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e1e-mail: [email protected], [email protected]

The author of the article SY , recently carried out an investigation to find out the theoretical spectra of a model which was resulted after imposition of a chiral constraint by Miao MIA , in the phase-space of the generalized chiral Schwinger model (GCSM) BASS ; WOT ; SAR . The article MIA however was not cited in SY . The innovative idea of imposing chiral constraint was put forward by Harada in his seminal work KH where he imposed a chiral constraint in the Chiral Schwinger model (CSM) and expresed it in terms of chiral boson SIG ; JSON . The generating functional of the GCSM is given by

The bosonized version BASS ; SAR ; MIA corresponding to the fermionic Lagrangian density constituting Eqn. (1) reads

The GCSM contains both the vector and axial-vector interaction terms with the unequal wight. For both the choices $r=\pm 1$ the GCSM corresponds to CSM JR ; RB ; MIT ; ROT1 ; ROT2 . Fermion of left/ right chirality takes part in interaction with the gauge field for $r=+1$/$r=-1$ respectively. Here $\tilde{\partial}_{\mu}=\epsilon_{\mu\nu}\partial^{\nu}$ and $\epsilon^{01}=+1$ and $\psi,\phi$, and $A_{\mu}$ respectively represent fermion, boson, and gauge field . The Lagrangian (2) after the imposition of the chiral constraint $\Omega=\pi_{\phi}-\phi^{\prime}=0$, along with the kinetic term of the electromagnetic background $\frac{1}{2}(\dot{A_{1}}-A_{0}^{\prime})^{2}$ turns into

The Lagrangian and Hamiltonian formulation of the model (3) is attempted in SY to study the theoretical spectrum of the model. The author claimed that the model was exactly solvable for the generic $r$ and theoretical spectrum contained only a massive boson with the square of the mass $m^{2}=ae^{2}\frac{r^{2}+a-1}{a-r^{2}}$ which showed a sharp disagreement with the result found in KH . For instance setting $r=1$ in SY although a massive boson was found the massless chiral boson was found absent. This absence of massless chiral boson in SY stood as a grave disparity so far counting of number of physical degrees of freedom is consented. Landing onto this erroneous result the author claimed that the fermion got confined SY . How the article SY is suffering from erroneous result and to what extent it is remediable that we would like to present through this note.

To determine the theoretical spectrum transparently through Hamiltonian analysis of the Lagrangian density (3 we compute the momenta corresponding to the field $\phi$, $A_{0}$, and $A_{1}$:

Note that $\Omega_{1}=\pi_{0}\approx 0$, is a primary constraint of the theory. The canonical Hamiltonian density is now obtained using Eqns. (4) by exploiting a Legendre transformation:

The preservation of $\Omega_{1}$ results a secondary constraint

$\Omega_{1}$ and $\Omega_{1}$ are weak conditions DIR and form a second class set. From Eqn. (6) we have a solution for $A_{0}$:

and it leads us to evaluate the reduced Hamiltonian density by plugging in Eqn. (7) in the Eqn. (5)

However, Poisson brackets needs to be replaced by Dirac brackets DIR since the weak condition (6) is used as strong condition in 8. The non-vanishing Dirac bracelets for the fields describing the reduced Hamiltonian are:

The Dirac brackets are found non-canonical. The reduced Hamiltonian density (8) along with the Dirac brackets leads to the following coupled equations of motions.

The determination of theoretical spectra needs the decoupling of the above set of equations (10),(11) and (12). We observe that for the choice $r=\pm 1$, these equations leads to Lorentz invariant theoretical spectra that contains a massive boson with mass $m=\sqrt{\frac{a^{2}e^{2}}{a-1}}$ accompanied with a massless chiral boson having left/right chirality for $r=\mp 1$. It shows an exact agreement with the result reported in KH . The massless chiral boson can be thought of in terms of a chiral fermion in the $(1+1)$ dimension. Thus the presence of chiral fermion ensures that the fermion did not confine for $r=\pm 1$. We should mention that Lagrangian density for $r=-1$ can be obtained imposing the constraint $\tilde{\Omega}_{Ch}=\pi_{\phi}+\phi^{\prime}=0$ in the phase-space of the theory described by Eqn. (2) and in this situation the resulting lagrangian density reds

Comparing Eqn. (10) with the Eqn. $(36)$ of the article SY absence of the term $-\frac{1+r}{e(a-r^{2})}\phi^{\prime\prime}$ from Eqn. $(36)$ has been found. In addition to that, equation of motion corresponding to $\phi$, e.g. $[\phi,H_{R}]$ was not taken into consideration to come to the conclusion concerning the theoretical spectra. The model (3) is also found exactly solvable for $r=0$ and $r>>1$. Note that the lagrangian (2) lands into the celebrated vector Schwinger model with coupling strength $e$, for $r=0$ while for $r>>1$ it can approximately be considered as Schwinger model described with axial vector interaction with coupling strength $re$. The models (2) however exhibit exact solvability for $r=0$ and $r>>1$ with two specific regularization. The choice $a=0$ and $a=r^{2}$ works for $r=0$, and $r>>1$ respectively. In the second care, it can be thought that the coupling constant gets scaled by a factor $r$, and the vector interaction ceases to zero. For these two cases, if chiral constraint $\Omega_{Ch}=\pi_{\phi}-\phi^{\prime}=0$ is imposed in (2), the resulting model describes confinement of fermion of a definite chirality, and the theoretical spectrum contains only a missive boson with mass $2e$ and $2re$ for $r=0$ and $r>>1$ respectively ARMPLA . So for the set of values $r=0$, $r=\pm 1$, and $r>>1$ although the model (2) is found exactly solvable, for a generic $r$ after the imposition of chiral constraint the issue of exact solvability along with confinement aspect of fermion remained un settled.

Let us now point out the inconsistencies in the Lagrangian formulation. Here the author introduced two ansatzes: Eqn. $(15)$, and $(16)$ for the fields $\phi$ and $A_{\mu}$ respectively. However, none of these two satisfy the Eqns. $(12)/(17)$, $(13)$, and $(14)$. it is surprising tat Eqn. $(16)$ is designed in an ambiguous manner to satisfy Eqn. $(18)$ without paying heed to the Eqn. $(12)/(17)$, $(13)$, and $(14)$ and ambiguously Eqn. $(18)$ is identified as a massive boson!

Therefore, it transpires that from the beginning the author spoiled both Hamiltonian and Lagrangian formulation in SY . We are afraid that it is irremediable for generic $r$. Albeit, it is known that the GCSM model (2) remains exactly solvable for a generic $r$, BASS ; MIA ; SAR , but the question of solvability and physical sensibility of (3) stood open for the generic $r$. Our investigation reveals clearly that exact solvability are manifested only for $r=0$, $r=\pm 1$, and $r>>1$, subject to the chiral constraint that fits with these permissible values of $r$ which ensures physical sensibility and confinement of fermion takes place only for $r=0$ and $r>>1$ (for very large $r$).

To summarize we reiterate that only the values $r=0,\pm 1$, and $r>>1$ the model (2) may lead to physically sensible results after the imposition of chiral constraint suitable for the values of $r$ that renders exact solvability. However confinement of fermion occurs only for $r=0$ and $r>>1$. The set of values $r=\pm 1$ although renders exact solvability, fermion of a left chirality for $r=-1$ and a fermion of of right chirality for $r=+1$ remain unconfined. We would also like to add that the attempt to establish the physical Lorentz invariance in SY for generic $r$ through Poincare algebra is not trustworthy which could be understood from the work previous work SAR . However, for the set of values values $r=0$, $r=\pm 1$, and $r>>1$, physical Lorentz invariance does maintained, although the Lagrangian densities does not have manifestly Lorentz covariant structure.