Relativistic interactions for meson-nucleon systems: applications in the theory of nuclear reactions

It is shown that the method of unitary clothing transformations (UCT) developed in [1, 2] and applied to nuclear physics problems [3, 4], gives a fresh look at constructing interactions between the "clothed'' nucleons, these quasiparticles with the properties of physical nucleons.

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Datum:	2012
Hauptverfasser:	Shebeko, A.V., Frolov, P.A., Dubovik, E.A.
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Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2012
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Section C. Theory of Elementary Particles. Cosmology
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spelling	irk-123456789-1070932016-10-14T03:02:18Z Relativistic interactions for meson-nucleon systems: applications in the theory of nuclear reactions Shebeko, A.V. Frolov, P.A. Dubovik, E.A. Section C. Theory of Elementary Particles. Cosmology It is shown that the method of unitary clothing transformations (UCT) developed in [1, 2] and applied to nuclear physics problems [3, 4], gives a fresh look at constructing interactions between the "clothed'' nucleons, these quasiparticles with the properties of physical nucleons. Показано, что метод унитарных одевающих преобразований (УОП), развитый в работах [1,2] и применённый к задачам ядерной физики [3,4], позволяет по-новому взглянуть на построение взаимодействий между "одетыми'' нуклонами, этими квазичастицами со свойствами физических нуклонов. Показано, що метод унітарних одягаючих перетворень (УОП), розроблений в працях [1,2] та застосований до задач ядерної фізики [3,4], дозволяє по-новому поглянути на побудову взаємодії між "одягненими'' нуклонами, цими квазічастинками з властивостями фізичних нуклонів. 2012 Article Relativistic interactions for meson-nucleon systems: applications in the theory of nuclear reactions / A.V. Shebeko, P.A. Frolov, E.A. Dubovik // Вопросы атомной науки и техники. — 2012. — № 1. — С. 188-192. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 21.45.Bc, 11.30.Cp, 13.75.Cs http://dspace.nbuv.gov.ua/handle/123456789/107093 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Section C. Theory of Elementary Particles. Cosmology Section C. Theory of Elementary Particles. Cosmology
spellingShingle	Section C. Theory of Elementary Particles. Cosmology Section C. Theory of Elementary Particles. Cosmology Shebeko, A.V. Frolov, P.A. Dubovik, E.A. Relativistic interactions for meson-nucleon systems: applications in the theory of nuclear reactions Вопросы атомной науки и техники
description	It is shown that the method of unitary clothing transformations (UCT) developed in [1, 2] and applied to nuclear physics problems [3, 4], gives a fresh look at constructing interactions between the "clothed'' nucleons, these quasiparticles with the properties of physical nucleons.
format	Article
author	Shebeko, A.V. Frolov, P.A. Dubovik, E.A.
author_facet	Shebeko, A.V. Frolov, P.A. Dubovik, E.A.
author_sort	Shebeko, A.V.
title	Relativistic interactions for meson-nucleon systems: applications in the theory of nuclear reactions
title_short	Relativistic interactions for meson-nucleon systems: applications in the theory of nuclear reactions
title_full	Relativistic interactions for meson-nucleon systems: applications in the theory of nuclear reactions
title_fullStr	Relativistic interactions for meson-nucleon systems: applications in the theory of nuclear reactions
title_full_unstemmed	Relativistic interactions for meson-nucleon systems: applications in the theory of nuclear reactions
title_sort	relativistic interactions for meson-nucleon systems: applications in the theory of nuclear reactions
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2012
topic_facet	Section C. Theory of Elementary Particles. Cosmology
url	http://dspace.nbuv.gov.ua/handle/123456789/107093
citation_txt	Relativistic interactions for meson-nucleon systems: applications in the theory of nuclear reactions / A.V. Shebeko, P.A. Frolov, E.A. Dubovik // Вопросы атомной науки и техники. — 2012. — № 1. — С. 188-192. — Бібліогр.: 5 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT shebekoav relativisticinteractionsformesonnucleonsystemsapplicationsinthetheoryofnuclearreactions AT frolovpa relativisticinteractionsformesonnucleonsystemsapplicationsinthetheoryofnuclearreactions AT dubovikea relativisticinteractionsformesonnucleonsystemsapplicationsinthetheoryofnuclearreactions
first_indexed	2025-07-07T19:29:09Z
last_indexed	2025-07-07T19:29:09Z
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fulltext	RELATIVISTIC INTERACTIONS FOR MESON-NUCLEON SYSTEMS: APPLICATIONS IN THE THEORY OF NUCLEAR REACTIONS A.V. Shebeko 1∗, P.A. Frolov 2, E.A. Dubovik 2 1National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine 2Institute of Electrophysics and Radiation Technologies, 61002, Kharkov, Ukraine (Received October 31, 2011) It is shown that the method of unitary clothing transformations (UCT) developed in [1, 2] and applied to nuclear physics problems [3, 4], gives a fresh look at constructing interactions between the “clothed” nucleons, these quasi- particles with the properties of physical nucleons. PACS: 21.45.Bc, 11.30.Cp, 13.75.Cs 1. SOME RECOLLECTIONS Our departure point is the Hamiltonian of a system of interacting mesons and nucleons that can be written as H = ∞∑ C=0 ∞∑ A=0 HCA, (1) HCA = ∫∑ HCA(1′, 2′, ..., n′ C ; 1,2,...,nA) × a†(1′)a†(2′)...a†(n′ C)a(nA)...a(2)a(1), (2) where the capital C(A) denotes particle-creation (an- nihilation) number for operator substructure HCA and HCA(1′, ..., C; 1,..., A) = δ(�p′1 + ...+ �p′C − �p1 − ...− �pA) × hCA(p′1μ ′ 1ξ ′ 1...p ′ Cμ ′ Cξ ′ C ; p1μ1ξ1...pAμAξA), (3) where c-number coefficients hCA do not contain delta function and a†[a](n) = a†[a](�pn, μn, ξn) is a cre- ation [annihilation] operator for particle of species ξn with momentum �pn and polarization μn. In turn, HCA = ∫ HCA(�x)d�x ⇒ H = ∫ H(�x)d�x (4) with density H(�x) = ∞∑ C=0 ∞∑ A=0 HCA(�x). (5) For example, in case with C = A = 2 we have H22(1′, 2′; 1, 2) = δ(�p′1 + �p′2 − �p1 − �p2)h(1′2′; 12) (6) and H22(�x) = 1 (2π)3 ∫∑ exp[−i(�p′1 + �p′2 − �p1 − �p2)�x] × h(1′2′; 12)a† (1′) a† (2′) a (2)a (1) . (7) The operator H , being divided into the no- interaction part HF and the interaction HI , owing to its translational invariance allowsHI to be written as HI = ∫ HI(�x)d�x. (8) Our consideration is focused upon various field mod- els (local and nonlocal) in which the interaction den- sity HI(�x) consists of scalar Hsc(�x) and nonscalar Hnsc(�x) contributions: HI(�x) = Hsc(�x) +Hnsc(�x), (9) where the property to be a scalar means UF (Λ, b)Hsc(x)U−1 F (Λ, b) = Hsc(Λx+ b), ∀ x = (t, �x) (10) for all Lorentz transformations Λ and spacetime shifts b. As an illustration, in case of the vector mesons (ρ and ω) we have Vv = V (1) v + V (2) v , V (1) v = ∫ d�xHsc(�x), V (2) v = ∫ d�xHnsc(�x), Hsc(�x) = gvψ̄(�x)γμψ(�x)ϕμ v (�x) + fv 4m ψ̄(�x)σμνψ(�x)ϕμν v (�x), ∗Corresponding author E-mail address: shebeko@kipt.kharkov.ua 188 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1. Series: Nuclear Physics Investigations (57), p. 188-192. Hnonsc(�x) = g2 v 2m2 v ψ̄(�x)γ0ψ(�x)ψ̄(�x)γ0ψ(�x) + f2 v 4m2 ψ̄(�x)σ0iψ(�x)ψ̄(�x)σ0iψ(�x), where ϕμν v (�x) = ∂μϕν v(�x) − ∂νϕμ v (�x) is a tensor of a vector field in Schrödinger (S) picture. Such a situ- ation is typical of theories with derivative couplings or spins j ≥ 1. 2. BOOST GENERATORS. RELATIVISTIC INVARIANCE (RI) AS A WHOLE To free ourselves from any dependence on pre- existing field theories, the three boost operators �N = (N1, N2, N3) can be written as: �N = ∞∑ C=0 ∞∑ A=0 �NCA, (11) �NCA = ∫∑ �NCA(1′, 2′, ..., n′ C ; 1, 2, ..., nA) × a†(1′)a†(2′)...a†(n′ C)a(nA)...a(2)a(1). (12) We have developed an algebraic procedure to find links between coefficients HCA and �NCA, compatible with commutations [Pi, Pj ] = 0, [Ji, Jj ] = iεijkJk, [Ji, Pj ] = iεijkPk, [�P ,H ] = 0, [ �J,H ] = 0, [Ji, Nj] = iεijkNk, [Pi, Nj] = iδijH, [H, �N ] = i �P , [Ni,Nj ] = −iεijkJk, (i, j, k = 1, 2, 3), where �P = (P 1, P 2, P 3) and �J = (J1, J2, J3) are lin- ear and angular momentum operators. For instant form of the relativistic dynamics after Dirac only Hamiltonian and boost operators carry in- teractions, viz., H = HF +HI , �N = �NF + �NI , while �P = �PF and �J = �JF . How one can build up operators HI and �NI is shown in [4]. Here we would like to present only a free part of the fermion boost operator �Nferm = �Norb ferm + �Nspin ferm, where �Norb ferm = i 2 ∫∑ d�pE�p ( ∂b†(�pμ) ∂�p b(�pμ) −b†(�pμ) ∂b(�pμ) ∂�p ∂d†(�pμ) ∂�p d(�pμ) − d†(�pμ) ∂d(�pμ) ∂�p ) , �Nspin ferm = −1 2 ∫∑ d�p �p× χ†(μ)�σχ(μ) E�p +m × ( b†(�pμ)b(�pμ) + d†(�pμ)d(�pμ) ) . Here E�p = √ �p2 +m2 is the nucleon energy and χ(μ) is the Pauli spinor. 3. METHOD OF UCTs IN ACTION Method in question is aimed at expressing a field Hamiltonian through the so-called clothed-particle creation (annihilation) operators αc, e.g., a†c(ac), b†c(bc) and d†c(dc) via UCTs W (αc) = W (α) = expR, R = −R† in similarity transformation α = W (αc)αcW †(αc) (13) that connect primary set α in bare-particle represen- tation (BPR) with the new operators in CPR. A key point of the clothing procedure is to remove the so-called bad terms from Hamiltonian H ≡ H(α) = HF (α) +HI(α) = W (αc)H(αc)W †(αc) ≡ K(αc). (14) By definition, such terms prevent physical vacuum \|Ω〉 (H lowest eigenstate) and one-clothed-particle states \|n〉c = a†c(n)\|Ω〉 to be H eigenvectors for all n included. Bad terms occur every time when any normally ordered product a†(1′)a†(2′)...a†(n′ C)a(nA)...a(2)a(1) of class [C.A] embodies, at least, one substructure ∈ [k.0] (k = 1, 2...) or/and [k.1] (k = 2, 3, ...). In this context all primary Yukawa-type (trilinear) couplings shown above should be eliminated. Respectively, let us write for boson–fermion sys- tem HI(α) = V (α) + Vren(α) (15) with primary (trial) interaction V (α) = Vbad + Vgood “good” (e.g., ∈ [k.2]) as antithesis of “bad” while Vren(α) ∼ [1.1] + [0.2] + [2.0] “mass renormalization counterterms”. It turns out that latter are important to ensure RI as a whole, i.e., in Dirac sense. In order to compare our calculations with those by Bonn group [5] we have employed V (α) = Vs + Vps + Vv. Then clothing itself is prompted by H(α) = K(αc) ≡W (αc)[HF (αc) + Vv(αc) + Vren(αc)]W †(αc) (16) or K(αc) = HF (αc) + V (1) v (αc) + [R,HF ] + V (2) v (αc) + [R, V (1) v ] + 1 2 [R, [R,HF ]] + [R, V (2) v ] + 1 2 [R, [R, V (1) v ] + ... (17) and requiring [R,HF ] = −V (1) v for the operator R of interest to get H = K(αc) = KF +KI (18) 189 with a new free part KF = HF (αc) ∼ a†cac and inter- action KI = 1 2 [R, V (1) v ] + V (2) v + 1 3 [R, [R, V (1) v ]] + ... (19) between clothed particles. Moreover, after modest effort we have 1 2 [ R, V (1) v ] (NN → NN) = Kv(NN → NN) +Kcont(NN → NN), (20) where the operator Kcont(NN → NN) may be as- sociated with a contact interaction since it does not contain any propagators (details see in Refs. [3]). It has turned out that this operator cancels completely non-scalar operator V (2). In parallel, we have �N(α) = �B(αc) = W (αc){ �NF (α) + �NI(α) + �Nren(α)}W †(αc) (21) with �NI = − ∫ �xVv(�x)d�x = − ∫ �x{V (1) v (�x) + V (2) v (�x)}d�x = �N (1) I + �N (2) I . (22) As before (see Refs. [2,3]) we find that the boost generator in CPR acquires the structure similar to K(αc): �B(αc) = �BF + �BI . (23) Here �BF = �NF (αc) boost operator for noninteract- ing clothed particles (in our case fermions and vector mesons) and �BI incorporates contributions induced by interactions between them �BI = + 1 2 [R, �N (1) I ] + 1 3 [R, [R, �N (1) I ]] + ... 4. RELATIVISTIC INTERACTIONS Operator KI contains only interactions responsible for physical processes, these quasipotentials between the clothed particles, e.g., KI ∼ a†cb † cacbc(πN → πN)+ b†cb † cbcbc(NN → NN) + d†cd † cdcdc(N̄N̄ → N̄N̄) + ... + [a†ca † cbcdc +H.c.](NN̄ ↔ 2π) + ... + [a†cb † cb † cbcbc +H.c.](NN ↔ πNN) + ... (24) After normal ordering of fermion operators we de- rive NN → NN interaction operator (mediated, for instance, by pions) KNN = ∫ d�p1d�p2d�p ′ 1d�p ′ 2VNN (�p ′ 1, �p ′ 2; �p1, �p2) × b†c(�p ′ 1)b † c(�p ′ 2)bc(�p1)bc(�p2), (25) VNN (�p ′ 1, �p ′ 2; �p1, �p2) = −1 2 g2 (2π)3 m2√ E�p1E�p2E�p ′ 1 E�p ′ 2 × δ(�p ′ 1 + �p ′ 2 − �p1 − �p2) × ū(�p ′ 1)γ5u(�p1) 1 (p1 − p′1)2 − μ2 ū(�p ′ 2)γ5u(�p2). (26) The corresponding relativistic and properly sym- metrized NN quasipotential is ṼNN (�p ′ 1, �p ′ 2; �p1, �p2) = 〈 b†c(�p ′ 1)b † c(�p ′ 2)Ω \| KNN \| b†c(�p1)b†c(�p2)Ω 〉 , (27) or through covariant (Feynman-like) “propagators”: ṼNN (�p ′ 1, �p ′ 2; �p1, �p2) = −1 2 g2 (2π)3 m2 2 √ E�p1E�p2E�p ′ 1 E�p ′ 2 × δ(�p ′ 1 + �p ′ 2 − �p1 − �p2) × ū(�p ′ 1)γ5u(�p1) 1 2 { 1 (p1 − p′1)2 − μ2 + 1 (p2 − p′2)2 − μ2 } ū(�p ′ 2)γ5u(�p2) − (1 ↔ 2). (28) Distinctive feature of potential (28) is the pres- ence of covariant (Feynman-like) “propagator”: 1 2 { 1 (p1 − p′1)2 − μ2 + 1 (p2 − p′2)2 − μ2 } . On the energy shell for NN scattering, that is Ei ≡ E�p1 + E�p2 = E�p ′ 1 + E�p ′ 2 ≡ Ef , this expression is converted into the genuine Feynman propagator. These quasipotentials form the kernel of the in- tegral equation for the nucleon-nucleon scattering R- matrix: 〈1′2′\| R̄(E) \|12〉 = 〈1′2′\| K̄NN \|12〉 + ∫ 34 ∑ 〈1′2′\| K̄NN \|34〉〈34\| R̄(E) \|12〉 E − E3 − E4 (29) with R̄(E) = R(E)/2 (K̄NN = KNN/2), symbol ∫ 34 ∑ implies the p.v. integration. 5. DEUTERON PROPERTIES The deuteron state \|Ψd(�P )〉 ∈ H2N in the CPR sat- isfies the eigenvalue equation [KF (αc) +KI(αc)]\|Ψd(�P )〉 = Ed\|Ψd(�P )〉 (30) with Ed = √ m2 d + �P 2, where �P is the total deuteron momentum, md = mp +mn−εd is the deuteron mass and εd represents the binding energy of the deuteron. Using the approximation with KI(αc) = K(NN → NN) = KNN we arrive to a simpler eigen- value problem [ KN F +KNN ] \|�P ;M〉 = Ed\|�P ;M〉 (31) 190 in the subspace H2N spanned onto the basis b†cb†c\|Ω〉 with KNN ∼ b†cb † cbcbc. Here M denotes the deuteron spin projection on the quantization axis. The solution of this equation can be represented as \|�P ;M〉 = ∫ d�p1d�p2DM (�P ; �p1μ1, �p2μ2) × b†c(�p1μ1)b†c(�p2μ2)\|Ω〉 (32) with the coefficients DM (�P ; �p1μ1, �p2μ2) = δ(�P − �p1 − �p2)ψM (�p1μ1, �p2μ2) that have the property ψM (1, 2) = −ψM (2, 1). In the deuteron rest frame the equation (31) takes the form \|ψM 〉 = [ md −KN F ]−1 KNN \|ψM 〉, (33) where \|ψM 〉 ≡ \|�P = 0;M〉 = ∫ d�pψM (�pμ1,−�pμ2)b†c(�pμ1)b†c(−�pμ2)\|Ω〉. (34) Using the basis vectors \|p(lS)JMJ , TMT 〉 introduced in our previous paper [3] (see Appendix B) the vector \|ψM 〉 can be written as \|ψM,TMT 〉 = 1√ 2 ∑∫ p2dp \|p(lS)1M,TMT 〉ψlST (p), (35) since the deuteron has the invariant spin equal J = 1. In Eq. (35) the permissible values of the quantum numbers l, S and T are restricted to the property Pferm\|ψM,TMT 〉 = \|ψM,TMT 〉 (36) with respect to the space inversion (see Appendix B in [1], where one can find formula (114) for the parity operator Pferm of the nucleon field in the CPR). In fact, there are only the two combinations of T , S and l, namely, T = 0, S = 1 and l = 0, 2. Respectively, \|ψM,00〉 ≡ \|ψM 〉 = 1√ 2 ∑ l=0,2 ∫ p2dp \|p(l1)1M〉ψl(p). (37) At this point, we accept the normalization condi- tion 〈ψM ′ \|ψM 〉 = δM ′M (38) that implies ∞∫ 0 p2dp [ ψ2 0(p) + ψ2 2(p) ] = 1. (39) Substituting the decomposition (37) into the equation (33) we get the set of homogeneous integral equations for “radial” components ψl(p) (l = 0, 2): ψl(p) = 1 md − 2E�p × ∑ l′ ∫ k2dkV J=S=1,T=0 l l′ (p, k)ψl′(k). (40) In a moving frame the corresponding eigenvector that belongs to the value Ed = √ �P 2 +m2 d can be de- termined either by solving directly the equation (31) or using the relation \|�P ;M〉 = exp[i�β �B(αc)]\|ψM 〉. (41) The boost operator �B(αc) = �BF (αc) + �BI(αc), de- termined in the CPR by �B(αc) = W (αc) �N (αc)W †(αc), (42) consists of the free �BF and interaction �BI parts. Here �N is the total boost operator for interacting fields. Perhaps, one should note that the required P̂μ\|�P ;M〉 = Pμ\|�P ;M〉 (43) follows from the property of the energy-momentum operator P̂μ = (H, P̂ 1, P̂ 2, P̂ 3) to be the four-vector. The parameters (β1, β2, β3) = �β for the Lorentz transformation md(1, 0, 0, 0) ⇒ (P 0, P 1, P 2, P 3) = P are related to the “velocity” �v = �P/P 0 of the moving frame as �β = β�n, �n = �v/v, tanhβ = v. (44) As in our previous paper [3] we continue compar- ison of UCT approach with results of the Bonn group [5]. In particular, the low-energy pa- rameters of NN scattering and deuteron prop- erties are presented in Table 1 and the fig- ure.The best-fit parameters are collected in Table 2. Table 1. Deuteron and low-energy parameters. The experimental values are from Table 4.2 of Ref. [5] Parameter Bonn B UCT Experiment as (fm) −23.71 −23.57 −23.748±0.010 rs (fm) 2.71 2.65 2.75±0.05 at (fm) 5.426 5.44 5.419±0.007 rt (fm) 1.761 1.79 1.754±0.008 εd (MeV) 2.223 2.224 2.224575 PD (%) 4.99 4.89 Deuteron wave function components ψd 0(p) = u(p) and ψd 2(p) = w(p). Solid (dotted) curves calculated for the Bonn B (UCT) potential 191 Table 2. The best-fit parameters for the two models. The row Potential B (UCT ) taken from Table A.1 in [5] (obtained by least squares fitting to OBEP values in Table 1 of Ref. [3] including deuteron binding energy and low-energy parameters). All masses are in MeV , and nb = 1 except for nρ = nω = 2 Model Meson π η ρ ω δ σ, T = 0 (T = 1) g2/4π [f/g] 14.4 3 0.9 [6.1] 24.5 2.488 18.3773 (8.9437) Potential B Λ 1700 1500 1850 1850 2000 2000 (1900) m 138.03 548.8 769 782.6 938 720 (550) g2/4π [f/g] 13.395 5.0 1.2 [6.1] 17.349 5.0 22.015 (5.514) UCT Λ 2500 1219 1593 2494 2169 1200 (2500) m 138.03 548.8 769 782.6 938 720 (550) 6. CONCLUSIONS Starting from a total Hamiltonian for interacting me- son and nucleon fields, we come to the Hamiltonian and boost generator in the CPR whose interaction parts consist of new relativistic interactions responsi- ble for physical (not virtual) processes, particularly, in the system of bosons (π−, η−, ρ−, ω−, δ− and σ−mesons) and fermions (nucleons and antinucle- ons). Using the unitary equivalence of CPR to BPR, we have seen how the NN scattering problem in QFT can be reduced to the three-dimensional LS- type equation for the T -matrix in momentum space. The equation kernel is given by clothed two-nucleon interaction of class [2.2]. Special attention has been paid to the elimina- tion of auxiliary field components. We encounter such a necessity for interacting vector and fermion fields when in accordance with the canonical formalism the interaction Hamiltonian density embodies not only a scalar contribution but nonscalar terms too. Being concerned with constructing two-nucleon states and their angular-momentum decomposition we have not used the so-called separable ansatz. The clothed two-nucleon partial waves have been built up as common eigenstates of the field total angular- momentum generator and its polarization (fermionic) part. As a whole, persistent clouds of virtual particles are no longer explicitly contained in CPR, and their influence is included in properties of clothed particles (these quasiparticles of the UCT method). In addi- tion, we would like to stress that problem of the mass and vertex renormalizations is intimately interwoven with constructing the interactions between clothed nucleons. References 1. A. Shebeko, M. Shirokov. Unitary transforma- tions in quantum field theory and bound states // Phys. Part. Nucl. 2001, v. 32, p. 31-95. 2. V. Korda, L. Canton, A. Shebeko. Relativistic interactions for the meson-two-nucleon system in the clothed- particle unitary representation // Ann. Phys. 2007, v. 322, p. 736-768. 3. I. Dubovyk, O. Shebeko. The method of unitary clothing transformations in the theory of nucleon- nucleon scattering // Few Body Syst. 2010, v. 48, p. 109-142. 4. A.V. Shebeko, P.A. Frolov. A possible way for constructing generators of the Poincaré group in quantum field theory // Few Body Syst. 2011, DOI:10.1007/s00601-011-0262-5. 5. R. Machleidt. The meson theory of nuclear forces and nuclear structure // Adv. Nucl. Phys. 1989, v. 19, p. 189-376. �� !�� "�� #�� $�%�� &"� �� '&"� ( �� ) �� ) �� � +�� , � �&�� - � � ��) �.� � )� )��) �� #��)�� &��/ �� 0 �� "'�� ( �� &�� ."� �� ) �)� �� "��1 #0�� $�%�� &"2 �� '&"� � �� 2 ��01 0 +��"' �� , � �&�� .� � ��0��) �� &�) ��) " � #0�� &��0�/ �3�

Relativistic interactions for meson-nucleon systems: applications in the theory of nuclear reactions

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