Study of exotic hadrons in s-wave chiral dynamics
Study of exotic hadrons in s-wave chiral dynamics
Tetsuo Hyodo1,∗), Daisuke Jido1 and Atsushi Hosaka2
1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502,
Japan
2Research Center for Nuclear Physics (RCNP), Ibaraki, 567-0047 Japan
We study the exotic hadrons in s-wave scattering of the Nambu-Goldstone boson with a
target hadron based on chiral dynamics. Utilizing the low energy theorem of chiral symme-
try, we show that the s-wave interaction is not strong enough to generate bound states in
exotic channels in flavor SU(3) symmetric limit, although the interaction is responsible for
generating some nonexotic hadron resonances dynamically. We discuss the renormalization
condition adopted in this analysis.
§1. Introduction
One of the nontrivial issues in hadron physics is almost complete absence of
flavor exotic hadrons. Experimentally, we have been observing more than hundred
of hadrons,1) whose flavor quantum numbers can be expressed by minimal valence
quark contents of q̄q or qqq. The only one exception is the exotic baryon Θ+ with
S = +1,2) which is composed of at lease five valence quarks. In this way, the exotic
hadrons are indeed “exotic” as an experimental fact. On the other hand, there is
no clear theoretical explanation for the nonobservation of the exotic hadrons. Our
current knowledge does not forbid to construct four or five quark states in QCD and
in effective models. Moreover, the multiquark components in nonexotic hadrons are
evident, as seen in the antiquark distribution (or pion cloud) in nucleon and successful
descriptions of some excited hadrons as resonances in two-hadron scatterings. In view
of these facts, it is fair to say that the nonobservation of the exotic hadrons is not
fully understood theoretically.
§2. Exotic hadrons in s-wave chiral dynamics
In chiral coupled-channel dynamics, some hadron resonances have been success-
fully described in s-wave scattering of a hadron and the Nambu-Goldstone (NG)
boson,3), 4), 5), 6) along the same line with the old studies with phenomenological vec-
tor meson exchange interaction.7), 8) It was found that the generated resonances
turned into bound states in flavor SU(3) symmetric limit.9) We therefore conjecture
that the bound states in the SU(3) limit are the origin of a certain class of physical
resonances, and we examine the possible existence of exotic hadrons as hadron-NG
boson bound states.10), 11)
The low energy interaction of the NG boson (Ad) with a target hadron (T ) in
∗) e-mail address: hyodo@yukawa.kyoto-u.ac.jp
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.1527v1
2 T. Hyodo, D. Jido and A. Hosaka
s-wave is given by
Vα = −
Cα,T , (2.1)
with the decay constant (f) and the energy (ω) of the NG boson. The factor Cα,T is
determined by specifying the flavor representations of the target T and the scattering
system α ∈ T⊗Ad:
Cα,T = −〈2FT · FAd〉α = C2(T )− C2(α) + 3, (2.2)
where C2(R) is the quadratic Casimir of SU(3) for the representation R. Eq. (2.1)
is the model-independent consequence of chiral symmetry, known as the Weinberg-
Tomozawa theorem.12), 13)
We have written down the general expression of the coupling strengths (2.2)
for arbitrary representations of target hadrons in SU(3). In order to specify the
exotic channels, we introduced the exoticness quantum number, as the number of
valence antiquarks to construct the given flavor multiplet for the states with positive
baryon number. Then we find that the Weinberg-Tomozawa interaction in the exotic
channels is repulsive in most cases, and that possible strength of the attractive
interaction is given by a universal value
Cexotic = 1, (2.3)
with α = [p− 1, 2] for T = [p, 0] and p ≥ 3B.10), 11)
Next we construct the scattering amplitude with unitarity condition using the
N/D method.5) The unitarized amplitude is given by
1− Vα(
as a function of the center-of-mass energy
s. The loop function G(
s) is regularized
by the once subtraction as
s) = −ã(s0)−
ρ(s′)
s′ − s
− ρ(s
s′ − s0
, (2.4)
where the phase space integrand is ρ(s) = 2MT
(s− s+)(s− s−)/(8πs), s± =
(m±MT )2, and m and MT are the masses of the target hadron and the NG boson.
In order to determine the subtraction constant ã(s0) and the subtraction point
s0, we adopt the renormalization condition given in Refs. 14), 6),
G(µ) = 0, µ = MT , (2.5)
which is equivalent to tα(µ) = Vα(µ) at this scale. We will discuss the implication
of this prescription in section 3. With the condition (2.5), we show that the bound
state can be obtained if the coupling strength (2.2) is larger than the critical value
Ccrit =
−G(MT +m)
Exotic hadrons in s-wave chiral dynamics 3
Varying the parameters m, MT , and f in the physically allowed region, we show
that the attraction in the exotic channels (2.3) is always smaller than the critical
value Ccrit. Thus, it is not possible to generate bound states in exotic channels in
the SU(3) symmetric limit.
We would like to emphasize that this conclusion is model independent in the
SU(3) limit, as far as we respect chiral symmetry. In this study, we only consider
the exotic hadrons composed of the NG boson and a hadron, so the existence of
exotic states generated by quark dynamics or rotational excitations of chiral solitons
is not excluded. In practice, one should bear in mind that the SU(3) breaking
effect and higher order terms in the chiral Lagrangian would play a substantial role.
Nevertheless, the study of exotic hadrons in a simple extension of a successful model
of hadron resonances as we have done in the present work can partly explain difficulty
of observation of the exotic hadrons.
§3. Interpretation of the renormalization condition
Here we discuss the renormalization condition (2.5) in this analysis. The inter-
action kernel Vα(
s) is constructed from chiral perturbation theory so as to satisfy
the low energy theorem. The low energy theorem also constrains the behavior of the
full unitarized amplitude tα(
s) at a scale
s = µm where the chiral expansion is
valid. Therefore we can match the unitarized amplitude tα(
s) with the tree level
one Vα(
s) at the scale µm:
tα(µm) = Vα(µm) + Vα(µm)G(µm)Vα(µm) + · · · = Vα(µm), (3.1)
This condition determines the subtraction constant such that the loop function
G(µm) vanishes. This is only possible within the region
MT −m ≤ µm ≤ MT +m, (3.2)
since the loop function has an imaginary part outside this region and the subtraction
constant is a real number. We consider that by employing this renormalization
condition, a natural unitarization of the kernel interaction based on chiral symmetry
is realized. Interestingly, if we apply this prescription for the case of the octet baryon
target, the subtraction constant turns out to be “natural size” which was found in
the comparison with three-momentum cutoff,5) and the experimental observables in
the S = −1 meson-baryon channel are successfully reproduced.
We take µm = MT in the present study. The dependence on µm within the
region (3.2) is found that the binding energy of the bound state increases if we
shift the matching scale µm to the lower energy region. This is discussed also in
Refs. 17),18) by varying the subtraction constant. The region µm ≤ MT corresponds
to the u-channel scattering. Thus µm = MT is the most favorable to generate a bound
state within the s-channel regime.
The unitarized amplitudes in the above prescription do not always reproduce ex-
perimental data. In such a case, the subtraction constants ã(s0) should be adjusted
in order to satisfy experimental data. The subtraction constants determined in this
4 T. Hyodo, D. Jido and A. Hosaka
way supplement the role of the higher order chiral Lagrangians, which is lacking in
the kernel interaction. As shown for the ρ meson effect in the meson-meson scatter-
ing,16) the higher order terms may contain the effect of the resonances. Therefore,
if the natural condition (3.1) is badly violated, one may speculate that the seeds
of resonances in the higher order Lagrangian, which are possibly the genuine quark
states, appear in the unitarized amplitude, as in the study of Ref. 19).
In summary, we have argued the following issues.
• In order for the unitarized amplitude tα(
s) to satisfy the low energy theorem,
the loop function should vanish in the region (3.2).
• This requirement can be regarded as a natural unitarization, without introduc-
ing the effect of resonances in the higher order Lagrangian.
• Lower matching scale µm is more favorable to generate a bound state.
Turning to the problem of exotic hadrons, what we have shown is the nonexistence
of the exotic bound states with the most favorable condition to generate a bound
state, without introducing the seed of genuine quark state.
Acknowledgements
T. H. thanks the Japan Society for the Promotion of Science (JSPS) for finan-
cial support. This work is supported in part by the Grant for Scientific Research
(No. 17959600, No. 18042001, and No. 16540252) and by Grant-in-Aid for the 21st
Century COE ”Center for Diversity and Universality in Physics” from the Ministry
of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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http://arxiv.org/abs/hep-ph/0612333
	Introduction
	Exotic hadrons in s-wave chiral dynamics
	Interpretation of the renormalization condition
