The existence of mirror particles was proposed by Lee and Yang, in the same paper where the possibility of parity violation was put forward [1], for restoring parity in more extended sense: for our particles being left-handed, Parity can be interpreted as a discrete symmetry which exchanges them with their right-handed mirror partners. Hence the parity, violated in each of ordinary and mirror sectors separately, would remain as an exact symmetry between two sectors. Kobzarev, Okun and Pomeranchuk [2] observed that mirror particles cannot have ordinary strong, weak or electromagnetic interactions, and so they must form a hidden parallel world as an exact duplicate of ordinary one interacting with normal matter via gravity. This idea was further expanded, with different twists, in many subsequent papers [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. (See reviews [26,27,28]; for a historical overview, see also Ref. [29]).
At the basic level, one can consider a theory based on a direct product \(G \times G'\) of identical gauge factors which can naturally emerge e.g. in the \(E_8 \times E'_8\) string theory. Ordinary particles belong to the Standard Model \(G = SU(3)\times SU(2)\times U(1)\) or its grand unified extension, while the gauge interactions \(G' = SU(3)'\times SU(2)'\times U(1)'\) (or its respective extension) describes mirror particles. The total Lagrangian must have a form \(\mathcal{L}_\mathrm{tot} = \mathcal{L}\, +\, \mathcal{L}'\, +\, \mathcal{L}_\mathrm{mix}\) where the Lagrangians \(\mathcal{L}\) and \(\mathcal{L}'\), which describe the particle interactions respectively in observable and mirror sectors, can be rendered identical by imposing a mirror parity \(G \leftrightarrow G'\) exchanging ordinary and mirror fermions modulo their chirality. Thus, if mirror sector exists, then all our particles: the electron e, proton p, neutron n, photon \(\gamma \), neutrinos \(\nu \) etc. must have invisible mass degenerate mirror twins: \(e'\), \(p'\), \(n'\), \(\gamma '\), \(\nu '\) etc. which are sterile to our strong and electroweak interactions but interact with ordinary particles via universal gravity.
Mirror matter can be a viable candidate for dark matter [19,20,21,22,23,24,25,26,27,28]. The possible interactions between the particles of two sectors (encoded in \(\mathcal{L}_\mathrm{mix}\)), as the kinetic mixing between photon and mirror photon [9,10,11,12,13] or interactions mediated by heavy messengers coupled to both sectors, as gauge bosons/gauginos of common flavor symmetry [30] or common \(B-L\) symmetry [31, 32], can induce mixing phenomena between ordinary and mirror particles. In fact, any neutral particle, elementary or composite, might have a mixing with its mirror twin. E.g. the photon kinetic mixing [9,10,11,12,13] can be searched experimentally via the positronium – mirror positronium oscillation [33,34,35,36,37] and also via direct detection of dark matter [38,39,40]. The gauge bosons of common flavor symmetry [30] can induce the mixing between the neutral pions and Kaons and their mirror partners, also with implications for dark matter direct search [41]. Three ordinary neutrinos \(\nu _{e,\mu \,\tau }\) can oscillate into their (sterile) mirror partners \(\nu '_{e,\mu \,\tau }\) [14,15,16,17,18]. The respective mass-mixing terms can emerge via the effective interactions which violate \(B-L\) symmetries of both sectors. These interactions can be induced via the seesaw mechanism by heavy gauge singlet “right-handed” neutrinos [23,24,25] which interact with both ordinary and mirror leptons. On the other hand, the same \(B-L\) non-conserving interactions would induce CP violating processes between ordinary and mirror particles and thus generate the baryon asymmetries in both sectors [23,24,25]. In this way, the relation between the dark and observable matter fractions in the Universe, \(\Omega '_B /\Omega _B \simeq 5\), can be naturally explained [23,24,25,26,27,28].
As it was shown in Refs. [42, 43], the present probes do not exclude the possibility that oscillation between the neutron n and its mirror twin \(n'\) is a rather fast process, in fact faster than the neutron decay. The mass mixing, \(\varepsilon (\overline{n} n' + \overline{n}' n)\), emerges from B-violating six-fermion effective operators of the type \((udd)(u'd'd')/M^5\) involving ordinary u, d and mirror \(u',d'\) quarks, with M being a cutoff scale related to new physics beyond the Fermi scale. As far as the masses of n and \(n'\) are exactly equal, they must have maximal mixing in vacuum and oscillate with timescale \(\tau _{nn'} = \varepsilon ^{-1}\sim (M/10\, \mathrm{TeV})^5\) s. Existing experimental limits or cosmological/astrophysical bounds cannot exclude oscillation time \(\tau _{nn'} = \tau \) of few seconds [42].Footnote 1 It is of key importance that in nuclei \(n\rightarrow n'\) transition is forbidden by energy conservation and thus nuclear stability bounds give no limitations on \(\tau \), while for free neutrons \(n{-}n'\) oscillation is affected by magnetic fields and coherent interactions with matter which makes this phenomenon rather elusive [42, 43]. On the other hand, it is striking that \(n\rightarrow n'\) transitions faster than the neutron decay can have far going implications for the propagation of ultra-high energy cosmic rays at cosmological distances [47,48,49], for the neutrons from solar flares [50], for primordial nucleosynthesis [51] and for neutron stars [25].Footnote 2
The possibility of fast \(n{-}n'\) oscillations can be tested in experiments searching for neutron disappearance \(n\rightarrow n'\) and regeneration \(n\rightarrow n' \rightarrow n\) [42] as well as via non-linear effects on the neutron spin precession [43]. In the ultra-cold neutron (UCN) traps \(n\rightarrow n'\) conversion can be manifested via the magnetic field dependence of the neutron loss rates. For the UCN flight times between wall collisions \(t\sim 0.1\) s, the experimental sensitivity can reach \(\tau \sim 500\) s [55] (see also Ref. [56] for a recent status of the UCN sources for fundamental physics measurements).
Several experiments searched for \(n{-}n'\) oscillation with the UCN traps [57,58,59,60,61]. Following the naive assumption [42] that the Earth has no mirror magnetic field, these experiments compared the UCN loss rates in zero (i.e. small enough) and non-zero (large enough) magnetic fields. In this case the probability of \(n{-}n'\) oscillation after a time t depends on the applied magnetic field B as \(P_{B}(t) = \sin ^2(\omega t)/(\omega \tau )^2\), \( \omega = \frac{1}{2} | \mu \mathbf {B} | = (B/1\,\mathrm{mG}) \times 4.5\,\mathrm{s}^{-1}\), where \(\mu = -6 \cdot 10^{-12}\,\)eV/G is the neutron magnetic moment.Footnote 3 For small fields (\(B < 1\) mG or so, when \(\omega t < 1\)) one has \(P_{0} = (t/\tau )^2\), while for large fields (\(B > 20\) mG or so, when \(\omega t \gg 1\)) oscillations are suppressed, \(P_{B} < (1/\tau \omega )^2 \ll (t/\tau )^2\). In this way, lower bounds on the oscillation time were obtained under the no mirror field hypothesis, the strongest being \(\tau > 414\) s at 90% CL [58] adopted by the Particle Data Group [62].
However, the above limits become invalid in the presence of mirror matter and/or mirror magnetic field [43]. If the Earth possesses mirror magnetic field \(B'\), then it would show up as uncontrollable background suppressing n-\(n'\) oscillation even if the ordinary magnetic field is screened in the experiments, i.e. \(B=0\). However, if experimental magnetic field is tuned as \(B\approx B'\), then n-\(n'\) oscillation would be resonantly amplified. In addition, in this case one could observe the strong dependence of the UCN losses on the direction of magnetic field [43].
Interestingly, some of the experiments have shown that the UCN loss rates depend on the magnetic field direction at certain values of magnetic fields, in particular, the measurements performed with vertical magnetic fields \(B\simeq 0.2\) G reported in Ref. [60]. The detailed analysis of these experimental data indicates towards \(5.2\sigma \) deviation from the null hypothesis [63] which can be interpreted as a signal for \(n{-}n'\) oscillation in the presence of mirror magnetic field \(B'\sim 0.1\) G at the Earth.
A dedicated experiment [61] tested \(n{-}n'\) oscillation in the presence of mirror magnetic field, with a series of measurements at different values of applied (vertical) magnetic field varied from 0 to 0.125 G with a step of 0.025 G, also altering its direction from up to down. Its results, yielding the limit \(\tau > 12\) s for any \(B'\) less than 0.13 G, restrict the parameter space (\(\tau , B',\beta \)) which can be responsible for the above \(5.2\sigma \) anomaly but do not cover it completely.
In this paper we report the results of additional measurements aiming to test the parameter space related to \(5.2\sigma \) anomaly [63]. We essentially repeated the experiment [60] with different values of the applied magnetic field. New limits on \(n{-}n'\) oscillation time were obtained as a function of mirror magnetic field \(B'\), which strongly restrict the parameter space relevant for the above anomaly, however still leave some margins for it. The paper is organized as follows. First we discuss \(n{-}n'\) oscillation in the presence of mirror magnetic field. Then we describe the experiment and show its results. At the end we confront our findings with the results of previous experiments and draw our conclusions.