Trinucleon form factors with relativistic multirankseparable kernels
aa r X i v : . [ nu c l - t h ] F e b Trinucleon form factors with relativistic multirankseparable kernels
Serge Bondarenko a , Valery Burov a , Sergey Yurev a a BLTP, Joint Institute for Nuclear Research, Dubna, 141980, Russia
Abstract
This paper studies elastic electron-trinucleon scattering in the relativistic im-pulse approximation. The amplitudes for a trinucleon have been obtainedby solving the relativistic generalization of the Faddeev equations with amultirank separable kernel of the nucleon-nucleon interactions. The staticapproximation and additional relativistic corrections for the trinucleon elec-tromagnetic form factors have been calculated for the momentum transfersquared up to 50 fm − . Keywords:
Elastic electron-trinucleon scattering, Bethe-Salpeter equation, Faddeevequation, relativistic approach
1. Introduction
In the previous article [1] we used the solutions of the Bethe-Salpeter-Faddeev equation (BSF) with the rank-one Yamaguchi and Tabakin sepa-rable kernels of the nucleon-nucleon (NN) interactions to calculate the Hecharge form factor ( F C ). Two approximations were considered: the staticapproximation (SA) and relativistic corrections (RC). The choice of the sim-ple rank-one kernel did not allow us to reproduce the diffraction minimumof the form factor F C .Authors of [2] have found that the calculations with multirank separableNN kernel of interactions in the SA can produce the first diffraction minimumof the form factor F C although at the higher value of the momentum transfer. Email address: [email protected] (Serge Bondarenko)
Preprint submitted to Nuclear Physics A February 12, 2021 ut nevertheless, this article has focused us on the idea that the RC can movethe first diffraction minimum closer to the one experimentally observed.In the current paper we have considered several multirank kernels: purelyphenomenological covariant generalization of the nonrelativistic (NR) Graz-II [3] potential and the derived one from separable approximation of theParis [4, 5] NN potential. The parameters of the NR separable kernels wererefitted in [2] to use them in the Bethe-Salpeter approach.Today there are several relativistic approaches known to calculate trin-ucleon form factors (see, review [6]). The first one is the conventional ap-proach: it views the nucleus as made up of nucleons interacting among them-selves (via two- and many-body realistic potentials), and with external elec-tromagnetic (EM) fields (via one- and many-body currents), including rela-tivistic corrections. The second approach is the chiral effective field theory(ChEFT) which uses the methods of quantum field theory with Lagrangianincluding the nucleon, pion and photon fields to obtain one- and many-bodyEM currents. The third approach applies the covariant spectator theory(CST) to construct the three-nucleon wave function. Within the frameworkof the latest approach, the corresponding covariant equation is solved bymeans of the one-boson-exchange kernel of the interactions including severalmesons.One should also mention the light-front relativistic Hamiltonian dynamics(RHD) which takes the three-body forces [7] into account.The main difference of the BSF formalism from the approaches mentionedabove is that all the nucleons of the trinucleon BSF amplitude are off-mass-shell and there are two more free variables p , q used in the calculations.These variables introduce additional difficulties in obtaining the solution ofthe BSF equations and calculating the form factors, namely, the peculiaritiesin the p , q complex plane.One of the important physical problems in the study of the trinucleonform factors is to describe the right location of the first diffraction minimum.For the He and H charge and magnetic form factors the result for theconventional and ChEFT approaches successfully reproduces the measuredform factors up to momentum transfers squared t ≃ − . The ChEFTresults seem to underestimate the experimental data beyond t & − , inparticular, they predict the form factor zeros at significantly lower valuesof t . On the other hand, the CST calculations are limited to the approx-imation which omits the interaction current contribution. Therefore, onlythe isoscalar magnetic contribution should be compared to the experimental2ata. For this isoscalar magnetic observable the agreement is good.The article follows the ideas given in [1] where the rank-one kernels wereapplied. In contrast to it, in this work the calculations with the multirankrelativistic separable kernels of NN interactions Graz-II and Paris have beenperformed. The results of the calculations better agree with the experimentaldata than the results obtained with the rank-one kernels – Yamagychi andTabakin.The paper is organized as follows: Sec. 2 gives the expressions for thetrinucleon EM form factors, Sec. 3 defines the static approximation andrelativistic corrections, Sec. 4 discusses the calculations and results, and,finally, Sec. 5 gives the conclusion.
2. Trinucleon form factors
As a system with one-half spin, the EM current of He or H can be pa-rameterized by two elastic form factors: charge (electric) F C and magnetic F M (see, for example, [8]). In the calculations below, we apply a straight-forward relativistic generalization of the NR expressions for the He and Hcharge and magnetic form factors [8, 9, 10, 11]:2 F C ( He) = (2 F p C + F n C ) F −
23 ( F p C − F n C ) F , (1) F C ( H) = (2 F n C + F p C ) F + 23 ( F p C − F n C ) F , (2) µ ( He) F M ( He) = µ n F n M F + 23 ( µ n F n M + µ p F p M ) F , (3) µ ( H) F M ( H) = µ p F p M F + 23 ( µ n F n M + µ p F p M ) F , (4)where F p,n C , F p,n M are the charge and magnetic form factors of the proton andneutron, µ ( He), µ ( H), µ p , µ n are magnetic moments of the He, H, protonand neutron, respectively.The functions F , can be expressed in terms of the wave functions ofthe trinucleon which can be represented as linear combinations of the BSFequation solutions with different spin-isospin states [1, 8].In our previous work we performed the calculations for the rank-one in-teraction kernels [1, 12]. There we also analyzed the influence of the type ofnucleon form factors on the He form factor [12]. In the current paper we3eal with the multirank separable kernels Graz-II (with p d =4,5,6%), Paris-1and Paris-2.To calculate the functions F , , it is convenient to use the Breit referencesystem. The solutions of the BSF equation, however, have been found inthe c.m. (rest) frames of the corresponding trinucleon. To relate the Breitand initial (final) particle c.m. frames, the Lorentz transformations shouldbe applied to the four-momenta.Thus, the arguments of the initial and final particle wave functions andpropagators were expressed in terms of the momenta calculated in the corre-sponding c.m. frames and related to each other using the Lorentz transfor-mations.
3. Static approximation and relativistic corrections
Below we remind the general approximations and corrections (see detailedformulae in [1]). Since the solutions of the BSF equations have been obtainedin the Euclidean space [13] and are known only for real values of q and p ,the simplest way to calculate F , is to apply the Wick rotation procedure p → ip , q → iq , if it is possible. Thus, one needs to investigate theanalytic structure of the integrand on the complex-valued variables p , q .The location of the variable p singularities allows one to apply the Wickrotation procedure, although this is not justified for the variable q in general.It is convenient to start with the so-called static approximation. It as-sumes that all the terms in the Lorentz transformations proportional to η arecanceled and the relativistic covariance for the EM current matrix elementis violated. Analyzing integrands of F , in this case, one can see that thepoles on q do not cross the imaginary q axis and always stay in the secondand fourth quadrants. Therefore, the Wick rotation procedure q → iq canbe applied.To recover the relativistic covariance, we consider the relativistic correc-tions to SA. They consist of three parts. First, we calculate the Lorentz boostin the one-particle propagator G ′ ( k ′ ) arguments, which gives a boost contri-bution (BC). Second, we take into account a simple pole on q , which gives anadditional term in integrals – a pole contribution (PC). Third, we computethe Lorentz boost of the arguments of the final trinucleon wave function bycarrying out the first term of the Taylor series expansion contribution (EC)on the parameter η . 4 . Results and discussion As in the case of the rank-one kernel, we have used numerical solutionsfor the trinucleon amplitudes obtained by solving the system of homogeneousintegral BSF equations by means of the Gaussian quadratures. The solutionsmentioned above were interpolated to the ( q , q ) points of integration toperform multiple integration in equations F , using the Vegas algorithm ofthe Monte-Carlo integrator. The stability of the result was tested by changingthe ( q , q )-meshes while solving the BSF equations. The best mesh was takenas N = 35 , N = 25 where N ( N ) stands for the number of q ( q ) points.The multiple integrals were calculated with the relative accuracy equal to0.01. -6 -5 -4 -3 -2 -1
0 5 10 15 20 25 30 35 40 45 50 He, Graz-II, SA F C (t) t, fm -2 int, F , pd=4%pd=4%int, F , pd=5%pd=5%int, F , pd=6%pd=6% Figure 1: Static approximation for the charge form factor of He as a function of t for themultirank separable kernel Graz-II at different values of the probability of the two-nucleon D -state p d = 4,5,6%. Complete calculations (the dotted line – for p d =4%, the solid line– for p d =5% and the long-dashed line – for p d =6%) are compared to the [2] calculationswithin certain approximations (the dashed line – for p d =4%, the dashed-dotted line – for p d =5% and the dashed-dotted-dotted line – for p d =6%). It should be emphasized that the considered separable covariant Graz-II kernels were obtained from the purely phenomenological nonrelativistic5 -5 -4 -3 -2 -1
0 5 10 15 20 25 30 35 40 45 50 He, Paris, SA F C (t) t, fm -2 int, F , Paris-1Paris-1int, F , Paris-2Paris-2 Figure 2: Static approximation for the charge form factor of He as a function of t forthe multirank separable kernel Paris. Complete calculations (the dotted line – for Paris-1 and the solid line – for Paris-2) are compared to the [2] calculations within certainapproximation (the dashed line – for Paris-1, the dashed-dotted line – for Paris-2). Graz-II potential, while the covariant Paris-1(2) kernels were received fromthe NR realistic one-boson-exchange Paris model.In this section we use the dipole fit for nucleon form factors if not statedotherwise.Now we discuss the comparison of the SA results with the ones obtainedin paper [2] within the following approximation: the functions ψ , whichdepend on angular integration x = cos θ qp were averaged numerically whilethe integration in the propagators was performed analytically. The angularaveraging leads to ψ = ψ ∼ ψ which consequently gives the followingcondition: v ∼
0. To imitate the [2] calculations, we have used the analyticalexpressions for the propagators and taken into account the F function onlywith the integral calculated numerically without averaging on the x variable.The obtained results are very similar to those shown in [2] but they differfrom the complete calculations. The reason is that in complete integrationthe function v is not equal to zero and gives a noticeable contribution intofunction F which dominates in the region of the form factor zero.6 -7 -6 -5 -4 -3 -2 -1
0 10 20 30 40 50 He, Graz-II, pd=4% F C (t) t, fm -2 BCPCECRC
Figure 3: Parts of contribution to relativistic corrections for the He charge form factoras a function of t for the multirank separable kernel Graz-II at the probability of thetwo-nucleon D -state p d = 4%. The solid line displays the sum of all considered relativisticcorrections, the dashed line – Lorentz transformations of arguments of a propagator, thedotted line displays the contribution of the simple pole of the one-particle propagator, andthe dashed-dotted line – Lorentz transformations of the arguments of the final particlewave function. Figure 1 shows the result of the calculations of the He charge form factorfor the cases of complete integrations and the approximation from [2] for thekernel Graz-II. As mentioned above, there is a big difference between thesetwo assumptions. Figure 2 illustrates the same as in Figure 1 but for theParis-1 and Paris-2 kernels.Now we discuss the contribution of the RC to the He and H chargeand magnetic form factors. As it is shown in [1] neither the SA nor RCcalculated with the rank-one NN kernel give the diffraction minimum inthe form factors. The static approximation within the multirank kernelsproduces the minimum/zero in the form factors which is shifted to the highermomentum transfer squared region in comparison to the experimental data.We expect the RC to shift the minimum/zero closer to the position of theexperimental data.Figure 3 shows the partial contributions of the RC to the He charge form7 -6 -5 -4 -3 -2 -1
0 10 20 30 40 50 He, Paris-1 F C (t) t, fm -2 BCPCECRC
Figure 4: The same as in Fig. 3 for the multirank separable kernel Paris-1. factor, namely, the BC, PC and EC for the Graz-II multirank kernel with p d = 4%. Figure 4 illustrates the same for the Paris-1 kernel. At the lowvalues of t =1-2 fm − the PC and EC are small due to the small correspondingvalue of variable η and the RC are defined by the BC. Starting approximatelyfrom t = 5 fm − the values of the BC and EC are equal to each other andtheir sum makes the major contribution to RC. As it is seen from the Figuresabove the form factor zero at value t = 33 fm − for the Graz-II kernel (at t = 14 fm − for Paris-1) is defined by zero of the EC at t = 30 fm − forGraz-II (at t = 11 fm − for Paris-1) and shifted by the positive contributionof the BC. In the region after the form factor zero the BC sharply decreasesand the RC are totally determined by the EC. The PC is rather small incomparison to the BC and EC and does not play a significant role except forthe case when the sum of the BC and EC is close to the value of PC at small t and near the form factor zero.Figure 5 shows the SA and RC for the He charge form factor as a functionof t for the multirank separable kernel Graz-II at different values of theprobability of the two-nucleon D -state p d = 4 , , -6 -5 -4 -3 -2 -1
0 10 20 30 40 50 He, Graz-II F C (t) t, fm -2 pd=4%, SApd=4%, RCpd=5%, SApd=5%, RCpd=6%, SApd=6%, RC Figure 5: Static approximation and relativistic corrections for the He charge form factoras a function of t for the multirank separable kernel Graz-II at different values of theprobability of the two-nucleon D -state p d = 4 , , p d =4%,the dotted line – SA for p d =5%, the dashed-dotted-dotted line – SA for p d =6%, the solidline – RC for p d =4%, the dashed-dotted line – RC for p d =5% and the long-dashed line –RC for p d =6%. The experimental data are taken from [14, 15, 16, 17, 18]. p d = 5 ,
6% where the minimum without changing the sign is found. Therelativistic corrections for all the values of p d shift zeros to the region of thesmaller t closer to the experimental data. The difference between the RCand experimental zeros is about 18-23 fm − .Figure 6 presents the SA and RC for the H charge form factor. In thiscase the SA gives the first form factor zero at about 39 fm − for p d =4%,34 fm − for p d =5% and 29 fm − for p d =6%. For all p d values the RC shiftthe results closer to the experimental data of the diffraction minima in com-parison to the SA. The difference between the RC and experimental zeros isless than in the He charge form factor and is about 10-14 fm − .Figure 7 shows the SA and RC for the He magnetic form factor. Theresults are similar to the He charge form factor. The difference between theRC and experimental zeros is less than in the He charge form factor and isabout 8-18 fm − . 9 -6 -5 -4 -3 -2 -1
0 10 20 30 40 50 H, Graz-II F C (t) t, fm -2 pd=4%, SApd=4%, RCpd=5%, SApd=5%, RCpd=6%, SApd=6%, RC Figure 6: The same as in Fig. 5 but for the H charge form factor. The experimental dataare taken from [14, 22, 23, 24, 25].
Figure 8 displays the SA and RC for the H magnetic form factor. Theresults are similar to the H charge form factor (Fig. 5), however, in thiscase the difference between the RC and experimental zeros is less than in the H charge form factor and is about 4-8 fm − for p d =4,5%. The relativisticcorrections for p d = 6% agree with the experimental data zero and theirbehaviour is rather well described.The results for the He and H charge and magnetic form factors withthe Paris-1 and Paris-2 kernels are presented in Figures 9–12. The calcula-tion results indicate that in this case as well as in the case of the Graz-IIkernel the RC shift the curves towards the experimental data. The differencebetween the data and calculation results is generally smaller than for theGraz-II kernel. Figure 9 demonstrates that for the He charge form factorthe RC are rather close to the data and the difference is really small – about3 fm − for Paris-1 and about 1.5 fm − for Paris-2 kernel. The zero positionand behaviour for the He and H charge form factors till t = 50 fm − for Heand about 24 fm − for H are in rather good agreement with the experimen-tal data (Figures 9-10). Opposite to the charge form factors the magneticform factors (Figures 11-12) have zeros shifted to the smaller values of t in10 -6 -5 -4 -3 -2 -1
0 10 20 30 40 50 He, Graz-II F M (t) t, fm -2 pd=4%, SApd=4%, RCpd=5%, SApd=5%, RCpd=6%, SApd=6%, RC Figure 7: The same as in Fig. 5 but for the He magnetic form factor. The experimentaldata are taken from [14, 15, 16, 18, 19, 20, 21]. comparison to the data. The difference between the RC and the experimen-tal data is about 3 fm − for Paris-1 (4 fm − for Paris-2) kernel for He andabout 12 fm − for Paris-1 (about 13 fm − for Paris-2) kernel for H.It should be stressed that for all the cases of the above-mentioned multi-rank kernels the RC also increase the value of the form factors by an orderof magnitude 0.5-1.5 at the value of t = 50 fm − in comparison to the SAcalculations.Finally, we consider the contribution of different models of the nucleonEM form factors: the dipole model (DIPOLE), relativistic harmonic oscilla-tor model (RHOM) [26] and vector meson dominance model (VMDM) [27].The VMDM has used the latest experimental data to refit the parameters.The results for the above three models are given for the He and H chargeform factors in Fig. 13 for Graz-II and in Fig. 14 for Paris kernels. The Fig-ures demonstrate that the models change the magnitudes of the form factorsby a factor of 1.5-2 at t = 50 fm − and also shift slightly the form factor zero.The main effect of the above changes happens due to the non-zero electricneutron form factor G nE in the RHOM and VMDM.We have to emphasize that the considered RC are nothing but the restora-11 -6 -5 -4 -3 -2 -1
0 10 20 30 40 50 H, Graz-II F M (t) t, fm -2 pd=4%, SApd=4%, RCpd=5%, SApd=5%, RCpd=6%, SApd=6%, RC Figure 8: The same as in Fig. 5 but for the H magnetic form factor. The experimentaldata are taken from [14, 22, 23, 24, 25]. tion of the covariance of the trinuleon EM current matrix element althoughat the first order of the Taylor series expansion of the final trinucleon wavefunction arguments. The effect of RC is large and important as it is seenfrom the results. These RC change the static approximation results in theproper direction. We have calculated the relativistic impulse approximation(RIA) which takes into account only three contributions from the individualphoton-nucleon diagrams and also considers the nucleon EM form factors on-mass-shell (as CIA discussed above). The RIA obviously violates the gaugeinvariance of the reaction. To recover this invariance, one should introducethe corresponding EM interaction currents which together with RIA providethe gauge invariance. Hopefully, this contribution will improve the agreementbetween the RC and experimental data.
5. Summary
In this paper the BSF equations solved for trinucleon have been used tocalculate the trinucleon electromagnetic form factors. The expressions forthe form factors have been obtained by a straightforward relativistic general-12 -5 -4 -3 -2 -1
0 10 20 30 40 50 He, Paris F C (t) t, fm -2 SA, Paris-1RC, Paris-1SA, Paris-2RC, Paris-2
Figure 9: Static approximation and relativistic corrections for the He charge form factoras a function of t for the multirank separable kernel Paris. The Dashed line – SA for Paris-1, the dotted line – SA for Paris-2, the solid line – RC for Paris-1, and the dashed-dottedline – RC for Paris-2. The experimental data coincide with Fig. 5. ization of the NR expressions. The multiple integration has been performedby means of the Monte Carlo algorithm. The Lorentz transformations ofthe propagator arguments and the final trinucleon wave function have beentaken into account while calculating.The obtained solutions with multirank kernels Graz-II and Paris-1(2) un-like the rank-one kernels have given the diffraction minimum. It is importantto stress that the strong dependence on the type of the NN interaction kernelhas been found in the calculations.Two approximations have been considered: the static approximation andrelativistic corrections. The relativistic corrections provide the diffractionminimum in the form factors and move it in the proper direction even if itdoes not appear in the static approximation.For the Graz-II with p d = 6% and Paris-1 kernels a good agreement hasfound for the position of the diffraction minimum in F M of H (Graz-II) and F C of He and H.Finally, we can also conclude that taking into account the EM interaction13 -5 -4 -3 -2 -1
0 10 20 30 40 50 H, Paris F C (t) t, fm -2 SA, Paris-1RC, Paris-1SA, Paris-2RC, Paris-2
Figure 10: The same as in Fig. 9 but for the H charge form factor. The experimentaldata coincide with Fig. 6. currents will be justified to provide the gauge invariance and further progressin the study of elastic electron-trinucleon scattering.
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0 10 20 30 40 50 He, Paris-1, RC F C (t) t, fm -2 DIPOLERHOMVMDM
Figure 15: The same as in Fig. 13 but for the Paris-1 kernel.
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0 10 20 30 40 50 H, Paris-1, RC F C (t) t, fm -2 DIPOLERHOMVMDM
Figure 16: The same as in Fig. 15 but for the Paris-1 kernel.
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