Time reversal invariance violation in neutron-nucleus scattering

TL;DR

This paper calculates the expected size of T and P violation effects in neutron-nucleus scattering using statistical nuclear models, providing bounds based on fundamental physics parameters.

Contribution

It introduces a statistical approach to estimate T,P-violating matrix elements in chaotic nuclear states, linking them to fundamental constants.

Findings

Calculated root mean square values of T,P-violating matrix elements.

Derived upper bounds on matrix element ratios using current fundamental parameter limits.

Expressed results in terms of nuclear interaction constants, meson exchange, QCD theta-term, and quark chromo-EDMs.

Abstract

Planning and interpretation of the experiments searching for the time reversal (T) and parity (P) violation in neutron reactions require values of the matrix elements of the T,P-violating nuclear forces between nuclear compound states. We calculate the root mean square values and the ratio of the matrix elements of the T,P-violating and P-violating interactions using statistical theory based on the properties of chaotic compound states and present the results in terms of the fundamental parameters in four different forms: in terms of the constants of the contact nuclear interaction, meson exchange constants, QCD theta-term and quark chromo-EDMs. Using current limits on these parameters, we obtain the upper bounds on the ratio of the matrix elements.

Figures

Equations

$$P\sim \frac{W_{sp}}{E_s-E_p}\sqrt{\frac{{\Gamma }_s^n}{{\Gamma }_p^n}},$$

$$\overline{(C_i^n{)}^2}=\frac{1}{\bar{N}}\Delta ({\Gamma }_{spr},E^n-E_i),$$

$$\Delta ({\Gamma }_{spr},E^n-E_i)=\frac{{\Gamma }_{spr}^2/4}{(E^n-E_i{)}^2+{\Gamma }_{spr}^2/4},$$

$$\frac{\Delta {\sigma }_{PT}}{\Delta {\sigma }_P}=\kappa \frac{⟨{\psi }_p\left|W_{PT}\right|{\psi }_s⟩}{⟨{\psi }_p\left|W_P\right|{\psi }_s⟩}.$$

$$W_P=\frac{Gg}{2\sqrt{2}m}\left\{(\sigma \mathbf{p}),\rho \right\},$$

$$W_{PT}=\frac{G\eta }{2\sqrt{2}m}(\sigma \nabla )\rho .$$

$$v=\sqrt{\overline{⟨{\psi }_p\left|W_P\right|{\psi }_s⟩⟨{\psi }_s\left|W_P\right|{\psi }_p⟩}},$$

$$w=\sqrt{\overline{⟨{\psi }_p\left|W_{PT}\right|{\psi }_s⟩⟨{\psi }_s\left|W_{PT}\right|{\psi }_p⟩}}.$$

$$C$$

$$\mid C\mid$$

$$\nabla \rho \sim \frac{\rho }{R_N}\sim \frac{\rho }{r_0{A}^{1/3}},$$

$$\frac{w}{v}\sim \frac{\eta }{g{A}^{1/3}}.$$

$$v=\frac{1}{\sqrt{\bar{N}}}{\left\{\sum _{abcd}{\nu }_a\left(1-{\nu }_b\right){\nu }_c\left(1-{\nu }_d\right)\frac{1}{4}{\left|V_{ab,cd}-V_{ad,cb}\right|}^2\Delta \left({\Gamma }_{spr},{ϵ}_a-{ϵ}_b+{ϵ}_c-{ϵ}_d\right)\right\}}^{\frac{1}{2}}.$$

$$v=\frac{1}{\sqrt{\bar{N}}}\sqrt{{\left({\Sigma }_{pp}^{(P)}{g}_p\right)}^2+{\left({\Sigma }_{nn}^{(P)}{g}_n\right)}^2+\left({\Sigma }_{pn}^{(P)}{g}_p{g}_n\right)},$$

$$w=\frac{1}{\sqrt{\bar{N}}}\sqrt{{\left({\Sigma }_{pp}^{(PT)}{\eta }_p\right)}^2+{\left({\Sigma }_{nn}^{(PT)}{\eta }_n\right)}^2+\left({\Sigma }_{pn}^{(PT)}{\eta }_p{\eta }_n\right)},$$

$$v=K_P\sqrt{g_n^2+k{g}_p^2},$$

$$w=K_{PT}\sqrt{{\eta }_n^2+k{\eta }_p^2}.$$

$$k=\frac{Z^2+2ZN}{N^2+2ZN}=0.76.$$

$$g_p$$

$$\displaystyle-11.3(h^{0}_{\omega}+h^{1}_{\omega})\Bigg{]}\,,$$

$$g_n$$

$$\displaystyle-12.8(h^{0}_{\omega}+h^{1}_{\omega})\Bigg{]}\,,$$

$$v=$$

$$v_{updated}$$

$$w_{updated}=$$

$$\frac{w}{v}$$

$$\mathcal{W}\left({\mathbf{r}}_1-{\mathbf{r}}_2\right)$$

$$\times \left[{\bar{g}}_0{\mathbf{\tau }}_1\cdot {\mathbf{\tau }}_2+{\bar{g}}_2\left({\mathbf{\tau }}_1\cdot {\mathbf{\tau }}_2-3{\tau }_{1z}{\tau }_{2z}\right)\right]$$

$$+{\bar{g}}_1\left({\tau }_{1z}{\mathbf{\sigma }}_1-{\tau }_{2z}{\mathbf{\sigma }}_2\right)\},$$

$$-{\eta }_p={\eta }_n=5\times 10^6\bar{g}\left({\bar{g}}_1+0.4{\bar{g}}_2-0.2{\bar{g}}_0\right).$$

$$\frac{w}{v}$$

Full text

I Introduction

A very popular way to search for time-reversal (T) and parity (P) violation and to test unification theories is based on the measurements of electric dipole moments (EDMs) of elementary particles and atomic systems. So far this method has produced stringent limits on EDMs which exclude or bound many models (see reviews in Refs. Ginges2004 ; Pospelov2005 ; Engel2017 ; Yamanaka2017 ; Chupp2019 ). Studies of T,P-violating (also known as T,P-odd) effects via EDM also give limits on the axion and relaxion interactions Stadnik2017 . An efficient alternative method is measurement of T,P-odd effects in neutron-nucleus scattering. This method is motivated by the millionfold enhancement of parity violation in neutron reactions near $p$-wave nuclear compound resonances, which was predicted in Refs. Sushkov80 ; Sushkov82 ; Flambaum84 ; Flambaum85 . The first confirmation was obtained in experiments performed at the Joint Institute for Nuclear Research in Dubna Alfimenkov1983 ; Alfimenkov1984 ; then a very extensive experimental study was done in several laboratories, including the Joint Institute for Nuclear Research (Dubna), Petersburg Institute of Nuclear Physics, KEK (Tsukuba), and especially in Los Alamos (see reviews in Refs. MitchellReview99 ; MitchellReview2001 ). This activity continues now (see, for example, the recent experimental paper Okudaira2018 and references therein). A similar mechanism of enhancement should work for the T,P-odd effects Bunakov1983 ; TRIV87 ; Gudkov1992 ; Flambaum_Gribakin95 . An unusual statistics of P-violating and T,P-violating effects, namely random-sign observables not vanishing upon averaging, was demonstrated in Refs. FlamGrib94 ; BerengutFlamGrib . Experiments searching for T,P-violating effects are in progress in Japan and the United States Snow2017 ; Okudaira2018 ; Palos2018 ; Kitaguchi2018 ; Beda2007 ; Bowman96 .

Without any enhancement, the effects of P violation in low-energy nuclear reactions are extremely small, $\sim 10^{-7}$ (e.g., in the proton scattering on hydrogen and helium, and neutron radiative capture by protons) Gericke2011 . The formula for a P-violation effect near a $p$-wave compound resonance may be presented as Sushkov80 ; Sushkov82 ; Flambaum84 ; Flambaum85 111We omit the numerical coefficient which depends on the specific process induced by the neutron capture.

[TABLE]

where $W_{sp}$ is the matrix element of the parity-violating interaction mixing $s$ and $p$ resonances, $E_{s}-E_{p}$ is the energy interval between these resonances, and $\Gamma_{s}^{n},\Gamma_{p}^{n}$ are the neutron widths of these resonances. We see that there are two reasons for the enhancement of P violation near $p$-wave compound resonances. First, in a nucleus excited by neutron capture the interval $E_{s}-E_{p}$ between the chaotic compound states (resonances) of opposite parity is very small, and this enhances by three orders of magnitude the mixing of these states by the weak P-violating interaction between nucleons. The second reason is that the admixture of opposite-parity states allows neutron capture in the $s$ wave to contribute to the $p$-wave resonance. At small neutron energies the $s$-wave amplitude is three orders of magnitude larger than the p-wave amplitude ($\sqrt{\Gamma_{s}^{n}/\Gamma_{p}^{n}}\sim 10^{3}$). As a result of these two $10^{3}$ factors, the P-violating parts reach 1–10% of reactions’ cross sections and become accessible to experimental scrutiny. T,P-violating effects are also produced by the parity-violating interaction; therefore, Eq. (1) and the enhancement mechanism works for them too Bunakov1983 ; TRIV87 ; Gudkov1992 ; Flambaum_Gribakin95 .

For the experiments to produce useful results we need theory for their interpretation. At first glance, it seems impossible, since chaotic compound states are very complicated. However, chaos allows us to develop a statistical theory, similar to the Maxwell-Boltzmann theory for macroscopic systems, which actually gives very accurate predictions. We developed such a theory, including a method to calculate matrix elements between chaotic states in finite systems (in excited nuclei, atoms, and molecules) Flam93 ; Flam93PRL ; Flambaum1994V ; Flambaum1995a ; Flambaum1995b ; FlamGrib2000 ; Bowman96V . We briefly present the ideas below.

An increase in the excitation energy of a nucleus increases the number of its active particles $k$ and available orbitals $p$, leads to an exponential increase of the density of energy levels $\sim p!/[(p-k)!k!]$, and brings the system into a state where the residual interaction between particles exceeds the intervals between the energy levels. The eigenstates $\left.\middle|n\right\rangle=\sum_{i}C_{i}^{n}\left.\middle|i\right\rangle$ become chaotic superpositions of thousands or even millions of Hartree-Fock basic states $\left.\middle|i\right\rangle$. All medium and heavy nuclei and atoms with an open $f$ shell have chaotic excited compound states in the discrete spectrum and/or chaotic compound resonances. The idea of Refs. Flam93 ; Flam93PRL is to treat the expansion coefficients $C_{i}^{n}$ as Gaussian random variables, with the average values $\overline{C_{i}^{n}}=0$ and variance

[TABLE]

[TABLE]

where $\bar{N}=\pi\Gamma_{spr}/2d$ is the normalization constant found from $\sum_{i}(C_{i}^{n})^{2}=1$, $d$ is the average energy distance between the compound states (resonances) with the same angular momentum and parity, and $\Gamma_{spr}$ is the spreading width of the component calculated using the Fermi golden rule ABohr ; $\bar{N}$ is called the number of principal components.222 Basis states $\left.\middle|i\right\rangle$ with shell-model energies $E_{i}$ close to the energy of a compound state $E^{n}$ (within the spreading width $\Gamma_{spr}$) have the highest weight ($\sim 1/\bar{N}$) and dominate in the normalization sum $\sum_{i}(C_{i}^{n})^{2}=1$. The number of such states is $\bar{N}$.

We have tested this distribution of $C_{i}^{n}$ by the numerical calculations of chaotic compound states in cerium and protactinium atoms Flambaum1994 ; FlambaumGribakina1995 ; FlambaumGribakina1996 ; FlambaumGribakina1998E ; FlambaumGribakina1998A ; FlambaumGribakina1999 ; Viatkina , in highly charged ions with an open $f$ shell GribakinGribakina1999b ; FlambaumGribakina2002 ; DzubaFlambaum2002 ; DzubaFlambaum2013 ; Berengut ; Harabati , in the two-body random interaction model Izrailev1996 ; Izrailev1997a ; Izrailev1997b ; Izrailev2000 and using an analytical approach FlamGrib2000 ; IzrailevGribakin .

The function $\overline{(C_{i}^{n})^{2}}=\Delta(\Gamma_{spr},E^{n}-E_{i})/\bar{N}$ gives the probability to find the basis component $\left.\middle|i\right\rangle$ in the compound state $\left.\middle|n\right\rangle$; i.e. it plays the role of the statistical partition function. The difference from the conventional statistical theory is that the partition function depends on the total energy of the isolated system $E^{n}$ instead of on the temperature of a system in a thermostat [recall the Boltzmann factor $\exp{(-E_{i}/T)}$]. One may compare this with the microcanonical distribution where the equipartition is assumed within the shell of the states with fixed energy $E_{i}$.

Expectation values of matrix elements of any operator $W$ in a chaotic compound state are found as $\overline{\left.\middle|\left\langle n\middle|\right.W\left.\middle|n\right\rangle\middle|\right.^{2}}=\sum_{i}\overline{(C_{i}^{n})^{2}}\left.\middle|\left\langle i\middle|\right.W\left.\middle|i\right\rangle\middle|\right.^{2}$. For example, this formula with $W=a^{+}_{k}a_{k}$ (the occupation-number operator) gives the distribution of the orbital occupation numbers in finite chaotic systems which replaces the Fermi-Dirac (or Bose-Einstein) distribution.333However, numerical calculations Flambaum1994 ; GribakinGribakina1999b ; FlambaumGribakina2002 ; Izrailev1996 give occupation numbers which are close to the Fermi-Dirac distribution.

Average values of the non-diagonal matrix elements are equal to zero, $\overline{\left\langle n\middle|\right.W\left.\middle|m\right\rangle}=0$, while the average values of the squared matrix elements $W^{2}\equiv\overline{\left.\middle|\left\langle n\middle|\right.W\left.\middle|m\right\rangle\middle|\right.^{2}}=\sum_{i,j}\overline{(C_{i}^{n})^{2}}\overline{(C_{j}^{m})^{2}}\left.\middle|\left\langle i\middle|\right.W\left.\middle|j\right\rangle\middle|\right.^{2}$ are reduced to the sum of matrix elements between the Hartree-Fock basis states $\left.\middle|\left\langle i\middle|\right.W\left.\middle|j\right\rangle\middle|\right.^{2}$, where $W$ is any perturbation operator. The distribution of the matrix elements $\left\langle n\middle|\right.W\left.\middle|m\right\rangle$ is Gaussian with the variance given by the $W^{2}$.

For the correlator between two different operators (e.g., P-violating and T,P-violating) we obtain $\overline{\left\langle n\middle|\right.W_{P}\left.\middle|m\right\rangle\left\langle m\middle|\right.W_{T,P}\left.\middle|n\right\rangle}=\sum_{i,j}\overline{(C_{i}^{n})^{2}}\overline{(C_{j}^{m})^{2}}\left\langle i\middle|\right.W_{P}\left.\middle|j\right\rangle\left\langle j\middle|\right.W_{T,P}\left.\middle|i\right\rangle$ Flam93 ; Flam93PRL ; Flambaum1994V ; Flambaum1995a ; Flambaum1995b . Note that our theory predicts the results averaged over several compound resonances.

We have done many tests comparing the statistical theory results for electromagnetic amplitudes FlambaumGribakina1998A , electron recombination rates GribakinGribakina1999b ; FlambaumGribakina2002 ; DzubaFlambaum2002 ; DzubaFlambaum2013 ; FlambaumKozlov2015 ; Berengut ; Harabati and parity-violation effects Flam93 ; Flam93PRL with the experimental data and with numerical simulations. For example, we obtained a thousandfold enhancement of the electron recombination rate with many highly charged tungsten ions due to the very dense spectrum of chaotic compound resonances FlambaumGribakina2002 ; DzubaFlambaum2002 ; DzubaFlambaum2013 ; FlambaumKozlov2015 ; Berengut ; Harabati . These results agree with all available experimental data and predict recombination rates for ions with a high ionization degree, where experiments are limited by existing techniques. Our results are important for thermonuclear reactors which are made from tungsten. Tungsten ions contaminate plasma and significantly affect the energy output.

Using the theory of chaotic nuclear compound resonances, we calculate in this paper the ratio $w/v$ of the root-mean-square values of the matrix elements of the T,P-odd ($w$) and P-odd ($v$) matrix elements. We show the results in terms of the fundamental parameters in five different forms: in terms of the constants of the contact nuclear interaction, meson exchange constants, QCD $\theta$-term, quark chromo-EDMs $\tilde{d}_{u}$ and $\tilde{d}_{d}$, and axion interaction constants. Using latest bounds on $\theta,\tilde{d}_{u}$ and $\tilde{d}_{d}$, and axion interaction constants we arrive at bounds on the magnitude of possible T violation. In the Conclusion section the results are compared with the expected experimental sensitivity to the T,P-violating effects.

II P- and T,P-violating interactions

The ratio of the time-reversal-invariance violating (TRIV) and parity violating (PV) parts of the neutron nuclear cross sections induced by mixing of $s$- and $p$-wave nuclear compound resonances, $\Delta\sigma_{PT}/\Delta\sigma_{P}$, can be expressed as Gudkov2018 ; Gudkov1990 ; Okudaira2018 :

[TABLE]

Here the factor $\kappa$ includes amplitudes of the partial neutron widths which depend on spin channels $J=I\pm 1/2$, where $I$ is the spin of the target nucleus and $J$ is the spin of the compound resonance. For example, for $J=0$, one obtains $\kappa=1$, as in this case $\kappa$ does not depend on neutron partial widths Gudkov1992 ; Bunakov1983 ; TRIV87 .

The ratio $\Delta\sigma_{PT}/\Delta\sigma_{P}$ for the neutron-deuterium scattering was calculated in Ref. Song2011 . However, experiments are planned for heavier nuclei where we expect a millionfold enhancement of the T,P-odd and P-odd effects.

In the short-range interaction limit, the PV operator $W_{P}$ and TRIV operator $W_{PT}$ are

[TABLE]

[TABLE]

Here $G$ is the weak-interaction Fermi constant, $m$ is the nucleon mass, ${\bf p}$ and $\sigma$ are nucleon momentum and spin respectively, and $\rho$ is the nucleon density. Nucleon dimensionless constants $g_{p,n}$ and $\eta_{p,n}$ characterize the strength of the interactions. Note that in the standard definition of angular wavefunctions the matrix element of $W_{P}$ between bound states is imaginary (since the momentum operator ${\bf p}=-i\nabla$) and the matrix element of TRIV operator $W_{PT}$ is real.

We define $v^{2}$ to be the average of the absolute value of the squared PV matrix element, and $w^{2}$ to be the average value of the squared TRIV matrix element between the $s$ and $p$ compound resonances, such that

[TABLE]

[TABLE]

Correlations might exist between the matrix elements of PV and TRIV interactions. The quantity parametrizing such correlations, the correlator, is defined as

[TABLE]

The correlator, which takes values between zero and one, can be useful to deduce the values and signs of TRIV effects, since much is already known about the PV effects. The correlator $C$ was calculated by the same technique as the mean-square matrix element and was found to be Flambaum1995b :

[TABLE]

This result tells us that the correlations between the matrix elements are relatively small so we may neglect them.

II.1 Rough estimate of w$/$v

Naively one would expect from Eqs. (5) and (6) the following relation: $w/v\sim\eta/g$. However, this ratio is actually $A^{1/3}$ times smaller than the ratio of interaction constants Flambaum1995a , where $A$ is the number of nucleons. Indeed, for $\nabla\rho$ in Eq. (6),

[TABLE]

where $r_{0}$ is the internucleon distance, and $R_{N}=r_{0}A^{1/3}$ is the nuclear radius. The momentum in Eq. (5) is approximated as $p\sim p_{F}\sim\hbar/r_{0}$. Thus, the ratio of matrix elements is smaller than the ratio of interaction constants in Eqs. (5) and (6) by a factor of $A^{1/3}$:

[TABLE]

For elements with the number of nucleons in the range 100–250, $A^{1/3}\approx 5$. A detailed discussion of this suppression factor including many-body effects can be found in Ref. Flambaum1995a .

II.2 Dependence of matrix elements on nucleon interaction constants

A general expression for the root-mean-square value of the matrix element $v$ the PV operator (and the matrix element $w$ of the TRIV operator) was derived in Ref. Flam93PRL :

[TABLE]

Here $\nu$ are the orbital occupation numbers given by the Fermi-Dirac distribution in an excited nucleus, numerical values of the matrix elements of the two-nucleon interaction $V_{ab,cd}$ (see Fig. 1) are presented in Refs. Flam93PRL ; Flambaum1995b , and $\Delta\left(\Gamma_{spr},\epsilon_{a}-\epsilon_{b}+\epsilon_{c}-\epsilon_{d}\right)$ is the “spread” $\delta$ function [Eq. (3)] of the change in energy $\epsilon_{a}-\epsilon_{b}+\epsilon_{c}-\epsilon_{d}$. Equation (13) has a clear interpretation. The $\Delta$ function means in fact an approximate energy conservation with an accuracy up to the spreading width $\Gamma_{spr}$ (since the single-particle states are not stationary states in this problem). In the case $\Gamma_{spr}\to 0$ we have $\Delta\left(\Gamma_{spr},\epsilon_{a}-\epsilon_{b}+\epsilon_{c}-\epsilon_{d}\right)\to(\pi\Gamma_{spr}/2)\delta(\epsilon_{a}-\epsilon_{b}+\epsilon_{c}-\epsilon_{d})$. To have a transition, initial states must be occupied (this gives $\nu_{a}$ and $\nu_{c}$) and final states empty (this gives $1-\nu_{b}$ and $1-\nu_{d}$).

The dependence of $v$ and $w$ on the nucleon interaction constants $g$ and $\eta$ [which appear in Eqs. (5) and (6)] can be presented in the following form Flam93PRL ; Flambaum1995b :

[TABLE]

[TABLE]

where $g_{p}$ and $g_{n}$ are proton and neutron weak constants — they characterize the strength of the P-odd weak potential; $\eta_{p},\eta_{n}$ are constants that characterize the strength of the T,P-odd potential, and $\Sigma$ are sums of the weighted squared matrix elements of the weak interaction between nucleon orbitals defined in Eq. (13). Contributions of the cross terms $\Sigma_{pn}^{(P)}g_{p}g_{n}$ and $\Sigma_{pn}^{(PT)}\eta_{p}\eta_{n}$ are small compared to the other terms since they contain products of different matrix elements which have random signs, while in the terms containing squared interaction constants all contributions are positive.

Therefore, we can present $v$ and $w$ in the following form:

[TABLE]

[TABLE]

The coefficient $k$ should be slightly smaller than 1 since in heavy nuclei the number of neutrons $N=1.5Z$, where $Z$ is the number of protons. To make a simple estimate of the sensitivity of $v$ and $w$ to changes in the interaction constants, we assume in the next step that $\Sigma$ from Eqs. (14) and (15) are proportional to the number of interaction terms in the nucleus. There are $Z^{2}/2$ interaction terms between protons, $N^{2}/2$ such terms between neutrons, and $ZN$ terms between a proton and a neutron (Fig. 1). Thus we can write

[TABLE]

Numerical calculations of $v$ and $w$ have been done in Refs. Flam93PRL ; Flambaum1994V ; Flambaum1995a ; Flambaum1995b for specific values of the interaction constants $g_{p},g_{n},\eta_{p}$ and $\eta_{n}$. The values of these constants have been updated since those calculations. Therefore, we would like to find updated values of these constants to insert into Eqs. (16) and (17). The general expressions for $g_{p}$ and $g_{n}$ are Flam97 ; FKS84 ; Flam93 ; Desplanques80

[TABLE]

where $f$ and $h$ are the weak $NN$-meson couplings, and $W_{\rho}$ and $W_{\pi}$ are constants which account for the repulsion between nucleons at small distances as well as for the finite range of the interaction potential. We take $W_{\rho}=0.4$ and $W_{\pi}=0.16$ as in Refs. Flam97 ; FKS84 .

For the choice of constants $g_{p}=4,g_{n}=1$ Noguera86 , numerical calculations give Flam93PRL

[TABLE]

We calculate updated values for $g_{p}$ and $g_{n}$, using the best values of the constants $h$ from Desplanques, Donoghue, and Holstein (DDH) Desplanques80 with an updated $f_{\pi}\equiv h^{1}_{\pi}$, which was recently derived by lattice QCD methods Wasem2012 ; Haxton2013 . Such calculations give $g_{p}=2.6\,,g_{n}=1.5$ (Table 1). Using these values with Eq. (21), we have

[TABLE]

where in the last step we used $k=0.76$.444For $k=1$ we would get $1.51$ meV. Thus we see that our result is not very sensitive to the value of $k$. This theoretical estimate is in excellent agreement with the experimental value $1.39^{+0.55}_{-0.38}\,\,\text{meV}$ Bowman90 ; Frankle91 .

Numerical calculations were done for $\eta_{p}=\eta_{n}$ and gave $w=0.2|\eta_{n}|$ meV Flambaum1995b . Using this result and Eq. (17) we obtain

[TABLE]

II.3 The ratio $w/v$ expressed via meson exchange constants, QCD $\theta$-term, quark chromo-EDMs $\tilde{d}_{u}$ and $\tilde{d}_{d}$, and axion exchange constants

Now we can express the ratio $w/v$ in five different ways: as a function of $\eta_{p},\eta_{n}$, by $\pi_{0}$ meson-exchange coupling constants with the nuclei, by QCD CP-violation parameter $\theta$, by quark chromo-EDMs, and finally by axion exchange constants.

First, to express the ratio $w/v$ as a function of $\eta$, we use Eqs. (22) and (23) to obtain

[TABLE]

If, following Refs. Flambaum1995b ; Flambaum2014 , we take $|\eta_{p}|=|\eta_{n}|$, the ratio in Eq. (24) becomes

[TABLE]

Second, the T,P-odd nuclear forces are dominated by $\pi_{0}$ meson exchange. Such an exchange is described by the interaction Haxton83 ; Khriplovich2000 ; Dmitriev03

[TABLE]

where $\bar{g}=13.6$ is the strong-force T,P-conserving $\pi NN$ coupling constant, $\bar{g}_{0},\bar{g}_{1}$, and $\bar{g}_{2}$, are the strengths of the isoscalar, isovector, and isotensor T,P-violating couplings, respectively, $m_{N}$ is the nucleon mass, $m_{\pi}$ is the pion mass, $\boldsymbol{\sigma}$ is the nucleon spin, $\boldsymbol{\tau}$ is the nucleon Pauli isospin matrix in vector form, and $r_{12}$ is the separation between nucleons. The coupling constants $\eta$ can be expressed in terms of $\bar{g}$ Flambaum2014 :

[TABLE]

Then we have

[TABLE]

Third, using the previous results $\bar{g}\bar{g}_{0}=-0.37\theta$ Crewther1980 ,where $\theta$ is the QCD CP-violation parameter, and $\bar{g}\bar{g}_{1}=\bar{g}\bar{g}_{2}=0$, we can write the ratio $w/v$ as a function of $\theta$:

[TABLE]

Using updated results Yamanaka2017 ; Vries2015

[TABLE]

we can write, still with $\bar{g}\bar{g}_{2}=0$,

[TABLE]

Using the current limit on $\theta$, obtained from constraints on neutron EDM, $|\theta|<10^{-10}$ Yamanaka2017 , we obtain

[TABLE]

Fourth, we can connect our result to the quark chromo-EDM $\tilde{d}$ Pospelov2005 :

[TABLE]

Then

[TABLE]

Using the current limits (Table IV in Swallows2013 ; see also Ref. Graner2016 )

[TABLE]

we obtain

[TABLE]

Finally, a T,P-violating interaction, similar to the pion-exchange-induced Eq. (II.3), may be due to exchange by any scalar particle which has both scalar (with the interaction constant $g^{s}$) and pseudoscalar (with the interaction constant $g^{p}$) couplings to nucleons. The most popular examples are the dark-matter candidates axion Moody ; Marsh and relaxion Graham ; Gupta ; Flacke , which have very small masses.555The limits on the T,P-violating electron-nucleon interactions mediated by the axion or relaxion exchange from EDM measurements were obtained in Ref. Stadnik2017 , where more references may be found. A numerical estimate shows that due to the long range of the interaction the matrix elements in the small-mass case ($e^{-mr}\approx 1)$ are $\sim 1.5$ times larger than the pion exchange matrix elements; i.e., we have instead of Eq. (28) the following estimate:

[TABLE]

The limit on $g^{s}g^{p}$ may be obtained from the proton EDM calculation,666The calculation is similar to that for electron EDM Stadnik2017 .

[TABLE]

and measurement Graner2016 , $|d_{p}|<2\times 10^{-25}e$ cm, $|g^{s}g^{p}|<1\times 10^{-9}$. Using limits from the proton EDM and the 199Hg nuclear-Schiff-moment measurements in Ref. Swallows2013 , the authors of Ref. musolf concluded that the limit on $|g^{s}g^{p}|$ is between $10^{-9}$ and $10^{-11}$. This gives a rather weak limit on $w/v$ induced by axion exchange:

[TABLE]

III Conclusion

Using the bound in Eq. (38), and assuming $\kappa\approx 1$ in Eq (4) (which is reasonable Gudkov2018 and matches experimental results Okudaira2018 ; Palos2018 ), we arrive at

[TABLE]

The limit based on the axion exchange in Eq. (41) is weaker. The vurrent expected experimental sensitivity is Gudkov2014 ; Palos2018

[TABLE]

Thus we confirm that the expected experimental sensitivity in neutron reactions may be sufficient to improve the limits on the TRIV interactions, or possibly to detect them.

Acknowledgements‏

We thank Anne Fabricant for editing the manuscript. We are grateful to William Michael Snow for informing us about the new measurement of the constant $h_{\pi}^{1}$ npdgamma . This work is supported by the Australian Research Council and Gutenberg Fellowship.