POLARIZATION PHENOMENA IN PROCESS e e N N g

The emission of the hard photon from the initial state is considered. The nucleon polarization and the differential cross sections for some experimental conditions have been calculated. The case of the emission of the collinear (with respect to the direction of the electron beam momentum) photon is considered separately. The differential cross section, the nucleon polarization, the correlation coefficients for both polarized nucleons (provided the electron beam is unpolarized or longitudinally polarized), the transfer polarization from the longitudinally polarized electron beam to the nucleon have been calculated. The photon energy distribution for the reaction 1 2 e e h h + - ﬁ g , where 1 h and 2 h are some hadrons for the case of the collinear photon, emitted in the initial state, has been calculated. As 1 2 h h final state we considered some channels, namely: two spinless mesons (for example, K K + -+ - , pp ), two spin–one particles (for example, dd + - , r r ), and the channels 1 (1260) a p and (1232) N D . The photon energy distributions are calculated in terms of the form factors of the 1 2 h h * ﬁ g transition ( * g is the virtual photon).


INTRODUCTION
The investigation of the electromagnetic form factors of the proton and neutron in both the space-like and timelike regions of the momentum transfer squared is important for the understanding of the internal structure of these particles and for the interpretation of many data on reactions with participation of the nucleons. The knowledge of the nucleon form factors is also required for the interpretation of the nuclear structure and various measurements of the reactions involving nuclei. So, the experimental determination of the elastic nucleon electromagnetic form factors in the region of small and large momentum transfer squared is one of the major fields of research in hadron physics (see the review [1]).
The measurement of the nucleon electromagnetic form factors in the space-like region of the momentum transfer squared has a long history. The electric and magnetic form factors were determined both for the proton and neutron using two different techniques: the Rosenbluth separation [2,3] and polarization transfer method [4,5,6,7]. It turned out that the measurements of the ratio of the proton electric and magnetic form factors using these two methods lead to the appreciably different results, and this difference is increasing when 2 Q (the four-momentum transfer squared) grows. The ratio p p E M G G / is monotonically decreasing with increasing 2 Q suggesting crossing zero at 2 2 (8 9) Q GeV ≈ − [8]. These unexpected results revived an experimental and theoretical investigations of this problem (see reviews [1,9]). One possible mechanism suggested for the explanation of this discrepancy is the two-photonexchange contribution to the elastic electron-nucleon scattering [10,11]. Other considerations lead to the conclusion that the contribution from the two-photon term is too small at the 2 Q values of interest [12] and/or lead to a definite non-linearity in the Rosenbluth plot which has not been seen in the data so far [13]. A model independent study of the two-photon-exchange mechanism in the elastic electron-nucleon scattering and its consequences on the experimental observables has been carried on in Refs. [14,15,16], and in the crossed channels: proton-antiproton annihilation into the lepton pair [17] and annihilation of the electron-positron pair into the nucleon-antinucleon [18].
The data on nucleon form factors in the time-like region are not numerous. So, the separation the electric and magnetic form factors in this region has not yet been done. One of the reasons is the limitation in the intensity of antiproton beams and of the luminosity of electron-positron colliders.
Nevertheless, a few unexpected results have been observed in the measurements of the nucleon form factors in the time-like region (note that the accuracy of the data set is not sufficiently good to do definite statements). Despite of the relation E M G G | |=| | , which must be valid at the threshold of the e e NN + − → reaction, the neutron electric form factor is negligible near the threshold as may be suggested from the measurement of the differential cross section. The general behavior of the neutron time-like form factors is rather unexpected. The proton magnetic form factor is smaller than the neutron one at 2 2 6 q GeV ≤ (where experiments were done). The review of the present status in this field of investigations is given in [1]. Note also that in the time-like region the proton magnetic form factor is considerably bigger than the corresponding space-like quantity.
Recent experimental data on the nucleon form factors (both in the space-and time-like regions) together with new theoretical developments [19] (where the analytic continuation of the QCD formulas from the space-like region of momentum transfer to the time-like one was discussed) show the necessity of a global description of the nucleon form factors in the full region of the 2 q variable. Some papers were already appeared [20,21,22]. The experimental data on the time-like form factors may turn out to be very sensitive to the details of existing models. For example, the analysis, performed in Ref. [23], taking into account the combined space-like and time-like data on the proton and neutron form factors leads to a good fit to the space-like form factors but cannot describe neutron time-like data.
So, the experimental investigation of the nucleon form factors in the time-like region may give additional valuable information about the internal nucleon structure and can test the existing models.
In the time-like region, the nucleon form factors can be measured using the reactions e e NN + − → or pp e e + − → .
In this region only a small set of data exists. [25].
Some experiments are planning to study this region of 2 q . A new experiment at an asymmetric collider is proposed at SLAC with the ambitious goal to measure the nucleon form factors from threshold up to 3 GeV with the same accuracy currently available in the space-like region [26].
The chance of measuring these form factors with higher precision will be given by a suitable upgrade of DA NE Φ energy [27]. The number of good detectable events per day is about, or exceeding, the total amount of events collected by FENICE in all its data takings. FINUDA planned to offer a unique possibility -a measurement of the nucleon polarization [27]. This kind of measurement would be of great interest, as it would be a handle to infer something about the relative phases of M G and E G form factors.
As it is known, the e e NN + − → reaction is the cross channel for the reaction of the elastic electron-nucleon scattering. The form factors describing the annihilation channel are assumed to be the analytical continuation of the space-like ones. So, one may expect that the problems existed in the scattering channel will also manifest itself in the annihilation channel. It concerns, in particular, the problem of the two-photon-exchange contribution.
Theoretically, the reaction e e NN + − → was studied in a number of papers. The dependence of polarization states of created one-half spin baryons in the e e BB + − → reaction on the polarization of colliding e e + − -beams was investigated in Ref. [28]. The formulae obtained in this paper exhaust all polarization effects of baryons with spin 1/2 in the e e BB + − → reaction. Numerical estimates of polarization effects were presented only for the nucleons. The polarization effects appear to be very sensitive to the choice of the nucleon form factors parametrization and are rather large in absolute value. The pronounced energy dependence measured in the cross section of the reactions e e pp + − → investigated in Ref. [29] in the near-threshold region. The authors considered the role of the antinucleon-nucleon interaction in the initial-or final state using NN potential derived within chiral effective field theory.
The existence of the T-odd single-spin asymmetry normal to the reaction scattering plane requires a non-zero phase difference between the electric and magnetic form factors. The measurement of the polarization of one of the outgoing nucleons allows to determine the phase of the ratio E M G G / . In Ref. [30] it was shown that measurements of the proton polarization in e e pp + − → reaction strongly discriminate between the analytic forms of models suggested to fit the proton data in the space-like region.
As it is known, the problem of taking into account the radiative corrections in the elastic electron-nucleon scattering is important for the reliable extraction of the nucleon form factors. The same is valid for the crossed channel. The importance of the e e N N − + + → + + γ reaction is not only due to the fact that it is a part of the radiative corrections to the e e N N − + + → + reaction but rather because it allows to measure the nucleon form factors by the radiative return method [31].
In this paper we investigate the polarization phenomena in the reaction where four-momenta of the corresponding particles are given in the brackets. We consider here the emission of the additional hard photon by the initial electron or positron since the emission of the photon by the final state particles is model dependent and suppressed with respect to the initial state radiation due to the large nucleon mass as compared with electron one and perhaps by the nucleon form factors.
Here we derive the expressions for the differential cross section and various polarization observables taking into account the nucleon form factors.
We consider a particular case of the high-energy photon emission at small angles (the radiative return). The differential cross section and various polarization observables (the nucleon polarization, the correlation coefficients for the nucleon-antinucleon pair and polarization transfer from the longitudinally polarized electron to the nucleon), when the angular distribution of the nucleon and energy of the emitted photon are measured, have been calculated for the case of the photon emitted at small angles relative to the electron beam momentum.
The standard analysis of the experimental data requires the account for all possible systematic uncertainties. One of the important source of such uncertainties are the electromagnetic radiative effects caused by physical processes which take place in higher orders of the perturbation theory with respect to the electromagnetic interaction. In present paper we calculate the model-independent QED radiative corrections to the observables (both polarized and unpolarized). Our approach is based on the covariant parametrization of the nucleon or antinucleon spin four-vectors in terms of the four-momenta of the particles in process (1) [32,33].
The photon energy distribution for the reaction + → + are given. In Section "HARD-PHOTON EMISSION" the emission of the hard photon by the initial state is considered. The nucleon polarization and the differential cross sections for some experimental conditions have been calculated. In Section "RADIATIVE RETURN. SMALL ANGLES" the emission of the collinear photon is considered in details. The differential cross section and various polarization observables have been calculated. In Section "PHOTON ENERGY DISTRIBUTION" we have calculated the photon energy distribution for the reaction Let us consider first the production of NN -pair without emission of additional photons: where four-momenta of the corresponding particles are given in the brackets. The matrix element of this reaction can be written as follows q k k p p = + = + is the virtual photon four-momentum. The leptonic and hadronic currents can be written as Then, the differential cross section of the e e N N − + + → + reaction, for the case of the polarized electron beam and unpolarized positron beam, can be written as follows in the reaction centre of mass system (CMS) where β is the nucleon velocity in CMS, 2 2 1 4M q = − / , β and the leptonic and hadronic tensors are defined as The leptonic tensor for the case of longitudinally polarized electron beam has the form (other components of the electron polarization lead to the observables suppressed by a factor m M / , where m is the electron mass) where ab a b < >= µνρσ ρ σ µν ε and e λ is the degree of the electron longitudinal polarization (we use the following definition for the antisymmetric tensor 1230 1 = ε ). Taking into account the polarization states of the produced nucleon and antinucleon, the hadronic tensor can be written as a sum of four contributions as follows: where the tensor (0) H µν describes the production of unpolarized nucleon and antinucleon, the tensor (1)( (1)) H H µν µν describes the production of polarized nucleon (antinucleon) and the tensor (2) H µν corresponds to the production of polarized particles, nucleon and antinucleon. Let us consider the production of unpolarized NN − pair as a result of annihilation of unpolarized e e + − − pair. In this case the general structure of the hadronic tensor can be written as One can get the following expressions for these structure functions for the case of the hadronic current given by Eq. (4) Then, the contraction of the leptonic and hadronic tensors, in the case of unpolarized initial beams and produced nucleon and antinucleon, can be written as The differential cross section of the e e N N − + + → + reaction, for the case of unpolarized particles, has the form in CMS π α β σ (13) Now, let us consider the single polarization observables. To do this, it is necessary to calculate the hadronic tensor for the case when produced nucleon is polarized. We can write this tensor as a sum of two terms: one is symmetrical and another one is antisymmetrical (over µ and ν indices) θ The polarization 4-vector 1 s µ of a nucleon in the system where it has momentum p is connected with the polarization vector 1 χ in its rest frame by a Lorentz boost Let us note that four-vector 1 s µ can be written down as  Note that the polarization 4-vectors of the particles can be parameterized in terms of the four-momenta of these particles in the reaction under study (it is very convenient when calculating the radiative corrections to this reaction).
Let us write the chosen axes in a covariant form in terms of the four-momenta. So, in the reaction CMS we choose the longitudinal direction l ( z axis) along the nucleon momentum and the transverse one t in the plane ( ) p k , ( x axis) and perpendicular to l ( y axis), then It can be verified that the set of the four-vectors ( ) l t n P , , µ has the properties (17) and that in the reaction CMS we have It is easy to show that the following relations are valid Note that, unlike the elastic electron-nucleon scattering in the Born approximation, the hadronic tensor (1) H µν in the time-like region contains the symmetric part even in the Born approximation due to the complexity of the nucleon form factors. So, this term leads to the non-zero polarization of the outgoing nucleon (the initial state is unpolarized) in the e e N N − + + → + reaction and it can be written as θ τ (20) This expression gives the well known result for the polarization y P obtained in Ref. [28]. One can see also that: -The polarization of the outgoing nucleon, in this case, is determined by the polarization component which is perpendicular to the reaction plane.
-The polarization, being T-odd quantity, does not vanish even in the one-photon-exchange approximation due to the complexity of the nucleon form factors in the time-like region (to say more exactly, due to the non-zero difference of the phases of these form factors). This is principal difference with the elastic electron-nucleon scattering.
-In the Born approximation this polarization becomes equal to zero at the scattering angle In the threshold region we can conclude that in the Born approximation this polarization must be zero due to the relation E which is valid at the threshold. If one of the colliding beam is longitudinally polarized then nucleon acquires x − and z − components of the polarization, which lie in the e e N N + − + → + reaction plane. These components can be written as (we assume 100% polarization of the electron beam) θ θ τ (21) These polarization components are T-even observables and they are non-zero in the Born approximation even for the elastic electron-nucleon scattering. Note that in the Born approximation we obtain the result of Ref. [28]. The polarization component z P equals to zero at the scattering angle 0 90 = θ in the Born approximation. Transversally polarized electron beam leads to the nucleon polarization which is smaller by factor ( ) m M / than for the case of the longitudinal polarization of the electron beam.
Let us consider the case when the produced antinucleon and nucleon are both polarized. The corresponding hadronic tensor can be written as  ( 1) Using previous formulae one can obtain the following expressions for the components of the polarization correlation tensor ( ) ik P i k x y z , = , , of the nucleon and antinucleon, created by the one-photon-exchange mechanism in the e e N N + − + → + process: where the first index of the tensor ik P refers to the component of the nucleon polarization vector, whereas the second index refers to the component of the antinucleon polarization vector.
The antinucleon polarization four-vector, 2 s µ , is described by the formula (15) where it is necessary to do the following substitution: p p → − and 1 2 → χ χ ( 2 χ is the polarization vector of the antinucleon in its rest frame). The antinucleon polarization 4-vectors ( ) i i l t n P , = , , µ (in terms of the particles four-momenta) can be written down as It is easy to show that the following relations are valid And for the completeness we give here the non-zero coefficients for the case of the longitudinally polarized electron beam 0 ( ) θ τ (27) The following relation exists for these coefficients 1 xx yy zz P P P + + = .
One can see that: -The components of the tensor describing the polarization correlations xx P , yy P , zz P , xz P , and zx P are the T-even observables, whereas the components yz P , and zy P are the T-odd ones.
-In the Born approximation the expressions for the T-odd polarization correlations coincide with the corresponding components of the polarization correlation tensor of baryon B and antibaryon B created by the one-photon-exchange mechanism in the e e B B + − + → + process [28]. The expressions for the T-even polarization correlations calculated in this paper have some misprints. πα πα (28) where the leptonic current with emission of additional photon has the form

HARD-PHOTON EMISSION
where 1 where the hadronic tensor has the same form as in the Born approximation but the structure functions defining this tensor depend on the shifted momentum transfer 1 2 q k k k = + − .
Let us represent the leptonic tensor L γ µν as a sum of the spin-independent and spin-dependent part (we consider only the case of the longitudinally polarized electron beam) where the spin-independent part of this tensor can be written as The spin-independent part of the leptonic tensor (0) L γ µν coincides with the one obtained in Ref. [34] and if we neglect Let us consider the case of unpolarized initial beams and when final state is unpolarized or the final nucleon has polarization. Then the contraction of the spin-independent leptonic tensor and hadronic tensor which corresponds to the polarized nucleon can be written as The contraction of the unpolarized lepton tensor and the hadron tensor corresponding to the polarized nucleon has the form Fig. 1. The angles defining the kinematics of the reaction (1) in its CMS.  Let us choose in the reaction CMS the following coordinate system: z axis is directed along the nucleon momentum 1 p , the momentum of the initial electron beam 1 k forms the xz plane (the angle between these two momenta is ϑ ), y axis is directed along the vector 1 1 k p × The momentum of the emitted photon k is defined by the polar and azimuthal angles, γ ϑ and γ φ , respectively. The angles defining the kinematics of the reaction (1) in its CMS are given in Fig.1.
Then the cross section of the process (1) can be written as where 1 2 ( ) E E and ω are the energies of the nucleon (antinucleon) and photon, respectively; 2 p is the antinucleon momentum.
On the basis of this expression we can obtain the different distributions depending on the experimental conditions. If we measure the nucleon scattering angle and variables of the emitted photon, we can obtain the following distribution The polar angle of the emitted photon γ ϑ can be also expressed in terms of the energies of the final hadrons. We have Let us parameterize the nucleon spin four-vector 1 s µ in terms of the four-momenta of the particles participating in the reaction under study. When measuring the polarization of the produced particle the z -axis is usually chosen along the momentum of this particle. So, in the reaction CMS we choose the longitudinal direction l ( z axis) along the nucleon momentum and the transverse one t in the plane 1 1 ( ) p k , and perpendicular to l , and denote these polarization fourvectors as ( ) where for the coefficients i L we have ω ω ϑ Let us integrate the expressions ( ) 3 i Q i l t n , = , , , over the angular variable γ φ . We have the following integrals If we integrate over the whole possible region of the angle γ φ , i.e., over (0 2 ) , π , we have that all these integrals are equal to zero. So, in this case only perpendicular (to the reaction plane) polarization of the nucleon gives nonzero contribution (as well as in the Born approximation). We have the following integrals So, we have after integration over the angle γ φ ( ) ( ) 12 3 0 1

RADIATIVE RETURN. SMALL ANGLES
Since the main contribution, proportional to the large logarithm, comes from the integration of the integrand in the case of collinear kinematics of photon emission, we consider this case. For definiteness let us consider the case when the emitted photon moves close to the initial electron direction: The differential cross section can be written as Integrating the leptonic tensor over the photon angular variables we obtain the following result for the case of unpolarized initial beams 2 ( ) 2 1 1 2 where the factor ( ) F L x , has the form [36] After the integration over the photon variables the differential cross section can be written as where the factor ( ) D x is The nucleon energy 1 E , the scattering angle ϑ and x variable are connected by the following relation 1 1 ϑ Using this relation we can determine the nucleon energy as a function of two variables: x and cosϑ . We have When calculating the radiative corrections to the polarization observables it is convenient to parameterize the nucleon polarization 4-vector in terms of the four-momenta of the particles participating in the reaction under study. Any fourvector ( ) i U µ which parameterize the polarization state of the particle can be read as PHOTON ENERGY DISTRIBUTION Let us calculate the differential cross section of the reaction (1) for the experimental conditions when only the energy of the collinear photon is measured. To do this, it is necessary to calculate the quantity d dx / γ σ , i.e., we have to integrate the differential cross section (62) over the nucleon angular variables. The invariant integration of the hadronic tensor is the simplest method to do this.
Let us define the following quantity