Analysis of the structure of \(\Xi (1690)\) through its decays
Eur. Phys. J. C
(1690) through its decays
T. M. Aliev 2
K. Azizi 0 1
H. Sundu 3
0 Department of Physics, Dogˇus ̧ University , AcibademKadiköy, 34722 Istanbul , Turkey
1 School of Physics, Institute for Research in Fundamental Sciences (IPM) , P.O. Box 193955531, Tehran , Iran
2 Physics Department, Middle East Technical University , 06531 Ankara , Turkey
3 Department of Physics, Kocaeli University , 41380 Izmit , Turkey
The mass and pole residue of the first orbitally and radially excited state as well as the ground state residue are calculated by means of the twopoint QCD sum rules. Using the obtained results for the spectroscopic parameters, the strong coupling constants relevant to the decays (1690) → K and (1690) → K are calculated within the lightcone QCD sum rules and width of these decay channels are estimated. The obtained results for the mass of and ratio of the Br ( → K )/Br ( → K ), with representing the orbitally excited state in channel, are in nice agreement with the experimental data of the Belle Collaboration. This allows us to conclude that the (1690) state, most probably, has negative parity.

1 Introduction
Understanding the spectrum of baryons and looking for new
baryonic states constitute one of the main research
directions in hadron physics. Impressive developments of
experimental techniques allow discovery of many new hadrons.
Despite these developments, the spectrum of baryon is
still not well established. This is due to the absence of high
intensity antikaon beams and small production rate of the
resonances. At present time only the ground state octet and
decuplet baryons as well as (1320) and (1530) baryons
are well established. Up to present time the quantum
numbers of (1690), (1820) and (1950) have not been
determined. Theoretically, the spectrum of baryon, within
different approaches, have been studied intensively (see [
1–9
]
and references therein).
The main results of these studies are that different
phenomenological models explain successfully the nature of
(1320) and (1530) states. However, these approaches
predict controversially results for other excitations of
baryons. In [
8
] using the nonrelativistic quark model the
mass of (1690) is calculated and it is obtained that it might
be radial excitation of with J P = 21 +. This result was
then supported by the quark model calculations in [
5
].
However within the relativistic quark model in [
9
] it was
established that the first radial excitation should have mass around
1840 MeV. In [
4
] the authors suggested that the (1690)
state might be orbital excitation of with J P = 21 −.
This point of view was supported by calculations performed
within Skyrme model [
2
] and chiral quark model [
7
]. The
controversy results suggests independent analysis for
establishing the nature of (1690) state.
In the present study, within the light cone QCD sum rules,
we estimate the widths of the → K and → K
transitions. We suggest that (1690) state may be radial ( )
or orbital ( ) excitation of baryon. For establishing these
decays we need to know the residue of (1690) as well as the
strong coupling constants for these decays. For calculation
of the mass and residue of the states as the main inputs
of the calculations we employ the two point QCD sum rule
method.
The paper is arranged as follows. In Sect. 2 the mass and
residue of (1690) baryon within both scenarios, namely
considering (1690) as the orbital and radial excitations
of baryon, are calculated. In Sect. 3 we present the
calculations of the strong coupling constants defining the
(1690) → ( )K transitions within both scenarios. By
using the obtained results for the coupling constants we
estimate the relevant decay widths and compare our
predictions on decay widths with the existing experimental data
in this section, as well. We reserve Sect. 1 for the
concluding remarks and some lengthy expressions are moved to the
Appendix.
2 Mass and pole residue of the first orbitally and
radially excited state
For calculation of the widths of → K and → K
decays we need to know the residues of , and baryons.
In present work we consider two possible scenarios about
nature of the (1690): (a) it is represented as radial excitation
of the ground state . In other words it carries the same
quantum numbers as the ground state , i.e. J P = 21 +. (b)
The (1690) state is considered as first orbital excitation of
the ground state , i.e. it is negative parity baryon with J P =
21 −. In the following we will try to answer the question that
which scenario is realized in nature? To answer this question
we will calculate the mass of (1690) state and decay width
of the → K and → K transitions and then compare
the ratio of these decays as well as the prediction on the
mass with existing experimental data. Note that the BABAR
Collaboration has measured the mass (m = 1684.7±1.3+2.2)
−1.6
MeV and width ( = 8.1+−33..95+−10..09) MeV of (1690) [
10,11
]
and Belle Collaboration has measured the mass (m = 1688±
2) MeV and width ( = 11 ± 4) MeV of this state as well as
the ratio BB(( ((11669900))00→→KK¯−0 0+) . The experimental value for this
)
ratio measured by Belle is 0.50 ± 0.26 [12].
For determination of the mass and residue of baryon,
we start with the following two point correlation function:
(q) = i
d4x eiq·x 0T η (x )η¯ (0) 0 ,
where η (x ) is the interpolating current for state with spin
J P = 21 + and T indicates the time ordering operator. The
general form of the interpolating current for the spin 21
baryon can be written as [
13,14
]:
η =
abc
sT,a (x )C ub(x ) γ5sc(x )
+β sT,a (x )C γ5ub(x ) sc(x ) ,
where a, b, c are the color indices and β is an arbitrary
parameter with β = −1 corresponding to the Ioffe current. C is the
charge conjugation operator.
According to the general philosophy of QCD sum rules
method, for calculation of the mass and residue of baryons
the correlation function needs to be calculated in two
different ways: (a) in terms of hadronic degrees of freedom
and (b) in terms of perturbative and vacuumcondensates
contributions expressed as functions of QCD degrees of
freedom in deep Euclidian domain q2 0. After equating
these two representations, the desired QCD sum rules for
the physical quantities of the baryons under consideration
are obtained. As already noted, the quantum numbers J P
of (1690) state have not been determined via experiments
yet. Therefore, firstly we consider the case when (1690)
represents a negative parity baryon. The hadronic side of the
correlation function is obtained by inserting complete sets of
relevant intermediate states. For calculation of the hadronic
side of the correlation function, we would like to note that
the above interpolating current has nonzero matrix element
with baryons of both parities. Taking into account this fact
and saturating the correlation function by complete sets of
intermediate states with both parities we obtain:
Phys(q) =
+
+ . . . ,
0η (q, s) (q, s)η0
m2 − q2
0η (q, s) (q, s)η0
m2 − q2
(3)
(4)
where m, m and s, s are the masses and spins of the ground
and first orbitally excited baryons, respectively. Here dots
represent the contributions of higher states and continuum.
The matrix elements in Eq. (3) are determined as
0η (q, s) = λu(q, s),
0η (q, s) = λγ5u(q, s).
Here λ and λ are the residues of the ground and first orbitally
excited baryons, respectively. Using Eqs. (3) and (4)
and performing summation over the spins of corresponding
baryons, we obtain
Phys(q) =
λ2(q/ + m)
m2 − q2 +
λ2(q/ − m)
m2 − q2 + · · · .
We perform Borel transformation in order to suppress the
contribution of higher state and continuum,
B
m2 m2
Phys(q) = λ2e− M2 (q/ + m) + λ2e− M2 (q/ − m).
(5)
where M 2 is the square of Borel mass parameter.
The correlation function from QCD side can be calculated
by inserting Eq. (2) to Eq. (1) and usage of Wick’s theorem
to contract the quark fields. As a result we have an
expression in terms of the involved quark propagators having
perturbative and nonperturbative contributions. For calculation
of these contributions we need explicit expressions of the
light quark propagators. By using the light quark
propagators in the coordinate space and performing the Fourier and
Borel transformations, as well as performing the continuum
subtraction by using the hadronquark duality ansatz, after
lengthy calculations, for the correlation function we obtain
B
QCD(q) = B 1QCDq/ + B 2QCD I,
(6)
where, the expressions for B 1QCD and B 2QCD are
presented in Appendix.
Having calculated both the hadronic and QCD sides of the
correlation function, we match the coefficients of the
corresponding structures q/ and I from these representations to
find the following sum rules:
m2
where B 1Q(C2)D = − d(1/dM2) B 1Q(C2)D.
The sum rules for mass and residue of the radially excited
state are obtained from Eq. (8) by replacements m → −m
and λ → λ . Note that, there are other approaches to separate
the contributions of the positive and negative parity baryons
(for instance see [
15–18
]).
The sum rules for the mass and residue of the orbitally
and radially excited state of the baryon as well as the
residue of the ground state contain many input parameters.
Their values are presented in Table 1. For performing
analysis of widths of the → K and → K decays in next
section, we also need the residues of the and baryons.
We use the values of these residues calculated via QCD sum
rules [
20
]. The mass of the ground state is taken as input
parameter, as well. Besides these input parameters, QCD sum
rules contains three auxiliary parameters, namely the value
of continuum threshold s0, Borel mass square M 2 and β
arbitrary parameter. Obviously any measurable physical quantity
must be independent of these parameters. Hence we need to
find the working regions of these parameters, where physical
quantities demonstrate good stability agains the variations of
these parameters. The window for M 2 is obtained by
requiring that the series of operator product expansion (OPE) in
QCD side is convergent and the contribution of higher states
and continuum is sufficiently suppressed. Numerical
analyses lead to the conclusion that both conditions are satisfied
in the domain
The considerations of the pole dominance and OPE
convergence lead to the following working window for the
continuum threshold:
1.92 GeV2 ≤ s0 ≤ 2.12 GeV2.
In Figs. 1, 2, 3 and 4 we present, as examples, the
dependence of the mass of the state and residues of the , and
baryons on M 2 and s0 at fixed value of cos θ = 0.7, with
β = tan θ . From these figures we observe that the results
shows quite good stability with respect to the variations of
M 2 and s0.
In order to find the working region of β, as an example
in Fig. 5 we present the dependence of the groundstate
baryon’s residue on the cos θ . From this figure we see that
the residue is practically insensitive to the variations of cos θ
in the domains
−1 ≤ cos θ ≤ −0.3, 0.3 ≤ cos θ ≤ 1.
(11)
We depict the numerical results of the masses and residues
of the first orbitally and radially excited baryon as well as
the ground state residue in Table 2. The errors in the presented
results are due to the uncertainties in determinations of the
working regions of the auxiliary parameters as well as the
errors of other input parameters. From this table we see that
although consistent with the experimental data [
10–12
], the
radial and orbital excitation of receive the same mass,
which prevent us to assign any quantum numbers to (1690)
only via mass calculations. The residue of these two states are
obtained to be differ from each other by a factor of roughly
three.
(9)
(10)
3
and
transitions To
K and
K
→
→
In present section we calculate the strong couplings g
g K , g K and g K defining the → K , →
K and K transitions.
K ,
K ,
m
m
s
s
s
where η (x ) and η (x ) are the interpolating currents for the
and baryons, respectively. The general forms of these
currents are taken as [
13,14
]
1
η (x ) = − √
2
abc
1
η (x ) = √
6
abc
2
i=1
2
i=1
− d T,c(x )C Ai1sb(x ) Ai2ua (x ) ,
+ uT,a (x )C Ai1sb(x ) Ai2dc(x )
+ d T,c(x )C Ai1sb(x ) Ai2ua (x ) ,
uT,a (x )C Ai1sb(x ) Ai2dc(x )
2 uT,a (x )C Ai1db(x ) Ai2sc(x )
(13)
where a, b, c are color indices, C is the charge conjugation
operator and A11 = I , A12 = A12 = γ5, A22 = β. According to
the method used, we again calculate the aforesaid correlation
function in two representations: hadronic and QCD.
Matching these two sides through a dispersion relation leads to the
sum rules for the coupling constants under consideration.
Firstly let us consider the → K transition. As we
already noted, the interpolating currents for baryons can
interact with both the positive and negative parity baryons. In
what follows, we denote the ground state positive (negative)
parity baryons with ( ) and ( ). Taking into account
this fact, inserting complete sets of hadrons with the same
quantum numbers as the interpolating currents and isolating
the ground states, we obtain
Phys( p, q) =
×
+
+
×
+
+
1OPE m2K + m m
− m m
2OPE m
3OPE m
− m
− m
− m
−
4OPE ,
where 1OPE, 2OPE, 3OPE and 4OPE are the invariant
amplitudes corresponding to the structures q/ /pγ5, /pγ5, q/γ5
and γ5 for → K decay, respectively.
If we carry out the same procedures for → K decay,
for the coupling constant g K we obtain:
g
K = λ λ (m
+
×
where p = p + q, p and q are the momenta of the ,
baryons and K meson, respectively. In this expression m
is the mass of the baryon. The dots in Eq. (14) stand for
contributions of the higher resonances and continuum states.
The matrix elements in Eq. (14) are determined as
0η  ( p, s) = λ u( p, s),
0η  ( p, s) = λ γ5u( p, s),
K (q) ( p, s) ( p , s ) = g
K (q) ( p, s) ( p , s ) = g
K (q) ( p, s) ( p , s ) = g
K (q) ( p, s) ( p , s ) = g
K u( p, s)u( p , s ),
K u( p, s)u( p , s ),
K u( p, s)γ5u( p , s ).
K u( p, s)γ5u( p , s ),
(15)
where gi are the strong coupling constants for the
corresponding transitions.
Using the matrix elements given in Eq. (15) and
performing summation over spins of and baryons and applying
the double Borel transformations with respect p2 and p 2 for
physical side of the correlation function we get
B Phys( p, q) = g
−g
× /p + m
+g
×γ5 /p + m
K λ λ e−m2 /M12 e−m2 /M22
× /p + m
γ5 /p + q/ + m
−g
K λ λ e−m2 /M12 e−m2 /M22
×γ5 /p + m
γ5 /p + q/ + m
γ5
K λ λ e−m2 /M12 e−m2 /M22
/p + q/ + m
γ5
K λ λ e−m2 /M12 e−m2 /M22
/p + q/ + m
,
(16)
where M12 and M22 are the Borel parameters.
From Eq. (16) it follows that we have different structures
which can be used to obtain sum rules for the strong
coupling constant of → K channel. We have four couplings
(see Eq. 16), and in order to determine the coupling g K
we need four equations. Therefore we select the structures
q/ /pγ5, /pγ5, q/γ5 and γ5. Solving four algebraic equations for
g K , finally we get
g
K = λ λ (m
+ m )
(17)
(18)
(19)
where 1OPE, 2OPE, 3OPE and 4OPE are the invariant
amplitudes corresponding to the structures q/ /pγ5, /pγ5, q/γ5
and γ5 for → K decay, respectively.
The general expressions obtained above contain two Borel
parameters M12 and M12. In our analysis we choose
M12 = M22 = 2M 2, M 2 = M12 + M22
M12 M22 ,
since the masses of the involved and ( ) are close to
each other.
The sum rules for the coupling constants for → K
and → K transitions can be easily obtained from Eqs.
(17) and (18), by replacing m → −m and λ → λ .
The OPE side of the correlation function OPE( p, q) can
be obtained by inserting the corresponding interpolating
currents to the correlation function, using Wick’s theorem to
contract the quark fields, and inserting into the obtained
expression the relevant quark propagators. The
nonperturbative contributions in light cone QCD sum rules, which are
described in terms of the K meson distribution amplitudes
(DAs), can be obtained by using Fierz rearrangement formula
α β = 41 βiα(sa i ub),
a b
s u
where i = 1, γ5, γμ, i γ5γμ, σμν /√2 is the full set of
Dirac matrices. The matrix elements of these terms between
the K meson and vacuum states, as well as ones generated
by insertion of the gluon field strength tensor Gλρ (uv) from
quark propagators, are determined in terms of the K meson
DAs with definite twists. The DAs are main nonperturbative
Table 3 The sum rule results for the strong coupling constants and
decay widths of the first orbitally and radially excited baryon
→
→
→
→
inputs of light cone QCD sum rules. The K meson
distribution amplitudes are derived in [
21–23
] which will be used in
our numerical analysis. All of these steps summarized above
result in lengthy expression for the OPE side of correlation
function. In order not to overwhelm the study with overlong
mathematical expressions we prefer not to present them here.
Apart from parameters in the distribution amplitudes, the sum
rules for the couplings depend also on numerical values of
the and baryon’s mass and pole residue, which are given
in Table 1. Note that the working region of the Borel mass
M 2, threshold s0 and β parameters for calculations of the
relevant couplings are chosen the same as in the residue and
mass computations.
Performing numerical analysis for the relevant coupling
constants we get values presented in Table 3. Using the
couplings g K , g K g K and g K we can easily
calculate the width of → K , → K , → K and
→ K decays. After some computations we obtain:
→
K
g2 K
= 16π m3
(m
+ m )2 − m2K
×λ1/2 m2 , m2 , m2K ,
and
→
K
g2 K
= 16π m3
(m
− m )2 − m2K
×λ1/2 m2 , m2 , m2K .
(20)
(21)
In expressions above the function λ(x 2, y2, z2) is given as:
λ(x 2, y2, z2) = x 4 + y4 + z4 − 2x 2 y2 − 2x 2z2 − 2y2z2.
The expressions for the widths of the → K and →
K can be easily obtained from Eqs. (20) and (21), by the
replacement m → m .
Using the values of coupling constants and formulas for
the decay widths we obtain the values of the partial width at
different decay channels presented in Table 3.
Using the values of the partial decay widths from Table
3, we finally obtain the ratio of the branching fractions in
channel as
(22)
(23)
(24)
Br
Br
Br
Br
Br
Br
and for
channel we get
→
→
→
→
K −
K 0
K −
K 0
Note that in [
24
], within the coupled channel approach, a very
similar results has been found. The authors have concluded
that the (1690) has spin1/2, but its parity has not been
established. Our prediction for the corresponding ratio in
channel is considerably small compared to the
experimental data. From these results and those for the values of the
corresponding masses we conclude that the (1690) state,
most probably, has quantum numbers 21 −, i.e. it represents a
negative parity spin1/2 baryon.
Acknowledgements K. A. thanks Dogus University for the partial
financial support through the grant BAP 201516D1B04.
Open Access This article is distributed under the terms of the Creative
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Funded by SCOAP3.
Appendix: The QCD side of the correlation function in
mass sum rules
B 1QCD(q) =
In present Appendix we present explicit forms of the
functions in QCD side of the two point correlation function used
in mass sum rules:
s0
1
ds e− Ms2 27π 2
s2(5β2 + 2β + 5)
24π 2
+ms s¯s (5β2 + 2β + 5) + 6 u¯u + d¯d (β2 − 1)
0
+
−
+
gs2GG (5β2 + 2β + 5)
24π 2
3m20 u¯u + d¯d ms (β2 − 1)
M2
log
gs2GG
uu + d¯d
¯
3M4
and
B 2QCD(q) =
×ms (β2 − 1) log
β − 1
+ 24
+ u¯u d¯d (β − 1)
2
3m20 u¯u + d¯d
gs2GG
− 3 · 24π 2 ms (β − 1)2 8 + 3γE − 3 log
⎫
⎬
+
gs2GG
(β2 − 1)⎭ + 210π 4
s
×γE ms M2(β − 1)2 1 − e− M02
ms
− 3 · 24 3 s¯s d¯d + s¯s u¯u (β2 − 1)
β − 1
+ 3 · 28π 2 gs2GG 3 u¯u + d¯d (β + 1)
− s¯s (β − 1) +
×ms (5β2 + 2β + 5) +
+
gs2GG m20 u¯u d¯d
32 · 26 M6
gs2GG u¯u d¯d
32 · 26 M4
3 · 24 M2
ms (5β2 + 2β + 5),
(A.2)
where, to shorten the expressions, the terms proportional to
mu and md are not presented, although their contributions
are taken into account in performing numerical analysis.
1. N. Isgur , G. Karl, Phys. Rev. D 18 , 4187 ( 1978 )
2. Y. Oh , Phys. Rev. D 75 , 074002 ( 2007 ). arXiv:hepph/0702126
3. F.X. Lee , X.Y. Liu , Phys. Rev. D 66 , 014014 ( 2002 ). arXiv:nuclth/0203051
4. M. Pervin , W. Roberts, Phys. Rev. C 77 , 025202 ( 2008 ). arXiv: 0709 .4000 [nuclth]
5. T. Melde , W. Plessas , B. Sengl , Phys. Rev. D 77 , 114002 ( 2008 ). arXiv: 0806 .1454 [hepph]
6. C.L. Schat , J.L. Goity , N.N. Scoccola , Phys. Rev. Lett . 88 , 102002 ( 2002 ). arXiv:hepph/0111082
7. L.Y. Xiao , X.H. Zhong , Phys. Rev. D 87 ( 9 ), 094002 ( 2013 ). arXiv: 1302 .0079 [hepph]
8. K.T. Chao , N. Isgur , G. Karl, Phys. Rev. D 23 , 155 ( 1981 )
9. S. Capstick , N. Isgur , Phys. Rev. D 34 , 2809 ( 1986 )
10. B. Aubert et al., BaBar Collaboration. Phys. Rev. D 78 , 034008 ( 2008 ). arXiv:0803 . 1863 [hepex]
11. B. Aubert et al. [ BaBar Collaboration] . arXiv: hepex/0607043 (V. Ziegler, SLACR 868)
12. K. Abe et al. [Belle Collaboration], Phys. Lett. B 524 , 33 ( 2002 ). arXiv:hepex/0111032
13. V. Chung , H.G. Dosch , M. Kremer , D. Scholl , Nucl. Phys. B 197 , 55 ( 1982 )
14. H.G. Dosch, M. Jamin , S. Narison, Phys. Lett. B 220 , 251 ( 1989 )
15. E. Bagan, M. Chabab , H.G. Dosch , S. Narison, Phys. Lett. B 301 , 243 ( 1993 )
16. D. Jido , N. Kodama , M. Oka , Phys. Rev. D 54 , 4532 ( 1996 )
17. Z.G. Wang , Phys. Lett. B 685 , 59 ( 2010 )
18. Z.G. Wang , Eur. Phys. J. A 45 , 267 ( 2010 )
19. C. Patrignani et al. (Particle Data Group), Chin. Phys. C 40 , 100001 ( 2016 ) (and 2017 update )
20. T.M. Aliev , A. Ozpineci , M. Savci , Phys. Rev. D 66 , 016002 ( 2002 ). arXiv:hepph/0204035 [Erratum: Phys. Rev. D 67 , 039901 ( 2003 )]
21. P. Ball , V.M. Braun , A. Lenz , JHEP 0605 , 004 ( 2006 )
22. V.M. Belyaev , V.M. Braun , A. Khodjamirian , R. Ruckl , Phys. Rev. D 51 , 6177 ( 1995 )
23. P. Ball , R. Zwicky , Phys. Rev. D 71 , 014015 ( 2005 )
24. K.P. Khemchandani , A. Martnez Torres , A. Hosaka , H. Nagahiro , F.S. Navarra , M. Nielsen . arXiv: 1712 .09465 [hepph]