CosySEL: Improving SAT Solving Using Local Symmetries

Solving necklace constraint problems

We propose a novel SAT-based approach to graph generation. Our approach utilizes the interaction between a CDCL SAT solver and a special symmetry propagator where the SAT solver runs on an encoding of the desired graph property. The symmetry propagator checks partially generated graphs for minimality with respect to a lexicographic ordering during the solving process. This approach has several advantages over a static symmetry breaking: (i) symmetries are detected early in the generation process, (ii) symmetry breaking is seamlessly integrated into the CDCL procedure, and (iii) the propagator performs a complete symmetry breaking without causing a prohibitively large initial encoding. We instantiate our approach by generating extremal graphs with certain restrictions in terms of forbidden subgraphs and diameter. In particular, we could confirm the Murty–Simon Conjecture (1979) on diameter-2-critical graphs for graphs up to 19 vertices and prove the exact number of Ramsey graphs \(\mathcal{R}(3,5,n)\) and \(\mathcal{R}(4,4,n)\) .

A modification of perturbation theory, known as the delta expansion (variationally improved perturbation), gave rigorously convergent series in some D=1 models (oscillator energy levels) with factorially divergent ordinary perturbative expansions. In a generalization of the variationally improved perturbation technique appropriate to renormalizable asymptotically free theories, we show that the large expansion orders of certain physical quantities are similarly improved, and prove the Borel convergence of the corresponding series for $m_{\mathrm {v}} \lessim 0$ , with $m_{\mathrm {v}}$ the new (arbitrary) mass perturbation parameter. We argue that non-ambiguous estimates of quantities relevant to dynamical (chiral) symmetry breaking in QCD are possible in this resummation framework.

We address the problem of accounting for light neutrino masses in theories with dynamical electroweak symmetry breaking. We discuss this in the context of a class of extended technicolor (ETC) models. As a possible solution, we propose a combination of suppressed Dirac masses and a seesaw involving dynamically generated condensates of standard‐model singlet, ETC‐nonsinglet fermions. We show how this can be realized in an explicit ETC model. An important feature of this proposal is that, because of the suppression of Dirac neutrino mass terms, a seesaw yielding realistic neutrino masses does not require superheavy Majorana masses. We then generalize this mechanism to theories with extended strong‐electroweak gauge symmetries.

Adding a fourth generation to the standard model and assuming it to be valid up to some cutoff $\ensuremath{\Lambda}$, we show that electroweak symmetry is broken by radiative corrections due to the fourth generation. The effects of the fourth generation are isolated using a Lagrangian with a genuine scalar without self-interactions at the classical level. For masses of the fourth generation consistent with electroweak precision data (including the $B\ensuremath{\rightarrow}K\ensuremath{\pi}$ $CP$ asymmetries), we obtain a Higgs mass of the order of a few hundreds GeV and a cutoff $\ensuremath{\Lambda}$ around 1--2 TeV. We study the reliability of the perturbative treatment used to obtain these results taking into account the running of the Yukawa couplings of the fourth quark generation with the aid of the renormalization group equations, finding similar allowed values for the Higgs mass but a slightly lower cutoff due to the breaking of the perturbative regime. Such low cutoff means that the effects of new physics needed to describe electroweak interactions at energy above $\ensuremath{\Lambda}$ should be measurable at the LHC. We use the minimal supersymmetric extension of the standard model with four generations as an explicit example of models realizing the dynamical electroweak symmetry breaking by radiative corrections and containing new physics. Here, the cutoff is replaced by the masses of the squarks and electroweak symmetry breaking by radiative corrections requires the squark masses to be of the order of 1 TeV.

Dynamical symmetry breaking can happen through a Higgs mechanism but also spontaneously within a strongly-enough coupled theory. We treat a field-theoretical quark model of QCD based on a linear+Coulomb Cornell potential (to account for the longitudinal interaction), together with a transverse interaction (to account for Coulomb-gauge gluons) in BCS approximation. After extracting the well-known BCS ground state on which abundant hadron phenomenology implementing dynamical chiral symmetry breaking has been built, we find two excited replica vacuum like states shown in figure 1. At the BCS level they are classically unstable, but second quantization blocks transitions between them in volumes much larger than a hadron size. Mesons built over the replica vacua have relative masses similar to normal mesons on the ordinary vacuum; we find no negative-mass mode, confirming their stability found earlier with a harmonic oscillator potential.

Proper vertex in studies of dynamical chiral symmetry breaking

Charge symmetry breaking (CSB) effects associated with the $u$ and $d$ quark mass difference are investigated in the quark distribution functions and spacelike electromagnetic form factors of the pion and kaon. We use a confining version of the Nambu--Jona-Lasinio model, where CSB effects at the infrared scale associated with the model are driven by the dressed $u$ and $d$ quark mass ratio, which because of dynamical chiral symmetry breaking is much closer to unity than the associated current quark mass ratio. The pion and kaon are given as bound states of a dressed quark and a dressed antiquark governed by the Bethe-Salpeter equation, and exhibit the properties of Goldstone bosons, with a pion mass difference given by $m_{\pi^+}^2 - m_{\pi^0}^2 \propto (m_u - m_d)^2$ as demanded by dynamical chiral symmetry breaking. We find significant CSB effects for realistic current quark mass ratios ($m_u/m_d \sim 0.5$) in the quark flavor-sector electromagnetic form factors of both the pion and kaon. For example, the difference between the $u$ and $d$ quark contributions to the $\pi^+$ electromagnetic form factors is about 8\% at a momentum transfer of $Q^2 \simeq 10\,$GeV$^2$, while the analogous effect for the light quark sector form factors in the $K^+$ and $K^0$ is about twice as large. For the Parton distribution functions, we find CSB effects which are considerably smaller than those found in the electromagnetic form factors.

It is known that the development and deep understanding of modern interaction theory and classical mechanics are made through geometry and symmetry. Yet, quantum mechanics which was regarded to be the microscopic theory of classical mechanics and achieved the crowning success in interpreting the entire microscopic world was developed purely from algebraic methods. In this thesis, we will study the geometry and dynamical symmetry in quantum systems, from which the question of integrability and chaos are explicitly addressed. First of all, the quantum dynamical degrees of freedom and quantum integrability are precisely defined and the inherent geometrical structure of quantum systems is explored from the fundamental structure of quantum theory. Such a geometrical structure can provide a framework to simultaneously build quantum and classical mechanics. The quantum-classical correspondence is then explicitly deduced. The dynamics of quantum system before it reaches the classical limit is formulated. Thus, the classical chaos is proven to be a special limiting phenomena of quantum systems and the dynamics before the system reaches its classical chaos is explored. The latter is the first step to seek the quantum manifestation of chaos. The relationship between integrability and dynamical symmetry are studied and some universal properties are discovered: a dynamical system (both quantum and classical) is integrable if it possesses a dynamical symmetry. Chaos will occur if the system undergoes a dynamical symmetry breaking and is accompanied by a structural phase transition. Thus, the concept of dynamical symmetry can be used to predict the general behaviors of a system. The theoretical underpinnings developed in this thesis are verified by many basic quantum mechanical examples. In the second part, we have applied the theory to construct the geometrical description for fermionic systems and then use this general formulation to study the geometrical properties, integrability and chaos in nuclear collective motions. The study of geometry for the fermion systems clearly show how the pure quantum properties (e.g. Pauli principle) are exhibited in geometry and how these geometrical manifestations can interpret experimental facts in nuclear collective motion (e.g. the satiation of nuclear deformation and narrow oblate window). The dynamical symmetry and dynamical symmetry breaking in nuclear collective motion are extensively studied.

Dimensional reduction and catalysis of dynamical symmetry breaking by a magnetic field

This report briefly describes the following research topics: aspects of dynamical symmetry breaking in gauge field theory; supersymmetric quantum electrodynamics and dynamical chiral symmetry breaking; anti-symmetric tensor gauge theories; extensions of the standard model; and neutrino oscillations in matter. (LSP)

We are going to analyze through a first order perturbative formulation the local loss of symmetry when a source of non-Abelian Yang-Mills and gravitational fields interacts with an external agent that perturbes the original geometry associated to the source. Then, as the symmetry in Abelian and non-Abelian field structures in four-dimensional Lorentzian spacetimes is displayed through the existence of local orthogonal planes of symmetry that we previously called blades one and two, the loss of symmetry will be manifested by the tilting of these planes under the influence of the external agent. It was found already that there is an algorithm to block diagonalize the Yang-Mills field strength isospace projections in a local gauge invariant way. Independently, it was also found an algorithm to diagonalize the Yang-Mills stress-energy tensor in a gauge invariant way. Using these results and perturbative analysis from a previous manuscript dealing with the Abelian case, we are going to demonstrate how to develop an algorithm for constructing local energy-momentum conserved currents inside both local orthogonal planes. As the interaction proceeds, the planes are going to tilt perturbatively, and in this strict sense the original local symmetries will be lost. But we will prove that the new blades at the same point will correspond after the tilting generated by perturbation, to new symmetries, with associated new local currents, both on each new local planes of symmetry. Old symmetries will be broken, new will arise. There will be a local symmetry evolution in the non-Abelian case as well.

With the advent of unified gauge theories, quantum chromodynamics, dynamical symmetry breaking and the like, the old and, as yet, unsolved problem of the n-p mass difference has been cast in a new light. Traditional attempts to explain the infamous sign reversal have invariably been based upon the assumption that the mass splitting was electromagnetic in origin.1 However, all attempts along these lines agree with naive Coulomb energy considerations and, therefore, fail. Nowadays one has an equally important origin of isospin symmetry breaking, namely, the quark masses themselves.2 Thus, the problem of the n-p mass difference becomes intimately related to the problem of quark mass splitting and these presumably arise from the dynamical breaking of an assumed unified symmetry. From this point of view, the smallness of isospin splitting appears accidental, being merely a reflection of the smallness of the current quark masses relative to more typical hadronic mass scales. This view leads to a remarkable conclusion, namely, that the formidable macroscopic consequences of the neutron being heavier than the proton have their origins at energies relevant to the breaking of a grand unified group. Regardless of the role played by the masses themselves, the conventional electromagnetic contribution has to be reckoned with; indeed, all quark mass splittings have two such contributions, namely those from dynamical symmetry breaking and those from gauge boson radiative corrections.

We investigate the dynamical mass generation resulted from interaction terms with four chiral superfields. The kind of interactions maybe considered a supersymmetric generalization of the four-fermion interactions of the classic Nambu--Jona-Lasinio model. A four-superfield interaction that contains a four-fermion interaction as one of its component terms has been the standard supersymmetrization of the NJL model for decades. Recently, we introduced a holomorphic variant with a dimension five interaction term instead. The latter is a main target of the present analysis. With the introduction of a new perspective on the superfield gap equation, we derive it for each one of the four-superfield interactions, using the supergraph technique. Through analyzing solutions to the gap equations, we illustrate the dynamical generation of superfield Dirac mass, including a supersymmetry breaking part. A dynamical symmetry breaking generally goes along with the dynamical mass generation, for which a bi-superfield condensate is responsible. The explicit illustration of dynamical symmetry breaking from the holomorphic dimension five interaction is reported for the first time. It has rich and novel features, which would be easily missed without the superfield approach developed here. We also discuss the nature of the bi-superfield condensate and its role of the effective Higgs superfield picture for both cases, illustrating their difference. Note that such a holomorphic quark superfield interaction term can successful account for the electroweak symmetry breaking with Higgs superfields as composites.

Symmetry breaking has emerged as a novel strategy to enhance energy conversion and storage performance, which refers to changes in the atomic configurations within a material reducing its internal symmetry. According to the location of the symmetry breaking, it can be classified into spontaneous symmetry breaking within the material, local symmetry breaking on the surface of the material, and symmetry breaking caused by external fields outside the material. However, there are currently few summaries in this field, so it is necessary to summarize how symmetry breaking improves energy conversion and storage performance. In this review, the fundamentals of symmetry breaking are first introduced, which allows for a deeper understanding of its meaning. Then the applications of symmetry breaking in energy conversion and storage are systematically summarized, providing various mechanisms in energy conversion and storage, as well as how to improve energy conversion performance and storage efficiency. Last but not least, the current applications of symmetry breaking are summarized and provide an outlook on its future development. It is hoped that this review can provide new insights into the applications of symmetry breaking and promote its further development.