Proton-Deuteron Scattering and Test of Time-Reversal Invariance

The integrated proton-deuteron scattering cross section σ̃ for transversely polarized protons (P y ) and tensor polarized deuterons (Pxz) constitutes a null test signal for timereversal invariance violating but P-parity conserving effects. This cross section will be measured at COSY. Using the generalized optical theorem and Glauber theory we study the null-test observable σ̃ for different types of T-odd P-even NN-interactions.The formalism includes full spin dependence of elementary pN-amplitudes and Sand Dcomponents of the deuteron wave function.


Introduction
Time-invariance-violating (T-odd) P-parity conserving (P-even) (TVPC) interactions do not arise at the fundamental level within the standard model.This type of interaction can be generated by radiative corrections to the T-odd P-odd interaction discovered in the physics of kaons and B-mesons.However, in this case its intensity is too low to be observed in experiments at present [1].Thus, observation of TVPC effects would be considered as an indication of physics beyond the standard model.
As was shown in Ref. [2], 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 nulltest observable for TVPC effects.The dedicated experiment is planned at COSY [3] at proton beam energy 135 MeV.The first analysis of the TVPC null-test signal [4] was done within the nonmesonic deuteron breakup channel pd → ppn estimated in the single scattering approximation.Recently we used the spin-dependent formalism [5] of the Glauber theory to calculate the cross section σ [6] and "null-combinations" of some differential spin observables of the pd elastic scattering [7] which deviate from zero if the TVPC effects occur.The formalism includes full spin dependence of elementary pNamplitudes 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 at 135 MeV [8].Here we consider some qualitative arguments concerning the ρ-meson contribution to σ and briefly explain the role of the deuteron D-wave.
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.The matrix elements of the operators (1), ( 2) are where μ (μ ) and λ (λ ) are spin projections of the initial (final) proton and deuteron on the beam direction, respectively.All diagonal matrix elements of the M T VPC operator are zeros.The total cross section of the pd scattering has the form [8] 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. ( 8) 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.Using the generalized optical theorem we find σ = −4 √ πIm 2 3 g.Hadronic amplitudes of pN scattering are taken as [5] where q, k and n 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. [4]: Here σ (σ N ) 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.

g -term
The g term contributes only to the charge exchange transitions, because the non-zero matrix elements of the isospin-operator connected with the g term in Eq. ( 10) are the following The isospin matrix element of the C-odd isospin operator T z = [τ × τ N ] z in Eq. ( 11) changes the sign under replacement p ↔ n.In contrast, the similar matrix elements for T-even P-even (strong) NN-interaction are equal one to other.This difference is one cause for the vanishing of the amplitude g for the double scattering mechanism of the process pd → pd.As was shown in Ref. [6], the g -term gives zero contribution to g within the Glauber model.Below we discuss this observation briefly.
The TVPC charge-exchange Glauber operator of the double scattering has a form [6] [M np→pn (q 2 )t pn→np (q 1 ) + t np→pn (q 2 )M pn→np (q 1 )], where q 1 = q/2 + q is the transferred momentum in the first and q 2 = q/2 − q in the second collision and q is the total transferred momentum.For the next step one has to calculate the matrix element of the operator (12) over the deuteron states ψ(r) with the factor exp (iq r) and integrate over q .Under the sign of this integral the operator ( 12) is not changed after the substitution q 1 ↔ q 2 [5,6].Therefore, one may add to the right side of Eq. ( 12) the term O c T VPC (1 ↔ 2) and divide the obtained sum by a factor of 2. In collinear kinematics (q = 0), this symmetry and linear dependence of g -term on [q × k] lead to cancellation of the spin-independent term A N in the transition operator (12).The same is true for the B N terms in Eq. ( 9).Thus, only C N and C N terms may contribute as [6] where n 1 = [k × q ], n1 = n 1 /|n 1 | and q = q 2 = −q 1 ; Π is a constant.In Eq. ( 13) the C N -terms do not contain the proton beam spin σ.We can show that this is a consequence of Eqs.(11).According to Eqs. ( 6), (7), it means that the contribution of the C N ×g term to the amplitude g is zero.Furthermore, due to Eqs. (11) the remaining terms with C N in Eq. ( 13) contain the difference σ n − σ p , but not the sum.These terms can be rewritten as . Thus, the contribution of the operator (13) to the amplitude g vanishes and this fact is directly connected with Eqs.(11).
Strong suppression of the contribution of the ρ-meson as compared to the axial h 1 meson was found numerically in the Faddeev calculations [10] of the null-test signal for the nd scattering at 100 keV, but no explanation of this result was offered.We suppose that the cause for this suppression is the same spin-isospin structure of the scattering amplitude which leads to the vanishing ρ-meson contribution within the Glauber approach.

h-and g-terms
For the single scattering mechanism the amplitude g vanishes within the Glauber theory.Using Eqs.
(3)- (7) for the double scattering mechanism with pN-amplitudes (9) and (10) we find for the h-and g-terms in Eq.( 10) that all T-even P-even amplitudes of the pd-scattering are zeros: g 1 = g 2 = g 3 = g 4 = 0. Furthermore, we find for the TVPC amplitude 21 st International Conference on Few-Body Problems in Physics where S (0) 0 = ∞ 0 dru 2 (r) j 0 (qr) and S (1)  2 (q) = 2 ∞ 0 dru(r)w(r) j 2 (qr) are the elastic form factors of the deuteron, and u(r) and w(r) are the S-wave and D-wave w(r) of the deuteron, respectively, [6].We can show that the S-D wave interference, not considered in Ref. [6], considerably diminishes the null-test signal σ at the energies of the planned COSY experiment [3] ∼100 MeV as compared to the pure S-wave contribution and provides an enhancement at 700-800 MeV.

Summary
In contrast to Ref. [4] we show, using the optical theorem, that within the single scattering approximation the null-test observable σ is zero.Our result obtained within the Glauber theory is formulated by Eq. ( 14).Only the amplitude C N appears in Eq.( 14) whereas other T-even P-even pN amplitudes, which were found in Ref. [4] to contribute to the TVPC null-test signal, are absent in Eq. ( 14).Furthermore, we find the deuteron D-wave gives a valuable contribution to the null-test signal for the case of the h-and g-type of interaction.The g -term caused by the ρ− meson exchange in the TVPC NN-interaction makes a zero contribution to g and this result is true in the case when both the S-and D-components of the deuteron wave function are taken into account.We discuss some symmetry arguments to clarify a cause for the vanishing contribution of the g -term.The g -optical potential [11] and the corresponding coupling constant of the ρ−meson to the nucleon ḡρ is widely used as a measure of intensity of the TVPC effects [12,13].Since the g -term gives zero contribution to σ within the Glauber theory, this parameter cannot be applied straightforwardly for the nucleondeuteron scattering as a scale of the TVPC interactions at large enough energies.However, the g -term can give contribution to the null-test signal σ if this interaction is included into the deuteron bound state [6].One-pion exchange is excluded from the TVPC NN-interaction [12], however, two-pion exchange probably contributes similarly to the P-violating pp-interaction [14].Finally, TVPC NN forces can contribute to the electromagnetic p-d interaction due to the toroidal quadrupole form factor of the deuteron [15].