Theoretical Status of Pentaquarks
Theoretical Status of Pentaquarks
Takumi Doi1,2,∗)
1 Dept. of Physics and Astronomy, Univ. of Kentucky, Lexington, KY 40506, USA
2 RIKEN BNL Research Center, BNL, Upton, NY 11973, USA
We review the current status of the theoretical pentaquark search from the direct QCD
calculation. The works from the QCD sum rule and the lattice QCD in the literature are
carefully examined. The importance of the framework which can distinguish the exotic
pentaquark state (if any) from the NK scattering state is emphasized.
§1. Introduction
While QCD was established as a fundamental theory of the strong interaction
a few decades ago, its realization in hadron physics has not been understood com-
pletely. For instance, (apparent) absence of “exotic” states, which are different from
ordinary qq̄ mesons and qqq baryons, has been a long standing problem. Therefore,
the announcement1) of the discovery of Θ+ (1540), whose minimal configuration is
uudds̄, was quite striking. For the current experimental status, we refer to Ref.2)
In this report, we review the theoretical effort to search the Θ+ pentaquark
state. The main issue here is whether QCD favors its existence or not, and the
determination of possible quantum numbers for the pentaquark families (if any). In
particular, in order to understand the narrow width of Θ+ observed in the experi-
ment, it is crucial to determine the spin and parity directly from QCD.
For this purpose, we employ two frameworks, the QCD sum rule and the lattice
QCD, where both allow the nonperturbative QCD calculation without models, and
have achieved a great success for ordinary mesons/baryons. Note, however, that
neither of them is infallible, and we consider them as complementary to each other.
For instance, the lattice simulation cannot be performed at completely realistic setup,
i.e., there often exists the artifact stemming from discretization error, finite volume,
heavy u,d-quark masses and neglection of dynamical quark effect (quenching), etc.
On the other hand, the sum rule can be constructed at realistic situation, and is free
from such artifacts in lattice. Unfortunately, it suffer from another type of artifact.
Because a sum rule yields only the dispersion integral of spectrum, an interpretive
model function have to be assumed phenomenologically. Compared to the ordinary
hadron analyses, this procedure may weaken the predictability for the experimentally
uncertain system, such as pentaquarks. Another artifact in the sum rule is the OPE
truncation: one have to evaluate whether the OPE convergence is enough or not.
We also comment on the important issue common to both of the methods. Recall
that the decay channel Θ+ → N +K is open experimentally. Considering also that
both methods calculate a two-point correlator and seek for a pentaquark signal in it,
it is essential to develop a framework which can distinguish the pentaquark from the
NK state in the correlator. In the subsequent sections, we examine the literatures
and see how the above-described issues have been resolved or remain unresolved.
∗) e-mail address: doi@pa.uky.edu
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.0959v1
2 T. Doi
§2. The QCD Sum Rule Work
More than ten sum rule analyses forΘ+ spectroscopy exist for J = 1/2.3), 4), 5), 6), 7), 8), 9), 10), 11), 12)
The first parity projected sum rule was studied by us5) for I = 0. The posi-
tivity of the pole residue in the spectral function is proposed as a signature of
the pentaquark signal. This is superior criterion to the consistency check of pre-
dicted/experimental masses, because it is difficult to achieve the mass prediction
within 100MeV (∼ [m(Θ+) −m(NK)]) accuracy. We also propose the diquark ex-
otic current J5q = ǫ
abcǫdef ǫcfg(uTaCdb)(u
d Cγ5de)Cs̄
g , in order to suppress the NK
state contamination. The OPE is calculated up to dimension 6, checking that the
highest dimensional contribution is reasonably small. We obtain a possible signal in
negative parity.
In the treatment of the NK state, improvement is proposed in Ref.7) There,
NK contamination is evaluated using the soft-Kaon theorem. Note here that the
NK contamination calculated by two-hadron reducible (2HR) diagrams in the OPE
level6) is invalid because what have to be calculated is the 2HR part in the hadronic
level, not in the QCD (OPE) level. The reanalysis7) of sum rule up to dimension 6
shows that the subtraction of the NK state does not change the result of Ref.5)
Yet, as described in Sec.1, the above sum rules may suffer from the OPE trun-
cation artifact. In fact, the explicit calculation up to higher dimension have shown
that this is indeed the case.10), 11), 12) Here, we refer to the elaborated work in Ref.12)
They calculate the OPE for I(JP ) = 0(1/2±) up to dimension D = 15. It is shown
that the terms with D > 6 are important as well, while further high dimensional
terms D > 15 are not significant. Another idea in Ref.12) is the use of the combi-
nation of two independent pentaquark sum rules. In fact, the proper combination
is found to suppress the continuum contamination drastically, which corresponds to
reducing the uncertainty in the phenomenological model function. Examining the
positivity of the pole residue, they conclude the pentaquark exists in positive parity.
Does the result12) definitely predict the JP = 1/2+ pentaquark ? At this mo-
ment, we conservatively point out remaining issues. The first problem is still the
NK contamination. While such contamination is expected to be partly suppressed
through the continuum suppression, it is possible that the obtained signal corre-
sponds to just scattering states. In this point, Ref.12) argues that the signal has
different dependence on the parameter 〈q̄q〉 from the NK state. We, however, con-
sider this discussion uncertain, because 〈q̄q〉 is not a free parameter independent of
other condensates. For further study, the explicit estimate in the soft-Kaon limit7) is
interesting check, but the calculation up to high dimension has not been worked out
yet. Second issue is related to the OPE. In the evaluation of the high dimensional
condensates, one have to rely on the vacuum saturation approximation practically,
while the uncertainty originating from this procedure is not known. Furthermore,
there exists an issue for the validity of the OPE itself when considering the sum rule
with high dimensionality. In fact, rough analysis of the gluonic condensates shows13)
that the nonperturbative OPE may break down around D >∼ 11− 16. One may have
to consider this effect as well, through, for instance, the instanton picture.11)
So far, we have reviewed J = 1/2 sum rules. While there are J = 3/2 works,14), 15)
Theoretical Status of Pentaquarks 3
it is likely that they suffer from slow OPE convergence. Further progress is awaited.
§3. The Lattice QCD Work
There are a dozen of quenched lattice calculations:16), 17), 18), 19), 20), 21), 22), 23), 24), 25), 26), 27), 28)
some of them16), 17), 23), 25) report the positive signal, while others18), 19), 20), 21), 26), 24)
report null results. This apparent inconsistency, however, can be understood in a
unified way, by taking a closer look at the “interpretation” of the numerical results
and the pending lattice artifact.
As discussed in Sec.1, the question is how to identify the pentaquark signal in
the correlator, because the correlator at large Euclidian time is dominated by the
ground state, the NK scattering state. In this point, we develop a new method in
Ref.19), 26) Intuitively, this method makes use of that a scattering state is sensitive to
the spacial boundary condition (BC), while a compact one-particle state is expected
to be insensitive. Practically, we calculate the correlator under two spacial BCs: (1)
periodic BC (PBC) for all u,d,s-quarks, (2) hybrid BC (HBC) where anti-periodic
BC for u,d-quarks and periodic BC for s-quark. The consequences are as follows. In
PBC, all of Θ+, N, K are subject to periodic BC. In HBC, while Θ+(uudds̄) remains
subject to periodic BC, N(uud,udd) and K(s̄d,s̄u) are subject to anti-periodic BC.
Therefore, the energy of NK will shift by PBC → HBC due to the momentum of
N and K, while there is no energy shift for Θ+. (Recall that the momentum is
quantized on lattice as 2~nπ/L for periodic BC and (2~n + 1)π/L for anti-periodic
BC, with spatial lattice extent L and ~n ∈ ZZ3.) In this way, the different behavior
between NK and Θ+ can be used to identify whether the signal is NK or Θ+. We
simulate the anisotropic lattice, β = 5.75, V = 123 × 96, aσ/aτ = 4, with the clover
fermion. The conclusion is: (1) the signal in 1/2− is found to be s-wave NK from
HBC analysis. No pentaquark is found up to ∼ 200MeV above the NK threshold.
(2) the 1/2+ state is too massive (> 2GeV) to be identified as Θ+(1540).
In comparison with other lattice results, we introduce another powerful method18)
to distinguish Θ+ from NK. This method makes use of that the volume dependence
of the spectral weight behaves as O(1) for one-particle state, and as O(1/L3) for
two-particle state. Intuitively, the latter factor O(1/L3) can be understood as the
encounter probability of the two particles. The calculation18) of the spectral weight
from 163 × 28 and 123 × 28 lattices reveals that the ground states of both the 1/2±
channels are not the pentaquark, but the scattering states. Further analysis is per-
formed in Ref.23) There, the 1st excited state in 1/2− is extracted with 2 × 2 vari-
ational method. The volume dependence of the spectral weight indicates that the
1st excited state is not a scattering state but a pentaquark state. This is consistent
with Ref.,27) where 19× 19 variational method is used to extract the excited states.
Note here that this results is consistent with the HBC analysis.19) In fact,
HBC analysis exclude the pentaquark up to ∼ 200MeV above threshold, while the
resonance observed in Ref.23) locates 200-300MeV above the threshold. The question
is whether the observed resonance is really Θ+ which experimentally locates 100MeV
above the threshold. To address this question, explicit simulation is necessary at
physically small quark mass without quenching. In particular, small quark mass
4 T. Doi
would be important considering that Refs.19), 23) are simulated at rather heavy quark
masses and expected to suffer from large uncertainty in the chiral extrapolation.
Finally, we discuss the JP = 3/2± lattice results. We performed the com-
prehensive study26) with three different operators and conclude that all the lattice
signals are too massive (> 2GeV) for Θ+, and are identified as not pentaquarks but
scattering states from the HBC analysis. On the other hand, Ref.25) claims that a
pentaquark candidate is found in 3/2+. We, however, observe that the latter result
are contaminated by significantly large statistical noise, which makes their result
quite uncertain. Note also that their criterion to distinguish Θ+ from scattering
states is based on rather limited argument compared to the HBC analysis.
§4. Conclusions
We have examined both of the QCD sum rule and lattice QCD works. In the
sum rule, progresses in OPE calculation and continuum suppression have achieved
stable analysis, while the subtraction of NK contamination remains a critical issue.
In the lattice, the framework which distinguish the pentaquark from NK have been
successfully established. In order to resolve the superficial inconsistency in the lattice
prediction, the calculation at small quark mass without quenching is highly desirable.
Acknowledgements
This work is completed in collaboration with Drs. H.Iida, N.Ishii, Y.Nemoto,
M.Oka, F.Okiharu, H.Suganuma and J.Sugiyama. T.D. is supported by Special Post-
doctoral Research Program of RIKEN and by U.S. DOE grant DE-FG05-84ER40154.
References
1) LEPS Collaboration, T. Nakano et al., Phys. Rev. Lett. 91 (2003), 012002.
2) T. Nakano, in these proceedings.
3) S.-L. Zhu, Phys. Rev. Lett. 91 (2003), 232002.
4) R.D. Matheus et al., Phys. Lett. B 578 (2004), 323.
5) J. Sugiyama, T. Doi, and M. Oka, Phys. Lett. B 581 (2004), 167.
6) Y. Kondo, O. Morimatsu, T. Nishikawa, Phys. Lett. B 611 (2005), 93.
7) S.H. Lee, H. Kim, Y. Kwon, Phys. Lett. B 609 (2005), 252.
8) Y. Kwon, A. Hosaka and S.H. Lee, hep-ph/0505040 (2005).
9) M. Eidemuller, Phys. Lett. B 597 (2004), 314.
10) B.L. Ioffe and A.G. Oganesian, JETP Lett. 80 (2004), 386, R.D. Matheus and S. Narison,
Nucl. Phys. Proc. Suppl. 152 (2006), 236, A.G. Oganesian, hep-ph/0510327 (2005).
11) H.-J. Lee et al., Phys. Rev. D 73 (2006), 014010, ibid., Phys. Lett. B 610 (2005), 50.
12) T. Kojo, A. Hayashigaki, D. Jido, Phys. Rev. C 74 (2006), 045206.
13) M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B 147 (1979), 385, 448.
14) T. Nishikawa et al., Phys. Rev. D 71 (2005), 016001, 076004.
15) W. Wei, P.-Z. Huang and H.-X. Chen and S.-L. Zhu, JHEP 0507 (2005), 015.
16) F. Csikor et al., JHEP 0311 (2003), 070, S. Sasaki, Phys. Rev. Lett. 93 (2004), 152001.
17) T.W. Chiu and T.H. Hsieh, Phys. Rev. D 72 (2005), 034505.
18) N. Mathur et al., Phys. Rev. D 70 (2004), 074508.
19) N. Ishii et al., Phys. Rev. D 71 (2005), 034001.
20) B.G. Lasscock et al., Phys. Rev. D 72 (2005), 014502.
21) F. Csikor et al., Phys. Rev. D 73 (2006), 034506.
22) C. Alexandrou and A. Tsapalis, Phys. Rev. D 73 (2006), 014507.
23) T.T. Takahashi, T. Kunihiro, T. Onogi, and T. Umeda, Phys. Rev. D 71 (2005), 114509.
24) K. Holland, and K.J. Juge, Phys. Rev. D 73 (2006), 074505.
http://arxiv.org/abs/hep-ph/0505040
http://arxiv.org/abs/hep-ph/0510327
Theoretical Status of Pentaquarks 5
25) B.G. Lasscock et al., Phys. Rev. D 72 (2005), 074507.
26) N. Ishii, T. Doi, Y. Nemoto, M. Oka and H. Suganuma, Phys. Rev. D 72 (2005), 074503.
27) O. Jahn, J.W. Negele and D. Sigaev PoS LAT2005 (2006), 069.
28) C. Hagen, D. Hierl and A. Schafer, Eur. Phys. J. A 29 (2006), 221.
	Introduction
	The QCD Sum Rule Work
	The Lattice QCD Work
	Conclusions
