$$\varvec{B^0\rightarrow K^{*0}\mu ^+\mu ^-}$$ 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.     

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 \(\mu > 0\).     

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.

     

Likelihood analysis of the pMSSM11 in light of LHC 13-TeV data

We use MasterCode to perform a frequentist analysis of the constraints on a phenomenological MSSM model with 11 parameters, the pMSSM11, including constraints from \(\sim 36\)/fb of LHC data at 13 TeV and PICO, XENON1T and PandaX-II searches for dark matter scattering, as well as previous accelerator and astrophysical measurements, presenting fits both with and without the \((g-2)_\mu \) constraint. The pMSSM11 is specified by the following parameters: 3 gaugino masses \(M_{1,2,3}\), a common mass for the first-and second-generation squarks \(m_{\tilde{q}}\) and a distinct third-generation squark mass \(m_{\tilde{q}_3}\), a common mass for the first-and second-generation sleptons \(m_{\tilde{\ell }}\) and a distinct third-generation slepton mass \(m_{\tilde{\tau }}\), a common trilinear mixing parameter A, the Higgs mixing parameter \(\mu \), the pseudoscalar Higgs mass \(M_A\) and \(\tan \beta \). In the fit including \((g-2)_\mu \), a Bino-like \(\tilde{\chi }^0_{1}\) is preferred, whereas a Higgsino-like \(\tilde{\chi }^0_{1}\) is mildly favoured when the \((g-2)_\mu \) constraint is dropped. We identify the mechanisms that operate in different regions of the pMSSM11 parameter space to bring the relic density of the lightest neutralino, \(\tilde{\chi }^0_{1}\), into the range indicated by cosmological data. In the fit including \((g-2)_\mu \), coannihilations with \(\tilde{\chi }^0_{2}\) and the Wino-like \(\tilde{\chi }^\pm _{1}\) or with nearly-degenerate first- and second-generation sleptons are active, whereas coannihilations with the \(\tilde{\chi }^0_{2}\) and the Higgsino-like \(\tilde{\chi }^\pm _{1}\) or with first- and second-generation squarks may be important when the \((g-2)_\mu \) constraint is dropped. In the two cases, we present \(\chi ^2\) functions in two-dimensional mass planes as well as their one-dimensional profile projections and best-fit spectra. Prospects remain for discovering strongly-interacting sparticles at the LHC, in both the scenarios with and without the \((g-2)_\mu \) constraint, as well as for discovering electroweakly-interacting sparticles at a future linear \(e^+ e^-\) collider such as the ILC or CLIC.     

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.     

Monojet signatures from heavy colored particles: future collider sensitivities and theoretical uncertainties

In models with colored particle \(\mathcal {Q}\) that can decay into a dark matter candidate X, the relevant collider process \(pp\rightarrow \mathcal {Q}\bar{\mathcal {Q}}\rightarrow X\bar{X}\,+\,\)jets gives rise to events with significant transverse momentum imbalance. When the masses of \(\mathcal {Q}\) and X are very close, the relevant signature becomes monojet-like, and Large Hadron Collider (LHC) search limits become much less constraining. In this paper, we study the current and anticipated experimental sensitivity to such particles at the High-Luminosity LHC at \(\sqrt{s}=14\) TeV with \(\mathcal {L}=3\) ab\(^{-1}\) of data and the proposed High-Energy LHC at \(\sqrt{s}=27\) TeV with \(\mathcal {L}=15\) ab\(^{-1}\) of data. We estimate the reach for various Lorentz and QCD color representations of \(\mathcal {Q}\). Identifying the nature of \(\mathcal {Q}\) is very important to understanding the physics behind the monojet signature. Therefore, we also study the dependence of the observables built from the \(pp\rightarrow \mathcal {Q}\bar{\mathcal {Q}} + j \) process on \(\mathcal {Q}\) itself. Using the state-of-the-art Monte Carlo suites MadGraph5_aMC@NLO+Pythia8 and Sherpa, we find that when these observables are calculated at NLO in QCD with parton shower matching and multijet merging, the residual theoretical uncertainties are comparable to differences observed when varying the quantum numbers of \(\mathcal {Q}\) itself. We find, however, that the precision achievable with NNLO calculations, where available, can resolve this dilemma.     

Scalar top quarks production in polarized photon-photon collisions at ILC

We study pair production of scalar top quarks (stop, \(\tilde t_1\)) in polarized photon-photon collisions with the subsequent decay of the top squarks into b quarks and charginos \(\tilde t_1 \to b\tilde \chi _1^ \pm\). We simulate this process by using PYTHIA6.4 for an electron beam energy \(2E_{beam}^e = \sqrt {s_{ee} } = 1000 GeV\). The energy spectrum of back-scattered photons is generated by CIRCE2 program. A set of criteria for physical variables is proposed which leads to a good separation of stop signal events from top quark pair production, being the main background. These criteria allow us to reconstruct the mass of the top squark, provided that the neutralino mass is known.     

Measurement of CP asymmetries in neutralino production at the ILC

We study the prospects to measure the CP-sensitive triple-product asymmetries in neutralino production \(e^{+} e^{-} \to\tilde{\chi}^{0}_{i}\tilde{\chi}^{0}_{1}\) and subsequent leptonic two-body decays \(\tilde{\chi}^{0}_{i} \to \tilde{\ell}_{R} \ell\), \(\tilde{\ell}_{R} \to \tilde{\chi}^{0}_{1} \ell\), for ?=e,μ, within the Minimal Supersymmetric Standard Model. We include a full detector simulation of the International Large Detector for the International Linear Collider. The simulation was performed at a center-of-mass energy of \(\sqrt{s}=500\) GeV, including the relevant Standard Model background processes, a realistic beam energy spectrum, beam backgrounds and a beam polarization of 80% and ?60% for the electron and positron beams, respectively. In order to effectively disentangle different signal samples and reduce SM and SUSY backgrounds we apply a method of kinematic reconstruction. Assuming an integrated luminosity of 500 fb^?1 collected by the experiment and the performance of the current ILD detector, we arrive at a relative measurement accuracy of 10% for the CP-sensitive asymmetry in our scenario. We demonstrate that our method of signal selection using kinematic reconstruction can be applied to a broad class of scenarios and it allows disentangling processes with similar kinematic properties.     

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 \(a_0(980)\).     

Pinning down neutrino oscillation parameters in the 2–3 sector with a magnetised atmospheric neutrino detector: a new study

We determine the sensitivity to neutrino oscillation parameters from a study of atmospheric neutrinos in a magnetised detector such as the ICAL at the proposed India-based Neutrino Observatory. In such a detector, which can separately count \(\nu _\mu \) and \(\overline{\nu }_\mu \)-induced events, the relatively smaller (about 5%) uncertainties on the neutrino–antineutrino flux ratios translate to a constraint in the \(\chi ^2\) analysis that results in a significant improvement in the precision with which neutrino oscillation parameters such as \(\sin ^2\theta _{23}\) can be determined. Such an effect is unique to all magnetisable detectors and constitutes a great advantage in determining neutrino oscillation parameters using such detectors. Such a study has been performed for the first time here. Along with an increase in the kinematic range compared to earlier analyses, this results in sensitivities to oscillation parameters in the 2–3 sector that are comparable to or better than those from accelerator experiments where the fluxes are significantly higher. For example, the \(1\sigma \) precisions on \(\sin ^2\theta _{23}\) and \(|\Delta {m^2_{32(31)}}|\) achievable for 500 kton year exposure of ICAL are \({\sim }9\) and \({\sim }2.5\)%, respectively, for both normal and inverted hierarchies. The mass hierarchy sensitivity achievable with this combination when the true hierarchy is normal (inverted) for the same exposure is \(\Delta \chi ^2\approx 8.5\) (\(\Delta \chi ^2\approx 9.5\)).     

Chargino and neutralino production at $$e^+e^-$$ colliders in the complex MSSM: a full one-loop analysis

For the search for charginos and neutralinos in the Minimal Supersymmetric Standard Model (MSSM) as well as for future precision analyses of these particles an accurate knowledge of their production and decay properties is mandatory. We evaluate the cross sections for the chargino and neutralino production at \(e^+e^-\) colliders in the MSSM with complex parameters (cMSSM). The evaluation is based on a full one-loop calculation of the production mechanisms \(e^+e^- \rightarrow {\tilde{\chi }}_{c}^\pm {\tilde{\chi }}_{c^\prime }^\mp \) and \(e^+e^- \rightarrow {\tilde{\chi }}_{n}^0 {\tilde{\chi }}_{n^\prime }^0\)  including soft and hard photon radiation. We mostly restricted ourselves to a version of our renormalization scheme which is valid for \(|M_1| < |M_2|, |\mu |\) and \(M_2 \ne \mu \) to simplify the analysis, even though we are able to switch to other parameter regions and correspondingly different renormalization schemes. The dependence of the chargino/neutralino cross sections on the relevant cMSSM parameters is analyzed numerically. We find sizable contributions to many production cross sections. They amount to roughly \({\pm }15\%\) of the tree-level results but can go up to \({\pm }40\%\) or higher in extreme cases. Also the complex phase dependence of the one-loop corrections was found non-negligible. The full one-loop contributions are thus crucial for physics analyses at a future linear \(e^+e^-\) collider such as the ILC or CLIC.     

Non-linear coupling in the dark sector as a running vacuum model

In this work we study a phenomenological non-gravitational interaction between dark matter and dark energy. The scenario studied in this work extends the usual interaction model proportional to the derivative of the dark component density adding to the coupling a non-linear term of the form \(Q = \rho '/3(\alpha + \beta \rho _{Dark})\) This dark sector interaction model could be interpreted as a particular case of a running vacuum model of the type \(\Lambda (H) = n_0 + n_1 H^2 + n_2 H^4\) in which the vacuum decays into dark matter. For a flat FRW Universe filled with dark energy, dark matter and decoupled baryonic matter and radiation we calculate the energy density evolution equations of the dark sector and solve them. The different sign combinations of the two parameters of the model show clear qualitative different cosmological scenarios, from basic cosmological insights we discard some of them. The linear scalar perturbation equations of the dark matter were calculated. Using the CAMB code we calculate the CMB and matter power spectra for some values of the parameters \(\alpha \) and \(\beta \) and compare it with \(\Lambda \)CDM. The model modify mainly the lower multipoles of the CMB power spectrum remaining almost the same the high ones. The matter power spectrum for low wave numbers is not modified by the interaction but after the maximum it is clearly different. Using observational data from Planck, and various galaxy surveys we obtain the constraints of the parameters, the best fit values obtained are the combinations \(\alpha = (3.7 \pm 7 )\times 10^{-4} \), \(-\,(1.5\times 10^{-5}\, \mathrm{eV}^{-1})^{4} \ll \beta < (0.07\,\mathrm{eV}^{-1})^4\).     

Fully constrained Majorana neutrino mass matrices using $$\varvec{\varSigma (72\times 3)}$$

In 2002, two neutrino mixing ansatze having trimaximally mixed middle (\(\nu _2\)) columns, namely tri-chi-maximal mixing (\(\text {T}\chi \text {M}\)) and tri-phi-maximal mixing (\(\text {T}\phi \text {M}\)), were proposed. In 2012, it was shown that \(\text {T}\chi \text {M}\) with \(\chi =\pm \,\frac{\pi }{16}\) as well as \(\text {T}\phi \text {M}\) with \(\phi = \pm \,\frac{\pi }{16}\) leads to the solution, \(\sin ^2 \theta _{13} = \frac{2}{3} \sin ^2 \frac{\pi }{16}\), consistent with the latest measurements of the reactor mixing angle, \(\theta _{13}\). To obtain \(\text {T}\chi \text {M}_{(\chi =\pm \,\frac{\pi }{16})}\) and \(\text {T}\phi \text {M}_{(\phi =\pm \,\frac{\pi }{16})}\), the type I see-saw framework with fully constrained Majorana neutrino mass matrices was utilised. These mass matrices also resulted in the neutrino mass ratios, \(m_1:m_2:m_3=\frac{\left( 2+\sqrt{2}\right) }{1+\sqrt{2(2+\sqrt{2})}}:1:\frac{\left( 2+\sqrt{2}\right) }{-1+\sqrt{2(2+\sqrt{2})}}\). In this paper we construct a flavour model based on the discrete group \(\varSigma (72\times 3)\) and obtain the aforementioned results. A Majorana neutrino mass matrix (a symmetric \(3\times 3\) matrix with six complex degrees of freedom) is conveniently mapped into a flavon field transforming as the complex six-dimensional representation of \(\varSigma (72\times 3)\). Specific vacuum alignments of the flavons are used to arrive at the desired mass matrices.     

Testing the cosmic conservation of photon number with type Ia supernovae and ages of old objects

In this paper, we obtain luminosity distances by using ages of 32 old passive galaxies distributed over the redshift interval \(0.11< z < 1.84\) and test the cosmic conservation of photon number by comparing them with 580 distance moduli of type Ia supernovae (SNe Ia) from the so-called Union 2.1 compilation. Our analyses are based on the fact that the method of obtaining ages of galaxies relies on the detailed shape of galaxy spectra but not on galaxy luminosity. Possible departures from cosmic conservation of photon number is parametrized by \(\tau (z) = 2 \upvarepsilon z\) and \(\tau (z) = \upvarepsilon z/(1+z)\) (for \(\upvarepsilon =0\) the conservation of photon number is recovered). We find \(\upvarepsilon =0.016^{+0.078}_{-0.075}\) from the first parametrization and \(\upvarepsilon =-\,0.18^{+0.25}_{-0.24}\) from the second parametrization, both limits at 95% c.l. In this way, no significant departure from cosmic conservation of photon number is verified. In addition, by considering the total age as inferred from Planck (2015) analysis, we find the incubation time \(t_{inc}=1.66\pm 0.29\) Gyr and \(t_{inc}=1.23\pm 0.27\) Gyr at 68% c.l. for each parametrization, respectively.     

Gluino reach and mass extraction at the LHC in radiatively-driven natural SUSY

Radiatively-driven natural SUSY (RNS) models enjoy electroweak naturalness at the 10% level while respecting LHC sparticle and Higgs mass constraints. Gluino and top-squark masses can range up to several TeV (with other squarks even heavier) but a set of light Higgsinos are required with mass not too far above \(m_h\sim 125\) GeV. Within the RNS framework, gluinos dominantly decay via \(\tilde{g}\rightarrow t\tilde{t}_1^{*},\ \bar{t}\tilde{t}_1 \rightarrow t\bar{t}\widetilde{Z}_{1,2}\) or \(t\bar{b}\widetilde{W}_1^-+c.c.\), where the decay products of the higgsino-like \(\widetilde{W}_1\) and \(\widetilde{Z}_2\) are very soft. Gluino pair production is, therefore, signaled by events with up to four hard b-jets and large \(\not \!\!{E_T}\). We devise a set of cuts to isolate a relatively pure gluino sample at the (high-luminosity) LHC and show that in the RNS model with very heavy squarks, the gluino signal will be accessible for \(m_{\tilde{g}} < 2400 \ (2800)\) GeV for an integrated luminosity of 300 (3000) fb\(^{-1}\). We also show that the measurement of the rate of gluino events in the clean sample mentioned above allows for a determination of \(m_{\tilde{g}}\) with a statistical precision of 2–5% (depending on the integrated luminosity and the gluino mass) over the range of gluino masses where a 5\(\sigma \) discovery is possible at the LHC.     

Scaling Limit of Symmetric Random Walk in High-Contrast Periodic Environment

It is shown that the deterministic infinite trigonometric products

$$\begin{aligned} \prod _{n\in \mathbb {N}}\left[ 1- p +p\cos \left( \textstyle n^{-s}_{_{}}t\right) \right] =: {\text{ Cl }_{p;s}^{}}(t) \end{aligned}$$

with parameters \( p\in (0,1]\ \& \ s>\frac{1}{2}\), and variable \(t\in \mathbb {R}\), are inverse Fourier transforms of the probability distributions for certain random series \(\Omega _{p}^\zeta (s)\) taking values in the real \(\omega \) line; i.e. the \({\text{ Cl }_{p;s}^{}}(t)\) are characteristic functions of the \(\Omega _{p}^\zeta (s)\). The special case \(p=1=s\) yields the familiar random harmonic series, while in general \(\Omega _{p}^\zeta (s)\) is a “random Riemann-\(\zeta \) function,” a notion which will be explained and illustrated—and connected to the Riemann hypothesis. It will be shown that \(\Omega _{p}^\zeta (s)\) is a very regular random variable, having a probability density function (PDF) on the \(\omega \) line which is a Schwartz function. More precisely, an elementary proof is given that there exists some \(K_{p;s}^{}>0\), and a function \(F_{p;s}^{}(|t|)\) bounded by \(|F_{p;s}^{}(|t|)|\!\le \! \exp \big (K_{p;s}^{} |t|^{1/(s+1)})\), and \(C_{p;s}^{}\!:=\!-\frac{1}{s}\int _0^\infty \ln |{1-p+p\cos \xi }|\frac{1}{\xi ^{1+1/s}}\mathrm{{d}}\xi \), such that

$$\begin{aligned} \forall \,t\in \mathbb {R}:\quad {\text{ Cl }_{p;s}^{}}(t) = \exp \bigl ({- C_{p;s}^{} \,|t|^{1/s}\bigr )F_{p;s}^{}(|t|)}; \end{aligned}$$

the regularity of \(\Omega _{p}^\zeta (s)\) follows. Incidentally, this theorem confirms a surmise by Benoit Cloitre, that \(\ln {\text{ Cl }_{{{1}/{3}};2}^{}}(t) \sim -C\sqrt{t}\; \left( t\rightarrow \infty \right) \) for some \(C>0\). Graphical evidence suggests that \({\text{ Cl }_{{{1}/{3}};2}^{}}(t)\) is an empirically unpredictable (chaotic) function of t. This is reflected in the rich structure of the pertinent PDF (the Fourier transform of \({\text{ Cl }_{{{1}/{3}};2}^{}}\)), and illustrated by random sampling of the Riemann-\(\zeta \) walks, whose branching rules allow the build-up of fractal-like structures.

     

Search of the neutrino-less double beta decay of $$^{}$$Se into the excited states of $$^{}$$Kr with CUPID-0

The CUPID-0 experiment searches for double beta decay using cryogenic calorimeters with double (heat and light) read-out. The detector, consisting of 24 ZnSe crystals 95\(\%\) enriched in \(^{82}\)Se and two natural ZnSe crystals, started data-taking in 2017 at Laboratori Nazionali del Gran Sasso. We present the search for the neutrino-less double beta decay of \(^{82}\)Se into the 0\(_1^+\), 2\(_1^+\) and 2\(_2^+\) excited states of \(^{82}\)Kr with an exposure of 5.74 kg\(\cdot \)yr (2.24\(\times \)10\(^{25}\) emitters\(\cdot \)yr). We found no evidence of the decays and set the most stringent limits on the widths of these processes: \(\varGamma \)(\(^{82}\)Se \(\rightarrow ^{82}\)Kr\(_{0_1^+}\))8.55\(\times \)10\(^{-24}\) yr\(^{-1}\), \(\varGamma \) (\(^{82}\) Se \(\rightarrow ^{82}\) Kr \(_{2_1^+}\))\(\,{<}\,6.25 \,{\times }\,10^{-24}\) yr\(^{-1}\), \(\varGamma \)(\(^{82}\)Se \(\rightarrow ^{82}\)Kr\(_{2_2^+}\))8.25\(\times \)10\(^{-24}\) yr\(^{-1}\) (90\(\%\) credible interval).     

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 \(\Omega =2^{-4},...,2^{7}\).     

Closing in on the radiative weak chiral couplings

We point out that, given the current experimental status of radiative kaon decays, a subclass of the \(\mathcal{O} (p^4)\) counterterms of the weak chiral lagrangian can be determined in closed form. This involves in a decisive way the decay \(K^\pm \rightarrow \pi ^\pm \pi ^0 l^+ l^-\), currently being measured at CERN by the NA48/2 and NA62 collaborations. We show that consistency with other radiative kaon decay measurements leads to a rather clean prediction for the \(\mathcal{{O}}(p^4)\) weak couplings entering this decay mode. This results in a characteristic pattern for the interference Dalitz plot, susceptible to be tested already with the limited statistics available at NA48/2. We also provide the first analysis of \(K_S\rightarrow \pi ^+\pi ^-\gamma ^*\), which will be measured by LHCb and will help reduce (together with the related \(K_L\) decay) the experimental uncertainty on the radiative weak chiral couplings. A precise experimental determination of the \(\mathcal{{O}}(p^4)\) weak couplings is important in order to assess the validity of the existing theoretical models in a conclusive way. We briefly comment on the current theoretical situation and discuss the merits of the different theoretical approaches.