$$\varvec{B^0\rightarrow K^{*0}\mu ^+\mu ^-}$$ decay in the aligned two-Higgs-doublet model

In the aligned two-Higgs-doublet model, we perform a complete one-loop computation of the short-distance Wilson coefficients \(C_{7,9,10}^{(\prime )}\), which are the most relevant ones for \(b\rightarrow s\ell ^+\ell ^-\) transitions. It is found that, when the model parameter \(\left| \varsigma _{u}\right| \) is much smaller than \(\left| \varsigma _{d}\right| \), the charged scalar contributes mainly to chirality-flipped \(C_{9,10}^\prime \), with the corresponding effects being proportional to \(\left| \varsigma _{d}\right| ^2\). Numerically, the charged-scalar effects fit into two categories: (A) \(C_{7,9,10}^\mathrm {H^\pm }\) are sizable, but \(C_{9,10}^{\prime \mathrm {H^\pm }}\simeq 0\), corresponding to the (large \(\left| \varsigma _{u}\right| \), small \(\left| \varsigma _{d}\right| \)) region; (B) \(C_7^\mathrm {H^\pm }\) and \(C_{9,10}^{\prime \mathrm {H^\pm }}\) are sizable, but \(C_{9,10}^\mathrm {H^\pm }\simeq 0\), corresponding to the (small \(\left| \varsigma _{u}\right| \), large \(\left| \varsigma _{d}\right| \)) region. Taking into account phenomenological constraints from the inclusive radiative decay \(B\rightarrow X_{s}{\gamma }\), as well as the latest model-independent global analysis of \(b\rightarrow s\ell ^+\ell ^-\) data, we obtain the much restricted parameter space of the model. We then study the impact of the allowed model parameters on the angular observables \(P_2\) and \(P_5'\) of \(B^0\rightarrow K^{*0}\mu ^+\mu ^-\) decay, and we find that \(P_5'\) could be increased significantly to be consistent with the experimental data in case B.     

Kinetically modified non-minimal inflation with exponential frame function

We consider supersymmetric (SUSY) and non-SUSY models of chaotic inflation based on the \(\phi ^n\) potential with \(n=2\) or 4. We show that the coexistence of an exponential non-minimal coupling to gravity \(f_\mathcal{R}=\mathrm{e}^{c_\mathcal{R}\phi ^{p}}\) with a kinetic mixing of the form \(f_{\mathrm{K}}=c_{\mathrm{K}}f_\mathcal{R}^m\) can accommodate inflationary observables favored by the Planck and Bicep2/Keck Array results for \(p=1\) and 2, \(1\le m\le 15\) and \(2.6\times 10^{-3}\le r_{\mathcal {R}\mathrm{K}}=c_\mathcal{R}/c_{\mathrm{K}}^{p/2}\le 1,\) where the upper limit is not imposed for \(p=1\). Inflation is of hilltop type and it can be attained for subplanckian inflaton values with the corresponding effective theories retaining the perturbative unitarity up to the Planck scale. The supergravity embedding of these models is achieved employing two chiral gauge singlet supefields, a monomial superpotential and several (semi)logarithmic or semi-polynomial Kähler potentials.     

Likelihood analysis of the sub-GUT MSSM in light of LHC 13-TeV data

We describe a likelihood analysis using MasterCode of variants of the MSSM in which the soft supersymmetry-breaking parameters are assumed to have universal values at some scale \(M_\mathrm{in}\) below the supersymmetric grand unification scale \(M_\mathrm{GUT}\), as can occur in mirage mediation and other models. In addition to \(M_\mathrm{in}\), such ‘sub-GUT’ models have the 4 parameters of the CMSSM, namely a common gaugino mass \(m_{1/2}\), a common soft supersymmetry-breaking scalar mass \(m_0\), a common trilinear mixing parameter A and the ratio of MSSM Higgs vevs \(\tan \beta \), assuming that the Higgs mixing parameter \(\mu > 0\). We take into account constraints on strongly- and electroweakly-interacting sparticles from \(\sim 36\)/fb of LHC data at 13 TeV and the LUX and 2017 PICO, XENON1T and PandaX-II searches for dark matter scattering, in addition to the previous LHC and dark matter constraints as well as full sets of flavour and electroweak constraints. We find a preference for \(M_\mathrm{in}\sim 10^5\) to \(10^9 \,\, \mathrm {GeV}\), with \(M_\mathrm{in}\sim M_\mathrm{GUT}\) disfavoured by \(\Delta \chi ^2 \sim 3\) due to the \(\mathrm{BR}(B_{s, d} \rightarrow \mu ^+\mu ^-)\) constraint. The lower limits on strongly-interacting sparticles are largely determined by LHC searches, and similar to those in the CMSSM. We find a preference for the LSP to be a Bino or Higgsino with \(m_{\tilde{\chi }^0_{1}} \sim 1 \,\, \mathrm {TeV}\), with annihilation via heavy Higgs bosons H / A and stop coannihilation, or chargino coannihilation, bringing the cold dark matter density into the cosmological range. We find that spin-independent dark matter scattering is likely to be within reach of the planned LUX-Zeplin and XENONnT experiments. We probe the impact of the \((g-2)_\mu \) constraint, finding similar results whether or not it is included.     

Likelihood analysis of the minimal AMSB model

We perform a likelihood analysis of the minimal anomaly-mediated supersymmetry-breaking (mAMSB) model using constraints from cosmology and accelerator experiments. We find that either a wino-like or a Higgsino-like neutralino LSP, \(\tilde{\chi }^0_{1}\), may provide the cold dark matter (DM), both with similar likelihoods. The upper limit on the DM density from Planck and other experiments enforces \(m_{\tilde{\chi }^0_{1}} \lesssim 3 \,\, \mathrm {TeV}\) after the inclusion of Sommerfeld enhancement in its annihilations. If most of the cold DM density is provided by the \(\tilde{\chi }^0_{1}\), the measured value of the Higgs mass favours a limited range of \(\tan \beta \sim 5\) (and also for \(\tan \beta \sim 45\) if \(\mu > 0\)) but the scalar mass \(m_0\) is poorly constrained. In the wino-LSP case, \(m_{3/2}\) is constrained to about \(900\,\, \mathrm {TeV}\) and \(m_{\tilde{\chi }^0_{1}}\) to \(2.9\pm 0.1\,\, \mathrm {TeV}\), whereas in the Higgsino-LSP case \(m_{3/2}\) has just a lower limit \(\gtrsim 650\,\, \mathrm {TeV}\) (\(\gtrsim 480\,\, \mathrm {TeV}\)) and \(m_{\tilde{\chi }^0_{1}}\) is constrained to \(1.12 ~(1.13) \pm 0.02\,\, \mathrm {TeV}\) in the \(\mu >0\) (\(\mu <0\)) scenario. In neither case can the anomalous magnetic moment of the muon, \((g-2)_\mu \), be improved significantly relative to its Standard Model (SM) value, nor do flavour measurements constrain the model significantly, and there are poor prospects for discovering supersymmetric particles at the LHC, though there are some prospects for direct DM detection. On the other hand, if the \(\tilde{\chi }^0_{1}\) contributes only a fraction of the cold DM density, future LHC Open image in new window -based searches for gluinos, squarks and heavier chargino and neutralino states as well as disappearing track searches in the wino-like LSP region will be relevant, and interference effects enable \(\mathrm{BR}(B_{s, d} \rightarrow \mu ^+\mu ^-)\) to agree with the data better than in the SM in the case of wino-like DM with \(\mu > 0\).     

Revisiting radiative decays of $$1^{+-}$$ heavy quarkonia in the covariant light-front approach

We revisit the calculation of the width for the radiative decay of a \(1^{+-}\) heavy \(Q \bar{Q}\) meson via the channel \(1^{+-} \rightarrow 0^{-+} +\gamma \) in the covariant light-front quark model. We carry out the reduction of the light-front amplitude in the non-relativistic limit, explicitly computing the leading and next-to-leading order relativistic corrections. This shows the consistency of the light-front approach with the non-relativistic formula for this electric dipole transition. Furthermore, the theoretical uncertainty in the predicted width is studied as a function of the inputs for the heavy-quark mass and wave function structure parameter. We analyze the specific decays \(h_{\mathrm{c}}(1P) \rightarrow \eta _{\mathrm{c}}(1S) + \gamma \) and \(h_{\mathrm{b}}(1P) \rightarrow \eta _{\mathrm{b}}(1S) + \gamma \). We compare our results with experimental data and with other theoretical predictions from calculations based on non-relativistic models and their extensions to include relativistic effects, finding reasonable agreement.     

Scattering of kinks of the sinh-deformed $$\varphi ^4$$ model

We consider the scattering of kinks of the sinh-deformed \(\varphi ^4\) model, which is obtained from the well-known \(\varphi ^4\) model by means of the deformation procedure. Depending on the initial velocity \(v_\mathrm {in}\) of the colliding kinks, different collision scenarios are realized. There is a critical value \(v_\mathrm {cr}\) of the initial velocity, which separates the regime of reflection (at \(v_\mathrm {in}>v_\mathrm {cr}\)) and that of a complicated interaction (at \(v_\mathrm {in}<v_\mathrm {cr}\)) with kinks’ capture and escape windows. Besides that, at \(v_\mathrm {in}\) below \(v_\mathrm {cr}\) we observe the formation of a bound state of two oscillons, as well as their escape at some values of \(v_\mathrm {in}\).     

On Blow-up Profile of Ground States of Boson Stars with External Potential

We study minimizers of the pseudo-relativistic Hartree functional \({\mathcal {E}}_{a}(u):=\Vert (-\varDelta +m^{2})^{1/4}u\Vert _{L^{2}}^{2}+\int _{{\mathbb {R}}^{3}}V(x)|u(x)|^{2}\mathrm{d}x-\frac{a}{2}\int _{{\mathbb {R}}^{3}}(\left| \cdot \right| ^{-1}\star |u|^{2})(x)|u(x)|^{2}\mathrm{d}x\) under the mass constraint \(\int _{{\mathbb {R}}^3}|u(x)|^2\mathrm{d}x=1\). Here \(m>0\) is the mass of particles and \(V\ge 0\) is an external potential. We prove that minimizers exist if and only if a satisfies \(0\le a<a^{*}\), and there is no minimizer if \(a\ge a^*\), where \(a^*\) is called the Chandrasekhar limit. When a approaches \(a^*\) from below, the blow-up behavior of minimizers is derived under some general external potentials V. Here we consider three cases of V: trapping potential, i.e. \(V\in L_{\mathrm{loc}}^{\infty }({\mathbb {R}}^3)\) satisfies \(\lim _{|x|\rightarrow \infty }V(x)=\infty \); periodic potential, i.e. \(V\in C({\mathbb {R}}^3)\) satisfies \(V(x+z)=V(x)\) for all \(z\in \mathbb {Z}^3\); and ring-shaped potential, e.g. \( V(x)=||x|-1|^p\) for some \(p>0\).     

CP violations in predictive neutrino mass structures

We study the CP-violation effects from two types of neutrino mass matrices with (i) \((M_\nu )_{ee}=0\), and (ii) \((M_\nu )_{ee}=(M_\nu )_{e\mu }=0\), which can be realized by the high-dimensional lepton number violating operators \(\bar{\ell }_R^c\gamma ^\mu L_L (D_\mu \Phi )\Phi ^2\) and \(\bar{\ell }_R^c l_R (D_\mu {\Phi })^2\Phi ^2\), respectively. In (i), the neutrino mass spectrum is in the normal ordering with the lightest neutrino mass within the range \(0.002\,\mathrm{eV}\lesssim m_0\lesssim 0.007\,\mathrm{eV}\). Furthermore, for a given value of \(m_0\), there are two solutions for the two Majorana phases \(\alpha _{21}\) and \(\alpha _{31}\), whereas the Dirac phase \(\delta \) is arbitrary. For (ii), the parameters of \(m_0\), \(\delta \), \(\alpha _{21}\), and \(\alpha _{31}\) can be completely determined. We calculate the CP-violating asymmetries in neutrino–antineutrino oscillations for both mass textures of (i) and (ii), which are closely related to the CP-violating Majorana phases.     

The decays of $$\bar{B}^0$$, $$\bar{B}^0_s$$B¯s0 and $$B^-$$B- into $$\eta _c$$ηc plus a scalar or vector meson

We investigate the decays of \(\bar{B}^0_s\), \(\bar{B}^0\) and \(B^-\) into \(\eta _c\) plus a scalar or vector meson in a theoretical framework by taking into account the dominant process for the weak decay of \(\bar{B}\) meson into \(\eta _c\) and a \(q\bar{q}\) pair. After hadronization of this \(q\bar{q}\) component into pairs of pseudoscalar mesons we obtain certain weights for the pseudoscalar meson-pseudoscalar meson components. In addition, the \(\bar{B}^0\) and \(\bar{B}^0_s\) decays into \(\eta _c\) and \(\rho ^0\), \(K^*\) are evaluated and compared to the \(\eta _c\) and \(\phi \) production. The calculation is based on the postulation that the scalar mesons \(f_0(500)\), \(f_0(980)\) and \(a_0(980)\) are dynamically generated states from the pseudoscalar meson-pseudoscalar meson interactions in S-wave. Up to a global normalization factor, the \(\pi \pi \), \(K \bar{K}\) and \(\pi \eta \) invariant mass distributions for the decays of \(\bar{B}^0_s \rightarrow \eta _c \pi ^+ \pi ^-\), \(\bar{B}^0_s \rightarrow \eta _c K^+ K^-\), \(\bar{B}^0 \rightarrow \eta _c \pi ^+ \pi ^-\), \(\bar{B}^0 \rightarrow \eta _c K^+ K^-\), \(\bar{B}^0 \rightarrow \eta _c \pi ^0 \eta \), \(B^- \rightarrow \eta _c K^0 K^-\) and \(B^- \rightarrow \eta _c \pi ^- \eta \) are predicted. Comparison is made with the limited experimental information available and other theoretical calcualtions. Further comparison of these results with coming LHCb measurements will be very valuable to make progress in our understanding of the nature of the low lying scalar mesons, \(f_0(500), f_0(980)\) and \(a_0(980)\).     

On quantum analogue of dynamical stabilisation of inverted harmonic oscillator by time periodical uniform field

Quantum analogue of stabilised forced oscillations around an unstable equilibrium position is explored by solving the non-stationary Schrödinger equation (NSE) of the inverted harmonic oscillator (IHO) driven periodically by spatial uniform field of frequency \(\Omega \), amplitude \(F_{0}\) and phase \(\phi \), i.e. the system with the Hamiltonian of \(\hat{{H}}=(\hat{{p}}^{2}/2m)-(m\omega ^{2}x^{2}/2)-F_0 x\sin \) \(\left( {\Omega t+\phi } \right) \). The NSE has been solved both analytically and numerically by Maple 15 in dimensionless variables \(\xi = x\sqrt{m\omega /\hbar }\hbox {, }f_0 =F_0 /\omega \sqrt{\hbar m\omega }\) and \(\tau =\omega t\). The initial condition (IC) has been specified by the wave function (w.f.) of a generalised Gaussian type which suits well the corresponding quantum IC operator. The solution obtained demonstrates the non-monotonous behaviour of the coordinate spreading \(\sigma \left( \tau \right) \hbox { =}\sqrt{\big ( {\overline{\Delta \xi ^{2}\big ( \tau \big )} } \big )}\) which decreases first from quite macroscopic values of \(\sigma _{0} =2^{12,\ldots ,25}\) to minimal one of \(\sim \!(1/\sqrt{2})\) at times \(\tau <\tau _0 =0.125\ln \!\left( {16\sigma _0^4 +1} \right) \) and then grows back unlimitedly. For certain phases \(\phi \) depending on the \(\Omega /\omega \) ratio and \(n=\log _2\!\sigma _0 \), the mass centre of the packet \(\xi _{\mathrm {av}}( \tau )= \overline{\hat{{x}}(\tau )} \cdot \sqrt{m\omega /\hbar }\) delays approximately two natural ‘periods’ \(\sim \!(4\pi /\omega )\) in the area of the stationary point and then escapes to ‘\(+\)’ or ‘?’ infinity in a bifurcating way.  For ‘resonant’ \(\Omega =\omega \), the bifurcation phases \(\phi \) fit well with the regression formula of Fermi–Dirac type of argument n with their asymptotic \(\phi ( {\Omega ,n\rightarrow \infty } )\) obeying the classical formula \(\phi _{\mathrm {cl}} ( \Omega )=-\hbox {arctg} \, \Omega \) for initial energy \(E = 0\) in the wide range of \(\Omega =2^{-4},...,2^{7}\).     

A Mysterious Cluster Expansion Associated to the Expectation Value of the Permanent of 0–1 Matrices

We consider two ensembles of \(0-1\) \(n\times n\) matrices. The first is the set of all \(n\times n\) matrices with entries zeroes and ones such that all column sums and all row sums equal r, uniformly weighted. The second is the set of \(n \times n\) matrices with zero and one entries where the probability that any given entry is one is r / n, the probabilities of the set of individual entries being i.i.d.’s. Calling the two expectation values E and \(E_B\) respectively, we develop a formal relation

$$\begin{aligned} E({{\mathrm{perm}}}(A)) = E_B({{\mathrm{perm}}}(A)) e^{\sum _2 T_i}.\quad \quad \quad \quad \mathrm{(A1)} \end{aligned}$$

We use two well-known approximating ensembles to E, \(E_1\) and \(E_2\). Replacing E by either \(E_1\) or \(E_2\) we can evaluate all terms in (A1). For either \(E_1\) or \(E_2\) the terms \(T_i\) have amazing properties. We conjecture that all these properties hold also for E. We carry through a similar development treating \(E({{\mathrm{perm}}}_m(A))\), with m proportional to n, in place of \(E({{\mathrm{perm}}}(A))\).

     

The S-wave resonance contributions in the $$B^{0}_{s}$$ decays into $$\psi (2S,3S)$$ψ(2S,3S) plus pion pair

The three-body decays \(B^0_s \rightarrow \psi (2S,3S) \pi ^+ \pi ^-\) are studied based on the perturbative QCD approach. With the help of the nonperturbative two-pion distribution amplitudes, the analysis is simplified into the quasi-two-body processes. Besides the traditional factorizable and nonfactorizable diagrams at the leading order, the next-to-leading order vertex corrections are also included to cancel the scale dependence. The \(f_0(980)\), \(f_0(1500)\) resonance contributions as well as the nonresonant contributions are taken into account using the presently known \(\pi \pi \) time-like scalar form factor for the \(s\bar{s}\) component. It is found that the predicted \(B^0_s \rightarrow \psi (2S) \pi ^+ \pi ^-\) decay spectra in the pion pair invariant mass shows a similar behavior as the experiment. The calculated S-wave contributions to the branching ratio of \(B^0_s \rightarrow \psi (2S) \pi ^+ \pi ^-\) is \(6.0\times 10^{-5}\), which is in agreement with the LHCb data \(\mathcal {B}(B^0_s \rightarrow \psi (2S) \pi ^+ \pi ^-)=(7.2\pm 1.2)\times 10^{-5} \) within errors. The estimate of \(\mathcal {B}(B^0_s \rightarrow \psi (3S) \pi ^+ \pi ^-)\) can reach the order of \(10^{-5}\), pending the corresponding measurements.     

Consequences of R-parity violating interactions for anomalies in $${\bar{B}}\rightarrow D^{(*)} \tau {\bar{\nu }}$$ and $$b\rightarrow s \mu ^+\mu ^-$$b→sμ+μ-

We investigate the possibility of explaining the enhancement in semileptonic decays of \({\bar{B}} \rightarrow D^{(*)} \tau {\bar{\nu }}\), the anomalies induced by \(b\rightarrow s\mu ^+\mu ^-\) in \({\bar{B}}\rightarrow (K, K^*, \phi )\mu ^+\mu ^-\) and violation of lepton universality in \(R_K = \mathrm{Br}({\bar{B}}\rightarrow K \mu ^+\mu ^-)/\mathrm{Br}({\bar{B}}\rightarrow K e^+e^-)\) within the framework of R-parity violating MSSM. The exchange of down type right-handed squark coupled to quarks and leptons yields interactions which are similar to leptoquark induced interactions that have been proposed to explain the \({\bar{B}} \rightarrow D^{(*)} \tau {\bar{\nu }}\) by tree level interactions and \(b\rightarrow s \mu ^+\mu ^-\) anomalies by loop induced interactions, simultaneously. However, the Yukawa couplings in such theories have severe constraints from other rare processes in B and D decays. Although this interaction can provide a viable solution to the \(R(D^{(*)})\) anomaly, we show that with the severe constraint from \({\bar{B}} \rightarrow K \nu {\bar{\nu }}\), it is impossible to solve the anomalies in the \(b\rightarrow s \mu ^+\mu ^-\) process simultaneously.     

Warm modified Chaplygin gas shaft inflation

In this paper, we examine the possible realization of a new inflation family called “shaft inflation” by assuming the modified Chaplygin gas model and a tachyon scalar field. We also consider the special form of the dissipative coefficient \(\Gamma ={a_0}\frac{T^{3}}{\phi ^{2 }}\) and calculate the various inflationary parameters in the scenario of strong and weak dissipative regimes. In order to examine the behavior of inflationary parameters, the \(n_s \)–\( \phi ,\, n_s \)–r, and \(n_s \)–\( \alpha _s\) planes (where \(n_s,\, \alpha _s,\, r\), and \(\phi \) represent the spectral index, its running, tensor-to-scalar ratio, and scalar field, respectively) are being developed, which lead to the constraints \(r< 0.11\), \(n_s=0.96 \pm 0.025\), and \(\alpha _s =-0.019 \pm 0.025\). It is quite interesting that these results of the inflationary parameters are compatible with BICEP2, WMAP \((7+9)\) and recent Planck data.     

Reexploration of interacting holographic dark energy model: cases of interaction term excluding the Hubble parameter

In this paper, we make a deep analysis for the five typical interacting holographic dark energy models with the interaction terms \(Q=3\beta H_{0}\rho _\mathrm{{de}}\), \(Q=3\beta H_{0}\rho _\mathrm{{c}}\), \(Q=3\beta H_{0}(\rho _\mathrm{{de}}+\rho _\mathrm{c})\), \(Q=3\beta H_{0}\sqrt{\rho _\mathrm{{de}}\rho _\mathrm{c}}\), and \(Q=3\beta H_{0}\frac{\rho _\mathrm{{de}}\rho _{c}}{\rho _\mathrm{{de}}+\rho _\mathrm{c}}\), respectively. We obtain observational constraints on these models by using the type Ia supernova data (the Joint Light-Curve Analysis sample), the cosmic microwave background data (Planck 2015 distance priors), the baryon acoustic oscillations data, and the direct measurement of the Hubble constant. We find that the values of \(\chi _\mathrm{min}^2\) for all the five models are almost equal (around 699), indicating that the current observational data equally favor these IHDE models. In addition, a comparison with the cases of an interaction term involving the Hubble parameter H is also made.     

A Theory of Hydrodynamic Turbulence Based on Non-equilibrium Statistical Mechanics

In earlier papers, we have studied the turbulent flow exponents \(\zeta _p\), where \(\langle |\Delta \mathbf{v}|^p\rangle \sim \ell ^{\zeta _p}\) and \(\Delta \mathbf{v}\) is the contribution to the fluid velocity at small scale \(\ell \). Using ideas of non-equilibrium statistical mechanics we have found

$$\begin{aligned} \zeta _p={p\over 3}-{1\over \ln \kappa }\ln \Gamma \left( {p\over 3}+1\right) \end{aligned}$$

where \(1/\ln \kappa \) is experimentally \(\approx \,0.32\,\pm \,0.01\). The purpose of the present note is to propose a somewhat more physical derivation of the formula for \(\zeta _p\). We also present an estimate \(\approx \,100\) for the Reynolds number at the onset of turbulence.

     

Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum

We study a spatial birth-and-death process on the phase space of locally finite configurations \({\varGamma }^+ \times {\varGamma }^-\) over \({\mathbb {R}}^d\). Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator \(L^+(\gamma ^-) + \frac{1}{\varepsilon }L^-\), \(\varepsilon > 0\). Here \(L^-\) describes the environment process on \({\varGamma }^-\) and \(L^+(\gamma ^-)\) describes the system process on \({\varGamma }^+\), where \(\gamma ^-\) indicates that the corresponding birth-and-death rates depend on another locally finite configuration \(\gamma ^- \in {\varGamma }^-\). We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states \(\mu _t^{\varepsilon }\) on \({\varGamma }^+ \times {\varGamma }^-\). Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let \(\mu _{\mathrm {inv}}\) be the invariant measure for the environment process on \({\varGamma }^-\). In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of \(\mu _t^{\varepsilon }\) onto \({\varGamma }^+\) converges weakly to an evolution of states on \({\varGamma }^+\) associated with the averaged Markov birth-and-death operator \({\overline{L}} = \int _{{\varGamma }^-}L^+(\gamma ^-)d \mu _{\mathrm {inv}}(\gamma ^-)\).