Determination of the $\Sigma$--$\Lambda$ mixing angle from QCD sum rules

The $\Sigma$--$\Lambda$ mixing angle is calculated in framework of the QCD sum rules. We find that our prediction for the mixing angle is $(1.00\pm 0.15)^0$ which is in good agreement with the quark model prediction, and approximately two times larger than the recent lattice QCD calculations.


Introduction
Flavor symmetry plays essential role in classification of the hadrons. The light hadronic states are successfully described by using SU (3) flavor symmetry. In the case this symmetry is exact, hadrons belonging to the same representation of SU(3) flavor group could be degenerate. Experimentally it is known that the hadrons belonging to the same representation have different masses, which leads to SU(3) flavor symmetry breaking. At quark level, this symmetry is broken due to the mass difference of the light u, d and s quarks.
The breaking of the SU(3) flavor symmetry might lead to mixing of hadrons. In other words, the definite flavor eigenstates can mix to form the physically observed states.
Long time ago, it is observed that the lowest lying hadrons Λ and Σ can be represented as the combination of the SU(3) octet, pure isospin I = 0 (Λ), and I = 1 (Σ 0 ) baryons in the following form [1], Λ = Λ 0 cos α − Σ 0 sin α , Σ = Λ 0 sin α + Σ 0 cos α . (1) The Σ − Λ mixing angle is estimated in framework of the quark model whose value is predicted to be 0.86 0 [1,2]. (see also [3]). Very recently, the lattice QCD (LQCD) group presented the latest estimate on Σ − Λ mixing angle to have the value α = 0.40 [4], which is approximately two times smaller compared to the prediction of the quark model. The aim of the present note is to determine the Σ-Λ mixing angle within the QCD sum rules, and compare this mixing angle with the predictions of the quark model and LQCD.
In determination of the Σ-Λ mixing angle within the QCD sum rules we follow the method suggested in [5], and for this goal we start by considering the following correlation function, where T is the time ordering operator, η H is the interpolating current, carrying the same quantum numbers as the corresponding hadron. If the bare H 0 1 and H 0 2 states are mixed, the corresponding physical states with definite mass should be the linear combinations of these bare states. In this case, the interpolating currents corresponding to the physical states could be represented as the superposition of the interpolating currents corresponding to the bare states, i.e., where α is the mixing angle between Λ 0 and Σ 0 states. In presence of only two physical states, Eq. (2) can be written as, It should be remembered that the general form of the correlator function is, Π(p) = Π 1 (p 2 ) p + Π 2 (p 2 )I , and coefficients of the p and I (unit operator) structures, i.e., Π 1 (p 2 ) and Π 2 (p 2 ) can both be used in determining the mixing angle. In order to construct the sum rules for the mixing angle α, the correlation function (4) is calculated in terms of hadrons, quarks and gluons. Using the duality ansatz these two representations are matched and the sum rules for the corresponding physical quantity is obtained.
The hadronic representation of the correlation function is obtained by saturating it with the full set of baryons having the same quantum numbers as the corresponding interpolating current. Since η H 1 and η H 2 can create only the states H 1 and H 2 , correspondingly, the hadronic part of the correlation function is obviously zero. Using Eq. (3) in Eq. (4), one can easily obtain the expression for the mixing angle for both structures, where Π 0 ij are the correlation functions corresponding to the unmixed states, i.e., where (i, j = Λ 0 or Σ 0 ). So the problem of determination of the mixing angle requires the calculation of the theoretical part of the correlation function, for which the expressions of the interpolating currents are needed. According to the SU(3) f classification the interpolating currents for the unmixed Λ 0 and Σ 0 are chosen as, where a, b, c are the color indices, C is the charge conjugation operator, and β is the arbitrary constant with β = −1 corresponding to the so-called Ioffe current. Using the operator product expansion at p 2 ≪ 0, one can easily obtain the expressions for the correlation functions Π 0 11 , Π 0 22 , and Π 0 12 from Eq. (3) from the QCD side for the p and I structures. The expressions of these correlation functions are presented in the Appendix.
In order proceed for the numerical calculations we need the values of the input parameters that are given as: qq ( [7]. For the masses of the light quarks we use their MS values given as: It follows from the expressions of the invariant functions that in order to determine the Σ-Λ mixing angle three arbitrary parameters are involved, namely, the continuum threshold s 0 , the Borel mass parameter M 2 , and the parameter β (see the expressions of the interpolating currents); and of course the mixing angle should be independent of them all. As is well known, the continuum threshold is related to the energy of the first excited state. The difference √ s 0 − m ground , where m ground is the mass of the ground state, is equal to the energy needed to excite the particle to its first energy state. This difference usually changes in the range between 0.3-0.8 GeV . It follows from the analysis of the mass sum rules that in order to reproduce the experimental values of the masses of the Σ and Λ baryons, the continuum threshold s 0 should lie in the range 2.5 GeV 2 ≤ s 0 ≤ 3.2 GeV 2 [8,9]. Moreover, the working region of the Borel mass parameter should be such that, the results for the Σ-Λ mixing angle should exhibit good stability with respect to the variation of M 2 at fixed values of s 0 . The upper bound of M 2 is obtained by demanding that the higher states and continuum contributions should be less than 30% of the total result. The lower bound of M 2 is determined from the condition that the contributions of higher dimensional operators should be less than the perturbative one. From these conditions the working region of M 2 is determined to be 1.4 In Figs. (1) and (2), we present the dependence of the mixing angle α on M 2 at the value of the continuum threshold s 0 = 3.2 GeV 2 and, at several fixed values of the auxiliary parameter β, for the coefficients of the structures p and I, respectively. We observe from Fig. (1) that in the range 1.4 GeV 2 ≤ M 2 ≤ 2.2 GeV 2 of the Borel parameter, the mixing angle α exhibits good stability for the values of the auxiliary parameter β = −3; ± 1 for the structure p. As can be traced from Fig. (2), the mixing angle α seems to be rather stable at all considered values of the auxiliary parameter β for the structure I at the fixed value of the continuum threshold s 0 = 3.2 GeV 2 .
Our final attempt for determination of the mixing angle is to find the region of β where the mixing angle exhibits insensitivity to its variation. For this aim we study the dependence of the mixing angle α on cos θ where β = tan θ, at several fixed values of M 2 and at s 0 = 3.2 GeV 2 , and presented them in Figs. (3) and (4) for the coefficients of the structures p and I, respectively. In this respect, the results of our numerical analysis depicted in Figs. (3) and (4) can be summarized as follows: • For the structure p, in the above-determined working regions of M 2 and s 0 , the best stability for the mixing angle is achieved when −1 ≤ cos θ ≤ −0.5, and the mixing angle is found to have the value α = (1.15 ± 0.05) 0 .
• For the structure I not only there is no stability region for the mixing angle, but also the mixing angle changes its sign. Therefore prediction for the value of the mixing angle from the structure I is not reliable.
Therefore we conclude that, the final result for the mixing angle is α = (1.15 ± 0.05) 0 which is obtained from the p structure. The error in determination of the mixing angle can be attributed to the uncertainties in the value of the continuum threshold s 0 , the quark condensates, and the scale parameter Λ. The results presented in this work can further be improved by taking O(α s ) corrections in to account.
Finally, we compare our result on the Σ-Λ mixing with the calculations of the quark and lattice QCD models, whose predictions are, α = 0.86 0 , [3] α = 0.40 0 ± 0.026 0 , [4] respectively. From these results we observe that, our prediction on the Σ-Λ mixing angle is very close to the result obtained in the quark model, and more than two times larger compared to that of the result obtained in the lattice QCD model. A reliable lattice QCD determination of the Σ-Λ mixing angle requires an equally highly accurate reproduction of the octet baryon mass differences, which has not yet been established.
In conclusion, the mixing angle between the Σ and Λ baryons is estimated within the framework of the light cone sum rules method. A comparison of our result with the predictions of the quark and lattice QCD models is presented.
where M 2 is the Borel parameter and Λ is the energy cut off separating perturbative and nonperturbative regimes; and γ E is the Euler constant. Note that the scale parameter Λ is calculated in [10] whose value is in the range 0.5 ÷ 1.0 GeV . Dependence of the Λ-Σ mixing angle on the Borel mass parameter M 2 at the fixed value of the continuum threshold s 0 = 3.2 GeV 2 , and at several fixed values of the auxiliary parameter β, for the structure p. Fig. 2 The same as in Fig. 1, but for the structure I. Fig. 3 Dependence of the Λ-Σ mixing angle on cos θ at the fixed value of the continuum threshold s 0 = 3.2 GeV 2 , and at several fixed values of the Borel mass parameter M 2 , for the structure p. Fig. 4 The same as in Fig. 1, but for the structure I.