Double-Polarized pd Scattering and Test of Time-reversal Invariance

The integrated proton-deuteron cross section σ˜ for the case of transversal polarization ppy of the proton and tensor polarization Pxz of the deuteron provides a null test signal for time-reversal invariance violating but P-parity conserving effects. The corresponding experiment is planned at COSY to measure the observable σ˜. Here we consider in general case the status of the null-test observable in pd scattering, calculate σ˜ within the Glauber theory of the double-polarized pd scattering at 100 - 1000 MeV and study the contribution of the deuteron S- and D-waves for several type of the T-odd NN interactions.


Introduction
Discrete symmetries play an important role in physics of fundamental interactions. In particular, the CP-violation (or T-invarinace violation under CPT symmetry) is responsible for the baryon asymmetry of the Universe [1]. However, within the standard model CP-violation effects observed in physics of kaons and B-mesons are far not sufficient to explain this asymmetry [2]. Therefore, other sources of CP violation have to be found. Time-invariance-violating (T-odd) P-parity conserving (P-even) (TVPC) interactions do not arise on the fundamental level within the standard model. This type of interaction can be generated by weak radiative corrections to the T-odd P-odd interaction discovered in physics of kaons and B-mesons. However in such a case its intensity is too small to be observed in experiments at present [3]. Thus, observation of TVPC effects would be considered as indication to physics beyond the standard model. The total polarized cross section σ of the proton-deuteron scattering with vector polarization of the proton p p y and tensor polarization of the deuteron P xz constitutes a null-test observable for TVPC effects [4]. The dedicated experiment is planned at COSY [5] at proton beam energy 135 MeV. The first analysis of the TVPC null-test signal [6] was done within the nonmesonic deuteron breakup channel pd → ppn estimated in the single scattering approximation. Recently we used the spin-dependent formalism [7] of the Glauber theory to calculate the cross section σ [8] and also "null-combinations" of some differential spin observables of the pd elastic scattering which deviate from zero only in case of presence of the TVPC effects [9]. The formalism includes full spin dependence of elementary pN-amplitudes and S-and D-components of the deuteron wave function. This formalism allows one to explain existing data on the non-polarized differential cross section and spin observables of the elastic pd scattering [10] at energies of the experiment planned at COSY [5]. We calculated energy dependence of σ for different type of TVPC interactions in the S-wave approximation for the deuteron [8]. Influence of the Coulomb interaction on the cross section σ was found to be negligible [8]. Here we consider in general case the status of the null-test observable in pd scattering and study for several type of the TVPC NN interactions the contribution of the D-wave of the deuteron.

Null-test signal of TVPC interactions
Time-reversal symmetry conserving and P-parity conserving (TCPC or T-even P-even) interactions lead to the following transition amplitude of the elastic pd scattering at zero degree [11] e ′ where e (e ′ ) is the polarization vector of the initial (final) deuteron, m is the unit vector along the beam momentum, σ is the Pauli matrix, g i (i = 1, . . . , 4) are complex amplitudes. To the right-hand side of Eq.(1) one can add the TVPC (T-odd P-even) term in a very general form where g is the TVPC transition amplitude.
Using the generalized optical theorem, the total cross section of the pd scattering can be written in the form [10] where p p (p d ) is the vector polarization of the initial proton (deuteron) and P zz and P xz are the tensor polarizations of the deuteron. The OZ axis is directed along the proton beam momentum m, OY↑↑ p p , OX ↑↑ [p p × m]. In Eq. (3) the terms σ i with i = 0, 1, 2, 3 are non-zero only for T-even P-even interactions corresponding to Eq. (1) and the last term σ constitutes a null-test signal of T-invariance violation with P-parity conservation. The generalized optical theorem gives σ = −4 √ πIm 2 3 g [8]. Hadronic amplitudes of pN scattering are taken as [7] M N (p, q; σ, σ N whereq,k andn are defined as unit vectors along the vectors q = p − p ′ , k = p + p ′ and n = [k × q], respectively; p (p ′ ) is the initial (final) proton momentum; σ N is the Pauli matrix acting on the spin state of the nucleon N . We consider the following terms of the TVPC NN interaction which were under discussion in Ref. [6]: is the Pauli matrix acting on the spin state of the proton (nucleon N = p, n), τ (τ N ) is the corresponding matrix acting on the isospin state; m p is the proton mass. In the framework of the phenomenological meson exchange interaction the term g ′ corresponds to ρ-meson exchange, and h-term provides the axial meson h 1 exchange.
The crucial point is the following. The strong (T-even P-even) background is excluded from the null-test observable σ. This means that when calculating (or measuring at ideal conditions) the observable σ one does not deal with a sum of the strong NN amplitude and the weak TVPC NN amplitude. Only products of these amplitudes may appear in σ (see below Eq. (6)). Furthermore, uncertainties in our knowledge of the strong amplitudes of the NN-scattering could not affect the final result for σ drastically.

g ′ term
As was shown in our papers [8] the g ′ -term give zero contribution to the null-test observable σ within the Glauber theory of the pd elastic scattering. The reason for this is the specific spin-isospin structure of the g ′ term given in Eq. (5) which allows only the charge-exchange double pN-scattering mechanism for the elastic pd scattering amplitude.

h-and g-terms and S-D interference
For the single scattering mechanism the amplitude g vanishes within the Glauber theory. This follows from Eq. (5) where all terms are zeros at zero transferred momentum q = 0. For double scattering mechanism considered for the h-and g-terms in Eq. (5) we derived in Ref. [8] a formula for g in the S-wave approximation. Taking into account the D-wave we find for the TVPC amplitude the following result where S

Numerical results
For the C ′ N amplitude of the pN scattering we use the SAID date base [12]. The deuteron wave function is taken in the CD Bonn model.  Fig.1 versus the proton beam energy T p . One can see that the D-wave of the deuteron taken into account changes considerably the result for the observbale σ as compared to the S-wave result obtained previously in [8,13]. The destructive S-D interference diminishes considerably the null-test signal σ at energies of the planned COSY experiment [5] ∼100 MeV as compared to the pure S-wave contribution and provides an enhancement at 700-800 MeV.

Summary
In the first theoretical analysis of the null-test observable σ performed in Ref. [6] only the single scattering mechanism was considered for the straightforward calculation of the inelastic and elastic pd-scattering. We use the optical theorem for calculation of the σ, that requires to get only the forward elastic pd scattering amplitude. On this way we find within the Glauber theory that the single scattering mechanism gives zero contribution to the null-test cross section σ. The double scattering mechanism for the h and g terms of the TVPC NN interaction leads to Eq. (6) for the TVPC amplitude of the pd scattering and shows that the only hadronic pN amplitude C ′ N from Eq. (4) modulates the TVPC observable σ. Furthermore, we have shown for the case of the hand g-type of interaction that the deuteron D-wave gives a valuable contribution to the null-test signal due to interference between the deuteron S-and D-waves. Of course, there are many other TVPC terms in the NN interaction [14] which we are going to consider later on.
The g ′ -term caused by the ρ− meson exchange in the TVPC NN-interaction does not contribute to σ within the Glauber theory with T-even P-even NN-interaction in the deuteron [8]. However, the g ′ -term can give contribution to the null-test signal σ if this interaction will be included into the deuteron bound state [8]. One-pion exchange is excluded from TVPC NN-interaction [15], however, two-pion exchange probably does contribution [16]. Furthermore, TVPC NN forces can contribute to the electromagnetic p-d interaction due to the toroidal quadrupole form factor of the deuteron [17]. Another T-odd P-even effect very similar to the σ observable in pd-scattering can be observed in inclusive interaction of unpolarized leptons with polarized nucleons [18].