119129530
Exact polynomial solutions of second order differential equations and their applications
We find all polynomials Z(z) such that the differential equation where X(z), Y(z), Z(z) are polynomials of degree at most 4, 3, 2, respectively, has polynomial solutions S(z) = ?ni = 1(z ? zi) of degree n with distinct roots zi. We derive a set of n algebraic equations which determine these roots. We also find all polynomials Z(z) which give polynomial solutions to the differential equation when the coefficients of X(z) and Y(z) are algebraically dependent. As applications to our general results, we obtain the exact (closed-form) solutions of the Schr?dinger-type differential equations describing: (1) two Coulombically repelling electrons on a sphere; (2) Schr?dinger equation from the kink stability analysis of ?6-type field theory; (3) static perturbations for the non-extremal Reissner?Nordstr?m solution; (4) planar Dirac electron in Coulomb and magnetic fields; and (5) O(N) invariant decatic anharmonic oscillator.
17047722
Infinite Families of Gauge-Equivalent R-Matrices and Gradations of Quantized Affine Algebras
Associated with the fundamental representation of a quantum algebra such as Uq(A1) or Uq(A2), there exist infinitely many gauge-equivalent R-matrices with different spectralparameter dependences. It is shown how these can be obtained by examining the infinitely many possible gradations of the corresponding quantum affine algebras, such as and , and explicit formulae are obtained for those two cases. Spectral-dependent similarity (gauge) transformations relate the R-matrices in different gradations, Nevertheless, the choice of gradation can be physically significant, as is illustrated in the case of quantum affine Toda field theories.
121731174
Some algebraic identities in anomalous gauge theories
Some useful algebraic relations between the anomaly of Dmu Jmu and anomalous commutators are derived using only the canonical commutators and the equation of motion. Their applications are discussed.
119694191
Construction of basis vectors for symmetric irreducible representations of O(5) $ \supset$ O(3)
Abstract.A recursive method for the construction of symmetric irreducible representations of $ O(2l+1)$ in the $ O(2l+1)\supset O(3)$ basis for identical boson systems is proposed. The formalism is realized based on the group chain $ U(2l+1)\supset U(2l-1)\otimes U(2)$ , of which the symmetric irreducible representations are simply reducible. The basis vectors for symmetric irreducible representations of the $ O(2l+1)\supset O(2l-1)\otimes U(1)$ can easily be constructed from those of U(2l + 1) $ \supset$U(2l - 1) ⊗ U(2) $ \supset$O(2l - 1) ⊗ U(1) with no l -boson pairs, namely with the total boson number exactly equal to the seniority number in the system, from which one can construct symmetric irreducible representations of $ O(2l+1)$ in the $ O(2l-1)\otimes U(1)$ basis when all symmetric irreducible representations of O(2l - 1) are known. As a starting point, basis vectors of symmetric irreducible representations of O(5) are constructed in the $ O_{1}(3)\otimes U(1)$ basis, where $ O_{1}(3)\equiv O(2l-1)$ , when l = 2 , which is generated not by the angular momentum operators of the d -boson system, but by the operators constructed from d -boson creation (annihilation) operators $ d^{\dagger}_{\mu}$ ( $ d_{\mu}$ with $ \mu=1$ , 0, -1 . Matrix representations of $ O(5)\supset O_{1}(3)\otimes U(1)$ , together with the elementary Wigner coefficients, are presented. After the angular momentum projection, a three-term relation in determining the expansion coefficients of the $ O(5)\supset O(3)$ basis vectors, where the O(3) group is generated by the angular momentum operators of the d -boson system, in terms of those of $ O_{1}(3)\otimes U(1)$ is derived. The eigenvectors of the projection matrix with zero eigenvalues constructed according to the three-term relation completely determine the basis vectors of $ O(5)\supset O(3)$ . Formulae for evaluating the elementary Wigner coefficients of $ O(5)\supset O(3)$ are derived explicitly. Analytical expressions of some elementary Wigner coefficients of $ O(5)\supset O(3)$ for the coupling $ (\tau 0)\otimes(1 0)$ with resultant angular momentum quantum number L = 2$ \tau$ + 2 - k for $ k=0,2,3,\ldots,6$ with a multiplicity 2 case for k = 6 are presented.
1264030
Elliptic Gaudin models and elliptic KZ equations
The Gaudin models based on the face-type elliptic quantum groups and the XYZ Gaudin models are studied. The Gaudin model Hamiltonians are constructed and are diagonalized by using the algebraic Bethe ansatz method. The corresponding face-type Knizhnik–Zamolodchikov equations and their solutions are given.
2107479
Local Quasitriangular Hopf Algebras
We find a new class of Hopf algebras, local quasitriangular Hopf algebras, which generalize quasitriangular Hopf algebras. Using these Hopf algebras, we obtain solutions of the Yang-Baxter equation in a systematic way. The category of modules with finite cycles over a local quasitriangular Hopf algebra is a braided tensor category.
118866595
Atmospheric Neutrino Mixing and b → s Transitions: Testing Lopsided SO(10) Flavor Structure in B physics
We point out that the correlation between the large atmospheric neutrino mixing and the b! s transition, if exists, comes from the lopsided flavor structure in SO(10) grand unification theory. We suggest testing the correlation by studying the deviation of SKS and S�'KS from the standard model predictions along with the constraints fromMBS and b! s in a realistic SO(10) model with lopsided flavor structure. We find a specific correlation between SKS and S�'KS that is intrinsic to the lopsided structure and discuss how to confirm or rule out this flavor pattern by the more accurate measurements of these CP violation quantities in B decay. The standard model (SM), although has passed many precision experimental tests, still has a number of out- standing problems which beg for a more fundamental the- ory. In terms of phenomenology, there are many places to test the more fundamental theory and, quite often, it is not one piece of phenomenology but a specific pattern of many pieces of phenomenology that fulfils this job. Neutrino oscillation clearly indicates the beyond SM physics in that it violates the accidental symmetry Le,µ,� of SM lepton flavor. Moreover, the tiny neutrino masses can be most naturally explained by the seesaw mech- anism (1), indicating the Majorana nature of neutrinos that leads to violation of another accidental symmetry B − L of SM at a very high scale close to the grand unification theory (GUT) scale. For this reason, study- ing neutrino oscillation in the GUT framework becomes particularly interesting. In SO(10) GUT, quarks and leptons are unified into a single 16 multiplet, yet experimental data shows that the 2-3 (generation) mixing is dramatically different in the quark and lepton sectors. While the quark mixing an- gle θ CKM is around 0.04, the lepton mixing angle θ PMNS (atmospheric mixing angle) is quite large and could be a maximum of π/4. This difference can be rather naturally explained by the lopsidedness in the 2-3 sector of mass matrices of down-type quarks and charged leptons:
118460040
Cosmological solutions of a nonlocal model with a perfect fluid
A nonlocal gravity model which does not assume the existence of a new dimensional parameter in the action and includes a function f(−1R), with the d'Alembertian operator, is studied. By specifying an exponential form for the function f and including a matter sector with a constant equation of state parameter, all available power-law solutions in the Jordan frame are obtained. New power-law solutions in the Einstein frame are also probed. Furthermore, the relationship between power-law solutions in both frames, established through conformal transformation, is substantially clarified. The correspondence between power-law solutions in these two frames is proven to be a very useful tool in order to obtain new solutions in the Einstein frame.
119268020
Integrand-level reduction of loop amplitudes by computational algebraic geometry methods
A bstractWe present an algorithm for the integrand-level reduction of multi-loop amplitudes of renormalizable field theories, based on computational algebraic geometry. This algorithm uses (1) the Gröbner basis method to determine the basis for integrand-level reduction, (2) the primary decomposition of an ideal to classify all inequivalent solutions of unitarity cuts. The resulting basis and cut solutions can be used to reconstruct the integrand from unitarity cuts, via polynomial fitting techniques. The basis determination part of the algorithm has been implemented in the Mathematica package, BasisDet. The primary decomposition part can be readily carried out by algebraic geometry softwares, with the output of the package BasisDet. The algorithm works in both D = 4 and D = 4 − 2ϵ dimensions, and we present some two and three-loop examples of applications of this algorithm.
119237527
SVOM gamma ray monitor
The space-based multi-band astronomical Variable Object Monitor (SVOM) mission is dedicated to the detection, localization and broad-band study of gamma-ray bursts (GRBs) and other high-energy transient phenomena. The gamma ray monitor (GRM) onboard is designed to observe GRBs up to 5 MeV. With this instrument, one of the key GRB parameters, Epeak, can be easily measured in the hard X-ray band. It can achieve a detection rate of 100 GRBs per year which ensures the scientific output of SVOM.
119294170
Topological charged black holes in generalized Hořava-Lifshitz gravity
As a candidate of quantum gravity in ultrahigh energy, the (3 + 1)-dimensional Horava-Lifshitz (HL) gravity with critical exponent z not equal 1 indicates anisotropy between time and space at short distance. In the paper, we investigate the most general z = d Horava-Lifshitz gravity in arbitrary spatial dimension d, with a generic dynamical Ricci flow parameter. and a detailed balance violation parameter epsilon. In arbitrary dimensional generalized HLd+1 gravity with z >= d at long distance, we study the topological neutral black hole solutions with general lambda in z = d HLd+1, as well as the topological charged black holes with lambda = 1 in z = d HLd+1. The HL gravity in the Lagrangian formulation is adopted, while in the Hamiltonian formulation, it reduces to Dirac-De Witt's canonical gravity with lambda = 1. In particular, the topological charged black holes in z = 5 HL6, z = 4 HL5, z = 3, 4 HL4, and z = 2 HL3 with lambda = 1 are solved. Their asymptotical behaviors near the infinite boundary and near the horizon are explored, respectively. We also study the behavior of the topological black holes in the (d + 1)-dimensional HL gravity with U(1) gauge field in the zero temperature limit and finite temperature limit, respectively. Thermodynamics of the topological charged black holes with lambda = 1, including temperature, entropy, heat capacity, and free energy are evaluated.