Search for time-reversal-invariance violation in double polarized antiproton-deuteron scattering

Apart from the $pd$ reaction also the scattering of antiprotons with transversal polarization $p_y^p$ on deuterons with tensor polarization $P_{xz}$ provides a null-test signal for time-reversal-invariance violating but parity conserving effects. Assuming that the time-reversal-invariance violating $\bar NN$ interaction contains the same operator structure as the $NN$ interaction, we discuss the energy dependence of the null-test signal in $\bar pd$ scattering on the basis of a calculation within the spin-dependent Glauber theory at beam energies of 50-300 MeV.


Introduction
Under CPT symmetry time-reversal-invariance violating but parity conserving (TVPC) forces are considered as a possible source of CP-invariance violation, which is required to account for the matterantimatter asymmetry in the universe [1]. In contrast to effects from time-reversal-invariance violation together with parity violation such as a permanent electric dipole moment (EDM) of elementary particles, so far much less attention was paid to TVPC effects. The reason why TVPC effects are interesting is that experimental limits on them are still rather weak, in particular, considerably weaker than those for the EDM.
Since the intensity of TVPC interactions within the standard model is extremely small [2], an observation of any effects at the present accuracy level of experiments would be a direct indication of physics beyond the standard model. Indeed a pertinent measurement is planned at the COSY accelerator in the Research Center in Jülich [3]. The observable in question is the integrated cross section for scattering of protons with transversal polarization p p y on deuterons with tensor polarization P xz . It provides a null-test signal for TVPC effects [4] and it will be measured in pd scattering at 135 MeV [3]. Theoretical studies of the energy dependence of the expected signal were performed at energies of the planned experiment [5][6][7][8][9][10][11][12] on the basis of the spin-dependent Glauber theory and demonstrate several unexpected effects. Among them are (i) the absense of the contribution from the lowest-mass meson-exchange (ρ meson) in the TVPC NN interaction, caused by its specific isospin, spin and momentum dependence; (ii) a strong impact of the deuteron D-wave on the null-test signal due to a destructive interference between the S -and D-wave contributions, even for zero transferred 3momentum; (iii) oscillating behaviour of the null-test signal as a function of the beam energy, i.e. the vanishing of the TVPC signal at some specific energies is possible even when the TVPC interaction itself is nonzero; (iv) a very small influence of the Coulomb interaction on the TVPC term of the pd forward scattering amplitude g. Furthermore, certain relations between differential observables of elastic pd scattering caused by time-reversal-invariance requirements were obtained and the degree of their violation by TVPC NN forces was studied [13,14].
Since the spin structure of the amplitude for pd-andpd elastic scattering is the same, it is obvious that the integrated cross section for scattering of a polarized (pp y ) antiproton on tensor polarized (P xz ) deuterons also provides a null-test signal for TVPC effects. Furthermore, the TVPCNN amplitude for elastic scattering contains the same operator structures as the one for TVPC NN elastic scattering, except for the charge-exchange terms. Therefore, the formalism developed in Refs. [7,8,11] within the Glauber theory for the calculation of the null-test signal in pd scattering can be straightforwardly applied topd scattering too. However, due to differences in the hadronic part of the pN andpN scattering amplitudes and also in the electromagnetic interactions, the energy dependence of the null test signal in pd andpd interaction has to be different. In the present work the energy dependence of the null-test signal inpd scattering is studied on the basis of calculations within the spin-dependent Glauber theory using the spin-dependentpN amplitudes from a recent partial wave analysis ofpp scattering [15].

Null-test signal for time-reversal-invariance violation
The total cross section forpd scattering with TVPC forces included can be written in the same form as for pd scattering [7] Here pp (p d ) is the vector polarization of the initial antiproton (deuteron), P zz and P xz are the tensor polarizations of the deuteron, and pp y is the transversal component of the antiproton vector polarization. The OZ axis is directed along the beam direction m, the OY axis is directed along the vector polarization of the antiproton beam pp and the OX axis is chosen to form a right-handed reference frame. The integrated cross sections σ t i (i = 0, 1, 2, 3) are those which arise from a standard timereversal invariant and parity conserving interaction, while the last term σ appears only in the presence of the TVPC interactions and constitutes the TVPC null-test signal. The result (1) can be derived using phenomenologicalpd forward scattering amplitudes and the generalized optical theorem.
The evaluation of the integrated cross sections σ t i and σ at beam energies > 100 MeV can be done on the basis of the spin-dependent Glauber theory ofpd scattering which is formulated similarly to the theory of pd scattering given in Ref. [16]. Indeed, as shown in Ref. [17], this theory allows one to describe rather well available data on differential spin observables of pd scattering in the forward hemisphere at beam energies of 135 − 200 MeV. For the antiproton-deuteron scattering this theory can be applied at even lower energies due to the presence of strong annihilation effects. In the Glauber theory one uses the elastic (on-shell)NN scattering amplitudes as input. Hadronic amplitudes of thē pN scattering are taken here in the same form as for pN scattering [16] M N (p, q; σ, σ N ) = A N + C N σn + C ′ N σ Nn + B N (σk)(σ Nk ) + 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) antiproton momentum.
In general, the TVPC NN interaction contains 18 different terms [18]. In the case of the onshell NN scattering amplitude there are only three terms with different (independent) spin-momentum structures. In the present study we consider the following two terms for the TVPC (on-shell) t-matrix of elasticpN scattering which have the same structure as those in TVPC pN scattering Here σ (σ N ) is the Pauli matrix acting on the spin state of the antiproton (nucleon N = p, n) and τ (τ N ) is the isospin matrix acting on the isospin state of the antiproton (nucleon). The momenta q and k were already defined above in the context of Eq. (2). Both terms in Eq. (3), h N and g N , occur in the TVPC pn interaction. The TVPC pN scattering amplitude contains also the charge-exchange term which describes the elastic transitions pn → np and np → pn. Within a picture of one-mesonexchange interaction this g ′ -term corresponds to the charged ρ-meson exchange [19]. The same term (4) corresponds to the charge-exchange processespp →nn ornn →pp. However, in contrast to pn scattering these processes are inelastic and therefore the operation of time-reversal invariance transforms, for example, thepp →nn amplitude to thenn →pp amplitude and does not impose any restrictions on these amplitudes. The h N -term in Eq. (3) can be associated with the axial h 1 -meson exchange. As shown in Ref. [19], contributions of the π-and σ-meson to the TVPC NN interaction are excluded, which is obviously true for the TVPCNN interaction as well.

TVPC amplitude ofpd forward scattering
One can write thepd forward elastic scattering amplitude in general form taking into account the TVPCNN interactions, as it was done for pd elastic scattering [7,17], and then apply the generalized optical theorem to derive Eq. (1) for the totalpd scattering cross section. As in Ref. [7], the integrated cross section σ is related to the TVPC term g of thepd forward elastic scattering amplitude by σ = −4 √ π Im 2 3 g. Furthermore, the TVPC forward amplitude ofpd elastic scattering g can be found within the Glauber theory [7]. We consider the h N -and g N -terms and take into account both the Sand D-wave components of the deuteron. Taking into account that the g N -term is excluded in the processpn →pn due to the isospin operator in Eq. (3), we obtain the following result for the TVPC forward amplitude from the corresponding equation in Ref. [11]: Here S ( j) i are the elastic form factors of the deuteron defined in Ref. [11]. The first term in the (big) squared brackets in Eq. (5), S (0) 0 (q), corresponds to the S -wave approximation, the second term, S (1) 2 (q), accounts for the S -D interference, and the last three terms contain the pure D-wave contributions. As was shown in Ref. [11], the contribution of the g ′ -term to the null-test signal vanishes in pd scattering due to the specific spin-isospin structure of the g ′ -interaction. Formally, for the same reason the charge-exchange g ′ -term given by Eq. (4) vanishes in thepd forward elastic scattering amplitude.
In the first theoretical work [5] where the null-test signal was calculated within the impulse approximation, the Coulomb interaction was not considered. In Ref. [6] Faddeev calculations were performed, but only for nd scattering and at rather low energies of ∼ 100 keV. The Coulomb interaction was taken into account for the first time in Ref. [7] in a calculation of the null-test signal of pd scattering within Glauber theory and found to be negligible. A similar result was found in Ref. [20] using Faddeev calculations.

Numerical results
Results of numerical calculations of the energy dependence of the null test-signal for the h-term are presented in Fig. 1, in units of the unknown TVPC coupling strength. One can see from this figure that the deuteron S -wave contribution (dashed line) leads to a smooth energy dependence and has a node at an antiproton beam energy of about 50 MeV. The inclusion of the D-wave changes this behaviour considerably (solid line) due to a destructive S -D interference (cf. dash-dotted line). As a result, a second zero of the null-test signal σ appears at higher energies, i.e. at T ≈ 300 MeV. The maximal value of σ is expected at 100 − 150 MeV. Note that the actual position of the nodes changes only slightly when deuteron wave functions from other NN models are used for the calculation.
Let us consider possible spurious effects that could mimic a TVPC signal. One source for a spurious signal is associated with a nonzero deuteron vector polarization p y d 0 (in the direction of the incident-proton-beam polarization p p ). In this case, the term σ 1 Pp y p d y in Eq. (1) contributes to the 4 EPJ Web of Conferences 181, 01015 (2018) https://doi.org/10.1051/epjconf/201818101015 EXA2017 asymmetry corresponding to the difference of the event counting rates for the cases of pp y P xz > 0 and p p y P xz < 0 (with the fixed sign of P xz ), which is planned to be measured at COSY [3]. According to our calculations, the integrated cross section σ 1 could be equal to zero at antiproton beam energies of ∼ 100 MeV (see results for the JülichNN interaction model in Refs. [21,22]). Therefore, at this energy the spurios signal caused by a nonzero value of the deuteron vector polariziation p d y could be minimized.

Concluding remarks
We have performed a study of time-reversal-invariance violating but parity conserving effects in antiproton-deuteron scattering. Specifically, we have evaluated the null-test TVPC signal for scattering of antiprotons with transversal polarization p p y on deuterons with tensor polarization P xz on the basis of the spin-dependent Glauber theory. The observed effects turned out to be similar to those in pd scattering: (i) There is a strong impact of the deuteron D-wave on the null-test signal that arises from a destructive interference between the S -and D-wave contributions; (ii) There is an oscillating behaviour of the null-test signal as a function of the beam energy. Accordingly, it is possible that the signal for TVPC effects is zero at some specific energies, even when the TVPC interaction itself is nonzero.