{+}\pi^{-}$ have distinct momenta from those in $K^{+}\pi^{-}$, misreconstructed signal events and $K/\pi$ particle misidentification are negligible. Furthermore, a vertex fit with the hypothesis that all tracks originate from the IP is performed, and the $\chi^{2}$ of the fit is required to be less than $200$.

For the $K^{+}\pi^{-}$ and $K^{-}\pi^{+}\pi^{+}\pi^{-}$ combinations, two variables, $M_{{\rm BC}}$ and $\Delta E$, are calculated:

[TABLE]

and

[TABLE]

where $\vec{p}_{D}$ and $E_{D}$ are the reconstructed momentum and energy of a $D$ candidate, $E_{{\rm beam}}$ is the calibrated beam energy. The signal events form a peak around zero in the $\Delta E$ distribution and around the $D^{0}$ mass in the $M_{{\rm BC}}$ distribution. We require $-0.03<\Delta E<0.03$ GeV for the $K^{+}\pi^{-}$ final state, $-0.033<\Delta E<0.033$ GeV for the $K^{-}\pi^{+}\pi^{+}\pi^{-}$ final state and $1.8575<M_{{\rm BC}}<1.8775$ GeV/$c^{2}$ for both of them. The corresponding $\Delta E$ and $M_{{\rm BC}}$ of selected candidate are shown in Fig. 1, where the background is negligible.

To ensure the $D^{0}$ meson is on shell and improve the resolution, the selected candidate events are further subjected to a five-constraint (5C) kinematic fit, which constrains the total four-momentum of all final state particles to the initial four-momentum of the $e^{+}e^{-}$ system, and the invariant mass of signal side $K^{-}\pi^{+}\pi^{+}\pi^{-}$ constrains to the $D^{0}$ mass in PDG PDG . We discard events with a $\chi^{2}$ of the 5C kinematic fit larger than $40$. In order to suppress the background of $D^{0}\rightarrow K_{S}^{0}K^{-}\pi^{+}$ with $K_{S}^{0}\rightarrow\pi^{+}\pi^{-}$, which has the same final state as our signal decay, we perform a vertex constrained fit on any $\pi^{+}\pi^{-}$ pair in the signal side if the $\pi^{+}\pi^{-}$ invariant mass falls into the mass window $|m_{\pi^{+}\pi^{-}}-m_{K_{S}^{0}}|<0.03$ GeV/$c^{2}$ ($m_{K_{S}^{0}}$ is the $K_{S}^{0}$ nominal mass PDG ), and reject the event if the corresponding significance of decay length ($e.g.$ the distance of the decay vertex to IP) is larger than 2$\sigma$. The $K_{S}^{0}$ veto eliminates about 80% $D^{0}\rightarrow K_{S}^{0}K^{-}\pi^{+}$ background while retaining about 99% of signal events. After applying all selection criteria, 15912 candidate events are obtained with a purity of 99.4$\%$, as estimated by MC simulation.

The MC studies indicate that the dominant background arises from the $D^{0}\rightarrow K_{S}^{0}K^{-}\pi^{+}$ decay, the corresponding produced number of events is estimated according to

[TABLE]

where $N(K_{S}^{0}K^{-}\pi^{+}|K^{+}\pi^{-})$ is the production of $\psi(3770)\rightarrow D^{0}\bar{D}^{0}$ with $D^{0}\rightarrow K_{S}^{0}K^{-}\pi^{+}$ and $\bar{D}^{0}\rightarrow K^{+}\pi^{-}$, $Y(K^{-}\pi^{+}\pi^{+}\pi^{-}|K^{+}\pi^{-})$ is the signal yield with background subtracted but without efficiency correction applied and $\epsilon$ is the corresponding efficiency obtained from the SIGNAL MC sample, which is generated according to the results of fit to data whose peaking background estimated from the generic MC sample. $\mathcal{B}(K^{-}\pi^{+}\pi^{+}\pi^{-})$ and $\mathcal{B}(K_{S}^{0}K^{-}\pi^{+})$ are the branching fractions for $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ and $D^{0}\rightarrow K_{S}^{0}K^{-}\pi^{+}$, respectively, which are quoted from the PDG PDG . According to Eq. (LABEL:Num_KsKPi), the number of peaking background events ($N_{\text{peaking}}$) is estimated to be $96.8\pm 14.5$.

All other backgrounds from $D\bar{D}$, $q\bar{q}$ and non-$D\bar{D}$ decays are studied with the generic MC sample. Their total contribution is estimated to be less than ten events, of which 5.5 and 2.0 are from the $D^{0}\bar{D}^{0}$ decays and the non-$D\bar{D}$ decays, respectively. These backgrounds are neglected in the following analysis and their effect is considered as a systematic uncertainty, as discussed in Sec. VI.0.2.

IV Amplitude Analysis

The decay modes which may contribute to the $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ decay are listed in Table 1, where the symbols S, P, V, A, and T denote a scalar, pseudoscalar, vector, axial-vector, and tensor state, respectively. The letters $S$, $P$, and $D$ in square brackets refer to the relative angular momentum between the daughter particles. The amplitudes and the relative phases between the different decay modes are determined with a maximum likelihood fit.

IV.1 Likelihood function construction

The likelihood function is the product of the probability density function (PDF) of the observed events. The signal PDF $f_{S}(p_{j})$ is given by

[TABLE]

where $\epsilon(p_{j})$ is the detection efficiency parametrized in terms of the final four-momenta $p_{j}$. The index $j$ refers to the different particles in the final state. $R_{4}(p_{j})dp_{j}$ is the standard element of the four-body phase space Zou , which is given by

[TABLE]

$M(p_{j})$ is the total decay amplitude which is modeled as a coherent sum over all contributing amplitudes

[TABLE]

where the complex coefficient $c_{n}=\rho_{n}e^{i\phi_{n}}$ ($\rho_{n}$ and $\phi_{n}$ are the magnitude and phase for the $n^{\text{th}}$ amplitude, respectively) and $A_{n}(p_{j})$ describe the relative contribution and the dynamics of the $n^{\text{th}}$ amplitude. In four-body decays, the intermediate amplitude can be a quasi-two-body decay or a cascade decay amplitude, and $A_{n}(p_{j})$ is given by

[TABLE]

where the indices 1 and 2 correspond to the two intermediate resonances. Here, $P_{n}^{\alpha}(m_{\alpha})$ and $F_{n}^{\alpha}(p_{j})$ ($\alpha=1,2$) are the propagator and the Blatt-Weisskopf barrier factor Blatt , respectively, and $F_{n}^{D}(p_{j})$ is the Blatt-Weisskopf barrier factor of the $D^{0}$ decay. The parameters $m_{1}$ and $m_{2}$ in the propagators are the invariant masses of the corresponding systems. For nonresonant states with orbital angular momentum between the daughters, we set the propagator to unity, which can be regarded as a very broad resonance. The spin factor $S_{n}(p_{j})$ is constructed with the covariant tensor formalism Zou . In practice, the presence of the two $\pi^{+}$ mesons imposes a Bose symmetry in the $K^{-}\pi^{+}\pi^{+}\pi^{-}$ final state. This symmetry is explicitly accounted for in the amplitude by exchange of the two pions with the same charge.

The contribution from the background is subtracted in the likelihood calculation by assigning a negative weight to the background events

[TABLE]

where $N_{{\rm data}}$ is the number of candidate events in data, $w_{k^{\prime}}^{{\rm bkg}}$ and $N_{{\rm bkg}}$ are the weight and the number of events from the background MC sample, respectively. In the nominal fit, only the peaking background $D^{0}\rightarrow K_{S}^{0}K^{-}\pi^{+}$ is considered, and the weight $w_{k^{\prime}}^{{\rm bkg}}$ is fixed to ${N_{\text{peaking}}}/{N_{\text{bkg}}}$. $p_{j}^{k}$ and $p_{j}^{k^{\prime}}$ are the four-momenta of the $j^{{\rm th}}$ final particle in the $k^{{\rm th}}$ event of the data sample and in the $k^{\prime{\rm th}}$ event of the background MC sample, respectively.

The normalization integral is determined by a MC technique taking into account the difference of detector efficiencies for PID and tracking between data and MC simulation. The weight for a given MC event is defined as

[TABLE]

where $\epsilon_{j,{\rm data}}(p_{j})$ and $\epsilon_{j,{\rm MC}}(p_{j})$ are the PID or tracking efficiencies for charged tracks as a function of $p_{j}$ for the data and MC sample, respectively. The efficiencies $\epsilon_{j,{\rm data}}(p_{j})$ and $\epsilon_{j,{\rm MC}}(p_{j})$ are determined by studying the $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ sample for data and the MC sample respectively. The MC integration is then given by

[TABLE]

where $k_{{\rm MC}}$ is the index of the $k_{{\rm MC}}^{{\rm th}}$ event of the MC sample and $N_{{\rm MC}}$ is the number of the selected MC events. $M^{{\rm gen}}(p_{j})$ is the PDF function used to generate the MC samples in MC integration. In the numerator of Eq. (4), $\epsilon(p_{j})$ is independent of the fitted variables, so it is regarded as a constant term in the fit.

IV.1.1 Spin factors

Due to the limited phase space available in the decay, we only consider the states with angular momenta up to 2. As discussed in Ref. Zou , we define the spin projection operator $P^{(S)}_{\mu_{1}\cdot\cdot\cdot\mu_{S}\nu_{1}\cdot\cdot\cdot\nu_{S}}$ for a process $a\rightarrow bc$ as

[TABLE]

for spin 1,

[TABLE]

for spin 2. The covariant tensors $\tilde{t}^{L}_{\mu_{1}\cdot\cdot\cdot\mu_{l}}$ for the final states of pure orbital angular momentum $L$ are constructed from relevant momenta $p_{a}$, $p_{b}$, $p_{c}$ Zou

[TABLE]

where $r=p_{b}-p_{c}$.

Ten kinds of decay modes used in the analysis are listed in Table 1. We use $\tilde{T}^{(L)}_{\mu_{1}...\mu_{L}}$ to represent the decay from the $D$ meson and $\tilde{t}^{(L)}_{\mu_{1}...\mu_{L}}$ to represent the decay from the intermediate state.

IV.1.2 Blatt-Weisskopf barrier factors

The Blatt-Weisskopf barrier factor Blatt $F_{L}(p_{j})$ is a function of the angular momentum $L$ and the four-momenta $p_{j}$ of the daughter particles. For a process $a\rightarrow bc$, the magnitude of the momentum $q$ of the daughter $b$ or $c$ in the rest system of $a$ is given by

[TABLE]

with $s_{\beta}=E_{\beta}^{2}-\vec{p}_{\beta}^{2},\beta=a,b,c$. The Blatt-Weisskopf barrier factor is then given by

[TABLE]

where $z=qR$. $R$ is the effective radius of the barrier, which is fixed to $3.0{\mbox{\,GeV}^{-1}}$ for intermediate resonances and $5.0{\mbox{\,GeV}^{-1}}$ for the $D^{0}$ meson. $X_{L}(q)$ is given by

[TABLE]

[TABLE]

[TABLE]

IV.1.3 Propagator

The resonances $\bar{K}^{*0}$ and $a_{1}^{+}(1260)$ are parametrized as relativistic Breit-Wigner function with a mass depended width

[TABLE]

where $m_{0}$ is the mass of resonance to be determined. $\Gamma(m)$ is given by

[TABLE]

where $q_{0}$ denotes the value of $q$ at $m=m_{0}$. The $K_{1}^{-}(1270)$ is parametrized as a relativistic Breit-Wigner function with a constant width $\Gamma(m)=\Gamma_{0}$, and the $\rho^{0}$ is parametrized with the Gounaris-Sakurai line shape GS , which is given by

[TABLE]

where

[TABLE]

and the function $h(m)$ is defined as

[TABLE]

with

[TABLE]

where $m_{\pi}$ is the charged pion mass. The normalization condition at $P_{\text{GS}}(0)$ fixes the parameter $d=f(0)/(\Gamma_{0}m_{0})$. It is found to be GS

[TABLE]

IV.1.4 Parametrization of the $K\pi$ $S$-wave

For the $K\pi$ $S$-wave [denoted as $(K\pi)_{{\rm S-wave}}$], we use the same parametrization as $BABAR$ KPiS , which is extracted from scattering data LASS . The model is built from a Breit-Wigner shape for the $\bar{K}^{*}_{0}(1430)^{0}$ combined with an effective range parametrization for the nonresonant component given by

[TABLE]

with

[TABLE]

where $a$ and $r$ denote the scattering length and effective interaction length. $F~{}(\phi_{F})$ and $R~{}(\phi_{R})$ are the relative magnitudes (phases) for the nonresonant and resonant terms, respectively. $q$ and $\Gamma(m_{K\pi})$ are defined as in Eq. (14) and Eq. (20), respectively. In the fit, the parameters $M$, $\Gamma$, $F$, $\phi_{F}$, $R$, $\phi_{R}$, $a$ and $r$ are fixed to the values obtained from the fit to the $D^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}$ Dalitz plot KPiS , as summarized in Table  2. These fixed parameters will be varied within their uncertainties to estimate the corresponding systematic uncertainties, which is discussed in detail in Sec. VI.0.1.

IV.2 Fit fraction and the statistical uncertainty

We divide the fit model into several components according to the intermediate resonances, which can be found in Sec. V. The fit fractions of the individual components (amplitudes) are calculated according to the fit results and are compared to other measurements. In the calculation, a large phase space (PHSP) MC sample with neither detector acceptance nor resolution involved is used. The fit fraction for an amplitude or a component (a certain subset of amplitudes) is defined as

[TABLE]

where $\tilde{A}_{\bf n}(p_{j}^{k})$ is either the $n^{{\rm th}}$ amplitude [$\tilde{A}_{{\bf n}}(p_{j}^{k})=c_{n}A_{n}(p_{j}^{k})$] or the ${\bf n}^{{\rm th}}$ component of a coherent sum of amplitudes [$\tilde{A}_{{\bf n}}(p_{j}^{k})=\sum{c_{n_{i}}A_{n_{i}}(p_{j}^{k})}$], $N_{{\rm gen}}$ is the number of the PHSP MC events.

To estimate the statistical uncertainties of the fit fractions, we repeat the calculation of fit fractions by randomly varying the fitted parameters according to the error matrix. Then, for every amplitude or component, we fit the resulting distribution with a Gaussian function, whose width gives the corresponding statistical uncertainty.

IV.3 Goodness of fit

To examine the performance of the fit process, the goodness of fit is defined as follows. Since the $D^{0}$ and all four final states particles have spin zero, the phase space of the decay $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ can be completely described by five linearly independent Lorentz invariant variables. Denoting as $\pi^{+}_{1}$ the one of the two identical pions which results in a higher $\pi^{+}\pi^{-}$ invariant mass and the other pion as $\pi^{+}_{2}$, we choose the five invariant masses $m_{\pi^{+}_{1}\pi^{-}}$, $m_{\pi^{+}_{2}\pi^{-}}$, $m_{K^{-}\pi^{+}_{1}\pi^{-}}$, $m_{\pi^{+}_{1}\pi^{+}_{2}\pi^{-}}$ and $m_{K^{-}\pi^{+}_{1}\pi^{+}_{2}}$. To calculate the goodness of fit, the five-dimensional phase space is first divided into cells with equal size. Then, adjacent cells are combined until the number of events in each cell is larger than 20. The deviation of the fit in each cell is calculated, $\chi_{p}=\frac{N_{p}-N_{p}^{{\rm exp}}}{\sqrt{N_{p}^{{\rm exp}}}}$, and the goodness of fit is quantified as $\chi^{2}=\sum_{p=1}^{n}\chi_{p}^{2}$, where $N_{p}$ and $N_{p}^{{\rm exp}}$ are the number of the observed events and the expected number determined from the fit results in the $p^{th}$ cell, respectively, and $n$ is the total number of cells. The number of degrees of freedom (NDF) $\nu$ is given by $\nu=(n-1)-n_{{\rm par}}$, where $n_{{\rm par}}$ is the number of the free parameters in the fit.

V Results

In order to determine the optimal set of amplitude that contribute to the decay $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$, considering the results in PDG PDG , we start with the fit including the components with significant contribution and add more amplitude in the fit one by one. The corresponding statistical significance for the new amplitude is calculated with the change of the log-likelihood value $\Delta{\ln{L}}$, taking the change of the degrees of freedom $\Delta\nu$ into account.

In the $K^{-}\pi^{+}$ and $\pi^{+}\pi^{-}$ invariant mass spectra, there are clear structures for $\bar{K}^{*0}$ and $\rho^{0}$. The intermediate resonance $K_{1}^{-}(1270)$ is observed with $K_{1}^{-}(1270)\rightarrow\bar{K}^{*0}\pi^{-}$ or $K^{-}\rho^{0}$. In the $\pi^{+}\pi^{+}\pi^{-}$ invariant mass spectrum, a broad bump appears. We find this bump can be fitted as $a_{1}^{+}(1260)$, which was also observed by the Mark III MarKIII experiment. If it is fitted with a nonresonant $(\rho^{0}\pi^{+})_{{\rm A}}$ amplitude instead, we find that the significance for $a_{1}^{+}(1260)$ with respect to $(\rho^{0}\pi^{+})_{{\rm A}}$ is larger than $10\sigma$. The three-body nonresonant states come from two kinds of contributions, $K^{-}\pi^{+}\rho^{0}$ and $\bar{K}^{*0}\pi^{+}\pi^{-}$. The $\bar{K}^{*0}\pi^{-}/K^{-}\rho^{0}$ can be in a pseudoscalar, a vector or an axial-vector state, while the $K^{-}\pi^{+}/\pi^{+}\pi^{-}$ can be in a scalar state. The four-body nonresonant states are relatively complex, such as $D\rightarrow$ VV, $D\rightarrow$ VS, $D\rightarrow$ TS, $D\rightarrow$ TV, $D\rightarrow$ AP with $\mbox{\,A}\rightarrow$ VP or SP, all of which may contribute to the decay. Since the process $D^{0}\rightarrow K^{-}a_{1}^{+}(1260)$, $a_{1}^{+}(1260)[S]\rightarrow\rho^{0}\pi^{+}$ has the largest fit fraction, we fix the corresponding magnitude and phase to $1.0$ and $0.0$ and allow the magnitudes and phases of the other processes to vary in the fit.

We keep the processes with significance larger than $5\sigma$ for the next iteration. The fit involving both the $K^{-}a_{1}^{+}(1260)$ and the nonresonant $K^{-}(\rho^{0}\pi^{+})_{{\rm A}}$ contribution does not result in a significantly improvement of fit, but the fit fractions of the two amplitudes are much different with the assumption of only $K^{-}a_{1}^{+}(1260)$ and are nearly 100% correlated. We avoid this kind of case and only consider the resonant term, in agreement with the analysis of Mark III MarKIII . For the process $D^{0}\rightarrow K^{-}_{1}(1270)\pi^{+}$ with $K^{-}_{1}(1270)[S]\rightarrow\bar{K}^{*0}\pi^{-}$, the corresponding significance is found to be $4.3\sigma$ only, but we still include it in the fit since the corresponding $D$-wave process is found to have a statistical significance of larger than $9\sigma$. Better projections in the invariant mass spectra and an improved fit quality $\chi^{2}$ are also seen with this $S$-wave process included.

Finally, we retain 23 processes categorized into seven components. The other processes, not used in our nominal results but have been tested when determining the nominal fit model, are listed in Appendix A. The widths and masses of $\bar{K}^{*0}$ and $\rho^{0}$ are determined by the fit. The results of are listed in Table 3.

The $K_{1}^{-}(1270)$ has a small fit fraction, and we fix its mass and width to the PDG values PDG . The $a_{1}^{+}(1260)$ has a mass close to the upper boundary of the $\pi^{+}\pi^{+}\pi^{-}$ invariant mass spectrum. Therefore, we determine its mass and width with a likelihood scan, as shown in Fig. 2. The scan results are

[TABLE]

where the uncertainties are statistical only. The mass and width of $a_{1}^{+}(1260)$ are fixed to the scanned values in the nominal fit.

Our nominal fit yields a goodness of fit value of $\chi^{2}/\nu=843.445/748=1.128$. To calculate the statistical significance of a process, we repeat the fit process without the corresponding process included, and the changes of log-likelihood value and the number of free degree are taken into consideration. The projections for eight invariant mass and the distribution of $\chi$ are shown in Fig. 3. All of the components, amplitudes and the significance of amplitudes are listed in Table 4. The fit fractions of all components are given in Table 5. The phases and fit fractions of all amplitudes are given in Table 6.

VI Systematic Uncertainties

The source of systematic uncertainties are divided into four categories: (I) amplitude model, (II) background estimation, (III) experimental effects and (IV) fitter performance. The systematic uncertainties of the free parameters in the fit and the fit fractions due to different contributions are given in units of the statistical standard deviations $\sigma_{stat}$ in Tables 7–9. These uncertainties are added in quadrature, as they are uncorrelated, to obtain the total systematic uncertainties.

VI.0.1 Amplitude model

Three sources are considered for the systematic uncertainty due to the amplitude model: the masses and widths of the $K_{1}^{-}(1270)$ and the $a_{1}^{+}(1260)$, the barrier effective radius $R$ and the fixed parameters in the $K\pi$ $S$-wave model. The uncertainty associated with the mass and width of $K_{1}^{-}(1270)$ and the $a_{1}^{+}(1260)$ are estimated by varying the corresponding masses and widths with 1$\sigma$ of errors quoted in PDG PDG , respectively. The uncertainty related to the barrier effective radius $R$ is estimated by varying $R$ within $1.5-4.5{\mbox{\,GeV}}^{-1}$ for the intermediate resonances and $3.0-7.0{\mbox{\,GeV}}^{-1}$ for the $D^{0}$ in the fit. The uncertainty from the input parameters of the $K\pi$ $S$-wave model are evaluated by varying the input values within their uncertainties. All the change of the results with respect to the nominal one are taken as the systematic uncertainties.

VI.0.2 Background estimation

The sources of systematic uncertainty related to the background include the amplitude and shape of the background $D^{0}\rightarrow K_{S}^{0}K^{-}\pi^{+}$, and the other potential backgrounds. The uncertainties related to the background $D^{0}\rightarrow K_{S}^{0}K^{-}\pi^{+}$ is estimated by varying the number of background events within 1$\sigma$ of uncertainties and changing the shape according to the uncertainties in PDF parameters from CLEO KsKPi . The uncertainty due to the the other potential background is estimated by including the corresponding background (estimated from generic MC sample) in the fit.

VI.0.3 Experimental effects

The uncertainty related to the experimental effects includes two separate components: the acceptance difference between MC simulations and data caused by tracking and PID efficiencies, and the detector resolution. To determine the systematic uncertainty due to tracking and PID efficiencies, we alter the fit by shifting the $\gamma_{\epsilon}(p)$ in Eq. (9) within its uncertainty, and the changes of the nominal results is taken as the systematic uncertainty. The uncertainty caused by resolution is determined as the difference between the pull distribution results obtained from simulated data using generated and fitted four-momenta, as described in Sec. VI.0.4.

VI.0.4 Fitter performance

The uncertainty from the fit process is evaluated by studying toy MC samples. An ensemble of 250 sets of SIGNAL MC samples with a size equal to the data sample are generated according to the nominal results in this analysis. The SIGNAL MC samples are fed into the event selection, and the same amplitude analysis is performed on each simulated sample. The pull variables, $\frac{V_{{\rm input}}-V_{{\rm fit}}}{\sigma_{{\rm fit}}}$, are defined to evaluate the corresponding uncertainty, where $V_{{\rm input}}$ is the input value in the generator, $V_{{\rm fit}}$ and $\sigma_{{\rm fit}}$ are the output value and the corresponding statistical uncertainty, respectively. The distribution of pull values for the 250 sets of sample are expected to be a normal Gaussian distribution, and any shift on mean and widths indicate the bias on the fit values and its statistical uncertainty, respectively.

Small biases for some fitted parameters and fit fractions are observed. For the pull mean, the largest bias is about 19% of a statistical uncertainty with a deviation of about 3.0$\sigma$ from zero. For the pull width, the largest shift is $0.87\pm 0.04$, about 3.0 standard deviations from 1.0. We add in quadrature the mean and the mean error in the pull and multiply this number with the statistical error to get the systematic error. The fit results are given in Tables 12$\sim$12. The uncertainties in Tables 12$\sim$12 are the statistical uncertainties of the fits to the pull distributions.

VII Conclusion

An amplitude analysis of the decay $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ has been performed with the 2.93 ${\mbox{\,fb}^{-1}}$ of $e^{+}e^{-}$ collision data at the $\psi(3770)$ resonance collected by the BESIII detector. The dominant components, $D^{0}\rightarrow K^{-}a_{1}^{+}(1260)$, $D^{0}\rightarrow\bar{K}^{*0}\rho^{0}$, $D^{0}\rightarrow$ four-body nonresonant decay and three-body nonresonant $D^{0}\rightarrow K^{-}\pi^{+}\rho^{0}$ improve upon the earlier results from Mark III and are consistent with them within corresponding uncertainties. The resonance $K_{1}^{-}(1270)$ observed by Mark III is also confirmed in this analysis. The detailed results are listed in Table 5.

About 40% of components comes from the nonresonant four-body ($D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$) and three-body ($D^{0}\rightarrow K^{-}\pi^{+}\rho^{0}$ and $D^{0}\rightarrow\bar{K}^{*0}\pi^{+}\pi^{-}$) decays. A detailed study considering the different orbital angular momentum is performed, which was not included in the analyses of Mark III and E691. An especially interesting process involving the $K\pi$ S-wave is described by an effective range parametrization.

By using the inclusive branching fraction $\mathcal{B}(D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-})=(8.07\pm 0.23)\%$ taken from the PDG PDG and the fit fraction for the different components $FF(n)$ obtained in this analysis, we calculate the exclusive absolute branching fractions for the individual components with $\mathcal{B}(n)=\mathcal{B}(D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-})\times FF(n)$. The results are summarized in Table 13 and are compared with the values quoted in PDG. Our results have much improved precision; they may shed light in a theoretical calculation. The knowledge of $D^{0}\rightarrow\bar{K}^{*0}\rho^{0}$ and $D^{0}\rightarrow K^{-}a_{1}^{+}(1260)$ increase our understanding of the decay $D^{0}\rightarrow VV$ and $D\rightarrow AP$, both of which are lacking in experimental measurements, but have large contributions to the $D^{0}$ decays. Furthermore, knowledge of the submodes in the decay $D^{0}\rightarrow K^{-}\pi^{+}\pi^{+}\pi^{-}$ will improve the determination of the reconstruction efficiency when this mode is used to tag the $D^{0}$ as part of other measurements, like measurements of branching fractions, the strong phase or the angle $\gamma$.

Acknowledgements.

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key Basic Research Program of China under Contract No. 2015CB856700; National Natural Science Foundation of China (NSFC) under Contracts No. 11075174, No. 11121092, No. 11125525, No. 11235011, No. 11322544, No. 11335008, No. 11375221, No. 11425524, No. 11475185, No. 11635010; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts No. 11179007, No. U1232201, No. U1332201; CAS under Contracts No. KJCX2-YW-N29, No. KJCX2-YW-N45; 100 Talents Program of CAS; INPAC and Shanghai Key Laboratory for Particle Physics and Cosmology; German Research Foundation DFG under Contract No. Collaborative Research Center CRC-1044; Istituto Nazionale di Fisica Nucleare, Italy; Ministry of Development of Turkey under Contract No. DPT2006K-120470; Russian Foundation for Basic Research under Contract No. 14-07-91152; U. S. Department of Energy under Contracts No. DE-FG02-04ER41291, No. DE-FG02-05ER41374, No. DE-FG02-94ER40823, No. DESC0010118; U.S. National Science Foundation; University of Groningen (RuG) and the Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt; WCU Program of National Research Foundation of Korea under Contract No. R32-2008-000-10155-0.

VIII Appendix A: Amplitudes Tested

The amplitudes listed below are tested when determining the nominal fit model, but not used in our final fit result.

**Cascade amplitudes

$K_{1}^{-}(1270)(\rho^{0}K^{-})\pi^{+}$**, $\rho^{0}K^{-}$ $D$-wave

$K_{1}^{-}(1400)(\bar{K}^{*0}\pi^{-})\pi^{+}$, $\bar{K}^{*0}\pi^{-}$ $S$ and $D$-waves

$K^{*-}(1410)(\bar{K}^{*0}\pi^{-})\pi^{+}$

$K^{*-}_{2}(1430)(\bar{K}^{*0}\pi^{-})\pi^{+}$, $K^{*-}_{2}(1430)(K^{-}\rho^{0})\pi^{+}$

$K^{*-}(1680)(\bar{K}^{*0}\pi^{-})\pi^{+}$, $K^{*-}(1680)(K^{-}\rho^{0})\pi^{+}$

$K^{*-}_{2}(1770)(\bar{K}^{*0}\pi^{-})\pi^{+}$, $K^{*-}_{2}(1770)(K^{-}\rho^{0})\pi^{+}$

$K^{-}a_{2}^{+}(1320)(\rho^{0}\pi^{+})$

$K^{-}\pi^{+}(1300)(\rho^{0}\pi^{+})$

$K^{-}a_{1}^{+}(1260)(f_{0}(500)\pi^{+})$

**Quasi-two-body amplitudes

$\bar{K}^{*0}f_{0}(500)$

$\bar{K}^{*0}f_{0}(980)$

Three-body amplitudes

$\bar{K}^{*0}(\pi^{+}\pi^{-})_{{\rm V}}$** $S$, $P$- and $D$-waves

$(K^{-}\pi^{+})_{{\rm V}}\rho^{0}$ $S$, $P$ and $D$-waves

$\bar{K}^{*0}_{2}(1430)(\pi^{+}\pi^{-})_{{\rm S}}$

$\bar{K}^{*0}_{2}(1430)\rho^{0}$

$\bar{K}^{*0}f_{2}(1270)$

$(K^{-}\pi^{+})_{{\rm S}}f_{2}(1270)$

$K^{-}(\rho^{0}\pi^{+})_{{\rm V}}$

$K^{-}(\rho^{0}\pi^{+})_{{\rm P}}$

$K^{-}(\rho^{0}\pi^{+})_{{\rm A}}$

$K^{-}(\rho^{0}\pi^{+})_{{\rm T}}$

$(\bar{K}^{*0}\pi^{-})_{{\rm T}}\pi^{+}$

$(K^{-}\rho^{0})_{{\rm T}}\pi^{+}$

$(\bar{K}^{*0}\pi^{-})_{{\rm A}}\pi^{+}$, $\bar{K}^{*0}\pi^{-}$ $S$ and $D$-waves

**Four-body nonresonance amplitudes

$(K^{-}\pi^{+})_{{\rm T}}(\pi^{+}\pi^{-})_{{\rm V}}$** $P$- and $D$-waves

$(K^{-}\pi^{+})_{{\rm V}}(\pi^{+}\pi^{-})_{{\rm T}}$ $P$- and $D$-waves

$(K^{-}\pi^{+})_{{\rm V}}(\pi^{+}\pi^{-})_{{\rm V}}$ $P$- and $D$-waves

$(K^{-}(\pi^{+}\pi^{-})_{{\rm S}})_{{\rm A}}\pi^{+}$