**Scientific Basis for Time Reversal Violation in Astrophysics**

Time reversal (T) symmetry of the fundamental particles and interactions of Nature would imply that replaying any motion backwards from the final conditions would cause the system to “retrace its path” back to a time-reversed version of its starting condition. For several decades physicists assumed time reversal symmetry, understood in this sense, applied to all microscopic physical processes. It was therefore a major shock when indirect evidence was discovered in 1964 for the violation of time reversal symmetry [3]. So far we have no fundamental explanation for this violation of time reversal symmetry. We do know, however, that the formalism of quantum mechanics can accommodate time reversal violation without difficulty.

Today the search for new sources of T violation is one of the highest intellectual priorities in nuclear/particle/astrophysics due to its deep connection with a so-far-unexplained puzzle: the apparent lack of large amounts of antimatter in the universe. A. Sakharov [4] showed that the equivalent condition, so-called CP violation (charge and parity reversal) is an essential requirement to produce the matter-antimatter asymmetry from equal proportions generated in the Big Bang. Since Sakharov’s proposal, the evidence in support of Big Bang cosmology has become good enough that researchers use the phrase “precision cosmology” without embarrassment. The strength of T violation observed in weak decays of subatomic particles (K, B, and now D mesons) is several orders of magnitude too small to explain this cosmological matter-antimatter asymmetry. New sources of CP/T violation are therefore needed to explain the observed matter-antimatter asymmetry of the universe within the Big Bang theory. We can testify from our experience that students find this story fascinating and they are excited to learn that they have the possibility to participate in work that can address such a fundamental cosmological question.

Many physicists have proposed theories with T violation from Beyond Standard Model (BSM) particles and interactions, and the search for new sources of T violation provides important insight into this field. T violation searches are known to be one of the the most effective methods to extend the search for new physics to higher energy scales [5]. Theoretical progress to identify the large number of possible sources for T violation and to trace how such new sources would manifest themselves in experiments has made it very clear that no single type of T violation search can be equally sensitive to all possibilities. Since T violation involving neutrons and protons has not yet been observed, a nonzero observation of T violation in any nuclear system would be of fundamental importance [6, 5]. The contribution from known sources of T violation in quark interactions in the proton and neutron is 5-6 orders of magnitude smaller than the present experimental upper bounds, so searches for T violation in these systems are essentially background-free, unambiguous signatures of new physics.

**Neutron Optics Parity and Time Reversal EXperiment (NOPTREX)**

The NOPTREX experiment can realize the motion-reversed condition directly corresponding to a T transformation by effectively passing the neutron beam backwards through the target with all spins reversed. The difference in the transmitted neutron intensity in the forward and reversed conditions measures a T-odd phase shift in the transmitted neutron wave function. This experiment has the potential to push the mass scale of new particles to >100 TeV [5, 7], beyond the reach of the Large Hadron Collider (LHC) at the European Organization for Nuclear Research (CERN).

The basic idea of this experiment is to investigate T-violation in neutron interactions with heavy nuclei at certain narrow neutron-nuclear resonances with orbital angular momentum L=1 by searching for a phase shift in the transmitted neutron wave function from a term of the form S = σn · (kn × I), where σn is the spin of the neutron, kn is the neutron momentum, and I is the spin of the nucleus. This term is both P-odd and T-odd. Because the neutron spends about 10^6 more time inside the nucleus than normal in this resonant condition, it can amplify both P-odd and T-odd effects by the same factor. P-odd amplitudes of 2-10% have already been measured in certain nuclei (139La, 131Xe, 81Br), which are indeed 10^6 times larger than expected from P-odd interactions between nucleons. The mechanism for this amplification has been understood theoretically for decades: in fact the amplified effects were predicted theoretically before they were measured in experiment, and the extensive measurements performed on many nuclei [8] are all understandable within this resonance amplification model. The resonant amplification mechanism works both for P violation and for the P-odd/T-odd effect of interest. Furthermore, the ratio of the P-odd/T-odd amplitude to the P-odd amplitude measured on the same L=1 resonance is quite insensitive to unknown properties of the compound resonant states involved.

The transmission difference can be expressed using the theory of neutron optics as the T-odd, P-odd scattering amplitude ∆σTP = 4π/k ℑ(f+ − f−), where f± are the forward scattering amplitudes for configurations with positive or negative values of S, respectively. The neutron fluence at the Japanese Spallation Neutron Source (JSNS) at J-PARC is high enough to measure the ratio of the P-odd/T-odd amplitude to P-odd amplitude to 1×10^−5 , which would improve the sensitivity to P-odd/T-odd neutron-nucleus interactions by about 2 orders of magnitude [9, 10, 11, 12, 13]. New theoretical work has sharpened our understanding of the potential reach of this experiment [14] and clarified the range of possible sources of T violation in the nucleon sector. In principle, this null test for T violation is free from the effects of so-called “final state interactions” [15, 9, 10] which can sometimes generate “fake” T-odd effects. We have demonstrated that our new approach [10] is robust against systematic errors from misalignments between the various spin and momentum vectors. New technology in neutron and nucleus polarization has made this experiment possible. The separation of neutron energies by neutron time-of-flight at pulsed neutron sources allows a powerful search for systematic errors by looking on and off the neutron resonance energy for both the transmitted and scattered neutrons.

While the ultimate goal is the search for T-violation, there are two intellectual motivations to search for large P-odd effects in other nuclei. One is to find additional candidate nuclei for the NOPTREX T violation search. The sensitivity of our search for a P-odd and T-odd term in neutron transmission is directly proportional to the size of the P-odd polarized neutron transmission asymmetry. Since there are many nuclei which have never been probed for large P-odd effects, we may discover new candidate nuclei and resonances for NOPTREX. A search for P violation in unexplored regions of nucleon number A can also test our understanding of P violation in neutron-nucleus resonances [8]. Second, we can test the statistical theories which have described the global features of neutron-nucleus parity violation in the regions of A probed so far, including an interesting recent speculation [16]. The neutron beamtime awarded in this long-term proposal is enough to perform sensitive searches for P-odd effects in several unmeasured heavy nuclei, and even null results contribute information to the statistical analysis. The proposed work will therefore produce publications and address important scientific issues no matter what the results.

**References**

[1] W.M. Snow, “Transmission Measurements with Polarized Epithermal Neutrons and Unpolar- ized (n,γ) Angular Distribution Measurements for Parity Violation and Time Reversal Vio- lation Physics,” approved proposal for the J-PARC Long-Term proposal competition (2020), 2020-02-28.

[2] D. Schaper, C. Auton, L. Barron-Palos, M. Borrego, A. Chavez, L. Cole, C. B. Crawford, J. Curole, H. Dhahri, J. Doskow, W. Fox, M.H. Gervais, B. M. Goodson, K. Knickerbocker, P. Jiang, P. M. King, H. Lu, M. Mocko, D. Olivera-Velarde, S.I. Penttila, A. Perez-Martin, B. Short, W. M. Snow, K. Steffen, J. Vanderwerp, and G. Visser, “A modular apparatus for use in high-precision measurements of parity violation in polarized eV neutron transmission”, Nucl. Inst. Meth. A 969, 163961 (2020).

[3] Christenson, J. H., Cronin, J. W., Fitch, V. L., Turlay, R. (1964). “Evidence for the 2π Decay of the K0 2 Meson System”. Phys. Rev. Lett. 13, 4 (1964).

[4] A. D. Sakharov, “Violation of CP Invariance, C Asymmetry, and Baryon Asymmetry of the Universe”, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967).

[5] J. Engel, M. J. Ramsey-Musolf, U. van Kolck, “Electric Dipole Moments of Nucleons, Nuclei, and Atoms: The Standard Model and Beyond”, Prog. Part. Nucl. Phys. 71, 21 (2013).

[6] M. Pospelov and A. Ritz, “Electric dipole moments as probes of new physics”, Ann. Phys. 318, 119 (2005).

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[8] G. E. Mitchell, J. D. Bowman, H. A. Weidenmüller, “Parity violation in the compound nucleus”, Rev. Mod. Phys. 71, 445 (1999).

[9] V. P. Gudkov, Y-H. Song, “Discovery potential in a search for time-reversal invariance violation in nuclei”, Hyperfine Interact. 214, 105 (2013).

[10] J. D. Bowman, V. P. Gudkov, “Search for time reversal invariance violation in neutron transmission”, Phys. Rev. C90, 065503 (2014).

[11] V. E. Bunakov and V. P. Gudkov, “Parity violation and related effects in neutron induced reactions”, Nucl. Phys. A401, 93 (1983).

[12] V. P. Gudkov, “On CP violation in nuclear reactions”, Phys. Rept. 212, 77 (1992).

[13] A. G. Beda and V. R. Skoy, “Current status of research on T invariance in neutron-nuclear reactions”, Physics of Particles and Nuclei 38, 775 (2007).

[14] P. Fadeev and V. V. Flambaum, Phys. Rev. C 100, 015504 (2019).

[15] V. P. Gudkov, “On the test of CP violation models in neutron reactions”, Phys. Lett. B243, 319 (1990).

[16] V. V. Flambaum and A. J. Mansour, “Parity and time-reversal invariance violation in neutron-nucleus scattering”, Phys. Rev. C 105, 015501 (2022).