Precision Measurement of the Decay $\Sigma^{+} \rightarrow p\gamma$ in the Process $J/\psi\rightarrow \Sigma^{+}\bar{\Sigma}^{-}$

TL;DR

This paper reports the first measurement of the $ ext{Sigma}^+ ightarrow p ext{gamma}$ decay in $J/\psi$ decays, providing the most precise values for its branching fraction and decay asymmetry parameter, with significant improvements over previous data.

Contribution

First measurement of $ ext{Sigma}^+ ightarrow p ext{gamma}$ decay in $J/\psi$ decays, with highly precise branching fraction and asymmetry parameter values.

Findings

Branching fraction measured as (0.996 ± 0.021 ± 0.018)×10^{-3}

Decay asymmetry parameter determined as -0.652 ± 0.056 ± 0.020

Results are the most precise to date, with significant improvements in accuracy.

Abstract

Using $(10\:087\pm44)\times10^{6}$ $J/\psi$ events collected with the BESIII detector, the radiative hyperon decay $\Sigma\rightarrow p\gamma$ is studied at an electron-positron collider experiment for the first time. The absolute branching fraction is measured to be $\left(0.996 \pm 0.021_{\rm stat.}\pm 0.018_{\rm syst.}\right)\times 10^{-3}$, which is lower than its world average value by $4.2$ standard deviations. Its decay asymmetry parameter is determined to be $-0.652 \pm 0.056_{\rm stat.}\pm 0.020_{\rm syst.}$. The branching fraction and decay asymmetry parameter are the most precise to date, and the accuracies are improved by $78\%$ and $34\%$, respectively.

Tables

Table 1. Table 1: The values of N ST obs superscript subscript 𝑁 ST obs N_{\rm{ST}}^{\rm{obs}} , ε ST subscript 𝜀 ST \varepsilon_{\rm{ST}} , N DT obs superscript subscript 𝑁 DT obs N_{\rm{DT}}^{\rm{obs}} , and ε DT subscript 𝜀 DT \varepsilon_{\rm{DT}} for the decays Σ + → p ​ γ → superscript Σ 𝑝 𝛾 \Sigma^{+}\to p\gamma and Σ ¯ − → p ¯ ​ γ → superscript ¯ Σ ¯ 𝑝 𝛾 \bar{\Sigma}^{-}\to\bar{p}\gamma . The BF and α γ subscript 𝛼 𝛾 \alpha_{\gamma} are obtained from both individual and simultaneous fits. The first uncertainties are statistical and the second systematic.

Mode	${\Sigma }^{+}\to p\gamma$	${\overline{\Sigma }}^{-}\to \overline{p}\gamma$
$N_{\mathrm{ST}}^{\mathrm{obs}}$	$\mathrm{2\hspace{0.17em}177\hspace{0.17em}771}\pm 2285$	$\mathrm{2\hspace{0.17em}509\hspace{0.17em}380}\pm 2301$
${\epsilon }_{\mathrm{ST}}$ ($\%$)	$39.00\pm 0.04$	$44.31\pm 0.04$
$N_{\mathrm{DT}}^{\mathrm{obs}}$	$1189\pm 38$	$1306\pm 39$
${\epsilon }_{\mathrm{DT}}$ ($\%$)	$21.16\pm 0.03$	$23.20\pm 0.03$
Individual BF (${10}^{-3}$)	$1.005\pm 0.032$	$0.993\pm 0.030$
Simultaneous BF (${10}^{-3}$)	$0.996\pm 0.021\pm 0.018$
Individual ${\alpha }_{\gamma }$	$-0.587\pm 0.082$	$0.710\pm 0.076$
Simultaneous ${\alpha }_{\gamma }$	$-0.651\pm 0.056\pm 0.020$

Equations

$$\mathrm{BF}({\Sigma }^{+}\to p\gamma )=\frac{N_{\mathrm{DT}}^{\mathrm{obs}}{\epsilon }_{\mathrm{ST}}}{N_{\mathrm{ST}}^{\mathrm{obs}}{\epsilon }_{\mathrm{DT}}},$$

$$\mathcal{W}(\xi )=$$

$$\times \left({\mathcal{F}}_1(\xi )+\sqrt{1-{\alpha }_{\psi }^2}\cos (\Delta \Phi ){\mathcal{F}}_2(\xi )+{\alpha }_{\psi }{\mathcal{F}}_6(\xi )\right)$$

$$+\sqrt{1-{\alpha }_{\psi }^2}\sin (\Delta \Phi )({\alpha }_{\gamma }{\mathcal{F}}_3(\xi )+{\bar{\alpha }}_0{\mathcal{F}}_4(\xi )),$$

$$M_{\mathrm{rec}}=\sqrt{{\left(E_{\mathrm{cms}}-E_{\bar{p}}-E_{{\pi }^0}\right)}^2/c^4-{\left({\mathbf{P}}_{\bar{p}}+{\mathbf{P}}_{{\pi }^0}\right)}^2/c^2}.$$

$$\mathcal{L}\left(H\right)=\prod _{i=1}^N\frac{\mathcal{W}\left({\xi }_i,H\right)\epsilon \left({\xi }_i\right)}{C\left(H\right)},$$

$$M_1(\cos {\theta }_{{\Sigma }^{+}})=\frac{m}{N}\sum _{i=1}^{N_k}\cos {\theta }_{\bar{p}}^i\cos {\theta }_p^i,$$

$$M_2(\cos {\theta }_{{\Sigma }^{+}})=\frac{m}{N}\sum _{i=1}^{N_k}\sin {\theta }_p^i\sin {\varphi }_p^i,$$

$$\displaystyle\Delta_{CP}=\frac{\mathcal{B}_{+}-\mathcal{B}_{-}}{\mathcal{B}_{+}+\mathcal{B}_{-}}=$0.006\pm 0.011_{\rm{stat.}}\pm 0.004_{\rm{syst.}}$,$$

$$\displaystyle A_{CP}=\frac{\alpha_{-}+\alpha_{+}}{\alpha_{-}-\alpha_{+}}=$0.095\pm 0.087_{\rm{stat.}}\pm 0.018_{\rm{syst.}}$.$$

Full text

Radiative hyperon decays provide valuable insight into the nature of nonleptonic weak interactions [1]. In general, the radiative decays of a spin-$\frac{1}{2}$ hyperon are described by a parity conserving (P-wave) and a parity violating (S-wave) amplitude. The non-vanishing parity violating amplitude produces an asymmetric angular distribution of the daughter baryon in the hyperon rest frame, as $\mathop{}\!\mathrm{d}N/\mathop{}\!\mathrm{d}\Omega=(N/4\pi)(1+\alpha_{\gamma}\mathbf{P}_{i}\cdot\hat{\mathrm{p}})$. Here $\mathbf{P}_{i}$ is the polarization of the decaying hyperon, $\hat{\mathrm{p}}$ is the unit vector of the daughter baryon’s momenta in the hyperon rest frame, and $\alpha_{\gamma}$ is the decay asymmetry parameter characterizing the mixing of S- and P-waves. In the limit of unitary symmetry, the parity violating amplitude of radiative hyperon decays is predicted to be small, resulting in $\alpha_{\gamma}=0$ [2]. However, the $\alpha_{\gamma}$ values measured in experiments are large [3, 4, *PhysRevLett.64.843, *PhysRevLett.72.808, *PhysRevLett.86.3239, 8]. Of all radiative hyperon decays, the $\Sigma^{+}\to p\gamma$ decay was observed first and stimulated continued controversy over several decades. It was first observed in bubble chamber experiments [9, *PhysRev.188.2077, *MANZ1980217, *Ang:1969hg], and was studied later at modern particle physics spectrometers [3, 13, *Bristol-Geneva-Heidelberg-Lausanne-QueenMaryColl-Rutherford:1985ksh]. The current world average value [15] is dominated by precise measurements using a polarized charged hyperon beam at Fermilab [8, 16], where both branching fraction (BF) and $\alpha_{\gamma}$ are obtained as ratios to those of the $\Sigma^{+}\to p\pi^{0}$ decay.

Various phenomenological models have been proposed to explain the experimental results of radiative hyperon decays [17, 18, 19, 20, *chang_challenges_2000, *dubovik_weak_2008, *Zenczykowski:2020hmg], but none of them gives a unified picture in describing all radiative hyperon decays. The theoretical predictions given by Ref. [24] in the framework of chiral perturbation theory (ChPT) are consistent with the experimental results of almost all radiative hyperon decays except for $\Sigma^{+}\to p\gamma$. Therefore, more precise measurements on $\Sigma^{+}\to p\gamma$ are crucial to test theoretical approaches such as ChPT. In addition, it has long been argued that $\Sigma^{+}\rightarrow pl^{+}l^{-}~{}(l=e,\mu)$ decays are good probes for physics beyond the standard model (SM) [25, 26, 27]. More precise BF and $\alpha_{\gamma}$ measurements of $\Sigma^{+}\to p\gamma$ offer critical information on the form factor of $\Sigma^{+}\rightarrow pl^{+}l^{-}$ [27], and thus provide better constraints on the SM predictions for these decays.

The unique properties of $J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$ events produced in electron-positron collisions, e.g. the spin correlation and transverse polarization of hyperon and anti-hyperon [28, 29], and the well defined kinematics, provide ideal conditions to study the decay $\Sigma^{+}\to p\gamma$. A double-tagged technique is applied to determine the absolute BF [30]. This approach was applied in a recent work on the $\Lambda\rightarrow n\gamma$ decay investigated in the $J/\psi\rightarrow\Lambda\bar{\Lambda}$ process by BESIII [31], which achieved a better precision than fixed target experiments. Moreover, the decay $\Sigma^{+}\to p\gamma$ and the corresponding charge conjugate one provide a good opportunity to search for charge-parity ($CP$) violation [32, *Tandean:2002vy], whose experimental information is absent for $\Sigma^{+}\to p\gamma$ [34, *PhysRevD.51.2271] currently.

In this Letter, using $J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$ decays from $(10\,087\pm 44)\text{\times}{10}^{6}$ $J/\psi$ events [36] collected at BESIII, we report the measurements of the absolute BF and decay asymmetry parameter of $\Sigma^{+}\to p\gamma$, as well as a test of $CP$ violation in hyperon decays. Throughout this Letter, charge conjugation is always implied unless noted otherwise.

A detailed description of the design and performance of the BESIII detector can be found in Ref. [37, *HUANG2022142], while the simulation and analysis software framework for BESIII are described in Refs. [39, *Asner:2009zza, *zhang2018low]. First, events of $J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$ are selected with a single-tag (ST) approach, i.e. reconstructing a $\bar{\Sigma}^{-}$ candidate in one of its dominant decay modes $\bar{\Sigma}^{-}\to\bar{p}\pi^{0}$ ($\mathrm{BF}=$51.57\pm 0.30\text{,}\mathrm{\char 37\relax}$$) [15]. Then, the double-tag (DT) decay $\Sigma^{+}\to p\gamma$ is searched for in the system recoiling against the ST $\bar{\Sigma}^{-}$ hyperon. The corresponding absolute BF is calculated by

[TABLE]

where $N^{\mathrm{obs}}_{\mathrm{ST(DT)}}$ and $\varepsilon_{\mathrm{ST(DT)}}$ are the ST (DT) yields and the corresponding detection efficiencies, respectively.

The helicity decay amplitudes of the processes [42, *PhysRevD.99.056008], which will be used for the measurement of $\alpha_{\gamma}$, are functions of the five observables $\xi=(\theta_{\Sigma^{+}},\theta_{p},\phi_{p},\theta_{\bar{p}},\phi_{\bar{p}})$. Here, $\theta_{\Sigma^{+}}$ is the angle between the $\Sigma^{+}$ hyperon and the electron beam in the $J/\psi$ rest frame, $\theta_{p}$ ($\theta_{\bar{p}}$) and $\phi_{p}$ ($\phi_{\bar{p}}$) are the polar and azimuthal angles of the proton (anti-proton) with respect to the $\Sigma^{+}$ helicity frame, respectively. The differential cross-section is given as $\mathop{}\!\mathrm{d}\sigma\propto\mathcal{W}(\xi)\mathop{}\!\mathrm{d}\xi$ with

[TABLE]

where $\alpha_{\psi}$ and $\Delta\Phi$ are the hyperon production parameters of the process $J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$, $\bar{\alpha}_{0}$ is the decay asymmetry parameter for the $\bar{\Sigma}^{-}\to\bar{p}\pi^{0}$ decays, and $\mathcal{F}_{i}(\xi)$ ($i=0,1,...,6$) are the angular functions as described in detail in Ref. [42]. The parameters used here are fixed to the values in Ref. [44], except for $\alpha_{\gamma}$ to be determined in this analysis.

A sample of Monte Carlo (MC) simulated events of generic $J/\psi$ decays corresponding to the luminosity of data is used to study possible background reactions. On the ST side, the signal MC sample of $J/\psi\rightarrow\Sigma^{+}\left(\rightarrow\mathrm{anything}\right)\bar{\Sigma}^{-}\left(\rightarrow\bar{p}\pi^{0}\right)$ is generated with its helicity decay amplitude. A phase space (PHSP) MC sample of $J/\psi\rightarrow\Delta(1232)^{+}(\rightarrow{\mathrm{anything}})\bar{\Delta}(1232)^{-}(\rightarrow\bar{p}\pi^{0})$ is generated to study ST background. On the DT side, MC samples for the $J/\psi\to\Sigma^{+}\left(\rightarrow p\gamma\right)\bar{\Sigma}^{-}\left(\rightarrow\bar{p}\pi^{0}\right)$ signal and the dominant background $J/\psi\to\Sigma^{+}\left(\rightarrow p\pi^{0}\right)\bar{\Sigma}^{-}\left(\rightarrow\bar{p}\pi^{0}\right)$ are generated according to their helicity decay amplitudes.

Charged tracks and photon candidates are detected in the main drift chamber and electromagnetic calorimeter (EMC), respectively. Requirements on the polar angle, distance of closest approach to the interaction point, particle identification probability of charged tracks, the polar angle, energy, EMC time and distance to the nearest charged track of photon candidates are applied as in Ref. [44]. The $\pi^{0}$ candidates are reconstructed from pairs of photon candidates with an invariant mass $M_{\gamma\gamma}$ satisfying 116\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}^{2}$<M_{\gamma\gamma}<$148\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}^{2}. A kinematic fit constraining their invariant mass to the $\pi^{0}$ mass [15] is performed on these photon pairs, and the updated four-momentum is used in further analysis.

The candidate ST $\bar{\Sigma}^{-}$ events are required to have at least one $\bar{p}$ and one $\pi^{0}$, and the $\bar{p}\pi^{0}$ combination is required to have an invariant mass of $\left|M_{\bar{p}\pi^{0}}-M_{\bar{\Sigma}^{-}}\right|<$13.5\text{,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}^{2}$$, where $M_{\bar{\Sigma}^{-}}$ is the nominal $\bar{\Sigma}^{-}$ mass [15]. If there is more than one combination satisfying this requirement, the one with the minimum $\left|M_{\bar{p}\pi^{0}}-M_{\bar{\Sigma}^{-}}\right|$ is kept for further analysis. Finally, the ST yield is obtained from the mass distribution recoiling against the reconstructed $\bar{\Sigma}^{-}$ candidate, which is defined as

[TABLE]

Here, $E_{\mathrm{cms}}$ is the center-of-mass energy, $E_{\bar{p}}$ ($\mathbf{P}_{\bar{p}}$) and $E_{\pi^{0}}$ ($\mathbf{P}_{\pi^{0}}$) are the energies (momenta) of the anti-proton and $\pi^{0}$ in the $J/\psi$ rest frame, respectively.

With the above selection criteria, the distribution of $M_{\rm{rec}}$ of the ST candidates is shown in Fig. 1, where there is a clear peak representing the signal of $J/\psi\to\Sigma^{+}\bar{\Sigma}^{-}$. For ST candidates, the background events are dominated by mis-combinations and $J/\psi\rightarrow\Delta(1232)^{+}(\rightarrow{\mathrm{anything}})\bar{\Delta}(1232)^{-}(\rightarrow\bar{p}\pi^{0})$, which contributes as a broad peak. An unbinned maximum likelihood fit is performed on the $M_{\rm{rec}}$ distribution to determine the ST yield, with the fit result shown in Fig. 1. In the fit, the signal and $J/\psi\rightarrow\Delta(1232)^{+}(\rightarrow{\mathrm{anything}})\bar{\Delta}(1232)^{-}(\rightarrow\bar{p}\pi^{0})$ background are described by their MC simulated shapes convolved with a Gaussian function, which represents the resolution difference between data and MC simulation. Other non-peaking background is described by a third order polynomial function. The ST yield and the detection efficiency evaluated using the signal MC sample are summarized in Table 1.

The signal process $\Sigma^{+}\to p\gamma$ is searched for in the remaining charged tracks and photons recoiling against the selected ST $\bar{\Sigma}^{-}$ candidates. One remaining $p$ and at least one remaining photon candidate are required for DT candidate events. A five-constraint (5C) kinematic fit under the hypothesis $J/\psi\rightarrow p\bar{p}\pi^{0}\gamma$ is performed. The fit constrains the total energy-momentum of the final state particles to the four-momentum of $J/\psi$ and the invariant mass of the photon pair on the ST side to the $\pi^{0}$ nominal mass [15]. The $\chi^{2}$ of the 5C kinematic fit ($\chi^{2}_{\mathrm{5C}}$) is required to be less than $30$. In case of multiple photon candidates on the DT side, the one with the minimum $\chi^{2}_{\mathrm{5C}}$ is kept. This requirement removes \sim$$68\text{\,}\mathrm{\char 37\relax} of the background at the cost of $13.9\text{\,}\mathrm{\char 37\relax}$ of the signal.

MC studies with a generic event type analysis tool [45] indicate that the dominant background events on the DT side are from the processes $J/\psi\rightarrow\Sigma^{+}(\rightarrow p\pi^{0})\bar{\Sigma}^{-}(\rightarrow\bar{p}\pi^{0})$ and $J/\psi\rightarrow\Delta(1232)^{+}(\rightarrow p\pi^{0})\bar{\Delta}(1232)^{-}(\rightarrow\bar{p}\pi^{0})$, denoted as $\Sigma^{+}\rightarrow p\pi^{0}$ and $\Delta(1232)^{+}\rightarrow p\pi^{0}$ background. To remove the $\Delta(1232)^{+}\rightarrow p\pi^{0}$ background, we make use of the life time difference between $\Sigma^{+}$ and $\Delta(1232)^{+}$. A secondary vertex fit [46] is performed for the $p$ and $\bar{p}$ combination. The length ($L$) from the intersection point to the interaction point shows significantly different distribution for the $\Delta(1232)^{+}\rightarrow p\pi^{0}$ background and the signal, as shown in Section 2 of the supplemental material [47]. The $L/\sigma_{L}$ value of the $\Delta(1232)^{+}\rightarrow p\pi^{0}$ background is generally less than $1.5$, where $\sigma_{L}$ is the decay length resolution. By vetoing these events, the $\Delta(1232)^{+}\rightarrow p\pi^{0}$ background is reduced by $93\text{\,}\mathrm{\char 37\relax}$ while $78\text{\,}\mathrm{\char 37\relax}$ of the signal is preserved. The remaining $\Sigma^{+}\rightarrow p\pi^{0}$ background usually includes a high energy photon from an asymmetric $\pi^{0}$ decay. To suppress this background, a 5C kinematic fit under the $J/\psi\to p\bar{p}\pi^{0}\gamma\gamma$ hypothesis is performed for all photon pair combinations. A candidate is eliminated if any $\chi^{2}$ of the kinematic fit under the $J/\psi\to p\bar{p}\pi^{0}\gamma\gamma$ hypothesis is less than $\chi^{2}_{5C}$ under the signal hypothesis.

After applying all the above selection criteria, the distribution of the proton momentum in the rest frame of $\Sigma^{+}$ ($P_{p}$) is shown in Fig. 2. The prominent peak at 0.204 GeV/$c$ is the major background from $\Sigma^{+}\to p\pi^{0}$, and the peaking structure around 0.223 GeV/$c$ is from the $\Sigma^{+}\to p\gamma$ signal. The contribution from the $\Delta(1232)^{+}\rightarrow p\pi^{0}$ background is negligible.

To obtain the DT signal yield, an unbinned maximum likelihood fit is performed on the $P_{p}$ distribution. In the fit, the $\Sigma^{+}\to p\gamma$ signal and the $\Sigma^{+}\to p\pi^{0}$ background are described by MC simulated shapes convolved with a Gaussian function. A second order polynomial function is included to represent the residual background such as $\Delta(1232)^{+}\rightarrow p\pi^{0}$. The fit results are shown in Fig. 2, and the DT signal yields are summarized in Table 1. In the fit, the yields of the $\Sigma^{+}\to p\pi^{0}$ and $\bar{\Sigma}^{-}\to\bar{p}\pi^{0}$ background are $18\,227.7\pm 135.0$ and $20\,334.9\pm 142.6$, respectively. To account for the small difference in the selection efficiencies of charged tracks and photons between data and MC simulation, an efficiency correction has been performed on the DT side [44]. The resultant DT efficiencies and the BFs are also summarized in Table 1. The BFs for $\Sigma^{+}\to p\gamma$ and $\bar{\Sigma}^{-}\to\bar{p}\gamma$ are consistent with each other within their uncertainties. Therefore, a simultaneous fit assuming the same BF between the charge conjugate channels is performed. The resultant BF is also shown in Table 1.

The decay asymmetry parameter $\alpha_{\gamma}$ is determined by an angular analysis according to Eq. (2). To further improve the purity of the data sample used for the $\alpha_{\gamma}$ measurements, two additional kinematic fits under the hypotheses of $J/\psi\to p\bar{p}\pi^{0}+\pi^{0}$ and $J/\psi\to p\bar{p}\pi^{0}+\gamma$ are performed for the surviving DT candidates, where the momenta information of the $\pi^{0}$ and photon are missing in the fit. The DT candidates are removed if $\chi^{2}_{p\bar{p}\pi^{0}+\pi^{0}}<\chi^{2}_{p\bar{p}\pi^{0}+\gamma}$, where $\chi^{2}_{p\bar{p}\pi^{0}+\pi^{0}}$ and $\chi^{2}_{p\bar{p}\pi^{0}+\gamma}$ are the $\chi^{2}$ from the two kinematic fits. Finally, $2345$ events (including $243$ background events) for the two charge conjugate modes in the signal region $0.215<P_{p}<$0.235\text{,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{$c$}}$$ are used in the following analysis.

$\alpha_{\gamma}$ is obtained from an unbinned maximum likelihood fit with the likelihood function defined as

[TABLE]

where $H=\left(\alpha_{\psi},\Delta\Phi,\alpha_{\gamma},\bar{\alpha}_{0}\right)$ represents the decay parameters, $N$ is the total number of DT candidate events, $i$ is the corresponding event index, $\mathcal{W}\left(\xi_{i},H\right)$ is the square of the decay amplitude as defined in Eq. (2), $\varepsilon\left(\xi_{i}\right)$ is the detection efficiency and $C\left(H\right)$ is the normalization factor calculated with the PHSP MC sample [28]. Due to the small number of events, the parameters $\alpha_{\psi}$, $\Delta\Phi$ and $\bar{\alpha}_{0}$ are fixed to those in Ref. [44] in the fit.

The value of $\alpha_{\gamma}$ is obtained by minimizing the likelihood function $S=-\left(\ln\mathcal{L}_{\rm{data}}-\ln\mathcal{L}_{\rm{bkg}}\right)$, where $\mathcal{L}_{\rm{data}}$ is the likelihood value from the data sample, and $\mathcal{L}_{\rm{bkg}}$ represents the background contributions including $\Sigma^{+}\to p\pi^{0}$ and other background components. The former background contribution is estimated with a MC simulated event sample that is five times the size of data and is generated according to its decay amplitude. The latter is estimated with data events in the side-band region defined as 0.11\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{c}} $<P_{p}<$ 0.16\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{c}} and 0.24\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{c}} <P_{p}<$$0.29\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\text{c}}. $\mathcal{L}_{\rm{bkg}}$ is estimated with the above samples and normalized to data. The fits are performed for the $\Sigma^{+}\to p\gamma$ and $\bar{\Sigma}^{-}\to\bar{p}\gamma$ decays individually. The $\alpha_{\gamma}$ results are consistent within their uncertainties and summarized in Table 1. Further, a simultaneous fit, assuming the same magnitude but opposite sign for the decay asymmetry parameters of the charge conjugate channels, is also performed, and the result is also shown in Table 1.

To visualize the effect of the decay asymmetry, two moments are calculated for $m=8$ intervals in $\cos\theta_{\Sigma^{+}}$:

[TABLE]

where $N$ is the total number of events and $N_{k}$ is the number of events in the kth $\cos\theta_{\Sigma^{+}}$ interval. Figure 3 shows the comparison of moments between data and MC projection based on the fit. Good data-MC consistencies are observed.

The systematic uncertainties due to proton tracking and particle identification ($0.4\text{\,}\mathrm{\char 37\relax}$), as well as photon detection ($0.3\text{\,}\mathrm{\char 37\relax}$), are studied with $J/\psi\rightarrow p\bar{p}\pi^{+}\pi^{-}$ and $J/\psi\rightarrow\gamma\mu^{+}\mu^{-}$ control samples. The uncertainties of $\chi^{2}$ requirements on the kinematic fits ($0.9\text{\,}\mathrm{\char 37\relax}$) and decay length requirement ($0.4\text{\,}\mathrm{\char 37\relax}$) are obtained by applying corresponding requirements on a $J/\psi\to\Sigma^{+}\left(\rightarrow p\pi^{0}\right)\bar{\Sigma}^{-}\left(\rightarrow\bar{p}\pi^{0}\right)$ control sample. The uncertainties of ST ($0.4\text{\,}\mathrm{\char 37\relax}$) and DT ($1.2\text{\,}\mathrm{\char 37\relax}$) yields are estimated by changing the fit parameters. The decay parameters used for signal MC generation are varied by $\pm 1\sigma$ [44] to study the MC model related uncertainty ($0.6\text{\,}\mathrm{\char 37\relax}$). Details can be found in the supplemental material [47]. By assuming all the sources to be independent, the total systematic uncertainty in the BF measurement is $1.8\text{\,}\mathrm{\char 37\relax}$ by summing up all above values in quadrature.

The systematic uncertainties in the determination of the decay asymmetry parameter $\alpha_{\gamma}$ are separated into two categories: the fit-related uncertainties and event selection uncertainties. The fit-related uncertainties are estimated with alternative fits by shifting the sideband region ($0.004$), changing the signal region in the fit ($0.014$), varying the number of background events by $\pm 1\sigma$ ($0.002$) and varying the fixed decay parameters by $\pm 1\sigma$ [44] ($0.011$), individually.

For the event selection induced uncertainties, only the angular dependent event selection criteria are considered, since other effects are negligible. For each requirement, its dependence on the angular distribution, and effect on the systematic uncertainty are detailed in the supplemental material [47]. The systematic uncertainty due to the track detection efficency ($0.001$) is estimated by performing an efficiency correction on the PHSP MC. The decay length requirement ($0.005$) and $\chi^{2}_{\gamma}<\chi^{2}_{\pi^{0}}$ requirement ($0.006$) uncertainties are studied with a $J/\psi\to\Sigma^{+}\left(\rightarrow p\pi^{0}\right)\bar{\Sigma}^{-}\left(\rightarrow\bar{p}\pi^{0}\right)$ control sample. The total systematic uncertainty is assigned to be $0.020$ by adding all individual uncertainties quadratically.

Based on the above results, the $CP$ asymmetry is calculated using the BFs and decay asymmetry parameters between the charge conjugate channels:

[TABLE]

Here, $\mathcal{B}_{+}$ ($\alpha_{+}$) denotes the BF ($\alpha_{\gamma}$) of $\Sigma^{+}\to p\gamma$ and $\mathcal{B}_{-}$ ($\alpha_{-}$) is that of $\bar{\Sigma}^{-}\to\bar{p}\gamma$. The systematic uncertainties of $\Delta_{CP}$ and $A_{CP}$ only consider the uncorrelated uncertainties of $B_{+/-}$ and $\alpha_{+/-}$. This represents the first search for $CP$ violation in $\Sigma^{+}\to p\gamma$. No obvious $CP$ violation is observed.

In summary, using $(10\,087\pm 44)\text{\times}{10}^{6}$ $J/\psi$ events collected with the BESIII detector, the radiative hyperon decay $\Sigma^{+}\to p\gamma$ is studied at an electron-positron collider for the first time. The absolute BF of this decay is determined to be $($0.996\pm 0.021_{\rm stat.}\pm 0.018_{\rm syst.}$)\times 10^{-3}$, with a decay asymmetry parameter of $-0.651\pm 0.056_{\rm stat.}\pm 0.020_{\rm syst.}$. Two independent observables between the two charge conjugate channels, $A_{CP}$ and $\Delta_{CP}$, are used to search for $CP$ violation, and no evidence of $CP$ violation is found.

The accuracies of the BF and $\alpha_{\gamma}$ are improved by $78\text{\,}\mathrm{\char 37\relax}$ and $34\text{\,}\mathrm{\char 37\relax}$, respectively. The comparison between the measurement results with the PDG values and theoretical predictions are shown in Fig. 4. The measured BF is lower than the PDG value [15] by $4.2\sigma$, where all previous measurement results were obtained as relative ratios to $\Sigma^{+}\to p\pi^{0}$. The decay asymmetry parameter is consistent with the world average value within $1.1\sigma$. A similar BF difference is also observed in another radiative hyperon decay $\Lambda\to n\gamma$ [31] at BESIII. With the updated BF and $\alpha_{\gamma}$ results of $\Sigma^{+}\to p\gamma$, further theoretical efforts are needed to clarify the physics of radiative hyperon decays. The SM prediction of $\Sigma^{+}\rightarrow pl^{+}l^{-}$ decays can also be further improved [27] to search for new physics beyond the SM.