Radiative corrections to polarization observables in elastic electron-deuteron scattering in leptonic variables

# Radiative corrections to polarization observables in elastic electron-deuteron scattering in leptonic variables

G. I. Gakh    M. I. Konchatnij    N. P. Merenkov Kharkov Institute of Physics and Technology
###### Abstract

The model–independent QED radiative corrections to polarization observables in elastic scattering of unpolarized and longitudinally–polarized electron beam by the deuteron target have been calculated in leptonic variables. The experimental setup when the deuteron target is arbitrarily polarized is considered and the procedure for applying derived results to the vector or tensor polarization of the recoil deuteron is discussed. The basis of the calculations consists of the account for all essential Feynman diagrams which results in the form of the Drell-Yan representation for the cross-section and use of the covariant parametrization of the deuteron polarization state. The numerical estimates of the radiative corrections are given for the case when event selection allows the undetected particles (photons and electron-positron pairs) and the restriction on the lost invariant mass is used.

## I Introduction

The process of the elastic electron-deuteron scattering has been a long time a reaction which is used for the investigation of the electromagnetic structure of the deuteron. These investigations, both theoretical and experimental, can help to clarify a number of the important problems: the properties of the nucleon-nucleon interaction, non-nucleonic degrees of freedom in nuclei (such as the meson exchange currents, the isobar configurations), as well as the importance of the relativistic effects (see, for example, the recent reviews on the deuteron G99 (); GO02 (); S02 (); GR02 ()).

The electromagnetic structure of the deuteron as a bound two-nucleon system with spin-one is completely determined by three functions of one variable, the four-momentum transfer squared . These are the so-called electromagnetic form factors of the deuteron: (the charge monopole), (the magnetic dipole) and (the charge quadrupole). They are real functions in the space-like region of the four-momentum transfer squared (the scattering channel as, for example, the elastic electron-deuteron scattering) and complex functions in the time-like region (the annihilation channel as, for example, ). So, the main experimental problem is to determine the electromagnetic deuteron form factors with a high accuracy and in a wide range of the variable . A recent review of past and future measurements of the elastic electromagnetic deuteron form factors is given in Ref. K08 ().

Note also that the deuteron is used as an effective neutron target in studies of the neutron electromagnetic form factors PB95 () and the spin structure functions of the neutron in the deep-inelastic scattering Ac97 ().

The knowledge of the deuteron electromagnetic properties means that we can calculate its form factors from the first principles. To do it, in the non relativistic approach, we need the deuteron wave function and form factors of the nucleon. The last ones are known from the analysis of the experimental data on the elastic -scattering. It was found that for the agreement with experiment at small momentum transfers it is sufficient to take into account one-body current. The meson exchange currents and the isobar configurations become significant at large momentum transfers. The manifestation of the quarks inside the deuteron was not found at present. Note that each deuteron form factor may be sensitive to some specific contribution. Thus, for example, the deuteron charge form factor is particularly interesting for the understanding of the role of the meson exchange currents. So, it is necessary to separate the three deuteron form factors. Measurements of the unpolarized cross section yield the structure functions and : they can be separately determined by variation of the scattered electron angle for a given momentum transfer squared to the deuteron. So, all three form factors can be separated, when either the tensor analyzing power or the recoil deuteron polarization is also measured (in both cases the electron beam is unpolarized). This has prompted development of both polarized deuterium targets for use with internal or external beams and polarimeters for measuring the polarization of recoil hadrons FL98 (). Both types of experiment result in the same combinations of form factors.

The measurement of the polarization observables in elastic ed-scattering can be done at present with the help of the internal or external targets.

1. The internal targets were used at storage rings in form of the polarized deuteron gas targets D85 (); G90 (); FL96 (); B99 (); N03 (). In order to get the required luminosity it is necessary to have high-intensity electron beam since the density of such targets is very small.

2. In the experiments on elastic ed-scattering with external targets, the measurement of the polarization of the scattered deuteron is used S84 (); G94 (); A00 (). In this case the high-intensity electron beam is also necessary since the polarization measurement requires polarimeter. This procedure leads to the second scattering which decreases the event number very essentially.

Current experiments at modern accelerators reached a new level of precision and this circumstance requires a new approach to data analysis and inclusion of all possible systematic uncertainties. One of the important source of such uncertainties is the electromagnetic radiative effects caused by physical processes which take place in higher orders of the perturbation theory with respect to the electromagnetic interaction.

While the radiative corrections have been taken into account for the unpolarized cross section, the radiative corrections for the polarization observables in the elastic electron-deuteron scattering at large momentum transfer are not known at present TGR (). Thus, for example, in the experiment on precise measurement of the deuteron elastic structure function (at 0.66-1.8 GeV), the radiative corrections (about 20 ) due to loses in the radiative tail were calculated according to the paper MT69 (). On the other side, the authors of recent experiments B99 (); N03 (); A00 () on measuring the polarization observables did not present the evidence about taking into account the radiative corrections.

The importance of the taking into account the radiative corrections can be seen on the example of the discrepancy between the Rosenbluth ROS () and the polarization transfer methods AR () for determination of the ratio of the electric to magnetic proton form factors. For a given value of , it is sufficient to measure the unpolarized elastic electron-nucleon scattering cross section for two values of (virtual photon polarization parameter) to determine the and form factors (the Rosenbluth method). But the measurement of the polarization observables in this reaction (using the longitudinally polarized electron beam) allows to determine the ratio to AR (). Two experimental set ups were used, namely: measurement of the asymmetry on the polarized target or measurement of the recoil-proton polarization (the polarization transfer method).

Recent experiments show that the extracted ratio , using the Rosenbluth and polarization transfer methods, are incompatible at large AND (); JG (). This discrepancy is a serious problem as it generates confusion and doubt about the whole methodology of lepton scattering experiments GV (). One plausible explanation of this problem is two-photon exchange effects T2 (). The data are consistent with simple estimates of the two-photon contributions to explain the discrepancy (see, e.g., ABM () and references therein).

The precise calculation of the radiative corrections is also important for the study of the two-photon exchange effects in the elastic electron-deuteron scattering. Earlier it was observed Gu73 (); Fr73 (); Bo73 (); L75 () that the relative role of the two-photon exchange can increase significantly in the region of large due to the steep decrease of the deuteron form factors as functions of the variable. Since one- and two-photon amplitudes have very different spin structures, the polarization phenomena have to be more sensitive to the interference effects than the differential cross section (with unpolarized particles).

An attempt to evaluate the presence of two intermediate hard photon in box diagrams using the existing data on the elastic electron-deuteron scattering was done in Ref.RTP (). The authors searched a deviation from the linear dependence in of the cross section, using a Rosenbluth fit, which has been parameterized in a model independent way according to crossing symmetry considerations.

The recent calculations of the two-photon contribution to the structure functions and polarization observables in the elastic scattering of longitudinally polarized electron on polarized deuteron have been done in Ref. KKDD () (the references on earlier papers can be found here).

The radiative corrections to deep-inelastic scattering of unpolarized and longitudinally polarized electron beam on polarized deuteron target was considered in Ref. AS () for the particular case of the deuteron polarization (which can be obtained from the general covariant spin-density matrix CLS () when spin functions are the eigenvectors of the spin projection operator). The leading-log model-independent radiative corrections in deep-inelastic scattering of unpolarized electron beam off the tensor polarized deuteron target have been considered in Ref. GM (). The calculation was based on the covariant parametrization of the deuteron quadrupole polarization tensor and use of the Drell-Yan like representation in electrodynamics. The model-independent QED radiative corrections to the polarization observables in the elastic scattering of the unpolarized and longitudinally polarized electron beam by the polarized deuteron target in the hadronic variables have been done in Ref. GM04 ().

In present paper we calculate the model-independent QED radiative corrections in leptonic variables to the polarization observables in the elastic scattering of unpolarized and longitudinally-polarized electron beam by deuteron target

 e−(k1)+D(p1)→e−(k2)+D(p2), (1)

where the four-momenta of the corresponding particles are indicated in the brackets. The experimental setup when the deuteron target is arbitrarily polarized is considered and the procedure for applying derived results to the vector or tensor polarization of the recoil deuteron is discussed. The basis of the calculations consists of the account for all essential Feynman diagrams which results in the form of the Drell-Yan representation for the cross section and use of the covariant parametrization of the deuteron polarization state. The numerical estimates of the radiative corrections are given for the case when event selection allows the undetected particles (photons and electron-positron pairs) and the restriction on the lost invariant mass is used.

## Ii Born approximation

From the theoretical point of view, different polarization observables in the process of the elastic electron-deuteron scattering have been investigated in many papers (see, for example, Refs. GP (); G (); M (); ons (); SRG (); Arn (). The polarization observables were expressed in terms of the deuteron electromagnetic form factors. An up-to-date status of the experimental and theoretical research into the deuteron structure can be found in reviews GO02 (); GR02 (). Here, we reproduce most of these results using the method of covariant parametrization of the deuteron polarization state in terms of the particle four-momenta and demonstrate the advantage of such approach.

Consider the process of elastic scattering of polarized electron beam by polarized deuteron target. In the one-photon-exchange approximation, we define the cross section of the process (1), in terms of the contraction of the leptonic and hadronic tensors (we neglect the electron mass wherever possible), as follows

 dσ=α22Vq4LBμνHμνd3k2ε2d3p2E2δ(k1+p1−k2−p2) , (2)

where and are the energies of the scattered electron and recoil deuteron, respectively, and is the four-momentum of the heavy virtual photon that probes the deuteron structure. In the case of longitudinally polarized electron beam we have for the leptonic tensor, in the Born approximation, following expression

 LBμν=q2gμν+2(k1μk2ν+k2μk1ν)+2iPe(μνqk1) , (3)
 (μνab)=εμνλρaλbρ , ε1230=1,

where is the degree of the electron beam polarization (further we assume that the electron beam is completely polarized and consequently ).

The hadronic tensor can be expressed via the deuteron electromagnetic current , describing the transition , as

 Hμν=JμJ∗ν . (4)

Using requirements of the Lorentz invariance, current conservation, parity and time-reversal invariances of the hadron electromagnetic interaction, the general form of the electromagnetic current for the spin-one deuteron is completely described by three form factors and it can be written as AR ()

 Jμ=(p1+p2)μ[−G1(Q2)U1⋅U∗2+ G3(Q2)M2(U1⋅qU∗2⋅q−q22U1⋅U∗2)]+ G2(Q2)(U1μU∗2⋅q−U∗2μU1⋅q), (5)

where and are the polarization four-vectors for the initial and final deuteron states, is the deuteron mass. The functions are the deuteron electromagnetic form factors depending only upon the virtual photon four-momentum squared. Due to the current hermiticity the form factors are real functions in the region of the space-like momentum transfer. We use here the convention

These form factors can be related to the standard deuteron form factors: (the charge monopole), (the magnetic dipole) and (the charge quadrupole). These relations are

 GM=−G2, GQ=G1+G2+2G3, GC=23η(G2−G3)+(1+23η)G1,  η=Q24M2. (6)

The standard form factors have the following normalization:

 GC(0)=1, GM(0)=(M/mn)μd, GQ(0)=M2Qd,

where is the nucleon mass, is deuteron magnetic (quadrupole) moment and their values are: MT (), ERC ().

If we write down the electromagnetic current in the following form , then the tensor can be written as

 Hμν=JμαβJ∗νσγρiασρfγβ, (7)

where is the spin-density matrix of the initial (final) deuteron.

Since we consider the case of a polarized deuteron target and unpolarized recoil deuteron, the hadronic tensor can be expanded according to the polarization state of the initial deuteron as follows:

 Hμν=Hμν(0)+Hμν(V)+Hμν(T), (8)

where the spin-independent tensor corresponds to the case of unpolarized initial deuteron and the spin-dependent tensor describes the case when the deuteron target has vector (tensor) polarization.

We consider the general case of the initial deuteron polarization state which is described by the spin-density matrix. We use the following general expression for the deuteron spin-density matrix in the coordinate representation 1 ()

 ρiαβ=−13(gαβ−p1αp1βM2)+i2M(αβsp1)+Qαβ , (9)

where is the polarization four-vector describing the vector polarization of the deuteron target and is the tensor describing the tensor (quadrupole) polarization of the initial deuteron . In the laboratory system (initial deuteron rest frame) all time components of the tensor are zero and the tensor polarization of the deuteron target is described by five independent space components . In the Appendix B we give the relation between the elements of the deuteron spin-density matrix in the helicity and spherical tensor representations and the ones in the coordinate representation. We give also the relation between the polarization parameters and the population numbers and describing the polarized deuteron target which is often used in the spin experiments.

In this paper we assume that the polarization of the recoil deuteron is not measured. So, its spin-density matrix can be written as

 ρfαβ=−(gαβ−p2αp2βM2).

The spin-independent tensor describes unpolarized initial and final deuterons and it has the following general form

 Hμν(0)=−W1(Q2)~gμν+W2(Q2)M2~p1μ~p1ν , (10)
 ~gμν=gμν−qμqνq2 ,  ~p1μ=p1μ−p1⋅qq2qμ .

Two real structure functions have the following expressions in terms of the deuteron electromagnetic form factors

 W1(Q2)=23Q2(1+η)G2M, W2(Q2)=4M2(G2C+23ηG2M+89η2G2Q). (11)

In the considered case the spin-dependent tensor , that describes the vector polarized initial deuteron and unpolarized final deuteron, can be written as

 Hμν(V)=iMS1(μνsq)+iM3S2[~p1μ(νsqp1)− (12)
 −~p1ν(μsqp1)]+1M3S3[~p1μ(νsqp1)+~p1ν(μsqp1)],

where three real structure functions can be expressed in terms of the deuteron electromagnetic form factors. They are

 S1(Q2)=M2(1+η)G2M, S3(Q2)=0, S2(Q2)=M2[G2M−2(GC+η3GQ)GM]. (13)

The third structure function is zero since deuteron form factors are real functions in the elastic scattering (space-like momentum transfers). In the time-like region of the momentum transfers (for annihilation processes, for example, ), where the form factors are complex functions, the structure function is not zero and it is determined by the imaginary part of the form factors, namely: (in this case is the square of the virtual photon four-momentum).

In the case of tensor-polarized deuteron target the general structure of the spin-dependent tensor can be written in terms of five structure functions as follows

 Hμν(T)=V1(Q2)¯Q~gμν+V2(Q2)¯QM2~p1μ~p1ν+ V3(Q2)(~p1μ˜Qν+~p1ν˜Qμ)+V4(Q2)˜Qμν + iV5(Q2)(~p1μ˜Qν−~p1ν˜Qμ), (14)

where we introduce the following notations

 ˜Qμ=Qμνqν−qμq2¯Q ,  ˜Qμqμ=0 ,
 ˜Qμν=Qμν+qμqνq4¯Q−qνqαq2Qμα−qμqαq2Qνα,
 ˜Qμνqν=0, ¯Q=Qαβqαqβ. (15)

The structure functions , which describe the part of the hadronic tensor due to the tensor polarization of the deuteron target, have the following form in terms of the deuteron form factors

 V1(Q2)=−G2M,  V5(Q2)=0,
 V2(Q2)=G2M+41+η(GC+η3GQ+ηGM)GQ,
 V3(Q2)=−2η[G2M+2GQGM],
 V4(Q2)=4M2η(1+η)G2M, . (16)

The fifth structure function is zero since deuteron form factors are real functions in the considered kinematical region. In the time-like region of momentum transfers this structure function is not zero and it is determined by the expression (in this case is the square of the virtual photon four-momentum).

Using the definitions of the cross-section (2) as well as leptonic (3) and hadronic (8) tensors, one can easily derive the expression for the unpolarized differential cross section (in the Born (one-photon-exchange) approximation) in terms of the invariant variables suitable for the calculation of the radiative corrections

 dσunBdQ2=πα2VQ4{2ρW1+W2τ[1−ρ(1+τ)]} , (17)
 ρ=Q2V,  τ=M2V.

In the laboratory system this expression can be written in a more familiar form using the standard structure functions and . Thus, the unpolarized differential cross section for elastic electron-deuteron scattering takes the form

 dσunBdΩ=σM{A(Q2)+B(Q2)tan2(θe2)}, (18)
 σM=α2E′cos2(θe2)4E3sin4(θe2),

where is the Mott cross section. Here and are the incident and scattered electron energies, is the electron scattering angle

 E′=E[1+2(E/M)sin2(θe/2)]−1, Q2=4EE′sin2(θe/2).

The scattering angle in laboratory system can be written in terms of invariants

 cosθe=1−ρ−2ρτ1−ρ, sinθe=2√ρτ(1−ρ−ρτ)1−ρ.

Two structure functions and are quadratic combinations of three electromagnetic form factors describing the deuteron structure

 A(Q2)=G2C(Q2)+89η2G2Q(Q2)+23ηG2M(Q2), (19)
 B(Q2)=43η(1+η)G2M(Q2).

From Eq. (18) one can see that the measurement of the unpolarized cross section at various values of the electron scattering angle and the same value of allows to determine the structure functions and . Therefore, it is possible to determine the magnetic form factor and the following combination of the form factors . So, the separation of the charge and quadrupole form factors requires the polarization measurements.

Before to write similar distributions for the scattering of polarized particles, let us note that for such experimental conditions there may exists, in the general case, the azimuthal correlation between the reaction (electron scattering) plane and the plane if the initial deuteron is polarized. But in the Born approximation, taking into account the P- and T-invariance of the hadron electromagnetic interaction, such correlation is absent. Further in this section we consider the situation when the polarization 3-vector belongs to the reaction plane and corresponding azimuthal angle equals to zero. Therefore, there exist only two independent components of the polarization vector which we call as longitudinal and transverse ones.

To calculate the radiative corrections to the polarization observables it is convenient to parameterize the polarization state of the target (in our case it is the deuteron polarization four-vector (describing the deuteron vector polarization) and quadrupole polarization tensor (describing the deuteron tensor polarization)) in terms of the four-momenta of the particles in the reaction under consideration. This parametrization is not unique and depends on the directions along which one defines the longitudinal and transverse components of the deuteron polarization in its rest frame.

As it was mentioned above, we have to define the longitudinal and transverse polarization four-vectors. (Often the longitudinal and transverse components of the deuteron polarization are defined along and axes). In our case it is naturally to choose the longitudinal direction, in the laboratory system, along the three-momentum transferred (the virtual photon momentum) and the transverse direction is perpendicular to the longitudinal one in the reaction plane. The corresponding polarization four-vectors can be written as GM ()

 s(T)μ=(4τ+ρ)k1μ−(1+2τ)qμ−(2−ρ)p1μ√Vc(4τ+ρ),
 s(L)μ=2τqμ−ρp1μM√ρ(4τ+ρ), c=1−ρ−ρτ. (20)

These four-vectors satisfy the following conditions: and So, they have the necessary properties of the polarization four-vectors.

One can verify that the set of the four-vectors in the rest frame of the deuteron (the laboratory system) has the form

 s(L)μ=(0,L),  s(T)μ=(0,T), (21)
 L=k1−k2|k1−k2|,  T=n1−(n1L)L√1−(n1L)2,  n1=k1|k1| .

This leads to simple expressions for the spin-dependent hadronic tensors (due to the vector polarization of the deuteron target) corresponding to the longitudinal and transverse direction of the spin four-vector

 HTμν(V)=−iGMG4√(4τ+ρ)τc[(4τ+ρ)(μνqk1)−
 (2−ρ)(μνqp1)],
 HLμν(V)=iG2M4τ(μνqp1)√ρ(4τ+ρ), (22)

where

 G=2GC+23ηGQ.

The spin-dependent parts of the cross-section, due to the vector polarization of the initial deuteron and longitudinal polarization of the electron beam, can be written as

 dσLBdQ2=−πα24τV22−ρρ√ρ(4τ+ρ)G2M , (23)
 dσTBdQ2=−πα2VQ2√(4τ+ρ)cτGMG , (24)

where we assume that in Eq.(3) equals to one and the degree of the vector polarization (longitudinal or transverse) of the deuteron target is 100 percent.

In the laboratory system these expressions lead to the following asymmetries (or the spin correlation coefficients) in the elastic electron-deuteron scattering in the Born approximation. These asymmetries are due to the vector polarization of the deuteron target, corresponding to the longitudinal and transverse direction of the spin four-vectors and , and longitudinal polarization of the electron beam

 I0ALB=−η√(1+η)(1+ηsin2(θe2))tan(θe2)sec(θe2)G2M , (25)
 I0ATB=−2tan(θe2)√η(1+η)GM(GC+η3GQ) , (26)

where

 I0=A(Q2)+B(Q2)tan2(θe2).

It is worth to note that the ratio of the longitudinal polarization asymmetry to the transverse one is

 ALBATB=√η(1+ηsin2(θe2))sec(θe2)GMG. (27)

This ratio is expressed in terms of the deuteron form factors and in the same way as the corresponding ratio in the case of the elastic electron-proton scattering is expressed via proton electromagnetic form factors and respectively AR (); AAM (). This is direct consequence of the relation between the proton and deuteron spin-dependent hadronic tensors which depend on the proton polarization and deuteron vector polarization, respectively

 Hμν(V)(GM,G)=−4τ+ρ8τHpμν(V)(GpM,GpE). (28)

Let us consider now the tensor polarized deuteron target. If we introduce for the completeness the orthogonal (to the reaction plane) four-vector defined as

 s(N)μ=2εμλρσp1λk1ρk2σV√Vcρ, (29)

then one can verify that the set of the four-vectors satisfies the conditions

 s(α)μs(β)μ=−δαβ,  s(α)μp1μ=0,  α,β=L,T,N.

In the rest frame of the deuteron (the laboratory system) the four-vector has the form

 s(N)μ=(0,N),  N=n1×n2√1−(n1n2)2,  n2=k2|k2|,

then the vector is directed along axis. If to add one more four-vector to the set of the four-vectors defined by the Eqs. (20) and (29), we receive the complete set of the orthogonal four-vectors with the following properties

 s(m)μs(m)ν=gμν,  s(m)μs(n)μ=gmn,  m,n=0,L,T,N.

This set of the four-vectors allows to express the deuteron quadrupole polarization tensor, in general case, as follows

 Qμν=s(m)μs(n)νRmn≡s(α)μs(β)νRαβ,
 Rαβ=Rβα, Rαα=0, (30)

because the time components and equal identically to zero due to the condition Quantities are, in fact, the degrees of the tensor polarization of the deuteron target in its rest system (laboratory system). In the Born approximation the components and do not contribute to the observables and this expansion can be rewritten in the following standard form

 Qμν=[s(L)μs(L)ν−12s(T)μs(T)ν]RLL+12s(T)μs(T)ν(RTT−
 RNN)+(s(L)μs(T)ν+s(T)μs(L)ν)RLT , (31)

where we took into account that

The part of the cross section in the Born approximation that depends on the tensor polarization of the deuteron target can be written as

 dσQBdQ2=dσLLBdQ2RLL+dσTTBdQ2(RTT−RNN)+dσLTBdQ2RLT,

where

 dσTTBdQ2=πα2Q42cηG2M,
 dσLTBdQ2=−πα2Q44η(2−ρ)√cρτGQGM. (32)

In the laboratory system these expressions lead to the following asymmetries (or analyzing powers) in the elastic electron-deuteron scattering caused by the tensor polarization of the deuteron target and unpolarized electron beam (in the Born approximation)

 I0AQB=ALLBRLL+ATTB(RTT−RNN)+ALTBRLT, (33)

where

 I0ALLB=12{8ηGCGQ+83η2G2Q+η[1+2(1+η)tan2(θe2)]G2M},
 I0ATTB=12ηG2M,  I0ALTB=−4η√η+η2sin2(θe2)sec(θe2)GQGM. (34)

Using the P-invariance of the hadron electromagnetic interaction, one can parameterize the differential cross section for elastic scattering of longitudinally polarized electron beam on the polarized deuteron target as follows (for the case of the coordinate representation of the deuteron and electron spin-density matrices)

 dσdQ2=dσundQ2[1+ANsy+ALLRLL+
 ALTRLT+ATT(RTT−RNN)+
 Pe(ALsz+ATsx+ALNRLN+ATNRTN)], (35)

where is the differential cross section for unpolarized particles, is the asymmetry (analyzing power) due to the normal component of the deuteron vector polarization (), and are the asymmetries (analyzing powers) due to the deuteron tensor polarization which correspond to the , and components of the quadrupole tensor; are the correlation parameters due to the longitudinal polarization of the electron beam and components of the deuteron vector polarization and , are the correlation parameters due to the longitudinal polarization of the electron beam and , components of the quadrupole tensor. Note that the amplitude of the elastic electron-deuteron scattering is real in the Born (one-photon-exchange) approximation. This fact leads to zero values of the following polarization observables in this approximation: , and .

The formalism of the spherical tensors is also used for the parametrization of the deuteron spin-density matrix (for the details see Appendix B). In this case the equation (35) can be written as

 dσdQ2=dσundQ2[1+2Imt11T11+
 t20T20+2Ret21T21+2Ret22T22+
 Pe(t10C10+2Ret11C11+2Imt21C21+2Imt22C22)], (36)

where are the polarization tensor describing the polarization state of the deuteron target, and are the analyzing powers and correlation parameters of the reaction, respectively.

The relations between the polarization observables in the coordinate representation and approach of the spherical tensors are the following

 T11=−1√3Ay, T20=−√23Azz, T21=12√3Axz,
 T22=−1√3Axx, C10=√23Az, C11=−1√3Ax,
 C21=12√3Ayz, C22=−12√3Axy. (37)

If the longitudinal direction is determined by the recoil deuteron three-momentum, the relations (21) do not affected by hard photon radiation in the lepton part of interaction (this corresponds to use of the so-called hadronic variables) because . But when this direction is reconstructed from the experiment using the three-momentum of the detected scattered electron (lepton variables), these relations break down because in this case. It means that in the leptonic variables the parametrization (20) is unstable and radiation of hard photon by electron leads to mixture of the longitudinal and transverse polarizations.

One can get rid of such mixture if we choose, in the laboratory system of the reaction (1), the longitudinal direction l along the electron beam momentum and the transverse one t - in the plane and perpendicular to l. Then the corresponding parametrization of the polarization four-vectors is GM ()

 s(l)μ=2τk1μ−p1μM ,  s(n)μ=s(N)μ,
 s(t)μ=k2μ−(1−ρ−2ρτ)k1μ−ρp1μ√Vcρ. (38)

One can verify that the set of these polarization four-vectors in the rest frame of the deuteron (the laboratory system) has the form

 s(l)μ=(0,l),  s(t)μ=(0,t),  s(n)μ=(0,n) , (39)
 l=n1,  t=n2−(n1n2)n1√1−(n1n2)2,  n=n1×n2√1−(n1n2)2.

And this set of the polarization four-vectors (together with four-vector ) is also a complete set of orthogonal four-vectors with the properties

 s(m)μs(m)ν=gμν,  s(m)μs(n)μ=gmn,  m,n=0,l,t,n.

Hadronic tensors corresponding to the longitudinal and transverse directions of the new spin four-vectors have the following form

 Hlμν=i4τ+ρ4τ{G[−2τ(μνqk1)+2τ(2−ρ)4τ+ρ(μνqp1)]+GMρ(1+2τ)4τ+ρ(μνqp1)}GM , (40)
 Htμν=i√ρτc{G(1+2τ)[2−ρ4τ(μνqp1)−4τ+ρ4τ(μνqk1)]−GMc2τ(μνqp1)}GM . (41)

In the case of the scattering off vector-polarized deuteron target the tensors and corresponding to two choices of the spin four-vectors are connected by trivial relations

 HLμν=cosθHlμν+sinθHtμν, HTμν=−sinθHlμν+cosθHtμν,

where

 cosθ=−(s(L)s(l)), sinθ=−(s(L)s(t)).

 cosθ=ρ(1+2τ)√ρ(4τ+ρ) ,  sinθ=−2√cτ4τ+ρ. (42)

These relations are the consequence of the fact that two sets of the spin four-vectors are connected by means of orthogonal matrix which describes the rotation in the plane perpendicular to the direction

where

Using this rotation matrix one can write the spin-dependent parts (due to the vector polarization of the target) of the Born cross-section, which correspond to parametrization (38), in the simple way

 dσβBdQ2=VβA(−θ)dσABdQ2, (43)

where the quantities and are defined by Eqs.(23) and (24). Therefore, we can write

 dσlBdQ2=−πα2V2[1+2τ4τ(2−ρ)GM+2cρG]GM , (44)
 dσtBdQ2=πα2VQ2√cρτ[12(2−ρ)GM−(1+2τ)G]GM. (45)

In the case of the tensor polarization of the deuteron target, the relations which are an analogue of Eq.(43) read

 dσβBdQ2=TβA(−θ)dσABdQ2, (46)

where now respectively. The rotation matrix in this case can be written as

 T(θ)=⎛⎜ ⎜ ⎜⎝14(1+3