1 Introduction

The forward scattering amplitude of a vector meson on a nucleon averaged over the nucleon spin is determined by two quantities: \(\sigma '_T=\sigma _T(1-i \alpha _T)\) and \(\sigma '_L=\sigma _T(1-i\alpha _L)\), where \(\sigma _{T(L)}\) is the total cross sections for the interaction of a transversely (longitudinally) polarized vector meson with the nucleon and \(\alpha _{T(L)}= Re f_{T(L)}(0)/Im f_{T(L)}(0)\) is the ratio of the real to imaginary part of the corresponding amplitudes at zero angle.

The vector meson forward scattering amplitude off spinless target reads:

$$\begin{aligned} f(\vec k,\vec k')=f_0(0) +f_1(0) (\vec S\vec n)^2 \end{aligned}$$
(1)

with \(\vec S\) the spin of the vector meson and \(\vec n=\vec k/k\) the unit vector in the direction \(\vec k\). According to the optical theorem imaginary parts of complex functions \(f_0(0),f_1(0)\) can be expressed in terms of the corresponding total cross sections \(\sigma _T,\sigma _L\):

$$\begin{aligned} Im f_0(0)=\frac{k}{4\pi } \sigma _L, \quad Im f_1(0)=\frac{k}{4\pi } \left( \sigma _T-\sigma _L\right) . \end{aligned}$$
(2)

The dependence of vector particle interaction on its polarization has been known for many years in the case when the constituents of the particle are in the D-wave state. A good example of such dependence is the deuteron interaction with matter [1, 2]. The D-wave component in the deuteron wave function leads to the different absorption in the matter for transversely and longitudinally polarized deuterons [3, 4].

Spin dichroism (dependence of interaction on particle polarization) leads to the appearance of tensor polarization [5]. The intensity of unpolarized deuteron beam (\(I^0_{+1}=I^0_{-1} =I^0_0=1/3\)) after it passage the distance z in the target with density \(\rho \) depends on the value of total cross sections of deuteron interaction with atoms of the target \(\sigma _{\pm 1},\sigma _0\)

$$\begin{aligned} I_{\pm 1}(z)=I^0_{\pm 1}e^{-\sigma _{\pm 1}\rho z}; I_0(z)=I^0_0 e^{-\sigma _0\rho z}. \end{aligned}$$
(3)

The tensor polarization of the deuteron beam is determined by the difference \(\sigma _0-\sigma _{\pm 1}\):

$$\begin{aligned} p_{zz}(z)=\frac{ I_{+1}(z)+I_{-1}(z) -2I_0(z)}{ I_{+1}(z)+I_{-1}(z) +I_0(z)}\approx \frac{2}{3}(\sigma _0-\sigma _{\pm 1})\rho z.\nonumber \\ \end{aligned}$$
(4)

Thus the difference of tensor polarization from the zero indicates that interaction of deuterons with target atoms depends on the deuteron polarization.

For instance, in the case of scattering of the deuteron beam with energy E = 5 GeV on carbon target [3] this difference is about 5%, which is a result of the presence of the D-wave in the deuteron wave function [6].

As to the case of vector mesons interaction the presence of nonzero orbital momentum between quarks is strongly correlated with the meson polarization [7, 8]. Unlike the deuteron where the contribution of the D-wave in the deuteron wave function is small, the D-wave contribution in the wave function of relativistic vector meson is noticeably. Exactly the dominance of the D-wave part in the virtual photon wave function at high virtuality \(Q^2\) allows to restore the scaling in deep inelastic scattering [8].

The quarks distribution in transversely and longitudinally polarized vector mesons are very different. For instance the authors of  [9] using the generalized QCD sum rules showed that the distribution of valence quarks in the longitudinally and transversely polarized \(\rho \) meson diverse strongly depending on meson polarization. The similar distinction takes place for the distribution of constituent quarks in the vector mesons polarized transversely and longitudinally  [10, 11].

The difference in quark distributions in vector mesons depending on its polarization should lead to impact of vector meson polarization on their interaction with the matter [12, 13].

To estimate the values of \(\sigma _T,\sigma _L\) we consider the vector meson scattering on nucleon in color dipole model [14, 15]. In any QCD description of photon or vector meson interaction with the target the first step is the conversion of the initial particle into a quark-antiquark pair. In color dipole model this pair interacts with a target as a color dipole, whose cross section depends on the transverse separation r in the \(q\bar{q}\) pair. The total cross section of the transverse and longitudinal polarized vector mesons with a target in mixed representation (z-share of the meson light-cone momentum carried by the quark) reads:

$$\begin{aligned}{} & {} \sigma ^{L(T)}(VN)=\int |\Psi _V^{L(T)}(r,z)|^2\sigma (r)d^2rdz\nonumber \\{} & {} \sigma (r)=\sigma _0(s) \left( 1-e^{-\frac{r^2}{R^2(s)}}\right) . \end{aligned}$$
(5)

Here \(\Psi _V^{L(T)}(r,z)\) is the wave function (quark distribution) in the vector meson polarized transversally or longitudinally, whereas the simple parametrization of dipole cross section \(\sigma (r)\) in (5) allows to describe not only mesons photoproduction, but also the interaction of high energy hadrons with nucleons and nuclei [16].

The dependence of \(\sigma _L(\rho N)\) and \(\sigma _T(\rho N)\) on invariant energy \(W=\sqrt{s}\) calculated by above expression using parameterizations of the vector meson wave function in Boosted Gaussian approach [17] and ADS/QCD Holographic model [10] is shown in Fig. 1. The impact of the polarization on the total cross section of the vector meson interaction with nucleon is large leading to the different absorption of vector mesons produced on nuclei depending on their polarization [12, 13].

Fig. 1
figure 1

Longitudinal \(\sigma _L\) and transverse \(\sigma _T\) \(\rho N\) total cross sections as a function of the invariant energy for different parameterizations of the \(\rho \) meson wave function: (left) Boosted Gaussian model [17]; (right) ADS/QCD Holographic model [10]

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In view of vector mesons beams absence the best way to measure \(\sigma _L(VN)\) and \(\sigma _T(VN)\) are the measurement of vector mesons production on nuclei. For many years such processes are the source of unique information on interaction of unstable particles with nucleons [18].

In vector mesons photoproduction due to the s-channel helicity conservation \(\rho ,\phi \) mesons are produced mainly transversely polarized and only a part of \(\omega \) mesons at JLAB energies (due to the pion exchange) is longitudinally polarized [19,20,21]. Thus in experiments on vector mesons production by real photons off nuclei one extracts the total cross section of interaction of transversely polarized vector mesons with nucleons. To get information on interaction of longitudinally polarized vector mesons with nucleon one should measure the vector mesons production in processes where the longitudinally polarized vector mesons are mainly produced. Such kind of processes are vector mesons production on nuclei by pion or kaon beams \(\pi ^\pm (K^\pm )+A\rightarrow V^0(K^{*0})+A'\), where due to dominating pion exchange the produced vector mesons are mainly longitudinally polarized [22, 23]. Later on we consider the vector mesons production off nuclei by pion and kaon beams with the aim to get the information on the value of \(\sigma _L(VN)\) from such type of reaction. Before discussing this issue let us explain why the knowledge of \(\sigma _L(VN)\) is important and in some cases even crucial.

2 Vector meson polarization vs color transparency

The knowledge of the cross section \(\sigma _L(VN)\) is an important task for instance in interpreting the color screening effect in the vector meson leptoproduction [13]. The idea of the color transparency (CT) is that a hadron produced in certain hard-scattering processes has a smaller probability to interact in the nuclear matter due to its smaller size compared to the physical hadron [25]. The vivid example of CT is the well known effect [18] of “shrinking photon” in leptoproduction of vector mesons. At high photon virtuality \(Q^2\) the transverse size of its hadronic component \(r\sim 1/Q^2\) is smaller than the size of a normal hadron. This would account for the pointlike behavior and the diminished absorption of virtual photons in nuclear matter. The increase with \(Q^2\) the nuclear transparency defined as \(Tr = \frac{\sigma _A}{A\; \sigma _N}\), where \(\sigma _A\) and \(\sigma _N\) are vector mesons production cross sections off nuclei and nucleon respectively considered as a direct result of CT effect [26, 27].

On the other hand the increase of nuclear transparency with \(Q^2\) can be partially a result of the large difference (see Fig. 1) between \(\sigma _L(VN)\) and \(\sigma _T(VN)\). Really the fraction of longitudinally polarized vector mesons in leptoproduction rises with \(Q^2\) [28]. Accounting that \(\sigma _L(\rho )\) is much less than \(\sigma _T(\rho )\) the absorption of longitudinally polarized \(\rho \) meson in nuclear matter should be weaker than for transverse one.

To estimate this effect we use the expression for nuclear transparency accounting for absorption of vector mesons with different polarization [20]

$$\begin{aligned}{} & {} Tr=\rho _{00}N(0,\sigma _L)+(1-\rho _{00})N(0,\sigma _T)\nonumber \\{} & {} N(0,\sigma )=\int \frac{1-\exp (-\sigma \int {\rho (b,z)dz})}{\sigma }d^2b, \end{aligned}$$
(6)

where \(\rho _{00}\) is the spin density matrix element corresponding to fraction of longitudinal polarized vector mesons production. In Fig. 2 we cited the experimental data for nuclear transparency [27] and our calculations denoted by stars accounting for different absorption of \(\rho \) meson in nuclei depending on its polarization.In calculations we put \(\sigma _T(\rho N)=25\) mb, \(\sigma _L(\rho N)=10\) mb. For the dependence of \(\rho _{00}\) on \(Q^2\) we used the relation \(\rho _{00}=\frac{\epsilon R}{(1+\epsilon R)}\) and the fit [28] for ratio of longitudinal to transverse cross section on the nucleon \(R(Q^2)=c_0\left( \frac{Q^2}{m^2}\right) ^{c_1}\) with constants \(\epsilon =0.8, c_0=0.56, c_1=0.47\).

Fig. 2
figure 2

Nuclear transparency of \(\rho ^0\) electroproduction as a function of \(Q^2\) compared to the experimental data from CLAS [27]

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As seen from the Fig. 2 the considered effect has to be separated from the effect of color transparency as it is not small and exhibit the same behavior. Thus the challenge of the impact of vector meson polarization on it interaction with nuclei and nucleon can be crucial for determination of such fundamental effect as color transparency.

3 Vector mesons production by pions

For the best of our knowledge, the only attempt to study the impact of the vector meson polarization on its absorption in nuclei was made many years ago  [29, 30] using the charge exchange reaction \(\pi ^- + A\rightarrow \rho ^0+A^\prime \). The incoherent cross section and spin density matrix elements of \(\rho \) mesons were measured for different nuclei: C, Al, Cu, Pb.

Due to the dominance of the pion exchange in this process, a large fraction of longitudinally polarized \(\rho \) mesons was produced. At the first glance, the experimental data support the assumption that \(\sigma _T(\rho N)\approx \sigma _L(\rho N)\). However, there are strong reasons against such a conclusion. It was shown [31] that, due to the relatively low energy of the primary pion beam (\(E_{\pi } = 3.7\;\textrm{GeV}\)) and large decay width of the \(\rho \) meson, the significant part of mesons decaying inside the nucleus complicates the interpretation of the experimental data.

The total cross sections of the \(\rho \) and f(1270) mesons with a nucleon were measured at Argonne [32] using the charge exchange process on neon nuclei \(\pi ^++Ne\rightarrow \rho (f)+Ne'\). Accounting for the possibility of the \(\rho \) mesons decay in nuclei (\(p_{\pi } = 3.5\) GeV/c), the \(\rho N\) total cross section is required to be \(\sigma (\rho N)\approx 12\) mb, which would contradicts to the value \(\sigma (\rho N)\approx 27\) mb obtained from the \(\rho \) meson photoproduction on nuclei. From our point of view this difference is a direct result of distinction between \(\sigma _T(\rho N)\) and \(\sigma _L(\rho N)\) as in charge exchange process mainly longitudinally polarized \(\rho \) mesons are produced, whereas in the photoproduction \(\rho \) mesons are transversely polarized due to s-channel helicity conservation.

Taking into account that the vector meson decay mean free path in the laboratory system grows with energy \(l=\frac{p}{m_V\Gamma _V}\), the vector mesons produced by pion and kaon beams with energies of tens GeV available at the M2 beam line at CERN [23], would help to determine uniquely the value of longitudinal VN total cross section \(\sigma _L(V N)\) while the value of the total cross section for transversely polarized vector mesons \(\sigma _T(V N)\) is known from photoproduction on nuclei [18] and can be cross-checked at SPS energies.

Vector mesons production on protons by pions beams has been measured at high energies with the goal to check the Regge model predictions. It was shown that in \(\omega \) and \(\varphi \) mesons production  [33, 34] the main contribution at high energies give the natural parity exchange leading to production of transversely polarized mesons. As to the \(\rho \) mesons production the dominance of pion exchange at modest transfer momenta leads to production predominantly the longitudinally polarized \(\rho \) mesons [22]. Unlike the pion exchange leads to the cross section falling as \(1/p^2\), at energies relevant to AMBER experiment region the cross section of the reaction \(\pi ^-+A\rightarrow \rho + A'\) is measurable and can give the unique information on the value of \(\sigma _L(\rho N)\).

There are some plans to produce the RF-separated kaon beam for future measurements at the experimental area of SPS [23, 24]. If so it is interesting to measure the charge exchange process \(K^{\pm }+A\rightarrow K^*(892)+A'\) to get information on \(K^*\) absorption in nuclear matter depending on vector meson polarization.

Recently ALICE collaboration presented the data [35] on \(K^*\) and \(\varphi \) mesons production in peripheral nucleus-nucleus collisions, which show that vector mesons polarization unlike their production in pp collisions depends on their transverse momenta and centrality and can’t be explained by conventional effects. Thus the investigation of \(K^*\) production off nuclei becomes a topical issue