Resonances, scattering theory, and rigged Hilbert spaces

The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,''in,'' and ''out'' eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ''out'' eigenvector family are the same as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ''complete'' sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ''out'' eigenvectors. The free, ''in'' and ''out'' eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee–Friedrichs model and to the scattering of a spinless particle by a local central potential. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Limiting two-Higgs-doublet models

We update the constraints on two-Higgs-doublet models (2HDMs) focusing on the parameter space relevant to explain the present muon g - 2 anomaly, Delta alpha(mu), in four different types of models, type I, II, ``lepton specific'' (or X) and ``flipped'' (or Y). We show that the strong constraints provided by the electroweak precision data on the mass of the pseudoscalar Higgs, whose contribution may account for Delta alpha(mu), are evaded in regions where the charged scalar is degenerate with the heavy neutral one and the mixing angles alpha and beta satisfy the Standard Model limit beta - alpha approximate to pi/2. We combine theoretical constraints from vacuum stability and perturbativity with direct and indirect bounds arising from collider and B physics. Possible future constraints from the electron g - 2 are also considered. If the 126 GeV resonance discovered at the LHC is interpreted as the light CP-even Higgs boson of the 2HDM, we find that only models of type X can satisfy all the considered theoretical and experimental constraints.

Predicting clinical response to anticancer drugs using an ex vivo platform that captures tumour heterogeneity

Predicting clinical response to anticancer drugs remains a major challenge in cancer treatment. Emerging reports indicate that the tumour microenvironment and heterogeneity can limit the predictive power of current biomarker-guided strategies for chemotherapy. Here we report the engineering of personalized tumour ecosystems that contextually conserve the tumour heterogeneity, and phenocopy the tumour microenvironment using tumour explants maintained in defined tumour grade-matched matrix support and autologous patient serum. The functional response of tumour ecosystems, engineered from 109 patients, to anticancer drugs, together with the corresponding clinical outcomes, is used to train a machine learning algorithm; the learned model is then applied to predict the clinical response in an independent validation group of 55 patients, where we achieve 100% sensitivity in predictions while keeping specificity in a desired high range. The tumour ecosystem and algorithm, together termed the CANScript technology, can emerge as a powerful platform for enabling personalized medicine.