Long-Time Asymptotics of the Modified KdV Equation in Weighted Sobolev Spaces

Gong Chen  and  Jiaqi Liu Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada In memory of Walter Craig
January 10, 2022

The long time behavior of solutions to the defocussing modified Korteweg-de vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of Deift and Zhou and its reformulation by Dieng and McLaughlin through -derivatives. To extend the asymptotics to solutions with initial data in lower regularity spaces, we apply a global approximation via PDE techniques.

1. Introduction

In this paper we calculate the long-time asymptotics of solutions to the defocussing modifed KdV equation (MKdV):

There is a vast body of literature regarding the MKdV equation, in particular with the local and global well-posedness of the Cauchy problem. For a summary of known results we refer the reader to Linares-Ponce [27]. Without trying to be exhaustive, we mention the works on the local and global well-posedness by Kato [23], Kenig-Ponce-Vega [24], Colliander-Keel-Staffilani-Takaoka-Tao [6], Guo [17] and Kishimoto [26]. In particular, we know that the MKdV for both the focussing and defocussing cases on the line is locally well-posed, see Kenig-Ponce-Vega [24], and globally well-posed as well, see Colliander-Keel-Staffilani-Takaoka-Tao [6], Guo [17] and Kishimoto [26], in for . These results are complemented by several ill-posedness results; see for example Christ-Colliander-Tao [5] and references therein.

Besides well-posedness, another fundamental question for dispersive PDEs is the long-time asymptotics. Using the complete integrability of the MKdV equation, Deift and Zhou in their seminal work [9] developed the celebrated nonlinear steepest descent method for oscillatory Riemann-Hilbert problems. In the same paper, the authors give explicit asymptotic formulae and error terms for Schwartz class initial data. Since then, analysis of long-time behavior of integrable systems have been extensively treated by many authors. The nonlinear steepest descent method provides a systematic way to reduce the original RHP to a canonical model RHP whose solution is calculated in terms of special functions. This reduction is done through a sequence of transformations whose effects do not change the long–time behavior of the recovered solution at leading order. In this way, one obtains the asymptotic behavior of the solution in terms of the spectral data (thus in terms of the initial conditions).

A natural question to ask is whether it is possible to study the asymptotic behavior of the MKdV without relying on the completely integrable structure. A proof of global existence and a (partial) derivation of the asymptotic behavior for small localized solutions, without making use of complete integrability, was later given by Hayashi and Naumkin [19, 20] using the method of factorization of operators. Recently, Germain-Pusateri-Rousset [14] use the idea of the space-time resonance to study the long-time asymptotics of small data and soliton stability problem. Also a precise derivation of asymptotics and a proof of asymptotic completeness, was given by Harrop-Griffiths [18] using wave packets analysis. Overall, although PDE techniques do not rely on the complete integrability, to our best knowledge, certain smallness assumptions on the initial data are required.

In the present paper, we use the inverse scattering transform/nonlinear steepest descent to study the long-time asymptotics of solution to the MKdV equation without smallness assumption on the initial data. We give a full description of the long-time behavior of solutions in the weighted Sobolev space which is necessary to construct the Beals-Coifman solution and extend these results to other Sobolev spaces including and via a global approximation argument. Moreover we also point out that the error terms estimates are sharper than those given by [14] and [18].

In Deift-Zhou [9], a key step in the nonlinear steepest descent method consists of deforming the contour associated to the RHP in such a way that the phase function with oscillatory dependence on parameters become exponential decay. In general the entries of the jump matrix are not analytic, so direct analytic extension off the real axis is not possible. Instead they must be approximated by rational functions and this results in some error term in the recovered solution. Therefore, in the context of nonlinear steepest descent, most works are carried out under the assumptions that the initial data belong to the Schwartz space.

In [35], Xin Zhou developed a rigorous analysis of the direct and inverse scattering transform of the AKNS system for a class of initial conditions belonging to the space . Here, denotes the completion of in the norm

Recently, much effort has been devoted to relax the regularities of the initial data. In particular, among the most celebrated results concerning nonlinear Schrödinger equations, we point out the work of Deift-Zhou [11] where they provide the asymptotics for the NLS in the weighted space . This topology is more or less optimal from the views of PDE and inverse scattering transformations. The global existence of the cubic NLS can be carried out by the Strichartz estimate and the conservation of the norm. But in order to obtain the precise asymptotics, one needs to “pay the price of weights”, i.e. working with the weighted space .

Dieng and McLaughlin in [12] (see also an extended version [13]) developed a variant of Deift-Zhou method. In their approach rational approximation of the reflection coefficient is replaced by some non-analytic extension of the jump matrices off the real axis, which leads to a -problem to be solved in some regions of the complex plane. The new -problem can be reduced to an integral equation and is solvable through Neumann series. These ideas were originally implemented by Miller and McLaughlin [29] to the study the asymptotic stability of orthogonal polynomials. This method has shown its robustness in its application to other integrable models. Notably, for focussing NLS and derivative NLS, they were successfully applied to address the soliton resolution in [4] and [22] respectively. In this paper, we incorporate this approach into the framework of [9] to calculate the long time behavior of the defocussing MKdV equation in weighted Sobolev spaces. The soliton resolution of the focussing MKdV equation will be addressed in a forthcoming article [7].

1.1. The Riemann–Hilbert problem and inverse scattering

To describe our approach, we recall that (1.1) generates an isospectral flow for the problem

This is a standard AKNS system. If , equation (1.2) admits bounded solutions for . There exist unique solutions of (1.2) obeying the the following space asymptotic conditions

and there is a matrix , the transition matrix, with . The matrix takes the form

and the determinant relation gives

Combining this with the symmetry relations

we arrive at

and conclude that is zero-free.

By the standard inverse scattering theory, we formulate the reflection coefficient:

The functions is called the scattering data for the initial data .

We also have the following identity

In [35], it is shown that the maps is Lipschitz continuous from potentials into . Since we are dealing with the defocussing MKdV, only the reflection coefficient is needed for the reconstruction of the solution. The long-time behavior of the solution to the MKdV is obtained through a sequence of transformations of the following RHP:

Problem 1.1.

Given for , find a matrix-valued function on with the following properties:

  1. as ,

  2. is analytic for with continuous boundary values

  3. The jump relation holds, where

    and the real phase function is given by

    with stationary points

Note that the jump matrix admits the following factorization on :

We define

then it is well known that solvability of the RHP above is equivalent to the solvability of the following Beals-Coifman integral equation:

From the solution of Problem 1.1, we recover

where the limit is taken in along any direction not tangent to .

1.2. Main results

The central results of this paper are the following theorems that give the long-time behavior of the solutions of (1.1) in different regions respectively.

Figure 1.1. Four Regions
Theorem 1.2.

Given initial data . Let be the solution to the MKdV

given by the reconstruction formula (1.10) (the Beals-Coifman solution). Denote by the stationary phase point of the phase function (1.7). Let be any real number with , ,

Then we have the following asymptotics

  1. In Region I,

  2. In Region II,

  3. In Region III,

  4. In Region IV,

  5. In Region V,

In the above asymptotics for Regions II, III, IV, is a solution of the Painlevé II equation

determined by .

Note that in all asymptotics above, the implicit constants in the remainder terms depend only on .

Before procceding to sketch our main steps and discuss more details, we first give several remarks for the above statements.

Remark 1.3.

In this paper, to derive asymptotics, our main focus is to establish the uniform estimates for the error terms which only depend on . All leading order terms from the asymptotic formulae in all regions are obtained from special functions, namely parabolic cylinder functions and Painlevé II, independent of the regularity of initial data. For brevity, we do not repeat lengthy identical steps. We refer Deift-Zhou [9] for full details.

Remark 1.4.

In the above statment, as we claimed, is a solution of the Painlevé II equation

determined by . More precisely, the reduced factorization problem in Regions IV, is related to the Painlevé II equation by an isomonodromy problem associated to the linear problem

Here over six sections, one has the jump relations

where ’s are determined by three parameters satisfying

Then one can reconstruct from as the analysis of the parabolic cylinder. In our setting, the solution is constructed as above and determined by three parameters

Since this isomonodromy problem is standard, we refer Deift-Zhou [9, Sec.5] for full details.

Remark 1.5.

In [9] Region II, we take . The purpose of this is to ensure that the asymptotic formulae match up in the overlap of adjacent regions. In the overlap of Region I and II, we have in fact

which decays faster than both of the leading order terms in Region I and II. Also in Region the overlap of I and II, the difference of leading order terms from parabolic cylinders and the Painlevé equation becomes part of error terms due to the choice of . The rest part of the matching follows directly from Deift-Zhou [9, Sec. 6].

Remark 1.6.

In the asymptotics for Regions IV, in the error terms which can be arbitrarily large. This resemble estimates in Deift-Zhou [11, Lemma 4.8].

Remark 1.7.

From the view of the scattering theory, it is natural to ask if one can determine the initial data uniquely from the asymptotics of a solution. Here we point out that in our asymptotics formulae, the solution to the Painlevé II equation only depends on , the reflection coefficient evaluated at the origin. Therefore, if one only looks at the asymptotics in regions and V, these pieces of information are not sufficient to determine the initial data which produce this solution. To obtain the full information of the initial data, we have to go to Region I from which one can determine the phase and modulus of the reflection coefficient from the formulae given by the parabolic cylinder. For more details, see Deift-Zhou [9]. Hence in some sense, in this defocussing case, the region I is the most physically interesting. But in the focussing problem, breathers can appear in all regions. For more details, see our forthcoming article [7].

We give a general description of the derivation of the long-time asymptotics and performing nonlinear steepest descent. The major part of this paper is devoted to the study of the Region I whose leading behavior is given by parabolic cylinder functions.

The first step (Section 2), is to conjugate the matrix with a scalar function which solves the scalar model RHP Problem 2.1. This conjugation leads to a new RHP, Problem 2.3. The purpose of this is to prepare for the lower/upper factorization of the jump matrix on the part of the real axis between two stationary point. This is needed in the contour deformation described in Section 3.

The second step ( Section 3) is a deformation of contour from to a new contour (Figure 4.3). It is to guarantee that the phase factors in the jump matrix (2.4) have the desired exponential decay in time along the deformed contours. Inevitably this transformation will results in certain non-analyticity in the sectors , which leads to a mixed –RHP-problem, Problem 3.2.

The third step is a ‘factorization’ of in the form where is solution of a localized RHP, Problem 4.1, and a solution of problem, Problem 5.1. The term ”localized” means the reflection coefficient is fixed at along the deformed contours. We then solve this localized RHP whose solution is given by parabolic cylinder functions. Since we have to separate the contribution from two stationary points , some error terms appear alongside and their decay rate are estimated.

The fourth step (Section 5) is the solution of the -problem through solving an integral equation. The integral operator has small -norm at large allowing the use of Neumann series. The contribution of this -problem is another higher order error term.

The fifth step (Section 6) is to group together all the previous transformations to derive the long time asymptotics of the solution of the MKdV in Region I, using the large- behavior of the RHP solutions. These five steps above are more or less standard, during the proof of which we mainly follow the outline of [28].

The sixth step is the study of Region II-V. The leading order term in these region are given by a solution to the Painlevé II equation and error estimates are obtained from scaling.

The paper ends with a section to extend the asymptotics from Theorem 1.2 to rougher solutions. First of all, we notice that one can trace all the details in our implementing of the nonlinear steepest descent and notice that actually it suffices to require the weights in to be with . Then we apply approximation arguments to study solutions in and with some weights. Using the lowest regularity for the local well-posedness in with obtained by Kenig-Ponce-Vega see [24], the growth estimates for the norm due to Colliander-Keel-Staffilani-Takaoka-Tao [6], Guo [17] and Kishimoto [26], we can use a global approximation argument to extend our long-time asymptotics to with and . Then we can extend the results in the previous theorem and obtain the following:

Theorem 1.8.

For any initial data with and , the solution to the MKdV

given by the integral representation

has the same asymptotics as in our main Theorem 1.2.

Hereinafter, for the sake of simplicity, we just focus on the case where . For the general case, we just notice that is sufficient for us to apply the Sobolev embedding and the estimate of modulus of continuity of the reflection coefficients in the Riemann-Hilbert problem.

1.3. Notations

Let be the third Pauli matrix:

and define the matrix operation

We define Fourier transforms as

Using the Fourier transform, one can define the fractional weighted Sobolev spaces:

As usual, or is the definition of by means of the expression . We use the notation . For positive quantities and , we write for where is some prescribed constant. Also for and . Throughout, we use , .

1.4. Some discussion

To finish the introduction, we highlight certain features of this paper.

Firstly, compared with the analysis of the nonlinear Schödinger equation in weighted Sobolev spaces [11], the defocussing MKdV exhibits more complicated behavior in terms of long-time asymptotics. This follows from the fact that phase function for the nonlinear Schödinger equation has one single stationary point while the phase function for the MKdV equation has two stationary points. The MKdV equation has the oscillatory region (Region I), the self-similar region (Region II-IV) and the soliton region (Region V), each of which has different leading order terms and error terms. The interaction of these two stationary points is not negligible. Taking the oscillatory region for example, we could not directly carry out steepest descent for each stationary point separately and treat the asymptotic formula as the sum of two separate parabolic cylinder functions. We instead proceed as in Deift-Zhou [9], using their ”restriction of resolvent operators” lemma [9, Lemma 2.56] under much weaker regularity assumptions. This will lead to an extra error term in the asymptotic formula. These two stationary points, due to symmetry, will lead to a real-valued solution to the equation. Meanwhile in certain self similar regions, the two stationary points will approach each other as . In this case, the decay in time results from a scaling factor instead of oscillation. As , these two stationary points can also appear in pairs along the imaginary axis. Thus, due to the complicated structure of the MKdV, we will explore some new applications of the -steepest descent method, namely breaking down the interaction between two stationary points as and obtaining the asymptotics in the self-similar region through scaling. Successfully obtaining the asymptotics in the Painlevé regions seems to be a new application of the -steepest descent method which might be useful in other models involving Painleve regions. We also point out that in [30] while studying the long time asymptotics of the massive Thirring model, the author also treated the interaction between two stationary points from a different perspective.

Secondly, we extend the asymptotics of the MKdV to solutions with initial data in lower regularity spaces using a global approximation via PDE techniques. In Deift-Zhou [11], due to the Strichartz estimates for the linear Schödinger equation and the conservation of the norm, the authors can globally approximate the solution to the nonlinear Schödinger equation with data in using the Beals-Coifman representation of solutions directly. Unlike the Schödinger equation, the smoothing estimates and Strichartz estimates for the Airy equation and the MKdV are much more involved. For example, one needs which behaves like a maximal operator. To directly work on the Beals-Coifman solution to the MKdV to establish the smoothing estimates and Strichartz estimates, one needs estimates for pesudo-differential operators with very rough symbols. To avoid these technicalities, we first identify the Beals-Coifman solution with the solution given by the Duhamel formula, which we call a strong solution. The equivalence of these two notations of solutions in is not transparent since they do not have enough smoothness to differentiate. Relying on smoothing estimates and the bijectivity of the scattering and inverse scattering transforms by Zhou [35] which plays the role of Plancherel theorem in Fourier analysis, we show these two notations are the same at the level of which is necessary to construct the Beals-Coifman solutions. Since the strong solutions by construction enjoy Strichartz estimates and smoothing estimates, by our identification, the Beals-Coifman solutions also satisfy these estimates. Then we can use Strichartz estimates and smoothing estimates to pass limits of Beals-Coifman solutions to obtain the asymptotics for rougher initial data in and . To illustrate the importance of and , we note that in , the MKdV has the energy conservation. On the other hand, is the optimal space to use iterations to construct the solution to the MKdV, for details, see later sections.

1.5. Acknowledgement

We would like to thank Professor Jean-Claude Saut for pointing out the references [31], [32] and [33].

2. Conjugation

We introduce a new matrix-valued function

where solves the scalar RHP Problem 2.1 below:

Problem 2.1.

Given and , find a scalar function , analytic for with the following properties:

  1. as ,

  2. has continuous boundary values for ,

  3. obey the jump relation

Lemma 2.2.

Suppose and that is given by (1.13). Then

  • Problem 2.1 has the unique solution

    where is given by equation (1.13) and

    here we choose the branch of the logarithm with .

  • For , ; for ,

  • Along any ray of the form with or ,

    The implied constant depends on through its -norm and is independent of .

The proofs of (i)-(iii) can be found in [9]. To establish (iv), we first note that

To bound the difference , notice that

where the third inequality follows from [3, Lemma 23]. ∎

It is straightforward to check that if solves Problem 1.1, then the new matrix-valued function is the solution to the following RHP.

Problem 2.3.

Given , find a matrix-valued function on with the following properties:

  1. as ,

  2. is analytic for with continuous boundary values

  3. The jump relation

    holds, where

    The jump matrix is factorized as

3. Contour deformation

We now perform contour deformation on Problem 2.3, following the standard procedure outlined in [28, Section 4]. Since the phase function (1.7) has two critical points at , our new contour is chosen to be

shown in Figure 3.1 and consists of rays of the form where .

Figure 3.1. Deformation from to

We now introduce another matrix-valued function :

Here is chosen to remove the jump on the real axis and brings about new analytic jump matrices with the desired exponential decay along the contour . Straight forward computation gives

We want to make sure that the following condition is satisfied

where are the boundary values of as . In this case the jump matrix associated to will be the identity matrix on .

From the signature table [9, Figure 0.1] we find that the function is exponentially decreasing on , , and increasing on , , , while the reverse is true of . Letting

we define as follows (Figure 3.2-3.3): the functions , , , , , , , satisfy

Figure 3.2. The Matrix for Region I, near


Correlation effects in electronic structure of PuCoGa

L. V. Pourovskii, M. I. Katsnelson, and A. I. Lichtenstein Institute for Molecules and Materials, Radboud University of Nijmegen, NL-6525 ED Nijmegen, The Netherlands
Institut für Theoretische Physik, Universität Hamburg, 20355 Hamburg, Germany
June 2, 2022

We report on results of the first realistic electronic structure calculations of the Pu-based PuCoGa superconductor based on the dynamical mean field theory. We find that dynamical correlations due to the local Coulomb interaction between Pu -electrons lead to substantial modification of the electronic structure with a narrow peak being formed in vicinity of , in agreement with the experimental photoemission spectra, and in contrast with the recent calculations within the LDA+U method Shick2005 , where only static electronic correlations have been included. Both Pu and Co contribute in equal footing to the narrow peak on the density of states at the Fermi level, the Co partial density of states being prominently affected by electronic correlations on the Pu sites. The -resolved spectral density is calculated and the theoretical spectral function resolved extended Van Hove singularity near . This singularity may lead to enchancement of the magnetic susceptebility and favour wave superconductivity.

71.27.+a, 71.15.Rf, 74.25.Jb, 74.70.Tx

The recent discovery of superconductivity in PuCoGa Sarr2002 and PuRhGa Wast2003 has attracted a lot of attention to electronic structure of these Pu-based compounds. They have the same HoCoGa-type tetragonal structure as the Ce-based superconductors CeGa ( is transition metal) Mathur98 , however PuCoGa has an order of magnitude higher transition temperature 18.5 K, the highest among known -electron superconducting materials. Both CeGa and PuGa are unconventional superconductors Curro2005 with the nodal wave symmetry of superconducting gap and antiferromagnetic fluctuations being a likely mechanism for their unconventional superconductivity. It has been suggested Bauer2004 that the increase of in the PuGa compounds in comparison with CeGa might be due to more itinerant character of the 5 band in Pu.

The experimental data Sarr2002 show no evidence of a long-range magnetic structure down to temperature of 1 K, whereas the magnetic susceptibility obeys the modified Curie-Weiss law with an effective local moment 0.68 . The specific heat coefficient =77 mJ molK Sarr2002 is slightly larger than the value observed in -Pu (50 mJ molK Wick80 ) indicating possible enhancement by strong dynamical spin fluctuations. The experimental photoemission spectra (PES) of PuCoGa Joyce2003 exhibits a broad feature centered 1.2 eV below the Fermi level, which is made up of Co - and Pu -bands, as well as a narrow peak at the Fermi level. The latter feature is similar to narrow resonances observed in the PES of -Pu Havel2003 as well as in the spectra of some Pu-based compounds Goud2000 ; Dur2004 .

The electronic structure of PuCoGa has been investigated in several papers on the basis of the local spin density approximation (LSDA) Opahle2003 ; Maeh2003 ; Soder2004 ; Opahle2004 . In the non-spin-polarized band structure calculations of Refs. Opahle2003, ; Maeh2003, for nonmagnetic state the main -state spectral weight is located in the vicinity of the Fermi level in contradiction with experiment. The spin polarization pushes occupied part of the -band below the Fermi level Opahle2003 ; Soder2004 , with the density of states (DOS) for the antiferromagnetic phase Soder2004 being in good agreement with the experimental PES Joyce2003 . However, the Pu total magnetic moment obtained within the LSDA is too large (around 2 ) in comparison with experimentally measured as well as the ordered magnetic structure in PuCoGa predicted by the LSDA calculations has not been observed in experiment. Joyce et al. Joyce2003 obtained theoretical PES in a very good agreement with the experimental one using the mixed level model (MLM). In the MLM 5 electrons of Pu is divided in four localized and one itinerant, however the energy position of the localized states is not calculated ab initio but rather treated as a parameter which value is chosen to match experiment.

The strong enhancement of the specific heat coefficient , as well as the characteristic narrow resonance at the Fermi level, hint on possible importance of dynamical electronic correlations in PuCoGa. In -Pu kotliarNature and Pu monochalcogenides Pour2005 a good agreement between the theoretical and measured PES for non-magnetic phase (which is the ground state for these materials in accordance with the experiment) has been obtained only when dynamical correlations between -electrons were taken into account by means of the dynamical mean-field theory (DMFT) Geor96 in a framework of the LDA+DMFT AnisDMFT ; lda++ method. For PuCoGa Shick et al. Shick2005 have taken into account only static correlations by means of the LDA+U method. They obtained the antiferromagnetic ground state with the reduced (in comparison with the LSDA) Pu effective magnetic moment around 1 . At the same time, the density of states reported in Ref. Shick2005, demonstrates some discrepancy with the experimental PES. While the position of the broad manifold reproduced quite well, the narrow feature at is absent. It seems reasonable to suggest that by analogy with -Pu and other Pu-based compounds an adequate description of the electronic structure of PuCoGa requires dynamical correlations to be taken into account. It is important to notice that in the Pu-based compounds the dynamical correlations affect most profoundly the states in vicinity of the Fermi level Pour2005 , which are involved in superconductivity. It is the purpose of this Letter to present electronic structure of PuCoGa with the dynamical electronic correlations explicitly included within the LDA+DMFT approach.

We start with the LSDA+U calculations of PuCoGa. Following Ref. Shick2005, to allow for the antiferromagnetic ground state we double the unit cell along the and crystallographic axes and point magnetic moments of the nearest-neighbor Pu atoms in the Pu planes in opposite directions, the magnetic moments of Pu atoms being aligned along the crystallographic axis. The calculations have been carried out by the full-potential linear MT-orbitals method (FPLMTO) Savr96 at experimental lattice parameter of PuCoGa (=7.845 Å, =1.603 for the unit cell) with the spin-orbit coupling included in a self-consistent second-order variational procedure. We include the full matrix of on-site Coulomb interaction between Pu 5-electrons parametrized by the average Coulomb interactions 3 eV and the Hunds-rule exchange =0.55 eV. These values of and are in the range of commonly accepted values for Pu. For the double counting (DC) term we have employed so-called “around mean-field” formulation of the LDA+U method Anis91 ; Czyz94 . This version of the LDA+U method was recently successfully used to explain non-magnetic ground state of the Pu -phase ShickEL . We used previously the same values of and as well as the same choice of the DC in the DMFT simulations of Pu monochalcogenides Pour2005 .

The total DOS (solid line) as well as the partial DOS for
Pu (dashed line) and Co (dot-dashed line) atoms obtained within
the DMFT. The thick solid and dot-dashed lines are the
Figure 1: The total DOS (solid line) as well as the partial DOS for Pu (dashed line) and Co (dot-dashed line) atoms obtained within the DMFT. The thick solid and dot-dashed lines are the experimental Joyce2003 and theoretical PES in arbitrary units, respectively.

The one-particle Hamiltonian obtained by the converged LDA+U calculations was then orthogonalized by the Löwdin transformation Lowd50 and used to compute the local Green’s function by means of the Brillouin zone (BZ) integration

where are the fermionic Matsubara frequencies for a given temperature , is the chemical potential and is the local self-energy with a “double counting” term subtracted. Within the DMFT scheme the local self-energy has been obtained by the solution of the many-body problem for a single quantum impurity coupled to an effective electronic “bath” through the Weiss field function Geor96

The spectral function of PuCoGa
Figure 2: The spectral function of PuCoGa. The sequence of colors in order of increasing spectral density is blue, green, yellow, and red. In the inset the BZ of PuCoGa is shown together with our designations for the high-symmetry points.

As a quantum impurity solver we have employed the recently developed spin-orbit version Pour2005 of the -matrix and fluctuating exchange technique (SPTF) Kats99 ; Kats2002 . In the spin-orbit SPTF the local self-energy is written as a sum of three contributions:

where and are the -matrix “Hartree” and “Fock” contributions, respectively, is the particle-hole contribution. and are obtained by substitution in the corresponding Hartree and Fock diagrams the bare Coulomb interaction with the frequency dependent -matrix, the latter is given by summation of the ladder diagrams in the particle-particle channel. is obtained by the RPA-type summation in the particle-hole channel with the bare vertex being substituted by static limit of the -matrix Kats2002 ; Pour2005 . While being of perturbative type the spin-orbit SPTF solver has been shown to provide an adequate description of correlations in 3 metals Kats99 ; Kats2002 , half-metallic ferromagets lulu , and some actinide systems Pour2005 .

For the double counting term in DMFT simulations we have used the static limit of the self energy Kats2002 . For the spin-orbit SPTF quantum impurity solver we employed 10 Matsubara frequencies and temperature 500 K. It is important to notice that with our choice of the double counting term the Hartree-Fock contribution is already included into the LDA+U one-particle Hamiltonian, therefore making magnetization almost temperature independent. We have carried out DMTF iterations until convergence in both the chemical potential and the local self-energy was achieved. Then the Pade approximant is used Vidb77 for analytical continuation of the local self-energy to the real axis in order to obtain the DOS and spectral function .

We started from the antiferromagnetic state of PuCoGa but the self-consistent calculations converged to practically zero magnetic moments both on the Pu and Co sites. The long-range magnetic order is suppressed due to combined influence of a strong relativistic spin-orbit coupling and the local Coulomb interaction effects. The states of Pu in PuCoGa are almost completely filled (the occupation number for -band =5.5) whereas the empty states are located around 3 eV above the Fermi level. It is this splitting that suppress the magnetism of Pu in comparison with the LSDA electronic structure. The Co 3 band is located between -2 eV and the Fermi energy.

The DMFT results for PuCoGa DOS is shown on the Fig. 1 together with the experimental PES spectrum Joyce2003 and the theoretical PES obtained from the DMFT DOS. In order to compute the theoretical PES spectrum the calculated DOS data have been multiplied by the Fermi function corresponding to the experimental temperature of 77 K Joyce2003 and then convoluted with a Gaussian to simulate for the instrumental resolution and lifetime broadening of the core hole Joyce2003 ; Arko2000 . One may see a pronounced quasiparticle peak at the Fermi level in the DMFT DOS. The theoretical PES agrees quite well with the experimental one, the shape of the feature at the Fermi level as well as relative spectral weight of the broad manifold and the resonance at being reproduced very accurately. However, the manifold is somewhat wider in the theoretical PES compare with the experiment, with a peak at -0.6 eV and a shoulder at -1.8 eV.

Interestingly, while in our DMFT calculations the correlation effects have been taken into account only on the Pu sites, both the Pu - and Co -bands are substantially modified in comparison with the LDA+U result and they both contribute in equal footing to the narrow peak at . The LSDA Opahle2004 as well as LSDA+U Shick2005 calculations for the antiferromagnetic phase predict that the Pu -band gives by far the dominant contribution to DOS at . We have performed also the LDA+DMFT calculations with the local Coulomb interaction between Co -electrons being included as well (with =2 eV and =0.9 eV), however that leads to rather minor changes in the electronic structure.

The spectral function of PuCoGa calculated within the DMFT is displayed in Fig. 2. The maximums in spectral function around , and points correspond to both the Pu -band and Co -band, contributing mainly to the manifold at energies between -2 and -1 eV. The increase of the spectral weight near the the Fermi level due to van Hove singularities (saddle points) close to and points is mainly due to the Co -band, and in between and points is due to contributions from both Co and Pu. The flat Co band located exactly at the Fermi level is clearly seen in the spectral function along the direction. The extended van Hove singularity near point together with the van Hove singularity at point result in strong -dependence of magnetic susceptibility (cf., e.g., the “parquet” results for the two-dimensional Hubbard model IKK ). This enhanced susceptibility can lead to wave superconductivity BSW89 . Relevance of the van Hove singularities for the superconductivity has been discussed for high-temperature cuprates IKK ; mark , as well as for SrRuO ruth . However, in comparison with these cases, the electron spectrum of PuCoGa is a bit less anisotropic and probably can not be considered as a purely two-dimensional one even in a zeroth-order approximation (see the results for direction in Fig. 2).

In conclusion, we have shown that the dynamical correlation effects modify electronic structure of the PuCoGa superconductor in essential way. We have taken the dynamical correlations into account by means of the dynamical mean-field theory using the spin-orbit -matrix and fluctuating exchange (SPTF) approximation for the quantum impurity solver. in agreement with the experimental data, the self-consistent solution turned out to be non-magnetic. The DOS obtained by the DMFT simulations is in a good agreement with the experimental PES, with the narrow peak at the Fermi level being very well reproduced. While only Pu sites have been treated as correlated, the dynamical correlations modify the Co bands as well. On the other hand, taking into account local correlations on Co sites does not effect noticeably on the electronic structure. Both the Co and Pu bands contribute almost on equal footing to the quasiparticle resonance at , in contrast with the LSDA picture, where only Pu bands contribute at DOS on the Fermi level. Theoretical spectral function demonstrates the extended Van Hove singularity near point which is due to the Co band. New ARPES experiments are highly desirable to investigate the energy spectrum of PuCoGa in the vicinity of the Fermi level and check the theoretical predictions.

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Flavor and CP violating exchange and the rate asymmetry in

[10mm] N.G. Deshpande111[email protected] and Dilip Kumar Ghosh222

[4mm] Institute of Theoretical Science

University of Oregon, Eugene, OR 97403

Recent measurements of time dependent CP asymmetry in , if confirmed, would indicate a new source of CP violation. We examine flavor violating tree-level currents in models with extra down-type quark singlets that arise naturally in string compactified gauge groups like . We evaluate the new operators at the scale in NLO, and using QCD improved factorization to describe , find the allowed range of parameters for and , the magnitude and phase of the flavor violating parameter . This allowed range does satisfy the constraint from flavor changing process . However, further improvement in measurement of these rates could severely constrain the model.

1 Introduction

The ongoing physics experiments by BaBar and Belle collaborations  [1, 2] provide a unique opportunity to study the flavor structure of the standard model quark sector and also the origin of CP violation. In addition to this, any new physics effects in physics can also be tested in these experiments. Recent time dependent asymmetries measured in the decay both by BaBar and Belle collaborations [References-References] show significant deviation from the standard model and this has generated much theoretical speculation regarding physics beyond the standard model [5]. In the standard model, the process is purely penguin dominated and the leading contribution has no weak phase. The coefficient of in the asymmetry therefore should measure , the same quantity that is involved in in the standard model. The most recent measured average values of asymmetries are  [4, 6] and . The value for agrees with the standard model expectation. The deviation in the is intriguing because a penguin process being a loop induced process is particularly sensitive to new physics which can manifest itself through exchange of heavy particles. In this article we will consider an extension of the standard model, with extra down type singlet quarks. These extra down type singlet quarks appear naturally in each 27-plet fermion generation of Grand Unification Theories (GUTs) [References-References]. The mixing of these singlet quarks with with the three SM down type quarks, provides a framework to study the deviations from the unitarity constraints of CKM matrix. This model has been previously studied in connection with and F-B asymmetry at the pole as it provides a framework for violation of the unitarity of the CKM matrix [References-References]. This mixing also induces tree-level flavor changing neutral currents (FCNC). These tree-level FCNC couplings can have a significant effects on different conserving as well as CP violating B processes [References, References - References].

In this article we study the FCNC effect arising from the coupling to the process. This new FCNC coupling can have a phase, which can generate the additional source of CP violation in the process, and thus affect measured values of and . We parameterize this coupling by . We then study taking into account the new interactions in the QCD improved factorization scheme (BBNS approach ) [25]. This method incorporates elements of naive factorization approach (as its leading term ) and perturbative QCD corrections (as sub-leading contributions) and allows one to compute systematic radiative corrections to the naive factorization for the hadronic decays. Recently, several studies of , and specifically have been performed within the frame work of QCD improved factorization scheme [References-References]. In our analysis of , we follow [30] which is based on the original paper [25]. In our analysis, we only consider the contribution of the leading twist meson wave functions, and also neglect the weak annihilation contribution which is expected to be small. Inclusion of these would introduce more model dependence in the calculation through the parameterization of an integral, which is otherwise infrared divergent.

The time dependent CP asymmetry of is described by :

where and are given by

and can be expressed in terms of decay amplitudes:

The branching ratio and the direct CP asymmetries of both the charged and neutral modes of have been measured [1, 2, 3, 4, 6, 31] 333Latest results were reported at XXXIX Rencontres de Moriond, Electroweak Interactions and Unified Theories, Italy, March 2004. See talks [32, 33, 34]. The Belle result on is unchanged [32] while BaBar finds [33] which is very close to the result in Eq.(7). Hence, our observations remain unchanged.

1.137 -0.295 0.021 -0.051 0.010 -0.065
1.081 -0.190 0.014 -0.036 0.009 -0.042
-0.024 0.096 -1.325 0.331 -0.364 -0.169
-0.011 0.060 -1.254 0.223 -0.318 -0.151
Table 1: Standard model Wilson coefficients in NDR scheme.

2 in the QCDF Approach

In the standard model, the effective Hamiltonian for charmless decay is given by [25]

where the Wilson coefficients are obtained from the weak scale down to scale by running the renormalization group equations. The definitions of the operators can be found in Ref.[25]. The Wilson coefficients can be computed using different schemes [35]. In this paper we will use the NDR scheme. The NLO values of and LO values of respectively at and used by us based on Ref.[25] are shown in Table 1.

In the QCD improved factorization scheme, the decay amplitude due to a particular operator can be represented in following form :

where denotes the naive factorization result.The second and third term in the bracket represent higher order and correction to the hadronic transition amplitude. Following the scheme and notations presented in Ref.[30], we write down the total amplitude, which is the sum of the standard model as well as exchange tree-level contribution from extra down-type quark singlets (EDQS) model in the heavy quark limit

where is summed over and . The coefficients are given by

with and . The effective Wilson coefficients is the sum of the standard model and the EDQS model Wilson coefficients. The quantities and are hadronic parameters that contain all nonperturbative dynamics, are given in Ref. [25, 36].

For the sake of completeness, we give the branching ratio for decay channel in the rest frame of the meson.

where, represents the meson lifetime and the kinematical factor is written as

3 Fcnc couplings in EDQS model

Models with extra down-type quarks (EDQS) have a long history. The earliest consideration of such models was in the context of the grand unification group which arises from string compactification. The quarks and leptons of each generation belong to the representation [References-References]. Each generation has one extra quark singlet of the down type, and also one extra lepton of the electron type. The group also has extra bosons, which we will assume to be too heavy to have any effect on process. The down type mass matrix is then a by-unitary matrix, and in general when we rotate the quarks to their mass basis, off-diagonal couplings arise. In EDQS model, the mediated FCNC interactions are given by

In general for copies of extra down-type quark singlet model, is :

where, represents the number of down type quark states, and is the neutral current mixing matrix for the down quark sector. The non vanishing components of will lead to FCNC process at tree level, generating new physics contribution to the measured CP asymmetries. The new tree level FCNC mediated contribution to the process is shown in Figure 1. The new operators arising from this tree-level FCNC process have been shown to lead to the following effective Hamiltonian for process in this model  [22]:

where, the new Wilson coefficients and at the scale are given by:

where, , and operators in Eq. (20) are given in Ref.[25]. We now evolve these new Wilson coefficients from the scale to the scale using the renormalization group equation. While doing this we have considered NLO QCD correction [37], neglecting the order electroweak contributions to the RG evolution equation which are tiny. At the low energy, after the RG evolution the above three Wilson coefficients generate new set of Wilson coefficients in this model. The values of Wilson coefficients ( without taking the overall factor ) at scales are shown in Table 2.

0.195 -0.088 0.0180 -0.053 0.133 0.108 -0.604 0.174
0.182 -0.0629 0.0157 -0.0370 0.136 0.0732 -0.574 0.122
Table 2: Wilson coefficients of EDQS model in NDR scheme, without the overall multiplicative factor .

4 physics constraints on

In this section, we review the constraints on the flavor violating parameter from different flavor changing processes. These processes can be classified into two classes, CP conserving and CP violating. Among the CP conserving processes, , and can put constraints on [13, 15, 16]. Using recent Belle [38] measurement of the authors in Ref.[21] had shown that . However, this bound has recently been updated in Ref.[22] to

which also updates their previous bounds in Ref.[17, 39] The bound in Eq. (24) is based on inclusive decays at NNLO [40]. We shall adopt this bound in our analysis. This bound is valid in both general extra down-type singlet quark model and in the model with a single extra down-type quark singlet [16]. Similarly, the branching ratio provides comparable limits [14, 16].

It has been shown in Ref. [20, 41] that in the presence of tree-level FCNC coupling and /or , the standard box diagram for is not gauge invariant by itself, but requires exchange penguin diagram as well as tree level FCNC diagram. The additional Feynman diagrams are given in Ref. [41]. Following the paper [20], with slight change in the notations, we write down the expression for the mixing:

where, and

where, the definitions of different parameters used above can be found in Ref.[20].

We have found that to satisfy the measured 444The experimental value for [20]. within one sigma, where we consider both the theory error of arising from the value of and the experimental error taken in quadrature, the FCNC coupling should be less than . This is a very stringent limit.

The has not been measured yet, and so only lower limit on the mass difference is available. We have found that the new contribution to from EDQS model is less than when compared to the standard model. It can be shown that for similar values of and , the FCNC effects on will be suppressed by a factor when compared with the effects on . This implies that in EDQS model, FCNC effects are hard to detect in mixing [41].

5 analysis

In the last section we have discussed the allowed range of the FCNC parameter from different processes. In this section we will study the effect of this FCNC parameter in the process. For this we express in the following form: . We will then vary and  555 in units of radian in range such that Eq. (24) is satisfied. We then study the allowed region of parameters in the plane from the three measured quantities , and . To get the allowed parameter space, from the branching ratio, we allow it to vary by respectively from its central value. This band contains both experimental and theoretical errors. The main source of theoretical error is the form factor . In our analysis we have considered error on this parameter. Similarly, we vary and by and from their central value to get the allowed region in the plane.

In Figure we show such allowed region in plane for the scale . The whole area left of the dotted contour is allowed by saturating Eq. (24). The area outside the thick contour labeled by BR is allowed region from the branching ratio measurement. The parameter space enclosed by the thin contour marked by is allowed by from data on the . This whole parameter space is allowed by from measurement, The regions (marked by ) is the only allowed parameter space in plane with and which satisfy the experimentally measured , and within the errors described above. We note that only negative values of give acceptable range of . The Figure correspond to the scale . In this case though we have larger allowed area from the measurement, but the branching ratio contour pushes the allowed range towards higher values of and somewhat lower range of the phase . This particular behavior of the branching ratio contour can be understood from that fact that for , the SM branching ratio is , which is much smaller than the lower end of the band of the experimental number. Hence, one needs larger values of to push the total branching ratio within the limit. For this reason the allowed region shrinks to a point in this case.

6 Conclusions

In this paper, we have studied the tree-level flavor violating contribution to process in models with extra down-type quark singlets which arise naturally in the context of the Grand Unification group . In the presence of such flavor violating interactions, process receives additional contributions, governed by a set of new operators which can be expressed in terms of the standard operators . We then evolved these new Wilson coefficients from the scale to the scale relevant for our process using the renormalization group equation. We have found that, at the lower scale these Wilson coefficients significantly modified from their initial values at the scale . We have found that this new flavor violating interaction can modify the standard model Wilson coefficients significantly. We have used following experimentally measured quantities: and to constrain the flavor violating parameter . We have shown that in the model with an arbitrary number of down type singlet quarks, the value of and can be well explained by the values of and in the region marked by in Figure . Improvements in measurements of can tighten the constraints in Eq. (24) and either rule in or rule out this model.

This work was supported in part by US DOE contract numbers DE-FG03-96ER40969. We would like to thank A. K. Giri for helpful correspondence. We would particularly like to thank G. Hiller for clarifying the operator structure of the mediated FCNC interactions.

7 APPENDIX : Input parameters and different form factors

In this Appendix we list all the input parameters, decay constants and form factors used for the calculation of .

  1. Coupling constants and masses ( in units of GeV ):

  2. Wolfenstein parameters :

  3. Constituent quark masses ( in units of GeV):

  4. The decay constants (in units of GeV):

  5. The form factors at zero momentum transfer :

Feynman diagram for

Figure 1: Feynman diagram for exchange tree-level contribution to process.

Contour plots of

Figure 2: Contour plots of , and in plane for two values of and respectively. The area leftside of the dotted contour is the allowed region of from the inclusive process. In figure , the area marked by is the allowed regions from the measurement of and . In figure , such allowed region is a point where the three curves intersect.

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The Northwest Territories

Flag of the Northwest Territories
Flag of the Northwest Territories. The territorial shield of arms is centered on a white field, with two vertical blue panels on either side. The white symbolizes the snow and ice of the winter, while the blue represents the territory’s lakes and waters.

Formation: 1870. The origin of the name: The name “the North-Western Territories,” initially assigned by the British government, once referred to all the lands held by the Hudson’s Bay Company.

TIME: 5 AM MST = noon GMT.

A vast area of forests and tundra, Northwest Territories is one-third of Canada’s total landmass, stretching from the eastern to western extremities of the country. It has a rich natural environment but is home to more than 60% of Canada’s biodiversity due to its largely uninhabited regions. The creation in 1999 of Nunavut, out of the eastern portion of the Northwest Territories, reduced the latter’s size by more than half.

The Northwest Territories are the northern expansions of Canada, reaching beyond the Arctic Circle and are bordered by Nunavut in the East, British Columbia, Alberta, and Saskatchewan in the south, and Yukon to the west. Yellowknife is the capital city of this territory. The Northwest Territories region is 1,346,106 square km / 519,735 square miles. The population is 41,070.

Two main types of logging are seen on the timberline. One type is the taiga or boreal forest, which spans from near the Mackenzie River delta in Canada to Manitoba and Yukon. It’s roughly parallel to the border with Nunavut. The barren grounds north and East of the timberline are flat, poorly drained lowlands that are more than a billion years old. The conditions support surface vegetation depending on the soil and climate. The Mackenzie Mountains in the west and southwest contain the highest and most-rugged relief in the territories; elevations reach 2,730 meters / 9,062 feet near Mount Sir James McBrien.

Map of the Northwest Territories

Map of the Northwest Territories
Map of the Northwest Territories

In the west-central portion of provinces and territories, the environment in the Mackenzie Lowlands is favorable for herbaceous plants when there’s no frost. The environment consists of black and white spruce mixed with deciduous species. Approximately 70 days are frost-free, but that doesn’t stop wildflowers from flourishing. The region sustains many valuable fur-bearing animals, such as muskrat and beaver. Moose, wolves, black bears, and grizzly bears are commonly found. Mountain sheep and mountain goats also live there.

The climate of the Mackenzie Lowlands is still cold, but not as cold as in other places. Due to the climate, there’s only a limited amount of time that you could use to navigate the Mackenzie River, and there’s also permafrost which causes construction problems.

North and East of the Mackenzie Lowlands and the tree line, the terrain changes to that of the ancient and rocky Precambrian mass known as the Canadian Shield, the western edge of which is straddled by the two largest lakes in the territories – Great Bear Lake (12,096 square miles [31,328 square km]) and Great Slave Lake (11,030 square miles [28,568 square km]). The Arctic Islands are remnant mountains formed over 300 to 400 million years ago. The growth of trees is stunted and eventually disappears, so a plant-less wilderness called the tundra takes its place. The soil in these barren lands tends to be sand or ice, with few plants but providing many types of animal life. The musk ox was at risk of extinction until the Canadian government placed it under protection in the early 20th century. There are now caribou subspecies that are also at risk. Seal, walrus, and polar bears live along coasts, and birdlife is plentiful during summer. Mosquitoes and other insects are prevalent during summer.

The climate in the Mackenzie Lowlands is relatively mild because of its warm and dry summers. The winters are long and cold, with an average temperature in January of −16 °F (-27°C) at Yellowknife, on the northern shore of Great Slave Lake. The north and East of the Arctic region have an Arctic climate. Here, annual precipitation is in light amounts, and winter temperatures are similar to those in the subarctic region to the west. However, summer temperatures do not rise above an average of 50 °F (10 °C) even in July.

Google maps of the Northwest Territories

American Indians comprise over one-third of the Canadian population, including the Dene and the Métis. The Dene are concentrated in the Mackenzie Valley area, and their languages come from the Athabaskan language family. The tribal organization was never strong among them, so small bands led by individuals chosen for their skill in hunting were the effective social unit. This arrangement was easily molded to the needs of a fur trade when it reached the Mackenzie Area in the 18th century. The Dene economy was based on the exchange of furs for imported goods. In 1899, the government established treaties with groups living south of Great Slave Lake. The Canadian government only made treaties with groups living north of Great Slave Lake in 1921. There is no reservation, but many small indigenous settlements have the same status as reservations elsewhere. With the decline of the fur trade in the 20th century, many Dene were unemployed. In 2003, the Métis (people of mixed Indian and European ancestry) were granted legal recognition by the Canadian government as a native group. The Inuit are a group that constitutes just 1/10th of the population. They form an isolated ethnic group who speak their language and tend not to mix with other groups, especially the Dene. Most of Appalachia’s population is of European descent, with most living in the more economically advanced Fort Smith region. They are employed in mining, transportation, and public service.

Previously, the aboriginals of the territory lived nomadic lives. They lived off of what they found in the land and learned to adapt to it but were forced to adopt new ways when Europeans came with guns. The introduction of guns was toxic for this community because many relied on caribou as a food source. They also tried to introduce domesticated animals, but there hasn’t been much success. Most people live in towns now and struggle with imported goods and goods that are no longer available locally. Nearly all the population lives in small settlements along the Mackenzie River, with smaller numbers along the Arctic coastlines of the mainland and northern islands. The main towns are Hay River, Fort Smith, and Inuvik; all are in the Mackenzie area. In contrast to provinces, territories tend to lose more residents than they gain due to migration.

The territories depend primarily on natural resources for income. These resources often cannot be fully exploited due to the high costs and transportation issues. Services have a significant impact on the economy, but manufacturing does not. The federal government collects money from natural resource royalties and transfers funds to the territorial governments, dependent on this income. Many resources in the territories have been developed with government help. Government agencies provide certain transportation services and electric power, create maps of the territories, and assist in developing significant resources such as roads.

The Mackenzie Valley has geographic areas of arable land, but some obstacles exist. Most foodstuff needs to be imported, increasing their price. Fort Smith has most of the arable and forested land in the region, but timber for lumber is not plentiful. Trapping still employs some aboriginal people. Muskrat, beaver, marten, mink, lynx, and arctic fox are the most prestigious furs in the Mackenzie area. In the Arctic regions, the arctic fox is most popular. Seals and whales are hunted to provide food, while sealskins are marketed commercially.

Mining has been the principal nonrenewable resource in territories. Gold has been mined on the north shore of Great Slave Lake since the late 1930s. However, in the early 1930s, they also mined radioactive ores. Uranium was later mined there in World War II and afterward until it was depleted in 1980. Lead, and zinc mining had an enormous scope at Pine Point from 1967 through the 1980s but has slowed down because it all ran out. Smaller plots of metals were also mined at various sites, some done up north. Various metals can be used to create jewelry. Diamonds were first discovered in Lac de Gras, creating a prospecting boom. Several other diamond-bearing properties were found shortly after, and production began in 1998. Exploration for petroleum took place at the Norman Wells and Pointed Mountain fields, and the Mackenzie delta has substantial oil reserves. Electrical generation of hydropower has not been successful, but local industries are conducting small-scale exploration.

Services account for an important sector of the Northern economy. Services include industries like tourism and public administration. About half the labor force is employed in service industries and a fifth in public administration. Services like tourism and natural phenomena such as long summer days make up the service industry.

Flights are the core means of transportation for air traffic, freight, and many significant settlements. Flight routes connect Yellowknife and other remote communities throughout the Mackenzie valley to Edmonton. Waterways are supplemented in the southern region for heavy freight transportation by the Mackenzie Highway and a railway connecting Hay River to Alberta through a highway. The delta is connected to Dawson, Yukon, by the Dempster Highway. Tractor trains and other overland vehicles also carry freight, while snowmobiles’ lighter winter travel is accomplished.

The Canadian federal government has ultimate responsibility for the territories, but most of those responsibilities have been delegated to the administration in Yellowknife. The administration consists of a commissioner, who the Canadian federal government appoints, and members of the Legislative Assembly, who are both directly elected to four-year terms. There is no system of parties that makes decision-making by consensus. Some members are not elected to the executive council and compose an “unofficial opposition” whose purpose is to represent other views. Those on the executive council make decisions about the overall direction of policy, manage the legislative agenda of the territorial administration, and oversee voting on legislation. All legislation technically has to be confirmed by the commissioner, but they mainly take a ceremonial role. There are elected and appointed members in the Canadian House of Commons. The justice system is handled by territorial courts, a police magistrate, and some Justices of the Peace. Natural resources are administered by the federal government department Indian and Northern Affairs Canada.

The history of the Northwest Territories

There is no record of Viking exploration in Canada before the 16th century, but there are records of English exploration. Explorers after Martin Frobisher went on to explore and record new findings of the Arctic. Once, there was a big focus on finding this passage, but it became less popular after 1750. Whaling ships made journeys to the Arctic commonplace during this period. The first recorded account of an exploration of mainland Canada is from Samuel Hearne in 1770-1772. Alexander Mackenzie traveled down the river named for him to find his way through the Arctic Ocean in 1789. In the 19th century, there was renewed interest in finding the Northwest Passage. John Franklin and his colleagues explored parts of the Mackenzie District, mapping their northern coastline. Searching for the lost expedition of 1845-1848, other arctic explorers mapped more parts of this region in the following decade. They had surpassed any further efforts to reach the North Pole in terms of practicalities; instead, they identified its resource potential.

After white settlers first arrived in Canada during the 17th and 18th centuries, the Northwest Territory became of utmost importance when whaling fleets visited the area. The Mackenzie Valley also attracted traders and missionaries in 1852. The Northwest Territories had no administrative authorities until the 20th century, when Hudson Bay Company was given power in Rupert’s Land. Canada’s North-Western Territory was ceded to Canada in 1870, and the Arctic islands were claimed by Britain and placed under Canadian jurisdiction. In 1880, the Northwest Territories were reduced to their pre-1999 limits by 1912. The Royal Canadian Mounted Police were responsible for maintaining law and order and providing whatever governmental administration required.

Fur traders, missionaries, and the police directed the life of the Northwest Territories until the 1920s when the discovery of oil near Fort Norman on the Mackenzie River prompted the Canadian government to establish a territorial administration for the area. Mining replaced the fur trade as the most crucial industry in Mackenzie District in 1948. World War II brought much government-financed construction activity to the territories. In southern Mackenzie, the Canol pipeline linked Norman Wells with a refinery in Whitehorse, and airfields were built in the East. The construction of the Mackenzie Highway to Great Slave Lake and the building of the Distant Early Warning radar network continued after World War II. The Canadian North saw a significant expansion of government-sponsored health, education, and welfare services.

As a result of environmental concerns, opposition among aboriginal groups to the commercial exploitation of resources, and an inquiry by Thomas R. Berger into a proposed Mackenzie Valley natural gas pipeline, development slowed in the 1970s. Subsequent negotiations with aboriginal groups led to several new agreements that addressed their concerns. The most fundamental change was embodied in the Nunavut Act, which created a territory out of the eastern portion of the Northwest Territories. After a transitional period, Nunavut was established on April 1, 1999.

In the early 21st century, natural resource exploitation furthered the economic development of the Northwest Territories, with energy and diamond-mining industries playing prominent roles. Meanwhile, the environment remained essential, and the territorial government increased efforts to resolve aboriginal land claims and related issues of aboriginal self-government. In addition, many social problems were confronted, including high rates of suicide and substance abuse–especially among youth–which were countered by increased efforts from local authorities.

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Signed, among others, by Medha Patkar of the Narmada Bachao Andolan (NBA), Aruna Roy of the National Campaign for People’s Right to Information, Dr Binayak Sen of the Peoples’ Union for Civil Liberties, Prafulla Samantaraof the Lok Shakti Abhiyan of the Niyamgiri Suraksha Samiti, among tens of others, the statement said, by addressing the two sides as ‘Hindu’ and ‘Muslim’ the judgment violates the secular character of the state.
"There were no parties with these names before the Court. The Uttar Pradesh Sunni Central Waqf Board was a petitioner representing that specific disputed land (and not the entire Muslim community of the country). Nor is ‘Ram Lalla Virajman’ (on behalf of Vishwa Hindu Parishad) a representative body of all Hindus", the statement insists.
NAPM believe, despite all this, "if the Apex Court was of the view that such a judgement was the ‘need of the hour to let peace prevail and move ahead’, the judgment should have said so in so many words."

The National Alliance of People’s Movements condemns the ‘unanimous’ verdict by the 5-judge Bench of the Supreme Court in the Ayodhya matter. The judgement, instead of holding accountable before law all those who criminally demolished the 450 year old Babri Masjid has rewarded the violators.
The judgement legitimizes majoritarianism and mobocracy and strikes at the very secular fabric of our Constitution. Full of contradictions, the verdict only pays sermons to the values of equality and fraternity, but ends up violating these very principles in its relief.
The judgement also does not pass the test of logic. Despite the Court stating at the outset of the hearing that the judgement in the matter of title dispute would be pronounced based on facts and not faith, the final judgement has ended up privileging faith in a disproportionate way and opening a dangerous flood gate.
On the one hand, the Court upheld that a masjid stood at the ‘disputed site’, that there was no evidence of a temple below the mosque (the Court accepted presence of a non-muslim structure), that idols were unlawfully sneaked into the mosque in 1949 and that the mosque itself was unlawfully demolished in 1992.
However, on the other hand, while issuing directions, the Court ordered that a temple be constructed at the very site of demolition of the mosque, which not only defies logic but is also a gross abuse of the due principles of law and natural justice.
To camouflage this, the Court directs that 5 acres land be given for ‘rehabilitation of the mosque’. There is no clarity as to the legal and constitutional basis for arriving at such a decision!
The invocation of Article 142 of the Constitution in this context would in fact give rise to many new challenges in the future. This was a clear title suit where the Uttar Pradesh Sunni Central Wakf Board was agitating for its legal rights and the question was not about ‘finding alternative land for the mosque’.
The judgement conveys an impression to Muslims that, despite being equal citizens of the country, their rights are not equal before the law. This is an extremely worrying message from the highest court in today’s times when Muslims are already being targeted and marginalized in many ways.
It is as if the Court is granting them a favour by offering the land. Further, granting the land (beneath the mosque) as a legal right to the same party which demolished the mosque is a historical insult to the values and principles enshrined in the Constitution.
Another unprecedented and unconstitutional aspect of the judgement is addressing the two sides as ‘Hindu’ and ‘Muslim’ which is clearly in violation to the secular character of the state and can prove quite dangerous.
Despite the verdict mandating that the central and state government must strictly implement the Places of Worship (Special Provisions) Act, 1991 which stipulates maintenance of status quo on the character of disputed sites as it existed in 1947, the judgement sets in motion a very dangerous precedent of allowing majoritarian faith over facts. Based on this, any community / sect can use their faith to contest the ownership of any structure and forcibly take over and demolish it by vicious campaigning and support from highest powers of state machinery. And it is no exaggeration to state that in the delicate circumstances that our nation is in today, the majoritarian political and religious forces would fully abuse such a judicial precedent and indulge in persecution of minorities.
If the Apex Court was of the view that such a judgement was the need of the hour to let peace prevail and move ahead, it should have said so in so many words
Be this as it may, the ‘Ayodhya matter’ itself is far from resolved and accountability is yet to be fixed on those who indulged in the criminal act of demolition. There are many unanswered questions. What happened of the Liberhan Commission’s Report? Why has the CBI been dragging its feet even 27 years after the demolition? Why haven’t the perpetrators of the crime been prosecuted and brought to justice?
Within days of the verdict, we also observe that the All India Hindu Mahasabha has now written to the Prime Minister and the Home Minister to withdraw all cases against the karsevaks responsible for the Babri demolition!
It is also a harsh reality that the political leaders, party and their violent affiliates who were behind the ‘Rath Yatra’ and demolition of the masjid in 1992 and the Gujarat pogrom of 2002, are in power in the centre and the state, to who the Apex Court’s verdict hands over the ‘disputed land’ on a platter despite admitting that they were guilty of criminal acts.
Ironically, these communal elements are now preaching to others that the ‘bitter past must be left behind’ in the interest of a ’new India’, seeking peace and brotherhood!
With a combination of draconian measures including the abrogation of Article 370 and lockdown of Kashmir for over 100 days, the communal criminalization of Triple Talaq, the proposed Citizenship Amendment Bill and proposals to extend the National Register of Citizens (NRC) across the country, the present verdict not only emboldens and encourages these powers but also gives them legal sanctity in the pursuit of their ‘Hindu Rashtra’ project.
If the Apex Court was of the view that such a judgement was the ‘need of the hour to let peace prevail and move ahead’, the judgement should have said so in so many words that it was depriving one party of their rightful ownership of the disputed site for pragmatic reasons, placing the imposed ‘compromise’ on record and acknowledging the sacrifice expected of the Muslim community.
Without such an express acknowledgement, it would be very unjust to call this decision, justice. We demand that the Supreme Court reviews this faulty judgement within the framework of the Constitution and decides it again on the basis of fact, law and logic and not on the basis of ‘faith’.
This is important since this is not just the case of one ‘disputed religious site’, but is necessary to ensure that hate-mongering elements do not get an opportunity to carry forward their divisive agendas and violate the Constitution.
We also demand that the state stop registration of false cases and persecution of anybody who critiques this judgement on the basis of law and logic. All those responsible for the criminal act of demolition must be brought to book at least now.
We hope that in this challenging climate of struggle for justice, the common citizens of the country and in particular, people belonging to both Hindu and Muslim faiths maintain peace. Democracy-loving citizens and groups need to be ever vigilant to ensure that in the garb of ‘temple construction’, there is no further hate and bloodbath.
At a time when the very right of citizens to question, reason and expression are trampled upon and all dissent is termed as anti –national, we must be on our toes to defend our democracy.
NAPM deeply appreciates the courage and peace-loving nature of the common masses, in particular the Muslim community and resolves to stand by them in these trying times. NAPM commits itself to fight alongside those who are wronged and continue its collective struggle for a just society based on peace, equality, reason, freedom, liberty and fraternity.
Let to organize to save our country from right-wing majoritarianism and strive to ensure that the values of the Constitution as well as the aspirations of Gandhi, Ambedkar, Maulala Azad, Bhagat Singh, Periyar, Shubash, Phule, Savitribai, and others are kept alive.

Whither Govt of India strategy to reduce import dependence on crude oil, natural gas?

By NS Venkataraman*  India presently imports around 80% of it’s crude oil requirement and around 50% of its natural gas requirements . As the domestic production of crude oil and natural gas are virtually stagnant and the domestic demand is increasing at around 7% per annum, India’s steadily increasing dependence on import of the vital energy source is a matter of high energy security concern. This is particularly so, since the price of crude oil and natural gas are considerably fluctuating / increasing in the global market due to geo political factors, which are beyond the control of India. India has promised to achieve zero emission by the year 2070, which mean that the level of emission has to start declining at slow and steady rate from now onwards. It is now well recognized that global emission is caused largely due to use of coal as fuel and natural gas as fuel and feedstock. While burning of coal as fuel cause emission of global warming carbon dioxide gas and sulphur

Muslim intellectuals met Bhagwat, extra-constitutional authority 'like Sanjay Gandhi'

By Shamsul Islam*  In a significant development a delegation of five Muslim intellectuals namely former chief election commissioner SY Quraishi; former senior bureaucrat Najeeb Jung; former AMU vice-chancellor and Lt Gen (retd) Zameer U Shah; politician-cum-journalist Shahid Siddiqui (presently with RLD); and businessman Saeed Shervani [Samajvadi Party] met RSS Supremo Mohan Bhagwat at RSS Delhi headquarters. The meeting was kept secret for reasons known to the participants and was held in August. According to the Muslim intellectuals the meeting held in “a very cordial” atmosphere continued for 75 minutes whereas time allotted was 30 minutes! In a post-meeting justification of the parleys Quraishi stated that their main concern was “the insecurity being increasingly felt by the Muslim community in the wake of recurring incidents of lynching of innocents, calls by Hindutva hotheads for genocide and the marginalisation of the community in almost every sphere”. This delegation consistin

'Massive concern for people': Modi seeking to turn India into global manufacturing hub

By Shankar Sharma*  The news item quoting Narendra Modi at the Shanghai Cooperation Organisation (SCO) meet, "Want to turn India into a manufacturing hub: PM Modi at SCO Summit" should be of massive concern to our people. One can only continue to be shocked by such policies, which can be termed as ill-conceived to say the least. Without objectively considering the environmental and social impacts on our communities in the medium to long term, such policies will also result in massive economic impacts because a lack of environmental and social perspective cannot be economically attractive either. In order to become the global manufacturing hub, India will have to meet an enormous demand for energy of various kinds, and in order to meet this much energy demand the economy has to manufacture enormous number of appliances/ gadgets/ machineries (to generate and distribute commercial forms of energy such as coal, nuclear, gas, hydro, and renewable energy (RE) sources such as so

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By Sandeep Pandey*  A padyatra (foot march) was to be taken out between 26 September and 4 October, 2022 from Randhikpur village in Dahod district of Gujarat to Ahmedabad to apologise to Bilkis Bano. Randhikpur is Bilkis Bano’s village. In 2002 Gujarat communal violence she was gang raped, her 3 years old daughter, another child in womb and a total of 14 family members were killed. 11 people were convicted and sentenced for life in 2008. However, on 15 August, 2022 after Prime Minister Narendra Modi made a speech from Red Fort appealing to people to change their attitude towards women and treat them with respect, a district level committee of Panchmahal decided to release the 11 rapists and murderers. A Bhartiya Janata Party leader described four of these criminals as virtuous Brahmins. Before the padyatra could begin from Randhikpur, on 25 September night, 7 activists were picked up from Godhra corporator Hanif Kalandar’s house where they had gone for d

Pesticide companies' lobbying 'seriously impairing' basics of governance, regulation

Dr Narasimha Reddy Donthi*  The Indian agricultural sector is grappling with low incomes, shortage of natural resources, increasing pest incidence and low public investments in research and extension. Pest attacks are increasing. Previously unknown pests are attacking crops. Farmers, indebted as they are due to various market mechanisms, are finding it hard to protect their crop investments. Thus, farmers are pushed into the conundrum of pesticide usage by pesticide markets and companies. Pesticide usage in India is increasingly becoming a regulatory problem. Regulation has not been effective in the face of such challenges. Scientific expertise on pesticides is often subsumed in the policy tradeoffs that, in the ultimate scenario, encourage production and marketing of Highly Hazardous Pesticides (HHPs). Expert Committee reports, which are recommending withdrawal of certain HHPs, are not being acted upon. Lobbying by pesticide companies has seriously impaired the basics of governance an

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By Sandeep Pandey*  A Muslim youth was beaten recently when he tried to witness the famous garba performance during the Hindu religious nine days festival of Navratri in Gujarat. There was a time when Muslims could easily participate in Garbha events in an atmosphere of cordiality. Bilkis Bano was gang raped in 2002 Gujarat communal violence, her 3 years old daughter, the child in womb and a total of 14 family members were killed. 11 accused were awarded life term. However, recently a District level committee has decided to release all the culprits. A ruling Bhartiya Janata Party leader has described some of these criminals as virtuous Brahmins, the highest among the Hindu hierarchical caste system. In a communally polarized Gujarat today most Muslims feel offended by the decision of the government and BJP supporters either justify the release of rapists and murderers or just ignore the ignominious decision. Mahatma Gandhi came from the Guj

GoI 'feeling threatened' by forces which can potentially fight 'Brahmanical fascism'

Counterview Desk  A network of civil rights and people’s organisations , Campaign Against State Repression (CASR)*, has characterised the recently-imposed ban on Popular Front of India (PFI), National Confederation of Human Rights Organizations (NCHRO) and other organisations as “Brahmanical Hindutva fascist” move of the Government of India (GoI), calling it “onslaught on democratic dissent”. In a statement, CASR said, the move is aimed at terrorizing and vilifying the Muslim community, adding, at the same time, the GoI is curbing any protest and demonstration against the “fascist diktat of ban”, with peoplebeing “detained and arrested.” It added, “This kind of attack on right to oppose or criticize any step of government should be conceived as an attack on the very democratic values of the people.” Text : On 28 September 2022, Central Government led by BJP-RSS banned the Popular Front of India, National Confederation of Human Rights Organizations, Campus Front of India, National Wom


Content: Home, Recipes

Soy sauce eggs

Added 21/Mar/2015

source: Hamish and Anatomy of Ramen

Add all of the ingredients (minus the eggs) into a saucepan and bring to a boil over high heat.

Remove from the heat and let the marinade come to room temperature.

Hard boiling Eggs

Fill another saucepan with water and bring to a boil over high heat. Lower the heat a little once it starts boiling.

Use a pushpin to gently open a small hole on the bottom of each egg (the rounded side is the bottom, not the pointy side). This helps loosen the egg white from the shell during cooking, making it easier to peel.

Gently place the eggs in the boiling water.

Set a timer for 6 minutes.

During the first 2 minutes: grab a chopstick or a spoon and spin the eggs around in the saucepan for the entirety of 2 minutes.

During minutes 3 to 6, stare lovingly at your boiling eggs.

Once your timer beeps, pour out the boiling water and run cool water to cool the eggs. Peel.

Pour your marinade into a bowl along with your peeled eggs. Marinate them in your fridge for up to 12 hours. The longer they marinate, the more pronounced the soy sauce flavors.


How to set up transactional campaigns through Maropost

How To Set Up Transactional Campaigns Through Maropost

What are transactional emails?

Transactional emails are emails intended to notify or inform recepients about activity on their account or a monetary transaction. Its purpose is to keep users posted about their interactions with an application or service. The difference between a transactional email and a marketing email is that although some transactional emails can be considered as marketing, their main purpose is to inform, rather than sell.

Why transactional emails are so important?

Transactional emails are very important because they allow you to interact with your target audience. You can send transactional emails to users that aren’t even on your marketing lists. Most importantly, transactional emails help with lead engagement. Hence the importance of making high-quality transactional emails.

What can I send through transactional emails?

You can send them to notify or inform users about anything related to their accounts’ interactions and events. Transactional emails are typically sent immediately after the trigger actions occur. Some examples for transactional emails are:

Onboarding emails:

These are considered by some as marketing emails, as they serve as a welcome email when a user subscribes to a list. Their primary purpose is to inform the new users of their subscription to a list. It’s commonly used as a way to welcome them and let them know what to expect about their subscription. They can also provide some content related to the business brand.

Account notifications

The sending for these happens after user accounts’ actions and requests. For example, if the user requests to change the password, it’d receive an email to notify about the action.

Transaction notifications

During purchase transactions, the app or website sends an email notifying the user if the payment was accepted or if there was a problem with it. Their purpose is to inform the users about monetary transactions happening in its account.

Bills and receipts

Sent after successful payments or when requesting a payment. They include file attachments and their purpose is to allow the user to keep a record of its finances.

Notification emails that are sent when a user requests a ticket for support, or when the support has finalized. These allow the user to know if the petition has been taken or the issue was addressed.

Shopping notifications

Notifications for events occurring after a session on an eCommerce application or website. These may apply to abandoned carts, order confirmation, and shipping.

Social notifications

These function as a way to inform users about interactions other people had with its account. For example, the notifications a user receives for a new comment reply in a forum, or when a user tags its name on a post.

What is a transactional campaign?

A transactional campaign refers to the segmenting and transactional emails so they can be set to be sent automatically. By creating transactional campaigns, you’ll be sure that your email is properly configured so the users will be receiving the correct notifications and confirmations. The creation of campaigns lets you keep track of the sending of emails and get performance-related reports.

How to set up transactional campaigns on Maropost?

Maropost lets you create and send transactional email campaigns so you have a way to send these emails outside your application. It’s a great way to keep the management of all your communications with your contacts on a single dashboard.

There are 2 ways you can connect your application to Maropost: Through SMTP and by using API.  You will be using JetSend as the bridge between your application and Maropost account and Maropost will be in charge of sending your transactional emails. This is very important as you won’t face some of the most recurring issues when sending transactional emails. Contrary, you will get the following:

-Good deliverability. You will be using a high-quality email provider service.

-Reports: you will get analytics data for each of your transactional campaigns. You can use this in conjunction with your marketing campaigns to build outstanding experiences for your users.

-Reliable service. You can make sure Maropost takes the customer very seriously. Its email sending service is one of the best of its kind. You won’t face sudden disconnections between your app and Maropost.

To start sending transactional emails for your campaigns check the following documentation. It’s highly recommended that you review it with your development team to make sure the connection is properly configured.

Use the following documentation to properly set up your transactional email campaigns:

Manage all of your workflows through Maropost

As you can see, Maropost is a great tool for administrating all of the stages within your sales workflows. From creating content to automating your total email sending by connecting your eCommerce application to Maropost.  We hope you find this article helpful. If you have any issues connecting your application we can give you a hand through our managed services. We cal also help you with anything related to your email campaigns. If you are interested don’t hesitate to contact us and we’ll reach to you as soon as possible.