Relativistic composite-particle theory of the gravitational form factors of pion: quantitative results

We use a version of the instant-form relativistic quantum mechanics of composite systems to obtain the gravitational form factors of the pion in a common approach to its electroweak and gravitational properties. In the preceding work [arXiv:2010.11640] we formulated the mathematical background, presented the principal scheme of calculation and testified the obtained qualitative results to satisfy the general constraints given by the principles of the theory of hadron structure. In the present work we give the detailed calculation of the gravitational form factors in large range of momentum transfer, their static limits and the slopes at zero value, the mean-square mass and mechanical radii of the pion. Now we take into account the qravitational structure of the constituent quarks. We show that the results are almost insensitive to the type of the model two-quark wave function in a close analogy to the case of the pion electromagnetic form factor. We present a correct calculation of the form factor $D$ and corresponding matrix element of the energy-momentum tensor, going beyond the scope of the modified impulse approximation. Most of the parameters that we use for the calculation had been fixed even earlier in our works on the pion electromagnetic form factors. The only free parameter is the $D$-term of the constituent quark, which we fix by fitting the result for the slope at zero of the normalized to pion $D$-term form factor $D$ of pion, to a choosen experimental value.

We use a version of the instant-form relativistic quantum mechanics of composite systems to obtain the gravitational form factors of the pion in a common approach to its electroweak and gravitational properties. In the preceding work [Phys.Rev. D 103, 014029 (2021)] we formulated the mathematical background, presented the principal scheme of calculation and testified the obtained qualitative results to satisfy the general constraints given by the principles of the theory of hadron structure. In the present work we give the detailed calculation of the gravitational form factors in large range of momentum transfer, their static limits and the slopes at zero value, the mean-square mass and mechanical radii of the pion. Now we take into account the qravitational structure of the constituent quarks. We show that the results are almost insensitive to the type of the model two-quark wave function in a close analogy to the case of the pion electromagnetic form factor. We present a correct calculation of the form factor D and corresponding matrix element of the energy-momentum tensor, going beyond the scope of the modified impulse approximation. Most of the parameters that we use for the calculation had been fixed even earlier in our works on the pion electromagnetic form factors. The only free parameter is the D-term of the constituent quark, which we fix by fitting the result for the slope at zero of the normalized to pion D-term form factor D of pion, to a choosen experimental value.

I. INTRODUCTION
The understanding of the gravitational structure of hadrons is a fundamental problem of particle physics. To consider this problem one needs to involve specific mathematical objects: the gravitational form factors (GFFs) of hadron (and in general the energy-momentum tensor (EMT)). These functions are in the focus of investigation in numerous recent works (see, e.g., the reviews [1][2][3][4][5]). The mathematical background of these investigations was laid in the sixties of the last century [6][7][8] but becomes of large use only during the last decade.
It is the formalism of Ref. [7] that we use in our investigations. In the preceding work [9] we formulated the mathematical statements and described the principal scheme of the method; we testified that the derived gravitational characteristics of the pion satisfy the constraints given by the general principles of the theory of hadron structure. In the present work we refine the approach as to give a detailed technique of calculation, thus transforming it into a quantitative method. The refinement includes three main points. First, we take into account the gravitational structure of the constituent quarks. Second, we analyze different types of the two-quark wave functions in the pion. Third, we formulate a minimal way to overstep our modified impulse approximation (MIA) * a krutov@rambler.ru † troitsky@theory.sinp.msu.ru for correct description of the pion form factor D. Note that the pion EMT contains two form factors, and the second form factor A is described well in the frame of MIA.
The majority of papers concerning closely the problem under consideration were reviewed in [9] and we do not repeat the review in the present paper. Here we discuss only some recent results. Although the papers published during last year are dealing mainly with the proton, they show the modern trends and perspectives in general. At present, there is no possibility to obtain directly the data on gravitational characteristics, including the D-term. The information about the GFFs is usually extracted from the hard-exclusive processes described in terms of unpolarized generalized parton distribution (GPD) [10,11]. The GPD gives the information on the space distribution of strong forces that act on quarks and gluons inside hadrons. In this connection, in works [12][13][14][15] the processes of deeply virtual Compton scattering (DVCS) on nucleons were investigated. In the paper [12] new technique of artificial neural networks was used for the reduction of model dependence of the data handling. The distribution of shear forces inside the proton was obtained for the first time in [13] and also the proton form factor D was constructed by fitting of three parameters in the multipole-type decomposition. The kinematic corrections to the cross-section appearing as a consequence of non-uniqueness of description of the photon in final state on the light cone, is discussed in the work [14]. It is shown that these corrections can be significant in the kinematic regions of future experiments.
The experimental study of DVCS is realized in time-like region for the first time in [15].
In another set of works (Refs. [16][17][18][19][20]), the gravitational characteristics of hadrons are extracted from the data on near-threshold vector mesons photoproduction. In the study [16], the mass radius is calculated not only for the proton, but for the deuteron, too. The analyzis of the possible sources of, systematically obtained in numerous recent calculations, difference between charge and mass radii of the hadrons is given in [19].
The low-energy chiral effective field theory(chiral EFT), was used in [21] to calculate the ρ-meson GFFs and in [22] to describe the mechanical stresses inside the nucleon.
Constituent quark models are exploited in the papers [23][24][25]. In particular, the spin-orbital correlations in pion in a quark model on the light cone are studied in [23] in terms of GPD and of generalized transverse momentum distributions. In [24], the proton GFFs are derived in a light-front quark-diquark model. The authors of [25] study general problems of the interpretation of the experimental data and theoretical methods of calculation for nucleon GGFs. The investigation is focused on space distributions of the energy, shear stresses, momentums, angular momentums in different frame systems. The authors use the quark bag model in large-N c limit in various cases -from non-relativistic to ultrarelativistic.
As to non-hadronic systems, the form factor D, the distribution of matter, and internal stresses in the electron are obtained in [26] in the one-loop approximation of QED.
It seems obvious that because of extreme weakness of the gravitational interaction at hadron scale, the information on GFFs will be extracted from electroweak processes as yet. So, a theory which describes electroweak and gravitational properties simultaneously, based on the unique foundations, and uses common model parameters, is welcome. The approach that we present here possesses these features.
We use a particular variant of the instant-form Dirac [27] relativistic quantum mechanics (RQM) (see, e.g., [28][29][30][31]) extended for composite systems. The approach was successfully used to describe the pion electromagnetic form factor (see., e.g., [32][33][34]). We have shown [9] that the pion GFFs can be derived in the same formalism using the same approximations and the same model parameters, adding only one new parameter fixed by fitting the slope at zero of the normalized to pion D-term form factor D of pion.
In the present paper, we take into account the gravitational structure of the constituent quarks and different forms of the quark-antiquark wave function. It is worth noting that the including of the gravitational structure of costituents in our relativistic model means an implicit accounting for gluons. Their degrees of freedom are incorporated in the parameters describing constituent quarks.
Let us emphasize that it was just the account of the structure of the constituents [32] that made it possible to predict the behavior of the pion charge form factor at intermediate and high momentum transfers. We have shown that, when two model parameters were fixed by the charge mean square radius and the lepton decay constant, then the form factors only weakly depended on the choice of model interaction of quarks in the pion. The curves corresponding to different interactions but one and the same value of quark mass were agglomerated into narrow groups, or bunches. The chosen group of curves with the constituent mass M =0.22 GeV had predicted, with surprising accuracy, the values of the pion charge form factor which were measured a decade later in JLab experiments. By the way, this value of the constituent-quark mass has been admitted and confirmed by other authors (from the well known work [35] to recent result [36]).
Let us list some other advantages of our model with fixed parameters. The obtained pion charge form factor at high momentum transfer coincides with the QCD predictions in the ultraviolet limit, reproducing correctly not only the functional form of the QCD asymptotics, but also the numerical coefficient [37][38][39] (analogous results were obtained for the kaon [40]). The method allows for an analytic continuation of the pion electromagnetic form factor from the space-like region to the complex plane of momentum transfer squares and gives an adequate description of the pion form factor in the time-like region [41]. Interesting results of the approach in the case of its generalization to vector mesons, which required to add the anomalous magnetic moment of quark, were obtained for the ρ meson [42,43].
As was mentioned above, we need to overstep the frame of MIA for calculating correctly the pion form factor D.
Here we present a possible minimal extension based on the non-relativistic limit of MIA. In the relativistic series expansion of the ill-defined terms of the form factor we preserve the main contributions only.
In the present paper we extend our model with nonpoint-like quarks to obtain the pion GFFs while making an essential assumption, which looks natural. We suppose that the functional forms of electroweak and gravitational form factors of constituent quark are the same. Also we suppose that charge and mass radii of quark, and the slope at zero of the normalized form factor D, -all these three values are equal. From one hand, this gives an opportunity to add to the model only one new parameter. From the other hand, the results of calculation are rather reasonable and plausible.
We fix the new free parameter, the quark D-term (instead of quark anomalous magnetic moment in vector meson case of electroweak processes) by fitting the slope at zero of the normalized to pion D-term form factor D of pion to the value obtained in Ref. [44]. The scheme of narrow bunches of curves, corresponding to different wave functions, works in the case of pion GFFs, too. As we show below it is possible to choose the quark D-term as the main characteristic of the bunch and so derive the value of the free parameter.
Other model parameters, which describe the mass and the gravitational structure of the constituent quarks and the quark interaction in the pion, are directly transported from our electroweak model. It is important to emphasize that the interval of the values of the quark D-term obtained by this fixation gives the value of the slope at zero of the normalized form factor D of pion [44] for any quark interaction (for each of three two-quark wave functions) without additional variation of the previous values of the parameters. This set of parameters -those inherited from the electroweak calculation plus the quark D-term -allow us to describe all other gravitational characteristics of the pion: the GFFs in a large range of momentum transfers, their static limits, including the pion D-term and the mean square mass radius. So, the results of calculations of the present paper can be considered as firm predictions in the following sense. If a value of, for example, the slope at zero of the normalized to D-term form factor D of pion extracted from a future experiment, is admitted by the community of specialists, then other gravitational characteristics can be predicted following our prescription. The success demonstrated previously by our method inspires hope on its validity in describing the pion GFFs that can be obtained in future experiments.
Let us remind that the first estimations of the pion GFFs were extracted from the data of Belle collaboration program KEKB [44,45]. Today, the real perspective of GPD estimation for light mesons, including the pion, is connected with the future experiments on the SuperKEKB collider. In general, we look forward for the experiments at the electron-ion collider (EIC) [46,47], Chinese electron-ion collider (EIcC) [48,49], and Large hadron-electron collider (LHeC) [50].
Moreover, it is possible that our unified approach can construct another bridge between electroweak and gravitational properties of composite systems, complementary to GPD.
The rest of the paper is organized as follows. In Sect.II we present the main basic points of IF RQM and the equations for the pion GFFs. Then in Sect. III we describe the gravitational structure of the constituent quarks and present the details of calculation. In Sect.IV we give the results of the calculation of the pion GFFs up to 10 GeV 2 , their static limits and the mean square mass and mechanical radii. We briefly discuss the results and conclude in Sect.V.

II. MAIN BASIC POINTS OF IF RQM AND THE CALCULATION OF THE PION GFFS
Let us recall briefly some important features of the IF RQM algebraic structure. The basic point is the direct realization of the Poincaré algebra on the set of dynamical observables of a composite system (see the reviews [28][29][30][31]). In this context, the fundamental property of the Poincare algebra, as compared to the algebra of the Galilean group, is the following. The adding of the op-erator of constituent interaction to the total energy operator (that is to zero component of total momentum) requires the including of the interaction also in the operators of other observables to preserve the algebraic structure. Different forms of relativistic Dirac dynamics correspond to different ways of realizing the interaction including, which are characterized by different kinematic subgroups, namely subgroups of interaction-independent observables. The kinematic subgroup in the case of IF RQM contains rotations and translations of three-dimensional space. From the point of view of the principles underlying the RQM theory, it occupies an intermediate position between local quantum field theory and non-relativistic quantum mechanical models. In particular, constituents of composite system are assumed to lie on the mass shell, and corresponding wave function is defined as eigenfunction of the complete set of commuting operators. In the case of IF RQM this set is: hereM I the mass operator for the system with interaction,Ĵ 2 is the operator of the square of the total angular moment,Ĵ 3 is the operator of the projection of the total angular moment on the z axis andˆ P is the operator of the total momentum. In the IF RQM the operatorsĴ 2 ,Ĵ 3 ,ˆ P coincide with corresponding operators for the composite system without interaction and only the termM 2 I (M I ) is interaction depending.
To solve the eigenfunction problem for the set (1) it is necessary to choose an appropriate basis in the Hilbert space of the state of the composite system. In the case of system of two constituent quarks one can use, first, the basis of individual spins and momenta (see [9] for details): where p 1 , p 2 are the 3-momenta of particles, m 1 , m 2 are the projections of spins to the z axis. Second, it is possible to use the basis in which the motion of the center of mass of two particles is separated: where s is the invariant mass of the system of two particles, l is the orbital momentum in the center-of-mass system (c.m.s.), S 2 = ( S 1 + S 2 ) 2 = S(S + 1) , S is the total spin in c.m.s., J is the total angular momentum, m J is the projection of the total angular momentum.
The bases (2) and (3) are linked by the Clebsch-Gordan decomposition of a direct product (2) of two irreducible representations of the Poincaré group into irreducible representations (3) [31].
In the basis (3) three out of four operators in the complete set (1) (exceptM I ) are diagonal. So, the two-quark wave function for the pion in the basis (3) has the following form: where p π is the pion 3-mometum. Here we do not exploit the implicit form of the normalization constant N C that can be found in [9]. The zero values of pion quantum numbers are omitted in the notation of basis vectors (3). The wave function of intrinsic motion is the eigenfunction of the operatorM 2 I (M I ) and in the case of two particles of equal masses is (see, e.g., [33]) where u(k) is a model quark-antiquark wave function of the pion and M is the mass of the constituent quarks. Let us construct now the pion EMT in IF RQM. Using the general method of the relativistic invariant parametrization of the matrix elements of the local operators we have obtained in [9] the followig form: where G (π) 10 , G 60 are gravitational form factors, g µν is the metric pseudotensor and We present the decomposition of the l.h.s. of (6) in terms of the basis (3) as a superposition of the same tensors as in the r.h.s. of (6), and so obtain (see [9] for details) the pion gravitational form factors in the following form of the functionals, given on two-quark wave functions (4), (5): hereG i0 (s, t, s ), i = 1, 6 are the Lorentz-invariant regular distributions.
To calculate the invariant distributions in r.h.s. of (7) one can use MIA. Let us discuss this problem in more detail. Consider the commonly used standard impulse approximation (IA). In general, the EMT of a composite system has the following form [9]: where the first term presents the sum of one-particle EMTs, the second term presents the sum of two-particle EMT, and so on. The first sum describes the scattering of a projectile by each independent constituent, the second sum describes the scattering by two constituents simultaneously and so on. The standard IA leaves in (8) only the first term: To construct the pion GFFs we use a modified impulse approximation that we first formulated earlier (see, e.g., Refs. [33,34] and the review [31]) In contrast to the baseline impulse approximation, MIA is formulated in terms of the reduced matrix elements, that is form factors, and not in terms of the operators itself. So, in MIA there appears important objects -the free gravitational form factors presenting the gravitational characteristics of systems without interaction.
Consider the system of two free constituent quarks [9]. Note, that in the work [51] it was shown that the form factor D is zero in the case of point-like free fermions. In contrast, our constituent quarks have all properties of realistic particles with internal structure that is described by a set of form factors including form factor D.
The corresponding relation based on the Clebsch-Gordan decomposition of the EMT of a system of noninteracting fermions with total quantum numbers of pion J = l = S = 0 has the following form [9]: where P, √ s | p 1 , m 1 ; p 2 , m 2 is the Clebsch-Gordan coefficient, the sums are over the variables m 1 , m 2 , m 1 , m 2 . In l.h.s. zero discrete quantum numbers in state vectors are ignored.
Using the general method of parametrization of localoperators matrix elements [7,9] we write the matrix element in l.h.s. as: where G (0) i0 (s, t, s ), i = 1, 6 are free two-particle GFFs, The method gives for the one-particle matrix elements in r.h.s. of (10) the form: q = u,d, D j w (p, p ) is the transformation operator from the small group, the matrix of three-dimensional rotation, g (u,d) i0 , i = 1, 4, 6 are the constituent-quark GFFs. Their links with conventional notations is given later in Sect.III.
Here Γ ρ (p ) is well known 4-vector of spin (see, e.g., [7,9,31,34]), µ ν λ ρ is the absolutely antisymmetric pseudotensor of rank 4, 0 1 2 3 = −1. Substituting (11), (12) into (10), multiplying both sides by scalars, first, by A µ A ν , and second by g µν , performing the integrations and summations, we obtain the system of equations for the free two-particle form factors which enter (11). MIA means replacing the invariant distributions in r.h.s. of (7) with free two-particle form factors from the equations (10), (11). The physical meaning of MIA is equivalent to that of the universally accepted IA (9) because the free two-particle form factors are given in terms of one-particle currents (10), (12). So, we obtain the expressions for the pion gravitational form factors in MIA. For convenience, we present them in a slightly different form than in [9]: 10 (t) + g G Here g (q) i0 (t) , q = u,d , i = 1, 4, 6 are the GFFs of the constituent quarks, also introduced previously in (12).
To calculate the form factors in the r.h.s. of the equations (14), (15), we use the modified impulse approximation (MIA) [9] and, so, write them in terms of the integrals Here i = 1, 4; k = 1, 4, 6; G 6k0 (s , t , s ) are components of the so-called free GFFs which desribe the system of two free particles with total quantum numbers of pion [9], ϕ(s) is the pion wave function in the sense of RQM (5), s , s are the invariant masses of the free twoparticle system in the initial and final states, respectively. Now we have under integrals in (16) -(17) (compare [9]): where R(s, t, s ) = (s + s − t) ξ(s, t, s ) = −(M 2 λ(s, t, s ) + ss t) , ω 1 and ω 2 are the Wigner spin-rotation parameters: λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc) .
Recall that the form factors G (0) k0 (s, t, s ), k = 1, 6 describe gravitational features of a system of two particles without interaction. Free two-particle form factors are regular generalized functions (distributions) given by the corresponding functionals, defined on the space of test functions depending on the variables (s, s ). The functionals, in turn, is a function of the variable t, a square of momentum transfer. This variable is to be considered as a parameter.
In the frames of MIA the pion GFFs are functionals (14) - (17), generated by the free two-particle GFFs of (11) on test functions which are the products of the twoquark wave functions (see ((14) - (22)). There is a following difficulty in calculating the form factor G Strictly speakig, the calculation of this additional term should include the construction of a theory of interaction of graviton with two and more quarks simultaneously. However, we avoid this difficult problem, choosing a simple "minimal" extension, based on the fact that the non-relativistic limit of G (0) 60 (s, t, s ) is not singular. It is important to note that the proximity of relativistic and non-relativistic descriptions of GFFs at t → 0 is largely adopted [3]. In fact, in our "minimal" variant the dynamical part of possible additional contribution, the analog of meson exchange currents in nuclear physics, remains out of scope.
The structure of the form factor G (0) 60 (s, t, s ), which is the analog of form factor D in the case of two free particles, is defined by the following relation, obtained using (11), (12) and (10): whereK µ ,R µ -4-vectors in (12) appearing after integration and summation in (10), A = A(s, t, s ) -a multiplier defined by normalization of the Clebsh-Gordan coefficients in (10). The first two terms in r.h.s. have purely relativistic origin and are zero in the non-relativistic limit, they do not depend on the quark form factors g (q) 60 from (12) (an analogue of the quark form factor D) and exactly these terms contain the singularity. So, to advance in the simplest way, we require that the additional term in (23) has zero for the non-relativistic limit and contributes only to two first terms in (24), compensating the singularity. An important point, in our opinion, is the fact that the mentioned dangerous terms depend on the vectors of oneparticle parametrization in r.h.s. of (10), (12), while the additional form factor in (23), according to its meaning, contributes to r.h.s. of (10), which contains the oneparticle currents. So, it seems natural that G (a) 60 (s, t, s ) deforms the terms with the vectors of parametrization of one-particle currents (13), (24).
We define the additional term in (23) as to compensate the main contribution of the relativistic series expansion of scalar products in the first two terms in (24). So, the divergent terms become closest to their non-relativistic limit. The series expansion is carried out using as parameters the quantities k/M, k /M, (−t)/M (the variable k is defined in (5)). Thus we choose for G (a) 60 (s, t, s ) the following relation: where [. . .] lt are the main terms of corresponding relativistic series. Because of the relations (23) - (25), only the main terms of the relativistic expansion remain in diverging terms of (24) so making them finite. It is seen from the equations (23) -(25) that the chosen scheme for going beyond MIA does not contain arbitrariness. Taking into account the additional term constracted with the use of (23) - (25), leads to the following replacements in the equations (20), (21): After this procedure is carried out the purely relativistic terms of (17) take the form: The new form factors G (R) 6k0 , k = 1, 4 are obtained using the equations (20), (21) and (26) - (29).

III. THE MODEL OF QUARK GRAVITATIONAL STRUCTURE AND DETAILS OF CALCULATION
In [9] we used the general method of the relativistic invariant parametrization of the matrix elements of the local operators established in Ref. [7] for systems with arbitrary spin. Although now we consider actually the pion, we preserve for reasons of convenience the notations of the preceding paper. The pion GFFs in this parametrization are connected with commonly used (see, e.g., [3]) by the following relations: where t = (p π − p π ) 2 , and p π , p π are the pion 4momenta in the initial and the final states, respectively, and G (π) 10 , G (π) 60 are given by the equations (14) - (19), (22), (30) - (32).
The quark GFFs in our approach are connected with GFFs that are commonly used for particles with spin 1/2 (see, e.g., [3]) by the following relations [9]: We assume that the GFFs of u-andd-quarks are equal: g To define the explicit form of the quark GFFs we recall our calculation of the electroweak structure of the pion. We derived the functional form of the electromagnetic form factor of the quark from the behavior of our charge form factor of pion at (−t) → ∞ [32], which turned out to coincide with that of QCD, as mentioned above: where r 2 q is a mean square charge radius of the constituent quark. The charge form factor of the constituent coincides with the function (38) and the magnetic form factor is equal to this function multiplied by the corresponding magnetic moment [32]. Now we admit a similar definition for the quark GFFs (35) - (37) in term of (38): where D q is the D-term of the constituent quark. These equations give standard static limits (see, e.g., [3]): We set the parameter r 2 q in (38), to be equal to the mass MSR of the quark and to define the slope of the quark form factor D at zero. The corresponding actual value is given by the following form [52][53][54][55]: For the calculation of the pion GFFs according to (14) - (17), (30), (34) we use in (5) consequently one of the following model wave functions: We use the parameter b in (42) - (44) fixed previously in the works [9,32,56] on the electroweak properties of the pion. The different type of functions correspond to the different type of confinement. So, in the model (42) quadratic confinement is carried out, the model (43), as our calculations of the electroweak properties of a pion in the work [32] show, gives predictions very close to the model with linear confinement [57], and finally, the wave function (44) corresponds to a confinement weaker than a linear one. We calculate the pion GFFs using the same value of the constituent-quark mass, M = 0.22 GeV, as in our previous works (see, e.g., [32]).
So, now only one free parameter remains -the D-term of the constituent quark D q in the form factor (39), (40). In Sect. IV we fix this free parameter and present the gravitational properties of pion calculated taking into account the constituent-quark gravitational structure.

IV. THE CALCULATION OF THE PION GFFS AND THEIR STATIC MOMENTS
To calculate numerically the pion GFFs we need, first of all, to fix the remaining free parameter D q which enters the pion form factor D. Namely, we use the static moments of D (π) (t). For the time being there is no strict experimental estimation of the value D (π) (0) [3]. However, the first data for the slope at zero of the normalized to pion D-term form factor D of pion extracted from the process γ * γ → π 0 π 0 were presented in Ref. [44]: We use this estimation to fix D q taking into account the following definition [3]: As all but one parameters of the model were fixed previously in the works on electroweak structure of the pion, then S (π) D (46) is a function of the D-term of the constituent quark, D q , only. This function is given in Fig. 1 for three types of model quark interaction in pion (42) Note that this interval is rather narrow (< 0.5%). The existence of this interval recalls the fact that when describing the electroweak properties of pion in RQM we obtained the same values of the physical characteristics of the constituent quark for all model wave functions [32]. Now we add the quark D-term to the previous set (the mass, the charge MSR, the anomalous magnetic moment). Now, with (47), all the parameters are fixed and we do calculate the pion GFFs. Note that the pion form factor A, (34), (14), does not depend on D q and is determined by the parameters obtained previously, so, it is predicted. The standard condition A (π) (0) = 1 is fulfilled automatically for arbitrary values of parameters, if the quark form factors satisfy Eq. (40).
The results of calculation of the static gravitational moments of the pion are presented in the Table I. The mass MSR is defined following the relation [3]: In the last line of the table we present the results for the pion mechanical radius using the standard expression  (42) - (44). The mass of constituent quarks is M = 0.22 GeV, their MSR is given in (41) and the D-term Dq is taken from (47). The deviation from the given values of S For the calculation it is important to have the form factor D in the large range of momentum transfer including its asymptotics. This is a complicated problem, which is out of scope of this paper. It will be considered elsewhere.
A thorough correct full-fledged approach can be realized in the spirit of papers [37][38][39][40]. The GFF obtained in the present paper predictably gives underestimated values of the mechanical radius as compared with mass and electromagnetic radii. We believe that the detailed calculation which is in progress will give the ultimate truth.
Note that the deviations of the values S (π) D from that given in the Table I when the parameter D q is changed in the interval (47) is approximately 0.1%.
Let us do some remarks about our values of the pion D-term. As is well known, in theories with broken chiral symmetry (see, e.g., [58,59]) the pion D-term is equal to −1. However, if EMT contains only the contribution of quarks, without taking gluons into account, then its value is approximately −0.75 obtained in Ref. [44] in agreement with the soft pion theorem D = −1, given that the extracted value does not include the gluon contribution [3,60] Our values given in the Table I are close to this value. The same is true about the comparison of our slope at zero of the pion D-form factor with that of chiral theory, where −D (π) (0) = 2.40 GeV −2 . The results for pion GFFs satisfy the well-known relation (see, e.g., [3]) −D (π) (0) > A (π) (0). In addition, our values of A (π) (0) are close to the estimations given, for example, in the same review: Let us compare the mass and charge MSRs of the pion. The experimental pion charge MSR is (see, e.g., [44]): The mass MSR was estimated in [44] and the following interval of the values was obtained: Our result for the mass radius (see Table I) lays much closer to the charge radius (51) than (52). Curiously enough, our difference between the mass and the charge radii is the same as obtained in the work [18] for the proton (∆R cm = 0.1709 ± 0.0304 fm). We recall that our result for r 2 1/2 mass (48) is a direct consequence of our model of electroweak properties of the pion without new fittings.  Table I Figs. 2 and 3. For comparison, the results for GFFs as extracted [44] from the experimental data are given, too. Note that the slope of our curves is close to that of the experimental one at large Q 2 . The results for the lower and higher bounds of D q from (47) Table I (53); double-dotted dashed line (green) presents the result of the extraction of the form factor D given in the paper [44].
for the values of the quark D-term, D q , in the interval (47), form a rather narrow bunch corresponding to the quark mass M = 0.22 GeV. The curves remain agglomerated in the bunch up to squares of momentum transfer ∼ 10 GeV 2 . The schema of bunches works anew.
As the dependence of the pion GFFs on the choice of model wave functions (42) -(44) is weak, it seems us reasonable to construct an averaged fit for these form factors. We take the fitting function in the following form, which is used, for example in [61]: Least square fitting gives the following values of parameters. For the form factor A (π) we obtain: F (0) = 1, a = 0.882 GeV 2 , n = 1.007 and for the form factor D (π) : F (0) = −0.657, a = 0.0949 GeV 2 , n = 0.293. The value of the parameter F (0) for D (π) is equal to average over models value of the pion D-term (see Table I) and the slopes of the formfactors A and D at zero -to the corresponding averaged slopes. Obviously, there is no such correspondence for the pion mechanical radius (see the remarks above, after (49)). The results of the fitting are shown in figures 2 and 3.

V. CONCLUSIONS
In this work we extend our relativistic theory of composite-particle systems which describe simultaneously their electroweak and gravitational properties to calculate the gravitational characteristics of the pion. The approach is based on a version of the instant-form relativistic quantum mechanics. In the preceding work [9] we formulated the mathematical base and described the principal schema of deriving the gravitational characteristics in the approach. In the present work we give a detailed technique of calculation, thus transforming the approach into a quantitative method. The extension includes three main points: the constituent quarks are no longer point-like, we analyze different types of the quark interaction in pion, and we formulate a minimal way to overstep the frame of our modified impulse approximation (MIA) for correct description of the pion form factor D.
We derive the equations for the pion GFFs, the mass radius and the slope at zero of the normalized to pion D-term form factor D of pion in term of parameters describing the quark structure. Most of the parameters have been fixed even earlier in our works on the pion electromagnetic form factors. The only free parameter is the D-term of the constituent quark, which we fix by fitting the result for the slope at zero of the normalized to pion D-term form factor D of pion, to a choosen experimental value [44]. The results for the form factor A and for the mass MSR of pion do not depend on this new parameter and, so, are direct predictions of our approach. As a whole, the obtained now results for the gravitational characteristics of pion can be considered as firm predictions if calculated with a new future value of the mentioned slope.
Using these parameters we calculate the pion GFFs, their derivatives and static moments. We present the pion GFFs in the range of low and intermediated momentum transfers up to ∼ 10 GeV 2 . At large Q 2 the slope of our curves is close to that of [44]. The curves for different model quark-antiquark wave functions are agglomerated to form bunches corresponding to the quark mass M = 0.22 GeV, as it is in the electroweak case. The fits of the calculated pion GFFs, averaged over obtained with three different model wave functions, are constructed. The values of the static moments, mass MSR and the D-term of pion are consistent with the values given in the literature.
It seems obvious that because of extreme weakness of the gravitational interaction at hadron scale, the information on GFFs will be extracted from electroweak processes as yet. So, we believe that our theory which describes electroweak and gravitational properties simultaneously, is based on the unique foundations, and exploits common model parameters, will be useful. Moreover, it is possible that our unified approach can construct another bridge between electroweak and gravitational properties of composite systems, complementary to GPD.
The work was supported in part by the Ministry of Science and Higher Education of the Russian Federation (grant N 0778-2020-0005)(A.K.).