Perspective Chapter: EPR Paradox – Experimental and Quantum Field Theoretical Status of Light Meson Resonances

Abstract

The inclusive reaction of the $V$-meson resonance production is studied in order to check consistency of the field-theoretical approach with formulation based on the Einstein-Podolsky-Rosen (EPR) concepts. For this aim we have constructed the relativistic field-theoretical amplitudes of the $V$-meson resonance decay $1+2\leftarrow V$ with structureless (pointlike) and composite (non-pointlike) resonance. Nonlocal composite states of a resonance are obtained using quark-gluon degrees of freedom in accordance with quantum chromodinamics (QCD). Particle 1 and 2 have opposite momentums ${\mathbf{p}}_1=-{\mathbf{p}}_2$ and strongly correlated spin states in the rest frame of $V$-meson decay. Moreover, for electron-positron decay of the $V$-meson resonance, electron and positron have the opposite helicity if the electron mass is neglected. Therefore, the decay of $V$-meson into two particles can be considered as the formation of the EPR-pair according to EPR gedanken experiment. In addition, the color quark-gluon states can be interpreted as hidden (non-observed) states within EPR formulation. As an example of such an approaches, we have considered papers with high-energy experimental data where for description of observable was used quark-antiquark states and concepts of EPR pairs and entanglement.

1. Introduction

Paradox of Einstein, Podolsky and Rosen [1] has motivated numerous experimental and theoretical investigations of basic concepts in quantum mechanics. In particular, in review papers [1, 2, 3, 4, 5, 6, 7] were studied the highly correlated so called EPR-pairs of particles with following possibility of determination of the location of one of these particles through the location of another particle. Schrödinger [8] showed that the EPR paradox lies in the possibility of separabelization of the two-particle wave function $\mid {\psi }_{A+B}>$ as product of the single-particle wave functions $\mid {\psi }_{A+B}>\iff \mid {\psi }_A>\mid {\psi }_B>$. The next principal result was obtained by Bell [9], where it is proved that EPR paradox in quantum mechanics should be supplemented by additional local hidden variables which enables to restore causality and locality. Subsequently, particles which satisfy the conditions of Schrödinger and Bell were called EPR pairs and interaction (correlation) of these particles in final states are defined as entanglement [2]. The notable exception of the Bell theory are given in [5] using nonlocal hidden variables and in [10, 11, 12] using the concept of superdeterminism. Superdeterministic hidden-variable theories can be local Superdetermine yet be compatible with observations. Theoretical and experimental evidences for the violation of Bell’s inequalities for n-particle systems of with spin are considered in [13].

On the other hand, Bell’s inequalities and modified Bell’s inequalities are based on simple, model-independent mathematical proofs [14]. Therefore, the mechanism and mathematical the justification for the violation of Bell’s inequalities is of particular interest.

At present, interest in the study of the EPR paradox and EPR pairs has grown due to the possibility of their practical application in different technologies and in the quantum computing. Review papers in this direction [2, 3, 4, 5, 6, 7] are constantly updated with new investigations (see e.g. [15, 16]).

Comprehensive study of the EPR paradox and its modifications involves analysis of locations of particles from EPR-pair in coordinate space. Measurements of cross sections and other characteristics of particle interactions in the high-energy region are usually carried out in momentum space since the time and radius of interaction of hadrons are very small and are not available for the direct measurement. The nonrelativistic quantum mechanical measurement of trajectory of particle is also strongly restricted via Heisenberg’s uncertainty principle. An alternative point of view on the theoretcal possibility of measuring the trajectory of quantum particles is discussed in the review article [17] using formulations of Einstein, De Broglie and Bohm. In relativistic quantum mechanics and relativistic quantum field theory interaction (entanglement) of particles and their trajectories are restricted also by strong interconnection between the temporal and spatial components of coordinate of each particle. This interconnection follows from the principles of relativity and causality. However, entanglement between two hadrons in coordinate space can be established on the basis of a theoretical model, which allows one to unambiguously determine the dependence of the observed characteristics from a two-hadron system. For example, entanglement of two atoms at a stand of 1 cm was found in [18] using one-photon exchange. Similarly, one can establish entanglement between two hadrons at microscopic distance using the theoretical model dependence of measurements of the two-hadron production.

Another aspect of the EPR paradox is problem of hidden variables to determine particle states and particle interactions. The color-charged quark-gluon states one can consider as hidden degrees of freedom because they can not be observed or isolated due to confinement in quantum chromodinamics (QCD). Hadronic and nuclear states are formed from colorless states. confinement was analytically reproduced using two-dimensional Schwinger model or Abelian gauge theories in 2 and 3 space-time dimensions. Moreover QCD as well as confinement obtained in the framework of the multi-dimensional String theory which is not yet justified experimentally. Note that the introduction of unobservable generalizations of particle states for convenient description of their interactions or reproduction of experimental data, is long-known and effective method in theoretical physic. For example, in the review article [19] are analyzed the five-dimensional theoretical models of many outstanding physicists from 1922 to 1991 years, where the fifth dimension has an auxiliary, mathematical meaning.

The above considerations can be applied to the analysis of high-energy experimental data using quark interaction models and with the emergence of the EPR-pairs. The production of the EPR pair of the neutral $B^o$-mesons mesons at the $\mathrm{\Upsilon }\left(4S\right)$-resonance was measured in [20] in order to study the non-local entanglement between $B^o$ and ${\overline{B}}^o$ mesons. For this aim was measured the time-dependent asymmetry between two $B^o$ states with different flavor, which is compared with prediction of quantum mechanical model and predictions of two realistic models. The flavor asymmetry of quarks for weak interacted particles as well as the mass difference and mixing (oscillating) of $B^o$ and ${\overline{B}}^o$ was predicted by standard model and was supported by many experiments (see e.g. [21]).

Another application of the concept of entanglement between quark states in the high energy particle scattering experiments is done in [22, 23] using LHC CERN data set collected by ATLAS detector for $\sqrt{s}=13$ TeV between 2013 and 2018 years. From this data were extracted differential cross section of the reaction $e^{+}+e^{-}\leftarrow t+\overline{t}$ with top diquark masses $m_{t\overline{t}}=350\mathit{GeV}$ and and $m_{t\overline{t}}\ge 1\mathit{TeV}$. It was shown that the separability of finite states $e^{+}$ and $e^{-}$ are violated due to interaction in the reaction $e^{+}+e^{-}\leftarrow t+\overline{t}$, which significantly depends on the scattering angle in this sub-process. Besides in these papers is considered application of suggested entanglement in quantum information theory and quantum technologies. Thus the entanglement of the electron-positron pair in [22, 23] is determined by inelastic scattering of a quark-antiquark pair $e^{+}+e^{-}\leftarrow t+\overline{t}$.

Most detailed and complete measurements of the exclusive ${\rho }^o$-meson resonance with the polarized virtual photons were fulfilled in numerous experiments during many years at CERN (see ref. [24] and papers quoted there), where the applications of various kinematic, quark-parton models were implemented.

The purpose of this paper is to consider the the implementation of the EPR-criteria for resonances in the framework of the relativistic quantum field theory. In particular, it will be considered the spin 1 meson ($V$-meson) resonance decay into hadronic or leptonic pairs in the high-energy experiments, where the measurement results are analyzed using concepts of EPR-pair and entanglement of particles in this pair.

Measuring the mass and width of the resonance $M_V$ and ${\mathrm{\Gamma }}_{V}2$ is usually performed using selection of the resonating pair of particles 1 and 2 by their four momentums $p_1=\sqrt{m_1^2+{\mathbf{p}}_1^2},{\mathbf{p}}_1\left)\right)$ and $p_2=\sqrt{m_2^2+{\mathbf{p}}_2^2},{\mathbf{p}}_2)$ which are located in the resonant region ${\left(M_V-\frac{{\mathrm{\Gamma }}_V}{2}\right)}^2\le {\left(p_1+p_2\right)}^2\le {\left(M_V+\frac{{\mathrm{\Gamma }}_V}{2}\right)}^2$. This condition determines the relationship or entanglement of particles 1 and 2 after the decay of the resonance. In the rest frame of the $V$-meson resonance, ${\mathbf{p}}_1$ and ${\mathbf{p}}_2$ have opposite direction and ${\mathbf{p}}_1=-{\mathbf{p}}_2$. Moreover, spin of these particles is strongly correlated. For instance, neglecting mass of electrons in vector meson decay one obtains that helicity of the electron and positron are also opposite. Therefore the $V$-meson resonance decay particle 1 and 2 form the EPR pair-type system.

For the relativistic description of the resonances we will use the field-theoretical formulation of Huang and Weldon [25, 26, 27, 28], where hadrons are bound states of quarks. In this formulation quarks and gluons are the point-like (structureless) particle which are described by local field operator. The field operators of quarks and gluons satisfy the QCD equation of motion and causality condition. Hadron states are constructed via colorless time-ordered product of the quark fields. In this approach is constructed also the asymptotic one hadron creation or annihilation operator and corresponding $\mathcal{S}$-matrix of the hadron-hadron interaction, which satisfies unitarity condition in hadronic sector. Therefore propagation of quarks in the intermediate states does not destroy the unitarity of the hadron-hadron $\mathcal{S}$-matrix. Therefore one can obtain the equation of motion for amplitudes of the hadron-hadron and hadron-nuclear interaction, where color states appear only between colorless hadronic or nuclear states. This enables to avoid problems with confinement in this approach. The 3D field-theoretical equations for the scattering amplitude as an alternate approach for the 4D Bethe-Salpeter equations is considered in Appendix A.

In contrast to the non-relativistic formulation, relativistic quantum mechanics requires strong connection between the temporal and spatial components of coordinates according to invariance under Lorentz transformation. Besides in the relativistic approach arise additional mechanisms of entanglement which are caused by off-energy and off mass shell effects and contributions from creation and annihilation of particles in intermediate states.

The organization of this papers is as follows. In Section 1 the general properties of the relativistic field-theoretical $\mathcal{S}$-matrix of reactions $1+2+X\leftarrow V+X\leftarrow A+B$ and corresponding construction of the resonance decay amplitude $1+2\leftarrow V$ are outlined. Section 2 is devoted to the $V$-meson resonance decay amplitude with intermediate quark-gluon degrees of freedom. In Section 3 angular distribution of the reaction $1+2+X\leftarrow V+X\leftarrow A+B$ is given in terms of the decay angles of the $V$-meson resonance and is considered the alignment of the particles 1 and 2 in this reaction. The last fourth part of the article provides a brief formulation of the results and examines the formation of an EPR pair and their entanglement based on relativistic field-theoretical description of the resonance. In particular are compared amplitudes with the local point-like resonance and with the nonlocal resonance consisting of quarks.

2. Amplitude of the structureless (point-like) resonance

In relativistic quantum field theory [29, 30, 31] $\mathcal{S}$-matrix of inclusive collision of two particles $A$ and $B$ which generated particles 1, 2 and subsystem $X$ is

$$\begin{array}{lll}{\mathcal{S}}_{1+2+X\leftarrow A+B}& \equiv & <\mathit{\text{out}};{\mathbf{p}}_1,{\mathbf{p}}_2,X\mid \mathcal{S}\mid {\mathbf{p}}_A,{\mathbf{p}}_B;\mathit{\text{in}}>\\ & =& <0\mid a_{{\mathbf{p}}_1}\left(\mathit{\text{out}}\right)a_{{\mathbf{p}}_2}\left(\mathit{\text{out}}\right),b_{{\mathbf{p}}_1}\left(\mathit{\text{out}}\right),\dots ,b_{{\mathbf{p}}_X}\left(\mathit{\text{out}}\right)a_{{\mathbf{p}}_A}^{+}\left(\mathit{\text{in}}\right),a_{{\mathbf{p}}_B}^{+}\left(\mathit{\text{in}}\right)\mid 0>\end{array}$$E1

where $a_{{\mathbf{p}}_1}\left(\mathit{\text{out}}\right)$, $a_{{\mathbf{p}}_2}\left(\mathit{\text{out}}\right)$ and $a_{{\mathbf{p}}_A}^{+}\left(\mathit{\text{in}}\right)$ and $a_{{\mathbf{p}}_B}^{+}\left(\mathit{\text{in}}\right)$ denotes corresponding particle creation operators in $\mathit{\text{out}}$ and $\mathit{\text{in}}$ states with momentum ${\mathbf{p}}_1$, ${\mathbf{p}}_2$, ${\mathbf{p}}_A$, ${\mathbf{p}}_B$ correspondingly. $b_{{\mathbf{p}}_1}\left(\mathit{\text{out}}\right)$,…, $b_{{\mathbf{p}}_X}\left(\mathit{\text{out}}\right)$ stands for particle from subsystem X. We shall suppose that particles 1 and 2 compose intermediate vector meson $V$ with spin 1 and we shall consider $V$-meson decay into electron-positron $e^{-}{e}^{+}$ and two pion ${\pi }^{+}{\pi }^{-}$ pairs. Therefore in the $V$-resonance energy region one gets following chain of subreactions $A+B\to V+X\to 1+2+X$. It is convenient to express $\mathcal{S}$-matrix (1) via corresponding amplitude

$${\mathcal{S}}_{1+2+X\leftarrow A+B}=i{\left(2\pi \right)}^4{\delta }^4\left(p_A+p_B-p_1-p_2-P_X\right){\mathcal{A}}_{1+2+X\leftarrow A+B}$$E2

where $p_A$, $p_B$, $p_1$, $p_2$ and $P_X$ are four momentums of on shell asymptotic particles $p_{A,B}=\left(\sqrt{m_{A,B}^2+{\mathbf{p}}_{A,B}^2},{\mathbf{p}}_{A,B}\right)$, $p_{1,2}=\left(\sqrt{m_{1,2}^2+{\mathbf{p}}_{1,2}^2},{\mathbf{p}}_{1,2}\right)$ with masses $m_{A,B}$, $m_{1,2}$ and three momentums ${\mathbf{p}}_{A,B}$, ${\mathbf{p}}_{1,2}$. Besides we shall use four-vector of $V$-meson resonance $P_V\equiv \left(P_V^o,{\mathbf{P}}_V\right)=\left(\sqrt{M_V^2+{\mathbf{P}}_V^2},{\mathbf{P}}_V\right)$.

Spin of relativistic particle or resonance is defined in the rest frame of this particle or resonance. Therefore we shall consider the rest frame of system of particles 1 and 2, where .

Resonance in quantum mechanics is defined as pole of the amplitude in the complex momentum plane at $\sqrt{s_{12}}\equiv \sqrt{{\left(p_1+p_2\right)}^2}\to M_V-i{\mathrm{\Gamma }}_V/2$, where $M_V$ and ${\mathrm{\Gamma }}_V$ denote the pole position and the $V$-resonance decay width in the particle system $1+2$. Therefore ${\mathcal{A}}_{1+2+X\leftarrow A+B}$ must have the following form

$${\mathit{lim}}_{s_{1,2}\to M_{12}-i{\mathrm{\Gamma }}_{12}/2}\left(\sqrt{s_{1,2}}-M_{12}+i{\mathrm{\Gamma }}_{12}/2\right){\mathcal{A}}_{1+2+X\leftarrow A+B}={\mathcal{A}}_{1+2\leftarrow V}{\mathcal{A}}_{V+X\leftarrow A+B}$$E3

or

$${\mathcal{A}}_{1+2+X\leftarrow A+B}={\mathcal{A}}_{1+2\leftarrow V}\frac{1}{\sqrt{s_{1,2}}-M_V+i\frac{{\mathrm{\Gamma }}_V}{2}}{\mathcal{A}}_{V+X\leftarrow A+B}+\mathit{\text{nonresonant}}\mathit{\text{part}}$$E4

where the pole part of the amplitude is represented by the $V$-meson exchange term. It is convenient to express ${\mathcal{A}}_{1+2\leftarrow V}$ via the product of the wave function $\mid {\mathrm{\Psi }}_V>$ and two-body potential $<1,2\mid V\mid 1^{\prime },2^{\prime }>$

$${\mathcal{A}}_{1+2\leftarrow V}\left(\sqrt{s_{1,2}}\right)=<1,2\mid V{\mathrm{\Psi }}_V>$$E5

where $\mid {\mathrm{\Psi }}_V>$ satisfies the equation

$$<1,2\mid {\mathrm{\Psi }}_V>=\frac{1}{\sqrt{s_{1,2}}-M_V+i\frac{{\mathrm{\Gamma }}_V}{2}}<1,2\mid V{\mathrm{\Psi }}_V>.$$E6

Unlike to the bound state, $<1,2\mid {\mathrm{\Psi }}_V>$ is not the eigen-function of the Hamiltonian of the system $1+2-1^{\prime }+2^{\prime }$.

Eqs. (4) and (6) enables to describe the resonant and non-resonant parts of ${\mathcal{A}}_{1+2+X\leftarrow A+B}$ using the same Lagrangian and same phenomenogical potential for all partial waves of $1+2$ subsystem. In this approach were reproduced low energy $\mathit{\pi N}-\mathit{\pi N}$, $\mathit{NN}-\mathit{NN}$ and $\mathit{\gamma N}-\mathit{\pi N}$ reactions [26, 32, 33]. In particular, description of low-energy $\mathit{\pi N}$ scattering amplitude in the $\mathrm{\Delta }$-resonance region was performed within the Born approximation in [30] (see eq. (10.67) and (10.79)).

In other relativistic approach resonance is treated as particle-type degrees of freedom with imaginer mass $M_V-i{\mathrm{\Gamma }}_V/2$, spin $s_V$ and with corresponding Lagrangian. Corresponding S,P,D,F pion-nucleon S,P,D,F phase shifts are calculated within third order chiral effective field theory in [34].

Cross section of the reaction $1+2+X\leftarrow A+B$ according to (4) is [31].

$$\frac{d{\sigma }_{1+2+X\leftarrow A+B}}{d{\mathrm{\Omega }}_{1+2+X\leftarrow A+B}}=\sum _{M{M}^{\prime }}\frac{d{\tilde{\sigma }}_{1+2\leftarrow V}^{M{M}^{\prime }}}{d{\mathrm{\Omega }}_{1+2\leftarrow V}}\frac{1}{{\left(m_V-\sqrt{s_{12}}\right)}^2+{\mathrm{\Gamma }}_V^2/4}\frac{d{\sigma }_{V+X\leftarrow A+B}^{M{M}^{\prime }}}{d{\mathrm{\Omega }}_{V+X\leftarrow A+B}}$$E7

where $M$, $M^{\prime }$ are magnetic quantum number of $V$ resonance

$$\frac{d{\sigma }_{V+X\leftarrow A+B}^{M{M}^{\prime }}}{d{\mathrm{\Omega }}_{V+X\leftarrow A+B}}={\mathcal{N}}_{A+B}\prod _i^X{\mathcal{A}}_{V+X\leftarrow A+B}^M{\left({\mathcal{A}}_{V+X\leftarrow A+B}^{\ast }\right)}^{M^{\prime }}$$E8

$$d{\mathrm{\Omega }}_{V+X\leftarrow A+B}={\left(2\pi \right)}^3\frac{d{\mathbf{P}}_{\mathbf{12}}}{{\left(2\pi \right)}^3}\prod _{i=1}^{X}d{\tilde{\mathbf{p}}}_i{\left(2\pi \right)}^4{\delta }^3\left({\mathbf{P}}_A+{\mathbf{P}}_B-{\mathbf{P}}_{12}-{\mathbf{P}}_X\right)\delta \left(P_A^o+P_B^o-P_{12}^o-P_X^o\right)$$

where ${\mathbf{P}}_{12}={\mathbf{p}}_1+{\mathbf{p}}_2$ and $P_{12}^o=p_1^o+p_2^o$,

$$\begin{array}{lll}\frac{d{\tilde{\sigma }}_{1+2\leftarrow V}^{M{M}^{\prime }}}{d{\mathrm{\Omega }}_{1+2\leftarrow V}}& =& {\mathcal{A}}_{1+2\leftarrow V}^M{\left({\mathcal{A}}_{1+2\leftarrow V}^{\ast }\right)}^{M^{\prime }}\\ d{\mathrm{\Omega }}_{1+2\leftarrow V}& =& d\widetilde{{\mathbf{p}}_{\mathbf{1}}}d\widetilde{{\mathbf{p}}_{\mathbf{2}}}{\delta }^3\left({{\mathbf{p}}_{\mathbf{1}}}^{\ast }+{\mathbf{p}}_2^{\ast }\right)\end{array}$$E9

where ${\mathrm{N}}_A=1/\left(2p_A^o\right)$ for bosons and ${\mathrm{N}}_A=m_A/p_A^o$ for fermions

$${\mathcal{N}}_{A+B}=\frac{1}{\left(2S_A+1\right)\left(2S_B+1\right)}\frac{{\mathrm{N}}_A{\mathrm{N}}_B}{\sqrt{{\left(P_A{P}_B\right)}^2-m_A^2{m}_B^2}};d{\tilde{\mathbf{p}}}_a={\mathrm{N}}_a\frac{d{\mathbf{p}}_a}{{\left(2\pi \right)}^3}$$E10

$d{\tilde{\sigma }}_{1+2\leftarrow V}^{M{M}^{\prime }}/d{\mathrm{\Omega }}_{1+2\leftarrow }$ does not contain $\delta \left(P_V^o-p_1^o-p_2^o\right)$ and ${\mathcal{A}}_{1+2\leftarrow V}^M$ depends on the decay width ${\mathrm{\Gamma }}_V$ of resonance according to (5) and (6). That’s why amplitude ${\mathcal{A}}_{1+2\leftarrow V}^M$ is nonlocal and off shell even for the point-like, structureless resonance. General field theoretical construction of the nonlocal $V$-resonance with quark degrees of freedom is given in next section.

According to the vector meson dominance model the main contribution into vector meson decay is made by diagram of the $V$-meson transformation into off shell photon ${\gamma }^{\ast }\leftarrow V$. Therefore in phenomenogical ${\mathcal{A}}_{1+2\leftarrow V}^M$ is used the form factor of electromagnetic-type (see e.g. [29] ch. 10.6), where current operator of the $V$-meson resonance is replaced by the current operator of the elementary particle with the mass $M_V$, spin 1 and the coupling constant $g_V$. Then ${\mathcal{A}}_{1+2\leftarrow V}^M$ for the leptonic decay takes the form

$${\mathcal{A}}_{{\mathrm{\ell }}^{+}+{\mathrm{\ell }}^{-}\leftarrow V}^M=g_{{\mathrm{\ell }}^{+}+{\mathrm{\ell }}^{-}\leftarrow V}{{\xi }_{\mu }}^{\ast }\left({\mathbf{P}}_V,M\right)\overline{u}\left({\mathbf{p}}_{\mathrm{\ell }}\right){\gamma }_{\mu }v\left({\mathbf{p}}_{\overline{\mathrm{\ell }}}\right)$$E11

where ${\xi }^{\mu \ast }\left({\mathbf{P}}_V,M\right)$ is the polarization vector of the spin 1 particle and coupling constant $g_V\equiv g_{{\mathrm{\ell }}^{+}+{\mathrm{\ell }}^{-}\leftarrow V}$ is defined according to vector meson dominance model.

In this approximation the hadronic decay amplitude is

$${\mathcal{A}}_{h+\overline{h}\leftarrow V}^M=g_{h+\overline{h}-V}\left({\left(p_h+p_{\overline{h}}\right)}^2\right)\left({{\xi }_{\mu }}^{\ast }\left({\mathbf{P}}_V,M\right){p_h^{\ast }}^{\mu }\right)$$E12

where for simplicity only spineless hadrons are considered. $g_{h+\overline{h}-V}\left({\left(p_h+p_{\overline{h}}\right)}^2\right)$ is form factor of $h+\overline{h}\leftarrow V$.

3. $V$-meson decay amplitude with the quark-gluon degrees of freedom

Amplitude ${\mathcal{A}}_{1+2\leftarrow V}$ (5) can be constructed with intermediate quark-antiquark-gluon degrees of freedom $q-\overline{q}-$ using the field-theoretical approach of the composite particle states [25, 26]. In this formulation the Heisenberg field operator of the composite particle is defined via the Bethe-Salpeter wave functions (see [31] ch. 10.2.1)

$${\chi }_{m_V,{\mathbf{P}}_V}\left(Y,y_{1,2}\right)=<0\mid \mathrm{T}\left({\psi }_1\left(y_1\right){\psi }_2\left(y_2\right)\right)\mid m_V,{\mathbf{P}}_V,M;\mathit{\text{in}}>,$$E13

where

$$Y=y_1+y_2,y_{1,2}={\eta }_2{y}_1-{\eta }_1{y}_2;{\eta }_1+{\eta }_2=1$$E14

are Jacobi coordinates. For the sake of simplicity in (1) we have introduced the complex mass $m_V=M-V-i{\mathrm{\Gamma }}_V/2$ for the resonance state. In alternate formulation the dependence on ${\mathrm{\Gamma }}_V$ of wave function implicitly through solution of the Bethe-Salpeter equation. In alternate formulation the dependence on ${\mathrm{\Gamma }}_V$ of wave function occurs implicitly through solution of Bethe-Salpeter equation.

One can introduce nonlocal operator of the resonance in analogy with bound-state field

$${\mathcal{B}}_{m_V,{\mathbf{P}}_V}\left(Y_o\right)=i\int d^3\mathbf{Y}{d}^4{y}_{12}exp\left({\mathit{iP}}_{V}Y\right)\left[\frac{\partial }{\partial Y_o}-{\mathit{iP}}_V^o\right]{\chi }_{m_V{\mathbf{P}}_V}\left(Y,y_{12}\right)\mathrm{T}\left({\psi }_1^{+}\left(x_1\right){\psi }_2^{+}\left(x_2\right)\right).$$E15

which transforms into single composite particle creation or annihilation operator in asymptotic regions $\pm \infty \Leftarrow Y_O$

$$<\mathrm{\Psi }\mid {\mathcal{B}}_{m_V,{\mathbf{P}}_V}\left(\mathit{\text{out}}\left(\mathit{\text{in}}\right)\right)\mid \mathrm{\Phi }>={\mathit{lim}}_{\pm \infty \Leftarrow Y_o}<\mathrm{\Psi }\mid {\mathcal{B}}_{m_V,{\mathbf{P}}_V}\left(Y_o\right)\mid \mathrm{\Phi }>,$$E16

where $\mid \mathrm{\Phi }>$ and $\mid \mathrm{\Psi }>$ denote an arbitrary wave functions.

The asymptotic operators satisfy the same commutation relation as elementary (point-like) particle

$$\begin{array}{lll}\left[{\mathcal{B}}_{m_V,{\mathbf{P}}_V}^{+}\left(\mathit{\text{out}}\left(\mathit{\text{in}}\right)\right),{\mathcal{B}}_{m_V,{\mathbf{P}}_V}\left(\mathit{\text{out}}\left(\mathit{\text{in}}\right)\right)\right]& =& {\left(2\pi \right)}^{3}2P_V^o\delta \left({\mathbf{P}}_{\mathbf{V}}^{\prime }-{\mathbf{P}}_{\mathbf{V}}\right);\\ \left[{\mathcal{B}}_{m_V,{\mathbf{P}}_V}\left(\mathit{\text{out}}\left(\mathit{\text{in}}\right)\right),{\mathcal{B}}_{m_V,{\mathbf{P}}_V}\left(\mathit{\text{out}}\left(\mathit{\text{in}}\right)\right)\right]& =& 0;\end{array}$$

But unlike elementary (point-like, structureless) particle, the same commutation relations for ${\mathcal{B}}_{m_V{\mathbf{P}}_V}\left(Y_o\right)$ (15) are violated

$$\begin{array}{lll}\left[{\mathcal{B}}_{m_V{\mathbf{P}}_V}^{+}\left(Y_o\right),{\mathcal{B}}_{m_V{\mathbf{P}}_V}\left(Y_o\right)\right]& \ne & {\left(2\pi \right)}^{3}2P_V^o\delta \left({\mathbf{P}}_{{\mathbf{V}}^{\prime }}-{\mathbf{P}}_{\mathbf{V}}\right);\\ \left[{\mathcal{B}}_{m_V{\mathbf{P}}_V}\left(Y_o\right),{\mathcal{B}}_{m_V{\mathbf{P}}_V}\left(Y_o\right)\right]& \ne & 0.\end{array}$$E18

Inequalities (18) are caused by nonlocality of ${\mathcal{B}}_{m_V{\mathbf{P}}_V}\left(Y_o\right)$ (15) which depends exactly on interacted fields ${\psi }_1\left(y_1\right)$, ${\psi }_2\left(y_2\right)$. and on the resonance decay width ${\mathrm{\Gamma }}_V/2$.

Using ${\mathcal{B}}_{m_V{\mathbf{P}}_V}\left(Y_o\right)$ (15) one can built the $V$ meson field ${\mathrm{\Upsilon }}_{m_V{\mathbf{P}}_V}\left(Y\right)$ and corresponding source operator ${\mathcal{J}}_{m_V{\mathbf{P}}_V}\left(Y\right)$

$${\mathrm{\Upsilon }}_{m_V{\mathbf{P}}_V}\left(Y\right)=\int \frac{d^3{\mathbf{P}}_V}{{\left(2\pi \right)}^{3}2P_V^o}\left[e^{-{\mathit{iP}}_{V}Y}{\mathcal{B}}_{m_V{\mathbf{P}}_V}\left(Y_o\right)+e^{{\mathit{iP}}_{V}Y}{\mathcal{B}}_{m_V{\mathbf{P}}_V}^{\ast }\left(Y_o\right)\right],$$E19

$${\mathcal{J}}_{m_V{\mathbf{P}}_V}\left(Y\right)=\left(W_Y+M_V^2\right){\mathrm{\Upsilon }}_{m_V{\mathbf{P}}_V}\left(Y\right)$$E20

which enables to obtain the general expression for amplitude of $V$ meson resonance

$${\mathcal{A}}_{1+2\leftarrow V}\left(\mathit{\text{nonlocal}}\right)=<\mathit{\text{out}};1,2\mid {\mathcal{J}}_{m_V{\mathbf{P}}_V}\left(0\right)\mid 0>$$E21

where $m_V=M_V-i{\mathrm{\Gamma }}_V/2$.

4. Spin density matrix and alignment for reaction $1+2+X\leftarrow V+X\leftarrow A+B$

The cross sections (7), (8) and (9) make it possible to obtain a convenient formula for the expansion of the angular distribution $W_{1+2+X\leftarrow A+B}$ in terms of the sum spin density matrices (see e.g. ch. 8 [35]) ${\rho }_{V+X\leftarrow A+B}^{M{M}^{\prime }}$ and ${\rho }_{1+2\leftarrow V}^{M{M}^{\prime }}$.

$$W_{1+2+X\leftarrow A+B}=\sum _{M,M^{\prime }}{\rho }_{1+2\leftarrow V}^{M{M}^{\prime }}{\rho }_{V+X\leftarrow A+B}^{M{M}^{\prime }},$$E22

where

$${\rho }_{V+X\leftarrow A+B}^{M{M}^{\prime }}=\frac{d{\sigma }_{V+X\leftarrow A+B}^{M{M}^{\prime }}/d{\mathrm{\Omega }}_{V+X\leftarrow A+B}^{\ast }}{\int d{\mathrm{\Omega }}_{V+X\leftarrow A+B}^{\ast }{\sum }_{M}d{\sigma }_{V+X\leftarrow A+B}^{\mathit{MM}}/d{\mathrm{\Omega }}_{V+X\leftarrow A+B}^{\ast }};$$E23

where

$${\rho }_{1+2\leftarrow V}^{M{M}^{\prime }}=\frac{d{\tilde{\sigma }}_{1+2\leftarrow V}^{M{M}^{\prime }}/d{\mathrm{\Omega }}_{1+2\leftarrow V}}{\int d{\mathrm{\Omega }}_{1+2\leftarrow V}^{\ast }{\sum }_{M}d{\tilde{\sigma }}_{1+2\leftarrow V}^{\mathit{MM}}/d{\mathrm{\Omega }}_{1+2\leftarrow V}^{\ast }}$$E24

Upper index ${}^{\ast }$ in (22)–(24) indicate the rest frame of $V$-meson, where ${\mathbf{p}}^{\ast }\equiv {\mathbf{p}}_1^{\ast }=-{\mathbf{p}}_2^{\ast }$ and decay angles ${\theta }^{\ast }$ и ${\varphi }^{\ast }$ are defined by momentum of the final particle

$${\mathbf{p}}^{\star }\equiv {\mathbf{p}}_1^{\star }=\mid {\mathbf{p}}^{\star }\mid \left(\mathit{sin}{\theta }^{\star }\mathit{cos}{\varphi }^{\star },\mathit{sin}{\theta }^{\star }sin{\varphi }^{\star },cos{\theta }^{\star }\right),$$E25

$d{\mathrm{\Omega }}_{V+X\leftarrow A+B}^{\star }$ and $d{\mathrm{\Omega }}_{1+2\leftarrow V}$ stands for Lorentz invariant phase space elements

$$d{\mathrm{\Omega }}_{V+X\leftarrow A+B}^{\star }=\frac{d^3{\mathbf{P}}_V^{\ast }}{{\left(2\pi \right)}^3}\prod _{i=1}^X{\mathrm{N}}_i\frac{d^3{\mathbf{p}}_{\mathbf{i}}^{\mathbf{\ast }}}{{\left(2\pi \right)}^3}$$E26

$$d{\mathrm{\Omega }}_{1+2\leftarrow V}={\mathrm{N}}_1{\mathrm{N}}_2\frac{d^3{\mathbf{p}}^{\ast }}{{\left(2\pi \right)}^3}$$E27

Cross sections (7) and decomposition of the corresponding angular distribution (22) through products of spin density matrices are often used to describe experimental data of exclusive and inclusive reactions using the helicity basis. Polarization effects in the corresponding generalizations of the cross sections (7) and corresponding spin density matrices are taking into account in [36, 37]. Beam polarization effects, decomposition of the spin density matrix by the transverse and longitudinal components of the virtual photon are considered in [24] for the exclusive reactions ${\mu }^{\prime }+p^{\prime }+{\rho }^o\leftarrow \mu +p$ and $p^{\prime }+{\rho }^o\leftarrow {\gamma }^{\ast }+p$.

Using vertex functions (11) and (12) and relationship between ${\xi }^{\mu }\left(0,M\right)$ and Wigner $D$-matrices $D_{\mathit{mn}}^1\left({\theta }^{\ast },{\varphi }^{\ast }\right)$

$$\left({\xi }^{\mu }\left(0,M\right)p_{\mu }^{\ast }\right)=\mid {\mathbf{p}}^{\ast }\mid D_{M0}^1\left({\theta }^{\ast },{\varphi }^{\ast }\right)$$E28

we obtain

$$W_{1+2+X\leftarrow A+B}=C+\sum _{M{M}^{\prime }}{D^{\ast }}_{M0}^1\left({\theta }^{\ast },{\varphi }^{\ast }\right){\rho }_{V+X\leftarrow A+B}^{M{M}^{\prime }}{D}_{M^{\prime }0}^1\left({\theta }^{\ast },{\varphi }^{\ast }\right)$$E29

where $C=0$ if $1,2$ are pions and $C=1$ if $1,2$ are electron-positron pair.

Angular distributions $W_{1+2+X\leftarrow A+B}$ (29) describe decay of the point-like (structureless) resonance $V$ on two particles 1 and 2, where dependence of $W_{1+2+X\leftarrow A+B}$ on the resonance decay angles is contained in $D_{M^{\prime }0}^1\left({\theta }^{\ast },{\varphi }^{\ast }\right)$. Therefore anisotropy regarding angles ${\theta }^{\ast },{\varphi }^{\ast }$ in $W_{1+2+X\leftarrow A+B}$ (29) appear due to spin density matrix ${\rho }_{V+X\leftarrow A+B}^{M{M}^{\prime }}$ of the sub-reaction $V+X\leftarrow A+B$. This anisotropy is called alignment of the decay particles. In particular, if ${\rho }_{V+X\leftarrow A+B}^{M=M^{\prime }}\simeq 1/3$ and ${\rho }_{V+X\leftarrow A+B}^{M\ne M^{\prime }}\simeq 0$, then $V$-meson decay is isotropic. In this case the decay particles 1 and 2 move independently from the interactions in the initial sub-reaction $V+X\leftarrow A+B$. Thus for isotropic decay entanglement between particles 1 and 2 caused only by location of effective mass of subsystem $1+2$ in resonance region

$${\left(M_V-\frac{{\mathrm{\Gamma }}_V}{2}\right)}^2\le {\left(p_1+p_2\right)}^2\le {\left(M_V+\frac{{\mathrm{\Gamma }}_V}{2}\right)}^2.$$E30

Vertex (21) for composite $V$-meson resonances can be presented as

$$\begin{array}{lll}{\mathcal{A}}_{1+2\leftarrow V}& =& <\mathit{\text{out}};{\mathbf{p}}^{\ast },S_1^{\ast }{M}_1^{\ast };-{\mathbf{p}}^{\ast },S_2^{\ast }{M}_2^{\ast }\mid {{\mathcal{J}}^{\ast }}_{m_V{\mathbf{0}}_V}^{1M}\left(0\right)\mid 0>=\\ & =& {\delta }_{J^{\ast },1}\sum _{L^{\ast },S^{\ast }}{\mathcal{Y}}_{L^{\ast },S^{\ast }}^{\mathit{JM}}\left({\widehat{\mathbf{p}}}^{\ast }\right){\mathrm{F}}_{S^{\ast }{L}^{\ast }}^{J^{\ast }}\left(\sqrt{s_{12}}\right),\end{array}$$E31

where ${\mathcal{Y}}_{L^{\ast },S^{\ast }}^{\mathit{JM}}\left({\widehat{\mathbf{p}}}^{\ast }\right)$ and $Y_{L^{\ast }M}^{\ast }\left({\widehat{\mathbf{p}}}^{\ast }\right)$ are tensor spherical harmonics and spherical harmonics

$${\mathcal{Y}}_{L^{\ast },S^{\ast }}^{\mathit{JM}}\left({\widehat{\mathbf{p}}}^{\ast }\right)=\sum _{M_L^{\ast }}<L^{\ast }{M}_L^{\ast }{S}^{\ast }{M}_S^{\ast }\mid 1M>Y_{L^{\ast }M}^{\ast }\left({\widehat{\mathbf{p}}}^{\ast }\right){\chi }_{S^{\ast }{M}_S^{\ast }}\left(S_1^{\ast }{M}_1^{\ast },S_2^{\ast }{M}_2^{\ast }\right),$$E32

and

$$\begin{array}{lll}{\mathrm{F}}_{S^{\ast }{L}^{\ast }}^1\left(\sqrt{s_{12}}\right)& =& \left({\xi }_{\mu }\left(0,M\right){\mathbf{p}}^{\mathbf{\ast }\mu }\right)\int d^3{\mathbf{q}}^{\ast }{\widetilde{{\mathrm{F}}^{\left(1\right)}}}_{S^{\ast }{L}^{\ast }}\left({\mathbf{p}}^{\mathbf{\ast }},{\mathbf{q}}^{\ast },\sqrt{s_{12}}\right)\\ & & +\int d^3{\mathbf{q}}^{\ast }\left({\xi }_{\mu }\left(0,M\right){\mathbf{q}}^{\mathbf{\ast }\mu }\right){\widetilde{{\mathrm{F}}^{\left(2\right)}}}_{S^{\ast }{L}^{\ast }}\left({\mathbf{p}}^{\ast },{\mathbf{q}}^{\ast },\sqrt{s_{12}}\right)\end{array}$$E33

is the radial part of ${\mathcal{A}}_{1+2\leftarrow V}$ with spin $S^{\ast }$, orbital momentum $L^{\ast }$ and their projections $M_S^{\ast }$ and $M_L^{\ast }$.

Using (31)–(33) one can represent the cross section of reaction $1+2+X\leftarrow A+B$ in the form

$$\frac{d{\sigma }_{1+2+X\leftarrow A+B}}{d{\mathrm{\Omega }}_{1+2+X\leftarrow A+B}^{\ast }}=\sum _{M{M}^{\prime }}\left(\sum _{L^{\ast }{S}^{\ast }}{\left[{\mathcal{Y}}_{\mathit{JM}}^{L^{\ast }{S}^{\ast }}\left({\widehat{\mathbf{p}}}_{12}^{\ast }\right){\mathrm{F}}_{S^{\ast }{L}^{\ast }}^1\left(\sqrt{s_{12}}\right)\right]}^{+}\frac{d{\sigma }_{A+B\Rightarrow V+}^{M{M}^{\prime }}}{d{\mathrm{\Omega }}_{A+B\Rightarrow V+X}^{\ast }}\frac{{\sum }_{L^{\prime }{S}^{\prime }}\left[{\mathcal{Y}}_{J{M}^{\prime }}^{L^{\prime }{S}^{\prime }}\left({\mathbf{p}}_{12}^{\ast }\right){\mathrm{F}}_{S^{\prime }{L}^{\prime }}^1\left(\sqrt{s_{12}}\right)\right]}{{\left(M_V^2-s_{12}\right)}^2+M_V^2{\mathrm{\Gamma }}_V^2}\right),$$E34

Thus cross sections (34) and (7) with the amplitude of the composite and structureless $V$-meson resonance decay (31), (11) and (12) have different angular momentum dependence and different meson decay form factors.

5. Conclusion and outlook

In this paper we have discussed the entanglement of two hadrons and two leptons, which occur in high energy collision $1+2+X\leftarrow V+X\leftarrow A+B$ after $V$ meson resonance decay $1+2\leftarrow V$. For the most general reproduction of entanglement mechanisms, we used a field-theoretic construction of the resonance decay amplitude with and without quark-gluon degrees of freedom.

In the both cases the two-body decay of any resonance automatically forms EPR-pair-type system in their rest frame, where these particles have opposite momentums and the strongly correlating spin states.

Big difference in the leptonic and hadronic resonance decay of $V$-meson produce corresponding difference between form factors (11) and (12). In particular, the hadronic and leptonic decay of the point-like (structureless) $V$-meson contains form factors $g_{h\overline{h}-V}\left({\left(p_h+p_{\overline{h}}\right)}^2\right)$ (12) and $g_{\mathrm{\ell }\overline{\mathrm{\ell }}-V}\simeq \mathit{\text{const}}$ (11). There are even more differences between the hadron decay form factors (33) and (12) with and without quark-gluon degrees of freedom for the non-point-like (non-local) and point-like (local) resonances correspondingly. Even more difference appear between the hadronic decay form factors (12) and (33) point-like (local) and non-point-like (nonlocal) resonances.

If decay of the $V$-meson resonance is approximately isotropic ${\rho }_{11}={\rho }_{00}={\rho }_{-1-1}\simeq 1/3$, then decay of it is identical with the decay of the isolated resonance. In this case decay of the $V$-meson is independent on the influence of the initial subprocess $V+X\leftarrow A+B$. After that, decay $1+2\leftarrow V$ can be considered as the decay of a quasi-free $V$-meson resonance and the entanglement between particles 1 and 2 can be determined only by the resonant decay mechanism.

Nonlocality of hadrons or nuclears in quantum field theory is a consequence of the composite (nonlocal) structure of hadrons or nuclei. The composite hadrons are built through the interactions of local (point-like) fields of quarks and gluons, and the composite (nonlocal) structure of nuclei in nuclear physics is determined through the interaction of local (point-like) nucleons. The equal-time commutators of the composite field operators composite particles (15) or (19) and (20) are not equal to zero even in a space-like region.

Cross sections (7) are valid for both local and non-local resonant decay of the $V$mesonÑŽ. One can rewrite this expression as

$$\frac{d{\tilde{\sigma }}_{1+2\leftarrow V}^{M{M}^{\prime }}}{d{\mathrm{\Omega }}_{1+2\leftarrow V}}={\mathcal{A}}_{1+2\leftarrow V}^M{\left({\mathcal{A}}_{1+2\leftarrow V}^{\ast }\right)}^{M^{\prime }}$$E35

Representation of (35) in the form of a product of cross sections of the reactions $1+2\leftarrow V$ and $V+X\leftarrow A+B$ is a consequence of the separability of scattering amplitudes with a single-particle $s$-channel exchange in quantum field theory or in nonrelativistic scattering theory [38].

Thus the nonlocality of the hadrons in quantum field theory [25] differs from Bell nonlocality (see [5] eq. (3)), where nonlocality arises due to integration over a certain variables $\lambda$, “having a joint causal influence on both outcomes, and which fully account for the dependence between” 1 and 2.

This discrepancy with Bell nonlocality is a consequence of the two reasons. First, nonlocality in quantum field theory and nonrelativistic nuclear and collision theories is introduced via construction of the composite particle states through degrees of freedom of local, structureless particles. And Bell’s nonlocality is determined through integration over $\lambda$ of the local probability in the Bell formulation.

Secondly, EPR thought experiments and similar subsequent formulations imply independent measurements of the location of a single particle from an EPR pair in coordinate space based on the uncertainty principle Heisenberg. But in real experiments in elementary particle physics and nuclear physics, the momentum of particles and the states of these particles in the momentum space are measured. Determining the position of particles in coordinate space requires not only a relativistic generalization of the Heisenberg uncertainty principle, but also model-independent calculations of the trajectory of quantum particles and other tasks beyond the scope of this article.

Appendix A: relativistic $4D$ and $3D$ field-theoretical equations

Field-theoretical Bethe-Salpeter equations for 4D amplitude

$${\mathcal{A}}^{4D}\equiv {\mathcal{A}}^{4D}\left({p^o}_A{\mathbf{p}}_A,{p^o}_B{\mathbf{p}}_B;p_C^o{\mathbf{p}}_C,p_D^o{\mathbf{p}}_D\right)$$E36

and relativistic Lippmann-Schwinger-type equations for 3D amplitude

$${\mathcal{A}}^{3D}\equiv {\mathcal{A}}^{3D}\left({\mathbf{p}}_A,{\mathbf{p}}_B;{\mathbf{p}}_C,{\mathbf{p}}_D;\right)$$E37

describe reactions $C+D\leftarrow A+B$ in framework of the relativistic field theory taking into account the absorption and appearance of particles. In (A1) ${\mathcal{A}}^{4D}$ depends on the off shell four-momentum of each particle , $p_B$, $p_C$ and $p_D$. In (A2) ${\mathcal{A}}^{3D}$ is function of on mass shell momentums $p_A^{\mathit{on}}=\left(\sqrt{m_A^2+{\mathbf{p}}_A^2},{\mathbf{p}}_A\right)$, $p_B^{\mathit{on}}=\left(\sqrt{m_B^2+{\mathbf{p}}_B^2},{\mathbf{p}}_B\right)$, $p_C^{\mathit{on}}=\left(\sqrt{m_C^2+{\mathbf{p}}_C^2},{\mathbf{p}}_C\right)$ and $p_D^{\mathit{on}}=\left(\sqrt{m_D^2+{\mathbf{p}}_D^2},{\mathbf{p}}_D\right)$. Observed particles of the reaction $C+D\leftarrow A+B$ are on shell, where $p_k=p_k^{\mathit{on}}$ $\left(k=A,B,C,D\right)$ and $p_C^{\mathit{on}}+p_D^{\mathit{on}}=p_A^{\mathit{on}}+p_B^{\mathit{on}}$. On shell 4D and 3D amplitudes coincide

$${\mathcal{A}}_{\mathit{on}}^{4D}={\mathcal{A}}_{\mathit{on}}^{3D}$$E38

In case of spin less particles Bethe-Salpeter equation for ${\mathcal{A}}^{4D}$ (A1) in coordinate space have the form (see e.g. [31] ch. 10)

$${\mathcal{A}}^{4D}\left(x_C{x}_D;x_A{x}_B\right)={\mathcal{V}}^{4D}\left(x_C{x}_D;x_A{x}_B\right)$$

$$+\int d^4{x}_{\alpha }{d}^4{x}_{\beta }{\mathcal{V}}^{4D}\left(x_C{x}_D;x_{\alpha }{x}_{\beta }\right){\left(W_{x_{\alpha }}+m_{\alpha }^2+i\mathrm{ϵ}\right)}^{-1}{\left(W_{x_{\beta }}+m_{\beta }^2+i\mathrm{ϵ}\right)}^{-1}{A}^{4D}\left(x_{\alpha }{x}_{\beta };x_A,x_B\right),$$E39

where ${\mathcal{V}}^{4D}\left(x_C{x}_D;x_A{x}_B\right)$ consist of sum of all Feynman diagrams with all possible intermediate states excluding asymptotic two-body states $\alpha +\beta \equiv C+D,A+B$.

General form of Bethe-Salpeter equations are derived in framework of the method of functional integrals [31] and graphical summation method for Feynman diagrams [39, 40]. These equations depend on non-physical relative time variables $x_A^o-x_B^o$, $x_C^o-x_D^o$ which greatly complicates their application and requires additional approximations. Therefore, were proposed so called 3D quasi-potential reductions of Bethe-Salpeter equations, where they are replaced by a set of four-dimensional equations for the relation between the 4D and 3D potentials and 3D reduced equation for the sought amplitude, 3D potential (so called quasipotential of Bethe-Salpeter equation) and the 3D Green’s functions [41, 42, 43]. However, there are an infinite number of three-dimensional reductions of same Bethe-Salpeter equation which produce different 3D equations with different potentials, different 3D off shell amplitudes and different Green function. Another version of the 3D reduction of the Bethe-Salpeter equations is given in [44].

From the beginning 3D field-theoretical equations ${\mathcal{A}}^{3D}$ (A2) are derived within modified “old perturbation theory” [45, 46, 47] where all particles are being on shell. In this formulation arose the additional quasi-particle degrees of freedom, which enables to take into account off-shellness of the intermediate states.

Another type of initially three-dimensional field-theoretic equations [26, 33, 48] was obtained within the $S$-matrix reduction method [29, 30, 31]. In this case

$${\mathcal{A}}^{3D}=<\mathit{\text{out}};{\mathbf{p}}_A\mid j_B\left(0\right)\mid {\mathbf{p}}_c{\mathbf{p}}_D;\mathit{\text{in}}>$$E40

where $j_B\left(x\right)=\left(W_x+m_B^2\right){\varphi }_B\left(x\right)$ is the source operator of particle $B$_. In this approach final equations have the form of the relativistic Lippmann Schwinger-type equations

$$\begin{array}{lll}{\mathcal{T}}^{3D}\left({\mathbf{p}}^{\prime },\mathbf{p};P_o\right)& =& {\mathcal{U}}^{3D}\left({\mathbf{p}}^{\prime },\mathbf{p};P_o\right)\\ & & +\int {\mathcal{U}}^{3D}\left({\mathbf{p}}^{\prime },{\mathbf{p}}^{″};P_o\right)\frac{d^3{\mathbf{p}}_1^{″}{d}^3{\mathbf{p}}_2^{″}\delta \left(\mathbf{P}-{\mathbf{P}}^{"}\right)}{2E_{{\mathbf{p}}_1^{″}}2E_{{\mathbf{p}}_2^{″}}\left(P_o-E_{{\mathbf{p}}_1^{″}}-E_{{\mathbf{p}}_2^{″}}+i\mathrm{ϵ}\right)}{\mathcal{T}}^{3D}\left({\mathbf{p}}^{″},\mathbf{p};P_o\right),\end{array}$$E41

where $P_o=E_{{\mathbf{p}}_1}+E_{{\mathbf{p}}_2}$_,$\mathbf{P}={\mathbf{p}}_1+{\mathbf{p}}_2$ denote total energy and total momentum of the system $1+2$, $E_{{\mathbf{p}}_{1,2}}=\sqrt{{\mathbf{p}}_{1,2}^2+m_{1,2}^2}$, and $\mathbf{p}=1/2\left({\mathbf{p}}_1-{\mathbf{p}}_2\right)$ are the relative momentum of particles 1, 2 in the rest frame $\mathbf{P}={\mathbf{p}}_1+{\mathbf{p}}_2=0$.

Potential ${\mathcal{U}}^{3D}\left({\mathbf{p}}^{\prime },\mathbf{p};P_o\right)$ of (A6) consists of all three-dimensional time ordered diagrams with on mass shell particle exchanges in $u$, $\overline{u}$ and $\overline{s}$. Besides ${\mathcal{U}}^{3D}$ contains so called seagull term which produces off mass shell one-particle exchange term and overlapping (contact) terms.

On shell ${\mathcal{T}}_{\mathit{on}}^{3D}$ and ${\mathcal{A}}_{\mathit{on}}^{3D}$ coincides.

Unlike the Bethe-Salpeter equations and their 3D reductions, the potential of three-dimensional field-theoretic Eq. (A6) is determined by vertex functions and amplitudes $<\mathit{\text{out}};n\mid j_a\left(0\right)\mid m;\mathit{\text{in}}>$, which contain only one particle $a$ with off shell source operator $j_a\left(0\right)$. Therefore, in these three-dimensional equations, it is not required to make approximations which simplify off shell behavior of the vertex functions and amplitudes.

Note that taking into account the quark-gluon degrees of freedom does not change the form of the Eq. (41) for the amplitudes of hadron interactions.