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Coherence in Scattering of Massive Weakly Interacting Neutral Particles of Nuclei

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Abstract

The paper presents a novel approach to the description of the nonrelativistic weak interaction of a massive neutral particle (lepton) and a nucleus, in which the latter retains its integrity. The cross section of such a process is a sum of the elastic (or coherent) contribution, when the nucleus remains in its original state, and the inelastic (incoherent) contribution, when the nucleus is in an excited state. Smooth transition from elastic scattering to inelastic scattering is governed by the dependence of the nuclear form factors on the momentum transferred to the nucleus. The intensity of the weak interaction is set by the parameters that determine the contributions to the probability amplitude from the scalar products of the leptonic and nucleon currents. The resulting expressions are of interest, at least in the problem of direct detection of neutral massive weakly interacting particles of dark matter, since in this case, in contrast to the generally accepted approach, both elastic and inelastic processes are simultaneously considered. It is shown that the presence of the inelastic contribution accompanied by emission of characteristic radiation (photons) from the deexcitation of the nucleus turns out to be decisive when the coherent cross section is strongly suppressed or cannot be detected. The former takes place if the corresponding interaction constant is close to zero or if the momentum transferred to the nucleus is too great and the coherence condition is not met. When the measurable recoil energy of the nucleus is below the detection threshold, the coherent cross section cannot be seen at all. In this situation, “inelastic” photons are the only detectable signal of the interaction between dark matter particles and matter. Therefore in order to extract maximum information about dark matter particles, one should plan experiments aimed at the direct detection of dark matter particles in a setting that allows one to detect both the recoil energy of the nucleus and the gamma quanta from the deexcitation of the nucleus

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Notes

  1. Except the DAMA/LIBRA results [11], which are ignored without adequate refutation.

  2. The square of the matrix element \({{\left| {{{\mathcal{M}}_{{mn}}}} \right|}^{2}}\) usually does not depend on the angle ϕ. Integration is performed over it.

  3. Derivation of formula does not depend on the explicit form of combinations of \(\gamma \)-matrices \(O_{k}^{\mu }\) (see [1–3]).

  4. Hereinafter, unless stated otherwise, the proton mass is taken to be equal to the neutron mass \(m = {{m}_{p}} = {{m}_{n}}\).

  5. To be in correspondence to [1–3], the product \(\chi _{m}^{*}(\{ {{r}^{{(k)}}}\} ){{\chi }_{n}}(\{ r\} )\), like \(\chi _{n}^{*}(\{ r\} ){{\chi }_{n}}(\{ r\} )\), will be considered equal to unity. Validity of this assumption is a subject for a separate study.

  6. The kinematic λ function is defined by the expression \({{\lambda }^{2}}(s,{{m}^{2}},m_{\chi }^{2}) \equiv {{s}^{2}} + {{m}^{4}} + m_{\chi }^{4} - 2s{{m}^{2}}\)\( - \,\,2sm_{\chi }^{2} - 2{{m}^{2}}m_{\chi }^{2} = (s - {{(m + {{m}_{\chi }})}^{2}})(s - {{(m - {{m}_{\chi }})}^{2}})\).

  7. Formally due to the condition \(\xi _{ - }^{2} = {{E}_{\chi }} - {{m}_{\chi }} \to 0\) and \(\lambda _{ - }^{2} = E - m \to 0\).

  8. For simplicity, we consider in all further calculations that \({{g}_{{\text{i}}}} \simeq {{g}_{{\text{c}}}} \simeq 1\).

  9. Constant quantities: \(1{\text{ fm}} = \frac{1}{{0.1973{\text{ GeV}}}} = \frac{1}{{197.3{\text{ MeV}}}}\), \({\kern 1pt} {\text{GeV}}\,{\kern 1pt} {{ = 10}^{3}}{\kern 1pt} {\text{MeV}}{\kern 1pt} ,\;{\kern 1pt} {\text{MeV}}{\kern 1pt} = 1\).

  10.  As was already mentioned, for each of the nuclei this moment determines \(T_{A}^{{\max }}\) in the corresponding plots.

  11.  Where it is stated that the scalar interaction is noticeably weaker than the spin-dependent one, because the initial and final nuclear states change, and only the spin-dependent interaction should be considered.

  12.  The first excited 3/2+ state of 129Xe is at 39.6 keV (half-life 0.97 ns) above the ground 1/2+ state [59]. The first excited 1/2+ state of 131Xe has the energy of 80.2 keV above the ground 3/2+ state (0.48 ns) [60].

  13.  This “classical” inelastic approach should not be mixed up with “inelasticity” caused by the transition of the incident χ1 lepton (of dark matter) to a more massive χ2 lepton (also from dark sector). The nucleus is thought of as being unchanged, i.e., interacting coherently. See, for example [14, 17, 18, 20, 23, 24, 35, 61].

  14.  In 127I, the first excited state 7/2+ is 57.6 keV above the ground state. In 133Cs, the lowest state 5/2+ has the energy of 81 keV.

  15.  Note that nucleons in [1] are not free, and the nuclear structure is considered in terms of nuclear form factors.

  16.  The relativistic version of this description is considered separately [75].

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ACKNOWLEDGMENTS

The author is grateful to V. Naumov, E. Yakushev, N. Russakovich, and I. Titkova for discussions and important comments.

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  1. Dzhelepov Laboratory of Nuclear Problems, Joint Institute for Nuclear Research, 141980, Dubna, Russia

    V. A. Bednyakov

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Bednyakov, V.A. Coherence in Scattering of Massive Weakly Interacting Neutral Particles of Nuclei. Phys. Part. Nuclei 54, 239–273 (2023). https://doi.org/10.1134/S1063779623020028

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