Hyperon I: Partial wave amplitudes for $K^-p$ scattering

TL;DR

This paper performs a coupled-channel partial wave analysis of $K^-$ proton scattering data, identifying new hyperon resonances and challenging some previously listed states, thereby advancing understanding of hyperon spectroscopy.

Contribution

It introduces a systematic search for new hyperon resonances in $K^-p$ scattering and evaluates the significance of known and new states, proposing five new hyperons.

Findings

Identified five new hyperon resonances.

Questioned the existence of 17 previously listed resonances.

Compared partial-wave amplitudes with other analyses.

Abstract

Early data on $K^-$ induced reactions off protons are collected and used in a coupled-channel partial wave analysis (PWA). Data which had been published in the form of Legendre coefficients are included in the PWA. In a {\it primary} fit using 3* and 4* resonances only, we observe some significant discrepancies with the data. In a systematic search for new $\Lambda$ and $\Sigma$ hyperon resonances, additional candidates are found. The significance of the known and of the additional resonances is evaluated. Seventeen resonances listed with 1* or 2* and one resonance listed with 3* in the Review of Particle Properties cannot be confirmed, five new hyperons are suggested. The partial-wave amplitudes deduced in this analysis are compared to those from other analyses.

Figures

Tables

Table 1. Table 1: List of reactions used in the partial wave analysis. Δ Δ \Delta denotes the Δ ​ ( 1232 ) ​ 3 / 2 + Δ 1232 3 superscript 2 \Delta(1232)3/2^{+} , Λ ∗ superscript Λ \Lambda^{*} the Λ ​ ( 1520 ) ​ 3 / 2 − Λ 1520 3 superscript 2 \Lambda(1520)3/2^{-} , Σ ∗ superscript Σ \Sigma^{*} the Σ ​ ( 1385 ) ​ 3 / 2 + Σ 1385 3 superscript 2 \Sigma(1385)3/2^{+} .

$K^{-}p\to K^{-}p$	$K^{-}p\to {\overline{K}}^{0}n$	$K^{-}p\to {\pi }^0\Lambda$
$K^{-}p\to \omega \Lambda$	$K^{-}p\to \eta \Lambda$	$K^{-}p\to {\pi }^0{\Sigma }^0,\eta {\Sigma }^0$
$K^{-}p\to {\pi }^{\mp }{\Sigma }^{\pm }$	$K^{-}p\to K^{+/0}{\Xi }^{-/0}$	$K^{-}p\to K^{-/0}{\Delta }^{+/0}$
$K^{-}p\to {\pi }^{\pm }{\Sigma }^{\ast \mp }$	$K^{-}p\to {\pi }^0{\Lambda }^{\ast }$	$K^{-}p\to K^{\ast -}p$
$K^{-}p\to K^{\ast 0}n$	$K^{-}p\to {\pi }^0{\pi }^0\Lambda$	$K^{-}p\to {\pi }^0{\pi }^0\Sigma$

Table 2. Table 2: Differential cross sections for K − ​ p → K − ​ p → superscript 𝐾 𝑝 superscript 𝐾 𝑝 K^{-}p\to K^{-}p elastic and K − ​ p → K ¯ 0 ​ n → superscript 𝐾 𝑝 superscript ¯ 𝐾 0 𝑛 K^{-}p\to\bar{K}^{0}n charge exchange scattering used in this analysis. Listed are the reaction, the momenta at which the data are given with a reference, the number of data points, and the χ 2 superscript 𝜒 2 \chi^{2} from the primary ( χ 1 2 superscript subscript 𝜒 1 2 \chi_{1}^{2} ) and from the final fit ( χ 2 2 superscript subscript 𝜒 2 2 \chi_{2}^{2} ).

Reaction	Mass	Ref.	$N_{\mathrm{data}}$	${\chi }_1^2$	${\chi }_2^2$
$K^{-}p\to K^{-}p$	1464 1466 1469 1472 1475 1478 1481 1484 1488 1491 1494 1498 1501 1505 1509
	1512 1516 1520 1524 1528 1532 1536 1540 1544 1548	Mast:1975pv	482	1313	990
	1732 1749 1758 1763 1768 1772 1777 1789	Cameron:1981qi	320	760	694
	1775 1796 1815 1833 1852 1870 1889 1907 1925 1941 1957	Conforto:1975nw	451	1216	852
	1858 1869 1877 1887 1902 1911 1921 1930	Griselin:1975pa	311	406	408
	2207 2246 2288 2328 2365 2397 2436	deBellefon:1976qr	75	189	124
	1696 1687 1681 1671 1662 1652 1642 1633 1624 1615 1606 1595 1586 1578 1569
	1561 1552 1544 1536	Armenteros:1970eg	752	1054	827
	1611 1626 1640 1654 1667 1680 1692 1704 1715 1726 1735 1746 1757 1767	Adams:1975mg	491	810	675
	1689 1702 1717 1724 1734 1744 1748 1754 1763 1772 1779 1789 1804 1814 1822
	1831 1841 1848 1856 1865 1875 1879 1898	Conforto:1971vy	896	969	907
	1735 1763 1798 1810 1819 1876 1909 1946	Albrow:1971yu	219	306	259
	1970 1992 2014 2037 2059 2080 2102 2123 2144 2186 2207 2227 2248 2268 2288	Daum:1968jey	327	833	656
	1785 1832 1860 1892 1924 2122	AnderssonAlmehed:1970nb	162	465	431
	2122 2137 2151 2166 2181 2196 2212 2229 2244 2260 2276 2292 2331 2348 2365
	2382 2400 2417	Barber:1975ci	268	1202	811
	1837 1849 1859 1869 1879 1889 1854 1911 1922 1933 1944 1956 1967	Barber:1975rg	416	1868	1687
$K^{-}p\to {\overline{K}}^{0}n$	1466 1469 1472 1475 1478 1481 1484 1488	Mast:1975pv	64	383	182
	1491 1494 1498 1501 1505 1509 1512 1516 1520 1524 1528 1532 1536 1540 1544	Mast:1975pv	297	696	475
	1732 1749 1758 1763 1768 1772 1777 1789	Cameron:1981qi	160	276	267
	1775 1796 1815 1833 1852 1870 1889 1907 1925 1941 1957	Conforto:1975nw	220	744	558
	1858 1869 1877 1887 1902 1911 1921 1930 1935 1949 1963 1976 1992	Griselin:1975pa	260	414	387
	2207 2246 2288 2328 2365 2397 2436	deBellefon:1976qr	75	177	102
	1696 1687 1681 1671 1662 1652 1642 1633 1624 1615 1606 1595 1586 1578 1569
	1561 1552 1544 1536	Armenteros:1970eg	380	618	572
	1569 1589 1598 1620 1634 1647 1659 1676	Prakhov:2008dc	128	325	216
	1729 1739 1741 1747 1755 1763 1775 1780 1794	Jones:1974at	333	400	398
	1915 1939 1963 1984 2006 2027 2042 2068 2088 2111 2125 2151 2169	Litchfield:1971ri	260	515	394
	1569 1578 1587 1597 1606 1615 1624 1633 1643 1652 1661 1670 1680 1689 1708
	1717 1726 1736 1745 1755 1763 1773	AlstonGarnjost:1977ct	808	1837	1237
	1689 1702 1717 1724 1734 1744 1748 1754 1763 1772 1779 1789 1804 1814 1822
	1831 1841 1848 1856 1865 1875 1879 1898	Armenteros:1969kn	460	498	534

Table 3. Table 3: Differential cross sections on K − ​ p superscript 𝐾 𝑝 K^{-}p scattering into two-body final states used in this analysis. Listed are the reaction, the momenta at which the data are given with a reference, the number of data points, and the χ 2 superscript 𝜒 2 \chi^{2} from the primary ( χ 1 2 superscript subscript 𝜒 1 2 \chi_{1}^{2} ) and from the final fit ( χ 2 2 superscript subscript 𝜒 2 2 \chi_{2}^{2} ). (LC: data given as Legendre coefficients).

$K^{-}p\to {\pi }^0\Lambda$	1732 1749 1758 1763 1768 1772 1777 1789	Cameron:1981qi	160	467	362
	1775 1796 1815 1833 1852 1870 1889 1907 1925 1941 1957	Conforto:1975nw	220	468	362
	1858 1869 1877 1887 1902 1911 1921 1930 1935 1949 1963 1976 1992	Griselin:1975pa	256	656	621
	1696 1687 1681 1671 1662 1652 1642 1633 1624 1615 1606 1595 1586 1578 1569
	1561 1552 1544 1536	Armenteros:1970eg	380	743	610
	1569 1589 1598 1620 1634 1647 1659 1676	Prakhov:2008dc	125	331	166
	1729 1739 1741 1747 1755 1763 1775 1780 1794	Jones:1974at	341	488	427
	1689 1702 1717 1724 1734 1744 1748 1754 1763 1772 1779 1789 1804 1814 1822
	1831 1841 1848 1856 1865 1875 1879 1898	Armenteros:1969kn	460	756	701
	1915 1939 1963 1984 2006 2027 2042 2068 2088 2111 2125 2151 2169	Berthon:1970fd	260	842	351
	1648 1657 1666 1675 1684 1692 1702 1711 1719 1728 1737 1746 1754 1763	Baxter:1974zs	126	230	235
	1600 1630 1648 1663 1678 1693 1708 1722 1740	London:1975zz	90	158	133
	2207 2246 2288 2328	deBellefon:1975lp	60	152	141
$K^{-}p\to {\pi }^0{\Sigma }^0$	1696 1687 1681 1671 1662 1652 1642 1633 1624 1615 1606 1595 1586 1578 1569
	1561 1552 1544 1536	Armenteros:1970eg	190	338	383
	1569 1589 1598 1620 1634 1647 1659 1676	Prakhov:2008dc	125	361	281
	1648 1657 1666 1675 1684 1692 1702 1711 1719 1728 1737 1746 1754 1763	Baxter:1974zs	140	235	229
	1605 1640 1660 1680 1700 1730	London:1975zz	54	88	77
	1569 1589 1598 1620 1634 1647 1659 1676	Manweiler:2008zz	72	238	169
$K^{-}p\to {\pi }^{-}{\Sigma }^{+}$	1732 1749 1758 1763 1768 1772 1777 1789	Cameron:1981qi	160	460	298
	1775 1796 1815 1833 1852 1870 1889 1907	Conforto:1975nw	220	490	400
	1858 1869 1877 1887 1902 1911 1921 1930 1935 1949 1963 1976 1992	Griselin:1975pa	259	2838	2224
	1696 1687 1681 1671 1662 1652 1642 1633 1624 1615 1606 1595 1586 1578 1569
	1561 1552 1544 1536	Armenteros:1970eg	380	551	439
	1729 1739 1741 1747 1755 1763 1775 1780 1794	Jones:1974at	304	446	400
	1689 1702 1717 1724 1734 1744 1748 1754 1763 1772 1779 1789 1804 1814 1822
	1831 1841 1848 1856 1865 1875 1879 1898	Armenteros:1969kn	460	784	627
	1915 1939 1963 1984 2005 2028 2042 2068 2088 2111 2126 2151 2169	Berthon:1970sp	252	866	577
	2207 2246 2288 2328 2365 2397 2436	deBellefon:1977sp	60	250	276
$K^{-}p\to {\pi }^{+}{\Sigma }^{-}$	1732 1749 1758 1763 1768 1772 1777 1789	Cameron:1981qi	160	271	205
	1775 1796 1815 1833 1852 1870 1889 1907 1925 1941 1957	Conforto:1975nw	220	684	381
	1858 1869 1877 1887 1902 1911 1921 1930 1935 1949 1963 1976 1992	Griselin:1975pa	259	383	234
	1696 1687 1681 1671 1662 1652 1642 1633 1624 1615 1606 1595 1586 1578 1569
	1561 1552 1544 1536	Armenteros:1970eg	380	863	889
	1729 1739 1741 1747 1755 1763 1775 1780 1794	Jones:1974at	306	395	342
	1689 1702 1717 1724 1734 1744 1748 1754 1763 1772 1779 1789 1804 1814 1822
	1831 1841 1848 1856 1865 1875 1879 1898	Armenteros:1969kn	460	666	597
	1915 1939 1963 1984 2005 2028 2042 2068 2088 2111 2126 2151 2169	Berthon:1970sp	240	740	421
	2207 2246 2288 2328 2365 2397 2436	deBellefon:1977sp	57	173	210
$K^{-}p\to \eta \Lambda$	1664 1665 1666 1667 1668 1669 1670 1672 1674 1676 1678 1680 1682 1683 1685	Starostin:2001zz	135	316	190
	1664 1671 1681 1687 1696	Armenteros:1970eg	25	125	50
$K^{-}p\to \eta \Sigma$	1750 1765 1780 1795	Jones:1974si	16	12	11
$K^{-}p\to K^0{\Xi }^0$	2020	Berge:1966jp	17	15	14
	1970 2070 2140	Burgun:1968g	26	24	15
	2110 2280 2470	Dauber:1969pm	16	20	20
	2150	Carlson:1973jr	8	24	22
	1263 1316 1368 1415 1462 1514 1546 1606 1653 1705 1741 1800 1843 1934
	2031 2135 2234 2331	deBellefon:1977sw	29	64	46
$K^{-}p\to K^{+}{\Xi }^{-}$	1138 1161 1179 1201 1233 1253 1276 1296 1305 1336 1367 1396 1434	Griselin:1975pa
	1970 2070 2140	Burgun:1968g	101	182	168
	2110 2280 2420 2480	Dauber:1969pm	60	77	66
	2240	Trippe:1967tg	20	25	18
	1950	Trower:1968wp	12	10	10
	1263 1316 1368 1415 1462 1514 1546 1606 1653 1705 1741 1800 1843 1934
	2031 2135 2234 2331 2412 2516	deBellefon:1977sw	16	24	16
$K^{-}p\to {\overline{K}}^{0}n$	1709 1738 1758 1767 1782 1803 1821 1846 1865 1887 1919 1937 1953
(LC)	1970 2001 2022 2051 2085 2106	Horn:1972aj	136	303	253
$K^{-}p\to {\pi }^0\Lambda$	1709 1738 1758 1767 1782 1803 1821 1846 1865 1887 1919 1937 1953
(LC)	1970 2001 2022 2051 2085 2106	Horn:1972aj	128	211	219
	1647 1656 1673 1677 1692 1698 1709 1715	Ponte:1975ra	40	138	97

Table 4. Table 4: Data on the polarization observable P 𝑃 P in K − ​ p superscript 𝐾 𝑝 K^{-}p scattering used in this analysis. Listed are the reaction, the momenta at which the data are given with a reference, the number of data points, and the χ 2 superscript 𝜒 2 \chi^{2} from the primary ( χ 1 2 superscript subscript 𝜒 1 2 \chi_{1}^{2} ) and from the final fit ( χ 2 2 superscript subscript 𝜒 2 2 \chi_{2}^{2} ).

Reaction	Mass	Ref.	$N_{\mathrm{data}}$	${\chi }_1^2$	${\chi }_2^2$
$K^{-}p\to K^{-}p$	1630 1652 1669 1687 1706 1720 1811 1827	Ehrlich:1977bs	134	245	285
	1735 1763 1798 1810 1819 1876 1909 1946	Albrow:1971yu	230	305	275
	1970 1992 2014 2037 2059 2080 2102 2123 2144 2186 2207 2227 2248 2268 2288	Daum:1968jey	327	801	485
	1785 1832 1860 1892 1924 2122	AnderssonAlmehed:1970nb	168	235	159
	1772 1791 1810 1828 1847 1865 1883 1902 1920	Bryant:1980yp	321	617	455
$K^{-}p\to {\pi }^0\Lambda$	1732 1749 1758 1763 1768 1772 1777 1789	Cameron:1981qi	160	236	162
	1775 1796 1815 1833 1852 1870 1889 1907 1925 1941 1957	Conforto:1975nw	187	371	238
	1569 1589 1598 1620 1634 1647 1659 1676	Prakhov:2008dc	123	346	140
	1729 1739 1741 1747 1755 1763 1775 1780 1794	Jones:1974at	153	183	155
	1689 1702 1717 1724 1734 1744 1748 1754 1763 1772 1779 1789 1804 1814
	1822 1831 1841 1848 1856 1865 1875 1879	Armenteros:1969kn	160	182	167
	1915 1939 1963 1984 2006 2027 2042 2068 2088 2111 2125 2151 2169	Berthon:1970fd	88	309	210
	2207 2246 2288 2328	deBellefon:1975lp	21	69	38
$K^{-}p\to {\pi }^0{\Sigma }^0$	1569 1589 1598 1620 1634 1647 1659 1676	Prakhov:2008dc	124	572	299
$K^{-}p\to {\pi }^{-}{\Sigma }^{+}$	1732 1749 1758 1763 1768 1772 1777 1789	Cameron:1981qi	160	663	672
	1775 1796 1815 1833 1852 1870 1889 1907 1925 1941 1957	Conforto:1975nw	146	236	161
	1729 1739 1741 1747 1755 1763 1775 1780 1794	Jones:1974at	92	114	116
	1689 1702 1717 1724 1734 1744 1748 1754 1763 1772 1779 1789 1804 1814
	1822 1831 1841 1848 1856 1865 1874 1879 1898	Armenteros:1969kn	180	236	232
	2207 2246 2288 2328	deBellefon:1977sp	15	35	61
$K^{-}p\to K^0{\Xi }^0$	1973 1973 2022 2065 2109 2138 2151 2274 2274 2468	Sharov:2011xq	11	22	19
$K^{-}p\to K^{-}{\Xi }^{+}$	1973 1973 2022 2065 2109 2138 2151 2274 2274 2468	Sharov:2011xq	18	17	17

Table 5. Table 5: Data on the K − ​ p superscript 𝐾 𝑝 K^{-}p induced reactions with three-body final states. For all reactions, differential cross sections and three ρ 𝜌 \rho -density matrix elements were expanded into associated Legendre polynomials and Legendre coefficients for l = 1 , ⋯ ​ 7 𝑙 1 ⋯ 7 l=1,\cdots 7 were determined. Listed are the reaction, the momenta at which the data are given with a reference, the number of data points, and the χ 2 superscript 𝜒 2 \chi^{2} from the primary ( χ 1 2 superscript subscript 𝜒 1 2 \chi_{1}^{2} ) and from the final fit ( χ 2 2 superscript subscript 𝜒 2 2 \chi_{2}^{2} ). Here, Δ Δ \Delta denotes Δ ​ ( 1232 ) ​ 3 / 2 + Δ 1232 3 superscript 2 \Delta(1232)3/2^{+} , Λ ∗ superscript Λ \Lambda^{*} is Λ ​ ( 1520 ) ​ 3 / 2 − Λ 1520 3 superscript 2 \Lambda(1520)3/2^{-} , and Σ ∗ superscript Σ \Sigma^{*} denotes Σ ​ ( 1385 ) ​ 3 / 2 + Σ 1385 3 superscript 2 \Sigma(1385)3/2^{+} . (LC: data given as Legendre coefficients).

	Mass	Ref.	$N_{\mathrm{data}}$	${\chi }_1^2$	${\chi }_2^2$
$K^{-}p\to \omega \Lambda$	1915 1940 1963 1984 2005 2028 2042 2068 2088 2111 2126 2151 2169	Brandstetter:1972xp	130	242	147
(LC)	1915 1940 1963 1984 2005 2028 2042 2068 2088 2111 2126 2151 2169	Brandstetter:1972xp	94	314	487
	1988 2012 2031 2051 2070 2088 2105 2120 2136 2151	Nakkasyan:1975yz	100	91	72
	2207 2246 2288 2328 2365 2397 2436	Baccari:1976ik	70	116	89
	1988 2012 2031 2051 2070 2088 2105 2120 2136 2151	Nakkasyan:1975yz	62	173	326
$K^{-}p\to 2{\pi }^0\Lambda$	1571 1589 1598 1620 1632 1650 1659	Prakhov:2004ri	26513	$\delta {\chi }^2=-{244}^a$
$K^{-}p\to 2{\pi }^0\Sigma$	1569 1589 1598 1620 1634 1647 1659 1676	Prakhov:2004an	3286	$\delta {\chi }^2=-{498}^a$
$K^{-}p\to {\pi }^0{\Lambda }^{\ast }$	1919 1942 1964 1986 2006 2027 2042 2068 2088 2111 2126 2151 2168	Litchfield:1973ap	377	766	444
(LC)	1988 2011 2030 2052 2070 2088 2104 2120 2134 2150	Litchfield:1973ap	290	1169	372
$K^{-}p\to {\pi }^0{\Lambda }^{\ast }$	1775 1796 1815 1833 1852 1870 1889 1907 1925 1941 1957	Cameron:1977jr	319	606	523
(LC)	1710 1728 1747 1759 1775	Cameron:1977jr	25	193	66
$K^{-}p\to {\pi }^{-}{\Sigma }^{\ast +}$	1775 1796 1815 1833 1852 1870 1889 1907 1926 1957	Cameron:1978en	319	547	440
(LC)	2005 2028 2042 2068 2088 2110 2126 2151 2167	Cameron:1978en	135	282	232
$K^{-}p\to {\pi }^{+}{\Sigma }^{\ast -}$	1775 1796 1815 1833 1852 1870 1889 1907 1926 1957	Cameron:1978en	319	524	538
(LC)	2005 2028 2042 2068 2088 2110 2126 2151 2167	Cameron:1978en	126	266	185
$K^{-}p\to K^{\ast -}p$	1815 1833 1852 1870 1889 1925 1941 1957	Cameron:1978qi	232	704	636
(LC)	1962 1985 2005 2025 2042 2067 2088 2111 2126 2151 2167	Cameron:1978qi	231	323	266
$K^{-}p\to K^{\ast 0}n$	1815 1833 1852 1870 1889 1907 1925 1941 1957	Cameron:1978qi	261	863	853
(LC)	1962 1985 2005 2025 2042 2067 2088 2111 2126 2151 2167	Cameron:1978qi	110	167	129
$K^{-}p\to \overline{K}\Delta$	1919 1942 1964 1986 2006 2027 2042 2068 2088 2111 2126 2151 2168	Litchfield:1973ey	377	547	484
(LC)	1988 2011 2030 2052 2070 2088 2104 2120 2134 2150	Litchfield:1973ey	290	499	384

Table 6. Table 6: Hyperons used in the primary fit to the data in which the Breit-Wigner masses and widths from the RPP  Tanabashi:2018oca are imposed.

	$J^P$	Status	Mass	Width
$\Lambda \left(1405\right)$	$1/2^{-}$	****	${1405}_{-1.0}^{+1.3}$	$50.5\pm 2$
$\Lambda \left(1670\right)$	$1/2^{-}$	****	$1660-1680$	$25-50$
$\Lambda \left(1800\right)$	$1/2^{-}$	***	$1720-1850$	$200-400$
$\Lambda \left(1520\right)$	$3/2^{-}$	****	$1519.5\pm 1.0$	$15.6\pm 1.0$
$\Lambda \left(1690\right)$	$3/2^{-}$	****	$1685-1695$	$50-70$
$\Lambda \left(1830\right)$	$5/2^{-}$	****	$1810-1830$	$60-110$
$\Lambda \left(2100\right)$	$7/2^{-}$	****	$2090-2110$	$100-250$
$\Lambda \left(1600\right)$	$1/2^{+}$	***	$1560-1700$	$50-250$
$\Lambda \left(1810\right)$	$1/2^{+}$	***	$1750-1850$	$50-250$
$\Lambda \left(1890\right)$	$3/2^{+}$	****	$1850-1910$	$60-200$
$\Lambda \left(1820\right)$	$5/2^{+}$	****	$1815-1825$	$70-90$
$\Lambda \left(2110\right)$	$5/2^{+}$	***	$2090-2140$	$150-250$
	$J^P$	Status	Mass	Width
$\Sigma \left(1750\right)$	$1/2^{-}$	***	$1730-1800$	$60-160$
$\Sigma \left(1670\right)$	$3/2^{-}$	****	$1665-1685$	$40-80$
$\Sigma \left(1940\right)$	$3/2^{-}$	***	$1900-1950$	$150-300$
$\Sigma \left(1775\right)$	$5/2^{-}$	****	$1770-1780$	$105-135$
$\Sigma \left(1660\right)$	$1/2^{+}$	***	$1630-1690$	$40-200$
$\Sigma \left(1385\right)$	$3/2^{+}$	****	$1382.80\pm 0.35$	$36.0\pm 0.7$
$\Sigma \left(1915\right)$	$5/2^{+}$	****	$1900-1935$	$80-160$
$\Sigma \left(2030\right)$	$7/2^{+}$	****	$2025-2040$	$150-200$

Table 7. Table 7: Comparison of the data base and the best χ 2 superscript 𝜒 2 \chi^{2} for the ANL-Osaka fits in their model A, their model B, and the final BnGa fit. The fits of the Legendre coefficients extracted from differential cross section and density matrix elements are marked with (LC).

	ANL-Os.	BnGa	Model A/B	BnGa
	$N_{\mathrm{data}}$		${\chi }^2/N_{\mathrm{data}}$
$K^{-}p\to K^{-}p$
$d\sigma /d\Omega$	3962	5170	3.07 / 2.98	1.80
$P$	510	1180	2.04 / 2.08	1.41
$K^{-}p\to {\overline{K}}^{0}n$
$d\sigma /d\Omega$	2950	3445	2.67 / 2.75	1.55
$d\sigma /d\Omega$ (LC)		134	/	1.86
$K^{-}p\to {\pi }^{-}{\Sigma }^{+}$
$d\sigma /d\Omega$	1792	2455	3.37 / 3.49	1.45
$P$	418	593	1.30 / 1.28	2.09
$K^{-}p\to {\pi }^0{\Sigma }^0$
$d\sigma /d\Omega$	580	691	3.68 / 3.50	1.96
$P$	196	124	6.39 / 5.80	2.41
$K^{-}p\to {\pi }^{+}{\Sigma }^{-}$
$d\sigma /d\Omega$	1786	2082	2.56 / 2.18	1.59
$K^{-}p\to {\pi }^0\Lambda$
$d\sigma /d\Omega$	2178	2478	2.59 / 3.71	1.66
$P$	693	892	1.41 / 1.73	1.25
$K^{-}p\to \eta \Lambda$
$d\sigma /d\Omega$	160	160	2.69 / 2.03	1.50
$P$	18	—	0.94 / 3.83	—
$K^{-}p\to K^0{\Xi }^0$
$d\sigma /d\Omega$	33	67	1.24 / 1.61	0.89
${\sigma }_{tot}$		16	/	1.00
$P$		11	/	1.70
$K^{-}p\to K^{+}{\Xi }^{-}$
$d\sigma /d\Omega$	92	193	2.05 / 1.74	1.31
${\sigma }_{tot}$		29	/	1.57
$P$		18	/	0.93
$K^{-}p\to \Lambda \omega$
$d\sigma /d\Omega$	–	300	— / —	1.03
${\rho }_{00},{\rho }_{10},{\rho }_{1-1}$(LC)	–	158	— / —	1.30

Table 8. Table 8: Resonances found to contribute to K − superscript 𝐾 K^{-} induced reactions. The columns give the Breit-Wigner mass and width, and the increase of χ 2 superscript 𝜒 2 \chi^{2} when a hyperon is removed from the fit, and our evaluation of the star rating. The small numbers give the PDG entries. Some 4* resonances cannot be removed from the fit without destroying the fit; δ ​ χ 2 > 𝛿 superscript 𝜒 2 absent \delta\chi^{2}> 10 000 is given for these resonances. A statistically significant very broad enhancements in the J P superscript 𝐽 𝑃 J^{P}  =  1 / 2 − 1 superscript 2 1/2^{-} wave are likely due to some background and not considered as true resonances. 1 : no estimate given in the RPP; our own estimate.

Mass Width $\Delta {\chi }^2$ Status $\Lambda \left(1405\right)1/2^{-}$ 1420$\pm$3 46$\pm$4 $4070$ **** 1405.1${}_{-1.0}{}^{}{}_{}{}^{+1.3}$ 50.5$\pm$2.0 **** $\Lambda \left(1670\right)1/2^{-}$ 1677$\pm$2 33$\pm$4 $3610$ **** 1660 to 1680 25 to 50 **** $\Lambda \left(1800\right)1/2^{-}$ 1811$\pm$10 209$\pm$18 $1896$ *** 1720 to 1850 200 to 400 *** $\Lambda \left(1520\right)3/2^{-}$ 1518.5$\pm$0.5 15.7$\pm$1.0 $>$10 000 **** 1519.5$\pm$1.0 15.6$\pm$1.0 **** $\Lambda \left(1690\right)3/2^{-}$ 1689$\pm$3 75$\pm$5 $>$10 000 **** 1685 to 1695 50 to 70 **** $\Lambda \left(1830\right)5/2^{-}$ 1821$\pm$3 64$\pm$7 $1790$ *** 1810 to 1830 60 to 110 **** $\Lambda \left(2080\right)5/2^{-}$ 2082$\pm$13 181$\pm$29 $770$ * - - new $\Lambda \left(2100\right)7/2^{-}$ 2090$\pm$15 290$\pm$30 $5412$ **** 2090 to 2110 100 to 250 **** $\Lambda \left(1600\right)1/2^{+}$ 1605$\pm$8 245$\pm$15 $>$10 000 **** 1560 to 1700 50 to 250 *** $\Lambda \left(1890\right)3/2^{+}$ 1873$\pm$5 103$\pm$10 $4480$ **** 1850 to 1910 60 to 200 **** $\Lambda \left(2070\right)3/2^{+}$ 2070$\pm$24 370$\pm$50 $1144$ * - - new $\Lambda \left(1820\right)5/2^{+}$ 1822$\pm$4 80$\pm$8 $>$10 000 **** 1815 to 1825 70 to 90 **** $\Lambda \left(2110\right)5/2^{+}$ 2086$\pm$12 274$\pm$25 $1418$ ** 2090 to 2140 150 to 250 ***

Mass Width $\Delta {\chi }^2$ Status $\Sigma \left(1620\right)1/2^{-}$ 1681$\pm$6 40$\pm$12 $386$ (*) $\approx$1620 10 to 400 * $\Sigma \left(1750\right)1/2^{-}$ 1692$\pm$11 208$\pm$18 $3032$ **** 1730 to 1800 60 to 160 *** $\Sigma \left(1900\right)1/2^{-}$ 1938$\pm$12 155$\pm$30 $1500$ ** 1900$\pm$21 191$\pm$47 * $\Sigma \left(2160\right)1/2^{-}$ 2165$\pm$23 320${}_{-60}{}^{}{}_{}{}^{+300}$ 1612 * new $\Sigma \left(1670\right)3/2^{-}$ 1665$\pm$3 54$\pm$6 $5894$ **** 1665 to 1685 40 to 80 **** $\Sigma \left(1940\right)3/2^{-}$ 1878$\pm$12 224$\pm$25 $1708$ *** 1900 to 1950 150 to 300 *** $\Sigma \left(2000\right)3/2^{-}$ 2005$\pm$14 178$\pm$23 $446$ * - - new $\Sigma \left(1775\right)5/2^{-}$ 1776$\pm$4 124$\pm$8 $>$10 000 **** 1770 to 1780 105 to 135 **** $\Sigma \left(2100\right)7/2^{-}$ 2146$\pm$17 260$\pm$40 666 * $\approx$2100 50 to 1501 * $\Sigma \left(1660\right)1/2^{+}$ 1665$\pm$20 300${}_{-40}{}^{}{}_{}{}^{+140}$ $1870$ *** 1630 to 1690 40 to 200 *** $\Sigma \left(2230\right)3/2^{+}$ 2240$\pm$27 345$\pm$50 $1200$ * new $\Sigma \left(1915\right)5/2^{+}$ 1918$\pm$6 102$\pm$12 $2002$ **** 1900 to 1935 80 to 160 **** $\Sigma \left(2030\right)7/2^{+}$ 2032$\pm$6 177$\pm$12 $2856$ **** 2025 to 2040 150 to 200 ****

Table 9. Table 9: Suggested changes in star rating (the octet and decuplet ground states are not included in the counting). Two states, Σ ​ ( 1620 ) ​ 1 / 2 − Σ 1620 1 superscript 2 \Sigma(1620)1/2^{-} and Λ ​ ( 1810 ) ​ 1 / 2 + Λ 1810 1 superscript 2 \Lambda(1810)1/2^{+} , we keep as one star resonances, even though they cannot be confirmed by our fits (see text and Hyperon-III ).

	0*	1*	2*	3*	4*
4* to	-	-	-	1	11
3* to	-	1	1	3	2
2* to	4	-	-	-	-
1* to	12	2	2	-	-
new	-	5	-	-	-

Equations

$$A(s,t)=\sum _{IJN}{C}_I{Q}_{JN}(s,t)A_{IJN}(s),$$

$$Q_{J\pm }(s,t)$$

$$k_{\mu }^{\perp }=\frac{1}{2}(k_1-k_2{)}_{\nu }{g}_{\mu \nu }^{\perp },$$

$$g_{\mu \nu }^{\perp }=g_{\mu \nu }-\frac{P_{\mu }{P}_{\nu }}{P^2}.$$

$$u(p)$$

$$\bar{u}(p)$$

$$\bar{u}(p)u(p)=1,\sum _{polarizations}u(p)\bar{u}(p)=\frac{m+\hat{p}}{2m}.$$

$$F_{{\nu }_1\dots {\nu }_n}^{{\mu }_1\dots {\mu }_n}=(-1{)}^n\frac{\sqrt{s}+\hat{P}}{2\sqrt{s}}{O}_{{\xi }_1\dots {\xi }_n}^{{\mu }_1\dots {\mu }_n}{T}_{{\beta }_1\dots {\beta }_n}^{{\xi }_1\dots {\xi }_n}{O}_{{\nu }_1\dots {\nu }_n}^{{\beta }_1\dots {\beta }_n},$$

$$T_{{\beta }_1\dots {\beta }_n}^{{\xi }_1\dots {\xi }_n}$$

$${\sigma }_{{\alpha }_i{\alpha }_j}$$

$$O$$

$$O_{{\nu }_1{\nu }_2}^{{\mu }_1{\mu }_2}$$

$$\displaystyle O^{\mu_{1}\ldots\mu_{L}}_{\nu_{1}\ldots\nu_{L}}=\frac{1}{L^{2}}\bigg{(}\sum\limits_{i,j=1}^{L}g^{\perp}_{\mu_{i}\nu_{j}}O^{\mu_{1}\ldots\mu_{i-1}\mu_{i+1}\ldots\mu_{L}}_{\nu_{1}\ldots\nu_{j-1}\nu_{j+1}\ldots\nu_{L}}-$$

$$\frac{4}{(2L-1)(2L-3)}\times$$

$$\displaystyle\sum\limits_{i<j\atop k<m}^{L}g^{\perp}_{\mu_{i}\mu_{j}}g^{\perp}_{\nu_{k}\nu_{m}}O^{\mu_{1}\ldots\mu_{i-1}\mu_{i+1}\ldots\mu_{j-1}\mu_{j+1}\ldots\mu_{L}}_{\nu_{1}\ldots\nu_{k-1}\nu_{k+1}\ldots\nu_{m-1}\nu_{m+1}\ldots\nu_{L}}\bigg{)}\,.$$

$$F_{{\nu }_1\dots {\nu }_n}^{{\mu }_1\dots {\mu }_n}{F}_{{\xi }_1\dots {\xi }_n}^{{\nu }_1\dots {\nu }_n}=(-1{)}^n{F}_{{\xi }_1\dots {\xi }_n}^{{\mu }_1\dots {\mu }_n}.$$

$$N_{{\mu }_1\dots {\mu }_n}^{(+)}(k^{\perp })$$

$$N_{{\mu }_1\dots {\mu }_n}^{(-)}(k^{\perp })$$

$$X^{(0)}$$

$$X_{{\mu }_1{\mu }_2}^{(2)}$$

$$X_{{\mu }_1{\mu }_2{\mu }_3}^{(3)}$$

$$X_{{\mu }_1\dots {\mu }_L}^{(L)}$$

$$Z_{{\mu }_1\dots {\mu }_L}^{\alpha }=\frac{2L-1}{L^2}\sum _{i=1}^L{X}_{{\mu }_1\dots {\mu }_{i-1}{\mu }_{i+1}\dots {\mu }_L}^{(L-1)}{g}_{{\mu }_i\alpha }^{\perp }-$$

$$\frac{2}{L^2}\sum _{\genfrac{}{}{0ex}{}{i,j=1}{i<j}}^L{g}_{{\mu }_i{\mu }_j}^{\perp }{X}_{{\mu }_1\dots {\mu }_{i-1}{\mu }_{i+1}\dots {\mu }_{j-1}{\mu }_{j+1}\dots {\mu }_L\alpha }^{(L-1)}.$$

$$\displaystyle\gamma_{0}=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right),\,\,\vec{\gamma}=\left(\begin{array}[]{cc}0&\vec{\sigma}\\ -\vec{\sigma}&0\end{array}\right),\,\,\gamma_{5}=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right).~{}$$

$$A(s,t)={\omega }^{\ast }\left[G(s,t)+H(s,t)i(\vec{\sigma }\vec{n})\right]{\omega }^{\prime },$$

$$\displaystyle G(s,t)=\sum\limits_{L}\big{[}(L\!+\!1)F_{L}^{+}(s)-LF_{L}^{-}(s)\big{]}P_{L}(z)\;,$$

$$\displaystyle H(s,t)=\sum\limits_{L}\big{[}F_{L}^{+}(s)+F_{L}^{-}(s)\big{]}P^{\prime}_{L}(z)\;,$$

$$F_L^{+}$$

$$F_L^{-}$$

$${\chi }_i$$

$${\alpha }_L$$

$$A(t)=g_1(t)g_2(t)\frac{1+\xi exp(-i\pi \alpha (t))}{\sin (\pi \alpha (t))}{\left(\frac{\nu }{{\nu }_0}\right)}^{\alpha (t)}.$$

$$1/(m_p-{\hat{k}}_u)\mathrm{where}u=k_u^2.$$

$$A(s)=\sum _{\beta }\frac{g_{in}^{\beta }{g}_{out}^{\beta }}{M_{\beta }^2-s-i{M}_{\beta }{\Gamma }_{tot}^{\beta }}$$

$$M_{\beta }{\Gamma }_{tot}^{\beta }=\sum _j{M}_{\beta }{\Gamma }_j^{\beta }=\sum _j{\rho }_j^{n\pm }(s)(g_j^{\beta }{)}^2,$$

$${\rho }_j^{n+}(s)$$

$${\rho }_j^{n-}(s)$$

Full text

1 Introduction

The spectrum of $N^{*}$’s and $\Delta^{*}$’s is presently studied vigorously in a number of photoproduction experiments at ELSA, JLab, MAMI, and SPring-8 (see Refs. Klempt:2009pi ; Crede:2013sze ). In the first excitation shell, seven negative-parity resonances (five $N^{*}$’s and two $\Delta^{*}$’s) are expected in the quark model Isgur:1977ef ; these are known since long. In the second excitation shell, 21 positive-parity states are predicted Isgur:1978wd , 16 of them have been observed, even though only 11 of them are considered to be established (with 3*’s or 4*’s in the notation of the Review of Particle Physics, RPP) Tanabashi:2018oca . But there is the chance that existing or new data with further analyses will add to our knowledge of the missing or not so well-known states.

In the hyperon spectrum, seven negative-parity $\Lambda^{*}$’s and seven negative-parity $\Sigma^{*}$’s resonances are expected in the first excitation shell. SU(3) links the seven $\Sigma^{*}$’s and five $\Lambda^{*}$’s to the spectrum of $N^{*}$’s and $\Delta^{*}$’s. Two further SU(3) singlet $\Lambda^{*}$’s are expected. Compared to this expectation, one $\Lambda^{*}$ resonance with spin-parity $J^{P}=3/2^{-}$ is missing but only four negative-parity $\Sigma^{*}$’s are established. Only a small fraction of the predicted spectrum of positive-parity hyperon resonances is known. Clearly, our knowledge on the hyperon spectrum needs to be improved.

In this paper we study the possibility to find missing hyperon ($\Lambda^{*}$ and $\Sigma^{*}$) resonances in existing data. The paper is motivated by recent advances of coupled-channel partial-wave analyses of the existing data of $K^{-}p$ reactions Zhang:2013cua ; Zhang:2013sva ; Kamano:2014zba ; Kamano:2015hxa ; Kamano:2016djv ; Fernandez-Ramirez:2015tfa , and by the prospects of new data on hyperon spectroscopy from J-PARC Sako:2013prop , JLAB Adhikari:2017prop , and the forthcoming PANDA experiment Iazzi:2016fzb .

The pioneering work of Ref. Zhang:2013cua ; Zhang:2013sva reported the first coupled-channel partial wave analysis of most available data on Kaon induced reactions. In a first step Zhang:2013cua , the authors constructed the partial-wave amplitudes for the reaction $\bar{K}N\to\bar{K}N,\pi\Sigma,\pi\Lambda$ in slices of the invariant mass in the range from $W=\sqrt{s}=1.48$ GeV to 2.1 GeV. In a second step Zhang:2013sva , these partial-wave amplitudes were fit using a multichannel parametrization in the form $S=B^{T}RB=I+2iT$, where $T$ is the partial-wave $T$-matrix, $R$ a generalized multichannel Breit-Wigner matrix, and $B$ and its transpose $B^{T}$ are unitary matrices describing non-resonant background. In these energy-dependent fits, the partial-wave amplitudes of two-body reactions in sliced bins in the invariant mass were exploited as well as the results of partial wave analyses on the reactions $K^{-}p\to\pi\Lambda(1520),\pi\Sigma(1385)$, $K^{*}N$, and $\bar{K}\Delta$.

The partial-wave amplitudes from Ref. Zhang:2013cua were also used by the authors of Ref. Fernandez-Ramirez:2015tfa exploiting a $K$-matrix formalism where poles are described as conventional pole terms and background contributions by poles at negative values of $s$. The energy-dependent amplitudes from their fit described reasonably well the energy-independent partial-wave amplitudes from Ref. Zhang:2013cua even though some significant discrepancies can be seen. When the observables were calculated from the energy-dependent amplitudes Fernandez-Ramirez:2015tfa , severe discrepancies showed up.

Kamano et al. Kamano:2014zba ; Kamano:2015hxa fitted a similar set of data on the reactions $\bar{K}N\to\bar{K}N,\pi\Sigma,\pi\Lambda$, $\eta\Lambda$, $K\Xi$ and quasi-two-body final states as reported in Zhang:2013cua . They tried two different models $A$ and $B$ (containing different sets of resonances) and compared their amplitudes with the energy independent amplitudes of Zhang:2013cua . The three sets of amplitudes (from Zhang:2013cua and from models $A$, $B$ of Ref. Kamano:2014zba ), are consistent for the dominant partial waves but show larger discrepancies for smaller partial-wave amplitudes. It is hence not surprising that the spectrum of hyperon resonances obtained in the three analyses Zhang:2013sva ; Kamano:2015hxa ; Fernandez-Ramirez:2015tfa agree only in the leading contributions. Technically, it often remains open why one resonance is included in a fit and another one is not.

Already in 2000, the Gießen group Keil:2000wr applied modern analysis techniques to study the reactions $K^{-}p\to\pi\Sigma$ and to $\pi\Lambda$. The authors solved the Bethe-Salpeter equation in an unitary K-matrix approximation and fitted the partial-wave amplitudes derived in Gopal:1976gs . The measurements of total cross sections compiled in Baldini:1988lb were imposed as constraints. Masses, widths and partial decay widths of the leading resonances below 1700 MeV were determined.

New data in the low-mass region stimulated further investigations. A new Crystal Ball collaboration was formed at BNL – exploiting a detector that had originally been built to study the charmonium spectrum Gaiser:1982ua and which was then transferred to DESY to study a wide range of particle physics including two-photon collisions Bienlein:1992en . Subsequently, the detector was exploited for hadron spectroscopy at BNL Nefkens:2004ui and is presently used at MAMI in Mainz Beck:2005aj for photoproduction experiments.

At BNL, the Crystal Ball collaboration made significant contributions to the spectroscopy of low-mass hyperons. Several reactions were studied at eight incident $K^{-}$ momenta between 514 and 750 MeV. Differential and total cross sections and the hyperon polarization were reported for the reaction $K^{-}p\to\eta\Lambda$ Starostin:2001zz and $K^{-}p\to\pi^{0}\Lambda$, $K^{-}p\to\pi^{0}\Sigma$, and $K^{-}p\to\bar{K}^{0}n$ Prakhov:2008dc (see also Manweiler:2008zz ), $K^{-}p\to\pi^{0}\pi^{0}\Lambda$ Prakhov:2004ri , $K^{-}p\to\pi^{0}\pi^{0}\Sigma$ Prakhov:2004an . A new path to hyperon spectroscopy was opened in Refs. Moriya:2013eb ; Moriya:2013hwg ; Moriya:2014kpv .

This paper is part of a comprehensive study of the hyperon spectrum. In this paper, a fit is presented to (nearly) all available data on $K^{-}p$ induced reactions. The reactions used in the fit are shown in Table 1. The emphasis of this paper lies on two-body final states and on a determination of the hyperon resonances needed to achieve a good fit to the data. In Hyperon-III , data on three-body and quasi-two-body final states are discussed. In that paper, we present properties of hyperon resonances in detail and compare the resulting spectrum with the Bonn quark model Loring:2001ky . In Hyperon-I , we present a fit of low-energy data on $K^{-}$ induced reactions, including data on $K^{-}p$ at rest, and found that only one pole is required to describe the $\Lambda(1405)1/2^{-}$ region. For the threshold region, data on $K^{-}p$ properties at rest are also very important: The decay ratios ${\Gamma_{K^{-}p\rightarrow\pi^{+}\Sigma^{-}}}/{\Gamma_{K^{-}p\rightarrow\pi^{-}\Sigma^{+}}}$, ${\Gamma_{K^{-}p\rightarrow\pi^{0}\Lambda}}/{\Gamma_{K^{-}p\rightarrow\text{neutral}}}$, and ${\Gamma_{K^{-}p\rightarrow\pi^{\pm}\Sigma^{\mp}}}/{\Gamma_{K^{-}p\rightarrow\text{inelastic}}}$ were taken from Refs. Tovee:1971ga ; Nowak:1978au , The SIDDHARTA experiment at DA$\Phi$NE determined the energy shift and width of the 1S level of the kaonic hydrogen atom Bazzi:2011zj ; Bazzi:2012eq . These data proved to be very important for the study of the $\Lambda(1405)$ region Oller:2000fj ; Oset:2001cn ; Jido:2003cb ; Cieply:2009ea ; Ikeda:2012au ; Guo:2012vv ; Mai:2012dt ; Mai:2014xna ; Roca:2013av ; Roca:2013cca ; Cieply:2016jby ; Miyahara:2018onh which suggested the existence of two isoscalar poles in the $\Lambda(1405)$ region. In Hyperon-IV we present a coupled-channel partial wave analysis of CLAS data on $\gamma p\to K^{+}(\pi\Sigma)$ Moriya:2013hwg and $K^{+}(\bar{K}N)$ Dey:2014tfa and argue that photoproduction may offer new chances to study hyperon resonances.

The paper is organized as follows: First, in Section 2, a short outline of the BnGa partial-wave-analysis method is presented. In Section 3, we list the data used in this analysis. The search for new or less established resonances is described in Section 4. The results of the final fit are presented in Section 5. In Section 6, we suggest a possible star rating for the resonances used in the final fit. Our partial wave amplitudes for $\bar{K}N\to\bar{K}N$ and $\bar{K}N\to\pi\Sigma$ scattering in isospin $I=0$ and $I=1$ and for $\bar{K}N\to\pi^{0}\Lambda$ scattering are compared to those from other analyses in Section 7. The paper concludes in Section 8 with a short summary.

2 The BnGa partial wave analysis

2.1 The scattering amplitude

The general form of the amplitude for meson-baryon scattering can be written as

[TABLE]

where $IJN$ are isospin, total spin and “naturality”. $C_{I}$ are the Clebsch-Gordan coefficients which depend on the isospin of all particles in the process (including intermediate states). $A_{IJN}(s)$ are partial wave amplitudes, and the $Q_{JN}(s,t)$ tensors describe the angular dependent part of the partial wave amplitudes. The naturality is an alternative way to describe the parity; it is given by $N=(-1)^{(n+1)}P$ where $n$ corresponds to the rank of the partial wave, $J=n+1/2$. The angular dependent part is constructed from the decay vertices, polarization vectors and tensors which describe the structure of the particle propagators (see Anisovich:2004zz ). For example, the angular part for the scattering of a pseudoscalar meson and a $J^{P}=1/2^{+}$ baryon is given by

[TABLE]

Here, the $k_{1}$ and $q_{1}$ are the momenta of the initial and final-state baryons, and $k_{2}$ and $q_{2}$ are the momenta of the initial and final-state mesons; the relation $P=(k_{1}+k_{2})=(q_{1}+q_{2}),\,s=P^{2}$ holds true. The momenta $k^{\perp}$ and $q^{\perp}$ are relative momenta in the initial and final states orthogonal to the total momentum $P$:

[TABLE]

The baryons are described with bispinors:

[TABLE]

Here, $\omega$ represents a 2-dimensional spinor and $\omega^{*}$ the conjugated and transposed spinor, and we use the normalization condition

[TABLE]

The structure of the resonance propagator corresponds to a convolution of the polarization vectors of the resonances and have the following covariant form:

[TABLE]

where

[TABLE]

The $O^{\mu_{1}\ldots\mu_{n}}_{\xi_{1}\ldots\xi_{n}}$ describe the structure of the boson propagator for the particle with spin $J=n$ and are constructed from the metrical tensors orthogonal to the momentum of the resonance, see Eqn. (3). For the lowest spin states,

[TABLE]

For higher states, the operator can be calculated using the recurrent expression

[TABLE]

The operator $O^{\mu_{1}\ldots\mu_{n}}_{\nu_{1}\ldots\nu_{n}}$ provides the symmetry and traceless condition for the indices within one group and the orthogonality to the particle momentum. The structure of the $T^{\mu_{1}\ldots\mu_{n}}_{\nu_{1}\ldots\nu_{n}}$ operator is unique and is defined by the orthogonality condition to the $\gamma$-matrix and the normalization condition

[TABLE]

The vertex functions $N^{(\pm)}_{\alpha_{1}\ldots\alpha_{n}}$ describe the spin structure of a resonance decaying into a baryon with $J^{P}=1/2^{+}$ and a pseudoscalar meson. The decay orbital momentum is connected with the total spin as $J=L+\frac{1}{2}$ for the ’+’ states and as $J=L-\frac{1}{2}$ for the ’-’ naturality states.

[TABLE]

where $X^{(n)}_{\mu_{1}\ldots\mu_{n}}(k^{\perp})$ is the orbital momentum operator which depends on the relative momentum orthogonal to the momentum of the decaying particle:

[TABLE]

The operators $X^{(L)}_{\mu_{1}\ldots\mu_{L}}$ for $L\geq 1$ can be written in the form of the recurrence expression

[TABLE]

where

[TABLE]

In our calculations we use the $\gamma$-matrices in the standard representation

[TABLE]

In the c.m.s. of the reaction the scattering amplitude (1) can be rewritten as:

[TABLE]

where $\omega$ and $\omega^{\prime}$ are nonrelativistic spinors and $\vec{n}$ is a unit vector normal to the decay plane. The $F$-functions are defined as follows:

[TABLE]

The approach through the standard $G$ and $H$ functions is absolutely identical to the our covariant approach, the covariant approach allows us to construct naturally the amplitudes with multibody final states and to perform a combined analysis of all available data sets.

2.2 t and u-channel exchange amplitudes

Non-resonance contributions to the reactions are described by constants in the $K$-matrix (see below) and by amplitudes for $t$ and $u$-channel exchanges (see Fig. 1).

The Reggeized amplitudes for $t$-channel exchange are given by

[TABLE]

Here $\nu=\frac{1}{2}(s-u)$, $\alpha(t)$ is the intercept of the Regge trajectory, and $\xi$ is its signature. We found a significant contribution only from the exchange by the $\rho$, $\omega$ and $K^{*}$ vector-meson trajectories and the $f_{0}$, $a_{0}$ and $K^{*}_{0}$ trajectories of scalar mesons.

The $u$-channel amplitudes are described as the exchange of the corresponding baryon. We observe significant contributions only from exchange of nucleons in the $Kp\to\Lambda\pi$ and $Kp\to\Sigma\pi$ reactions. This exchange is represented by a spin-1/2 propagator

[TABLE]

2.3 The structure of the partial wave amplitudes

As the first step we parameterized the partial wave amplitudes as a sum of Breit-Wigner resonances:

[TABLE]

The total width of a resonance is equal to the sum of its partial widths:

[TABLE]

where $J=n+\frac{1}{2}$; $\rho_{j}^{n\pm}(s)$ is the imaginary part of the loop diagram with vertices given in Eq. (16). In the case of the pseudoscalar meson-baryon channel these functions are given by

[TABLE]

where $k_{j}=(k_{0},\vec{k})$ is momentum of the final-state baryon in the channel $j$ calculated in the resonance rest system. The Blatt-Weiskopf form factor ${B(n,|\vec{k}_{j}|^{2},R)}$ (with $R=0.8$ fm) is introduced to provide the correct asymptotic behavior of the phase volume at large energies.

The phase volumes for the three particle final states are described by the spectral integral which has all corresponding cuts and branching points in the complex plane of the total energy. The formulae for these functions are given in Anisovich:2006bc .

In the case of overlapping resonances the sum of Breit-Wigner amplitudes can violate the unitarity condition. In this case one can use the $K$-matrix approach which satisfies unitarity. Here the partial wave amplitudes are represented by a matrix with elements which describe the transition between channels $i$ and $j$ ($i,j=pK,\pi\Lambda,\pi\Sigma\ldots$) in the form

[TABLE]

where $\hat{\rho}$ is the diagonal matrix of the phase volumes. The $K$-matrix is parameterized as a sum of $K$-matrix poles and nonresonant contributions,

[TABLE]

This parameterization does not take into account rescattering effects described, e.g., by triangle diagrams. For baryons this effects exists even for two-particle final states since the pseudoscalar meson can be re-absorbed by the baryon. To take into account such effects, we substitute the $K$-matrix approach by the so-called $D$-matrix approach Anisovich:2011zz . Here the amplitude has the form:

[TABLE]

where the $D$-matrix is given by

[TABLE]

In the present analysis we describe the decay parameters of the D-matrix by

[TABLE]

The (small) phases take into account the contribution of multiparticle scattering diagrams.

3 The data and the primary fit

A detailed survey of the data that are used here in a coupled-channel analysis is presented in Tables 2 to 5. The tables list the reactions, the invariant masses at which measurements have been performed, and the references to the data. $N_{\rm data}$ gives the total number of data points for a reaction. In the last line of Table 2, e.g., there are differential cross sections at 20 angles and for 23 momenta giving $N_{\rm data}$=460. The $\chi^{2}$ for these data achieved in the primary fit (using only established hyperons and a relativistic Breit-Wigner description of the contributing resonances) and in the final fit are given in the last two columns.

Some data are slightly incompatible with other data in the normalization. The following scaling factors were applied in the fits:

[TABLE]

Polarization data are important for partial wave analyses but often limited in statistics. Also other low-statistics data like $K^{-}p\to K\Xi$ can be described rather badly without significantly affecting the total $\chi^{2}_{\rm tot}$. To avoid this, data are given a weight. The differential cross section for the final states $\eta\Lambda$, $\omega\Lambda$, and $K\Xi$ get a weight of 3, the corresponding polarization data a weight of 6. Also the data on the quasi-two-body reactions get a weight: 2 for $K^{*}p$; 3 for $\bar{K}\Delta(1232)$; 10 for $\pi\Lambda(1520)$. The polarization data for the $\pi\Lambda$ final state get a weight factor 2, those for $\pi\Sigma$ of 4. The weights are chosen to get a reasonable fit to these data without significantly distorting the fit to the other data sets.

Most data are limited to the region below $W=2.0$ GeV in the invariant mass; a few data extend the mass region up to $W=2.4$ GeV. We limit our study to resonances below 2.25 GeV. The data of Tables 2 and 3 include elastic and charge-exchange scattering, the inelastic channels with a $\Lambda$ hyperon in the final state produced with a $\pi^{0}$ or an $\eta$ meson, inelastic scattering into the three $\pi\Sigma$ charge states or into $\eta\Sigma$, and the production of cascade hyperons. Table 4 gives references to publications on the polarization observable $P$. For $K^{-}p\to K^{-}p$ elastic scattering, $P$ was measured by scattering off a polarized target. Hyperons in the final state reveal their polarization $P$ via the asymmetry of their decay.

Table 5 lists the data on $K^{-}p\to\Lambda\omega$ and on reactions with three particles in the final state, including quasi-two-body final states. In the low-mass region, we include the data on $K^{-}p\to 2\pi^{0}\Lambda$ Prakhov:2004ri and $K^{-}p\to 2\pi^{0}\Sigma$ Prakhov:2004an . For these data, the individual events are available, and we include the data event-by-event in an event-based likelihood fit. We also fit data on the quasi-two-body reactions $K^{-}p\to\omega\Lambda$ $K^{-}p\to\pi^{0}\Lambda(1520)$ Litchfield:1973ap ; Cameron:1977jr , $K^{-}p\to\pi^{\mp}\Sigma^{\pm}(1385)$ and $\rho\Lambda$ Cameron:1978en , $\bar{K}N\to\bar{K}^{*}N$ Cameron:1978qi , and $K^{-}p\to\bar{K}\Delta(1232)$ Litchfield:1973ey . Details on these reactions are shown in an accompanying paper Hyperon-III .

We performed four types of fits: one primary fit, a series of exploratory fits called mass scans, and the final fit. At the end, we performed fits in which the significance of the contributing resonances is estimated, and a series of error defining fits in which resonances are added in all contributing partial waves.

The primary fit to the data listed in Tables 2 to 5 used only those hyperons which were listed in the Review of Particle Properties (RPP) Tanabashi:2018oca with three or four stars. A list of hyperons used in the primary fit and the ranges of masses and widths is given in Table 6. The resonances are mostly well separated and Breit-Wigner parametrisations were used for the resonances. Masses, widths and coupling constants are allowed to vary within the limits quoted in the RPP.

Figure 2 shows the angular distributions expected for $J=3/2,5/2$, and $7/2$. They are identical for both parities. Forward-backward asymmetries come from the interference of even and odd waves. This figure can serve as a guide when the data are interpreted.

Figures 3-19 show a comparison of the data with our primary (dashed curves) and our final (solid curves) fit. Mostly, the two fits show hardly any difference: The established states serve as a rather good approximation. In these figures, no scaling factors are applied, neither to the data nor to the fit.

First, we compare Figs. 3, 5 with Figs. 6, 8. Note that both isospins can contribute to these reactions, with defined Clebsch-Gordan coefficients. Thus, $\Lambda^{*}$ and $\Sigma^{*}$ resonances both contribute with interfering amplitudes. The $K^{-}p$ and $\bar{K}^{0}n$ decay amplitudes of $\Sigma^{*}$ resonances have the same, of $\Lambda^{*}$ resonances opposite signs: this allows one to separate the $\Sigma^{*}$ and $\Lambda^{*}$ contributions from the data shown in Figs. 3 - 8. Similar arguments hold true for measurements of $K^{-}p\to\pi^{-}\Sigma^{+}$ and $\pi^{+}\Sigma^{-}$ or $K^{+}\Xi^{-}$ and $K^{0}\Xi^{0}$.

The cross sections for elastic scattering are significantly larger than those for charge exchange. At $W=1500$ MeV, the former is 2-4 mb/sr while the latter is in the order of 1 mb/sr. At 2000 MeV the elastic cross section rises in forward direction and reaches 10 mb/sr while charge exchange leads to an oscillating cross section staying below 1 mb/sr. Obviously, there are significant $t$-channel contributions to the elastic channel – likely due to Pomeron exchange – while $t$-channel contributions to the charge-exchange reaction need $\rho$ exchange. The impact of hyperon resonances is better seen in the charge-exchange data.

At the lowest mass – 1465 MeV – the charge-exchange cross section falls off slightly in forward direction; it is largely due to the dominant $S$-wave scattering with a small $P$-wave contribution. $S$-wave or $P$-wave alone would both lead to a constant angular distribution. At some masses, the reactions are dominated by one resonance. It may be helpful to compare the experimental angular distributions with the theoretical distributions (see Fig. 2) for a few mass intervals.

Figure 6 shows the angular distribution for $K^{-}p\to\bar{K}^{0}n$. At 1520 MeV, there is a clear ($3\cos^{2}\theta+1$) distribution above a very small background. A comparison with Fig. 2 shows that $J=3/2$ is the dominant wave: of course, it is the well-known $\Lambda(1520)$. With increasing mass, the $J=3/2$ contribution gets smaller and the minimum shifts without additional wiggles and without a strong forward-backward asymmetry. This pattern signals additional contributions from $J^{P}=1/2^{+}$ and $J^{P}=1/2^{-}$ waves.

At 1680 MeV, a sharp minimum is seen in the total $K^{-}p\to\bar{K}^{0}n$ cross section (see Fig. 22) which is assigned to a sign change of the amplitude in the $N\bar{K}$ S-wave amplitude at about this mass. $\Lambda(1670)1/2^{-}$ appears as a dip rather than as a peak (like $f_{0}(980)$ in $\pi\pi$ scattering). The effect is enhanced by the sudden rising of the $\Lambda(1690)3/2^{-}$ contribution. Interestingly, the $K^{-}p\to\eta\Lambda$ cross section rises from threshold to a peak value of above 0.1 mb/sr at 1670 MeV and has fallen below 0.03 mb/sr in the highest-mass bin at 1696 MeV.

In the subsequent energy bins of the $K^{-}p\to\bar{K}^{0}n$ differential cross sections, a stronger forward-backward asymmetry develops which indicates the interference of odd and even partial waves. Gradually, the angular distribution develops a strong w-shaped distribution, best recognized in the 1820 to 1850 MeV mass bins. The comparison with Fig. 2 suggests significant $J=5/2$ contributions. The partial wave analyses assigns this to a strong $\Lambda(1820)5/2^{+}$ production, the forward-backward asymmetry to a smaller $\Sigma(1775)5/2^{-}$ amplitude and some smaller contributions from lower partial waves. Above 2000 MeV, the angular distributions are characterized by forward and backward maxima with two additional maxima: In this mass region, $J^{P}=7/2^{\pm}$ are the most prominent partial waves (see Fig. 2). Similar observations can be made for the reaction $K^{-}p\to\pi^{0}\Lambda$.

The differential cross sections for the processes $K^{-}p\to\pi^{0}\Lambda$ and the $\Lambda$ polarization are shown in Figs. 8, 10, and 10, those for $K^{-}p\to\pi^{0}\Sigma^{0}$ and the $\Sigma^{0}$ polarization in Figs. 14 and 19. Only $\Sigma^{*}$ resonances contribute to $K^{-}p\to\pi^{0}\Lambda$, and only $\Lambda^{*}$ resonances to $K^{-}p\to\pi^{0}\Sigma^{0}$.

Inspecting Figs. 3, 5, 6, 8 again, we notice a few discrepancies between data and fit. Often, the discrepancies are enforced by data in neighboring bins: the structure at $\cos\theta=0$ and 1935 MeV in Fig. 5 is, e.g., incompatible with the neighboring bins. The same statements holds for the data from Barber:1975rg above 2200 MeV which are incompatible with those from Daum:1968jey and deBellefon:1976qr .

The polarization data are shown in Fig. 5. Polarization data are available in the full energy range considered here even though above 2 GeV with limited statistical significance only. For a single partial wave, the polarization vanishes. The complicated angular dependence of $P$ indicates the presence of several partial waves which makes a direct interpretation difficult. However, these data provide important constraints for the partial-wave analysis.

When the cross sections for $\pi^{-}\Sigma^{+}$ and $\pi^{+}\Sigma^{-}$ production in Figs. 11 and 12 are compared, similar effects as in $\bar{K}N$ production are seen. The cross sections for $\pi^{-}\Sigma^{+}$ are larger and the angular distributions at high energies show forward peaking. Note that for $t$-channel exchange, $\pi^{+}\Sigma^{-}$ production requires an exotic particle to be exchanged. It is not included in the fits and not required by the data. The contributing resonances are discussed using the data on $K^{-}p\to\pi^{+}\Sigma^{-}$.

The branching ratios for $\Lambda(1520)\to\Lambda\pi\pi$, $\Sigma\pi\pi$ and $\Lambda\gamma$ of (12$\pm$1)% are imposed as missing width; the $\Lambda(1520)\to\pi\Sigma$ decay fraction is then fixed fixed by unitarity ($\Gamma_{\rm tot}=\sum\Gamma_{i}$) when the data on elastic and charge exchange $K^{-}p$ scattering are used; these data span the mass range down to $\sim$1465 MeV.

The $\Lambda(1690)3/2^{-}$ can be recognized by the ($3\cos^{2}\theta+~{}1$) angular distribution. The resonance makes a very significant contribution, jointly with $\Lambda(1670)1/2^{-}$. Both are even partial waves, and the angular distribution remains approximately symmetric. At masses above 1800 MeV, the w-shaped angular distribution turns up again, signalling $\Lambda(1820)5/2^{+}$ but with a forward-backward asymmetry due an odd partial wave from $\Lambda(1830)5/2^{-}$.

Data on the reaction $K^{-}p\to\pi^{0}\Sigma^{0}$ were difficult to extract with the experimental techniques of the 70ties of last century: The $\Sigma^{0}$ decays to $\Lambda\gamma$; with $\pi^{0}\to\gamma\gamma$, there are three $\gamma$ in the final state. In London:1975zz , a bubble chamber was used that was filled with propane (C3H8) and freon (CF3Br) in which $\gamma$ ray have a high chance to convert. The $\gamma$ conversion probability depends on the $\gamma$ energy which was difficult to simulate. In view of these difficulties, the agreement between data and fit seems acceptable (see Fig. 14). The polarization $P$ for this reaction (see Fig. 19) was determined using the Crystal Ball detector at BNL Prakhov:2008dc with its excellent photon detection capability but with a limited energy range.

The first mass bin in Fig. 14 seems to suggest that there should be more $\Lambda(1520)$ than the fit admits. A larger $\Lambda(1520)$ contribution would, however, worsen the fit to the data on the $\pi^{\mp}\Sigma^{\pm}$ final states. In the 1650 to 1700 MeV mass range, a ($3\cos^{2}\theta+1$) contribution signals $\Lambda(1690)$ $3/2^{-}$ contributions above a small even partial wave. The fit additionally identifies $\Lambda(1670)1/2^{-}$. Polarization data exist only over a very limited range.

The low-energy region of the $K^{-}p\to\pi^{0}\Lambda$ differential cross section is dominated by the interference of even and odd partial waves, of $S$- and $P$-waves. At higher energies similar structures show up as we have seen them before: first a ($3\cos^{2}\theta+1$), later a $9/4\cdot$($5\cos^{4}\theta-2\cos^{2}\theta+1)$ angular distribution distorted by contributions from other waves. At masses above 2000 MeV, more wiggles show up. The fit is sensitive to the polarization data but their statistical value is limited again.

The differential cross sections for $K^{-}p\to\eta\Lambda$ are shown in Figs. 19. They do not show striking structures.

The structure in the $K^{-}p\to K^{+}\Xi^{-}$ differential cross section (Fig. 19) identifies leading contributions in the $J^{P}=3/2^{+}$ and $3/2^{-}$ partial waves, the assignment to the $\Lambda$ or $\Sigma$ sector follows from the $K^{-}p\to K^{0}\Xi^{0}$ channel (Fig. 19): $K^{0}\Xi^{0}\to\Sigma^{0}$ is forbidden. Description of the recoil asymmetry for both reactions is shown in Fig. 20. The two reactions were studied in a single-channel analysis Landay:2018wgf . Possibly contributing hyperon resonances were tested in a blindfold identification process. Ten resonances were suggested to contribute to the reaction, among them four 1* resonances for which we find no evidence in any final state. The strongest evidence is seen for $\Sigma(2030)7/2^{+}$ which we do not observe in this decay mode.

The authors of Ref. Horn:1972aj reported measurements of the charge exchange reaction and $\pi^{0}\Lambda$ formation. The results were expanded into associated Legendre polynomials:

[TABLE]

The data were included in our fits. The data are shown in Fig. 21. Significant structures are seen, in particular for $K^{-}p\to\bar{K}^{0}n$ at 1800 MeV where the interference of $\Lambda(1800)1/2^{-}$, $\Lambda(1820)5/2^{+}$, $\Lambda(1830)5/2^{-}$ leads to a complicated pattern. The largest contributions in $K^{-}p\to\pi^{0}\Lambda$ are at about 1700 MeV which are strongly influenced by $\Sigma(1620)$ $1/2^{-}$, $\Sigma(1660)1/2^{+}$, $\Sigma(1670)3/2^{-}$. With $S$, $P$, and $D$-waves, the Legendre coefficients up to $A_{4}$ show traces of these resonances while $A_{5}$, $A_{6}$ and $A_{7}$ stay small.

4 Mass scans

In the second step, we searched for new resonances and performed scans. In these exploratory fits, the resonances were described by relativistic Breit-Wigner amplitudes. States rated with three and four stars in the RPP were assumed to exist and used in all fits. In the mass scans, the masses of further states were scanned one by one in the corresponding mass regions. Figure 24 shows the first series of scans. The primary fit contained five positive and seven negative-parity $\Lambda$ resonances and four negative and four positive-parity $\Sigma$ resonances in the $J^{P}=1/2^{\pm},3/2^{\pm}$, $5/2^{\pm}$ and $7/2^{\pm}$ partial waves. In the scan, we added one resonance and varied its mass in steps. At each step, a full fit to the data was performed and the $\chi^{2}$ of the fit recorded. The $\chi^{2}$’s as functions of the imposed mass vary from fit to fit; sometimes a significant improvement is observed. The minimum defines a possible candidate for an additional resonance.

In the case of scan for an additional positive-parity $\Lambda$ resonance, the deepest minimum is seen in the scan of the $J^{P}=1/2^{+}$-wave. The minimum is rather broad and indicates that more resonances could be required. In further fits, no significant structure in this partial waves survives. The minimum in the $3/2^{+}$ wave is fake: the minimum is entirely due to the data on $K^{-}p\to\pi^{0}\pi^{0}\Lambda$ Prakhov:2004ri at the transition from one momentum to the next momentum. In the scan for a $J^{P}=7/2^{-}$ $\Lambda$ resonance, a narrow minimum is seen at about 1930 MeV which would be a rather low mass for a $7/2^{-}$-resonance. It is kept for further investigations, and is finally not confirmed. Furthermore, there are indications for a $J^{P}=3/2^{-}$ state at about 1870 MeV. In the scans for positive-parity $\Sigma$ resonances, we keep a $1/2^{+}$ at 1780 MeV and a $3/2^{+}$ at 2000 MeV for further investigations. The scan of the $J^{P}=1/2^{-}$-wave for $\Sigma$ resonances, a very significant minimum at 1670 MeV is seen even though a $\Sigma(1670)1/2^{-}$ resonance is already included in the primary fit. In the $5/2^{-}$-wave, a second minimum is seen at 2080 MeV. However, this minimum faded away in further studies.

These preliminary fits demonstrate that significant minima can be seen in fits but that the minima do not necessarily correspond to true physical states. For the analysis presented here, we performed several thousand fits with different hypotheses. At the end, we had a set of resonances which improved the $\chi^{2}$ by more than 400 units. This is called our final fit. These resonances will be discussed below (see Table 8).

A search for a further resonance – beyond those listed in Table 8 – led to residual fluctuations of less than 300 in $\chi^{2}$. Figure 24 shows the scans for positive and negative-parity $\Lambda$ and $\Sigma$ resonances. Some candidates could exist which improve $\chi^{2}$ by about 200 or more: There might be one broad or two unresolved $\Lambda$ candidates with $3/2^{-}$ in the 1800 to 2100 MeV range, or a $7/2^{-}$ and a $3/2^{+}$ $\Lambda$ candidate above 2300 MeV. There are indications for a $\Lambda$ $3/2^{+}$ candidate at 1800 MeV, a $\Sigma(2180)1/2^{+}$ and a $\Sigma$ candidate with $5/2^{+}$ at 1775 MeV and a $5/2^{-}$ $\Sigma$ candidate above 2100 MeV. The $\chi^{2}$ minimum in the $\Sigma$ $1/2^{-}$ scan at 1900 MeV might indicate that the description of $\Sigma(1900)1/2^{-}$ is not perfect. We do not consider any of these residual minima as statistically significant.

Figure 25 shows the final scans of two new $\Lambda$ states after all resonances have been identified. The mass scan of $\Lambda(2080)5/2^{-}$, e.g., shows a deeper minimum than the $\Lambda(2070)3/2^{+}$ state although the drop of the latter state has a more significant effect for the data description. $\Lambda(2070)3/2^{+}$ is significantly broader and it influences the via interference with other waves over a wider range. There is no clear minmum of $\Lambda(2070)3/2^{+}$ in any reaction: the global minimum is due to a $\chi^{2}$ function decreasing with mass for $K^{-}p\to\bar{K}N$ and increasing for $K^{-}p\to\pi\Sigma$. When this behavior is observed, we assign at most 1* to the resonance. The 3* and 4* resonance show clear minima for several final states.

5 The final fit

While the primary fit uses a Breit-Wigner description for the contributing resonances, the final fit uses the multi-channel K-/D-matrix-formalism described in Section 2. Figure 22 exhibits the total cross sections and the partial wave contributions determined in the primary and the final fit. The total cross sections are shown twice: the contributions from $\Lambda$ resonances are shown on the left figures, those from $\Sigma$ resonances on the right figures. Interferences between different partial waves – which play an important role in the analysis – do not contribute to the total cross section. Of relevance for the total cross section are on the other hand interferences of different isospin contributions in the same partial wave, which are not shown. Contributions from $t$- and $u$-channel exchanges are also not shown: This is the reason why, e.g., the sum of the resonant contributions for $K^{-}p$ elastic scattering is much less than the total cross section. The $\chi^{2}$ contributions from the individual data sets are listed in Tables 2 to 5. While the $\chi^{2}$’s for the primary fit are already acceptable, they are considerably improved when additional resonances are included. Some data are perfectly described in the final fit, other data contribute with a large $\chi^{2}$. One has, however, to have in mind that the data often are not fully consistent. Thus a $\chi^{2}$ of one per degree of freedom cannot be expected.

In Table 7 we compare our $\chi^{2}$’s with those obtained by the ANL-Osaka group. We use the same data but in several cases, our data set is slightly extended. Yet, for the polarization observable in $K^{-}p\to\pi^{0}\Sigma^{0}$, we use only one of two existing data sets Prakhov:2008dc ; Manweiler:2008zz . When both data sets were excluded from the analysis, the predicted polarization was close to the values from Prakhov:2008dc and disagreed with Manweiler:2008zz . Hence we decided not to use the latter data.

Due to a larger number of resonances, the BnGa fit achieves a better $\chi^{2}$ even though more data are used. The Kent Zhang:2013sva and Carnegie-Mellon Fernandez-Ramirez:2015tfa groups do not report the $\chi^{2}$’s achieved in their fits.

Finally, we give in Table 8 a list of the hyperon resonances which we use in the fits. One $\Lambda$ resonance is not listed. We find a very broad $\Lambda$ resonance with $J^{P}=1/2^{-}$ at 2230 MeV with a width of $\approx 450$ MeV. It is statistically significant; according to the criteria described in Section 6, it would be listed with three stars. Due to its large width, we think the resonance may hide a number of resonances with different spin and parities. Hence we do not include it in Table 8. A similar resonance shows up in the $\Sigma$ sector. The $\Sigma(2160)1/2^{-}$ may have a width of up to 600 MeV and may also cover the contribution of several resonances. It could, however, also have a width of 260 MeV; that is the reason that we include this resonance in the listings. But due to this uncertainty, we assign only one star to it even this its statistical evidence would justify three stars.

There are several one-star resonances for which we see no evidence underlying their weakness in kaon-induced reactions. Most four and three-star resonances are confirmed.

Compared to our expectation of seven $\Lambda^{*}$ and seven $\Sigma^{*}$ resonances with negative parity and below $\approx 1900$ MeV, we were missing one resonance in both sectors. The expected $3/2^{-}$ companion of $\Lambda(1800)1/2^{-}$ and $\Lambda(1830)5/2^{-}$ is definitely not seen, see the discussion in Section 4. In the $\Sigma^{*}$ sector, a low-mass partner of $\Sigma(1670)3/2^{-}$ with $J^{P}=1/2^{-}$ is missing. There is a 1* candidate $\Sigma(1620)1/2^{-}$. If we include this in the fit, the $\chi^{2}$ gain is 386, just below our limit at 400 (see below). Thus we keep it as 1* resonance. Its mass of 1681 MeV is unexpectedly close to $\Sigma(1750)1/2^{-}$ which we find at 1692 MeV. We notice that the K-matrix poles of the three low-mass $\Sigma^{*}$ resonances with $J^{P}=1/2^{-}$ are 1610, 1695, and 1900 MeV. In the nucleon sector, it had turned out that the helicity amplitudes for photoexcitation of $N(1535)1/2^{-}$ and $N(1650)1/2^{-}$ off protons and neutrons determined at the K-matrix pole Anisovich:2015tla agree very well with expectations of the Single-Quark-Transition-Model Burkert:2002zz and are at variance with SU(3) relations at the T-matrix poles Boika:2014aha .

6 Hyperon resonances and their star rating

The final fit converged with a $\chi^{2}$ minimum at $\chi^{2}=40615$. Table 8 lists the resonances used in the fit, their masses and widths. From the full list of resonances used to fit the data we removed individual resonances one by one. The new fit with readjusted parameters deteriorated, the increase in $\chi^{2}$ is used to estimate the significance of the resonance. When some dominant 4* resonances were removed, the fit became very bad. In these cases we did not try to improve the fit but just left these resonances with their 4* rating.

In the RPP - not counting $\Lambda$(1116), $\Sigma$(1193), and $\Sigma$(1385), there are at present 12 $\Lambda$ and $\Sigma$ resonances with 4*’s. We use these resonances to define criteria to estimate the star rating of resonances. In our analysis, the least significant 4* resonances are $\Lambda(1830)5/2^{-}$ and $\Sigma(1915)5/2^{+}$; when removed they led to an increase in $\chi^{2}$ of 1790 or 2002, respectively. Since we do not wish to drastically alter the criteria for the star rating, we assign a 4* rating to resonances for which a $\chi^{2}$ change of more than 2000 is observed. Thus 11 of the 12 4*-star resonances kept their star rating. We defined 400 as the minimum $\chi^{2}$ change to accept a resonance with 1*. The ratings are thus defined by

[TABLE]

We do not observe any evidence for the low-mass $\Sigma$ “bumps” at 1480, 1560, 1620, 1670, or 1690 MeV, observed in production experiments, and no evidence for the 1* $\Sigma(1580)3/2^{-}$ in agreement with the authors of Ref. Olmsted:2003is . Below 2200 MeV, we find no evidence for the 1* resonances $\Sigma(1620)$ $1/2^{-}$, $\Sigma(1730)3/2^{+}$, $\Sigma(1770)1/2^{+}$, $\Sigma(1840)3/2^{+}$, $\Sigma(1940)3/2^{+}$, $\Sigma(2000)1/2^{-}$, $\Sigma(2070)5/2^{+}$, and no evidence for the 2* $\Sigma(1880)1/2^{+}$ and $\Sigma(2080)3/2^{+}$. We do not observe the 1* resonances $\Lambda(1710)1/2^{+}$, $\Lambda(2000)$, $\Lambda(2020)7/2^{+}$, and $\Lambda(2050)3/2^{-}$. $\Lambda(1810)1/2^{+}$ is the only 3* resonance for which we find no evidence. If we include it, the $\chi^{2}$ improves by 106 units only, and mass and width are fitted to $M$=1773$\pm$7; $\Gamma$= 39$\pm$15 MeV. Two resonances above 2230 MeV are included in the analysis but are not included in Table 8.

The changes in star rating suggested here are collected in Table 9.

The most significant changes are suggested for resonances which so far had a 1* or 2* rating. Most of them are not seen here. The changes suggested for 3* and 4* resonances are moderate. Five new 1*-resonances are suggested.

In most cases, the fits finds a minimum within the mass-boundaries of the RPP, except for four exceptions: We find $\Sigma(1620)1/2^{-}$ at $1681\pm 6$ MeV and $\Sigma(1750)1/2^{-}$ at $1692\pm 11$ MeV, these two resonances are very close in mass and the evidence for the lower mass state is weak as discussed in the previous section. $\Sigma(1940)3/2^{-}$ is found at $1878\pm 12$ MeV. In the $3/2^{-}$ partial waves, a new resonance is found, and it is not surprising that the masses of known resonances are shifted. In addition the $\Lambda(1405)$ is found at $1420\pm 3$ MeV. This state is discussed in detail in Hyperon-I .

Finally, we made error defining fits. We chose solutions with an additional resonance and a local minimum as shown in Figs. 24. From the spread of results we estimated the errors given to masses, widths and other properties (see Hyperon-III ).

7 Partial wave amplitudes

Our partial wave amplitudes are not derived from energy-independent fits (i.e. from fits in slices of energy). Measurements of the spin-rotation parameters – needed for a truly energy-independent analysis – do not exist. The Kent group succeeded nevertheless to construct the amplitudes by first determining the leading waves and then defining the smaller ones. Our amplitudes are determined from energy-dependent fits to the differential cross sections and polarization data. As discussed above, we have made numerous fits, in particular also a large number of fits with different resonance contents. Some additional poles led to a small $\chi^{2}$ reduction, of less than 400. For these fits, all partial wave amplitudes were determined as well. Thus, we derived a set of partial wave amplitudes which all are about consistent with the data. The spread of these results were used to determine a band for each partial wave amplitude.

Figures 26 and 27 show the real and imaginary parts of the partial wave amplitudes for $\bar{K}N$ elastic scattering in the two isospin channels, for the isospin 1 $\bar{K}N\to\pi\Lambda$ scattering and for the two isospins in $\bar{K}N\to\pi\Sigma$ determined in this analysis and in Refs. Zhang:2013cua and Kamano:2014zba . Giving the limitations of the data, the comparison shows reasonable consistency, at least for the leading contributions. The partial waves are very similar in all four solutions where strong resonances like $\Lambda(1520)3/2^{-}$, $\Lambda(1820)5/2^{+}$, $\Lambda(1830)5/2^{-}$, $\Sigma(1670)3/2^{-}$, $\Sigma(1775)5/2^{-}$, $\Sigma(1915)5/2^{+}$, or $\Sigma(2030)7/2^{+}$ dominate the partial waves. For $\Lambda(1600)$ $1/2^{+}$, the imaginary part is similar in all four analyses while the real part of analysis Zhang:2013cua deviates. The $\Lambda$ $1/2^{-}$ low-mass amplitudes for $\bar{K}N\to\pi\Sigma$ from Zhang:2013cua and Kamano:2014zba are consistent but inconsistent with our findings. However, a significant $\Lambda(1670)1/2^{-}$ structure is seen in all analyses.

Above the lowest-mass resonance, the structure of the amplitudes shows significant differences. This is to be expected since the resonance content of the four analyses is different. To resolve these discrepancies, new data are likely mandatory.

8 Summary

We have collected existing data on hyperon formation in $K^{-}p$ elastic and inelastic scattering. The data were fitted in a coupled-channel analysis within the BnGa framework. We looked systematically for contributing resonances in a large number of fits and mass scans. The statistical significance of all resonances was evaluated. We find five new resonances; some resonances are suggested to be upgraded others to be downgraded. For eighteen resonances – mostly listed as 1* or 2* resonances – we did not find any signature. The partial-wave amplitudes derived in our fits are compared to those from other analyses.

Acknowledgements

We are very grateful to Sergey Prakhov for providing to us the Crystal Ball data on the $K^{-}p\to\pi^{0}\pi^{0}\Lambda$ and $K^{-}p\to\pi^{0}\pi^{0}\Sigma$ reactions. Comments by D. M. Manley and H. Kamano are kindly acknowledged. This work was supported by the Deutsche Forschungsgemeinschaft (SFB/TR110) and the Russian Science Foundation (RSF 16-12-10267).