An improved model of vector mesons in holographic QCD

We analyze the sector of dimension-three vector meson operators in the "hard wall" model of holographic QCD, including the vector and axial currents, dual to gauge fields in the bulk, and the tensor operator $\bar{\psi}\sigma^{\mu\nu}\psi$, dual to a two-form field satisfying a complex self-duality condition. The model includes the effect of chiral symmetry breaking on vector mesons, that involves a coupling between the dual gauge field and the two-form field. We compute the leading logarithmic terms in the operator product expansion of two-point functions and the leading non-perturbative contribution to the tensor-vector correlator. The result is consistent with the operator product expansion of QCD. We also study the spectrum of vector mesons numerically.Comment: 19 page

Maximal quadratic modules on *-rings

We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to $\ast$-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The simplest example of a maximal proper quadratic module is the cone of all positive semidefinite complex matrices of a fixed dimension. We show that the support of a maximal proper quadratic module is the symmetric part of a prime $\ast$-ideal, that every maximal proper quadratic module in a Noetherian $\ast$-ring comes from a maximal proper quadratic module in a simple artinian ring with involution and that maximal proper quadratic modules satisfy an intersection theorem. As an application we obtain the following extension of Schm\" udgen's Strict Positivstellensatz for the Weyl algebra: Let $c$ be an element of the Weyl algebra $\mathcal{W}(d)$ which is not negative semidefinite in the Schr\" odinger representation. It is shown that under some conditions there exists an integer $k$ and elements $r_1,...,r_k \in \mathcal{W}(d)$ such that $\sum_{j=1}^k r_j c r_j^\ast$ is a finite sum of hermitian squares. This result is not a proper generalization however because we don't have the bound $k \le d$.Comment: 11 page

Quaternion-Octonion SU(3) Flavor Symmetry

Starting with the quaternionic formulation of isospin SU(2) group, we have derived the relations for different components of isospin with quark states. Extending this formalism to the case of SU(3) group we have considered the theory of octonion variables. Accordingly, the octonion splitting of SU(3) group have been reconsidered and various commutation relations for SU(3) group and its shift operators are also derived and verified for different iso-spin multiplets i.e. I, U and V- spins. Keywords: SU(3), Quaternions, Octonions and Gell Mann matrices PACS NO: 11.30.Hv: Flavor symmetries; 12.10-Dm: Unified field theories and models of strong and electroweak interaction

Quaternion Octonion Reformulation of Quantum Chromodynamics

We have made an attempt to develop the quaternionic formulation of Yang - Mill's field equations and octonion reformulation of quantum chromo dynamics (QCD). Starting with the Lagrangian density, we have discussed the field equations of SU(2) and SU(3) gauge fields for both cases of global and local gauge symmetries. It has been shown that the three quaternion units explain the structure of Yang- Mill's field while the seven octonion units provide the consistent structure of SU(3)_{C} gauge symmetry of quantum chromo dynamics

Semi-invariants of symmetric quivers of finite type

Let $(Q,\sigma)$ be a symmetric quiver, where $Q=(Q_0,Q_1)$ is a finite quiver without oriented cycles and $\sigma$ is a contravariant involution on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form on a representation $V$ of $Q$. We shall call the representation orthogonal if is symmetric and symplectic if $$ is skew-symmetric. Moreover we can define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. For symmetric quivers of finite type, we prove that the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type $c^V$ and, in the case when matrix defining $c^V$ is skew-symmetric, by the Pfaffians $pf^V$

Dynamical Chiral Symmetry Breaking on the Light Front.II. The Nambu--Jona-Lasinio Model

An investigation of dynamical chiral symmetry breaking on the light front is made in the Nambu--Jona-Lasinio model with one flavor and N colors. Analysis of the model suffers from extraordinary complexity due to the existence of a "fermionic constraint," i.e., a constraint equation for the bad spinor component. However, to solve this constraint is of special importance. In classical theory, we can exactly solve it and then explicitly check the property of ``light-front chiral transformation.'' In quantum theory, we introduce a bilocal formulation to solve the fermionic constraint by the 1/N expansion. Systematic 1/N expansion of the fermion bilocal operator is realized by the boson expansion method. The leading (bilocal) fermionic constraint becomes a gap equation for a chiral condensate and thus if we choose a nontrivial solution of the gap equation, we are in the broken phase. As a result of the nonzero chiral condensate, we find unusual chiral transformation of fields and nonvanishing of the light-front chiral charge. A leading order eigenvalue equation for a single bosonic state is equivalent to a leading order fermion-antifermion bound-state equation. We analytically solve it for scalar and pseudoscalar mesons and obtain their light-cone wavefunctions and masses. All of the results are entirely consistent with those of our previous analysis on the chiral Yukawa model.Comment: 23 pages, REVTEX, the version to be published in Phys.Rev.D; Some clarifications in discussion of the LC wavefunctions adde

Self-bound dense objects in holographic QCD

We study a self-bound dense object in the hard wall model. We consider a spherically symmetric dense object which is characterized by its radial density distribution and non-uniform but spherically symmetric chiral condensate. For this we analytically solve the partial differential equations in the hard wall model and read off the radial coordinate dependence of the density and chiral condensate according to the AdS/CFT correspondence. We then attempt to describe nucleon density profiles of a few nuclei within our framework and observe that the confinement scale changes from a free nucleon to a nucleus. We briefly discuss how to include the effect of higher dimensional operator into our study. We finally comment on possible extensions of our work.Comment: 17 pages, 5 figures, figures replaced, minor revision, to appear in JHE

Strongly Interacting Neutrinos and the Highest Energy Cosmic Rays

Cosmic rays of energies larger than the Greisen-Zatsepin-Kuzmin (GZK) cutoff may be neutrinos if they acquire strong interactions due to a ``precocious unification'' of forces. A scenario for this to happen is outlined. There is no contradiction with precision measurements carried out at LEP and SLAC. Observable consequences at LHC and future neutrino detectors are discussed.Comment: 9 pages, LaTeX2e, no macros, 2 eps. figures. Uses epsf.te