Exotic Hadron in Pole-dominated QCD Sum Rules
Toru Kojo 1,∗), Daisuke Jido, 2 and Arata Hayashigaki 3
1 Department of Physics, Kyoto University, Kyoto 606-8502, Japan
2 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502,
Japan
3Institut für Theoretische Physik, J.W. Goethe Universität, D-60438 Frankfurt am
Main, Germany
We study pentaquark Θ+(I = 0, J = 1/2) in the QCD sum rules emphasizing that we
can not extract any properties of the pentaquark outside of the Borel window. To find the
appropriate Borel window, we prepare a favorable set up of the correlators and carry out
the operator product expansion up to dimension 15 within factorization hypothesis. Our
procedures reduce the unwanted high energy contaminations and enhance the low energy
correlation. In the Borel window, independent analyses for the chiral-even and odd sum
rules give the consistent values of the Θ+ mass, 1.68±0.22 GeV, and the residue. The parity
is found to be positive.
§1. Introduction
The experimental announcement for the discovery of the pentaquark Θ+(1540)1)
triggered tremendous amount of theoretical and experimental works on the exotic
states. Although the existence of such exotic states is still not so obvious, the exotics
provide a good opportunity to get the deeper insight of the hadron structures and
their connection to QCD. One of approaches from QCD to exotics is the QCD sum
rule (QSR),2) which relates informations of QCD to the hadronic properties through
the correlator analysis for the interpolating fields of hadrons. The Borel transformed
sum rules with the simplest pole + continuum parametrization are given as (i = 0, 1
correspond to the chiral even and odd part, respectively)
(ope)
i (−Q
2) = λ2i e
−m2/M2 +
ds e−s/M
(ope)
i (s), (1
where the relation ±mλ20 = λ21 holds due to the spinor structure and the relative
sign of the residues λ2i represents with the parity of the resonance state. Using these
sum rules, we can derive the approximated expressions of the mass and residue as a
function of M and sth. To extract properties of the low energy excitations with good
accuracy, we need to treat sum rules in the appropriateM2-region, i.e., Borel window,
where the low energy correlation is large enough compared to the contaminations
from high energy components which have no relations with properties of low-lying
resonances. The setting the Borel window is the most important step in QSR and,
only within this window, we can evaluate the physical quantities.
In the exotic cases, as reported in Ref. 3), it is extremely difficult to find
the appropriate Borel window in contrast to the usual meson and baryon cases.
∗) e-mail address: torujj@ruby.scphys.kyoto-u.ac.jp
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.0413v1
2 Toru Kojo, Daisuke Jido and Arata Hayashigaki
This is because the OPE convergence is very slow and the unwanted high energy
components dominate the spectral integral. In addition, we often encounter the
artificial stability of the physical quantities against M2-variation. This is the case
that physical quantities depend too strongly on the threshold parameter sth and not
on the low energy correlations which we want to extract. To attack these serious
problems common to the exotics, we propose a new approach and apply it to the Θ+,
assuming its quantum number as I = 0, J = 1/2, as one example of the exotics.4)
§2. OPE and favorable set up of the correlators
To find the Borel window, it is necessary to increase low energy informations in
the spectral function efficiently and, at the same time, reduce high energy contami-
nations. For these purposes, we take the following treatments.
Inclusion of the higher dimension terms of OPE is particularly important be-
cause they strongly reflect the low energy dynamics. For example, in the case of the
sum rules for ρ and A1 mesons, the dim.0 and 4 terms are the same due to the chiral
symmetry realized in the high energy, and the splitting of the masses is explained
only after the inclusion of dim.6 terms, 〈q̄q〉2, which appear due to the chiral sym-
metry breaking. From these observations, we perform the OPE calculation up to
dim.15 within factorization hypothesis both for taking into account the low energy
correlations and for the confirmation of good OPE convergence.
As later shown, simple inclusion of the low energy correlations through the higher
dimension terms is found to be not sufficient to find the Borel window because
high energy contaminations are too large in the QSR for the exotics. To reduce
the high energy contaminations, we take the difference between correlators for two
interpolating fields with similar structure but different chirality each other, i.e.,
d4x eiq·x
∣T [P (x)P̄ (0) − t S(x)S̄(0)]
ds e−s/M
Im[ΠP0 (s)− tΠS0 (s)] q̂ + Im[ΠP1 (s)− tΠS1 (s)]
, (2.1)
where Π0, Π1 correspond to the chiral even and odd part respectively, and
P = ǫabcǫdef ǫcfg{uTaCdb}{uTd Cγµγ5de}γµCsTg , (2.2)
S = ǫabcǫdef ǫcfg{uTaCγ5db}{uTd Cγµγ5de}γµCsTg . (2.3)
Here the only difference in these interpolating fields is that the first diquark structures
have the opposite chirality.
Let us first explain in the case of the chiral even part. Since they show the same
behavior in high energy due to the chiral symmetry, after the subtraction of two
correlators (t = 1 case), the irrelevant high energy contributions are expected to be
canceled out in the similar way as the Weinberg sum rules.5) In terms of OPE, this
cancellation corresponds to the cancellation of the lower dimension terms. It is not
a priori evident whether the low energy correlations remain enough even after the
subtraction because the low energy contribution could also cancel out. Our Borel
Exotic Hadron in Pole-dominated QCD Sum Rules 3
analysis, however, reveals that, in the case of t = 1, the large low energy correlation
remains enough even after the subtraction. As a result, we can find the Borel window
in the relatively large M2-region.
On the other hand, for the chiral odd part, the subtraction procedure corre-
sponding to t = 1 case does not lead the cancellation of the high energy components
because chiral odd part is constructed of the chiral symmetry breaking terms. How-
ever, in the case of t = 1, the OPE convergence is found to be very good and then we
can find the Borel window in the small M2-region where high energy contaminations
are suppressed due to the Borel factor e−s/M
in the integral of the spectral function.
§3. Borel analysis for mass and residue
Here we explain our criterion for the Borel window. The lower bound of the
Borel window is given so that the highest-dimensional terms in the truncated OPE
are less than 10% of its whole OPE, while the upper bound is determined by the
region where the absolute value of the pole contribution is larger than the absolute
value of the integrated spectral function in the region s ≥ sth. Note that the 50%
pole contribution in our criterion is extremely large in comparison with any prior
pentaquark sum rules, where the pole contributions are not more than 20%.3)
To recognize the problems in the case of QSR for exotics, let us see Fig.1 for
M2-dependence of the mass in the cases of t = −1, 0, 10 corresponding to PP̄ +SS̄,
PP̄ , SS̄ cases respectively. The threshold value is fixed to typical value
sth = 2.2
GeV. In these cases, we fail to find stabilities of the mass in the M2-region lower
than the upper bound of the Borel window. The stabilities above the upper bound
are simply artifacts which often appear in QSR. Fig.1 shows that typical mass of PP̄
case is much smaller than that of SS̄, and then we can expect that the low energy
correlation of PP̄ is much larger than that of SS̄. This observation leads that even
after the subtraction PP̄ − SS̄ (t = 1 case), the low energy correlation can remain
enough.
Now we see the case of around t = 1. We tune the value of t around t = 1 to
get the widest Borel window. As expected, for even part (t = 0.9), the high energy
SS (t = 10)
PP (t = 0)
PP (t = 0)
SS (t = ‐1)
SS (t = ‐1)
SS (t = 10)
Fig. 1. The behavior of the mass as a function of M2 for t = −1, 0, 10. The left arrows represent
the upper bound of the Borel window. In the smaller M2-region than the upper bound, we can
not find stable region of the mass. The stabilities above the upper bound are simply artifacts
which often appear in QSR.
4 Toru Kojo, Daisuke Jido and Arata Hayashigaki
even odd
(t=1.1)(t=0.9)
Fig. 2. The behavior of the mass as a function of M2. The left (right) arrows represent the upper
(lower) bound of the Borel window. We succeed to find the Borel window and stabilities of
mass.
contaminations are canceled out due to chiral symmetry and we find the wide Borel
window in the relatively large M2-region. On the other hand, for odd part (t = 1.1),
thanks to the good OPE convergence, we also find the wide Borel window in the
small M2-region. The threshold values are taken to make the physical quantities
most stable in the Borel window.
The best stability is achieved with
sth = 2.2 GeV (even) and 2.1 GeV (odd),
giving mΘ+ = 1.64 GeV (even) and 1.72 GeV (odd) respectively. The values of the
residue are also obtained from the chiral even and odd sum rules as λ20 = (3.0±0.1)×
10−9 GeV12 and λ21/mΘ+ = (3.4 ± 0.2) × 10−9 GeV12. It is remarkable that these
numbers are quite similar with the close t. This implies that our analysis investigates
consistently the same state in the two independent sum rules. Note that from the
relative sign of the residues, we assign positive parity to the observed Θ+ state.
In conclusion, we perform the Borel analysis for Θ+ with special setup of the
correlators in order to find the Borel window. Within uncertainties of the condensate
value, independent analyses for the chiral-even and odd sum rules give the consistent
values of the Θ+ mass, 1.68± 0.22 GeV, and the residue. The parity is found to be
positive.
Acknowledgements
We thank Profs. M. Oka, A. Hosaka and S.H. Lee for useful discussions about
QSR for the exotics during the YKIS2006 on ”New Frontiers on QCD” held at
the Yukawa Institute for Theoretical Physics. This work is supported in part by
the Grant for Scientific Research (No.18042001) 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.
References
1) T. Nakano et al., Phys. Rev. Lett. 91 (2003), 012002.
2) M.A. Shifman, A.I. Vainshtein, and V.I. Zakharow, Nucl. Phys. B 147 (1979), 385.
3) R.D. Matheus and S. Narison, Nucl. Phys. Proc. Suppl. 152 (2006), 236.
4) T. Kojo, A. Hayashigaki, and D. Jido, Phys. Rev. C 74 (2006), 045206
5) S. Weinberg, Phys. Rev. Lett. 18 (1967), 507.
	Introduction
	OPE and favorable set up of the correlators
	Borel analysis for mass and residue

  We study pentaquark $\Theta^{+} (I=0,J=1/2)$ in the QCD sum rules emphasizing
that we can not extract any properties of the pentaquark outside of the Borel
window. To find the appropriate Borel window, we prepare a favorable set up of
the correlators and carry out the operator product expansion up to dimension 15
within factorization hypothesis. Our procedures reduce the unwanted high energy
contaminations and enhance the low energy correlation. In the Borel window,
independent analyses for the chiral-even and odd sum rules give the consistent
values of the $\Theta^+$ mass, $1.68\pm0.22$ GeV, and the residue. The parity
is found to be {\it positive}.

Introduction
The experimental announcement for the discovery of the pentaquark Θ+(1540)1)
triggered tremendous amount of theoretical and experimental works on the exotic
states. Although the existence of such exotic states is still not so obvious, the exotics
provide a good opportunity to get the deeper insight of the hadron structures and
their connection to QCD. One of approaches from QCD to exotics is the QCD sum
rule (QSR),2) which relates informations of QCD to the hadronic properties through
the correlator analysis for the interpolating fields of hadrons. The Borel transformed
sum rules with the simplest pole + continuum parametrization are given as (i = 0, 1
correspond to the chiral even and odd part, respectively)
(ope)
i (−Q
2) = λ2i e
−m2/M2 +
ds e−s/M
(ope)
i (s), (1
where the relation ±mλ20 = λ21 holds due to the spinor structure and the relative
sign of the residues λ2i represents with the parity of the resonance state. Using these
sum rules, we can derive the approximated expressions of the mass and residue as a
function of M and sth. To extract properties of the low energy excitations with good
accuracy, we need to treat sum rules in the appropriateM2-region, i.e., Borel window,
where the low energy correlation is large enough compared to the contaminations
from high energy components which have no relations with properties of low-lying
resonances. The setting the Borel window is the most important step in QSR and,
only within this window, we can evaluate the physical quantities.
In the exotic cases, as reported in Ref. 3), it is extremely difficult to find
the appropriate Borel window in contrast to the usual meson and baryon cases.
∗) e-mail address: torujj@ruby.scphys.kyoto-u.ac.jp
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.0413v1
2 Toru Kojo, Daisuke Jido and Arata Hayashigaki
This is because the OPE convergence is very slow and the unwanted high energy
components dominate the spectral integral. In addition, we often encounter the
artificial stability of the physical quantities against M2-variation. This is the case
that physical quantities depend too strongly on the threshold parameter sth and not
on the low energy correlations which we want to extract. To attack these serious
problems common to the exotics, we propose a new approach and apply it to the Θ+,
assuming its quantum number as I = 0, J = 1/2, as one example of the exotics.4)
§2. OPE and favorable set up of the correlators
To find the Borel window, it is necessary to increase low energy informations in
the spectral function efficiently and, at the same time, reduce high energy contami-
nations. For these purposes, we take the following treatments.
Inclusion of the higher dimension terms of OPE is particularly important be-
cause they strongly reflect the low energy dynamics. For example, in the case of the
sum rules for ρ and A1 mesons, the dim.0 and 4 terms are the same due to the chiral
symmetry realized in the high energy, and the splitting of the masses is explained
only after the inclusion of dim.6 terms, 〈q̄q〉2, which appear due to the chiral sym-
metry breaking. From these observations, we perform the OPE calculation up to
dim.15 within factorization hypothesis both for taking into account the low energy
correlations and for the confirmation of good OPE convergence.
As later shown, simple inclusion of the low energy correlations through the higher
dimension terms is found to be not sufficient to find the Borel window because
high energy contaminations are too large in the QSR for the exotics. To reduce
the high energy contaminations, we take the difference between correlators for two
interpolating fields with similar structure but different chirality each other, i.e.,
d4x eiq·x
∣T [P (x)P̄ (0) − t S(x)S̄(0)]
ds e−s/M
Im[ΠP0 (s)− tΠS0 (s)] q̂ + Im[ΠP1 (s)− tΠS1 (s)]
, (2.1)
where Π0, Π1 correspond to the chiral even and odd part respectively, and
P = ǫabcǫdef ǫcfg{uTaCdb}{uTd Cγµγ5de}γµCsTg , (2.2)
S = ǫabcǫdef ǫcfg{uTaCγ5db}{uTd Cγµγ5de}γµCsTg . (2.3)
Here the only difference in these interpolating fields is that the first diquark structures
have the opposite chirality.
Let us first explain in the case of the chiral even part. Since they show the same
behavior in high energy due to the chiral symmetry, after the subtraction of two
correlators (t = 1 case), the irrelevant high energy contributions are expected to be
canceled out in the similar way as the Weinberg sum rules.5) In terms of OPE, this
cancellation corresponds to the cancellation of the lower dimension terms. It is not
a priori evident whether the low energy correlations remain enough even after the
subtraction because the low energy contribution could also cancel out. Our Borel
Exotic Hadron in Pole-dominated QCD Sum Rules 3
analysis, however, reveals that, in the case of t = 1, the large low energy correlation
remains enough even after the subtraction. As a result, we can find the Borel window
in the relatively large M2-region.
On the other hand, for the chiral odd part, the subtraction procedure corre-
sponding to t = 1 case does not lead the cancellation of the high energy components
because chiral odd part is constructed of the chiral symmetry breaking terms. How-
ever, in the case of t = 1, the OPE convergence is found to be very good and then we
can find the Borel window in the small M2-region where high energy contaminations
are suppressed due to the Borel factor e−s/M
in the integral of the spectral function.
§3. Borel analysis for mass and residue
Here we explain our criterion for the Borel window. The lower bound of the
Borel window is given so that the highest-dimensional terms in the truncated OPE
are less than 10% of its whole OPE, while the upper bound is determined by the
region where the absolute value of the pole contribution is larger than the absolute
value of the integrated spectral function in the region s ≥ sth. Note that the 50%
pole contribution in our criterion is extremely large in comparison with any prior
pentaquark sum rules, where the pole contributions are not more than 20%.3)
To recognize the problems in the case of QSR for exotics, let us see Fig.1 for
M2-dependence of the mass in the cases of t = −1, 0, 10 corresponding to PP̄ +SS̄,
PP̄ , SS̄ cases respectively. The threshold value is fixed to typical value
sth = 2.2
GeV. In these cases, we fail to find stabilities of the mass in the M2-region lower
than the upper bound of the Borel window. The stabilities above the upper bound
are simply artifacts which often appear in QSR. Fig.1 shows that typical mass of PP̄
case is much smaller than that of SS̄, and then we can expect that the low energy
correlation of PP̄ is much larger than that of SS̄. This observation leads that even
after the subtraction PP̄ − SS̄ (t = 1 case), the low energy correlation can remain
enough.
Now we see the case of around t = 1. We tune the value of t around t = 1 to
get the widest Borel window. As expected, for even part (t = 0.9), the high energy
SS (t = 10)
PP (t = 0)
PP (t = 0)
SS (t = ‐1)
SS (t = ‐1)
SS (t = 10)
Fig. 1. The behavior of the mass as a function of M2 for t = −1, 0, 10. The left arrows represent
the upper bound of the Borel window. In the smaller M2-region than the upper bound, we can
not find stable region of the mass. The stabilities above the upper bound are simply artifacts
which often appear in QSR.
4 Toru Kojo, Daisuke Jido and Arata Hayashigaki
even odd
(t=1.1)(t=0.9)
Fig. 2. The behavior of the mass as a function of M2. The left (right) arrows represent the upper
(lower) bound of the Borel window. We succeed to find the Borel window and stabilities of
mass.
contaminations are canceled out due to chiral symmetry and we find the wide Borel
window in the relatively large M2-region. On the other hand, for odd part (t = 1.1),
thanks to the good OPE convergence, we also find the wide Borel window in the
small M2-region. The threshold values are taken to make the physical quantities
most stable in the Borel window.
The best stability is achieved with
sth = 2.2 GeV (even) and 2.1 GeV (odd),
giving mΘ+ = 1.64 GeV (even) and 1.72 GeV (odd) respectively. The values of the
residue are also obtained from the chiral even and odd sum rules as λ20 = (3.0±0.1)×
10−9 GeV12 and λ21/mΘ+ = (3.4 ± 0.2) × 10−9 GeV12. It is remarkable that these
numbers are quite similar with the close t. This implies that our analysis investigates
consistently the same state in the two independent sum rules. Note that from the
relative sign of the residues, we assign positive parity to the observed Θ+ state.
In conclusion, we perform the Borel analysis for Θ+ with special setup of the
correlators in order to find the Borel window. Within uncertainties of the condensate
value, independent analyses for the chiral-even and odd sum rules give the consistent
values of the Θ+ mass, 1.68± 0.22 GeV, and the residue. The parity is found to be
positive.
Acknowledgements
We thank Profs. M. Oka, A. Hosaka and S.H. Lee for useful discussions about
QSR for the exotics during the YKIS2006 on ”New Frontiers on QCD” held at
the Yukawa Institute for Theoretical Physics. This work is supported in part by
the Grant for Scientific Research (No.18042001) 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.
References
1) T. Nakano et al., Phys. Rev. Lett. 91 (2003), 012002.
2) M.A. Shifman, A.I. Vainshtein, and V.I. Zakharow, Nucl. Phys. B 147 (1979), 385.
3) R.D. Matheus and S. Narison, Nucl. Phys. Proc. Suppl. 152 (2006), 236.
4) T. Kojo, A. Hayashigaki, and D. Jido, Phys. Rev. C 74 (2006), 045206
5) S. Weinberg, Phys. Rev. Lett. 18 (1967), 507.
	Introduction
	OPE and favorable set up of the correlators
	Borel analysis for mass and residue
