Pion gravitational form factors in a relativistic theory of composite particles
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Pion gravitational form factors in a relativistic theory of composite particles.
A.F. Krutov
1, 2, ∗ and V.E. Troitsky † Samara State Technical University, 443100 Samara, Russia P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 443011 Samara, Russia D.V. Skobeltsyn Institute of Nuclear Physics,M. V. Lomonosov Moscow State University, Moscow 119991, Russia (Dated: October 26, 2020)We extend our relativistic theory of electroweak properties of composite systems to describesimultaneously the gravitational form factors of hadrons. The approach is based on a version of theinstant-form relativistic quantum mechanics and makes use of the modified impulse approximation.We exploit the general method of the relativistic invariant parametrizaton of local operators towrite the energy-momentum tensor of a particle with an arbitrary spin. We use the obtained resultsto calculate the gravitational form factors of the pion assuming point-like constituent quarks. Allbut one parameters of our first-principle model were fixed previously in works on electromagneticform factors. The only free parameter, D q , is a characteristic of the gravitational form factor ofa constituent quark. The derived form factors of the pion satisfy the constraints given by thegeneral principles of the quantum field theory of hadron structure. The calculated gravitationalform factors and gravitational mean-square radius are in a good agreement with known results.The results demonstrate self-consistency of a common relativistic approach to electromagnetic andgravitational form factors of pion. I. INTRODUCTION
The probably most fundamental information abouta particle is contained in the matrix elements of itsenergy-momentum tensor (EMT). So, it is clear that thegravitational form factors (GFFs) of hadrons that en-ter the EMT matrix elements and their dependence onthe square of the momentum transfer t are in the focusof investigations (see, e.g. [1–5] and references therein).These form factors contain the information about thedistribution of mass, spin and internal forces inside thehadron. These forces are connected with the additionalglobal characteristic of a particle, the so-called D term ofthe EMT matrix. The study [6] of the structure of the el-ements of the EMT matrix and of their Lorentz-covariantdecomposition in terms of the form factors gives the lim-itations at t → D ( t )is not constrained (not even at t = 0) by general princi-ples and the value D = D (0) is therefore not known for(nearly) any particle (see also [7, 8]).At present one obtains the information about theGFF mainly from the hard-exclusive processes describedin terms of unpolarized generalized parton distribution(GPD). Particularly, in Ref. [9] (see also Ref. [10]) it isshown that GPD, derived from processes, gives the infor-mation on the space distribution of strong forces that acton quarks and gluons inside hadrons. The link of grav-itational form factors with GPD gives a possibility ofobtaining the data on these form factors using the hard-exclusive processes. The first results for nucleon GFFs ∗ [email protected] † [email protected] were obtained through the analysis of JLab data [5]. Thedata for the pion form factors were extracted from theexperiment of the collaboration Belle at KEKB [11, 12].It is worth noting that the model-independent extrac-tion of GPD from the experimental data is a difficultlong-term problem. So, today the theoretical estimationof the GFF, including D -term, is usually obtained in theframework of different model approaches. We mentionhere only the publications that are strictly related tothe present paper while more general information canbe found, for example, in the reviews [1–4].The pion electromagnetic and gravitational form fac-tors are obtained from GPD in the Nambu–Jona-Lasino(NJL) model [13, 14]. The authors show, in particu-lar, that the light-cone mass radii for the pion are al-most twice smaller than the light-cone charge radii. Thecalculated light-cone mass radius agrees with the valueobtained through a phenomenological extraction fromKEKB data [12]. Note that the NJL model as well asthe model of our approach contains gluons only implic-itly.Different approaches based on various forms of disper-sion relations (see, e.g., [15, 16] and references therein),that is on quite general principles of the quantum-field theory, can be considered as ”maximal model-indepemdent” approaches. The D term was calculatedusing unsubtracted t -channel dispersion relations for thedeeply virtual Compton scattering amplitudes [15].In connection with the first experimental results forthe GFFs of nucleons [5], the gravitational characteris-tics of these particles were calculated. The dependenceof EMT on the long-ranged electromagnetic interactionwas investigated in Ref. [17]. Using a simple model itwas shown that in the case of the long-ranged forces inthe proton one needs a sophisticated theory of the D - a r X i v : . [ h e p - ph ] O c t term construction. There is a possibility that the D -term is ill-defined and even singular. They propose theexploiting of the fixed- t dispersion relations for deep vir-tual Compton scattering as in Ref. [10] to avoid thisdifficulty. It is of interest that in the free-field model, aswas shown in Ref. [18, 19], the D -term for the fermionwith spin 1/2 is of a dynamical nature and vanishes forthe free fermion. The interaction inclusion gives rise tothe D -term of a fermion with an internal structure, thenucleon. Recently [20] the results of [3] have been ex-tended to the different frames where the nucleon has anonvanishing average momentum.The authors of [21] use the Skyrme model which re-spects the chiral symmetry and provides a practical re-alization of the large- N c picture of baryons described assolitons of mesonic fields. The EMT form factors are con-sistently described (see, e.g., [22] and references therein)in the bag model in the large- N c limit. It is importantto mention the recent results for the GFFs as obtaibedfrom the lattice QCD (see, e.g., [23, 24] and the refer-ences therein), and the study using an approach basedon the light-cone sum rules [25], and in the light-conequark model [26].Different theoretical approaches to the GFFs ofhadrons give different results and the absence of themodel-independent extracted data makes it impossibleto choose between them. We believe that in such a situa-tion, a theory which is intrinsically self-consistent theoryand describes as large set of the physical charathacteris-tics and systems as possible is welcome. This motivatesthe extension of the relativistic theory of the electromag-netic properties of composite systems developed previ-ously to the calculation of the GFF of the pion.The goal of the present paper is twofold. First, we ex-tend our relativistic model of the electromagnetic struc-ture of composite systems to include their gravitationalcharacteristics. Second, we derive the pion GFFs us-ing our previous calculations of electroweak properties ofhadrons.The model [27, 28] was successfully used for variouscomposite two-particle systems, namely, the deuteron[29], the pion [30–33], the ρ meson [34–36] and the kaon[37]. This model had predicted, with surprising accuracy,the values of the form factor F π ( Q ), which were mea-sured later in JLab experiments (see the discussion inRef. [32]): all new measurements follpwed the predictedcurve. Another advantage of the approach is matchingwith the QCD predictions in the ultraviolet limit, whenconstituent-quark masses are switched off, as expected athigh energies. The model reproduces correctly not onlythe functional form of the QCD asymptotics, but alsothe numerical coefficient; see Refs. [31, 33, 37] for de-tails. The method allows for an analytic continuation ofthe pion electromagnetic form factor from the space-likeregion to the complex plane of momentum transfers andgives good results for the pion form factor in the time-likeregion [38].Now we show that besides electroweak properties of composite systems, our approach can be used to calcu-late their gravitational characteristics. Even in a simpleversion of our approach (with the point-like quarks andthe two-particle wave functions of the harmonic oscilla-tor) the results agree well with other calculations andwith scarce measurements. The only free parameter thatwe add to the model is the constituent-quark D (0) = D q .This parameter is constrained from the pion mean-squareradius. Despite uncertainties in the latter, D q is fixed toa narrow interval which makes it possible to predict theGFFs at nonzero momentum transfers. Using the ob-tained results we calculate the values of the static grav-itational characteristics of the pion and obtain A and D form factors as functions of momentum transfer up to1 GeV . Note that the new parameter is not used in thecalculation of the A term, its value is a direct predictionof our previous approach. The form factors calculatedthrough our nonperturbative method satisfy all the con-straints given by the general principles of the quantum-field theory of hadron structure [6–8].The approach that we use is a particular variant ofthe theory based on the classical paper by P. Dirac [39],so-called Relativistic Hamiltonian Dynamics or Relativis-tic Quantum Mechanics (RQM). It can be formulated indifferent ways or in different forms of dynamics. Themain forms are the instant form, point form and light-front dynamics. Here we are dealing with instant-form(IF) RQM. The properties of different forms of RQM dy-namics are discussed in the reviews [40–43]. Today thetheory is largely used as a basis of the nonperturbativeapproaches to the particles structure.The presentation of the matrix elements of EMT interms of form factors, the invariant parametrization, isan important part of model approaches to gravitationalcharacteristics of particles. The majority of authors usethe parametrization given by Pagels [44] (see also [45]).The parametrization [44] was constructed in an almostphenomenological way using an analogy with the inves-tigations performed in connection with the self-stress ofthe electron. It is valid for the simplest cases of spin0 and 1 / / /
2, momenta (cid:126)p , (cid:126)p andspin projections m , m , that is the two-particle systemhaving quantum numbers of the pion. We construct theEMT in the basis with separated center-of-mass motion[49] | (cid:126)P , √ s (cid:105) , where P = p + p , s = P is the invariantmass squared and P , p and p are 4-vectors. We refer tothe corresponding form factors as to the free two-particleGFFs. These form factors are the functions of the invari-ant masses of the two-particle system in the initial andthe final states and depend on the momentun-transfersquare as a parameter. They are the regular generalizedfunctions, the distributions corresponding to the func-tionals given by the the two-dimensional integrals overthe invariant masses [27].To construct the pion GFFs we use a modified im-pulse approximation (MIA) (see Refs. [27, 28] and thereview [43]). In contrast to the baseline impulse approx-imation, MIA is formulated in terms of the form factorsand not in terms of the EMT operator itself. So, in MIAthe pion GFFs are presented as functionals given by thefree two-particle form factors on the set of the two-quarkwave functions of the pion. The necessity of using thedistributions was justified in the case of the electroweakinteraction in [27, 28, 47] (see also [6, 43]).The rest of the paper is organized as follows. In Sect.IIwe construct the matrix elements of the EMT of the par-ticle with an arbitrary spin and, in particular, with spins0 and 1 /
2. Sect.III presents the construction of the EMTmatrix element for the system of two free spin 1 / . We brieflyconclude and discuss the results in Sect.VI. II. THE ENERGY-MOMENTUM TENSORMATRIX ELEMENTSFOR A PARTICLE WITH AN ARBITRARY SPIN
In this Section we describe the general procedure ofparametrization of the EMT matrix element for a particlewith mass M and spin j . To write EMT in terms ofgravitational form factors we make use of the method[46]. Because of translational invariance it is sufficient toconsider only the following matrix element: (cid:104) (cid:126)p, m | T µν (0) | (cid:126)p (cid:48) , m (cid:48) (cid:105) , (1)where (cid:126)p (cid:48) , (cid:126)p are the particle moments, m (cid:48) , m are the spinprojections in the initial and final states, respectively; p (cid:48) = p = M . The normalization condition for the state vectors in(1) is: (cid:104) (cid:126)p, m | (cid:126)p (cid:48) , m (cid:48) (cid:105) = 2 p δ ( (cid:126)p − (cid:126)p (cid:48) ) δ mm (cid:48) , (2)with p = (cid:112) M + (cid:126)p . We have exploited the generalmethod of parametrization of matrix elements of localoperators developed in [46] to construct the matrix ele-ments of the operator of the electromagnetic current (see,e.g., [27, 28, 43]). Upon formulation of this method thecanonical basis in the Hilbert space was used. From thepoint of view of group theory the parameterization pro-cedure represents the realization of the known Wigner– Eckart theorem on the Poincar´e group [47]. The pa-rameterization represents the procedure of separation ofthe reduced matrix elements (form factors) which are in-variant with respect to transformations of the Poincar´egroup. The main idea of the canonical parameteriza-tion can be formulated as follows. Objects of two typesshould be constructed from the variables in the vectorsin the Hilbert space in (1):1. The set of linearly independent matrices in spinprojections in the initial and final states. At the sametime this set represents the set of linearly independentLorentz scalars (scalars and pseudoscalars). This set de-scribes the EMT matrix elements non-diagonal with re-spect to m, m (cid:48) and the behavior of the matrix elementsunder discrete space-time transformations.2. The set of linearly independent objects with thesame tensor dimension as the operator. In our case (1)this is a 4-tensor of the rank two. This set describes thebehavior of the matrix elements under Lorentz transfor-mations.The matrix element of the operator is written as thesum of all possible products of objects of the first typeand objects of the second type. The coefficients of theelements of this sum are the desired reduced matrix ele-ments, that is form factors. The obtained linear combi-nation is modified if additional constraints, for example,conservation laws, are imposed on the EMT operator.To construct a Lorentz-invariant matrix in spin projec-tions we use the well-known 4-pseudovector of (see, e.g.,[50]): Γ ( p ) = ( (cid:126)p(cid:126)j ) , (cid:126) Γ( p ) = M (cid:126)j + (cid:126)p ( (cid:126)p(cid:126)j ) p + M , Γ = − M j ( j + 1) . (3)Under the Lorentz transformations p µ = Λ µν p (cid:48) ν , the op-erator of the 4-spin (3) is transformed according to therepresentation of the small group:Γ µ ( p ) = Λ µν D jw ( p, p (cid:48) ) Γ ν ( p (cid:48) ) D jw ( p (cid:48) , p ) , (4)where Λ µν is the matrix of a Lorentz transformation and D jw ( p, p (cid:48) ) is the transformation operator from the smallgroup, the matrix of three-dimensional rotation. TheLorentz-transformation matrix in our case is of the formΛ µν = δ µν + 2 M p µ p (cid:48) ν − ( p µ + p (cid:48) µ )( p ν + p (cid:48) ν ) M + p λ p (cid:48) λ . (5)It can be shown using (4) that matrix elements of the op-erator D jw ( p, p (cid:48) )Γ µ ( p (cid:48) ) transform as the 4-pseudovectorand matrix elements of the operators D jw ( p, p (cid:48) ) p µ Γ µ ( p (cid:48) )and p (cid:48) µ Γ µ ( p ) D jw ( p, p (cid:48) ) as 4-pseudoscalars. Thus, the setof linearly independent scalars composed of the vectors p µ , p (cid:48) µ and the pseudovector Γ µ ( p (cid:48) ) contains not onlydiagonal (with respect to spin projections) terms, butnon-diagonal terms, too. Note, that the pseudovectorΓ µ ( p ) D jw ( p, p (cid:48) ) does not enter the set of scalars. Its lin-ear dependence can be shown if we use relation (4) andthe explicit form of the matrix Λ µν (5). After simple cal-culations we obtainΓ µ ( p ) D jw ( p, p (cid:48) ) = D jw ( p, p (cid:48) ) [Γ µ ( p (cid:48) ) −− p µ + p (cid:48) µ M + p µ p (cid:48) µ [ p ν Γ ν ( p (cid:48) )] (cid:21) . (6)Since p (cid:48) µ Γ µ ( p (cid:48) ) = 0, the desired set of linear indepentmatrices (that is the set of independent Lorentz scalars)is given by 2 j + 1 elements D jw ( p, p (cid:48) ) ( ip µ Γ µ ( p (cid:48) )) n , n = 0 , , . . . , j . (7)The imaginary unit i = − p (cid:48) µ Γ µ ( p ) D jw ( p, p (cid:48) ) = − D jw ( p, p (cid:48) ) p µ Γ µ ( p (cid:48) ) . (8)The number of linearly independent scalars in (7) is lim-ited by the fact that the product containing more than2 j numbers of factors Γ µ ( p (cid:48) ) is reduced to the productsof smaller number of factors, i.e., is not linearly indepen-dent. For even n the obtained objects in (7) are scalars,and for odd n they are pseudoscalars.In the decomposition of the matrix element (1) wemake use of the metric pseudotensor g µν and the rank2 tensors, that should be constructed from the variableson which the state vectors in (1) do depend. Using theavailable variables in the state vectors of the particle, itis possible to construct one pseudovector Γ µ ( p (cid:48) ) (3) andthree independent vectors: K µ = ( p − p (cid:48) ) µ , K (cid:48) µ = ( p + p (cid:48) ) µ ,R µ = (cid:15) µ ν λ ρ p ν p (cid:48) λ Γ ρ ( p (cid:48) ) . (9)Here (cid:15) µ ν λ ρ is the absolutely antisymmetric pseudotensorof rank 4, (cid:15) = −
1. For the matrix elements of theoperators in (9) to transform as the 4-vector, it is nec-essary to multiply them by D jw ( p, p (cid:48) ) from the left (inanalogy with (7)). The matrix element (1) is written in terms of all pos-sible products of vectors (9), pseudovector Γ µ ( p (cid:48) ), andpseudotensor g µν . Each of these objects is multiplied bya sum of linearly independent scalars (7). The coeffi-cients in such a decomposition are just form factors, orreduced matrix elements.Taking into account the symmetry properties of theEMT the parametrization of the matrix element (1) canbe written in the form: (cid:104) (cid:126)p, m (cid:12)(cid:12)(cid:12) T ( π ) µν (0) (cid:12)(cid:12)(cid:12) (cid:126)p (cid:48) , m (cid:48) (cid:105) = (cid:88) m (cid:48)(cid:48) (cid:104) m | D jw ( p, p (cid:48) ) | m (cid:48)(cid:48) (cid:105)×× (cid:104) m (cid:48)(cid:48) | τ µν (0) | m (cid:48) (cid:105) , (10)where τ µν (0) = G K (cid:48) µ K (cid:48) ν + G Γ µ Γ ν + G ( K (cid:48) µ Γ ν + Γ µ K (cid:48) ν )++ G ( K (cid:48) µ R ν + R µ K (cid:48) ν ) + G ( R µ Γ ν + Γ µ R ν )++ G K µ K ν + G g µν + G ( K µ Γ ν + Γ µ K ν )++ G ( K (cid:48) µ K ν + K µ K (cid:48) ν ) + G ( K µ R ν + R µ K ν ) , (11) G i = (cid:88) n g in ( t )( ip µ Γ µ ( p (cid:48) )) n . (12)In (12), g in ( t ) are the invariant coefficients, form factors, t = K is momentum-transfer square and Γ µ = Γ µ ( p (cid:48) ).Let us impose some additional physical conditions onthe operator (10).1. The requirement of self-adjointness. It is easy toshow, making use of (8), that the self-adjointness forr.h.s. of (10) requires a modification of the pseudovectorΓ µ with the help of the quantities introduced by (7), (9);namely:Γ µ → ˜Γ µ = Γ µ ( p (cid:48) ) − (cid:18) K µ K + K (cid:48) µ K (cid:48) (cid:19) [ p µ Γ µ ( p (cid:48) )] . (13)Note that this modification (13) does not affect theLorentz scalars (7). The requirement of self-adjointnessalso results in the multiplication of the terms containing G , G , G , G by the imaginary unit.2. The conservation law for EMT, T µν K µ = 0, givesthe following conditions to be imposed on the Lorentzscalars, G = G = G = 0 . (14)The conservation law requires also the following changes: G → − G , G → tG ; . (15)3. The parity-conservation condition gives limitationsfor the summation in (7). Namely, in G , G , G , G thevalues of n are even while in G , G they are odd. Thelimits of summations are the following: for G , G theyare 0 ≤ n ≤ j ; for G , G they are 0 ≤ n ≤ j − G , G they are 0 ≤ n ≤ j −
2. Summing is limitedby the fact that each term in the decomposition (11)contains no more than 2 j factors Γ( p (cid:48) ).So, the most general parameterization of the matrix el-ement (10) has the following form if the above constraintsare taken into account: τ µν (0) = 12 G K (cid:48) µ K (cid:48) ν + G ˜Γ µ ˜Γ ν + G ( K (cid:48) µ ˜Γ ν + ˜Γ µ K (cid:48) ν )++ iG ( K (cid:48) µ R ν + R µ K (cid:48) ν ) + iG ( R µ ˜Γ ν + ˜Γ µ R ν )++ G ( tg µν − K µ K ν ) , (16)where the summation is limited as is pointed above andthe factor 1/2 before G is a result of the normalizationcondition: the static limit of EMT should be equal to themass.Let us use the obtained general parametrization in thecase of spin 0. Now for the pion EMT we have: (cid:104) (cid:126)p (cid:12)(cid:12)(cid:12) T ( π ) µν (0) (cid:12)(cid:12)(cid:12) (cid:126)p (cid:48) (cid:105) = 12 G ( π )10 ( t ) K (cid:48) µ K (cid:48) ν ++ G ( π )60 ( t ) [ tg µν − K µ K ν ] , (17)The pion GFFs in canonical parametrization (17) areconnected with commonly used (see, e.g., [3]) by the fol-lowing relations: G ( π )10 ( t ) = A ( π ) ( t ) , G ( π )60 ( t ) = − D ( π ) ( t ) . (18)In the case of spin 1/2, Eq. (16) gives the followingresult which we will use below as the constituent-quarkEMT canonical parametrization: (cid:104) p, m (cid:12)(cid:12)(cid:12) T ( q ) µν (0) (cid:12)(cid:12)(cid:12) p (cid:48) , m (cid:48) (cid:105) = (cid:88) m (cid:48)(cid:48) (cid:104) m (cid:12)(cid:12)(cid:12) D / w ( p, p (cid:48) ) (cid:12)(cid:12)(cid:12) m (cid:48)(cid:48) (cid:105)×(cid:104) m (cid:48)(cid:48) (cid:12)(cid:12)(cid:12) (1 / g ( q )10 ( t ) K (cid:48) µ K (cid:48) ν + ig ( q )40 ( t ) (cid:2) K (cid:48) µ R ν + R µ K (cid:48) ν (cid:3) ++ g ( q )60 ( t ) [ tg µν − K µ K ν ] (cid:12)(cid:12)(cid:12) m (cid:48) (cid:105) , (19)These GFFs in the canonical parametrization (19) canbe written in terms of commonly used GFFs for particlesof spin 1/2 in the form g ( q )10 ( t ) = 1 (cid:112) − t/ M × (cid:20)(cid:18) − t M (cid:19) A ( q ) ( t ) + 2 t M J ( q ) ( t ) (cid:21) , (20) g ( q )40 ( t ) = − M J ( q ) ( t ) (cid:112) − t/ M ) , (21) g ( q )60 ( t ) = − (cid:114) − t M D ( q ) ( t ) . (22)In the following Section we generalize the method ofconstruction of the EMT matrix elements, given above,to composite systems. III. THE EMT MATRIX ELEMENTS FOR ASYSTEM OF TWOFREE PARTICLES WITH PION QUANTUMNUMBERS
Our relativistic approach to form factors of compositesystems of interacting components makes use of form fac-tors of corresponding free systems. So, to obtain GFFs ofa composite system we need to construct GFFs that de-scribe the gravitational properties of a system of two freeconstituents, the two-particle system as a whole havingquantum numbers of the composite system under con-sideration. We call the GFFs of the two-particle systemwithout interaction the free gravitational two-particleform factors. The form factors of a composite systemof two interacting particles are written in our approachin terms of free two-particle form factors and wave func-tions exploiting modified impulse approximation (MIA).This approximation was first formulated in the case ofelectroweak properties of hadrons in our papers [27, 28](see also the review [43]).EMT operator T (0) µν (0) for a system of two free particlesis of the form T (0) µν (0) = T µν ⊗ I (2) ⊕ T µν ⊗ I (1) . (23)Here T , µν are EMTs of the particles, and I (1 , are theidentity operators in one-particle Hilbert-state spaces ofthe particles. The following set of two-particle vectorscan be chosen as the basis: | (cid:126)p , m ; (cid:126)p , m (cid:105) = | (cid:126)p m (cid:105) ⊗ | (cid:126)p m (cid:105) , (24)where (cid:126)p , (cid:126)p are the 3-momenta of particles, m , m arethe projections of spins to the z axis, the normalizationof one-particle vectors is given in (2). In terms of matrixelements in the basis (24), the relation (23) is rewrittenas the sum of matrix elements of one-particle EMT op-erators (cid:104) (cid:126)p , m ; (cid:126)p , m | T (0) µν (0) | (cid:126)p (cid:48) , m (cid:48) ; (cid:126)p (cid:48) , m (cid:48) (cid:105) == (cid:104) (cid:126)p , m | (cid:126)p (cid:48) , m (cid:48) (cid:105)(cid:104) (cid:126)p , m | T µν (0) | (cid:126)p (cid:48) , m (cid:48) (cid:105) ++ (1 ↔ . (25)Each of the matrix elements of the one-particle EMTin (25) can be written in terms of GFFs (19) (see, e.g.,[27, 28, 49]).Along with this basis (24), we consider the basis inwhich the motion of the center of mass of two particlesis separated ([27, 28, 49]): | (cid:126)P , √ s, J, l, S, m J (cid:105) , (cid:104) (cid:126)P , √ s, J, l, S, m J | (cid:126)P (cid:48) , √ s (cid:48) , J (cid:48) , l (cid:48) , S (cid:48) , m J (cid:48) (cid:105) == N CG δ (3) ( (cid:126)P − (cid:126)P (cid:48) ) δ ( √ s − √ s (cid:48) ) × δ JJ (cid:48) δ ll (cid:48) δ SS (cid:48) δ m J m J (cid:48) , (26) N CG = (2 P ) k √ s , k = (cid:112) λ ( s , M , M )2 √ s , where P µ = ( p + p ) µ , P µ = s , √ s is the invari-ant mass of the system of two particles, l is the or-bital momentum in the center-of-mass system (c.m.s.), (cid:126)S = ( (cid:126)S + (cid:126)S ) = S ( S +1) , S is the total spin in c.m.s., J is the total angular momentum, m J is the projectionof the total angular momentum, M is the constituentmass, and λ ( a, b, c ) = a + b + c − ab + ac + bc ). Thebasis (26) is related to the basis (24) by the Clebsch-Gordan decomposition for the Poincar´e group. The cor-responding decomposition of a direct product (24) of twoirreducible representations of the Poincar´e group into ir-reducible representations (26) for particles with spin 1/2has the form [49] (see also [43]): | (cid:126)p , m ; (cid:126)p , m (cid:105) = (cid:88) | (cid:126)P , √ s, J, l, S, m J (cid:105)×(cid:104) Jm J | S l m S m l (cid:105) Y ∗ lm l ( ϑ , ϕ ) (cid:104) S m S | / / m ˜ m (cid:105)× (cid:104) ˜ m | D / w ( P, p ) | m (cid:105)(cid:104) ˜ m | D / w ( P, p ) | m (cid:105) , (27)where (cid:126)p = ( (cid:126)p − (cid:126)p ) / p = | (cid:126)p | , ϑ , ϕ are the sphericalangles of the vector (cid:126)p in c.m.s., Y lm l is the sphericalfunction, (cid:104) S m S | / / m ˜ m (cid:105) and (cid:104) Jm J | S l m S m l (cid:105) are the Clebsh-Gordan coefficients of the group SU (2), (cid:104) ˜ m | D / w ( P, p ) | m (cid:105) is the matrix of the three-dimensionalspin rotation, that is necessary for the relativistic invari-ant summation of the particle spins. The sums go overall discrete variables , ˜ m , ˜ m , m l , m S , l , S , J , m J .To obtain the basis where the center-of-mass motion isseparated we invert the decomposition (27): | (cid:126)P , √ s, J, l, S, m J (cid:105) = (cid:88) m m (cid:90) d(cid:126)p p d(cid:126)p p ×| (cid:126)p , m ; (cid:126)p , m (cid:105)× × (cid:104) (cid:126)p , m ; (cid:126)p , m | (cid:126)P , √ s, J, l, S, m J (cid:105) , (28)with the Clebsh-Gordan coefficient (cid:104) (cid:126)p , m ; (cid:126)p , m | (cid:126)P , √ s, J, l, S, m J (cid:105) == √ s [ λ ( s, M , M )] − / P δ ( P − p − p ) ×× (cid:88) (cid:104) m | D / w ( p , P ) | ˜ m (cid:105)(cid:104) m | D / w ( p , P ) | ˜ m (cid:105)×(cid:104) / / m ˜ m | S m S (cid:105) Y lm l ( ϑ , ϕ ) (cid:104) S l m S m l | Jm J (cid:105) , the sum being over ˜ m , ˜ m , m l , m S .We use below the basis (28) with pion quantum num-bers J = l = S = 0: | (cid:126)P , √ s, , , , (cid:105) = | (cid:126)P , √ s (cid:105) . (29)We construct the EMT matrix element in the basis(28) for quantum numbers given above using the generalmethod of parametrization of Section II. Using (10)-(12),(16) we obtain the parametrization which is analogous tothat for zero spin (17): (cid:104) P, √ s (cid:12)(cid:12)(cid:12) T (0) µν (0) (cid:12)(cid:12)(cid:12) P (cid:48) , √ s (cid:48) (cid:105) == 12 G (0)10 ( s, t, s (cid:48) ) A (cid:48) µ A (cid:48) ν ++ G (0)60 ( s, t, s (cid:48) ) [ t g µν − A µ A ν ] , (30)where G (0) i ( s, t, s (cid:48) ) , i = 1 , A µ = ( P − P (cid:48) ) µ , A = t ,A (cid:48) µ = 1( − t ) (cid:2) ( s − s (cid:48) − t ) P µ + ( s (cid:48) − s − t ) P (cid:48) µ (cid:3) . It is easy to show that all the imposed constraints aresatisfied.It is possible to derive the equations analogous to (30)for free two-particle systems with different quantum num-bers. Such constructions were obtained in [27, 28, 34, 49]in the context of the parametrization of the matrix ele-ments of electroweak currents.Note that the objects G (0) i ( s, t, s (cid:48) ) , i = 1 ,
6, in gen-eral, are generalized functions (distributions), definedon a space of test functions (see, e.g., [51], and also[6, 27, 28, 47]), and so the static limits at t → (cid:104) P, √ s (cid:12)(cid:12)(cid:12) T (0) µν (0) (cid:12)(cid:12)(cid:12) P (cid:48) , √ s (cid:48) (cid:105) == (cid:88) (cid:90) d(cid:126)p p d(cid:126)p p d(cid:126)p (cid:48) p (cid:48) d(cid:126)p (cid:48) p (cid:48) (cid:104) P, √ s | (cid:126)p , m ; (cid:126)p , m (cid:105)× (cid:104) (cid:104) (cid:126)p , m | (cid:126)p (cid:48) , m (cid:48) (cid:105)(cid:104) p , m (cid:12)(cid:12)(cid:12) T (2) µν (0) (cid:12)(cid:12)(cid:12) p (cid:48) , m (cid:48) (cid:105) ++ (cid:104) (cid:126)p , m | (cid:126)p (cid:48) , m (cid:48) (cid:105)(cid:104) p , m (cid:12)(cid:12)(cid:12) T (1) µν (0) (cid:12)(cid:12)(cid:12) p (cid:48) , m (cid:48) (cid:105) (cid:105) ×(cid:104) (cid:126)p (cid:48) , m (cid:48) ; (cid:126)p (cid:48) , m (cid:48) | P (cid:48) , √ s (cid:48) (cid:105) , (31)where the sums are over the variables m , m , m (cid:48) , m (cid:48) .The substitution of (30), one-particle matrix elementsof EMT (19), and the Clebsh-Gordan coefficiemts (28) for J = l = S = 0 in (31) gives the desired free two-particleGFFs. The integrals in (31) are written in the coordinateframe with (cid:126)P (cid:48) = 0 , (cid:126)P = (0 , , P ). The D w -functions forspin 1/2 are of the form [52]: D / w ( p , p ) = cos( ω/ − i ( (cid:126)k ˆ (cid:126)j ) sin( ω/ ,(cid:126)k = [ (cid:126)p (cid:126)p ] | [ (cid:126)p (cid:126)p ] | ,ω = 2 arctan | [ (cid:126)p (cid:126)p ] | ( p + M )( p + M ) − ( (cid:126)p (cid:126)p ) , (32)where ˆ (cid:126)j is the operator of the particle spin written interms of the Pauli matrices. In the choosen coordinatesystem, two D w -functions in r.h.s. of (31) become unitymatrices and the other are written with the use of (32).The sum of the rotations around the same axis is obtainedfollowing the prescription: D / w ( ω ) D / w ( ω ) = D / w ( ω + ω ) . (33)After performing the convolution of both sides, first, withthe tensor A (cid:48) µ A (cid:48) ν , second, with g µν , and the integra-tions and summations, we obtain the system of two alge-braic equations for the free form factors G (0) i ( s, t, s (cid:48) ) , i =1 ,
6: 12 G (0)10 (cid:20) λ ( s, t, s (cid:48) ) t (cid:21) − λ ( s, t, s (cid:48) ) G (0)60 == 12 A (cid:26) (cid:104) g ( u )10 ( t ) + g ( ¯ d )10 ( t ) (cid:105) ( s + s (cid:48) − t ) cos( ω + ω ) − − M (cid:104) g ( u )40 ( t ) + g ( ¯ d )40 ( t ) (cid:105) ξ ( s, t, s (cid:48) )( s + s (cid:48) − t ) sin( ω + ω ) −− (cid:104) g ( u )60 ( t ) + g ( ¯ d )60 ( t ) (cid:105) λ ( s, t, s (cid:48) ) cos( ω + ω ) (cid:111) , (34)12 G (0)10 (cid:20) λ ( s, t, s (cid:48) )( − t ) (cid:21) + 3 t G (0)60 == 12 A (cid:26) (cid:104) g ( u )10 ( t ) + g ( ¯ d )10 ( t ) (cid:105) (4 M − t ) cos( ω + ω )++ 3 t (cid:104) g ( u )60 ( t ) + g ( ¯ d )60 ( t ) (cid:105) cos( ω + ω ) (cid:111) , (35)where A = A ( s, t, s (cid:48) ) = 2 R ( s, t, s (cid:48) ) λ ( s, t, s (cid:48) ) ,R ( s, t, s (cid:48) ) = ( s + s (cid:48) − t )2 (cid:112) ( s − M )( s (cid:48) − M ) × ϑ ( s, t, s (cid:48) )[ λ ( s, t, s (cid:48) )] / ,ξ ( s, t, s (cid:48) ) = (cid:112) − ( M λ ( s, t, s (cid:48) ) + ss (cid:48) t ) ,ω and ω are the Wigner spin-rotation parameters: ω = arctan ξ ( s, t, s (cid:48) ) M (cid:104) ( √ s + √ s (cid:48) ) − t (cid:105) + √ ss (cid:48) ( √ s + √ s (cid:48) ) ,ω = arctan α ( s, s (cid:48) ) ξ ( s, t, s (cid:48) ) M ( s + s (cid:48) − t ) α ( s, s (cid:48) ) + √ ss (cid:48) (4 M − t ) ,α ( s, s (cid:48) ) = 2 M + √ s + √ s (cid:48) , ϑ ( s, t, s (cid:48) ) = θ ( s (cid:48) − s ) − θ ( s (cid:48) − s ), θ is the Heaviside function. s , = 2 M + 12 M (2 M − t )( s − M ) ∓ M (cid:112) ( − t )(4 M − t ) s ( s − M ) ,g ( u, ¯ d ) i ( t ) , i = 1 , , u - and ¯ d - quarks, re-spectively. The cutting off by the Heaviside functions in(34), (35) gives the kinematically available region in theplane of invariant variables ( s, s (cid:48) ) (see, e.g., [43]).The formal solution of the system (34), (35) is of theform: G (0)10 ( s, t, s (cid:48) ) = R ( s, t, s (cid:48) ) tλ ( s, t, s (cid:48) ) × (cid:26) (cid:104) g ( u )10 ( t ) + g ( ¯ d )10 ( t ) (cid:105) (cid:2) (4 M − t ) λ ( s, t, s (cid:48) ) ++ 3 t ( s + s (cid:48) − t ) (cid:3) cos( ω + ω ) −− M t (cid:104) g ( u )40 ( t ) + g ( ¯ d )40 ( t ) (cid:105) × ξ ( s, t, s (cid:48) )( s + s (cid:48) − t ) sin( ω + ω ) } , (36) G (0)60 ( s, t, s (cid:48) ) = 12 R ( s, t, s (cid:48) ) × (cid:26) (cid:104) g ( u )10 ( t ) + g ( ¯ d )10 ( t ) (cid:105) (cid:2) ( s + s (cid:48) − t ) ++ (4 M − t ) λ ( s, t, s (cid:48) ) /t (cid:3) cos( ω + ω ) −− M (cid:104) g ( u )40 ( t ) + g ( ¯ d )40 ( t ) (cid:105) × ξ ( s, t, s (cid:48) )( s + s (cid:48) − t ) sin( ω + ω ) } ++ 2 (cid:104) g ( u )60 ( t ) + g ( ¯ d )60 ( t ) (cid:105) λ ( s, t, s (cid:48) ) cos( ω + ω ) . (37)Note that the system (34), (35) is ill-defined at t → t = 0. So,the solution for the form factor (37) cesses to exist inthe vicinity of the point t = 0 and the weak limit at t → t = 0 of thegravitational D -form factor of pion. The form (37) isvalid only for finite values of momentum-transfer square.We discuss the physical consequences of this singularityin the free form factor (37) in Sect. V. Now we showonly that a regular anzatz can be choosen by making useof the non-relativistic limit of the form factor (37) whenthe singularity disappears: G (0)60 nr ( k, t, k (cid:48) ) = 2 (cid:104) g ( u )60 ( t ) + g ( ¯ d )60 ( t ) (cid:105) ϑ ( k, t, k (cid:48) ) k k (cid:48) (cid:112) ( − t ) , (38) ϑ ( k, t, k (cid:48) ) = θ (cid:16) k (cid:48) − (cid:12)(cid:12)(cid:12) k − (cid:112) ( − t ) / (cid:12)(cid:12)(cid:12)(cid:17) − θ (cid:16) k (cid:48) − (cid:16) k + (cid:112) ( − t ) / (cid:17)(cid:17) , The quantity G (0)60 nr ( k, t, k (cid:48) ) is, in fact, a free non-relativistic two-particle form factor, the non-relativisticanalog of the form factor (30), (37). The weak limit ofthe form factor (38) at t → anzats , we construct the desired functionas the relativistic generalization of the non-relativisticform factor (38). The comparison of the expressions (38) and (37) allows to choose the free form factor in theneighborhood of the point t = 0 in the following form: G (0)60 ( s, t, s (cid:48) ) = G (0 a )60 ( s, t, s (cid:48) ) = R ( s, t, s (cid:48) ) λ ( s, t, s (cid:48) ) × (cid:104) g ( u )60 ( t ) + g ( ¯ d )60 ( t ) (cid:105) cos( ω + ω ) , (39)where G (0 a ) denotes the functions that appear as a re-sult of the supposition, while R ( s, t, s (cid:48) ) , andω , are givenby (34), (35). In non-relativistic limit, (39) transformsinto (38). In what follows we use (37) for the pion D -formfactor at finite values of t , but in the vicinity of the t = 0 point we make use of (39). IV. GRAVITATIONAL FORM FACTORS OFPION IN MODIFIED IMPULSEAPPROXIMATION
To obtain the GFFs of pion we use instant form (IF) ofRQM ([39–42]). The details of our version for compositesystems can be foud in the review ([43]). In RQM theinteraction operator is included in the generators of thePoincar´e group, the commutation relations of the alge-bra being preserved. We include the interaction in thealgebra of the Poincar´e group following the procedure of[53]: ˆ M → ˆ M I = ˆ M + ˆ V , (40)here ˆ M is the operator of the invariant mass for a freesystem, ˆ V is interaction operator, and ˆ M I the mass op-erator for the system with interaction.The wave function of the system of interating particlesin IF RQM is defined as the eigenfunction of the followingcomplete set of the operators:ˆ M I (or ˆ M I ) , ˆ J , ˆ J , ˆ (cid:126)P , (41)here ˆ J is the operator of the square of the total angularmoment, ˆ J is the operator of the projection of the totalangular moment on the z axis and ˆ (cid:126)P is the operator ofthe total momentum.In the IF RQM the operators ˆ J , ˆ J , ˆ (cid:126)P coincide withcorresponding operators for the composite system with-out interaction and only the term ˆ M I ( ˆ M I ) is interactiondepending. The two-quark wave function of pion in thebasis given by the complete set of vectors (26), (28), (29)diagonalizes (41) and has the form: (cid:104) (cid:126)P , √ s | (cid:126)p (cid:105) = N C δ ( (cid:126)P − (cid:126)p ) ϕ ( k ) ,N C = (cid:112) p (cid:114) N CG k , (42)The wave function of intrinsic motion is the eigenfunctionof the operator ˆ M I ( ˆ M I ) and in the case of two particlesof equal masses is ϕ ( k ( s )) = √ s u ( k ) k , (cid:90) u ( k ) k dk = 1 , (43)The normalization factors in (43) correspond to the tran-sition to the relativistic density of states k dk → k dk √ k + M . (44)The decomposition of the matrix element (17) of thepion EMT in terms of the complete set of the vectors(26), (28), (29) is (cid:104) (cid:126)p (cid:12)(cid:12)(cid:12) T ( π ) µν (0) (cid:12)(cid:12)(cid:12) (cid:126)p (cid:48) (cid:105) == (cid:90) d (cid:126)P d (cid:126)P (cid:48) N CG N (cid:48) CG d √ s d √ s (cid:48) (cid:104) (cid:126)p | (cid:126)P , √ s (cid:105)×(cid:104) (cid:126)P , √ s | T ( π ) µν (0) | (cid:126)P (cid:48) , √ s (cid:48) (cid:105)(cid:104) (cid:126)P (cid:48) , √ s (cid:48) | (cid:126)p (cid:48) (cid:105) , (45)where (cid:104) (cid:126)P (cid:48) , √ s (cid:48) | (cid:126)p (cid:48) (cid:105) is the wave function in the sense ofIF RQM (42).We obtain (cid:104) (cid:126)p (cid:12)(cid:12)(cid:12) T ( π ) µν (0) (cid:12)(cid:12)(cid:12) (cid:126)p (cid:48) (cid:105) == (cid:90) N C N (cid:48) C N CG N (cid:48) CG d √ s d √ s (cid:48) ϕ ( s ) ×(cid:104) (cid:126)p , √ s (cid:12)(cid:12)(cid:12) T ( π ) µν (0) (cid:12)(cid:12)(cid:12) (cid:126)p (cid:48) , √ s (cid:48) (cid:105) ϕ ( s (cid:48) ) . (46)The matrix element of the tensor in (46) is to beconsidered as a Lorentz-covariant generalized function[27, 28, 47, 51]), that has a meaning only under the in-tegral. The integral itself presents a functional giving aregular distribution. The decomposition of the tensor inthe integral in terms of tensors which were used in thedecomposition in l.h.s. of (46) entering (17) is: N C N (cid:48) C N CG N (cid:48) CG (cid:104) (cid:126)p , √ s (cid:12)(cid:12)(cid:12) T ( π ) µν (0) (cid:12)(cid:12)(cid:12) (cid:126)p (cid:48) , √ s (cid:48) (cid:105) ϕ ( s (cid:48) ) == 12 ˜ G ( s, t, s (cid:48) ) K (cid:48) µ K (cid:48) ν ++ ˜ G ( s, t, s (cid:48) ) [ tg µν − K µ K ν ] , (47)here ˜ G i ( s, t, s (cid:48) ) , i = 1 , G ( π ) i ( t ) = (cid:90) d √ s d √ s (cid:48) ϕ ( s ) ˜ G i ( s, t, s (cid:48) ) ϕ ( s (cid:48) ) ,i = 1 , . (48)The main point now is the calculation of the function˜ G i ( s, t, s (cid:48) ). To obtain similar form factors describingelectroweak structure of composite hadrons it is custom-ary exploit the so-called impulse approximation (IA) (see,e.g., the review [41]). Let us demonstrate the meaning ofIA extending the approach to GFF. These form factorscharacterize the scattering cross-section of a projectile bya composite system in the process of graviton exchange.So, the EMT in this case can be written in the followingform: T = (cid:88) k T ( k ) + (cid:88) k (cid:104) m T ( km ) + . . . , (49)where the first term presents the sum of one-particleEMTs, the second term presents the sum of two-particleEMT, and so on. The first sum describes the scatter-ing of a projectile by each independent constituent, thesecond sum describes the scattering by two constituentssimultaneously and so on. The standard IA leaves in (49)only the first term: T ≈ (cid:88) k T ( k ) . (50)Note that in the approximation (50) the operators inthe instant form RQM does not satisfy the Lorentz-covariance conditions and the conservation law [41].To study the electroweak structure of hadrons, we hadproposed [27, 28] the modified impulse approxination(MIA). Constructing MIA for GFFs we change the formfactors ˜ G i ( s, t, s (cid:48) ) in (48) for free two-particle GFFs (30):in the invariant part of the decomposition (46), (47) wethrow off the contribution of the simultaneous scatteringby two and more constituents and take into account onlyscattering by free two-constituent system.The covariant part of the decomposition (46) – (48)is not changed by MIA and so, the Lorentz-covarianceconditions and the conservation law for the EMT ma-trix element (46) are not broken. This happens becausein MIA the contribution of the second term in (49) ispartially taken into account in a self-consistent way.In MIA, the pion GFFs (48) are written in the form: G ( π ) i ( t ) = (cid:90) d √ s d √ s (cid:48) ϕ ( s ) G (0) i ( s, t, s (cid:48) ) ϕ ( s (cid:48) ) ,i = 1 , , (51)0where G (0) i ( s, t, s (cid:48) ) are free two-particle form factors (30),given by (36), (37), (39).In the following Section, the details of calculation ofpion GFFs using (51) and the corresponding results aregiven. V. RESULTS OF CALCULATUONS
In what follows we use the conventional notations of A , J and D form factors (see, e.g., [3]) and the linkingrelations (18), (20) – (22).To obtain the pion form factor A we use directly theequations (18), (51), (36) while in the case of the formfactor D , which is ill-defined, we involve an anzatz de-scribed in detail in Section III. We obtain the pion D form factor in the vicinity of t = 0 using (18), (51) andassuming (39) (the form factor D ( πa ) ( t )). For the over-all description of the pion D form factor we need to joinsmoothly this function D ( πa ) ( t ) with the solution for fi-nite values of t given by (18), (51), (37). We describethis procedure in detail later.Let us list first the relativistic effects contained in (51).The contributions of the J form factors of the constituentquarks ( g ( q )40 ( t )) to the pion A ( G ( π )10 ( t )) and D ( G ( π )60 ( t ))form factors are a consequence of pure relativistic effect ofspin rotation. These contributions vanish if we set ω , in (36), (37) equal to zero. The contribution of quark A form factor g ( q )10 ( t ) to pion D form factor (37) is ofrelativistic origin, too.To obtain numerical results for pion GFFs in our model(51) (36), (37), we need some parameters to be used asan input. We suppose that u - and d - quarks have oneand the same gravitational structure and so, we have toset three quark GFFs as functions of momentum transfersquare. It is also necessary to choose a model two-quarkwave function of pion (43), and to fix the mass of lightquark, M .In the present work we consider the simpliest case, thatof point-like constituent quarks. This means that insteadof quark form factors, we use their standard static mo-ments: A ( q ) ( t ) = A ( q ) (0) = 1 , J ( q ) ( t ) = J ( q ) (0) = 12 ,D ( q ) ( t ) = D ( q ) (0) = D q , q = u, ¯ d , (52)where D q is the D -term of the constituent quark.We had shown [30] that the results of calculations forelectromagnetic form factors depend weakly on the actualform of the two-quark wave function in pion. Here wechoose for (43) the wave function of the ground stateof harmonic oscillator which ensures square-law quarkconfinement, u ( k ) = (cid:18) √ π b (cid:19) / exp (cid:18) − k b (cid:19) . (53) Here b is the parameter of the model along with thequark-mass M . The best results for electroweak prop-erties of light mesons [30–38] were obtained for the fol-lowing values of these parameters: M = 0 .
22 GeV , b = 0 .
35 GeV . (54)In what follows we fix these values also for the calculationof pion GFFs. So, to derive the pion GFFs we need tofix only the constituent-quark D -term (52).The mean-square radius (MSR) of pion we define asfollows (see [3] and the original paper [46]): (cid:104) r π (cid:105) = 6 A ( π ) (cid:48) (0) − M π D ( π ) (0) , (55)where M π = 0 . A ( π ) (0) = 1 is fulfilled automatically.For the interval of possible data for the pion MSR weadopt the interval that can be calculated using the resultslisted in the review [3], namely: (cid:104) r π (cid:105) min = 65 .
38 GeV − , (cid:104) r π (cid:105) max = 69 .
52 GeV − . (56)To obtain this interval of MSR values in our approach,we require in addition the parameter D q (52) to be inthe following region of approximately the same relativespread D q = − . ± . . (57)The interval of values of the pion D -term correspondingto the chosen interval of the quark D -term (57) is D ( π ) (0) min = − . , D ( π ) (0) max = − . . (58)The equations for the form factor A ( π ) ( t ) do not con-tain the parameter D q . So, the derivative of the A formfactor of pion at t = 0 is defined by the parameters (54)fixed in our model approach to the pion electroweak formfactors and has a predictive nature. This value is ob-tained numerically using (18), (51): A ( π ) (cid:48) (0) = 0 . − . (59)The results of calculation of the pion A form factor arepresented in fig. 1 and fig. 2. Note that fig. 1 demon-strates, in particular, that the relativistic spin rotationeffect gives an essential contribution to A form factor.This effect is purely kinematical and thus takes place forany model wave function. The effect changes essentiallythe slope of A form factor at zero t and, as a consequence,the value of the pion gravitational radius (55), (59). Thisfact emphasizes the importance of the corresponding the-ory to be essentially relativistic.As we have mentioned above recently the data on thepion GFFs was extracted from the experiment [11] forthe first time in [12]. In fig. 2 we compare our results1 FIG. 1. Gravitational A form factor of pion. Full line (red) –the full result; dashed line (blue) – the contribution of the A form factors g ( q )10 of the constituent quarks; short-dashed line(magenta) – the contribution of the quark J form factors g ( q )40 (relativistic spin rotation effect).FIG. 2. Our A form factor of pion (full line (red)) in compari-son with the result of the authors of [12]. Their A form factorof pion is normalized to its value at zero t (double-dot-dashed,green) line. for pion A form factor with those given in [12]. The re-sults are in a qualitative agreement, however the slope ofour A form factor is smaller. Note that we choose herethe simpliest variant of the model confining ourseves topoint-like constituent quarks. If we depart from this con-dition, the quark form factors would give the additionaldecreasing of the pion form factor and would amelioratethe agreement.To calculate the pion D form factor we need first to joinsmoothly the function D ( πa ) ( t ) defined in the vicinity of t = 0 by (18), (51) and the suggestion (39) with thesolution for finite values of t given by (18), (51), (37).The smooth join is possible because the function D ( πa ) ( t )in the vicinity of zero is defined up to order of ∼ t . Let us give some details of the procedure. First, we add to D ( πa ) ( t ) a cubic polynom, which vanish at t →
0, withthe coefficients a, b, c that are to be defined by the jointconditions: D ( π a ) ( t ) + a ( − t ) + b ( − t ) + c ( − t ) , (60)We require the form factor (60) to be joint smoothlywith the form factor for finite t at a point t = t c . Thecoefficients a, b, c and the point t c can be calculated un-ambiguously if the following conditions are satisfied.1. The derivative of the function (60) satisfies the follow-ing constraints obtained in [12] (see also [3]): D ( π a ) (cid:48) (0) D ( π a ) (0) = 2 . ∼ .
31 GeV − . (61)2. The values of the two functions coincide at the point t = t c , as well as the values of their first derivatives.3. The form factor (60) satisfies the condition D ( π a ) ( t ) < D form factor defined for finite t does satisfy thiscondition.4. For t c we choose among all possible points satisfyingthe conditions 1–3 the point of maximal absolute value | t c | . This is necessary for the contribution of singularterm ∼ /t be minimized at small values of t .We demonstrate the procedure in fig. 3, using for thecalculation the minimal value from the interval (61) and D q = − . FIG. 3. The join of two functions for the pion D form factorfor the middle value from (57) and the minimal value from(61). The full line (red) – the D form factor (60); the dashedline (blue) - the solution of (18), (37), (51); the dot-dashedline (black) – D ( πa ) ( t ); ( − t c ) = 2.53 GeV . We present in fig. 4 the dependence of the procedureon the values of the parameter D q (57) and on the valuesfrom the interval (61). Fig. 4 demonstrates the stabilityof the procedure. As can be seen in fig. 4 the result ofjoining depends weakly on the value of the quark D -term2 FIG. 4. The dependence of the joint D form factor of pionprocedure on the values of the parameter D q (57) and onthe values from the interval (61). The full line (red) – D q = − . D q = − . D q = − . − t ) ∼ – for the minimal value from (61), the lowerset – for the maximal value from (61). from (57). However, the parameters in (60) and the pointof join t c do depend on the value from (61).The results of calculation of the pion D form factorusing the equations (18), (37), (51), D ( πa ) ( t ) and theseparate contributions of the quark A , J and D formfactors are presented in fig. 5. One can see from fig. FIG. 5. The pion gravitational D form factor calculated forthe parameters (52), (54) and D q = − . A formfactors g ( q )10 . Short-dashed line (magenta)– the spin rotationeffect (the contribution of the quark J form factor g ( q )40 ). Dot-dashed line (black) – the contribution of the quark D formfactor g ( q )60 ; this curve coincides with D ( πa ) ( t ). D form factor at the point t = 0 is caused by the term containing the A formfactors of the constituent quarks g ( q )10 (20), (37). Notealso, that, as well as the pion A form factor, the pion D form factor contains large contribution of the relativisticspin rotation effect through the contribution of J formfactors of the constituent quarks g ( q )40 (21). It is seen thatthe condition of mechanical stability of pion D ( π ) ( t ) < D form factor calculated with the use of (60)for D q = − . FIG. 6. The pion D form factor calculated for the quarkparameters (52), (54), D q = − . D form factor from [12]. Using the results for the pion GFFs we calculate themass radius and the mechanical radius of pion, definedas follows: (cid:104) r (cid:105) mass = 6 dA ( π ) dt (cid:12)(cid:12)(cid:12)(cid:12) t =0 , (cid:104) r (cid:105) mech = 6 D ( π ) ( t ) dD ( π ) dt (cid:12)(cid:12)(cid:12)(cid:12) t =0 (62)To calculate the mass radius we need only the parameters(52), (54) and so obtain the strictly fixed by our previousresults value (cid:112) (cid:104) r (cid:105) mass = 0 . D q gives for the pion mechanical radius the interval ofvalues (cid:112) (cid:104) r (cid:105) mech = 0 . − .
88 fm. It is highly probablethat the model with non-point-like quarks will give largervalues for the radii.Note, that the slopes of the form factors at t = 0 in fig.6 are different. Nevertheless our result for the mechanicalMSR as defined above coincides with that of [12].Let us make some remarks concerning a possibility ofcomparing our results with experimental data.First, we use an extremely rough approximation - thepoint-like constituent quarks. As it was pointed out and3argued in detail in [54], the accounting for the quarkstructure, the full quark form factor, is a necessary partof efficient describing of the electromagnetic form fac-tors of hadrons. We use here the simpliest model aim-ing to demonstrate that relativistic invariant canonicalparametrization together with MIA in the framework ofIF RQM does give a real possibility of obtaining the pionGFFs. The obtained results are reasonable and satisfyall standard constraints.Second, today there are no trustworthy results onpion GFFs unambiguously extracted from preciseexperimental data. Although the pion GFFs and gravi-tational radii were estimated [12], the errors of the Bellemeasurements are large (even at current stage), and theobtained results can be affected by the experimentalerrors. Belle II began data taking with the muchhigher luminosity SuperKEKB in 2018, and the precisemeasurements of γ ∗ γ → π π can be expected sincethe statistic errors are much larger than the systematicerrors in the previous Belle data [55]. One may expectmore quantitative insights from experiments CLAS atJefferson Lab [56], COMPASS at CERN [57] and theenvisioned future Electron-Ion-Collider [58]. VI. CONCLUSION
In this work we extend our relativistic theory ofelectroweak properties of composite systems, developedpreviously, to describe simultaneously the gravitationalstructure of hadrons. The approach is based on a ver-sion of the instant-form relativistic quantum mechanicsand makes use of the modified impulse approximation.We use the general method of the relativistic invariantparametrizaton of local operators to write the energy-momentum tensor of particle with an arbitrary spin.From the point of view of group theory the parameteri-zation procedure represents the realization of the known Wigner – Eckart theorem on the Poincar´e group. Wegive general formulae and use for the actual calculationthose for systems of spin 0(the pion), spin 1 / D q , is a characteris-tic of gravitational form factor of constituent quark, thequark D -term. This parameter is constrained from thepion mean-square radius despite large uncertainties inthe extraction of the latter from the experimental datathrough a phenomenological approach. We calculate thevalues of the static gravitational characteristics of thepion and obtain A and D form factors as functions ofmomentum transfer up to 1 GeV . Note that the newparameter is not used in the calculation of the A formfactor, its value is a direct prediction of our previous ap-proach. The important role of the relativistic effects inthe pion gravitational characteristics is discussed in de-tail. The calculated gravitational form factors and gravi-tational mean-square radii are in a reasonable agreementwith the known results. 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