Global Well-Posedness of the Energy-Critical Defocusing NLS on $${\mathbb{R} \times \mathbb{T}^3}$$

We prove global well-posedness in H ¹ for the energy-critical defocusing initial-value problem \({(i\partial_t+\Delta_x)u=u|u|^2,\quad u(0)=\phi,}\) in the semiperiodic setting \({x\in\mathbb{R} \times \mathbb{T}^3}\) .     

Study of the weak annihilation contributions in charmless $$\varvec{B_s\rightarrow VV}$$ decays

In this paper, in order to probe the spectator-scattering and weak annihilation contributions in charmless \(B_s\rightarrow VV\) (where V stands for a light vector meson) decays, we perform the \(\chi ^2\)-analyses for the endpoint parameters within the QCD factorization framework, under the constraints from the measured \(\bar{B}_{s}\rightarrow \) \(\rho ^0\phi \), \(\phi K^{*0}\), \(\phi \phi \) and \(K^{*0}\bar{K}^{*0}\) decays. The fitted results indicate that the endpoint parameters in the factorizable and nonfactorizable annihilation topologies are non-universal, which is also favored by the charmless \(B\rightarrow PP\) and PV (where P stands for a light pseudo-scalar meson) decays observed in previous work. Moreover, the abnormal polarization fractions \(f_{L,\bot }(\bar{B}_{s}\rightarrow K^{*0}\bar{K}^{*0})=(20.1\pm 7.0)\%,(58.4\pm 8.5)\%\) measured by the LHCb collaboration can be reconciled through the weak annihilation corrections. However, the branching ratio of \(\bar{B}_{s}\rightarrow \phi K^{*0}\) decay exhibits a tension between the data and theoretical result, which dominates the contributions to \(\chi _\mathrm{min}^2\) in the fits. Using the fitted endpoint parameters, we update the theoretical results for the charmless \(B_s\rightarrow VV\) decays, which will be further tested by the LHCb and Belle-II experiments in the near future.     

SU(3) symmetry breaking in charmed baryon decays

We explore the breaking effects of the SU(3) flavor symmetry in the singly Cabibbo-suppressed anti-triplet charmed baryon decays of \(\mathbf{B}_c\rightarrow \mathbf{B}_n M\), with \(\mathbf{B}_c=(\Xi _c^0,\Xi _c^+,\Lambda _c^+)\) and \(\mathbf{B}_n(M)\) the baryon (pseudo-scalar) octets. We find that these breaking effects can be used to account for the experimental data on the decay branching ratios of \({\mathcal {B}}(\Lambda _c^+\rightarrow \Sigma ^{0} K^{+},\Lambda ^{0} K^{+})\) and \(R'_{K/\pi }={\mathcal {B}}(\Xi ^0_c \rightarrow \Xi ^- K^+)\)/\({\mathcal {B}}(\Xi ^0_c \rightarrow \Xi ^- \pi ^+)\). In addition, we obtain that \({\mathcal {B}}(\Xi _{c}^{0} \rightarrow \Xi ^{-} K^{+},\Sigma ^{-} \pi ^{+})=(4.6 \pm 1.7,12.8 \pm 3.1)\times 10^{-4}\), \({\mathcal {B}}(\Xi _c^0\rightarrow pK^-,\Sigma ^+\pi ^-)=(3.0 \pm 1.0, 5.2 \pm 1.6)\times 10^{-4}\) and \({\mathcal {B}}(\Xi _c^+\rightarrow \Sigma ^{0(+)} \pi ^{+(0)})=(10.3 \pm 1.7)\times 10^{-4}\), which all receive significant contributions from the breaking effects, and can be tested by the BESIII and LHCb experiments.     

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

On the Convexity of the KdV Hamiltonian

Motivated by perturbation theory, we prove that the nonlinear part \({H^{*}}\) of the KdV Hamiltonian \({H^{kdv}}\), when expressed in action variables \({I = (I_{n})_{n \geqslant 1}}\), extends to a real analytic function on the positive quadrant \({\ell^{2}_{+}(\mathbb{N})}\) of \({\ell^{2}(\mathbb{N})}\) and is strictly concave near \({0}\). As a consequence, the differential of \({H^{*}}\) defines a local diffeomorphism near 0 of \({\ell_{\mathbb{C}}^{2}(\mathbb{N})}\). Furthermore, we prove that the Fourier-Lebesgue spaces \({\mathcal{F}\mathcal{L}^{s,p}}\) with \({-1/2 \leqslant s \leqslant 0}\) and \({2 \leqslant p < \infty}\), admit global KdV-Birkhoff coordinates. In particular, it means that \({\ell^{2}_+(\mathbb{N})}\) is the space of action variables of the underlying phase space \({\mathcal{F}\mathcal{L}^{-1/2,4}}\) and that the KdV equation is globally in time \({C^{0}}\)-well-posed on \({\mathcal{F}\mathcal{L}^{-1/2,4}}\).     

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.     

A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index { < 4}

It is well known that the quantum double \({D(N\subset M)}\) of a finite depth subfactor \({N\subset M}\), or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. It is big open conjecture that all (unitary) modular tensor categories arise from conformal field theory. We show that for every subfactor \({N\subset M}\) with index \({[M:N] < 4}\) the quantum double \({D(N\subset M)}\) is realized as the representation category of a completely rational conformal net. In particular, the quantum double of \({E_6}\) can be realized as a \({\mathbb{Z}_2}\)-simple current extension of \({{{\rm SU}(2)}_{10}\times {{\rm Spin}(11)}_1}\) and thus is not exotic in any sense. As a byproduct, we obtain a vertex operator algebra for every such subfactor. We obtain the result by showing that if a subfactor \({N\subset M }\) arises from \({\alpha}\)-induction of completely rational nets \({\mathcal{A}\subset \mathcal{B}}\) and there is a net \({\tilde{\mathcal{A}}}\) with the opposite braiding, then the quantum \({D(N\subset M)}\) is realized by completely rational net. We construct completely rational nets with the opposite braiding of \({{{\rm SU}(2)}_k}\) and use the well-known fact that all subfactors with index \({[M:N] < 4}\) arise by \({\alpha}\)-induction from \({{{\rm SU}(2)}_k}\).     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

Spectral Measures Associated to Rank two Lie Groups and Finite Subgroups of {GL(2,\mathbb{Z})}

Spectral measures for fundamental representations of the rank two Lie groups SU(3), Sp(2) and G₂ have been studied. Since these groups have rank two, these spectral measures can be defined as measures over their maximal torus \({\mathbb{T}^2}\) and are invariant under an action of the corresponding Weyl group, which is a subgroup of \({GL(2,\mathbb{Z})}\). Here we consider spectral measures invariant under an action of the other finite subgroups of \({GL(2,\mathbb{Z})}\). These spectral measures are all associated with fundamental representations of other rank two Lie groups, namely \({\mathbb{T}^2=U(1) \times U(1)}\), \({U(1) \times SU(2)}\), U(2), \({SU(2) \times SU(2)}\), SO(4) and PSU(3).     

Trigonometric and Elliptic Ruijsenaars–Schneider Systems on the Complex Projective Space

We present a direct construction of compact real forms of the trigonometric and elliptic \({n}\)-particle Ruijsenaars–Schneider systems whose completed center-of-mass phase space is the complex projective space \({{\mathbb{CP}}^{n-1}}\) with the Fubini–Study symplectic structure. These systems are labeled by an integer \({p\in\{1,\ldots,n-1\}}\) relative prime to \({n}\) and a coupling parameter \({y}\) varying in a certain punctured interval around \({p\pi/n}\). Our work extends Ruijsenaars’s pioneering study of compactifications that imposed the restriction \({0 < y < \pi/n}\), and also builds on an earlier derivation of more general compact trigonometric systems by Hamiltonian reduction.     

Noncommutative Lévy Processes for Generalized (Particularly Anyon) Statistics

Let \({T=\mathbb R^d}\) . Let a function \({QT^2\to\mathbb C}\) satisfy \({Q(s,t)=\overline{Q(t,s)}}\) and \({|Q(s,t)|=1}\). A generalized statistics is described by creation operators \({\partial_t^\dagger}\) and annihilation operators ? _t , \({t\in T}\), which satisfy the Q-commutation relations: \({\partial_s\partial^\dagger_t = Q(s, t)\partial^\dagger_t\partial_s+\delta(s, t)}\) , \({\partial_s\partial_t = Q(t, s)\partial_t\partial_s}\), \({\partial^\dagger_s\partial^\dagger_t = Q(t, s)\partial^\dagger_t\partial^\dagger_s}\). From the point of view of physics, the most important case of a generalized statistics is the anyon statistics, for which Q(s, t) is equal to q if s < t, and to \({\bar q}\) if s > t. Here \({q\in\mathbb C}\) , |q| = 1. We start the paper with a detailed discussion of a Q-Fock space and operators \({(\partial_t^\dagger,\partial_t)_{t\in T}}\) in it, which satisfy the Q-commutation relations. Next, we consider a noncommutative stochastic process (white noise) \({\omega(t)=\partial_t^\dagger+\partial_t+\lambda\partial_t^\dagger\partial_t}\) , \({t\in T}\) . Here \({\lambda\in\mathbb R}\) is a fixed parameter. The case λ = 0 corresponds to a Q-analog of Brownian motion, while λ ≠ 0 corresponds to a (centered) Q-Poisson process. We study Q-Hermite (Q-Charlier respectively) polynomials of infinitely many noncommutatative variables \({(\omega(t))_{t\in T}}\) . The main aim of the paper is to explain the notion of independence for a generalized statistics, and to derive corresponding Lévy processes. To this end, we recursively define Q-cumulants of a field \({(\xi(t))_{t\in T}}\). This allows us to define a Q-Lévy process as a field \({(\xi(t))_{t\in T}}\) whose values at different points of T are Q-independent and which possesses a stationarity of increments (in a certain sense). We present an explicit construction of a Q-Lévy process, and derive a Nualart–Schoutens-type chaotic decomposition for such a process.     

Exotic tetraquark states with the $$qq\bar{Q}\bar{Q}$$ configuration

In this work, we study systematically the mass splittings of the \(qq\bar{Q}\bar{Q}\) (\(q=u\), d, s and \(Q=c\), b) tetraquark states with the color-magnetic interaction by considering color mixing effects and estimate roughly their masses. We find that the color mixing effect is relatively important for the \(J^P=0^+\) states and possible stable tetraquarks exist in the \(nn\bar{Q}\bar{Q}\) (\(n=u\), d) and \(ns\bar{Q}\bar{Q}\) systems either with \(J=0\) or with \(J=1\). Possible decay patterns of the tetraquarks are briefly discussed.     

Two Higgs doublet models and $$b\rightarrow s$$ exclusive decays

We computed the leading order Wilson coefficients relevant to all the exclusive \(b\rightarrow s\ell ^+\ell ^-\) decays in the framework of the two Higgs doublet model (2HDM) with a softly broken \(\mathbb {Z}_2\) symmetry by including the \(\mathcal {O}(m_b)\) corrections. We elucidate the issue of appropriate matching between the full and the effective theory when dealing with the (pseudo-)scalar operators for which keeping the external momenta different from zero is necessary. We then make a phenomenological analysis by using the measured \({\mathcal {B}}(B_s\rightarrow \mu ^+\mu ^-)\) and \({\mathcal {B}}(B\rightarrow K \mu ^+\mu ^-)_{\mathrm {high}-q^2}\), for which the hadronic uncertainties are well controlled, and we discuss their impact on various types of 2HDM. A brief discussion of the decays with \(\tau \)-leptons in the final state is provided too.     

Infinitesimal Deformations of a Formal Symplectic Groupoid

Given a formal symplectic groupoid G over a Poisson manifold (M, π ₀), we define a new object, an infinitesimal deformation of G, which can be thought of as a formal symplectic groupoid over the manifold M equipped with an infinitesimal deformation \({\pi_0 + \varepsilon \pi_1}\) of the Poisson bivector field π ₀. To any pair of natural star products \({(\ast,\tilde\ast)}\) having the same formal symplectic groupoid G we relate an infinitesimal deformation of G. We call it the deformation groupoid of the pair \({(\ast,\tilde\ast)}\) . To each star product with separation of variables \({\ast}\) on a Kähler–Poisson manifold M we relate another star product with separation of variables \({\hat\ast}\) on M. We build an algorithm for calculating the principal symbols of the components of the logarithm of the formal Berezin transform of a star product with separation of variables \({\ast}\) . This algorithm is based upon the deformation groupoid of the pair \({(\ast,\hat\ast)}\) .     

Generic Rigidity for Circle Diffeomorphisms with Breaks

We prove that \({C^r}\)-smooth (\({r > 2}\)) circle diffeomorphisms with a break, i.e., circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity, are generically, i.e., for almost all irrational rotation numbers, not \({C^{1+\varepsilon}}\)-rigid, for any \({\varepsilon > 0}\). This result complements our recent proof, joint with Khanin (Geom Funct Anal 24:2002–2028, 2014), that such maps are generically \({C^1}\)-rigid. It stands in remarkable contrast to the result of Yoccoz (Ann Sci Ec Norm Sup 17:333–361, 1984) that \({C^r}\)-smooth circle diffeomorphisms are generically \({C^{r-1-\varkappa}}\)-rigid, for any \({\varkappa > 0}\).     

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 \(a_0(980)\).     

Deformations of Nearly Kähler Instantons

We characterize the absolutely continuous spectrum of the one-dimensional Schrödinger operators \({h = -\Delta + v}\) acting on \({\ell^2(\mathbb{Z}_+)}\) in terms of the limiting behaviour of the Landauer–Büttiker and Thouless conductances of the associated finite samples. The finite sample is defined by restricting h to a finite interval \({[1, L] \cap \mathbb{Z}_+}\) and the conductance refers to the charge current across the sample in the open quantum system obtained by attaching independent electronic reservoirs to the sample ends. Our main result is that the conductances associated to an energy interval \({I}\) are non-vanishing in the limit \({L \to \infty}\) iff \({{\rm sp}_{\rm ac}(h) \cap I \neq \emptyset}\). We also discuss the relationship between this result and the Schrödinger Conjecture (Avila, J Am Math Soc 28:579–616, 2015; Bruneau et al., Commun Math Phys 319:501–513, 2013).     

Two Dimensional Water Waves in Holomorphic Coordinates

In this article we investigate spectral properties of the coupling \({H + V_\lambda}\), where \({H = -i\alpha \cdot \nabla+m\beta}\) is the free Dirac operator in \({\mathbb{R}^3}\), \({m > 0}\) and \({V_\lambda}\) is an electrostatic shell potential (which depends on a parameter \({\lambda \in \mathbb{R}}\)) located on the boundary of a smooth domain in \({\mathbb{R}^3}\). Our main result is an isoperimetric-type inequality for the admissible range of \({\lambda}\)’s for which the coupling \({H + V_\lambda}\) generates pure point spectrum in \({(-m, m)}\). That the ball is the unique optimizer of this inequality is also shown. Regarding some ingredients of the proof, we make use of the Birman–Schwinger principle adapted to our setting in order to prove some monotonicity property of the admissible \({\lambda}\)’s, and we use this to relate the endpoints of the admissible range of \({\lambda}\)’s to the sharp constant of a quadratic form inequality, from which the isoperimetric-type inequality is derived.     

The Structure of the Kac–Wang–Yan Algebra

The Lie algebra \({\mathcal{D}}\) of regular differential operators on the circle has a universal central extension \({\hat{\mathcal{D}}}\). The invariant subalgebra \({\hat{\mathcal{D}}^+}\) under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum \({\hat{\mathcal{D}}^+}\)-module with central charge \({c \in \mathbb{C}}\), and its irreducible quotient \({\mathcal{V}_c}\), possess vertex algebra structures, and \({\mathcal{V}_c}\) has a nontrivial structure if and only if \({c \in \frac{1}{2}\mathbb{Z}}\). We show that for each integer \({n > 0}\), \({\mathcal{V}_{n/2}}\) and \({\mathcal{V}_{-n}}\) are \({\mathcal{W}}\)-algebras of types \({\mathcal{W}(2, 4,\dots,2n)}\) and \({\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}\), respectively. These results are formal consequences of Weyl’s first and second fundamental theorems of invariant theory for the orthogonal group \({{\rm O}(n)}\) and the symplectic group \({{\rm Sp}(2n)}\), respectively. Based on Sergeev’s theorems on the invariant theory of \({{\rm Osp}(1, 2n)}\) we conjecture that \({\mathcal{V}_{-n+1/2}}\) is of type \({\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}\), and we prove this for \({n = 1}\). As an application, we show that invariant subalgebras of \({\beta\gamma}\)-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.     

Off-Diagonal Decay of Toric Bergman Kernels

We study the off-diagonal decay of Bergman kernels \({\Pi_{h^k}(z,w)}\) and Berezin kernels \({P_{h^k}(z,w)}\) for ample invariant line bundles over compact toric projective kähler manifolds of dimension m. When the metric is real analytic, \({P_{h^k}(z,w) \simeq k^m {\rm exp} - k D(z,w)}\) where \({D(z,w)}\) is the diastasis. When the metric is only \({C^{\infty}}\) this asymptotic cannot hold for all \({(z,w)}\) since the diastasis is not even defined for all \({(z,w)}\) close to the diagonal. Our main result is that for general toric \({C^{\infty}}\) metrics, \({P_{h^k}(z,w) \simeq k^m {\rm exp} - k D(z,w)}\) as long as w lies on the \({\mathbb{R}_+^m}\)-orbit of z, and for general \({(z,w)}\), \({{\rm lim\,sup}_{k \to \infty} \frac{1}{k} {\rm log} P_{h^k}(z,w) \,\leq\, - D(z^*,w^*)}\) where \({D(z, w^*)}\) is the diastasis between z and the translate of w by \({(S^1)^m}\) to the \({\mathbb{R}_+^m}\) orbit of z. These results are complementary to Mike Christ’s negative results showing that \({P_{h^k}(z,w)}\) does not have off-diagonal exponential decay at “speed” k if \({(z,w)}\) lies on the same \({(S^1)^m}\)-orbit.