On the Neutron Transition Magnetic Moment

We discuss the possibility of the transition magnetic moments (TMM) between the neutron n and mirror neutron n', its hypothetical sterile twin from parallel particle"mirror"sector. The neutron can be spontaneously converted into mirror neutron via these TMM's (in addition to the more conventional transition channel due to n-n' mass mixing) interacting with the magnetic field B as well as with mirror magnetic field B'. We derive analytic formula for the average probability of n-n' oscillation and consider possible manifestations of the neutron TMM effects. In particular, we discuss potential role of these effects in the neutron lifetime measurement experiments leading us to new, testable predictions.


Introduction
In the Refs. [1,2] the idea was conjectured that the neutron n can be transformed into a sterile neutron n ′ that belongs to the hypothetical parallel mirror sector. Mirror sector is an exact copy of the ordinary particle sector with the identical fermion content and identical gauge forces, different in the respect that the strong and electroweak forces described by the Standard Model (SM) SU(3) × SU(2) × U(1) act only between ordinary particles, and gauge forces of the mirror Standard Model (SM ′ ) SU(3) ′ × SU ′ (2) × U(1) ′ act only between mirror particles (for a review, see e.g. [3]). The particle physics of two sectors are exactly the same due to a discrete symmetry P Z 2 under exchange of all ordinary and mirror particles modulo the fermion chirality which symmetry can be considered as a generalization of parity, namely for our sector being left-handed, mirror sector can be considered as right-handed. In fact, the parity restoration was the initial motivation for introducing the mirror sector [4]. However, in principle, this discrete symmetry can be simply Z 2 without the chirality change, in which case the parallel sector will have identical physics in left basis [3]. Most of physical applications do not depend on the type of this discrete symmetry, and we shall continue to coin parallel sector as a mirror sector in both cases. If Z 2 or P Z 2 is an exact symmetry, then each ordinary particle as the electron, photon, proton, neutron etc. must have a mirror twin, the electron ′ , photon ′ , proton ′ , neutron ′ etc. exactly degenerate in mass. Interactions between the two sectors are possible via the common gravitational force, but in principle, also via some very feeble interactions induced by new physics beyond the Standard Model. These new interactions, typically related to the higher order effective operators, can arise at some a priori unknown energy scale which in principle might be not very far, and can be even as small as few TeV. New interactions must respect gauge invariances of both sectors and can manifest itself in a mixing phenomena between the neutral particles of two sectors, in particular as the photon-mirror photon kinetic mixing [5], the mixing of (active) ordinary neutrinos and (sterile) mirror neutrinos [6], the above mentioned mixing of the neutron and mirror neutron [1,2], and similar mixings between other neutral particles as pions and Kaons induced e.g. by common gauge flavor symmetry between two sectors [7]. Mirror matter is a viable candidate for light dark matter, a sort of asymmetric atomic dark matter consisting dominantly of mirror hydrogen and helium [8]. Therefore, any transition of a neutral particle of ordinary sector into a neutral particle of another sector can be considered as a conversion to dark matter. However, in this paper we will not focus on the cosmological aspects of mirror matter as Dark Matter, and shall discuss essentially the features of a laboratory search for n → n ′ transition.
This transition was suggested to occur via the mass mixing term εnn ′ + h.c. in Refs. [1], and the masses of n and n ′ are exactly the same due to the initial assumption of mirror parity leading to the the degeneracy between the ordinary and mirror particles. Then, the phenomenon of free n−n ′ oscillation is essentially described in the same way as free neutronantineutron oscillation n −n due to the Majorana mass term ǫ nn n T Cn + h.c. [9]. Namely, for free non-relativistic neutrons in vacuum but in presence of magnetic fields B the time evolution of n −n and n − n ′ systems are described respectively by the Hamiltonianŝ where m, m ′ and µ, µ ′ are respectively the masses and magnetic moments of the neutron and mirror neutron, and σ = (σ 1 , σ 2 , σ 3 ) stands for Pauli matrices. However, between these two cases there are important differences: (i) First of all, n −n transition changes the baryon number B by two units, ∆B = 2, while n − n ′ transition changes B by one unit, ∆B = 1, but it changes also the mirror baryon number B ′ by one unit, ∆B ′ = −1. Moreover, n − n ′ conserves the combination of two baryon numbersB = B + B ′ , ∆B = 0, while n −n violates alsoB.
(ii) Exact degeneracy between the neutron and antineutron masses (and also magnetic moments) is based on fundamental CPT invariance, which cannot be violated in the frames of local relativistic field theories. (However, environmental energy splitting can be induced by some long range fifth-forces related e.g. to very light B − L baryophotons [10].) The degeneracy between the neutron and mirror neutron is related to mirror parity which in principle can be spontaneously broken [11]. The order parameters of this breaking can be naturally small and the mass splitting between n and n ′ states can be rather tiny, say as small as 10 −100 neV in which case it can have implications for the neutron lifetime problem [12]. Both n −n and n − n ′ oscillations are affected by the matter medium and magnetic fields. However, in the case of n − n ′ oscillation, the presence of mirror matter and mirror magnetic field B ′ will be manifested as uncontrollable background.
(iii) n −n transition can be experimentally manifested [9] as the antineutron appearance in the beam of free neutrons, or as nuclear disintegration (A, Z) → (A − 2, Z − ∆Z) + π's due to n →n conversion of a neutron bound in nuclei and its subsequent annihilation in the medium of other nucleons producing multiple pions. As for n → n ′ transition, it cannot occur for a bound neutron simply because of energy conservation and thus it has no influence for the stability of nuclei [1]. As for n − n ′ oscillation, it can be experimentally manifested as the neutron disappearance n → n ′ or regeneration n → n ′ → n [1].
(iv) There are severe experimental limits on n −n mixing mass ǫ nn , usually expressed as limits on free oscillation time τ nn = 1/ǫ nn . 1 Namely, direct experimental limit on free n − n ′ oscillation is τ nn > 0.86 × 10 8 s [13] while the limits from the nuclear stability yield τ nn > 2.7 × 10 8 s [14]. The latter corresponds to the upper bound ǫ nn < 2.5 × 10 −24 eV. As for n − n ′ oscillation, it is not excluded to be very fast, even faster than the neutron decay itself [1]. Several dedicated experiments were preformed for testing the ultra-cold neutron (UCN) disappearance due to n → n ′ transition [15,16,17,18,19,20]. Assuming the absence of the mirror magnetic field at the Earth the strongest limit was obtained in Ref. [17] and it implies τ nn ′ > 448 s. This limit, however, becomes invalid if the Earth possess a mirror magnetic field [2]. In fact, some of experimental data show significant anomalies, the strongest one of 5.2σ deviation from the null hypothesis [21], which can be interpreted by n − n ′ oscillation with τ nn ′ ∼ 10 s in the presence of mirror magnetic fields B ′ ∼ 0.1 G. The latter could be induced by a tiny fraction of captured mirror matter via the electron drag mechanism [22]. The summary of all UCN experimental limits on n − n ′ oscillation time can be found in Ref. [20]. In fact, for B ′ > 0.3 G the limits are practically absent and n − n ′ oscillation time can be as small as a second. Let us also remark that fast n − n ′ oscillation can have interesting implications for the propagation of ultra-high energy cosmic rays at cosmological distances [23], and it can be tested via n → n ′ → n regeneration experiments like these discussed in this paper and in [24,25].
(v) The neutron magnetic dipole moment µ determines the Larmor precession of the neutron spin in an external magnetic field B. It is known with high precision, µ = −1.91304273 ± 0.00000045 µ N [26] in units of the nuclear magneton µ N = e/2m p . It is convenient to use it as µ = 6.0 · 10 −12 eV/G. In principle, the neutron could have also an electric dipole moment which would violate P and CP-invariance, however there are severe experimental limits on it [26]. If mirror sector exists, then the same should be applied to the mirror neutron. Moreover, the transformation n − n ′ can be a reason for existence of transition magnetic moment (TMM). Notice that the existence of transition magnetic moment between the neutron and antineutron are forbidden by Lorentz invariance while between the neutron and mirror neutron it is not forbidden [27]. In other terms, there can exist the following non-diagonal operators between n and n ′ states where F µν and F ′ µν are respectively the ordinary and mirror electromagnetic fields. For the transition magnetic moments (TMM) we have η ′ = ±η depending on the type of exchange symmetry between two sectors, Z 2 or P Z 2 .
In this paper we are interested in exploring possible experimental manifestations of the neutron TMM. We do not discuss the particular mechanisms of its generation. Most generically, if mechanism exists that causes the n − n ′ mixing, it will involve the charges of the constituents of n and n ′ and the corresponding, possible interaction of these charges either with photons or with mirror photons. Moreover, the neutron TMM can be induced by loops involving charged particles, in the analogous way as transition magnetic moment between the neutrinos, and in particular between active and sterile neutrinos. In principle, such loops can induce both transition magnetic moment and electric dipole moment, which in fact correspond to real and imaginary parts of the same operator. However, here for brevity we concentrate on the case of transitional magnetic moment only.
2 Neutron TTM and n − n ′ system For describing the effect of nTMM on time evolution of the mixed (n, n ′ ) system in the background of uniform magnetic fields B and B ′ and possible presence of ordinary and/or mirror matter, the Hamiltonian H nn ′ of Eq. (1) has to be modified to the following form 2 where V and V ′ stand for n and n ′ optical potentials induced respectively by ordinary and mirror matter, σ is a set of Pauli matrices, and η = η nn ′ and η ′ = η ′ nn ′ , are the TMM's between n and n ′ related respectively to ordinary and mirror magnetic fields. In the following, in view of Z 2 or P Z 2 parities, we take µ ′ = µ for normal magnetic moments of n and n ′ while for the TMM's we consider two possibilities, η ′ = ±η.
The magnitude of nTMM η is unknown; it can be either positive or negative. We assume η is much smaller than µ, and parametrize it as η = κµ, κ ≪ 1. Therefore, in the absence of matter (V, V ′ = 0) this Hamiltonian can be conveniently rewritten aŝ where 2ω = µB and 2ω ′ = µB ′ . We have dropped the masses of n and n ′ assuming that they are equal, since equal additive diagonal terms will not affect the evolution of the system. We take the TMM's in units of the neutron magnetic moment itself, κ = η/µ being dimensionless parameter. The sign ± in the non-diagonal terms takes into account that the unknown parity of the mirror photon (B ′ ) can be the same or opposite to that of the ordinary photon. For simplicity, we coin these two cases as the cases of + and − parities. The Hamiltonian (4) is a 4 × 4 Hermitian matrix that includes two polarizations of n and n ′ states. Without the nTMM terms, the exact general solution for n − n ′ oscillation probability vs time for a constant and uniform magnetic fields was obtained in Refs. [2,21]. The probability of n ′ appearance in the neutron beam in this case has a resonance character at |B| = |B ′ | and depends on the spatial angle β between vectors B and B ′ . The direction and magnitude of the vector B ′ is a priori unknown but can be determined from scanning experiments, e.g. with the high-flux cold neutron beams as described in [24] and [25]. Limits on the on n − n ′ oscillation time τ = 1/ε for n → n ′ were obtained in several dedicated experiments with ultra-cold neutrons [15,16,17,18,19,20] adopted by the Particle Data Group [26]. In the assumption of B ′ = 0 the most stringent limit τ > 448 s was obtained in Ref. [17]. Limits on τ for non-zero B ′ are much weaker; they are summarized in Ref. [20]. Now, including also the nTMM in the Hamiltonian (4), the probability of n−n ′ transition P nn ′ depends on the values of ε as well as κ. First, we should note that magnitude of the resonance will be enhanced in case when |2κσ(ω ± ω ′ )| > ε. The exact expression for time dependent probability is difficult to extract in analytical form from the Hamiltonian represented by 4 × 4 matrix (4). However, if we focus on non-resonant region of ω by taking into account that mean time of flight of neutron in experiments is typically 0.1 s and by assuming that the difference of magnetic fields |B − B ′ | can be made larger than several mG, the mean oscillation probability P nn ′ (ε, κ, B, B ′ ) averaged over many oscillations can be readily calculated following the techniques of Ref. [2]. Interestingly, the interference terms between two effects simply cancel and one gets average probability in the form of a simple sum where is the average n − n ′ oscillation probability due to mass mixing ε in the absence of transition magnetic moments, κ = 0, given in slightly different form in Refs. [2,21]. Here ω = 1 2 |µB| = (B/1 G) × 4500 s −1 and analogously ω ′ = 1 2 |µB ′ |, and β is the angle between the vectors B and B ′ . The second term P η (B, B ′ ) instead describes the probability of n − n ′ transition only due to transition magnetic moment η, in the limit ε = 0, and it depends on the choice of ± parity. Namely, in the case of (−) parity, it does not depend on the magnetic field values and relative orientation and one simply gets As for the case of + parity, we get Let us explore possible manifestations of the neutron TMM η in the limit ε = 0, neglecting for time being n − n ′ mass mixing. Let us first consider the case when mirror magnetic field is negligibly small, B ′ ≈ 0 (e.g. mirror galactic magnetic fields ∼ 10 µG can be the case here). In this case the spin quantization axis can be taken as the magnetic field direction, B = (0, 0, B), and the Hamiltonian (4) reduces to 2 × 2 matrix for both polarizations (and independent on the parity of mirror photon): Then, taking into account that κ ≪ 1, n → n ′ transition probability after time t reads: where, once again, ω = 1 2 |µB| = (B/1 mG) × 4.5 s −1 . The typical time for free neutron propagation in experiments is t 0.1 s (e.g. time between bounces of ultra-cold neutrons (UCN) in the gravitational trap). Thus, for ωt ≫ 1, which means that B is larger than few mG, the time oscillation can be averaged and we get P η = 2κ 2 , in agreement with the formulas (7) and (8). On the other hand, for B < 1 mG we have ωt < 1 and thus P η (t) = 4κ 2 ω 2 t 2 < 2κ 2 . In this case the flight time t ∼ 0.1 is too short for completing the oscillation, and no time average can be taken. Thus, in very small magnetic fields, n − n ′ conversion due to nTMM should be suppressed, in difference from the case of mass mixing term with ε.
For the large fields B > few mG the conversion probability due to nTMM: does not depend on the value of magnetic field B. Reciprocally, if the Earth has a mirror magnetic field B ′ > few mG, we would get the same oscillation probability in the limit B = 0. The dependence on the magnetic field B and its orientation for + parity case can show up only close to the resonance B ≃ B ′ (8), while for − parity case it can be described by formula (7). The Earth magnetic field with magnitude B E ∼ 0.5 G in all UCN neutron lifetime gravity trap experiments is usually considered as a non-essential factor. Therefore, we can advocate that, in addition to possible resonance effect at |B − B ′ | ∼ 0, the contribution of nTMM (that does not depend on the magnitude of the magnetic field) to the n → n ′ transformation probability could be also essential as a source of disappearance of neutrons.
It above we assumed V, V ′ = 0, i.e. no low-density matter or mirror matter was present and the n → n ′ transformation occurs in vacuum in the presence of magnetic fields B, B ′ . Possible density of mirror matter, as a gas undetectable by experiments, is unknown and we will keep the assumption that V ′ = 0 together with B ′ = 0. However, we can consider that cold neutrons can propagate and oscillate in air or in another gas described by the positive Fermi quasi-potential V F [28]. Density of this gas can be low enough to neglect the decoherent scattering at a finite angle θ and the absorption. Thus, e.g. for air at STP this Fermi potential is V F ≡ V ≈ 0.12 neV. This potential should be added to the Hamiltonian (9). Then, assuming constant and uniform magnetic field B, we can re-write Hamiltonian (9) as follows:Ĥ We see from (12) that for one of polarizations of neutron it is possible to chose such a value of magnetic field B that will compensate V and will bring Hamiltonian to a pure oscillatory form. Value of the constant magnetic field for the air at STP as example is B ≅ 20 G, which justifies the approximation B ′ = 0. The probability of elastic scattering in air at STP is ≈ 0.05 per meter of path. It is low enough to consider few meters of cold neutron beam propagation in air in constant field of ∼ 20 G. With careful tuning of magnetic field B around the resonance value of ∼20 Gauss the effect of V will be compensated. Field tuning accuracy ≤ 1% will be required. Then, close to the resonance, for the given polarization of cold neutrons, one can have almost free oscillations of n n ′ P n ′ (t) ∼ = sin 2 (ηBt) wheret = L v −1 is average time of flight (averaged over velocity spectrum) of the cold neutron beam through the length L of the constant magnetic field. Maximum transformation probability, close to 100% (with some losses due to scattering on the gas), for one polarization of the neutron beam can be potentially achieved when κµBt = π 2 (14) 3 (n, n ′ ) system in non-uniform magnetic field If at t ≤ 0 the (n, n ′ ) system was in vacuum and in a constant magnetic field B = B 0 (B ′ = 0) then for initial condition at t = 0, the kinetic energies T = T ′ were the same, as it is assumed in Eq. (3). Two orthogonal eigenstates of the oscillating (n, n ′ ) system are formed corresponding to the different eigenvalues of energy. The system will be oscillating in time between these two states. State 1 will have interaction properties close to that of the ordinary neutron (n) and state 2 close to that of the mirror neutron (n ′ ). These two states would interact with ordinary matter differently. At t > 0, we assume, the system will enter a region of non-uniform magnetic field that is a function of coordinates with some gradients. The gradient of potential energy U = (µ · B) from semi-classical point of view generates a force. In some UCN experiments, e.g. in [29], this force can exceed the Earth's gravitational attraction force and cause vertical bouncing of neutrons (of a certain polarization) from the surface with an arranged strong magnetic field gradient. By virtue of the Mirror Model this gradient force is acting mainly on the neutron component of the system but not on the mirror component, while the gravity is the same for both components. Similar situation occurs in collision of neutrons with the wall in gravitational UCN traps, where gradient is due to the Fermi potential of the trap walls. Positive Fermi potential V F of the wall material will repulse neutrons with kinetic energy smaller than V F at the characteristic distance of the neutron wave packet, ∼ 1000Å for typical V F = 100 neV. In this way, the gradient of potential in the interaction with wall can be estimated as ∼ 1 eV/m. In the magnetic trap experiment UNCτ [29] the gradient of magnetic field is ∼1 T/cm near the bouncing surface that corresponds to the gradient of potential energy ∼ 6 × 10 −6 eV/m. Integrated over time ∆t this force is providing a relative momentum ∆p between two components that, if exceeds the ∆p p the width of the neutron wave packet, will lead to "separation of components" or to decoherent collapse of (n, n ′ ) system into either n or n ′ states. If this force would be exerted only for time ∆t and then reduced to zero, then finite relative momentum ∆p > ∆p p will continue in time the separation of the components. Same conclusion can be formulated in more appropriate for Quantum Mechanics language of potential energy variation different for two components of (n, n ′ ) wave function in nonuniform magnetic field.
At some point two components will be "pulled away," and the entanglement of (n, n ′ ) system moving in the non-uniform magnetic field will be broken. If "measured" at this moment the (n, n ′ ) system due to gradient separation will be found with probability P in the pure state of "mirror neutron" (0, 1) or with probability (1−P) in the state of "neutron" (1, 0). If not "measured", each of the states will be subject of further oscillation with the reset initial conditions at the time of "separation".
This action is similar to the interaction of the (n, n ′ ) system with the material wall, where entanglement will be broken with the pure n component being reflected or absorbed by the wall and the pure n ′ -state with small probability P will escape from the UCN trap through the wall.
More general, the motion of the oscillating (n, n ′ ) system in the adiabatically slow changing magnetic field can lead either to a change of the kinetic energy of the components of wavefunction and/or to their spin rotation. Both these effects can result in decoherence of the system. The decoherence problem expectedly can be properly treated by modern methods, e.g. [30], by considering master equation of the evolution of density matrix of (n, n ′ ) system in the environment of the external potential U(r), different for two components of the system (four components when spins included). We think that this technique is not yet sufficiently developed and tested in respect to particle oscillations and therefore will refrain from the construction and solution the density matrix evolution equation, but will try instead consider decoherence problem qualitatively. For these qualitative arguments we will ignore the spin rotation in a non-uniform magnetic field and will consider one-dimensional motion in a field with a gradient. Let us first assume a hypothetical simple case of a constant gradient of the magnetic field along the direction of the (n, n ′ ) system motion. The magnitude of B is linearly increasing with the distance x, but the direction of vector B remains practically the same. For a neutron with a certain polarization propagating along axis x from the initial condition (1, 0) and for the observation time ∆t, the energy width of the wave packet ∆E p of the (n, n ′ ) system is For path ∆x = x f − x i , passed by the neutron for the observation time ∆t, the change of potential energy produced by the magnetic field gradient will be ∆U = U f − U i . If ∆U will be much smaller than ∆E p , then the (n, n ′ ) system will remain entangled. For sufficiently large ∆B, the entanglement of the (n, n ′ ) system will be broken. This means that the system will collapse to pure states of either n ′ or n with probabilities P and (1 − P) correspondingly. Assuming that the velocity will not change significantly by magnetic gradient for the time ∆t (say, for the time of flight between two wall collisions in a UCN gravitational trap), then the gradient that should lead to the decoherence collapse of the system can be estimated in a following way: Thus, for the entangled evolution of the (n, n ′ ) system e.g. with velocity v = 3 m/s in a UCN gravitational trap with a time of flight ∆t ≃ 0.1 s between wall collisions, the magnetic field gradient should be ≪ 3.7 mG/m. Larger gradients can cause the collapse of the wavefunction at an earlier time and can cause the transformation of n to n ′ to occur in the volume of the trap rather than in collisions with the trap walls. Thus, the effect of a non-uniform magnetic field might result in the n → n ′ transformation in the trap volume with the same result as that which would be expected in collisions of the entangled (n, n ′ ) system with the trap walls. Usually in the UCN gravitational trap experiments used for neutron lifetime measurements, the number of wall collisions is experimentally extrapolated to the "zero number of collisions," e.g. in [31,32,33]. A magnetic field non-uniformity can produce a similar disappearance effect in the volume of UCN gravitational traps, but this effect is not removable by extrapolation to the zero number of wall collisions. Unfortunately, some of UCN gravitational trap experiments do not use magnetic shielding of the trap, and the actual maps of magnetic fields in these experiments are unknown. To obtain some idea of the possible non-uniformity of the Earth magnetic field in such conditions, we have measured some vertical gradients as high as 75 mG/m in an arbitrary general-purpose laboratory room of a typical university building.
Let us note that in the proposed cold-neutron-beam disappearance/regeneration experiments [24,25], with average neutron velocity v ∼ 800 m/s and flight path e.g. 16 m, for the entangled evolution of the (n, n ′ ) oscillating system, the gradients ≪ 0.4 mG/m would be required. Larger gradients can shorten the path of entangled evolution and effectively will lead to multiple shorter-in-time collapses, which will increase the production of n ′ states, since the probability (11) will remain constant.
If the initial state of the (n, n ′ ) system is (0, 1), describing the propagation of a mirror neutron in the environment with non-uniform magnetic field B (still assuming the mirror magnetic field absent B ′ = 0), the entanglement breaking decoherent conditions collapsing the (n, n ′ ) system as described by the equation (11) will take place with the same probability.

Possible magnitude of neutron TMM
It is a well known problem in UCN gravitational trap experiments that the measured wall losses are larger than these predicted from theoretical models using known scattering lengths of materials. (See discussion in [34] and references therein.) Lowering the temperature of the trap walls, although reducing the loss coefficient (per single collision), does not resolve the discrepancy between experiment and theoretical calculations. Only in one of a few experiments using fomblin-oil coated trap [33] measured and expected loss factors per wall collision were in agreement (∼ 2 × 10 −6 .) We will speculate that losses at least at the level of 1 × 10 −6 per collision can be due to losses in the trap volume caused by the neutron TMM in the non-uniform environmental magnetic field. The local Earth magnetic field in these experiments can be affected by magnetic constructional materials, platforms, etc., as well as by metal reinforced concrete walls of the industrial buildings. Let us assume that in a typical trap of the size ∼ 0.3 m the magnetic field uniformity is < 5% from wall to wall such that the moving neutron can see a constant gradient of about 75 mG/m. We are not considering the effect of neutron spin rotation due to possible change of the direction of the magnetic field -this might result in additional decoherence effect that is not being discussed here. For a typical neutron velocity 3 m/s, and assuming that the velocity will not essentially be changed by the gradient during the flight from wall to wall, from equation (17) we can find that the typical time for collapse to occur will be ∼ 0.02 s. Thus, we can estimate that ∼ 5 volume collapse events will occur per one collision with the trap wall. Attributing this totally to the neutron TMM transformations described by equation (11), we can estimate that losses per collision of 1 × 10 −6 correspond to a neutron TMM of κ 3 × 10 −4 (18) In the recent neutron lifetime measurement with large UCN gravitational trap [32], assuming similar as above magnetic field gradients, the number of decoherence events per second occurring in the volume of the storage trap can be estimated as ∼ 50 s −1 . With the probability of n → n ′ transformation per event from Eq. (11), this can generate a neutron disappearance rate that will be explaining the difference [35] of the result [32] of the neutron lifetime experiment from the beam appearance measurement [36]. The required value of nTMM in this case can be estimated as If magnetic field gradients in the gravitational trap are higher than we have assumed, then the magnitude of neutron TMM κ required to produce mentioned disappearance rate can be lower than in (19). Also, if |B ′ | < |B| is present it might increase the conversion probability thus reducing our estimate of of κ in (19). Would the UCN lifetime experiment with gravitational trap be magnetically shielded with residual field magnitude ≪ 1 mG (that means assuming that both B ≈ 0 and B ′ = 0) then n → n ′ disappearance effect due to nTMM will vanish and measured lifetime might be affected only by a much smaller effect of n → n ′ oscillations due to mixing mass ε as was discussed earlier.
In experiments with "UCN magnetic field trap" [29,37], where neutrons are repulsed from the strong gradient of the magnetic field, the number of decoherence events can be greatly increased. For a rough estimate we have attempted to reproduce in a simplified one-dimensional way the vertical bouncing of UCN in the magnetic field with a strong gradient described in the papers [29] up to the height of 50 cm and obtained with Eq. (17) approximately 3450 decoherence events per a second of the neutron motion. Thus, to produce a UCN disappearance rate ∼ 10 −5 per second in experiment [29], using equation (11) one can estimate the following magnitude of nTMM: Simulations with more experimental details (not available to us) will likely affect this estimate. Also, the potential reason for disappearance in [29] can be a change of the spin precession phase due to the presence of (n, n ′ ) oscillations [2] that can lead to unexpected depolarization effect in the magnetic trap and, thus, also might change the magnitude of extracted nTMM in (20). We also can get another nTMM estimate from the different interpretation of the limit on ε obtained in [17] at B = 0 and under assumption that B ′ = 0. We will assume instead that mirror magnetic field B ′ is present (although unknown) with magnitude > several mG and thus, the measured limit on disappearance probability in [17] should be taken as a limit determined by the magnitude of nTMM from Eq. (11). From this we can obtain the following estimate: We can conclude this section by noting that these estimates, although very rough, are suggesting the order of magnitude of the nTMM κ ∼ 10 −4 −10 −5 that doesn't contradict the existing experimental observations and might be the parameter of the mechanism responsible for the neutron disappearance in the UCN trap lifetime experiments, such as [29,32,37] and others.

How nTMM can be measured
Beyond the results obtained in n → n ′ searches with UCN [15,16,17,18,19,20] and summarized in [20] possible new searches with cold neutron, described in [24,25], might bring more evidence whether n → n ′ transformation exists. These new proposed measurements are based on the detection of cold neutrons in intense beam after coming through the "region A" of controlled uniform magnetic field (in disappearance mode) or on total absorption of the cold neutron beam after passing through the "region A", allowing only produced mirror neutron to pass through absorber and then regenerating them back to the detectable neutrons in the second "region B" with similar controlled uniform magnetic field (regeneration mode). Variation of the magnitude of uniform magnetic field B in the range 0 to 0.5 G by small-steps scan (also with variation of direction of vector B) can reveal the resonance in the n-counting rates corresponding to |B| |B ′ | with magnitude and width related to mass mixing parameter ε in Eq. (4). Presence of nTMM can enhance the resonance magnitude or, as follows from Eq. (5), can offer an alternative interpretation of the effect (if observed or excluded) in terms of magnitude of nTMM.
We would like to discuss here other two new methods that can be used for the detection of nTMM effect in the magnetic fields stronger than the Earth magnetic field.
First method is a variation of regeneration method where "region A" and "region B" mentioned above instead of uniform magnetic field are implementing strongly non-uniform magnetic field. Since nTMM transition probability in Eq. (11) remains constant in any sufficiently large magnetic field, strong gradients of the field can generate large number of decoherent collapses of (n, n ′ ) system into either n or with small probability (11) to n ′states; the latter will lead to accumulation of n ′ in the "region A". The absorber between "region A" and "region B" will remove all neutrons from the beam allowing only the n ′ to pass through the absorber. Strong gradients of the magnetic field in the "region B" will similarly enhance the transformation of n ′ back to detectable n to be counted above the natural background e.g. by the 3 He detector.
Strong magnetic field gradients in "region A" and "region B" can be implemented e.g. as a series of equally spaced coils around the neutron beam with alternating directions of constant current. Every coil will contribute opposite direction of magnetic field with zig-zag pattern and can provide practically constant strong |dB/dz| gradient along the beam axis. Simple estimates show that the gradient ∼ 100 G/m can be maintained along the vacuum tube that is e.g. 15 m long. For the cold neutron beam with average velocity ∼ 800 m/s using Eq. (17) we can estimate that the number of decoherent collapses in such "region A"or "region B"-devices will be around 500.
Use of superconducting solenoidal magnet with borehole along the beam axis and with maximum magnetic field of ∼ 10 T will be more compact approach for reaching high gradients. Total length of the magnet can be of the order of 1 m. Magnet front side with increasing field can serve as a "region A" and back side with decreasing field as a "region B", each with decoherence collapse factor ∼ 700. Beam absorber should be installed in the middle of the magnet, thus transforming it to the regeneration device.
With cold neutron beams available e.g. at ILL and at HFIR/ORNL reactors, or at SNS/ORNL or at future ESS spallation neutron sources [38] where cold beam intensities are in the range 10 10 − 10 11 n/s the effect of nTMM in the whole range of κ ∼ 10 −4 − 10 −5 can be explored in rather short and not expensive experiments [39].
Second method is based on the idea of compensation of Fermi quasi-potential of the gas media with constant uniform magnetic field discussed at the end of the Section 2 in this paper with oscillation probability given by Eq. 13). In this method the probability of quasi-free oscillation can coherently grow with cold neutrons propagating through the gas, e.g. through air at STP. This will work only for one of neutron polarizations. Regeneration scheme can be used again with "region A" and "region B" represented by the air-filled tube of length e.g. L = 2m in the solenoidal constant uniform field of ∼ 20 G. Effects of the beam scattering and absorption will reduce beam intensity by ≈ 20%. Probability of regeneration provided by this second method can be order of magnitude higher than for the first method discuss earlier. Second method can be further optimized for the use with a beam of collimated UCN propagating in the gas with low absorption. In this case the probability for n → n ′ transformation potentially can grow approaching the maximum value described by Eq. 14).

Conclusion
In the paradigm of the existence of n → n ′ oscillations [1], the mirror matter model should naturally include additional effects that are inherent to self-interacting mirror matter. These include the presence of a neutron transition magnetic moment (nTMM) discussed in this paper, the existence of mirror magnetic fields on Earth [2], the possible accumulation of mirror matter in the Solar system and inside the Earth with a presence in the Earth atmosphere [40], and possibly other effects that can be probed experimentally. The existing difference between the neutron lifetime measurements by beam (appearance) [36] and UCN trap (disappearance) methods [29,31,32,33,37] can be explained by n → n ′ oscillations with the effect of small neutron transition magnetic moment. The magnitudes of nTMM extracted from the difference in neutron lifetime results can be measured in rather simple experiments, such as one proposed in [25] for the GP-SANS beamline at HFIR. If mirror magnetic field B ′ will be present in experiment with non-zero value, the experimental strategy with controlled uniform magnetic field can be pursued, that was described in the papers [24,25].
YK acknowledge the hospitality provided by the Institute for Nuclear Theory, University of Washington, Seattle, where in October 2017 this work was initiated during the INT-17-69W Workshop "Neutron Oscillations: Appearance, Disappearance, and Baryogenesis".