Remarks on N_c dependence of decays of exotic baryons
TPJU-03/2007
Remarks on N
dependence of decays of exotic baryons
Karolina Pieściuk ∗) and Micha l Prasza lowicz ∗∗)
M. Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4,
30-049 Kraków, Poland
We calculate the Nc dependence of the decay widths of exotic eikosiheptaplet within the
framework of Chral Quark Soliton Model. We also discuss generalizations of regular baryon
representations for arbitrary Nc.
§1. Introduction
One of the most puzzling results of the chiral quark-soliton model (χQSM) for
exotic baryons consists in a very small hadronic decay width,1) governed by the
decay constant G10. While the small mass of exotic states is rather generic for all
chiral models1)–3) the smallness of the decay width appears as a subtle cancelation
of three different terms that contribute to G10. Decay width in solitonic models
4) is
calculated in terms of a matrix element M of the collective axial current operator
corresponding to the emission of a pseudoscalar meson ϕ1) – see Ref. 5) for criticism
of this approach:
Ô(8)ϕ = 3
ϕi −G1 dibcD
ϕb Ŝc −
ϕ8 Ŝi
× piϕ. (1.1)
For notation see Ref. 1). Constants G0,1,2 are constructed from the so called moments
of inertia that are calculable in χQSM. The decay width is given as
ΓB→B′+ϕ =
M M ′
M2 = 1
M M ′
A2. (1.2)
The “bar” over the amplitude squared denotes averaging over initial and summing
over final spin (and, if explicitly indicated, over isospin).
For B(10) → B′(8) + ϕ for spin ”up” and ~pϕ = (0, 0, pϕ) we have
81/2, B
∣ Ô(8)ϕ
∣101/2, B
3G10√
× pϕ (1.3)
G10 = G0 −G1 −
G2. (1.4)
∗) e-mail address: yessien@gmail.com
∗∗) e-mail address: michal@if.uj.edu.pl
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.0196v2
2 K. Pieściuk and M. Prasza lowicz
In order to have an estimate of the width (1.2) the authors of Ref. 1) calculated G10
in the nonrelativistic limit6) of χQSM and got G10 ≡ 0. It has been shown that this
cancelation between terms that scale differently with Nc (G0 ∼ N
c , G1,2 ∼ N
is in fact consistent with large Nc counting,
7) since
G10 = G0 −
Nc + 1
G2 (1.5)
where the Nc dependence comes from the SU(3) Clebsch-Gordan coefficients calcu-
lated for large Nc. In the nonrelativistic limit (NRL):
G0 = −(Nc + 2)G, G1 = −4G, G2 = −2G, G ∼ N1/2c . (1.6)
In this paper we ask whether the similar cancelation takes place for the decays of
27 of spin 1/2 and 3/2. We also discuss the possible modifications of the Nc depen-
dence of the decay width due to the different choice of the large Nc generalizations
of regular SU(3) multiplets.
§2. Baryons in large N
limit
Soliton is usually quantized as quantum mechanical symmetric top with two
moments of inertia I1,2:
B = Mcl +
S(S + 1) +
C2(R) − S(S + 1) −
B . (2
Here S denotes baryon spin, C2(R) the Casimir operator for the SU(3) representation
R = (p, q):
C2(R) =
p2 + q2 + pq + 3(p + q)
(2.2)
and quantities δ
B denote matrix elements of the SU(3) breaking hamiltonian:
Ĥ ′ =
σ + αD
88 ) + βY +
8A ĴA. (2
Model parameters that can be found in Ref. 8)
α = −Nc
(σ + β), β = −ms
, γ = 2ms
, σ =
mu + md
scale with Nc in the following way:
i1,2 = 3I1,2/Nc where i1,2 ∼ O(N0c ), σ, β, γ ∼ O(msN0c ). (2.4)
Here ΣπN is pion-nucleon sigma term and mq denote current quark masses. Numer-
ically σ > |β| , |γ|.
So far we have specified explicit Nc dependence (2.4) that follows from the fact
that model parameters are given in terms of the quark loop. Another type of the
Remarks on Nc dependence 3
Nc dependence comes from the constraint
9) that selects SU(3)flavor representations
R = (p, q) containing states with hypercharge YR = Nc/3. Therefore for arbitrary
Nc ordinary baryon representations have to be extended and one has to specify which
states correspond to the physical ones. Usual choice10)
”8” = (1, (Nc − 1)/2) , ”10” = (3, (Nc − 3)/2) , ”10” = (0, (Nc + 3)/2) , (2.5)
depicted in Fig. 1 corresponds – in the quark language – to the case when each time
when Nc is increased by 2, a spin-isospin singlet (but charged) 3 diquark is added,
as depicted in Fig. 2.
Fig. 1. Standard generalization of SU(3) flavor baryon representations for arbitrary Nc
Fig. 2. Adding 3 diquarks to regular SU(3) baryon representations 8, 10 and 10 corresponds to the
representation set of Fig.1.
Extension (2.5) leads to (1.5). It implies that mass differences between centers
of multiplets scale differently with Nc:
∆10−8 =
∼ O(1/Nc), ∆10−8 =
Nc + 3
∼ O(1). (2.6)
The fact that ∆10−8 6= 0 in large Nc limit triggered recently discussion on the
validity of the semiclassical quantization for exotic states.11) Since in the chiral
limit the momentum pϕ of the outgoing meson scales according to (2.6), overall Nc
dependence of the decay width is strongly affected by its third power (1.2):
ΓB→B′+ϕ ∼
O(A2)O(p3ϕ). (2.7)
Phenomenologically, however, scaling (2.6) is not sustained. Indeed, meson mo-
menta in ∆ and Θ decays are almost identical (assuming M
Θ ≃ 1540 MeV):
pπ ≃ 225 MeV, pK ≃ 268 MeV. (2.8)
Unfortunately, going off SU(3)flavor limit does not help. Explicitly:
δ(8) =
(Nc − 3)
(Nc − 2)α + 32γ
Nc + 7
3(Nc + 2)α− 12(2Nc + 9)γ
(Nc + 3)(Nc + 7)
4 K. Pieściuk and M. Prasza lowicz
(6α + (Nc + 6)γ)
(Nc + 3)(Nc + 7)
− I(I + 1)
= 3σ + 2β − σY + . . . (2.9)
δ(10) =
(Nc − 3)(Nc + 4)
(Nc + 1)(Nc + 9)
Nc − 3
5(Nc − 3)
2(Nc + 1)(Nc + 9)
3(Nc − 1)α − 52(Nc + 3)γ
(Nc + 1)(Nc + 9)
Y = 3σ + 2β − σY + . . . (2.10)
δ(10) =
Nc(Nc − 3)
(Nc + 3)(Nc + 9)
Nc − 3
β − 3(Nc − 3)
2(Nc + 3)(Nc + 9)
6Ncα− 9γ
2(Nc + 3)(Nc + 9)
Y = 5σ + 4β − σY + . . . (2.11)
where . . . denote terms O(1/Nc), Y and I denote physical hypercharge and isospin.
Interestingly in all cases in the large Nc limt, ms splittings are proportional to
the hypercharge differences only. In this limit Σ−Λ splitting in the octet is zero and
this degeneracy is lifted in the next order at O(1/Nc). This explains the smallness of
Σ − Λ mass difference. Additionally δ(8)N ≃ δ
∆ up to higher order terms O(1/N2c ),
however δ
Θ − δ
N ≃ σ + 2β > 0. This implies that
α + β − 3
γ → 3
+ σ + 2β + O(1/Nc),
γ → O(1/Nc). (2.12)
The first equation shows that the Θ − N 6= 0 in the large Nc limit even if ms
corrections are included. We will come back to this problem in the last section.
§3. Decay constants of twentysevenplet for large N
In this section we shall consider decays of eikosiheptaplet (27-plet)
”27” = (2, (Nc + 1)/2) (3.1)
that can have either spin 1/2 or 3/2, the latter being lighter. Mass differences read
∆273/2−8 =
Nc + 1
∼ O(1), ∆271/2−8 =
Nc + 7
∼ O(1),
∆273/2−10 =
Nc + 1
∼ O(1), ∆271/2−10 = −
Nc + 7
∼ O(1),
∆273/2−10 =
∼ O(1/Nc), ∆271/2−10 =
∼ O(1/Nc). (3.2)
Matrix elements for the decays of eikosiheptaplet (with S3 = 1/2) read:
A(B273/2 → B
8 + ϕ) = 3
8 ”8”
8(Nc + 5)
9(Nc + 3)(Nc + 9)
×G27,
Remarks on Nc dependence 5
A(B273/2 → B
10 + ϕ) = −3
8 ”10”
(Nc − 1)(Nc + 7)
9(Nc + 1)(Nc + 3)(Nc + 9)
× F27,
A(B273/2 → B
+ ϕ) = 3
8 ”10”
2(Nc + 1)(Nc + 7)
3(Nc + 3)(Nc + 9)
× E27,
(3.3)
Decay Large Nc NRL
Scaling
in NRL
273/2 → 81/2 G27 = G0 − Nc−14 G1 = −3G N
273/2 → 103/2 F27 = G0 − Nc−14 G1 −
G2 = 0 0
273/2 → 101/2 E27 = G0 + G1 = −(Nc + 6)G N
For S = 1/2 and S3 = 1/2 we have:
A(B271/2 → B
8 + ϕ) = −3
8 ”8”
(Nc + 1)(Nc + 5)
9(Nc + 3)(Nc + 7)(Nc + 9)
×H27,
A(B271/2 → B
10 + ϕ) = −3
8 ”10”
8(Nc − 1)
9(Nc + 3)(Nc + 9)
×G′27,
A(B271/2 → B
+ ϕ) = 3
8 ”10”
Nc + 4
9(Nc + 3)(Nc + 9)
×H ′27,
(3.4)
Decay Large Nc NRL
Scaling
in NRL
271/2 → 81/2 H27 = G0−Nc+54 G1 +
G2 = 0 0
271/2 → 103/2 G′27 = G0−Nc+54 G1 = 3G N
271/2 → 101/2 H ′27 = G0 + 2Nc+52Nc+8G1 +
2Nc+8
G2 = − (Nc+3)(Nc+7)Nc+4 G N
In order to calculate the Nc behavior of the width we have to know the Nc
dependence of the flavor Clebsch-Gordan coefficients that depend on the states in-
volved. For the decays into 8 and 10 the only possible channels are Θ27 → N(∆)+K,
and the pertinent Clebsches do not depend on Nc. For the decays into 10 we have
Θ27 → Θ10 + π that scales like O(1) and Θ27 → N10 +K that scales like O(1/
The resulting scaling of ΓΘ27→B′+ϕ calculated from Eq.(2
.7) reads as follows:
6 K. Pieściuk and M. Prasza lowicz
decay of Nc scaling decay of Nc scaling
Θ273/2 exact NRL Θ271/2 exact NRL
→ N8 + K O(1) O(1/N2c ) → N8 + K O(1) 0
→ ∆10 + K O(1) 0 → ∆10 + K O(1) O(1/N2c )
→ N10 + K O(1/N3c ) O(1/N3c ) → N10 + K O(1/N3c ) O(1/N3c )
→ Θ10 + π O(1/N2c ) O(1/N2c ) → Θ10 + π O(1/N2c ) O(1/N2c )
Interestingly, we see that whenever the exact scaling is O(1), the nonrelativistic
cancelation (exact or partial) lowers the power of Nc, whereas in the case when the
width has good behavior for large Nc, there is no NRL cancelation.
§4. Alternative choices for large N
multiplets
So far we have only considered the ”standard” generalization (2.5) of baryonic
SU(3)flavor representations for large Nc. This choice is based on the requirement that
generalized baryonic states have physical spin, isospin and strangeness, however their
hypercharge and charge are not physical.10) Moreover, the generalization of the octet
is not selfadjoint and antidecuplet is not complex conjugate of decuplet. Some years
ago it has been proposed to consider alternative schemes.12)
Fig. 3. Generalization of SU(3) flavor representations in which octet is selfadjoint
Fig. 4. Adding triquarks to regular SU(3) baryon representations 8, 10 and 10 corresponds to the
representation set of Fig.3.
If we require the generalized octet to be self-adjoint we are led to the following
set of representations
”8” = (Nc/3, Nc/3) , ”10” = ((Nc + 6)/3, (Nc − 3)/3) , ”10” = ”10”∗ (4.1)
that are depicted in Figs. 3 and 4. This means that we enlarge Nc in steps of 3 adding
each time a uds triquark. Generalized states have physical isospin, hypercharge (and
charge), but unphysical strangeness and spin that is of the order of Nc. With this
Remarks on Nc dependence 7
choice both ∆10−8, ∆10−8 6= 0 in large Nc limit:
∆10−8 = (Nc/6 − 1) /I1, ∆10−8 = (Nc/6 − 1) /I2. (4.2)
With this power counting we can calculate large Nc approximation of the meson
momenta in the decays of ∆ and Θ:
∆ → N pπ =
(M∆ −MN )2 −m2π = 256 MeV,
Θ → N pK =
(MΘ −MN )2 −m2K = 339 MeV (4.3)
that are much closer to the physical values (2.8) than (2.6).
Fig. 5. Generalization of SU(3) flavor representations in which decuplet is fully symmetric (0, q).
Fig. 6. Adding sextet diquarks to regular SU(3) baryon representations 8, 10 and 10 corresponds
to the representation set of Fig.5.
Finally let us mention a third possibility in which we require generalized decuplet
to be a completely symmetric SU(3)flavor representation for arbitrary Nc. This leads
to (see Figs. 5 and 6):
”8” = (Nc − 2, 1) ”10” = (Nc, 0) ”10” = (Nc − 3, 3) . (4.4)
Interestingly this choice has a smooth limit to the one flavor case. In the quark
language it amounts to adding a symmetric diquark to the original SU(3)flavor rep-
resentation when increasing Nc in steps of 2. As seen from Fig. 5 physical states are
situated at the bottom of infinite representations (4.4) and therefore have unphysical
strangeness, charge (hypercharge) and also spin.
The mass splittings for this choice read
∆10−8 = Nc/ 2I1, ∆10−8 = 3/ 2I2. (4
Here the generalized decuplet remains split from the ”8”, while ∆10−8 → 0 for large
Nc. The phase space factor for Θ decay is therefore suppressed with respect to the
one of ∆.
§5. Summary
In this short note we have shown that very small width of exotic baryons – if
they exist – cannot be explained by the standard Nc counting alone. Certain degree
8 K. Pieściuk and M. Prasza lowicz
of nonrelativisticity is needed to ensure cancelations between different terms in the
decay constants. This phenomenon observed firstly for antidecuplet, is also operative
for the decays of eikosiheptaplet. We have shown that in χQSM in the nonrelativistic
limit all decays are suppressed for large Nc. Exact cancelations occur for Θ273/2 →
∆10 + K and Θ271/2 → N8 + K, leading Nc terms cancel for Θ273/2 → N8 + K and
Θ271/2 → ∆10 + K. For 27 → 10 there are no cancelations, but the phase space is
N−3c suppressed.
We have also briefly discussed nonstandard generalizations of regular baryon
representations for arbitrary Nc. For Nc > 3 bayons are no longer composed from 3
quarks and therefore they form large SU(3)flavor representations that reduce to octet,
decuplet and antidecuplet for Nc = 3. The standard way to generalize regular baryon
representations is to add antisymmetric antitriplet diqaurk when Nc is increased in
intervals of 2. This choice fulfils many reasonable requirements; most importantly
for SU(2)flavor these representations form regular isospin multiplets. However, repre-
sentations (2.5) do not obey conjugation relations characteristic for regular represen-
tations. Therefore we have proposed generalization (4.1) that satisfies conjugation
relations. Most important drawback of (4.1) is that spin S ∼ Nc that contradicts
semiclassical quantization. Nevertheless as a result meson momenta emitted in 10
and 10 decays scale in the same way with Nc (4.3), consistently with ”experimental”
values (2.8), whereas for (2.5) the scaling is different (2.6).
Acknowledgements
One of us (MP) is grateful to the organizers of the Yukawa International Sym-
posium (YKIS2006) for hospitality during this very successful workshop.
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http://arxiv.org/abs/hep-ph/9703373
http://arxiv.org/abs/hep-ph/0308114
http://arxiv.org/abs/hep-ph/0703072
http://arxiv.org/abs/hep-ph/9505328
http://arxiv.org/abs/hep-ph/9806431
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http://arxiv.org/abs/hep-ph/0310221
http://arxiv.org/abs/hep-ph/0312191
	Introduction
	Baryons in large Nc limit
	Decay constants of twentysevenplet for large Nc
	Alternative choices for large Nc multiplets
	Summary
