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We study resonances for the Schr\"odinger operators with quadratic or sub-quadratic repulsive potential. In the present paper, we show the non-existence of resonances in some complex neighborhood of a fixed energy by employing schemes of Briet-Combes-Duclos and Hunziker and by introducing a proper distortion based on, but slightly modified, positive commutator method of Mourre., Comment: 16 pages

We investigate spectral theory for a large class of one-body Stark Hamiltonians using a commutator technique. Our results include Rellich's theorem, the limiting absorption principle, radiation condition bounds and Sommerfeld's uniqueness theorem, all stated and proved in the framework of Besov spaces.

We study the stationary scattering theory for a perturbed 1-body Stark operator. We prove existence and completeness of the stationary wave operators, construct the associated generalized Fourier transforms, and characterize asymptotics of the generalized eigenfunctions of minimal growths in terms of the stationary scattering matrix.

We develop a new scheme of proofs for spectral theory of the $N$-body Schr\"odinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich's theorem, limiting absorption principle bounds, microlocal resolvent bounds, H\"older continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich's theorem which is unified with exponential decay estimates studied previously only for $L^2$-eigenfunctions. Each pair-potential is a sum of a long-range term with first order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with `zeroth order' operators. In particular they do not rely on Mourre's differential inequality technique., Comment: 38 pages

In order to describe forward hadron productions in high-energy nuclear collisions, we propose a Monte-Carlo implementation of Dumitru-Hayashigaki-Jalilian-Marian formula with the unintegrated gluon distribution obtained numerically from the running-coupling BK equation. We discuss influence of initial conditions for the BK equation by comparing a model constrained by global fit of small-x HERA data and a newly proposed one from the running coupling MV model., Comment: Talk given at conference Quark Matter 2011, 4 pages

A homogeneous color magnetic field is known to be unstable for the fluctuations perpendicular to the field in the color space (the Nielsen-Olesen instability). We argue that these unstable modes, exponentially growing, generate an azimuthal magnetic field with the original field being in the z-direction, which causes the Nielsen-Olesen instability for another type of fluctuations. The growth rate of the latter unstable mode increases with the momentum p_z and can become larger than the former's growth rate which decreases with increasing p_z. These features may explain the interplay between the primary and secondary instabilities observed in the real-time simulation of a non-expanding glasma, i.e., stochastically generated anisotropic Yang-Mills fields without expansion., Comment: 18 pages, 4 figures (minor revision)

Proceedings of ISMD08, Comment: Edited by: J. Bartels, K. Borras, G. Gustafson, H. Jung, K. Kutak, S. Levonian, and J. Mnich

Strong longitudinal color flux fields will be created in the initial stage of high-energy nuclear collisions. We investigate analytically time evolution of such boost-invariant color fields from Abelian-like initial conditions, and next examine stability of the boost-invariant configurations against rapidity dependent fluctuations. We find that the magnetic background field has an instability induced by the lowest Landau level whose amplitude grows exponentially. For the electric background field there is no apparent instability although pair creations due to the Schwinger mechanism should be involved., Comment: 4p, 3figs; poster contribution to QM2008

We investigate the effects of baryon chemical potential \mu on the shear viscosity coefficient \eta and the viscosity to entropy density ratio \eta/s of a pion-nucleon gas mixture. We find that \eta is an increasing function of T and \mu, while the ratio \eta/s turns to a decreasing function in a wide region of T-\mu plane. In the kinematical region we studied, the smallest value of \eta/s is about 0.3., Comment: 4pages, 5 figures, presented at the 20th International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions (QM2008)

We present an analytic study of the physics of the glasma which is a strong classical gluon field created at early stage of high-energy heavy-ion collisions. Our analysis is based on the picture that the glasma just after the collision is made of color electric and magnetic flux tubes extending in the longitudinal direction with their diameters of the order of 1/Q_s (Q_s is the saturation scale of the colliding nuclei). We find that both the electric and magnetic flux tubes expand outwards and the field strength inside the flux tube decays rapidly in time. Next we investigate whether there exist instabilities against small rapidity-dependent perturbations for a fixed color configuration. We find that the magnetic background field exhibits an instability induced by the fluctuations in the lowest Landau level, and it grows in the time scale of 1/Q_s. For the electric background field we find no apparent instability while the possible relation to the Schwinger mechanism for particle pair creations is suggested., Comment: 29 pages, 5 figures; minor revision, references added

I discuss recent results on the Color Glass Condensate which is a dense saturated gluonic state and appears as the universal picture of hadrons or nuclei at very high energies., Comment: 10 pages, 4 figures. Plenary talk given at 18th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions: Quark Matter 2005, Budapest, Hungary, 4-9 Aug 2002

We propose an effective theory which governs Pomeron dynamics in QCD at high energy, in the leading logarithmic approximation, and in the limit where N_c, the number of colors, is large. In spite of its remarkably simple structure, this effective theory generates precisely the evolution equations for scattering amplitudes that have been recently deduced from a more complete microscopic analysis. It accounts for the BFKL evolution of the Pomerons together with their interactions: dissociation (one Pomeron splitting into two) and recombination (two Pomerons merging into one). It is constructed by exploiting a duality principle relating the evolutions in the target and the projectile, more precisely, splitting and merging processes, or fluctuations in the dilute regime and saturation effects in the dense regime. The simplest Pomeron loop calculated with the effective theory is free of both ultraviolet or infrared singularities., Comment: 13 pages, 1 figure

We discuss the definition and the energy evolution of scattering amplitudes with $C$-odd ("odderon") quantum numbers within the effective theory for the Color Glass Condensate (CGC) endowed with the functional, JIMWLK, evolution equation. We explicitly construct gauge-invariant amplitudes describing multiple odderon exchanges in the scattering between the CGC and two types of projectiles: a color--singlet quark--antiquark pair (or `color dipole') and a system of three quarks in a colorless state. We deduce the energy evolution of these amplitudes from the general JIMWLK equation, which for this purpose is recast in a more synthetic form, which is manifestly infrared finite. For the dipole odderon, we confirm and extend the non--linear evolution equations recently proposed by Kovchegov, Szymanowski and Wallon, which couple the evolution of the odderon to that of the pomeron, and predict the rapid suppression of the odderon exchanges in the saturation regime at high energy. For the 3--quark system, we focus on the linear regime at relatively low energy, where our general equations are shown to reduce to the Bartels--Kwiecinski--Praszalowicz equation. Our gauge--invariant amplitudes, and the associated evolution equations, stay explicitly outside the M\"obius representation, which is the Hilbert space where the BFKL Hamiltonian exhibits holomorphic separability., Comment: 43 pages, 1 figure

We study the properties of light mesons in the scalar, pseudo-scalar, and vector channels within the light-front quantization, by using the (one flavor) Nambu--Jona-Lasinio model with vector interaction. After taking into account the effects of chiral symmetry breaking, we derive the bound-state equation in each channel in the large N limit (N is the number of colors), which means that we consider the lowest q\bar{q} Fock state with the constituent quark and antiquark. By solving the bound-state equation, we simultaneously obtain a mass and a light-cone (LC) wavefunction of the meson. While we reproduce the previous results for the scalar and pseudo-scalar mesons, we find that, for a vector meson, the bound-state equations for the transverse and longitudinal polarizations look different from each other. However, eventually after imposing a cutoff which is invariant under the parity and boost transformations, one finds these two are identical, giving the same mass and the same (spin-independent) LC wavefunction. When the vector interaction becomes larger than a critical value, the vector state forms a bound state, whose mass decreases as the interaction becomes stronger. While the LC wavefunction of the pseudo-scalar meson is broadly distributed in longitudinal momentum (x) space, that of the vector meson is squeezed around x=1/2., Comment: 34 pages, 5 figures, minor modification, to appear in Phys. Rev. D

We present a systematic, and fully analytic, study of the ratio R_{pA} between the gluon distribution in a nucleus and that in a proton scaled up by the atomic number A. We consider initial conditions of the McLerran-Venugopalan type, and quantum evolution in the Color Glass Condensate, with both fixed and running coupling. We perform an analytic study of the Cronin effect in the initial conditions and point out an interesting difference between saturating effects and twist effects in the nuclear gluon distribution. We show that the distribution of the gluons which make up the condensate in the initial conditions is localized at low momenta, but this particular feature does not survive after the quantum evolution. We demonstrate that the rapid suppression of the ratio R_{pA} in the early stages of the evolution is due to the DGLAP-like evolution of the proton, whose gluon distribution grows much faster than that in the nucleus because of the large separation between the respective saturation momenta. The flattening of the Cronin peak, on the other hand, is due to the evolution of the nucleus. We show that the running coupling effects slow down the evolution, but eventually lead to a stronger suppression in R_{pA} at sufficiently large energies., Comment: 87 pages, 11 figures. More explanations added (especially on the A-dependence of the ratio R_{pA}), and also more acknowledgements and references. The discussion of the running coupling case has been considerably extended

We show that the HERA data for the inclusive structure function F_2(x,Q^2) for x < 0.01 and 0.045 < Q^2 < 45 GeV^2 can be well described within the color dipole picture, with a simple analytic expression for the dipole-proton scattering amplitude, which is an approximate solution to the non-linear evolution equations in QCD. For dipole sizes less than the inverse saturation momentum 1/Q_s(x), the scattering amplitude is the solution to the BFKL equation in the vicinity of the saturation line. It exhibits geometric scaling and scaling violations by the diffusion term. For dipole sizes larger than 1/Q_s(x), the scattering amplitude saturates to one. The fit involves three parameters: the proton radius R, the value x_0 of x at which the saturation scale Q_s equals 1GeV, and the logarithmic derivative of the saturation momentum \lambda. The value of \lambda extracted from the fit turns out to be consistent with a recent calculation using the next-to-leading order BFKL formalism., Comment: 15 pages, 3 figures, 3 tables

We apply the light-front quantization to the Nambu--Jona-Lasinio model with the vector interaction, and compute vector meson's mass and light-cone wavefunction in the large N limit. Following the same procedure as in the previous analyses for scalar and pseudo-scalar mesons, we derive the bound-state equations of a q-qbar system in the vector channel. We include the lowest order effects of the vector interaction. The resulting transverse and longitudinal components of the bound-state equation look different from each other. But eventually after imposing an appropriate cutoff, one finds these two are identical, giving the same mass and the same (spin-independent) light-cone wavefunction. Mass of the vector meson decreases as one increases the strength of the vector interaction., Comment: 11 pages, 3 figures, discussion on the cutoff scheme changed, Fig.3 replaced, and one reference added

We demonstrate that the dipole-hadron cross-section computed from the non-linear evolution equation for the Colour Glass Condensate saturates the Froissart bound in the case of a fixed coupling and for a small dipole (Q^2 >> Lambda_{QCD}^2). That is, the cross-section increases as the logarithm squared of the energy, with a proportionality coefficient involving the pion mass and the BFKL intercept (alpha_s N_c/pi)4 ln 2. The pion mass enters via the non-perturbative initial conditions at low energy. The BFKL equation emerges as a limit of the non-linear evolution equation valid in the tail of the hadron wavefunction. We provide a physical picture for the transverse expansion of the hadron with increasing energy, and emphasize the importance of the colour correlations among the saturated gluons in suppressing non-unitary contributions due to long-range Coulomb tails. We present the first calculation of the saturation scale including the impact parameter dependence. We show that the cross-section at high energy exhibits geometric scaling with a different scaling variable as compared to the intermediate energy regime., Comment: 42 pages, 5 included figures. A few references and explanations added. Some recent arguments concerning the role of the "white region" are clarified. Final version to appear in Nuclear Physics A

The two-flavor color superconductivity is examined over a wide range of baryon density with a single model. To study the structural change of Cooper pairs, quark correlation in the color superconductor is calculated both in the momentum space and in the coordinate space. At extremely high baryon density, our model becomes equivalent to the usual perturbative QCD treatment and the gap is shown to have a sharp peak near the Fermi surface due to the weak-coupling nature of QCD. On the other hand, the gap is a smooth function of the momentum at lower densities due to strong color magnetic and electric interactions. The size of the Cooper pair is shown to become comparable to the averaged inter-quark distance at low densities, which indicates a crossover from BCS to BEC (Bose-Einstein condensation) of tightly bound Cooper pairs may take place at low density., Comment: 6 pages, 5 figures. Invited talk at the Joint CSSM/JHF Workshop on Physics at Japan Hadron Facility (March 14-21, Adelaide, 2002)

Geometric scaling is a novel scaling phenomenon observed in deep inelastic scattering at small x: the total virtual photon-proton cross section depends upon the two kinematical variables Q^2 and x only via their combination Q^2 R_0^2(x), with R_0^2(x) \propto x^{\lambda}. At sufficiently low Q^2, below the saturation scale Q_s^2(x) (a few GeV^2), this phenomenon finds a natural explanation as a property of the Color Glass Condensate, the high-density matter made of saturated gluons. To explain the experimental observation of geometric scaling up to much higher values of Q^2, of the order of 100 GeV^2, we study the solution to the BFKL equation subjected to a saturation boundary condition at Q^2\sim Q_s^2(x). We find that the scaling extends indeed above the saturation scale, within a window 1 < \ln(Q^2/Q_s^2) << \ln(Q_s^2/\Lambda^2_{\rm QCD}), which is consistent with phenomenology., Comment: 4 pages, 1 figure, uses moriond.sty, included. Talk presented at the XXXVIIth Rencontres de Moriond "QCD and high energy hadronic interactions", Les Arcs, France, March 16-23, 2002

We show that the evolution equations in QCD predict geometric scaling for quark and gluon distribution functions in a large kinematical window, which extends above the saturation scale up to momenta $Q^2$ of order $100 {\rm GeV}^2$. For $Q^2 < Q^2_s$, with $Q_s$ the saturation momentum, this is the scaling predicted by the Colour Glass Condensate and by phenomenological saturation models. For $1 \simle \ln(Q^2/Q_s^2) \ll \ln(Q_s^2/\Lambda^2_{\rm QCD})$, we show that the solution to the BFKL equation shows approximate scaling, with the scale set by $Q_s$. At larger $Q^2$, this solution does not scale any longer. We argue that for the intermediate values of $Q^2$ where we find scaling, the BFKL rather than the double logarithmic approximation to the DGLAP equation properly describes the dynamics. We consider both fixed and running couplings, with the scale for running set by the saturation momentum. The anomalous dimension which characterizes the approach of the gluon distribution function towards saturation is found to be close to, but lower than, one half., Comment: 24 pages

We discuss a description of chiral symmetry breaking in the light-front (LF) formalism. Based on careful analyses of several models, we give clear answers to the following three fundamental questions: (i) What is the difference between the LF chiral transformation and the ordinary chiral transformation? (ii) How does a gap equation for the chiral condensate emerge? (iii) What is the consequence of the coexistence of a nonzero chiral condensate and the trivial Fock vacuum? The answer to Question (i) is given through a classical analysis of each model. Question (ii) is answered based on our recognition of the importance of characteristic constraints, such as the zero-mode and fermionic constraints. Question (iii) is intimately related to another important problem, reconciliation of the nonzero chiral condensate and the invariance of the vacuum under the LF chiral transformation. This and Question (iii) are understood in terms of the modified chiral transformation laws of the dependent variables. The characteristic ways in which the chiral symmetry breaking is realized are that the chiral charge is no longer conserved and that the transformation of the scalar and pseudoscalar fields is modified. We also discuss other outcomes, such as the light-cone wave function of the pseudoscalar meson in the Nambu-Jona-Lasinio model., Comment: 26 pages, no figure, REVTEX, Journal version

We study the diffractive photo- and leptoproductions of $J / \psi$ and $\psi '$ on the proton, and examine the validity of the small-size color dipole approximation to the production of radially excited heavy quarkonium. The dipole model predicts a small ratio of $\psi '$ to $J / \psi$ photoproduction cross sections, which does not agree with experimental data. We show that this discrepancy originates from a large transverse size of $\psi '$ which makes the convergence of the transverse size expansion questionable, and the calculation without the dipole approximation turns out to be consistent with the data. Productions of $\Upsilon '(2S,3S)$ are also discussed, and the dipole approximation is found to be reasonable for the $\Upsilon$-family., Comment: minor modifications are made, Phys. Rev. D in press

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 added

Dynamical chiral symmetry breaking in the DLCQ method is investigated in detail using a chiral Yukawa model closely related to the Nambu-Jona-Lasinio model. By classically solving three constraints characteristic of the light-front formalism, we show that the chiral transformation defined on the light front is equivalent to the usual one when bare mass is absent. A quantum analysis demonstrates that a nonperturbative mean-field solution to the ``zero-mode constraint'' for a scalar boson (sigma) can develop a nonzero condensate while a perturbative solution cannot. This description is due to our identification of the ``zero-mode constraint'' with the gap equation. The mean-field calculation clarifies unusual chiral transformation properties of fermionic field, which resolves a seemingly inconsistency between triviality of the null-plane chiral charge Q_5|0>=0 and nonzero condensate. We also calculate masses of scalar and pseudoscalar bosons for both symmetric and broken phases, and eventually derive the PCAC relation and nonconservation of Q_5 in the broken phase., Comment: Revised version to appear in Phys. Rev. D. 19 pages, 4 figures, REVTEX. Derivation of the PCAC relation is given. Its relation to the nonconservation of chiral charge is clarified. 1 figure and some references added

We investigate canonical structure of the Abelian Higgs model within the framework of DLCQ. Careful boundary analysis of differential equations, such as the Euler-Lagrange equations, leads us to a novel situation where the canonical structure changes in a drastic manner depending on whether the (light-front) spatial Wilson line is periodic or not. In the former case, the gauge-field ZM takes discrete values and we obtain the so-called ``Zero-Mode Constraints'' (ZMCs), whose semiclassical solutions give a nonzero vev to the scalar fields. Contrary, in the latter case, we have no ZMC and the scalar ZMs remain dynamical as well as the gauge-field ZM. In order to give classically nonzero vev to the scalar field, we work in a background field which minimizes the light-front energy., Comment: 10 pages, reference modified

Spontaneous symmetry breaking of the light-front Gross-Neveu model is studied in the framework of the discretized light-cone quantization. Introducing a scalar auxiliary field and adding its kinetic term, we obtain a constraint on the longitudinal zero mode of the scalar field. This zero-mode constraint is solved by using the $1/N$ expansion. In the leading order, we find a nontrivial solution which gives the fermion nonzero mass and thus breaks the discrete symmetry of the model. It is essential for obtaining the nontrivial solution to treat adequately an infrared divergence which appears in the continuum limit. We also discuss the constituent picture of the model. The Fock vacuum is trivial and an eigenstate of the light-cone Hamiltonian. In the large $N$ limit, the Hamiltonian consists of the kinetic term of the fermion with dressed mass and the interaction term of these fermions., Comment: 25 pages, Latex, no figures, to be published in Progress of Theoretical Physics

Correlation between instantons and QCD-monopoles is studied in the abelian-gauge-fixed QCD. From a simple topological consideration, instantons are expected to appear only around the QCD-monopole trajectory in the abelian-dominating system. The QCD-monopole in the multi-instanton solution is studied in the Polyakov-like gauge, where $A_4(x)$ is diagonalized. The world line of the QCD-monopole is found to be penetrate the center of each instanton. For the single-instanton solution, the QCD-monopole trajectory becomes a simple straight line. On the other hand, in the multi-instanton system, the QCD-monopole trajectory often has complicated topology including a loop or a folded structure, and is unstable against a small fluctuation of the location and the size of instantons. We also study the thermal instanton system in the Polyakov-like gauge. At the high-temperature limit, the monopole trajectory becomes straight lines in the temporal direction. The topology of the QCD-monopole trajectory is drastically changed at a high temperature., Comment: 17 pages, Plain Tex, uses PHYZZX (12 figures - available on request from suganuma@miho.rcnp.osaka-u.ac.jp )

The correlation between instantons and QCD-monopoles is studied both in the lattice gauge theory and in the continuum theory. From a simple topological consideration, instantons are expected to live only around the QCD-monopole trajectory in the abelian gauge. First, the instanton solution is analytically studied in the Polyakov-like gauge, where $A_4(x)$ is diagonalized. The world line of the QCD-monopole is found to be penetrate the center of each instanton inevitably. For the single-instanton solution, the QCD-monopole trajectory becomes a simple straight line. On the other hand, in the multi-instanton system, the QCD-monopole trajectory often has complicated topology including a loop or a folded structure, and is unstable against a small fluctuation of the location and the size of instantons. We also study the thermal instanton system in the Polyakov-like gauge. At the high-temperature limit, the monopole trajectory becomes straight lines in the temporal direction. The topology of the QCD-monopole trajectory is drastically changed at a high temperature. Second, the correlation between instantons and QCD-monopoles is studied in the maximally abelian (MA) gauge and/or the Polyakov gauge using the SU(2) lattice with $16^4$. The abelian link variable $u_\mu (s)$ is decomposed into the singular (monopole-dominating) part $u_\mu ^{Ds}(s)$ and the regular (photon-dominating) part $u_\mu ^{Ph}(s)$. The instanton numbers, $Q({\rm Ds})$ and $Q({\rm Ph})$, are measured using the SU(2) variables, $U_\mu ^{Ds}(s)$ and $U_\mu ^{Ph}(s)$, which are reconstructed by multiplying the off-diagonal matter factor to $u_\mu ^{Ds}(s)$ and $u_\mu ^{Ph}(s)$, respectively. A strong correlation is found between $Q({\rm Ds})$ in the singular part and the ordinary topological charge $Q({\rm SU(2)})$ even after the Cabibbo-Marinari, Comment: Talk presented by H. Suganuma at International Workshop on ``Nonperturbative Approaches to QCD'', ECT, Trento, 10 - 29 July 1995, 18 pages, Plain Tex, uses PHYZZX( 17 figures - available on request from suganuma@miho.rcnp.osaka-u.ac.jp )