Hamiltonian, Path Integral and BRST Formulations of the Vector Schwinger Model with a Photon Mass Term with Faddeevian Regularization

Abstract

Recently (in a series of papers) we have studied the vector Schwinger model with a photon mass term describing one-space one-time dimensional electrodynamics with mass-less fermions in the so-called standard regularization. In the present work, we study this model in the Faddeevian regularization (FR). This theory in the FR is seen to be gauge-non-invariant (GNI). We study the Hamiltonian and path integral quantization of this GNI theory. We then construct a gauge-invariant (GI) theory corresponding to this GNI theory using the Stueckelberg mechanism and recover the physical content of the original GNI theory from the newly constructed GI theory under some special gauge-choice. Further, we study the Hamiltonian, path integral and Becchi-Rouet-Stora and Tyutin formulations of the newly constructed GI theory under appropriate gauge-fixing conditions.

Appendix A: Mukhopadhyay and Mitra’s Derivation [21] of the Mass Term for the U(1) Gauge Field in the Faddeevian Regularization using Pauli-Villars Method (cf. Ref. [21])

In this Appendix, we reproduce (for a ready reference) the derivation due to Mukhopadhyay and Mitra [21] of the mass term for the U(1) gauge field in the so-called Faddeevian regularization, for the case of CSM [20], using Pauli-Villars method (cf. Ref. [21]). The same considerations apply not only to the case of the VSM-PMT being studied in the present paper, but they consistently hold for a large class of 2D field theory models.

We briefly recap here that the bosonized action of CSM in SR [4, 6–10, 13–17] (after setting g = e for harmonizing the notations with that of Refs. [19–21]) reads:

$$\begin{array}{@{}rcl@{}} S = \int d^{2}x \left[ \frac{1}{2}\partial_{\mu} \phi \partial^{\mu} \phi + e (g^{\mu\nu} - \epsilon^{\mu\nu}) \partial_{\mu} \phi A_{\nu} - \frac{1}{4}F_{\mu\nu} F^{\mu\nu} + \frac {1}{2} a e^{2} A_{\mu} A^{\mu} \right] \end{array} $$

(75)

and the above action in the FR due to Mitra [19, 20] and MM [21] reads:

$$\begin{array}{@{}rcl@{}} S = \int d^{2}x \left[ \frac{1}{2}\partial_{\mu} \phi \partial^{\mu} \phi + e (g^{\mu\nu} - \epsilon^{\mu\nu}) \partial_{\mu} \phi A_{\nu} - \frac{1}{4}F_{\mu\nu} F^{\mu\nu} + \frac{1}{2} e^{2} A_{\mu} M^{\mu\nu} A_{\nu} \right] \end{array} $$

(76)

where the mass term of the U(1) gauge field A ^μ in FR is given (as mentioned in the foregoing) by: $~\mathcal {L}_{m}^{FR} = \frac {1}{2} e^{2} \left (A_{\mu } M^{\mu \nu } A_{\nu } \right ),~$ with $~M^{\mu \nu } = \left(\begin {array}{cc}+1&-1\\-1&-3 \end {array} \right)$.

After the above brief recapitulation, we are now in a position to review the work of MM [21], on the Pauli-Villars method of regularization for obtaining the effective action of CSM with the above unconventional mass term for the U(1) gauge field in FR and used by us in our present work for the VSM with a PMT. MM in their work [21], follow the same technique that has been used by Frolov and Slavnov [6] for getting the conventional mass term in SR except for the fact that now the regulator fields couple with both the light-cone components of the gauge field (also, the interaction is not restricted by the requirement of manifest Lorentz covariance).

MM [21] start with the action expressed (in Fermi fields) by:

(77)

This above action is known [6] to give rise to the effective action [21]:

$$\begin{array}{@{}rcl@{}} Z_{0}[A] &=& \int (\mathcal{D} \bar{\psi}(x)) (\mathcal{D} \psi(x)) ~ exp ~\left[ iS[\bar{\psi}(x) , \psi(x), A(x)] \right] \\ &=& ~exp ~iS^{0}_{eff} [A(x)] \end{array} $$

(78)

where

$$ S^{0}_{eff} [A(x)] = - e^{2} \int d^{2}x ~ \left(A^{\mu}(x) (g_{\mu\alpha} + \epsilon_{\mu\alpha}) \right) ~ \left(\frac{\partial^{\alpha} \partial^{\beta}}{\partial^{2}} (g_{\beta\nu} - \epsilon_{\beta\nu})A^{\nu}(x) \right) $$

(79)

The derivation needs to be regularized and MM [21] add to (77), a regularizing action containing several parameters [21]:

(80)

where

$$ {\Gamma}^{\mu}_{r} = \left[ a_{r} K^{\mu\nu} (1 + i \gamma_{5}) + b_{r} {\Sigma}^{\mu\nu} (1 - i \gamma_{5}) \right] \gamma_{\nu} $$

(81)

The fields $~\bar {\psi }_{r}(x)~$ are the regulator fields with the mass m _r , coupling $~{\Gamma }^{\mu }_{r}~$ and statistics 𝜖 _r = ±1 . Here Σ^μν and K ^μν are the matrices to be determined later. At the end of the calculation, m _r is made to tend to infinity so that the regulator fields decouple. The contribution of the regulator fields to the vacuum functional is given by [21]:

$$\begin{array}{@{}rcl@{}} Z_{reg}[A] &=& \int ~{\prod}_{r} ~ (\mathcal{D}\bar{\psi}(x)) (\mathcal{D} \psi(x)) ~ exp ~\left[ iS_{reg} [\bar{\psi}_{r}(x) , \psi_{r}(x), A(x)] \right] \\ &=& ~exp ~iS^{eff}_{reg} [A(x)] \end{array} $$

(82)

Further it could be shown that [21]:

$$ S^{eff}_{reg} [A] = - \frac{e^{2}~\pi}{2} \int d^{2}x ~ \left(A_{\mu}(x) G^{\mu\nu}(x,y) ~A_{\nu}(y)\right) $$

(83)

with

$$ G^{\mu\nu}(x,y) = \int \frac{d^{2}p}{(2\pi)^{2}} ~ \bar{G}^{\mu\nu}(p) ~ exp [ -i p \cdot (x - y) ] $$

(84)

where

(85)

Now a calculation of the integral in the rth term of the above equation yields [21]:

$$\begin{array}{@{}rcl@{}} \bar{G}^{(r)}_{\mu\nu}(p)\!\! &=& \frac{1}{\pi} \left[\! \left((a_{r})^{2} T^{1}_{\mu\nu\lambda\kappa} + (b_{r})^{2} T^{2}_{\mu\nu\lambda\kappa} \right) \left(2\left(g^{\lambda\kappa} - \frac{p^{\lambda}p^{\kappa}}{p^{2}}\right) \left(2 \,-\, \frac{i}{y_{r}} ~ ln (-1) +\! O(y_{r})\right)\right.\right. \\ &&\left.\left. \!\!\!+ g^{\lambda\kappa} \!\left(\!1 \,-\, \frac{i}{y_{r}} ln (-1) \,+\, O(y_{r})\right)\!\right)\! \,+\, 2 a_{r} b_{r} M_{\mu\nu} \left(\!1\! - \frac{i}{y_{r}} ~ ln (-1) \,+\, O(y_{r})\right)\! \right] \end{array} $$

(86)

where

$$\begin{array}{@{}rcl@{}} T^{1}_{\mu\nu\lambda\kappa} &=& K_{\mu\rho} ~ (g^{\rho}_{\lambda} + \epsilon^{\rho}_{\lambda}) ~ K_{\nu\sigma} ~ (g^{\sigma}_{\kappa} + \epsilon^{\sigma}_{\kappa}) \\ T^{2}_{\mu\nu\lambda\kappa} &=& {\Sigma}_{\mu\rho} ~ (g^{\rho}_{\lambda} + \epsilon^{\rho}_{\lambda}) ~ {\Sigma}_{\nu\sigma} ~ (g^{\sigma}_{\kappa} + \epsilon^{\sigma}_{\kappa}) \\ M_{\mu\nu} &=& [K_{\mu\lambda} ~(g^{\lambda\kappa} - \epsilon^{\lambda\kappa}) ~{\Sigma}_{\nu\kappa} + {\Sigma}_{\mu\lambda} ~(g^{\lambda\kappa} - \epsilon^{\lambda\kappa}) ~K_{\nu\kappa}] \end{array} $$

(87)

and

$$ {y_{r}^{2}} = \frac{p^{2}}{{m_{r}^{2}}} $$

(88)

Further by imposing the conditions [6]:

$$\begin{array}{@{}rcl@{}} \sum\limits_{r} (\epsilon_{r} {a_{r}^{2}}) &=& \sum\limits_{r} (\epsilon_{r} {b_{r}^{2}}) = \sum\limits_{r} (\epsilon_{r} {a_{r}^{2}} m_{r}) = \sum\limits_{r} (\epsilon_{r} {b_{r}^{2}} m_{r}) = \sum\limits_{r} (\epsilon_{r} a_{r} b_{r} m_{r}) = 0 \\ 2 ~ \sum\limits_{r} (\epsilon_{r} a_{r} b_{r}) &=& 1 \end{array} $$

(89)

and then letting m _r tend to infinity, one gets [21]:

$$ S^{eff}_{reg} [A] = + \frac{e^{2}}{2} \int d^{2}x ~ \left(A_{\mu}(x) M^{\mu\nu}(x,y) ~A_{\nu}(y)\right) $$

(90)

Now there are several ways to show that the above general form accommodates the desired effective action of Mitra [20]. For example, one could choose:

$$\begin{array}{@{}rcl@{}} K^{00} = 2, ~~ K^{11} = 0, ~~K^{01} = - K^{10} = \frac{3}{2},~~~~{\Sigma}^{00} = {\Sigma}^{11} = 0, ~~ - {\Sigma}^{01}= {\Sigma}^{10} = 1 \end{array} $$

(91)

Then M ^μν finally becomes [21]:

$$ M^{\mu\nu} = \left(\begin{array}{cc} +1&-1\\ -1&-3 \end{array} \right)$$

(92)

This then is the desired mass term [20] of the U(1) gauge field in the so-called Faddeevian regularization. This completes the explicit construction of a generalized Pauli-Villars regularization which leads to this mass term in FR. We refer to the work of Ref. [21] for many further details on this topic.

We wish to mention here that this MM’s derivation [21] for the mass term of the U(1) gauge field for the case of a CSM also holds for a large class of 2D field theories, including the theory studied in the present work.