82. Λ and Σ Resonances

182. Λ and Σ Resonances
Revised August 2021 by V.D. Burkert (Jefferson Lab), V. Crede (Florida State U.), E. Klempt(Bonn U.), U. Thoma (Bonn U.), L. Tiator (KPH, JGU Mainz) and R.L. Workman (GeorgeWashington U.).

82.1 Introduction
For several decades, there has been very little new experimental data bearing on the propertiesof Λ and Σ resonances. An exception was the study at JLab of the reactions γp → K + Σ ± π ∓and γp → K + Σ 0 π 0 [1], which established the spin and parity of the Λ(1405) [2]. There was also
from BNL new data on the very low energy region of K − p scattering [3–7]. Otherwise, the field isstarved for data. Recent analyses (see below) have improved what we know about the properties of
the known Λ and Σ resonances, but the established resonances are the same ones that were listed
in our 1984 edition [8] except for the Σ(2250) which we consider 2-star only due to its unkown
spin-parity. The 1990 Review [9] gave a full report of the status of Lambda and Sigma hyperons,
and included Argand plots from the partial-wave analyses. The 2018 Review [10] has a short surveyof the Σ(1670)-region.
In the last few years, four groups have re-analyzed K − p reactions using more extensive collections of the old data. These analyses justify an update of the status of the Λ and Σ resonances.
Although they have not established any new resonances, they have provided at least some evidence
for new states and have given a better understanding of the old ones.
Tables 82.1 and 82.2 give our evaluation of the status, both overall and channel by channel, ofeach Λ and Σ resonance in the Particle Listings. In making these evaluations, we considered, inaddition to the four analyses [11–14], the ratings that predated them. The ratings use a 1- to 4-starsystem. For more details on the evaluation of the overall star ratings, see [15]. The main Summary
Table includes only established states with an overall status of 3 or 4 stars; as has already been
noted, these are the same fourteen Λ resonances (including Λ(1116)) and nine Σ resonances insteadof the former ten (including Σ(1193)3/2+ , and Σ(1385)3/2+ ) that had long been in the Table. Inaddition, there are seven 1-star and two 2-star Λ’s, and fourteen 1-star and three 2-star Σ’s in theParticle Listings.

82.2 New analyses
The new analysis progress was pioneered by the Kent group which collected a large fraction of
the available data and performed a comprehensive partial wave analysis [11, 16]. K − p scattering
into a pseudoscalar meson and an octet baryon is governed by two complex amplitudes; hence fourquantities need to be measured to fully construct the amplitudes (up to an arbitrary phase perenergy and angular bin). Discussions of complete experiments also generally assume perfect data(no experimental uncertainties); realistic uncertainties further complicate the task of amplitude
extraction. Here, the available data are limited to the differential cross section and the targetor hyperon recoil polarization P ; data on the polarization transfer do not exist. The authors ofRef. [16] overcame this difficulty by using start values for the partial wave amplitudes determinedin [17] and/or from an energy-dependent fit and by freezing or releasing sets of amplitudes. The
resulting amplitudes were fitted with a unitary multichannel parameterization [11].
The JPAC group presented a coupled-channel fit to the K̄N partial waves derived by the Kentgroup [13]. The JPAC approach was based on the K-matrix formalism. Special attention waspaid to the analytical properties of the amplitudes determined by the square-root unitary branch
points and the continuation to the complex angular momentum plane. The fit described the Kent
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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82. Λ and Σ Resonances2

partial waves reasonably well. However, when observables were calculated from their partial-wave
amplitudes, significant discrepancies became apparent. The results were therefore not included inthe RPP.
The ANL-Osaka group derived the energy-dependent amplitudes in fits to a large subset ofthe data collected in Ref. [16] and further data sets described in Ref. [18]. Their fits were basedon a phenomenological SU(3) Lagrangian [18]. The two ANL-Osaka models agree on the leading
contributions but differ significantly in cases with weaker candidates [12].
Table 82.1: The status of the Λ resonances. Only those with an overall
status of ∗∗∗ or ∗∗∗∗ are included in the main Baryon Summary Table.
Decay channels other than N K and Σπ are only given for ∗∗∗ and ∗∗∗∗resonances.

ParticleΛ(1116)
Λ(1380)Λ(1405)
Λ(1520)Λ(1600)Λ(1670)Λ(1690)Λ(1710)Λ(1800)Λ(1810)Λ(1820)Λ(1830)Λ(1890)Λ(2000)Λ(2050)Λ(2070)Λ(2080)Λ(2085)
Λ(2100)Λ(2110)Λ(2325)Λ(2350)Λ(2585)

JP
1/2+
1/2−
1/2−
3/2−
1/2+
1/2−
3/2−1/2+1/2−
1/2+5/2+5/2−3/2+1/2−
3/2−
3/2+5/2−
7/2+7/2−
5/2+3/2−
9/2+

Overallstatus∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

Status as seen in —NKΣπ∗∗∗∗∗∗∗∗∗∗∗∗∗
∗∗∗∗
∗∗∗∗∗
∗∗∗
∗∗
∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
∗∗
∗∗∗∗∗

∗∗
∗∗∗∗
∗∗∗∗∗∗∗∗
∗∗∗∗
∗∗∗
∗
∗∗
∗∗
∗∗∗∗
∗∗∗∗
∗∗
∗
∗
∗
∗
∗
∗∗
∗∗

Other channels
N π (weak decay)Λππ, Λγ, Σππ
Λππ, Σ(1385)πΛηΛππ, Σ(1385)π∗Λππ, N K∗NK
Σ(1385)πΣ(1385)π∗Σ(1385)π, N K

∗
NK
∗
NK

∗

The Bonn-Gatchina (BnGa) group added further (old) data to those analyzed in Ref. [16]. The
data set was fitted in a modified K-matrix approach and the resulting amplitudes were comparedwith those from Refs. [16, 18]. New resonances were found, other states, mostly one and twostar states could not be confirmed; all resonances were tested for their statistical significance.
Additional states with any set of quantum numbers were tested and were found to produce only
small improvements in the fit [14]. In Ref. [19], properties of the full set of contributing hyperons
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82. Λ and Σ Resonances

3Table 82.2: The status of the Σ resonances. Only those with an overall
status of ∗∗∗ or ∗∗∗∗ are included in the main Baryon Summary Table.
Decay channels other than N K, Λπ and Σπ are only given for ∗∗∗ and∗∗∗∗ resonances.

ParticleΣ(1193)
Σ(1385)
Σ(1580)Σ(1620)Σ(1660)Σ(1670)Σ(1750)Σ(1775)
Σ(1780)Σ(1880)Σ(1900)Σ(1910)Σ(1915)
Σ(1940)Σ(2010)Σ(2030)Σ(2070)Σ(2080)Σ(2100)Σ(2110)Σ(2230)Σ(2250)Σ(2455)
Σ(2620)Σ(3000)Σ(3170)JP
1/2+3/2+3/2−1/2−1/2+3/2−1/2−5/2−3/2+1/2+1/2−3/2−5/2+3/2+3/2−
7/2+5/2+3/2+7/2−1/2−3/2+

Overallstatus∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
∗
∗∗∗∗
∗∗∗
∗

Status as seen in —NK∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
∗∗
∗∗∗∗
∗
∗
∗∗∗∗∗
∗
∗
∗∗∗
∗
∗∗

ΛπΣπ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

Other channels
N π (weak decay)ΛγΣη∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∆(1232)K, N K , Σ(1385)π∗∗∗∗∗were reported.
The star ratings of Λ and Σ resonances given in our earlier editions, and the new results from theKent, ANL-Osaka and BnGa groups were used to update the star rating of the hyperon resonances.
In [19], the overall star rating is directly estimated, for [11] we estimate the star rating from thebranching-ratio uncertainties. In [12], two solutions are given but no uncertainties for branchingratios. The overall star ratings are based on the evidence for the resonances in the new analyses
as well as their consistency including also the results of earlier analyses given in [20]. For furtherdetails see also [15].
We decided to remove the three so-called bumps Σ(1480), Σ(1560), Σ(1670), and Σ(1690) as
well as Σ(1620) from production experiments. The entries from Σ(1770)1/2+ are now listed underΣ(1660)1/2+ and Σ(1880)1/2+ , the entries from Σ(1730)3/2+ and Σ(1840)3/2+ are now combined
to one Σ(1780)3/2+ . The one-star Σ(2000)1/2− is combined with a new one-star Σ(2160)1/2− to a
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82. Λ and Σ Resonances4

single one-star Σ(2110)1/2− . Apart from Λ(1380)1/2− , four further new resonances were includedin the Listings: Λ(2070)3/2+ , Λ(2080)5/2− , Σ(2010)3/2− , and Σ(2230)3/2+ , all with one star.

82.3 Sign conventions for resonance couplings
In terms of the isospin-0 and isospin-1 elastic scattering amplitudes A0 and A1 , the amplitude
0for K − p → K n scattering is ±(A1 − A0 )/2, where the sign depends on conventions used inconjunction with the Clebsch-Gordan coefficients (such as, is the baryon or the meson the “first”particle). If this reaction is partial-wave analyzed and if the overall phase is chosen so that, say, theΣ(1775)5/2− (D15 ) amplitude at resonance points along the positive imaginary axis (points “up”),
then any Σ at resonance will point “up” and any Λ at resonance will point “down” (along thenegative imaginary axis). Thus the phase at resonance determines the isospin. The above ignoresbackground amplitudes in the resonating partial waves.

{10}

{8}
{8}Λ (1670) Λ (1690)

Σ (1385)P13

S01{8}{8}Λ (1820) Λ (1830)

D03

F05{10} {1}ΛΣ (2030) Λ (2100)

D05

F17G07

ΣπS01

D03

D13{1}{1}{8}

Λ (1405) Λ (1520) Σ (1670)

S11

D15{8}{8}

S11D15

Σ (1750) Σ (1775){8} {8}Σ (1750) Σ (1775)

F15

Σ (1915){8}{8}{10}

F15

F17

Σ (1915) Σ (2030)

ΛπP13

Σ (1385){10}

D13

Σ (1670){8}

Figure 82.1: The signs of the imaginary parts of resonating amplitudes in the KN → Λπ andΣπ channels. The signs of the Σ(1385) and Λ(1405), marked with a •, are set by convention, andthen the others are determined relative to them. The signs required by the SU(3) assignments of
the resonances are shown with an arrow, and the experimentally determined signs are shown withan ×.
That is the basic idea. In a similar but somewhat more complicated way, the phases of theKN → Λπ and KN → Σπ amplitudes for a resonating wave help determine the SU(3) multiplet
to which the resonance belongs. Again, a convention has to be adopted for some overall arbitrary
phases: which way is “up”? Our convention is that of Levi-Setti [21] and is shown in Fig. 82.1, which
also compares experimental results with theoretical predictions for the signs of several resonances.
In the Listings, a + or − sign in front of a measurement of an inelastic resonance coupling indicatesthe sign (the absence of a sign means that the sign is not determined, not that it is positive).
Also other decay modes can be used to assign a hyperon to a SU(3) multiplet [22, 23]. Modern
analyses determine properties of resonances at the pole position. In these analyses, the + or −sign is replaced by a phase. Background amplitudes can lead to significant phase shifts, and anadditional phase shift due to rescattering is admitted in some analyses. In comparison to quark
model predictions [19, 24], three Λ spin doublets can be identified as being mainly SU(3) singlets:
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582. Λ and Σ Resonances

the well-known (Λ(1405)1/2− , Λ(1520)3/2− ), the (Λ(2080)5/2− , Λ(2100)7/2− ), and (Λ(2070)3/2+ ,Λ(2110)5/2+ ).

82.4 The Λ(1405)In coupled-channels calculations based on the chiral SU(3) effective field theory, the strongly
attractive forces between N K and Σπ generate five poles, one SU(3) singlet pole, two Λ octet
poles and two (or one) octet Σ poles (see Section 83). In quark models, these five states areΛ(1405), Λ(1670), Λ(1800), Σ(1620), and Σ(1750). The octet states are found 100 to 150 MeV
above the corresponding nucleon resonances. In chiral SU(3) effective field theories, at least threeof these states are seen in the 1300 to 1600 MeV mass range. The appearance of two Λ poles inthis mass range, a narrow SU(3) octet at ∼1420 MeV and a wider SU(3) singlet at ∼1380 MeV, wasunexpected. This approach has been pursued by a number of groups; for a summary of the results
see our Review 83, “Pole Structure of the Λ(1405) Region”. In the Listings, we have introduced theΛ(1380) as a new candidate resonance (with two stars), named in accordance with its approximate
pole position. The second SU(3) octet Λ state is the well-known Λ(1670). The masses of the twoassociated Σ states are uncertain so far, and no new entries are introduced in the Listings.
In traditional approaches only one resonance was seen in this mass region, the narrow state at1405 MeV. It was reported to be the SU(3) singlet state a long time ago in Ref. [25], in agreementwith the quark-model expectations but in contrast to the findings based on coupled-channels calculations within chiral SU(3) effective field theories. In the Listings, the Λ(1405) has been retainedwith its traditional name. In quark models, this state is identified with the SU(3) singlet state, thetwo Λ octet states with Λ(1670) and Λ(1800), and the two Σ states with Σ(1620) and Σ(1750).

82.5 Uncertainties on masses and widths
The uncertainties quoted on resonance parameters from partial-wave analyses are often only
statistical, and the parameters can change beyond these uncertainties when a different parametrization of the waves is used. Furthermore, the different analyses use more or less the same data, so
it is not really appropriate to treat the different determinations of the resonance parameters as
independent or to average them. In any case, the spread of the masses, widths, and branching
fractions from the different analyses is certainly a better indication of the uncertainties than are
the quoted uncertainties. In the Listings, we usually give a range reflecting the spread of the values
rather than a particular value with error.

82.6 Production experiments
Partial-wave analyses of course separate partial waves, whereas a peak in a cross section or aninvariant mass distribution usually cannot be disentangled from background and analyzed for its
quantum numbers; and more than one resonance may be contributing to the peak. The Σ(1385)
and Λ(1405) lie below the KN threshold and nearly everything about Σ(1385) is learned fromproduction experiments. Our knowledge on Λ(1405) benefits greatly from photoproduction of thethree Σπ charge states [1, 2] and from the precise measurement of the energy shift and width of
the kaonic hydrogen atom [26].
Production and formation experiments agree quite well in the case of Λ(1520) and results havebeen combined. Above this mass, no new results on peak hunting have been reported for about 40years. For these early results, we refer the reader to our earlier editions. In photoproduction withenergetic photons [27, 28] or at LHCb [29], hyperons are produced abundantly. So far, no attempt
has been made to extract hyperon properties from these data. New data on hyperon spectroscopy
can be expected from J-PARC [30], JLAB [31], and the forthcoming PANDA experiment [32].
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