Mixing effects of $\Sigma^0-\Lambda^0$ in $\Lambda_c^+$ decays

We analyze the mixing between $\Sigma^0$ and $\Lambda^0$ based on the baryon masses. We distinguish the contributions from QCD and QED in the baryon mass splittings. We find that the mixing angle between $\Sigma^0$ and $\Lambda^0$ is $(2.07\pm 0.03)\times 10^{-2} $, which leads to the decay branching fraction and up-down asymmetry of $\Lambda_c^+ \to \Sigma^0 e^+ \nu_e$ to be ${\cal B}(\Lambda_c^+ \to \Sigma^0 e^+ \nu_e)=(1.5\pm 0.2)\times 10^{-5}$ and $\alpha(\Lambda_c^+ \to \Sigma^0 e^+ \nu_e)=-0.86\pm 0.04$, respectively. Moreover, we obtain that $\Delta {\cal B}\equiv {\cal B}(\Lambda_c^+\to \Sigma^0 \pi^+) - {\cal B}(\Lambda_c^+\to \Sigma^+\pi^0)=(3.8\pm 0.5)\times 10^{-4}$ and $\Delta \alpha \equiv\alpha(\Lambda_c^+\to \Sigma^0 \pi^+) -\alpha(\Lambda_c^+\to \Sigma^+\pi^0)=(-1.6\pm 0.7)\times10^{-2}$, which should vanish without the mixing.


I. INTRODUCTION
The flavor content is one of the cornerstones in particle physics. The hadrons are often categorized and named in terms of their flavor states. For instance, the hadrons in the isospin group, such as pions, share the same name. Since the QCD energy scale is much larger than the mass difference of the hadrons with the same isospin, it is believed that the hadrons also have similar wave functions. The isospin group can be extended to SU(3) by including the strange quark, which is the so-called SU(3) F flavor symmetry and has been widely used in particle physics.
Based on the SU(3) F symmetry, the precision for the Gell-Mann-Okubo (GMO) mass formula is around one percent, indicating that the baryons indeed share the similar wave functions. Another well know mass formula based on SU(3) F for the octet baryons is the Coleman-Glashow (CG) one [1]. Note that the masses of the octet baryons have been intensively studied in the calculations of the Lattice QCD (LQCD) [2][3][4][5][6] as well as the theoretical models with the baryon wave functions [7][8][9][10][11][12][13].
The octet baryon states of Σ 0 and Λ 0 have the same quark components of uds. Originally, they are categorized by the isospin property with Σ 0 and Λ 0 being the triplet and singlet states under the SU(2) I isospin symmetry, respectively. This categorization is based on that the isospin symmetry is much better than SU(3) F . However, both SU(2) I and SU(3) F are not exact, resulting in a possible mixing between Σ 0 and Λ 0 . The physical baryons shall be made of the mixing of isospin triplet and singlet states. In general, the mixing angle is estimated to be the ratio of the SU(2) I and SU(3) F breaking energy scales. Note that the calculation in the LQCD gives the mixing angle θ = 0.006 ± 0.003 [14].
Recently, the BESIII Collaboration has announced the up-down asymmetries for Λ + c → Σπ, given by [15] Note that the corresponding branching ratios have been measured to be [16]: However, in the limit of the isospin symmetry, both asymmetries in Eq. (1) and branching ratios in Eq. (2) should be equal. On the other hand, the semi-leptonic decay of Λ + c → Σ 0 l + ν l is forbidden since Λ + c and Σ 0 belong to different isospin representations. In this study, we explore the isospin breaking effect in the Σ 0 −Λ 0 mixing and discuss the effects in Λ + c decays. Our paper is organized as follows. In Sec. II, we introduce the hadron representations under the SU(3) F flavor group. The mixing effects in Λ + c decays are studied in Sec. III. We present our conclusions in Sec. IV.

II. HADRON REPRESENTATIONS IN SU(3) F
In terms of SU(3) F , the matrix representations of the octet baryons can be written as [17] where the prime ′ denotes the un-mixed state. For instance, the proton's matrix representation and state correspond to (p) i j = δ i1 δ j3 and (p) i j | j i = | 3 1 , respectively. In the standard model, the SU(3) F symmetry is broken by the quark masses as well as the electromagnetic interaction. The matrix representations of the light quark masses and electric charges of the quark flavors are given as respectively. Note that both M and Q belong to 8 under the SU(3) F group. Consequently, the baryon mass must be the function of M and Q, given by where H and |B are the mass operator and state of the octet baryon, respectively.
Naively, one may write down the baryon mass operator as where the superscripts of n = 0, 1, 2 stand for the n-order approximations and the subscripts of m and q imply the breaking sources of the SU(3) F symmetry. However, the second-order correction from the strange quark mass can be the same size as the first-order one from the up and down quark masses, e.g. (m s /µ H ) 2 ∼ m q /µ H , where µ H is the typical hadronic scale.
A better way to do the approximation is to categorize the breaking effects according to their symmetry properties instead of the sources. We rewrite Eq. (4) as with where Here, we have decomposed the matrix representation into two different parts. Note that T 8 is invariant under the isospin transformation, whereas T 3 is not. Accordingly, the baryon mass operator in Eq.(6) is given by where we have used that the matrix representation is linear and the operators in the parentheses have the same representation, respectively. Since H 1 S contains the correction from m s , it is much larger than H 1 I . On the other hand, the second-order correction in O(H 2 ) caused by m s can be the same order as H 1 I . Explicitly, we have the hierarchy, given by Note that the correction from m s is invariant under the isospin transformation. If O(H 2 ) is neglected in the calculation of H 1 I , it is only reasonable to deal with the physical quantities, which are not affected by the correction from m s .
Notice that the baryon wave functions in Eq. (3) are chosen as the eigenvectors of H 0 +H 1 S . Explicitly, they have the following properties However, the physical baryon states are the eigenvectors of the full baryon mass operator instead. In general, the isospin breaking term H 1 I has a nonzero matrix element between Σ ′0 and Λ ′0 , i.e. Σ ′0 |H 1 I |Λ ′0 = 0, while the correction due to m s has no contribution to this matrix element due to the isospin symmetry. The physical baryon condition is given by with [18,19] |Λ 0 = cos θ|Λ 0′ − sin θ|Σ 0′ and |Σ 0 = cos θ|Σ 0′ + sin θ|Λ 0′ , where P denotes as the physical baryon and θ is the mixing angle to be determined through the octet baryon masses.
The baryon masses are evaluated from Eqs. (5) and (9). With the SU(3) F symmetry described in the beginning of the previous section, we can apply the WignerEckart theorem.
Consequently, the matrix element in Eq. (5) is parametrized as  Table I, where the results from QCD and QED are given at the end of this section. We have rounded the experimental data to the second decimal place due to that the SU(3) F symmetry is only an approximation. The higher precision is not expected.
From Table I, one can easily obtain the CG mass formula, given by along with additional one, given by Mass differences SU (3) F parameters data [16] QCD QED .08 ± 0.08 8.59 ± 0.01 −0.51 ± 0.08 where (Σ − , n), (p, Ξ − ) and (Ξ 0 , Σ + ) have the same V-spin representations. Likewise, Eq.  If the QED effect is ignored, the baryon mass depends on the quarks mass matrix only, resulting in that where M 1 and M 2 are the unknown parameters. By using Eqs. (7), (14) and (15), we obtain that where the subscript of "QCD" indicates that the QED effect is ignored. Since S 1 and S 2 are dominated by the strange quark mass correction, we can safely approximate Eq. (24) as The calculation in the LQCD is indeed consistent with Eq. (25), where the ratio is around 1.6 ∼ 2.2 [2][3][4][5][6]. In reality, I 1 /I 2 has the value of −5.25 ± 0.06, which clearly indicates that I 1 /I 2 = (I 1 /I 2 ) QCD . Moreover, from Eq. (23), we have where the mass ratios among the light quarks in Ref. [20] have been used. With Eqs. (20), (22) and (26) along with I i = (I i ) QCD + (I i ) QED , we can separate the contributions to the mass differences of the isospin breakings in QCD and QED as listed in Table I. The results are fairly closed to those in the literature based on LQCD and QED [4][5][6].
The mixing angle is determined by [18] tan θ = Σ ′0 |H 1 where the matrix element is given as By using , our formula is the same as the one in Ref. [19], in which only the mixing through the electromagnetic interaction is considered.
Consequently, we obtain the mixing angle Although the mixing has a little effect on the baryon masses, it plays a significant role in the decays associated with the isospin symmetry. We now study the mixing effects in the semileptonic and nonleptonic charmed baryon decays. In general, the states of Λ + c and Σ + c , corresponding to I = 0 and I = 1, are also mixed. However, the mixing is suppressed by the charmed quark mass [21]. As a result, it will be ignored in this work.
Since Λ + c (Σ ′0 ) is anti-symmetric (symmetric) in up and down quarks, the decay channel of Λ + c → Σ 0 e + ν e without the mixing is forbidden. The ratio between the decay rates of Λ + c → Λ 0 e + ν e and Λ + c → Σ 0 e + ν e is given by where we have approximated that M Σ 0 ≈ M Λ 0 in the kinematic phase space. In addition, the angular distributions of the Σ 0 and Λ 0 modes should be the same. With the experimental data for Λ + c → Λ 0 e + ν e [16] and the mixing angle in Eq. (29), the branching ratio and up-down asymmetry of Λ + c → Σ 0 e + ν e are given by respectively.
We now explore the non-leptonic charmed baryon decays of Λ + c → Σ + π 0 and Λ + c → Σ 0 π + . If there is no mixing between Λ 0 and Σ 0 , two decays should have the same decay width and up-down asymmetry parameter, given by [22] respectively, where A and B are associated with the P and S wave amplitudes, and m i , p i and E i are the mass, momentum and energy for the ith hadron in the CM frame, respectively.
According to the Eqs. (33), (34) and (35), the isospin breaking effects caused by the mixing are given by which are consistent with the current experimental data, given by (5 ± 12) × 10 −4 and (16 ± 22) × 10 −2 [16], respectively. Since the data are also consistent with zero, it is clear that future experiments with higher accuracy are needed.
The mixing effects in Λ + c → Σπ are in the first order of θ , while those in Λ + c → Σ 0 e + ν e and Λ 0 b → Σ 0 J/ψ [23] in the second one. Clearly, the experiments in Λ + c → Σπ are more promising for searching the Σ 0 − Λ 0 mixing.