# Strong Coupling Constants of Negative Parity Heavy Baryons withπ and K Mesons

Authors: T. M. Aliev; S. Bilmis; M. Savci
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

The strong coupling constants of negative parity heavy baryons belonging to sextet and antitriplet representations ofSUf(3) with light π and K mesons are estimated within the light cone QCD sum rules. It is observed that each class of the sextet-sextet, sextet-antitriplet, and antitriplet-antitriplet transitions can be described by only one corresponding function. The pollution arising from the positive to positive, positive to negative, and negative to positive parity baryons transitions is eliminated by constructing sum rules for different Lorentz structures. The obtained coupling constants are compared with the ones for the positive parity heavy baryons.


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## Body

## 1. Introduction

Recent years have been very productive in heavy baryon spectroscopy. The new states withJP=1/2+ (Λc+, Ξc+) and JP=1/2- (Λc+(2593), Ξc+(2790), Ξc0(2790)) as well as JP=1/2+ (Ωc(∗), Σc(∗)) and JP=3/2+ (Ξc(∗)) states have been discovered [1]. Lately, LHCb collaboration has reported the observation of the five extremely narrow excited Ωc baryons decaying into Ξc+K- [2]. One possible explanation for the existence of these baryons is that they could be the bound states of P wave ss-diquark and c-quark [3]. Up to now, the quantum numbers of newly observed Ωc states have not been identified yet. Therefore, the phenomenological studies play a crucial role in this subject. We hope that in the near future more detailed analysis of experiments will be carried out and the quantum numbers will be determined. We stress that the observation of new Ωc baryons paves the way for focusing on the studies of the properties of the negative parity baryons. A comprehensive theoretical study of weak, electromagnetic, and strong decay requires particular attention. A better way to distinguish the particles with different parity from the same final state is studying various decay channels, where one of the main parameters is the strong coupling constant.Present work is devoted to the calculation of the strong coupling constants of negative parity heavy baryons with pseudoscalarπ and K mesons within the light cone QCD sum rules (LCSR) (see, e.g., [4]). The strong coupling constants are the fundamental quantities to determine the strength of the strong interaction among the participated particles as well as playing an essential role for understanding the structure of hadrons. Note that the coupling constants of positive parity heavy baryons with π and K mesons are calculated in [5]. A comparison study of strong decay constants for positive and negative parity states gives insight into underlying QCD dynamics at large distance.The paper is organized as follows. In Section2, the sum rules for the strong coupling constants of negative parity heavy baryons with pseudoscalar mesons (BBP) are derived. In Section 3, the numerical analysis of the derived sum rules forBBP couplings is performed. Section 4 contains our conclusion.


## 2. Theoretical Framework

It is well known that the heavy baryons with single heavy quark belong to either a symmetric6F or antisymmetric 3¯F flavor representation of SU(3)F group. From spin, color, and flavor symmetry properties it follows that the particles belonging to 6F representation have total spin 1/2 or 3/2, while the particles belonging to 3¯F representation only have spin 1/2.In order to calculate theBBP coupling constants in the framework of the light cone QCD sum rules, we consider the following correlation function:(1)Παρ=i∫d4xeiqxPqηαxη¯ρ00,where α=ρ=S, α=S, ρ=A, and α=ρ=A cases correspond to sextet-sextet, sextet-antitriplet, and antitriplet-antitriplet transitions, respectively. The interpolating currents for the spin-1/2 sextet and antitriplet heavy baryons can be written as(2)ηS=-12ϵabcq1aTCQbγ5q2c+βq1aTCγ5Qbq2c-QaTCq2bq1c+βQaTCγ5q2bq1c,(3)ηA=16ϵabc2q1aTCq2bγ5Qc+2βq1aTCγ5q2bQc+q1aTCQbγ5q2c+βq1aTCγ5Qbq2c+QaTCq2bγ5q1c+βQaTCγ5q2bq1c,where a, b, c are the color indices and β is the arbitrary parameter, and β=-1 case corresponds to Ioffe current which we shall use in further discussions. The light quark content of sextet and antitriplet heavy baryons are given in Table 1.Table 1
The quark flavorsq1 and q2 for the baryons in the sextet and the antitriplet representations.
Σ
b
(
c
)
+
(
+
+
)
Σ
b
(
c
)
0
(
+
)
Σ
b
(
c
)
-
(
0
)
Ξ
b
(
c
)
-
(
0
)
′
Ξ
b
(
c
)
0
(
+
)
′
Ω
b
(
c
)
-
(
0
)
Λ
b
(
c
)
0
(
+
)
Ξ
b
(
c
)
-
(
0
)
Ξ
b
(
c
)
0
(
+
)
q
1
u
u
d
d
u
s
u
d
u
q
2
u
d
d
s
s
s
d
s
sThe first stage of the calculation ofΠ(αρ) within the light cone sum rules is writing it in terms of hadronic parameters. Saturating (1) by corresponding baryons we get,(4)Παρ=0ηαBfpBfpPq∣Bip+qBip+qη¯ρ0p2-mf2p+q2-mi2+⋯,where p (p+q) and mf (mi) are the four momenta of final (initial) baryon and mass, respectively.The interpolating currentη(S) or η(A) interacts with both positive and negative parity baryons. Under this circumstance the matrix elements in (2) are defined as(5)0ηαB2+p=λ+αu¯2p,0ηαB2-p=λ-αγ5u2p,B2±Pq∣Bi±=g±±u¯2pγ5u1p+q,B2±Pq∣Bi∓=g∓±u¯2piu1p+q,where λ+(-) is the residue of the positive (negative) parity baryon and gij is the coupling constant of the pseudoscalar meson with heavy baryon.Now let us first consider the sextet-sextet transition. Using (4) and (5) and performing summations over spins of initial and final baryons, we get(6)Παρ=A++αρp+m2+γ5p+q+m1++A--αργ5p+m2-γ5p+q+m1--γ5+A-+αρp+m2+p+q+m1--γ5+A+-αργ5p+m2-p+q+m1+in which(7)Aijαρ=gijαρλ2jαλ1iρm2j2-p2m1i2-p+q2, and i(j)=±(±).To construct the sum rules for the strong coupling constants ofSSP, SAP, and AAP (S, A, and P correspond to sextet, antitriplet states, and pseudoscalar meson, resp.), the expressions of the correlation functions from the QCD side are needed. Before calculating the SSP, SAP, and AAP coupling constants, we would like to make the following remark. Considering the approach presented in [6], one can easily show that each class of transitions can be described by only one invariant function. The main advantage of this approach is that relations among the invariant functions become structure independent although the expression of the invariant function is structure dependent. (For more details see [5].) Hence, to determine all the relevant coupling constants, it is enough to consider the ΣQ0→ΣQ0P, ΞQ′0→ΞQP and ΞQ0→ΞQ0P transitions. All the other possible strong coupling constants can be obtained from these couplings; that is, each class of transitions is described only by one coupling constant.Relations among the invariant functions can easily be obtained by using the approach given in [6]. It follows from (6) that the correlation function contains contributions from positive-negative, negative-positive, and negative-negative transitions. Among them, we need only one, namely, A--, which described the transition between negative parity heavy baryons. For this goal, we need to solve four equations obtained from four different Lorentz structures, pqγ5, pγ5, qγ5, and γ5.Now, let us turn our attention to the calculation of the correlation function from QCD side. As we have already noted, to construct the sum rules for strong coupling constants forSSP, SAP, and AAP transitions, calculations of the correlation functions from QCD side are required. This can be calculated by using the operator product expansion (OPE) over the twist of nonlocal operators in the deep Euclidean region, where -p2→∞ and -(p+q)2→∞.The QCD part of the correlation function can be obtained by contracting quark fields. In this case, two different contributions appear: perturbative and nonperturbative. The perturbative contribution can be obtained by inserting the free quark propagators into the expression of correlation functions. The expressions of the light and heavy quark propagators are(8)Sabx=ixδab2π2x4-mqδab4π2x2-i32π2gsGμνabσμνx+xσμν-ms32π2gsGμνnλnab2ln⁡-x2Λ24σμν-mqδab293π2x2ln⁡-x2Λ24gs2G2+⋯,SQabx=mQ3δab2π2mQixmQ2-x22K2mQ-x2+1mQ-x2K1mQ-x2-mQgsGμνab82π2mQiσμνx+xσμνK1mQ-x2mQ-x2+2σμνK0mQ-x2,where Ki are the modified Bessel functions of the second kind; mq and mQ are the light and heavy quark mass, respectively. Λ is the parameter for separating the perturbative and nonperturbative regions and for numerical calculations we use Λ=0.5GeV [7]. On the other hand, the nonperturbative contribution can be obtained by replacing one of the light quark propagators by(9)Sαβab⟶-14Γβαiq¯aΓiqb,where Γi is the full set of Dirac matrices. Then this expression is sandwiched between the pseudoscalar meson and vacuum states. These matrix elements as well as the ones 〈P(q)|q¯ΓGμνq|0〉, where Gμν is the gluon field strength tensor, are determined in terms of pseudoscalar meson distribution amplitudes (DA). Up to twist 4 the explicit expressions of the pseudoscalar meson DAs are presented in [8, 9]. In the present study we neglect the four-particle contributions due to their small values (see, e.g., [10].) Using these expressions for the propagators and DAs for the pseudoscalar mesons, the correlation functions from the QCD side can be calculated.Equating the coefficients of the Lorentz structurespqγ5, pγ5, qγ5, and γ5 in hadronic and QCD sides and performing the Borel transformations over p2 and (p+q)2 in order to suppress the contributions of higher states and continuum, we get the following four linear equations:(10)Π1αρB=-A++αρB+A--αρB-A-+αρB+A+-αρB,Π2αρB=m1+-m2+A++αρB+A--αρBm1--m2+-m2++m1-A-+αρB-m2-+m1+A+-αρB,Π3αρB=-m2+A++αρB-m2-A--αρB-m2+A-+αρB-m2-A+-αρB,Π4αρB=-m1+2+m1+m2+A++αρB+-m2-m1-+m1-2A--αρB-m2+m1-+m1-2A-+αρB+m2-m1++m1+2A+-αρB,where Aij(αρ)B=gij(αρ)λ1i(α)λ2j(ρ)e-m1i2/M12-m2j2/M22.Solving (10) for A--(αρ)B we get the desired sum rules for the strong coupling constant g--, responsible for the SSP, SAP, and AAP transitions:(11)g--αρ=em1-2/M12+m2-2/M222m1-m2-+m2+2m1++m2--m2+λ1-αλ2-ρ×m1-m1+Π1αρB-Π3αρBm1-+m2++m1-Π2αρB+m1+m2+Π1αρB-m1+Π2αρB+m2+Π2αρB-Π4αρB.The expressions ofΠi(αρ)B responsible for SSP, SAP, and AAP are very lengthy, and for this reason, we do not present all of them here.In (11), M12 and M22 are the Borel mass square parameters. Due to the fact that initial and final baryon masses are very close to each other, we take M12=M22=2M2. The residues of λ1i(α) and λ2j(ρ) of heavy baryons are calculated in [11]. The continuum subtraction procedure is carried out at M12=M22=2M2 and at u0=M12/M12+M22=1/2 point by using the expression(12)M2ne-m2/M2⟶1Γn∫mQ2s0dse-s/M2s-mQ2n-1.


## 3. Numerical Analysis

In the present section, we perform the numerical analysis of the light cone sum rules for the strong coupling constants of spin1/2 negative parity heavy baryons with π and K mesons obtained in the previous section. The main input parameters of the LCSR are the pseudoscalar meson DAs which are derived in [8, 9]. The values of other input parameters entering the sum rules are the lepton decay constants fπ and fK whose values are fπ=0.13 GeV and fK=0.16 GeV. For performing numerical analysis, the masses of the u and d quarks are taken as zero and for the mass of strange, charm and beauty quarks we used: ms=(96-4+8) MeV, mc=(1.27±0.03) GeV, and mb=(4.18-0.03+0.04) GeV, and we put renormalization scale μ=1 GeV. If the renormalization scale is chosen as μ=2 GeV, the results change by 5–8% at most.The sum rules for the coupling constants describingSSP, SAP, and AAP transitions contain two auxiliary parameters (when β=-1), continuum threshold s0 and Borel mass parameter M2. One of the main issues of the sum rules is to find the so-called working region of these parameters where the coupling constants are insensitive to their variation. The limits of M2 are obtained from the conditions that the pole contributions should be dominant over the higher states and continuum contributions as well as the convergence of operator product expansion. Our numerical analysis shows that these conditions are satisfied when M2 change in the regions.(i)
Forb-sector:(13)5GeV2≤M2≤10GeV2forΛb,Σb,5GeV2≤M2≤11GeV2forΞb,Ξb′,6GeV2≤M2≤12GeV2forΩb.(ii)
Forc-sector:(14)3GeV2≤M2≤6GeV2forΛc,Σc,3GeV2≤M2≤7GeV2forΞc,Ξc′,4GeV2≤M2≤8GeV2forΩc.The continuum threshold in sum rules is related to the first excited energy state,s0-mground=Δ. Analysis of the various sum rules leads to the result that usually Δ varies between 0.3 and 0.8 GeV. In our numerical analysis for Δ, we used Δ=0.4: 0.5 GeV. As an example, in Figures 1, 2, and 3, we present the dependence of the coupling constant for Ωc0→Ξc′0K, Ξc′+→Σc++K-, and Ξc0→Λc+K- transitions on M2 at two fixed values of s0. From these figures we observe that g-- exhibits good stability with respect to the variation of M2 in the “working region.” The results are weakly dependent on the s0 values and we get(15)g--=10±2,forΩb⟶Ξb′0K-10±2,forΞc′+⟶Ξc++K-10.5±1.0,forΞc0⟶Λc+K-.A similar analysis for strong coupling constants of the other channels is performed, and the obtained values are depicted in Tables 2 and 3. In this table, we also present the values of the strong coupling constants of the corresponding positive parity baryons for completeness.Table 2
Channel
Strong coupling constant
Negative parity
Positive parity
Ξ
b
′
-
→
Ξ
b
-
K
+
3
±
1
9.8
±
3.5
Ω
b
-
→
Ξ
b
-
K
-
0
5
±
1
13.5
±
4.8
Σ
b
0
→
Ξ
b
0
K
-
0
5
±
1
8.9
±
3.1
Ξ
b
′
0
→
Σ
b
+
K
-
9
±
1
10.0
±
3.6
Ω
b
-
→
Ξ
b
′
0
K
-
10
±
2
12.3
±
4.4
Σ
b
-
→
Λ
b
0
π
-
8
±
2
11.5
±
3.9
Ξ
b
′
0
→
Ξ
b
0
π
0
4
±
1
6.1
±
2.2
Σ
b
0
→
Σ
b
-
π
+
8
±
2
13
±
4.5
Ξ
b
′
0
→
Ξ
b
′
0
π
0
4
±
1
7.3
±
2.6Table 3
Channel
Strong coupling constant
Negative parity
Positive parity
Ξ
c
′
0
→
Ξ
c
0
K
+
1.5
±
0.5
2.1
±
0.8
Ω
c
0
→
Ξ
c
0
K
-
0
2.5
±
0.5
3.0
±
1.1
Σ
c
+
→
Ξ
c
+
K
0
2
±
1
3.7
±
1.3
Ξ
c
′
+
→
Ξ
c
+
+
K
-
10
±
2
9.0
±
1.0
Ω
c
0
→
Ξ
′
+
K
0
11
±
1
5.6
±
1.9
Σ
c
0
→
Λ
c
+
π
-
3
±
1
5.6
±
1.8
Ξ
c
′
+
→
Ξ
c
π
0
1.5
±
0.5
2.0
±
0.7
Σ
c
+
→
Σ
c
0
π
+
11
±
2
4.1
±
1.5
Ξ
c
′
+
→
Ξ
c
′
+
π
0
5
±
1
3.0
±
1.1Figure 1
The dependence of the strong coupling constant forΩc→Ξc′+K transition on M2 at two fixed values of s0.Figure 2
The same as in Figure1, but for Ξc′+→Σc++K- transition.Figure 3
The same as in Figure1, but for Ξc0→Λc+K- transition.The errors in the values of theg-- included uncertainties coming from the variation of s0, M2, and the values of the input parameters. Comparing these results, we see that the strong coupling constants of heavy hadrons with b-quark with pseudoscalar mesons for positive parity baryons are larger than similar couplings for negative parity ones. In the transitions between heavy hadrons with c-quark, the situation is analogous to the previous case. Only for a small number of transitions, the coupling constants of the negative parity heavy baryons with pseudoscalar mesons exceed similar couplings for positive parity ones.Our final remark is about the form of the interpolating current. As we already noted, the interpolating current contains auxiliary parametersβ. We perform calculations at β=-1 (Ioffe current). In order to compare the results for strong coupling constants when general current and Ioffe current are used, the results of this present work should be recalculated, and the numerical analysis needs to be reperformed.


## 4. Conclusion

In this work, we estimate the strong coupling constants of the negative parity heavy baryons with pseudoscalar mesons within the light cone QCD sum rules. It is emphasized that each class of transitions, namely, sextet-sextet, sextet-antitriplet, and antitriplet-antitriplet transitions, can be described by only one corresponding invariant function. The values of strong coupling constants for the transitions under consideration are presented, and the results are compared with similar couplings for positive parity heavy baryons.



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