Gravitational form factors of hyperons in light-cone QCD

The quark parts of the gravitational form factors of hyperons are calculated by means of the light-cone QCD sum rule. In the calculations, the distribution amplitudes (DAs) of $\Sigma$, $\Xi$ and $\Lambda$ together with the general forms of their interpolating currents as well as the quark part of the energy-momentum tensor current are used. These form factors can provide information on their mass and distributions of the angular momentum, energy and pressure inside the hyperons. It is obtained that the $Q^2$ dependencies of the hyperon gravitational form factors are nicely characterized by a multipole fit function. Using the fits of these form factors, some mechanical properties such as the mechanical radius of the hyperons and the pressure and energy distributions at the center of these particles are obtained. The obtained results can help us in better understanding of the internal structures of these baryons and the QCD as theory of the strong interaction.

The hyperon states are very interesting as they can be used to check the order of SU(3) flavor symmetry breaking at different interactions. Our knowledge on these particles is very limited because of unstable nature of these particles. This makes experimental studies very difficult and this situation increase the importance of theoretical studies on different aspects of hyperons. In this accordance, we investigate the gravitational form factors defining the Σ − Σ, Ξ − Ξ, Λ − Λ and the Σ 0 − Λ transitions within the light-cone QCD sum rule approach [37][38][39]. To our best knowledge, this is the first study on the gravitational form factors of the light octet hyperons in the literature. However, there have been attempts to extract the octet and decuplet hyperon's electromagnetic form factors [40][41][42][43][44][45][46][47][48][49][50], axial form factors [43,[51][52][53][54][55][56][57][58][59] as well as their tensor form factors [60,61].
The outline of the paper is as follows: In Sec. II, using LCSR, analytical results are obtained for hyperon's where is the two-sided covariant derivative, C is charge conjugation operator, t is arbitrary mixing parameter; an a, b, c, d, and e are color indexes. In the above equations, by Σ we mean Σ + .
To obtain the gravitational form factors in light-cone QCD sum rule, we need to calculate the correlation function in two different frameworks. First, we calculate the correlation function in terms of hadronic parameters, known as hadronic side. Second, we obtain the correlation function in terms of the quark-gluon parameters via distribution amplitudes of the related hadron, known as QCD side. In order to suppress undesired contributions, which are related to the excited and continuum states, the Borel transformations are applied. Then, we match both representations of the correlation function by the help of the quark-hadron duality ansatz.
To calculate the correlation function in terms of hadronic properties, a complete hadronic set with the same quantum numbers as the interpolation currents is inserted. Thus, the correlation function takes the form where with λ H and u H (p ′ , s ′ ) being the residue and Dirac spinor corresponding to the hyperon H, respectively. Substituting Eq. (1) and Eq. (6) into Eq. (5), we acquire the hadronic representation of the correlation function in terms of the hadronic observables as The next step is computation of the correlation function in terms of the QCD degrees of freedom as well as the hyperon's distribution amplitudes. Inserting the explicit expressions for the hyperon interpolating currents as well as the quark part of the energy-momentum current into Eq. (3), and performing all contractions among the quark fields with the help of Wick theorem, for the correlation function in x-space, we obtain for Σ-Σ transition, for Ξ-Ξ transition, for Λ-Λ transition and for Σ 0 -Λ transition. Here S q (x) is the light-quark propagator and it is given as Gq 192 The 0|ǫ abc u a σ (a 1 x)u b θ (a 2 x)d c φ (a 3 x)|Σ(p) , 0|ǫ abc s a σ (a 1 x)s b θ (a 2 x)u c φ (a 3 x)|Ξ(p) and 0|ǫ abc u a σ (a 1 x)d b θ (a 2 x)s c φ (a 3 x)|Λ(p) matrix elements are the expressions including the distribution amplitudes of the hyperon states and they are necessary for further evaluations. The explicit form of this matrix elements in terms of DAs together with the explicit forms of DAs for hyperons are presented in the Refs. [62,63]. After using the explicit forms of the above matrix elements and the light-quark propagator, we obtain an expressions in x-space. The next step is to perform the Fourier integrals. These procedures are standard in the method used but very lengthy. Thus, we do not present these steps here.
The desired sum rules are acquired by matching both representations of the correlation function. To do this, different and independent Lorentz structures are needed to be chosen. For this purpose, we choose p µ q ν , form factors, respectively. As a result, we obtain the sum rules, The Π QCD i (Q 2 ) functions appear in Eqs. (13) to (16) are very lengthy. Hence, as an example, we present the result of the Π QCD 1 (Q 2 ): Here, S i , P i , A i , V i and T i are DAs of different twists. They are expressed in terms of hyperon's wavefunctions. The wavefunctions are also functions of different parameters that all are presented in Refs. [62,63]. In order to remove the unwanted contributions corresponding to the excited and continuum states, the Borel transformation and continuum subtraction are applied. Following these procedures, the contributions of the higher states and continuum are exponentially suppressed. The Borel transformation and the continuum subtraction are carried out by employing the subsequent replacement rules [64]: where,

III. NUMERICAL RESULTS
This section is dedicated to the numerical analysis for gravitational form factors of hyperon states. The distribution amplitudes of the Σ, Ξ and Λ baryons have been evaluated in the framework of light-cone QCD sum rule in Refs. [62,63]. In Table I, we present the numerical values of the input parameters inside the DAs of hyperons. For the numerical computations, the hyperon masses are also taken as, m Σ = 1189.37 ± 0.07 MeV, m Ξ = 1314.86 ± 0.20 MeV, m Σ 0 = 1192.642 ± 0.024 MeV and m Λ = 1115.685 ± 0.006 MeV [65]. Additionally, we need the values of the residues of Σ, Ξ, Σ 0 and Λ baryons, which are borrowed from [46].
The estimations for the gravitational form factors of hyperon states depend on several auxiliary parameters: mixing parameter t, the Borel mass parameter M 2 and continuum threshold s 0 . Based on the standard prescriptions of the approach used, the physical observables should weakly depend on these auxiliary parameters. The arbitrary mixing parameter t is chosen such that, the predictions of the gravitational form factors are practically independent of the values of t. From the numerical analyzes, it is seen that in the region -0.19 ≤ cosθ ≤ -0.44 the gravitational form factors  weakly depend on t, where tanθ = t. The working interval for the continuum threshold s 0 is obtained considering the fact that the gravitational form factors also weakly depend on this parameter. It is not totally arbitrary, but depend on the energy of the first excited states in the channels under consideration. We follow the steps described below to get the working window of Borel mass parameter M 2 . The lower limit of M 2 is obtained requiring that the perturbative contribution exceeds over the nonperturbative one and the series of nonperturbative operators are convergent. The upper limit of M 2 is obtained by the requirement that the contributions of continuum and higher states should be less than the ground state contribution. Performed numerical calculations show that the regions 2.00 GeV 2 ≤ M 2 ≤ 3.00 GeV 2 for Σ − Σ, 2.50 GeV 2 ≤ M 2 ≤ 3.50 GeV 2 for Ξ − Ξ, 2.00 GeV 2 ≤ M 2 ≤ 3.00 GeV 2 for Λ − Λ, 2.00 GeV 2 ≤ M 2 ≤ 3.00 GeV 2 for Σ 0 − Λ, satisfy all the aforementioned constraints on M 2 and s 0 . To visualize effects of t, M 2 and s 0 on the gravitational form factors we depict the dependence of these form factors on the auxiliary parameters in Fig. 1. As it can be seen from this figure, the gravitational form factors are quite stable against the variation of these parameters in their working windows. Thus the selected working intervals for the auxiliary parameters satisfy the criteria of the method. In The behaviour of this form factor is not reliable. Therefore, we will not present the fit result of this form factor. Our numerical results for the transitional gravitational form factors of Σ 0 − Λ transition are very small and close to zero, hence, we do not show the Q 2 and M 2 dependence of the form factors corresponding to this transition. To extrapolate the gravitational form factors to low-momentum region, Q 2 = 0, we fit the light-cone QCD sum rule results to different forms of mono-pole and dipole, which unluckily fail to give reasonable characterizations of data with a two-parameter fit. However, our numerical studies indicate that the gravitational form factors of hyperons are nicely characterized employing the multipole fit function, where  Table II. To our best knowledge, this is the first study in the literature dedicated to the investigation of the hyperon's gravitational form factors. Therefore, experimental data or theoretical estimations are not yet available to compare  [66] and JLab data: D N −N (0) = −2.11 ± 0.46 [67]. It should be noted here that the results of other theoretical approaches have been obtained at different renormalization scale and these results have been shifted to the renormalization scale µ 2 = 1 GeV 2 (For details, see e.g., [34]). As one can see from these predictions, the numerical results for the D-term of hyperons obtained in the present work are close to the nucleon's D-term and we see a reasonable SU(3) flavor violation.
After obtaining the gravitational form factors, we can use them to calculate some mechanical properties associated with the hyperons such as mechanical radius square ( r 2 mech ) as well as energy (E) and pressure (p 0 ) distributions at the center of these particles. The related formulas are given as [2], where y = Q 2 .  The numerical results, acquired using the fit functions of form factors, for mechanical properties of hyperons are presented in Table III. Our results may be checked via different theoretical approaches.

IV. SUMMARY AND CONCLUDING REMARKS
We calculated the quark part of the gravitational form factors of hyperons by using the light-cone QCD sum rule method. We used the general forms of the interpolating currents for the hyperons and the quark part of the energymomentum current together with the DAs of the hyperonic states. These form factors carry information about the angular momentum and mass as well as energy and pressure distributions inside the baryons. We obtained that the Q 2 dependence of the hyperon's gravitational form factors are nicely characterized by a multipole fit function. Using these fit functions, we obtained the static form factors, i. e., their values at Q 2 = 0. The hyperons under study are unstable and have a small life-time, hence measurements on their properties are very difficult compared with the nucleons. We have a good experimental information on the electromagnetic form factors of nucleons, even for the D-term of the nucleons we have some data provided by JLab. We also have a good knowledge of theory on the electromagnetic, axial, tensor and gravitational form factors of the nucleons. However, the number of works dedicated to the electromagnetic, axial and tensor form factors of hyperons are very limited. In the case of gravitational form factors there are no theory predictions are available. Hence, our results may be checked via different methods in future. We hope we can have experimental data on the gravitational form factors of hyperons in future. Comparison of different results on these form factors can help us gain valuable information on the internal structures of hyperons.
As a by product, we calculated some mechanical properties of hyperons such as the mean mechanical radius square as well as the pressure and energy distributions at the center of these particles, as well. The results indicate that the r 2 mech H for H = Σ, Λ are considerably bigger than that of the nucleons, r 2 mech N = (0.52 − 0.54) ± 0.05 f m 2 , obtained via the same method in Ref. [34]. However, the mean redius square for Ξ hyperon is comparable with that of the nucleon. Our results on the mechanical properties of hyperons may be checked via different theoretical approaches as well as Lattice QCD. Any future experimental data and their comparison with our estimations can give valuable knowledge on the internal structure and geometrical shapes of hyperons.