Mixing and decays of the antidecuplet in the context of
approximate SU(3) symmetry
Vadim Guzey (Bochum) and M.V. Polyakov (Liege)
1. Motivation
2. Antidecuplet mixed with three octets: General expressions
for 10 partial decay widths
3. Emerging picture of N10 and Σ10 decays
4. Discussion (influence of the mixing with 27-plet, ideal mixing)
and conclusions
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
V. Guzey
Motivation
• Approximate flavor SU(3) symmetry of strong interactions
allows to group all hadrons into certain multiplets. Only
singlets, octets (8) and decuplets (10) were believed to be
realized in Nature.
The discoveries of the Θ+ and Ξ−− , if confirmed, mean the
existence of a new physical multiplet: the antidecuplet (10).
• Approximate SU(3) symmetry works surprisingly well: Mass
splittings and partial decay widths of all baryon multiplets
(singlets, octets, decuplets) are described and predicted with
good accuracy: Gell-Mann and Ne’eman 1964; Kokkedee 1969;
Samios, Goldberg, Meadows 1974.
• Approximate SU(3) suits best for establishing in a modelindependent way the overall structure of a given SU(3)
multiplet: Mass splittings, necessity of mixing with other
multiplets due to SU(3) breaking, correlations between partial
decay widths.
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
V. Guzey
• This is exactly what one needs for the antidecuplet: a reliable
overall picture of 10 and its mixing with other multiplets and
a way to systemize the present experimental info on the 10
decays.
• Since SU(3) is broken, states from different multiplets with
the same spin and parity can mix. Because of the small width
of Θ+, even small mixing dramatically affects predictions for
the 10 decays.
At the same time, small mixing with 10 affects very little
non-exotic multiplets. This means that one can use the
results of SU(3) analysis of the non-exotic multiplets (three
octets in our case) in the SU(3) analysis of 10 decays.
• After the SU(3) picture of 10 is established using the scarce
experimental info on 10 decays, one can make modelindependent predictions for unmeasured decays and assess
available models of the 10 mixing.
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
V. Guzey
Antidecuplet mixing with three octets
We consider the scenario that 10 mixes with three J P =
1/2+ octets:
the ground-state octet,
the octet containing N (1440) Λ(1600), Σ(1660) and Ξ(1690),
the octet containing N (1710), Λ(1800), Σ(1880) and Ξ(1950).
The mixing takes place through the N10 and Σ10 and the
corresponding N and Σ octet states:





|N1phys i
|N2phys i
|N3phys i
phys
|N10
i






=


1
0
0
− sin θ1
0
1
0
− sin θ2
0
0
1
− sin θ3
sin θ1
sin θ2
sin θ3
1

|N1i

  |N2i

  |N3i
|N10i
• We assume that θi mixing angles are small, θi = O(), where
is a small parameter of SU(3) breaking. We systematically
neglect O(2) terms.
• The |N1i, |N2i and |N3i states can mix among themselves,
i.e. they can belong to several different octets. Using the
χ2 fit to the measured decays, we find that |N2i and |N3i
states are slightly mixed (it is legitimate to neglect 10 at this
stage.
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
V. Guzey





After this is taken into account, it is sufficient to consider
phys
i.
only the mixing of each individual |Niphysi with |N10
• The mixing angles θi and θiΣ are related,
phys
phys
phys
phys
Σ
− Σ10
− N10
sin θi Ni
= sin θi Σi
,
which becomes θi = θiΣ ignoring O(2) terms.
• Gell-Mann–Okubo mass formulas, which describe the mass
splitting inside SU(3) multiplets, are not sensitive to small
mixing
phys
Ni
phys
≡ hNi
phys
|M̂ |Ni
2
2
i = Ni + sin θiN10 = Ni + O( ) ,
It is not legitimate to estimate the mixing angles from the
Gell-Mann–Okubo mass formula.
Instead, one has to consider decays which contain both O(1)
and O() terms.
• We assume that SU(3) symmetry is violated by non-equal
masses inside a given multiplet and mixing and that SU(3) is
exact in decay vertices → finite number of universal SU(3)
coupling constants.
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
V. Guzey
General expressions for 10 couplings: ΓΘ+ and G10
In our analysis, ΓΘ+ and Σπ N are external parameters,
which are varied in the following intervals: 1 ≤ ΓΘ+ ≤ 5 MeV;
45 ≤ Σπ N ≤ 75 MeV.
Σπ N determines the θ1 mixing angle with the ground state
octet; ΓΘ+ determines the G10 and H10 (H10 = 2 G10 − 18)
coupling constants using
G10
Praszalowicz, hep-ph/0402038; R.A. Arndt et al., PRC 69 (2004) 035208
√ !
1
5
gΘ + N K = √
G10 + sin θ1H10
4
5
10
8
6
4
2
0
-2
-4
ΓΘ+=1 MeV
-6
ΓΘ+=3 MeV
ΓΘ+=5 MeV
-8
-10
45
50
55
60
65
70
75
ΣπN, MeV
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
V. Guzey
General expressions for N10 couplings

1
gN N π = √ G10 + sin θ1
2 5
10

1
gN N η = √ −G10 + sin θ1
2 5
10
√
7
5
H10
− G8 √
4
5
√
!
5
1
H10
− G8 √
4
5
−
!
X
i=2,3
+
X

sin θi gN N π  ,
i
i=2,3


sin θi gN N η  ,
i

X
1
4
sin θi gN Λ K  ,
gN Λ K = √ G10 + sin θ1 G8 √ +
i
2 5
5
10
i=2,3


X
2
sin θi gN ∆ π 
gN ∆ π = √ sin θ1 G8 +
i
5
10
i=2,3
• The gNiB P coupling constants are determined by the χ2 fit
to the measured decays of the octets; the θ2,3 are left as free
parameters.
• Important correlation: Mixing with the octets can decrease
gN10N π and simultaneously increase gN10N η .
• The N10∆ π decay is possible only due to mixing.
• The partial decay widths are found from
M2
|~
p|3
Γ(B1 → B2 + P ) = 3|gB1B2P |
2π(M1 + M2 )2 M1
2
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
V. Guzey
Part. decay width Γ(N → N π), MeV
*
ΓΘ+=1 MeV, ΣπN=45 MeV
35
30
25
20
15
10
5
ΣπN=75 MeV
90
80
70
60
50
40
30
20
10
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.1
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
sin θ 2
-0.1
0
ΓΘ+=5 MeV, ΣπN=45 MeV
0.1
0.2
sin θ 2
ΣπN=75 MeV
20
17.5
15
12.5
10
7.5
5
2.5
60
50
40
30
20
10
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.1
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
sin θ 2
-0.1
0
Part. decay width Γ(N → N η), MeV
0.1
0.2
sin θ 2
*
ΓΘ+=1 MeV, ΣπN=45 MeV
2.4
ΣπN=75 MeV
5
4.8
4.6
4.4
4.2
4
3.8
3.6
2.2
2
1.8
1.6
1.4
0.2
0.2
0.1
sin
0
θ -0.1 -0.2
3
-0.2
-0.1
0
0.1
0.2
0.1
sin θ 2
sin
0
θ -0.1 -0.2
3
-0.2
-0.1
0
ΓΘ+=5 MeV, ΣπN=45 MeV
0.1
0.2
sin θ 2
ΣπN=75 MeV
5.4
5.2
5
4.8
4.6
4.4
4.2
4
3.8
8
7.75
7.5
7.25
7
6.75
6.5
6.25
6
0.2
0.2
0.1
sin
0
θ -0.1 -0.2
3
-0.2
-0.1
0
0.1
0.2
sin θ 2
0.1
sin
0
θ -0.1 -0.2
3
-0.2
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
-0.1
0
0.1
0.2
sin θ 2
V. Guzey
Part. decay width Γ(N → ΛK), MeV
*
ΓΘ+=1 MeV, ΣπN=45 MeV
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
ΣπN=75 MeV
3
2.5
2
1.5
1
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.2
0.1
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
sin θ 2
-0.1
0
ΓΘ+=5 MeV, ΣπN=45 MeV
0.1
0.2
sin θ 2
ΣπN=75 MeV
4.5
4
3.5
3
2.5
2
1.5
2.5
2.25
2
1.75
1.5
1.25
1
0.75
0.5
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.2
0.1
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
sin θ 2
-0.1
0
Part. decay width Γ(N → ∆π), MeV
0.1
0.2
sin θ 2
*
ΣπN=45 MeV
ΣπN=75 MeV
140
225
120
200
175
100
150
80
125
60
100
75
40
50
20
25
0.2
0.15
0.1
0.05
0.2
0.1
0
si
nθ2
0
-0.05
-0.1
-0.15 -0.2
-0.2
θ3
θ3
-0.1
sin
sin
0
-0.05
-0.1
-0.15 -0.2
-0.2
0.2
0.15
0.1
0.05
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
0.2
0.1
0
-0.1
sin
θ2
V. Guzey
What is presently known about N10 ?
• The PWA analysis of R.A. Arndt et al., PRC 69 (2004) 035208
gives two candidate states with masses 1680 MeV and 1730
MeV. Both states should have ΓN10→N π ≤ 0.5 MeV.
• GRAAL observes a narrow nucleon resonance near 1670 MeV
in the reaction γ n → n η V. Kuznetsov for the GRAAL Collab.,
hep-ex/0409032. Interpretation: ΓN10→N η should not be too
small.
• STAR observes a narrow peak at 1734 MeV and only a
weak indication of a narrow peak at 1693 MeV in the Λ KS
invariant mass S. Kabana for the STAR Collab., hep-ex/0406032.
Interpretation: ΓN10→Λ K is possibly suppressed.
We find that this picture of N10 decays can be
realized by suitable choice of θi.
In particular, we impose the ΓN10→N π ≤ 1 MeV cut and find
unsuppressed ΓN10→N η and somewhat suppressed ΓN10→Λ K .
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
V. Guzey
Part. decay width Γ(N → N π), MeV: ΓN*→ N π< 1 MeV
ΓΘ+=1 MeV, ΣπN=45 MeV
ΣπN=75 MeV
*
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.1
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
sin θ 2
-0.1
0
ΓΘ+=5 MeV, ΣπN=45 MeV
0.1
0.2
sin θ 2
ΣπN=75 MeV
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.1
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
sin θ 2
-0.1
0
0.1
0.2
sin θ 2
Part. decay width Γ(N → N η), MeV: ΓN*→ N π< 1 MeV
ΓΘ+=1 MeV, ΣπN=45 MeV
ΣπN=75 MeV
*
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
3
2.5
2
1.5
1
0.5
0
0.2
0.2
0.1
sin
0
θ -0.1 -0.2
3
-0.2
-0.1
0
0.1
0.2
0.1
sin θ 2
sin
0
θ -0.1 -0.2
3
-0.2
-0.1
0
ΓΘ+=5 MeV, ΣπN=45 MeV
0.1
0.2
sin θ 2
ΣπN=75 MeV
5
7
6
5
4
3
2
1
0
4
3
2
1
0
0.2
0.2
0.1
sin
0
θ -0.1 -0.2
3
-0.2
-0.1
0
0.1
0.2
sin θ 2
0.1
sin
0
θ -0.1 -0.2
3
-0.2
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
-0.1
0
0.1
0.2
sin θ 2
V. Guzey
Part. decay width Γ(N → ΛK), MeV: ΓN*→ N π< 1 MeV
ΓΘ+=1 MeV, ΣπN=45 MeV
ΣπN=75 MeV
*
0.6
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.5
0.4
0.3
0.2
0.1
0
0.2
0.2
0.1
sin
0
θ -0.1 -0.2
3
-0.2
-0.1
0
0.1
0.2
0.1
sin θ 2
sin
0
θ -0.1 -0.2
3
-0.2
-0.1
0
ΓΘ+=5 MeV, ΣπN=45 MeV
0.1
0.2
sin θ 2
ΣπN=75 MeV
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.2
0.2
0.1
sin
0
θ -0.1 -0.2
3
-0.2
-0.1
0
0.1
0.2
0.1
sin θ 2
sin
0
θ -0.1 -0.2
3
-0.2
-0.1
0
Part. decay width Γ(N → ∆π), MeV
0.1
0.2
sin θ 2
*
ΓΘ+=1 MeV, ΣπN=45 MeV
18
16
14
12
10
8
6
4
2
0
ΣπN=75 MeV
8
7
6
5
4
3
2
1
0
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.1
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
sin θ 2
-0.1
0
ΓΘ+=5 MeV, ΣπN=45 MeV
0.1
0.2
sin θ 2
ΣπN=75 MeV
40
35
30
25
20
15
10
5
0
25
20
15
10
5
0
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.1
0.2
sin θ 2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
-0.1
0
0.1
0.2
sin θ 2
V. Guzey
General expressions for Σ10 couplings


X
3
1
ΣG √ −
sin θiΣ gΣ Λ π  ,
gΣ Λ π = √ G10 − sin θ1
8
i
2 5
5
10
i=2,3


!
√
X
√
5
1
gΣ Σ π = √ G10 + sin θ1 H10
− G8 5 −
sin θiΣ gΣ Σ π  ,
i
2
30
10
i=2,3


√
1
5
Σ G √4 + X sin θ Σ g

= √ −G10 + sin θ1 H10
g
+ sin θ1
8
i Σ NK ,
Σ10 N K
2
30
20
i
i=2,3

√ 
X
30
G sin θ +
gΣ Σ π =
sin θiΣ gΣ Σ π 
8
1
10
i 10
15
10
i=2,3
• There are no distinct correlations among the partial decay
widths when θ2,3 are free
• Imposing the ΓN10→N π ≤ 1 MeV cut, we can have
ΓΣ N K > ΓΣ10Λ π , ΓΣ10Σ π .
10
Σ10 looks like a narrow Σ(1770)!
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
V. Guzey
Part. decay width Γ(Σ → Λπ), MeV: ΓN*→ N π< 1 MeV
ΓΘ+=1 MeV, ΣπN=45 MeV
ΣπN=75 MeV
*
3
0.5
2.5
0.4
2
0.3
1.5
0.2
1
0.5
0.1
0
0
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.1
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
sin θ 2
-0.1
0
ΓΘ+=5 MeV, ΣπN=45 MeV
0.1
0.2
sin θ 2
ΣπN=75 MeV
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
5
4
3
2
1
0
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.1
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
sin θ 2
-0.1
0
0.1
0.2
sin θ 2
Part. decay width Γ(Σ → Σπ), MeV: ΓN*→ N π< 1 MeV
ΓΘ+=1 MeV, ΣπN=45 MeV
ΣπN=75 MeV
*
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1
0.8
0.6
0.4
0.2
0
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.1
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
sin θ 2
-0.1
0
ΓΘ+=5 MeV, ΣπN=45 MeV
0.1
0.2
sin θ 2
ΣπN=75 MeV
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0.6
0.5
0.4
0.3
0.2
0.1
0
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.1
0.2
sin θ 2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
-0.1
0
0.1
0.2
sin θ 2
V. Guzey
Part. decay width Γ(Σ → NK), MeV: ΓN*→ N π< 1 MeV
ΓΘ+=1 MeV, ΣπN=45 MeV
ΣπN=75 MeV
*
4
3.5
3
2.5
2
1.5
1
0.5
0
3
2.5
2
1.5
1
0.5
0
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.2
0.1
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
sin θ 2
-0.1
0
ΓΘ+=5 MeV, ΣπN=45 MeV
0.1
0.2
sin θ 2
ΣπN=75 MeV
8
7
6
5
4
3
2
1
0
5
4
3
2
1
0
0.2
0.2
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
-0.1
0
0.2
0.1
0.1
0
sin
-0.1 -0.2
θ
-0.2
3
sin θ 2
-0.1
0
0.1
0.2
sin θ 2
Part. decay width Γ(Σ → Σ(1385)π), MeV: ΓN*→ N π< 1 MeV
ΓΘ+=1 MeV,ΣπN=45 MeV
ΣπN=75 MeV
*
0.5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.4
0.3
0.2
0.1
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Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
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V. Guzey
What is presently known about Σ10 ?
• This is the least known member of 10. We use mΣ10 = 1765
MeV (equally spaced between the N (1670) and Ξ−− (1862)).
• Among the experiments reporting the Θ+ signal, there were
four experiments, where the Θ+ was observed as a peak in
the p KS invariant mass. Since Σ10 decays in the same final
state (N K), the four experiments give direct information on
the Σ10 → N K decay!
• The analysis of A.E. Asratyan, A.G. Dolgolenko and M.A. Kubantsev,
Phys. At. Nucl. 67, 682 (2004) reveals a number of peaks the
1650 < Mp KS < 1850 MeV mass range.
• The analysis of SVD Collab., A. Aleev et al., hep-ex/0401024
contains at least two prominent peaks in the 1700-1800
MeV mass range, before the cuts.
• The HERMES and ZEUS p KS invariant mass spectra extend
only up to 1.7 MeV.
A possibly narrow Σ10(1765) should be searched for
in the available data!
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
V. Guzey
Discussion
• In order to estimate theoretical uncertainty of our predictions,
we add an additional mixing of the antidecuplet with a 27-plet
J. Ellis, M. Karliner, M. Praszalowicz, JHEP 0405 (2004) 002.
We arrive at two scenarios. In the first case, the qualitative
picture of the N10 decays does not change. The correlations
between the Σ10 partial decay widths do change, which
makes it impossible to identify Σ10 with Σ(1770).
In the second case, the picture of N10 decays becomes only
marginally compatible with experiment. The pattern of the
Σ10 decays is similar to Σ(1770). Both N10 and Σ10 states
are very narrow.
• We find that the scenario of large (ideal) mixing is
incompatible with the notion of approximate SU(3) symmetry
Jaffe and Wiczek, PRL 91 (2003) 042003; D.K. Hong, hep-ph/0412132.
Identifying N10 with N (1710), which is ideally mixed with
the Roper N (1440), the χ2 fit fails to simultaneously
accommodate ΓΘ+ ≤ 10 MeV and a large ΓN (1440)→N π .
An acceptably low χ2 can be only found assuming very small
mixing, |θN | ≈ 40.
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
V. Guzey
Conclusions
• We considered mixing of the antidecuplet with three J P =
1/2+ octets in the framework of approximate flavor SU(3)
symmetry and in the limit of small mixing angles.
• We presented general expressions for the 10 partial decay
widths as functions of the three mixing angles and ΓΘ+ .
• Identifying N10 with the N (1670) observed by GRAAL, we
arrive at the following picture of 10 decays: Θ+ could be as
narrow as 1 MeV; the N10 → N η decay is sizable, while the
N10 → N π decay is suppressed and the N10 → Λ K decay
is possibly suppressed.
• Constraining the mixing angles by the N10 decays, we make
predictions for the Σ10 decays. We point out that Σ10(1765)
could be searched for in the available data on KS p invariant
mass spectrum, which already revealed the Θ+ peak.
It is important to experimentally verify the decay properties
of Σ(1770) because its mass and J P = 1/2+ make it an
attractive candidate for Σ10.
Three Days of Hadronic Physics, 16.12.-18.12.2004, Spa, Belgium
V. Guzey