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Exotic baryon states in QCD sum rule
A.G.Oganesian
Institute of Theoretical and Experimental Physics
B.Cheremushkinskaya 25, 117218 Moscow,Russia
Abstract
It is shown, that the small decay width of # + = uudd•s baryon is suppressed by chirality violation.
It is shown that # + decay width # is proportional to # 2
s #0|•qq|0# 2 , for any pentaquark current without
derivatives.
In this talk we will discuss the the narrow exotic baryon resonance # + with quark content # + = uudd•s
and mass 1.54 GeV. This resonance had been discovered last year by two groups [1, 2]. Later, the existence
of this resonance was confirmed by many other groups, although some searches for it were unsuccessful.
(see [3] for the review). # + baryon was predicted in 1997 by D.Diakonov, V.Petrov and M.Polyakov
[4] in the Chiral Soliton Model as a member of antidecouplet with hypercharge Y = 2. The recent
theoretical reviews are given in [5, 6]. # + was observed as a resonance in the systems nK + and pK 0 .
No enhancement was found in pK + mass distributions, what indicates on isospin T = 0 of # + in accord
with theoretical predictions [4].
My talk is based mainly on our paper [7] and I try to explain some point of our paper more detailed.
One of the most interesting features of # + is the very narrow width. Experimentally, only an upper limit
was found, the stringer bound was presented in [2]: # < 9MeV . The phase analysis of KN scattering
results in the even stronger limit on # [8], # < 1MeV . A close to the latter limitation was found in [9]
from the analysis of Kd # ppK reaction and in [10] from K +Xe collisions data [2]. The Chiral Quark
Soliton Model gives the estimation [4]: #CQSM <
# 15MeV (R.E.Ja#e [11] claims that this estimation
has a numerical error and in fact #CQSM <
# 30MeV -- see, however, [12]). In any way, such extremely
narrow width of # + (less than 1MeV ) seems to be very interesting theoretical problem. My talk will be
organized in the following way: in the sect.1 I suggest the qualitative explanation of the narrow width of
pentaquark and show that it is strongly parametrically suppressed. It will be shown, that the conclusion
does not depend of the choice of the pentaquark current (without derivatives). In sect. 2 we will discuss
the possible pentaquark currents and consider two­point correlation function.
Part 1.
In this section we will estimate the pentaquark width. Let us consider 3­point correlator
# µ = # e i(p1 x-qy)
#0 | # # (x)j 5
µ (y)# n (0) | 0# (1)
where #n (x) is the neutron quark current [?], (# n = # abc (d a C#µ d b )# 5 #µ u c ),
#0 | #n | n# = #n vn , (v n and #n are nucleon spinor and nucleon transition constant into nucleon current
eta n ), # # is arbitrary pentaquark current (not only # + , but with other isospin also) #0 | # # | # +
# = # # v #
and j µ5 = • s#µ # 5 u is the strange axial current.
As an example of # # one can use the following one:
## (x) = [# abc (d a C# µ# d b )# # u c
· •
s#µ # 5 u - (u # d)]/ # 2, (2)
though all results in this section are the same for any pentaquark current (without derivatives). (Note,
that this current have isospin 1, and I glad to thank M. Nielsen, who pay my attention to this, but for
any current with isospin 0 and without derivatives results will be just the same).
As usual in QCD sum rule the physical representation of correlator (1) can be saturated by lower
resonances (both in ## and nucleon channel)
# Phys
µ = #0 | # # | # +
### +
| j µ | n##n | #n | 0# 1
p 2
1 -m 2
#
1
p 2
2 -m 2
+ cont. (3)
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where p 2 = p 1 - q is nucleon momentum, m and m # are nucleon and pentaquark masses.
Obviously,
## +
| j µ | n# = g A
#n •
v #
# g µ#
- q µ q #
q 2
# # v # 5 vn (4)
where axial transition constant g A
#n is just we are interesting in (the width is proportional to the square
of this value). Such a method for calculation the width in QCD sum rules is not new, see, e.g. [?].
Substituting all these in eq (3) one can easily see, that 3­point correlator (1) is proportional to g A
#n .
Let us neglect quark masses and perform the chiral transformation in (1) q # # 5 q. It is evident, that
#n and j µ5 are even under such transformation, while ## is odd. Therefore, the correlator (1) vanishes
in the limit of chiral symmetry. It is easy to see, that this statement is valid for any form of pentaquark
and nucleon quark currents (spinless and with no derivatives). In the real world the chiral symmetry
is spontaneously broken. The lowest dimension operator, corresponding to violation of chiral symmetry
is •
qq. So, the correlator (1) is proportional to quark condensate #0|•qq|0#. Just the same result one can
found from direct calculation of invariant amplitude (at convenient kinematical structure, for example
“ pp µ ). So we come to conclusion, that # + width is suppressed by chirality violation for any pentaquark
current without derivatives(i.e. axial transition constant should be proportional to quark condensate).
The second source of the suppression also does not depend on the form of the pentaquark current. Let
us again consider correlator (1). One can easily note, that unit operator contribution to this correlator
(bare diagram) is expressed in the terms of the following integrals
# e i(p1 x-qy) d 4 xd 4 y
((x - y) 2 ) n (x 2 ) m # # e ip1 x
(x 2 ) m
e -iqt
(t 2 ) n d 4 xd 4 t (5)
It is clear that such integrals have imaginary part on p 2
2 and q 2 ­ the momentum of nucleon and
axial current ­ but there is no imaginary part on p 2
1 ­ the momentum of pentaquark. So we come to the
conclusion that bare diagram correspond to the case, when there is no # + resonance in the pentaquark
current channel (this correspond to background of this decay). (Note, that this conclusion don't depend
on the fact that one of the quark propagators should be replaced by condensate, as we discuss before). The
imaginary part on p 2
1 (i.e. # + resonance)appears only if one take into account hard gluon exchange. So
we come to conclusion, that if # + is a genuine 5­quark state (not, say, the NK bound state), then in (2)
the hard gluon exchange is necessary, what leads to additional factor of # s . We come to the conclusion,
that ## # # 2
s #0|•qq|0# 2 , i.e., ## is strongly suppressed. This conclusion takes place for any genuine
5­quark states -- the states formed from 5 current quarks at small separation, but not for potentially
bounded NK­resonances, corresponding to large relative distances. There are no such suppression for
the latters. I want to repeat once more, that this conclusion don't depend on the choice of current
(without derivatives).
(I would like to add that recently (after this talk was given) in the paper of D.Melikhov and B.Stech
[13], the pentaquark in the Chiral symmetry limit was investigated and authors note that they results
agree with our conclusion about pentaquark width suppression due to chirality violation).
Author thanks K.Goeke and to M.Polyakov for useful discussions and for their kind hospitality in
Bochum university and M.Nielsen for significant note.
This work was supported in part by INTAS grant 2000­58 and by RFBR grant 03­02­16209.
References
[1] T.Nakano et al., Phys.Rev.Lett. 91 (2003) 012002
[2] V.V.Barmin, A.G.Dolgolenko et al., Yad.Fiz. 66 (2003) 1763 (Phys.At.Nucl. 66 (2003) 1715)
[3] K.Hicks,hep­ph/0408001.
[4] D.Diakonov, V.Petrov and M.Polaykov, Z.Phys. A359 (1997) 305
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[5] D.Diakonov, hep­ph/0406043.
[6] V.B.Kopeliovich, Uspekhi Fiz.Nauk, 174 (2004) 323
[7] B.L.Io#e, A.G.Oganesian, hep­ph/0408152 (2004)
[8] R.A.Arndt, I.I.Strakovsky and R.L.Workman, nucl­th/0311030.
[9] A.Sibirtsev, J.Heidenbauer , S.Krewald and lf­G.Meissner, hep­ph/0405099.
[10] A.Sibirtsev, J.Heidenbauer, S.Krewald and Ulf­G.Meussner, nucl.th/0407011.
[11] R.Ja#e, hep­ph/0401158,hep­ph/0405268.
[12] D.Diakonov, V.Petrov and M.Polyakov,hep­ph/0404212.
[13] D. Melikhov and B.Stech hep­ph/04090015 (2004).
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