Gravitational form factors of a baryon with spin-3/2

We define the form factors of the energy-momentum tensor (EMT) for a spin-3/2 hadron. The static EMT is related to the energy, angular momentum, pressure and shear force densities. We also studied the nucleon and $\Delta$ form factors of the energy-momentum tensor in the large $N_{c}$ limit within the SU(2) Skyrme model.


I. INTRODUCTION
established in Ref. [31,32,51] and the GFFs of the vector mesons are evaluated in Ref. [37,52,53]. For the higher spin hadrons, the works on the parametrization of the stress tensor are made in Ref. [27,54].
We sketch the present work as follows: In Section II we define the hadronic matrix elements of the EMT as the GFFs and reorganize the GFFs in terms of the multipole expansion. We also define the EMT densities of a baryon with spin-3/2 in terms of the multipole expansion and present the relations between the GFFs and the EMT densities. To verify the general requirements and relations proposed in Section II, the GFFs and the EMT multipole densities of the ∆ are obtained within the Skyrme model in Section III and the numerical results are showed and discussed in Section IV. In Section V, we make a summary.
In the Breit frame the average of the baryon momenta and the momentum transfer are respectively defined by P µ = (p µ + p µ )/2 = (E, 0, 0, 0) and ∆ µ = p µ − p µ = (0, ∆) with the initial (final) momentum p (p ). The momentum squared is defined as ∆ 2 = −∆ 2 = t = 4(m 2 − E 2 ) with the baryon mass m. The explicit expressions of the Rarita-Schwinger spinor and the polarization vetor are given in Appendix A. In this frame, the matrix elements of the EMT current are expressed in terms of the gravitational multipole form factors (GMFFs) as One can refer to Appendix A in detail. The sum of the quark and gluon contributions to the GMFFs is also scaleinvariant: The static EMT T µν (r, σ , σ) is given by the Fourier transform of the matrix element of the EMT current in momentum space:

B. Energy density
The temporal component of static EMT T 00 (r, σ , σ) is related to the energy density. The mutipole expansion of the energy density is defined by where the monopole and quadrupole densities ε 0,2 (r) are respectively given by At the same time, the energy multipole form factors E 0,2 (t) can be expressed in terms of the energy densities ε 0,2 (r) in coordinate space: For a particle of arbitrary spin, the general tensor quantity is introduced in Ref. [32,54] by The monopole moment corresponds to the mass of a hadron, accordingly one arrives at the apparent relation which gives the normalization The constraint F 1,0 (0) = 1 for spin-3/2 hadron coincides with that for the spin-1/2 and spin-1 hadrons. The gravitational quadrupole density of a hadron presents how the energy density is deformed from the spherically symmetric shape. This quantity does not appear in the spherically symmetric hadrons. It can be quantitatively estimated as Another interesting property is the mass radius of a hadron. It can be derived by the r 2 -weighted energy density in the Breit frame. The expression of the mass radius is found to be C. Angular momentum density The spin density is given by The angular momentum density is obtained from the 0k-components of the static EMT, which is decomposed into the 0-, 2-and 4-multipole components (see also Ref. [55,56]). The sum of the angular momentum contributions from quark and gluons to the spin-3/2 baryon is obtained by accordingly the averaged angular momentum density is given by with spin S = 3/2. The angular momentum form factor can be expressed in terms of the averaged angular momentum density as

D. Pressure and shear force densities
The pressure and shear force densities are related to the ij-components of the static EMT. These densities are firstly defined in Ref. [31,54] and newly parametrized in Ref. [27,53] to conveniently express the strong forces in a hadron acting on the radial area element. Following Ref. [27,53] the stress tensor can be expressed in terms of the pressure and shear forces densities by From the EMT conservation ∂ i T ij (r, σ , σ) = 0, the following equilibrium relations between the pressure and shear force densities are derived: ds n (r) dr + 2 s n (r) r + dp n (r) dr = 0, with n = 0, 2, 3.
This differential equation guarantees the stability condition. The functions p 0 (r) and s 0 (r) correspond to the pressure and shear force densities appearing in the spherically symmetric hadrons. The functions p 2 (r) and p 3 (r) are named the quadrupole pressure densities, and the s 2 (r) and s 3 (r) are called the quadrupole shear force densities according to Ref. [27]. These densities p n (r) and s n (r) are respectively written as Similarly, the form factors D 0,2,3, (t) can be expressed in terms of the pressure and shear force densities in coordinate space: The pressure densities p n (r) satisfying the relation given in Eq. (28) comply with the von Laue condition Note that the dimensionless constants (generalized D-terms) are defined by [27] The generalized D-terms D 0,2,3 introduced in Ref. [27] are related to the form factors D 0,2,3 (t) as follows: Interestingly, the strong forces carried by constituents can be interpreted as a certain combination of pressure and shear force densities [1]. The spherical components of the strong forces (dF r , dF θ and dF φ ) acting on the radial area element (dS = dS rêr + dS θêθ + dS φêφ ) are expressed as follows [27]: Here, as defined in Ref. [2], the mechanical radius can be given by As for the unpolarized spin-3/2 hadron, since the normal force acting on the radial area element (dF r /dS r ) is solely due to p 0 (r) + 2 3 s 0 (r), it should comply with the local stability criterion given in Ref. [2] as

III. GRAVITATIONAL FORM FACTORS OF THE ∆ IN THE SKYRME MODEL
The Skyrme Lagrangian density is given by where U is the SU(2) chiral field, and e stands for a dimensionless parameter and tr F is trace over flavors. The F π and the m π are the pion decay constant and the pion mass, respectively. In the large-N c limit, we have the parameter scales , so as to have L = O(N c ). In this limit, the chiral field is assumed to be a static one. Here, the static chiral field is written as U (r) = exp[r i τ i P (r)], withr i = r i /|r| and the isospin Pauli matrices τ i , by adopting the hedgehog ansatz where P (r) indicates a profile function with the boundary conditions P (0) → π and P (∞) → 0.
The classical soliton mass is defined by M sol = − d 3 r L, which is expressed with respect to the profile function and the model parameters by The profile function is obtained by minimizing the classical soliton mass and has an asymptotic behavior in the limiting case r → ∞, where the constant R 0 is determined by the profile function derived by minimizing the classical soliton mass and is given in terms of the axial coupling constant in the chiral limit [41,57].
In order to assign the quantum number to the soliton, the time-dependent chiral field should be considered as Having quantized the collective coordinates (Ω i →Ĵ i /2I), one obtains the collective Hamiltonian [58] with the moment of inertia The collective Hamiltonian acts on the collective baryon wave functions given in Ref. [58]. In principle F π and e are model parameters. However, the parameters are fixed to repoduce the following observables [41] In this work, we strictly follow the set of parameters used in Ref. [41] to keep consistency. The canonical EMT in the Skyrme model can be derived by where φ a is a time-dependent mesonic field with U (t, r) = φ 0 + iτ · φ. The degree of freedom of the mesonic field is reduced to three (a = 1, 2, 3) by the constraint φ 2 0 + φ 2 = 1. With the constraint, the canonical EMT in the Skyrme model is found to be symmetric. The respective components of EMT is expressed as (1 − cosP (r)) , The rotational corrections O(1/N c ) to the EMT are obtained by The rotational corrections of the 0k-components of the EMT is found to be null in the current Skyrme Lagrangian. The corrections appear in the higher-order derivative terms which generate Ω 3 ∼ O(1/N 3 c ). However, the corrections are strongly suppressed in the large-N c expansion. Thus, we can safely neglect the rotational corrections to the 0k-component of the EMT.
As given in Eqs. (12), (24) and (26), the multipole densities can be extracted from the EMT in Eq. (47): In the same manner, the rotational corrections are derived from Eq. (48): Interestingly, in the chiral soliton picture the rotational corrections of the densities in Eq. (50) are related to the quadrupole densities, which was firstly found in Ref. [27]. The quadrupole energy density ε 2 (r) is found to be In other words, the quadrupole energy density ε 2 (r) is related to the energy densities of the nucleon and the ∆ with Also, the usual sizes of the both densities δ rot ε ( 1 2 , 3 2 ) 0 (r) and ε 2 (r) are estimated by As for the quadrupole pressure and shear force densities, the remarkable general relation in a chiral soliton picture is derived in Ref. [27] p 2 (r) + 2 3 s 2 (r) = 0.
By comparing Eq. (26) with Eq. (48) we reproduce this relation in the Skyrme model. The above relation (55), together with the EMT conservation (27), implies the nullification of the quadrupole densities s 2 (r) and p 2 (r), and the similar relations to Eq. (51) for the pressure and shear force densities are obtained [27]: Therefore, we arrive at the similar expressions as Eq. (52) with Here, one has to bear in mind that due to the EMT densities with the induced rotational corrections the pressure p n (r) and the shear force s n (r) densities do not uniquely exist and should be reasonably determined. Let us first recall that the leading-order (LO) result of p 0 (r) satisfies the von Laue condition which is equivalent to satisfying the equation of motion [41]. However, once one considers the next-leading-order (NLO) result of p 0 (r), it breaks the von Laue condition. Thus, one might introduce the "variation after projection" method-minimizing the baryon mass after projecting the soliton on the quantum number-as a prescription. However, this method also has a drawback: the chiral symmetry in the large-r region is not satisfied [57]. To preserve the chiral symmetry and satisfy the von Laue condition, we first adopt the "projection after variation" method-projecting the soliton on the quantum number after minimizing the soliton mass-and treat the NLO result as a small perturbation. Of course, the von Laue condition is broken. Thus, instead of directly using the model result of p 0 (r), we solve the differential equation given in Eq. (27) with the shear force density s 0 (r) and reconstruct the pressure p 0 (r)| reconst , which then automatically complies with the stability condition [57]. We also derive the reconstructed quadrupole pressure p 3 (r)| reconst in the same manner. Before discussing GFFs, it is important to discuss the large distance properties of the EMT densities. The large distance behaviors of the EMT densities for the spherically symmetric baryon were investigated and presented within the Skyrme model in Ref. [41,57]. For completeness, we present the large distance properties of the quadrupole densities in the chiral limit as The · · · indicates the contributions strongly suppressed in the large-r region. Interestingly, the quadrupole densities ε 2 (r) and p 3 (r) are weakly suppressed in the large distance, which have the analogous behavior with the angular momentum density ρ J (r) ∝ 1 r 4 . Keep in mind that in order to respect the chiral physics and the stability condition, we discard the result of p 3 (r) and adopt the newly reconstructed p 3 (r)| reconst by solving the differential equation (27). As a result, the discrepancy between p 3 (r)| reconst and p 3 (r) arises from the fulfillment of both the chiral physics and the stability condition and is inevitable. For finite pion mass, the densities are exponentially suppressed.
The large-N c expansion is valid in the region |t| M 2 ∆ . In this limit, we have following large-N c behaviors and the scales of the GMFFs are found to be or according to Ref. [27] the generalized D-terms have the scales as It is found that the GMFFs have the order of ∼ N 0 c except for the form factors D 0 (t) and D 3 (t), which have the order of ∼ N 2 c , whereas the generalized D-terms D 0 , D 2 and D 3 have the orders of ∼ N 2 c , ∼ N 0 c and ∼ N 0 c , respectively.

IV. NUMERICAL RESULTS
In this section, we discuss the numerical results. We first examine the monopole and quadrupole energy densities appearing in the 00-component of the static EMT. The LO monopole energy densities of the nucleon and the ∆ are degenerate. To lift the degeneracy, we take into account the rotational corrections Ω 2 ∼ O(1/N 2 c ). Thus, the integrations of the NLO monopole energy densities over space yield masses of the nucleon (M N = 1159 MeV) and the ∆ (M ∆ = 1452 MeV) projected on quantum numbers S = I = 1 2 , 3 2 , respectively, and they satisfy the constraint given in Eq. (18), In the left panel of Fig. 1, the monopole energy densities of the classical soliton (LO), the nucleon (NLO) and the ∆ (NLO) as a function of radius r are presented. The center of monopole energy density is found to be ε 0 (0) = ε N,∆ 0 (0) = 2.27 GeV · fm −3 . The monopole density of the ∆ is slightly broader than that of the nucleon and the classical soliton. As expected, the rotational corrections to the monopole energy density are suppressed by O(1/N 2 c ). Those effects on the monopole energy densities can be quantitatively observed by calculating the mass radii given in Eq. (20), The rotational corrections make the monopole energy density broader as the spin quantum number increases. In the chiral limit, the NLO mass radii for the nucleon and the ∆ diverge, i.e., given in Eq. (56) and Eq. (59). In the right panel of Fig. 1, the quadrupole energy density of the ∆ is drawn as a function of radius r. It has a peak around 0.4 fm with a negative sign and its strength is marginal compared with the monopole energy density. With Eq. (54), one can obtain the numerical result as The value of the integration of the quadrupole energy density over r is approximately 10% ∼ O(1/N 2 c ) of that of the monopole energy density. This result exactly complies with the relation obtained in Ref. [27]. Another interesting property is the mass quadrupole moment given in Eq. (19). Its value exhibits how the energy density is deformed from the spherically symmetric shape and is found to be In the chiral limit, the mass quadrupole moment diverges for the same reason of the NLO mass radius, i.e, Q ij σ σ ∝ ∞ 0 dr r 4 ε 2 (r) with δ rot ε (J) 0 (r) ∝ 1 r 4 . Fig. 2 shows the averaged angular momentum density ρ J (r) normalized by the corresponding particle spin S. As shown in Eq. (21) and Eq. (24), ρ J (r) is related to the 0k-components of the static EMT. In QCD, the total angular momentum of a hadron comes from the orbital angular momentum and the spin of constituents, i.e., d 3 r ρ J (r) = S. For the mean square radius r 2 J , one gets the following degenerate result for the nucleon and the ∆ Since the absence of the higher-order corrections O(1/N 2 c ) given in Eq. (50), the averaged angular momentum density of the ∆ merely differs from that of the nucleon by factor three, and the mean square radius of the ∆ is the same as that of the nucleon. 4 As mentioned in the previous section, the higher-order corrections of the averaged angular momentum arise from the higher derivative terms generating Ω 3 ∼ O(1/N 3 c ) in the Skyrme Lagrangian. In principle, the corresponding expressions can be derived from the effective chiral Lagrangian incorporating the infinite order of derivatives terms [59]. However, we neglect the corrections in this work. The left panel of Fig. 3 shows the pressures of the classical soliton (LO), the nucleon (NLO), and the ∆ (NLO) as a function of radius r. The LO pressure naturally complies with the von Laue condition that is equivalent to taking the variation of the soliton mass. However, as discussed in the previous section the NLO pressures do not satisfy the stability condition. Thus, we adopt a strategy, preserving the chiral symmetry and satisfying the stability condition, that we first take the variation of the soliton mass, and then project the soliton on the quantum numbers. By treating the rotational corrections as a small perturbation, we afterward obtain the approximated shear force density s 0 (r). With s 0 (r), we reconstruct the pressure p 0 (r)| reconst from Eq. (27) so that the pressure meets the stability condition. Thus, the LO and NLO pressures in Fig. 3 satisfy the stability condition. In the meantime, the satisfaction of the stability condition implies that the pressure has at least one nodal piont (r 0 ) where the pressure vanishes. The nodal points of the pressures are located at The rotational corrections make the pressure density weaker and spreading more widely as the spin quantum number increases. In the right panel of Fig. 3, approximated shear forces are presented.  (r)| reconst were derived from Eq. (27). Thus, p 3 (r)| reconst complies with the stability condition given in Eq. (31), and has a nodal point located at r 0 = 0.56 fm. The shapes of p 3 (r)| reconst and s 3 (r) show a similar tendency with those of p 0 (r) and s 0 (r), respectively.
The left panel of Fig. 5 shows normal force components acting on the radial area element for the classical soliton, the nucleon and the unpolarized ∆. When it comes to a spherically symmetric baryon, the normal force can be directly related to a local stability criterion (36), which signifies the positivity of the normal force. In other words, the normal force should be directed outwards. As for the polarized ∆, it has the quadrupole contributions to the normal force as shown in Eq. (34). Except for the nullified densities s 2 (r) and p 2 (r), the quadrupole contribution to the normal force is shown in the right panel of Fig. 5. For the large distance in the chiral limit, the unpolarized normal force p N,∆ 0 (r)| reconst + 2 3 s N,∆ 0 (r) keep the positivity which means that the local stability condition is satisfied, and also the quadrupole normal force p 3 (r)| reconst + 2 3 s 3 (r) has the positive sign within the Skyrme model However, for the polarized ∆ we do not know how the quadrupole densities are related to the local stability conditions as of now. Another interesting quantity is the mechanical radius. As defined in Ref. [2], the mechanical radius is obtained by Finally, we discuss the GFFs and GMFFs. As shown in Sec. II, the GMFFs are expressed in terms of the GFFs. In the meanwhile, the GMFFs can also be expressed in terms of the EMT densities as shown in Eq. (15), Eq. (25) and Eq. (30). The results of the ∆ GMFFs and GFFs are shown in Fig. 6 and Fig. 7. In the Skyrme model, the energy E 0 (t) and angular momentum J 1 (t) form factors satisfy the constraints E 0 (0) = F 1,0 (0) = 1 and J 1 (0) = 1 3 F 4,0 (0) = 1 2 . Besides the octupole angular momentum form factor J 3 (t) is assume to be zero, i.e., J 3 (0) = − 1 6 [F 4,0 (0)+F 4,1 (0)] = 0, because the corresponding density is suppressed by Ω 3 ∼ O(1/N 3 c ) in the large-N c expansion. As a result, we get the relation F 4,1 (0) = −F 4,0 (0) = −3/2. There is no additional constraint on E 2 (t), D 0 (t), D 2 (t) and D 3 (t). Therefore, we determine the moments of the GMFFs from Eq.
The energy quadrupole form factor E 2 (0) is obtained as 0.34, and it can be related to the mass quadrupole moment given in Eq. (19). The value of the D-term 6 D 0 (0) of the ∆ is found to be −3.53. The quadrupole form factors D 2 (0) and D 3 (0) turn out to be −0.20 and 0.24, respectively. As introduced in Eq. (32), the generalized D-terms are determined as The generalized D-terms of the present work are comparable with that of the Ref. [27,57]. We restrict ourselves in the range of 0 < (−t) < 1 GeV 2 because of the validity of the large-N c expansion, i.e., |t| M 2 ∆ . We find that the quadrupole form factors E 2 (t) D 2 (t) and D 3 (t) are relatively small in comparison with the monopole form factors E 0 (t) and D 0 (t). It seems that there are conflicts with the large-N c behavior. For instance, the sizes of the numerical values for D 3 (0) and D 0 (0) with the same order of ∼ N 2 c are not comparable. However, in nature, the number of color N c is just three, and these conflicts arise from the different orders of the multipole structures. The fact indicates that the N c counting is not valid between the different orders of the multipole form factors. 5 By assuming that the pressure pn(r) and shear force sn(r) densities vanish at large-r faster than any power of r, one can arrive at the general relation, arising from the differential equation (27), between them as follows [39]: ∞ 0 dr r N sn(r) = − 3(N +1) 2(N −2) ∞ 0 dr r N pn(r) for N > −1. 6 The discrepancy of D-term between this work and Ref. [27,57] arises because of the different masses. They have used the LO mass, whereas we take the NLO masses to keep a consistency in this work.

V. SUMMARY AND CONCLUSIONS
In the present work, we aim at investigating the formalism for the GMFFs of the spin-3/2 baryon and verifying the general requirements of the GFFs based on the Skyrme model. The possible structure of the EMT current was sorted out, and the GFFs were defined in Ref. [26]. In addition, the multipole structure of the stress tensor for the spin-3/2 baryon was investigated, and the general relations between GFFs in the large-N c limit were derived in Ref. [27]. In this work, the GFFs defined in Ref. [26] are classified into the GMFFs, and also the energy, spin, pressure and shear force densities are given in terms of the multipole expansion.
As derived in Ref. [27,57], we have shown that all general requirements of the EMT are satisfied up to the leading order of N c expansion. However, in order to go beyond the leading order of N c the stability condition should be broken. To cure this problem, we first quantize the soliton after minimizing its mass while the rotational corrections are treated        as a small perturbation. With the approximated shear force densities with the induced rotational corrections, we then reconstruct the pressure densities. Utilizing this strategy, we can satisfy all the requirements. We obtain the large-N c behavior of the GMFFs, and it is found that the GMFFs have the order of ∼ N 0 c except for the form factors D 0 (t) and D 3 (t), which have the order of ∼ N 2 c . Since the large-N c approach is valid in the region |t| M 2 ∆ , we predict the t dependence of the GMFFs in the region of 0 < (−t) < 1 GeV 2 . The moments of the GMFFs are evaluated-the mass quadrupole moment, the generalized D-terms, the mass radius, etc. We find that the quadrupole form factors E 2 (t) D 2 (t) and D 3 (t) are relatively small in comparison with the monopole form factors E 0 (t) and D 0 (t).
As suggested in Ref. [27] the lattice measurements of ∆ GFFs can be used to check whether ∆-baryon is rotating soliton. Here, we provided the first numerical estimates of corresponding GFFs in the soliton picture using the Skyrme model. Thus, we expect that the results of the lattice QCD or the theoretical model will soon come out.