Bounds on Time Reversal Violation From Polarized Neutron Capture With Unpolarized Targets

We have analyzed constraints on parity-odd time-reversal noninvariant interactions derived from measurements of the energy dependence of parity-violating polarized neutron capture on unpolarized targets. As previous authors found, a perturbation in energy dependence due to a parity (P)-odd time (T)-odd interaction is present. However, the perturbation competes with T-even terms which can obscure the T-odd signature. We estimate the magnitudes of these competing terms and suggest strategies for a practicable experiment.


Introduction
The enhancement factor of a million observed in the 1980s in compound nucleus parity violating observables stimulated great interest in searching for time reversal violation. The enhancement is expected to be present for all symmetry breaking observables in compound nuclear systems, arising as it does from the close spacing and long lifetimes of the states. The largest enhancements were seen in transmission experiments with epithermal neutrons at resonances in nuclei A > 100. Despite considerable effort, however, no epithermal neutron transmission test of time reversal violation (P-even or P-odd) has been carried out, primarily due to difficulties in preparing a suitable spin polarized or aligned nuclear target. For general background on the proposed experiments and the difficulties see [1,2].
For P-even time reversal violation, tests with higher energy neutrons have been performed in holmium (A = 165) using a nuclear spin aligned target [3]. The experimental precision is high. However, there are no compound nuclear enhancement mechanisms at work, and a 1/A suppression factor arises since only the last valence nucleon contributes to the T-violating effect. Further improvement with heavy targets and MeVbeams of neutrons therefore appears unlikely. Use of a tensor polarized deuteron target avoids the 1/A suppression, and a test using a few hundred MeV polarized proton beam is planned for the COoler SYnchrotron storage ring facility (COSY) at the Institut fur Kernphysik (IKP) Juelich, Germany by the Time Reversal Invariance Test at COSY collaboration (TRIC). The experiment is still under development but does have the potential to make an order of magnitude improvement in sensitivity to the underlying T-violating meson exchange coupling constants [4].
Given the difficulties associated with the need for a polarized target in an on-resonance neutron transmission P-odd test, it is appropriate to investigate whether other experiments could investigate time reversal violation, taking advantage also of the intense fluxes of neutrons expected to be available from the next generation of spallation neutron sources in the US, Japan and Europe. In the early 1980s, Bunakov and Gudkov [5] and Flambaum and Sushkov [6] noted that measurements with unpolarized targets of the energy dependence near p-wave resonances of parity-violating correlations in polarized neutron capture could constrain Podd T-odd interactions. Although parity-violating asymmetries of the order of a few percent had earlier been observed in polarized neutron capture, the idea was not pursued further. Instead, in a separate development, the energy dependence of forward-backward asymmetries in unpolarized neutron capture was used [7] to look for evidence of parity-conserving timereversal noninvariance. The study was restricted to a single resonance, but demonstrated that the method could in principle yield a competitive bound on the strength of the P-even T-odd interaction among nucleons if extended to an appropriate sample.
In this paper, we expand on the analysis of P-odd Tviolation suggested in [5,6]. The purpose of the work is to establish to what extent T-even contributions may mask the perturbation due to the P-odd T-odd interaction of interest. Despite uncertainties in the precise values of resonance parameters, the theory of how to model neutron resonance reactions is well enough established to allow us to estimate the order of magnitude of these contributions. We follow the Flambaum and Sushkov model for the energy dependence of the relevant asymmetries.
Our results confirm that there is a shift in the zero of the capture correlation asymmetry from the resonance energy E p , of order (v PT /v P )Γ, where v PT is the rootmean-square (rms) value of compound nucleus matrix elements of the unknown P-odd T-odd interaction and v P is the rms value of compound nucleus matrix elements of the P-odd weak interaction. Our results also indicate that, in the epithermal regime, electromagnetic and weak interaction effects give rise to two T-even displacements of the zero crossing: one of order and the other of order (Γ /D)Γ, where Γ is the average width of resonances and D is the average spacing between them.
A fuller account, also including analysis of the effects of distant resonances, is published elsewhere [8].

Two Resonance Analysis
The P-odd asymmetry of interest to us measures the strength of the dependence of the differential cross section for the (n, γ) reaction on the pseudo-scalar σ σ · n γ , where σ σ is the transverse polarization of the neutron beam and n γ is the unit vector in the direction of observed photon's momentum. In the notation of the decomposition of the differential cross section for the (n, γ) reaction in Eq. (17) of [6], we study the energy dependence of the combination A ≡ a 9 -a 12 /3, which is precisely the coefficient of σ σ · n γ when all terms in Eq. (17) of [6] are considered. For the sake of definiteness, we restrict ourselves (as do Flambaum and Sushkov in section 3 of [6]) to radiative neutron capture reactions involving: a) a target nucleus with a ground state and a final nucleus with a 0 + ground state, and; b) gamma-quanta corresponding to transitions from 1 + or 1states of the intermediate compound nucleus to the 0 + ground state of the final nucleus. Then, the general expressions of Appendix A in [6] imply that A = A (13) + A (24) , where A (13) ≡ 2Re [V 1 (V 3 )*] and being abbreviations for the invariant amplitudes V 1 (E, 1 + ), V 2 (E, 1 -, j), V 3 (E, 1) and (E, 1, j) of Eq. (15) in [6], respectively.
In the two resonance approximation, only the terms corresponding to the p-wave resonance at which the measurement is performed and the nearest 1 + s-wave resonance (of energy E s and width Γ s ) are retained in the invariant amplitudes. Thus, Re , The notation for the partial width amplitudes , , etc) differs from that used in [6] (namely, T s , A sf , etc). More importantly, we take the interaction matrix element W sp to include both a P-odd perturbation U and a P-odd T-odd perturbation U, i.e. W sp = u sp + iu sp , where u sp and u sp are real.
Concerning the partial width amplitudes, we assume for the moment that they are all real: is the amplitude for capture by the s-wave [p-wave] resonance of a neutron [of angular momentum j]; is the amplitude for the M1 [E1] electromagnetic deexcitation of the s-wave [p-wave] resonance to the ground state. In terms of these partial width amplitudes, the neutron partial widths of the s-and p-wave resonances are respectively, and the partial gamma width for the M1 and E1 transitions to the ground state are and respectively. Below, the normalized partial width amplitudes are used. Substitution of Eqs. (2.1)-(2.4) into A (13) and A (24) yields (2.5) where, in terms of the coefficients a p , a p , and b p are Equation (2.5) demonstrates that a P-odd T-odd interaction does modify, as claimed in [6], the energy dependence of the P-odd asymmetry associated with the pseudoscalar σ σ · n γ .
A signature of this change is its effect on the location of the zero in the asymmetry (or, equivalently, A).
According to Eq. (2.5), the zero is offset from the resonance energy E p by an amount (2.6) If we suppose that |V 1 | and the 's are comparable when E ≈ E p (the parity-mixing essential to the asymmetry under consideration will not be substantial unless this is the case), then where Γ is the average width of resonances and D is the typical spacing between J = 1 resonances, and the following order of magnitude estimates apply: a p -1 = O(Γ 2 /D 2 ), a p -1 = O(Γ 2 /D 2 ), and b p = O(Γ /D). On omitting terms less than of order (Γ /D) 2 Γ p by at least one order of magnitude [9], the expression for the offset simplifies to (2.7) Observe that Eq. (2.7) implies that ∆E p << Γ p /2.
We can accommodate hard sphere phase shifts in our analysis by formally replacing in Eqs. (2.1)-(2.4) by respectively. We also have to allow for the fact that the radiative partial width amplitudes are, in principle, complex [10]. To this end, we make the substitutions and . In the present two resonance approximation, some of these phases cancel for the combinations of invariant amplitudes appearing in A so that, in fact, A depends only on the phase differences The coefficients a p , a p , and b p become  and ( ) ,