1 Introduction↩︎

In quantum field theory renormalizable models are divided into asymptotically free and asympltotically non free models. Two famous examples of asymptotically free and asymptotically non free models are QCD (quantum chromodynamics) and QED (quantum electrodynamics). In QCD the effective coupling constant decreases at small distances therefore we can use the perturbation theory in the ultraviolet region. The theory looks self-consistent at least at small distances. While in QED the effective coupling constant increases at small distances and the perturbation theory is not applicable at small distances. Moreover the “naive” use of the perturbation theory leads to the appearance of the famous Landau pole singularity with negative norm state [1]. There is convincing evidence that the four-dimensional \(\phi^4\) model does not exist as mathematically consistent quantum field theory [2]–[4]. All renormalizable models without nonabelian gauge fields are non asymptotically free models and probably such models are not self-consistent at small distances. Therefore it would be very interesting and important to investigate the problem of consistency or nonconsistency of non asymptotically free renormalizable field theories. In refs. [5], [6] some evidence but not proof has been presented that the WZ (Wess-Zumino) model [7] - [8] is not self-consistent local field theory. See, however refs.[9], [10]. In ref. [11] we proved that the regime of fixed point or finite wave function renormalization is not realized in the WZ model, see also [12].

The aim of this paper is more careful investigation of the (non)consistency problem in non asymptotically free models on the example of the WZ model. We prove that the WZ model is not self-consistent local field theory provided three conditions are valid. The first condition is that the structure of the counterterms in the WZ model coincides with the structure of the counterterms in the perturbation theory. The second condition is that commutation relations among generalized coordinates and momenta are valid in the renormalized field theory. The third condition is that the spectrum of the WZ model does not contain negative norm states.

The organization of the paper is the following. The next section contains short description of the WZ model. In the third section we describe commutation relations between the generalized coordinates and the momenta in quantum field theory. In section 4 we derive and analyze Schwinger’s equations for the WZ model. In sections 5 and 6 we consider the case of infinite and finite wave function renormalization correspondingly. Section 7 contains concluding remarks.

2 WZ model↩︎

The WZ model [7] - [8] describes the interaction of the scalar and Majorana fields. In the superspace \(x^M = (x^{\mu}, \theta_{\alpha}, \bar{\theta}_{\dot{\alpha}})\) the Lagrangian of the model has the form \[L = L_0 + \int W d^2\theta + \int W^* d^2{\bar{\theta}} \,, \label{WZ0}\tag{1}\] where \[L_0 = \int \sum_{k=1}^N \phi_k^*(x,\theta, \bar{\theta})\phi_k(x,\theta, \bar{\theta}) d^2\theta d^2\bar{\theta} \,, \label{WZ1}\tag{2}\] \[W = \sum_{i,j,k}[\frac{g_{ijk}}{3}\phi_i(x, \theta) \phi_j(x, \theta) \phi_k(x, \theta) + \frac{m_{ij}}{2}\phi_i(x,\theta)\phi_j(x,\theta)] \,, \label{WZ2}\tag{3}\] \[W^* = \sum_{i,j,k}[\frac{g^*_{ijk}}{3}\phi^*_i(x, \bar{\theta}) \phi^*_j(x, \bar{\theta}) \phi^*_k(x, \bar{\theta}) + \frac{m^*_{ij}}{2}\phi^*_i(x,\bar{\theta})\phi^*_j(x,\bar{\theta})] \,, \label{WZ242}\tag{4}\]

Here \(\phi_k(x, \theta, \bar{\theta}) = \phi_k(x^{\mu} + i\theta\sigma^{\mu}\bar{\theta}, \theta)\) and \(\phi_k(x, \theta) = \phi_k(x) + \sqrt{2}\psi_k(x)\theta + \theta \theta F_k(x)\) is chiral scalar superfield. In terms of component fields \((\phi_k(x), \psi_k(x), F_k(x))\) the Lagrangians have the form \[L_0 = \sum_{k=1}^N [\partial^{\mu}\phi_k \partial_{\mu}\phi^*_k -i\partial_{\mu}\bar{\psi}_k\bar{\sigma}^{\mu}\psi_k + F^*_kF_k] \,, \label{WZ2a}\tag{5}\] \[\int W d^2\theta = \sum_{i,j,k} g_{ijk}(F_i\phi_j\phi_k - \phi_i\psi_j\psi_k) + \sum_{i,j} m_{ij}(F_i\phi_j -\frac{1}{2}\psi_i\psi_j) \,. \label{WZ2b}\tag{6}\] \[\int W^* d^2\bar{\theta} = ( \int W d^2\theta)^* \,. \label{WZ2b42}\tag{7}\]

The WZ Lagrangian (1 ) is invariant under the supersymmetry transformations \[\delta\phi_k =\sqrt{2}\zeta \psi_k \,, \label{WZ2ba}\tag{8}\] \[\delta\psi_k =i\sqrt{2}\sigma^{\mu}\bar{\zeta}\partial_{\mu}\phi_k + \sqrt{2} \zeta F_k \,, \label{WZ2bb}\tag{9}\] \[\delta F_k =i\sqrt{2}\bar{\zeta}\bar{\sigma}^{\mu}\partial_{\mu}\psi_k \,. \label{WZ2bc}\tag{10}\]

The WZ model with the superpotential (3 ) is renormalizable model [7] - [8]. Moreover all ultraviolet divergencies can be eliminated by the introduction of the wave function renormalization counterterm [7] - [8] \[\Delta L = \int \sum_{k=1}^N \Delta Z_k \phi_k^*(x,\theta, \bar{\theta})\phi_k(x,\theta, \bar{\theta}) d^2\theta d^2\bar{\theta} \,. \label{WZ3}\tag{11}\] The WZ model has the simplest structure of the counterterms among all renormalizable \(d=4\) models with scalar and fermion fields. For instance, in renormalizable \(\phi^4\) model to make the Green’s functions ultraviolet finite we have to introduce three different countertems \(\Delta L = \Delta Z_1 \frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi - \frac{\delta m^2 }{2}\phi^2 - \Delta Z_2\lambda \phi^4\) .

2.1 Regularization of the WZ model↩︎

The Feynman diagrams for the WZ model are ultraviolet divergent and to make the model well defined at least within perturbation theory we have to introduce the regularization. The most convenient regularization is the supersymmetry invariant regularization. We shall use the generalization of the Pauli-Villars regularization [13]. For scalar propagator in the \(\phi^4\)-model the Pauli-Villars regularization consists in the replacement \[\frac{1}{p^2 - m^2 +i\epsilon} \rightarrow \frac{1}{p^2 - m^2 +i\epsilon} - \frac{1}{p^2 - M^2 +i\epsilon} \label{WZ4}\tag{12}\] that makes most Feynman diagrams ultraviolet finite. The Pauli-Villars regularization leads to the existence of negative norm states already at the level of free Lagrangian. We shall use the modification of the Pauli-Villars regularization which at the level of the perturbation theory is equivalent to the original Pauli-Villars regularization. For instance, for \(\phi^4\)-model we use the following regularized Lagrangian: \[L_{reg} = L_{0,reg} + L_{int,reg} \,, \label{WZ5a}\tag{13}\] where \[L_{0,reg} = \frac{1}{2}[\partial^{\mu}\phi\partial_{\mu}\phi -m^2 \phi^2] + \frac{1}{2}[\partial^{\mu}\Phi\partial_{\mu}\Phi - M^2 \Phi^2] \,, \label{WZ5b}\tag{14}\] \[L_{int,reg} = -g(\phi +i\Phi)^4 \,. \label{WZ5c}\tag{15}\] The Lagrangian (14 ) describes two free scalar fields \(\phi(x)\), \(\Phi(x)\) with masses \(m\), \(M\) and positively definite metric. The interaction Lagrangian (15 ) is nonhermitean and as a consequence the regularized model is not unitary. The propagator for the effective field \(\phi_{eff} = \phi +i\Phi\) coincides with Pauli-Villars propagator (12 ). The generalization of this regularization to the WZ model is straightforward. Namely we introduce N additional chiral superfields \(\Phi_k(x, \theta)\) with masses \(M_k\). The regularized Lagrangian (2 ) takes the form \[L_{0,reg } = L_0 + \int \sum_{k=1}^N \Phi_k^*(x,\theta, \bar{\theta})\Phi_k(x,\theta, \bar{\theta}) d^2\theta d^2\bar{\theta} +( \int \sum_{k} M_{k}\frac{\Phi_k(x,\theta)\Phi_k(x,\theta) }{2} d^2\theta + h.c.) \,. \label{WZ5cd}\tag{16}\] The regularization of the interaction Lagrangian \[W_{int} = \int \sum_{l,j,k}[g_{ljk}\phi_l(x, \theta) \phi_j(x, \theta) \phi_k(x, \theta) d^2\theta \, \label{WZ5d}\tag{17}\] and its hermitean conjugate \(W^*_{int}\) is the following: \[W_{int} \rightarrow \int \sum_{l,j,k}[g_{ljk}(\phi_l(x, \theta) + i\Phi_l(x, \theta)) (\phi_j(x, \theta) + i\Phi_j(x, \theta)) (\phi_k(x, \theta) + i \Phi_k(x, \theta)) d^2\theta \,, \label{WZ6a}\tag{18}\] \[W^*_{int} \rightarrow \int \sum_{l,j,k}[g^*_{ljk}(\phi^*_l(x, \bar{\theta}) + i\Phi^*_l(x, \bar{\theta})) (\phi^*_j(x, \bar{\theta}) + i\Phi^*_j(x, \bar{\theta})) (\phi^*_k(x, \bar{\theta}) + i \Phi^*_k(x, \bar{\theta})) d^2{\bar{\theta}} \,. \label{WZ6b}\tag{19}\] One can find that for modified superpotentials (18 , 19 ) all Feynman diagrams are ultraviolet finite. However the interaction Lagrangian is not hermitean as in the case of the \(\phi^4\)-model. So we find that it is possible to reformulate Pauli-Villars regularization in such a way that free Lagrangian has positive metric while the interaction Lagrangian is not hermitean and as a consequence the property of the unitarity is lost.

Another possible regularization consists in the nonlocal generalization of the vertex \(g_{ijk}\) in formula (17 ), namely \[g_{ijk} \rightarrow g_{ijk}\exp(-\frac{\vec{p}_i^2}{\Lambda^2} - \frac{\vec{p}_j^2}{\Lambda^2} - \frac{\vec{p}_k^2}{\Lambda^2} ) \,. \label{WZ7}\tag{20}\]

3 Canonical quantization↩︎

In classical mechanics with the Lagrangian \(L(q_k, \dot{q}_k )\) \(( ~k = 1,2,...N )\) canonical momenta are defined as \(p_k = \frac{\partial L(q_k, \dot{q}_k)}{d{\dot{q_k}}}\) and canonical quantization consists in the replacement of classical functions \(q_k(t)\), \(p_k(t)\) to the operators \(\hat{q}_k(t)\) and \(\hat{p}_k(t)\) acting in Hilbert space. The commutation relations¹ \[[\hat{q}_k(t), \hat{q}_l(t)] = [\hat{p}_k(t), \hat{p}_l(t) = 0 \, \label{Can1}\tag{21}\] \[i [\hat{p}_k(t),\hat{q}_l(t)] = \delta_{kl} \, \label{Can2}\tag{22}\] are postulated. In the field theory with the complex scalar fields \(\phi_k(x)\) and with the Lagrangian \(L = \sum_{k=1}^{N}\partial_{\mu}\phi_k(x)\partial^{\mu}\phi^*_k(x) - V(\phi_k(x), \phi^*_k(x))\) the density of the momenta is \(\pi_k(t, \vec{x}) = \frac{\partial \phi_k(t, \vec{x})}{dt}\) and the commutation relations are [1] \[[\phi_k(t,\vec{x}), ~\phi_l(t,\vec{y})] = [\phi^*_k(t,\vec{x}), ~\phi_l(t,\vec{y})] = 0 \,, \label{Can3a}\tag{23}\] \[[\frac{\partial_k\phi_k(t,\vec{x})}{dt}, ~\phi_l(t,\vec{y})] = 0 \,, \label{Can3b}\tag{24}\] \[[\frac{\partial_k\phi_k(t,\vec{x})}{dt}, ~\phi_l^*(t,\vec{y})] = 0 ~at ~k \neq l \,, \label{Can3c}\tag{25}\] \[i [\frac{\partial_k\phi_k(t,\vec{x})}{dt}, ~\phi_k^*(t,\vec{y})] = \delta(\vec{x} - \vec{y}) \,. \label{Can3d}\tag{26}\]

The renormalized field \(\phi_{kr\Lambda}(x)\) is proportional to the bare field \(\phi_k(x)\), namely [1] \(\phi_{kr\Lambda}(x) = Z^{-1/2}(\Lambda,...) \phi_k(x)\). Here \(\Lambda\) is ultraviolet cutoff. Note that the limit of the regularization removing \(\Lambda \rightarrow \infty\) exists at least in the perturbation theory. We shall assume that the limit \(\Lambda \rightarrow \infty\) exists irrespective of the perturbation theory, in other words we assume that the \(lim_{\Lambda \rightarrow \infty} ~\phi_{kr\Lambda}(x) = \phi_{kr}(x)\) exists. As a consequence the commutation relations (23  - 25 ) are valid for renormalized fields. The commutation relation (26 ) takes the form \[i[\frac{\partial_k\phi_{kr}(t,\vec{x})}{dt}, ~\phi_{kr}^*(t,\vec{y})] = \frac{1}{Z_k(\infty,...)} \delta(\vec{x} - \vec{y}) \,. \label{Can3dr}\tag{27}\] We assume that for all states \(|n>\) the metric is positive, i.e. \(<n|n> > 0\). As a consequence of this assumption the KL (Kallen -Lehmann) representation [14], [15] for two point vacuum commutator \(<0|[\phi_{kr}(x), \phi^*_{lr}(y)]|0>\) reads \[<0|[\phi_{kr}(x), \phi^*_{lr}(y)]|0> = \frac{1}{i}\int_{0}^{\infty} D(x-y,t)\rho_{kl}(t) dt \,, \label{Can4a}\tag{28}\] where \[D(x-y, m^2) = \frac{i}{(2\pi)^3}\int \exp{[-ik(x-y)]} \epsilon(k^0)\delta(k^2-m^2)d^4k \,, \label{Can4d}\tag{29}\] \[\rho_{kl}(t) = c^2_k\delta_{kl}\delta(t- m^2_k) + \Delta\rho_{kl}(t) \, \label{Can4d1}\tag{30}\] and \(c^2_k > 0\), \(\Delta\rho_{kk}(t) \geq 0\). The first term in the formula (30 ) describes one particle state contribution while the second term describes many particles contribution. As a consequence of the commutation relations (25 , 27 ) one can find that \[\int_{0}^{\infty}\rho_{kl}(t)dt = 0 ~at~ k \neq l \,, \label{Can5d}\tag{31}\] \[\int_{0}^{\infty}\rho_{kk}(t)dt = \frac{1}{Z_k(\infty,...)} \,. \label{Can5dd}\tag{32}\] For often used normalization condition \(c^2_k = 1\) well known inequality [14], [15] \[0 \leq Z(\infty,...) = [1 + \int_{0}^{\infty}\Delta \rho_{kk}(t)dt]^{-1} \leq 1 \,, \label{Can5e}\tag{33}\] is valid. For the propagator \(\frac{1}{i}\bar{D}_c(x-y) = <0|T(\phi_r(x), \phi_r^*(y))|0>\) of the renormalized scalar field \(\phi_r(x)\) the KL representation takes the form \[\bar{D}_c(x) = \frac{1}{(2\pi)^4}\int \exp({-ikx})D_c(k^2) \,, \label{Can5e1}\tag{34}\] \[D_c(k^2) = \frac{1}{m^2_{0} - k^2 - i\epsilon} + \int_{4m^2_o}^{\infty} dt \frac{\Delta\rho(t)}{t - k^2 -i\epsilon} \,, \label{Can5e2}\tag{35}\] where \(\Delta\rho(t) \geq 0\). As a consequence of the commutation relation () and the KL representation () one can find that only two possibilities are possible. Namely, we can have finite wave function renormalization \(Z(\infty, ...) \neq 0\) or infinite wave function renormalization with \(Z(\infty, ...) = 0\). The possibility with \(Z(\infty, ...) = \pm \infty\) is excluded due to assumed nonnegativity of \(\Delta \rho_{kk}(t)\). The infinite \(Z(\infty, ...)\) means the negativity of \(\Delta \rho_{kk}(t_0) < 0\) at some \(t = t_0\), i.e. the existence of negative norm states in the spectrum. For \(Z(\infty, ...) \neq 0\) the ultraviolet behaviour of the propagator () coincides up to some factor with free massless propagator \(\frac{1}{-k^2- i\epsilon}\) while for infinite renormalization with \(Z(\infty, ...) = 0\) the propagator decreases more slowly than free propagator, i.e. \(k^2 D_c(k^2) \rightarrow \infty\) at \(k^2 \rightarrow \infty\).

It should be noted that for the WZ model in the leading order of the perturbation theory \(Z(\Lambda, ..) = 1 - ag^2\ln(\Lambda) + o(g^2)\) \(( a >0)\) and \(Z \rightarrow - \infty\) at \(\Lambda \rightarrow \infty\). The manifestation of this fact is the existence of Landau pole singularity for the scalar field propagator in leading log approximation.

4 Schwinger’s equations↩︎

The Schwinger’s equations [16] are mostly conveniently obtained in the functional integral formalism by means of the identity \[\int\prod_{k} d[\phi_k] \prod_{i} \frac{\delta}{\delta \bar{\phi}_i(x_i)}[R(\phi_k) \exp(iS(\Phi_k)] = 0 \,. \label{Sch1}\tag{36}\] Here \(\phi_k\) denotes the multiplet \((\phi_k, F_k, \psi_k)\), the action \(S(\Phi_k) = \int d^4x L +\int J_k\phi_{k} d^4x\) and \(R[\phi_k]\) an arbitrary function; \(\bar{\phi_i}\) may be taken to be any of the fields \(\phi_k\) \(F_k\) and \(\psi_k\). The crusial point is that for renormalized Lagrangian (18 , 19 ) and for finite regularization \(\Lambda\) we can derive Schwinger’s equations for any \(\Lambda\). It should be stressed that we assume finite limit for \(Z(\Lambda,...)\) at \(\Lambda \rightarrow \infty\). This assumption is not valid in the perturbation theory where at one loop level \(Z = 1 - ag^2_r \ln(\frac{\Lambda}{\mu})\), \(a > 0\).

5 The infinite renormalization \(Z(\infty, ...) = 0\)↩︎

As an example consider the WZ model with three chiral superfields \(\phi_k(x, \theta) = \phi_{k}(x) + \sqrt{2}\psi_k(x)\theta + F_k(x)\theta\theta\) (\(k = 1,2,3\)) and with the superpotential \[W = W_2 + W_3 \,, \label{infinite1a}\tag{37}\] \[W_2 = \frac{m}{2}(\phi^2_1(x,\theta) + \phi^2_2(x, \theta) + \phi^2_3(x, \theta)) \,, \label{infinite01}\tag{38}\] \[W_3 = g\phi_1(x, \theta)\phi_2(x, \theta)\phi_3(x,\theta) \,. \label{infinite1}\tag{39}\] The kinetic term has standard form \[L_{0} = \int d^2\theta d^2{\bar{\theta}} \sum_{k=1}^3 \phi^*_k(x, \theta, \bar{\theta})\phi_k(x, \theta, \bar{\theta}) \,, \label{infinite1b}\tag{40}\] The Lagrangian with the superpotential (37 , 38 , 39 ) is invariant under the discrete transformations \(\phi_k \rightarrow \phi_l\), \(\phi_l \rightarrow \phi_k\). As a consequence the counterterm \[\Delta L = \int d^2\theta d^2{\bar{\theta}} \sum_{k=1}^{3}( Z(\Lambda, g_r, \mu, m_r) - 1)\phi^{*}_k(x, \theta, \bar{\theta}) \phi_k(x, \theta, \bar{\theta}) \label{infinite2}\tag{41}\] makes the Green’s functions ultraviolet finite in each order of the perturbation theory. Here we use the ultraviolet supersymmetric regularization with three additional chiral superfields \(\Phi_k(x, \theta)\) with additional superpotential \(\Delta W = \sum_{k=1}^3 \frac{\Lambda}{2}\Phi_k(x,\theta)\Phi_k(x, \theta)\) and kinetic term \(L_{0\Phi} = \int d^2\theta d^2{\bar{\theta}} \sum_{k=1}^3 \Phi^*_k(x, \theta, \bar{\theta})\Phi_k(x, \theta, \bar{\theta})\). The chiral superfields \(\Phi_k(x, \theta)\) describe massive scalar and Majorana fields with a mass \(\Lambda\). The regularization with additional chiral superfields \(\Phi_k(x, \theta)\) consists in the replacements \(\phi_k(x, \theta) \rightarrow \phi_k(x, \theta) + i \Phi_k(x, \theta), ~~ \phi_k^{*}(x, \bar{\theta}) \rightarrow \phi_k^{*}(x, \bar{\theta}) + i \Phi_k^{*}(x, \bar{\theta})\) for the superpotential \(W_3 = g\phi_1(x, \theta)\phi_2(x, \theta)\phi_3(x,\theta)\) and \(W_3^* = g\phi^*_1(x, \bar{\theta})\phi_2^*, \bar{\theta})\phi_3^*,\bar{\theta})\). The regularizarion with the introduction of additional massive chiral superfields \(\Phi_k(x, \theta)\) preserves the supersymmetry and at the level of the absence of the interaction \(g = 0\) there are no ghost states in the spectrum. However the interaction Lagrangian is not hermitean that leads to the violation of the unitarity for regularized Lagrangian. As a consequence of the Schwinger’s equations we find that \[- Z(\Lambda,...) <0|T(\phi^*_{r1}(x, \Lambda), F^*_{r1}(y, \Lambda)|0> = m_r<0|T(\phi^{*}_{r1}(x, \Lambda), \phi_{r1}(y, \Lambda)|0> + \label{Infinite4}\tag{42}\] \[g_r <0|T(\phi^{*}_{r1}(x, \Lambda), (\phi_{r2}(y, \Lambda)\phi_{r3}(y, \Lambda))_{reg})|0>,\]

where \(\phi_{rk}(x, \Lambda) = Z^{-1/2}( \Lambda, ...)\phi_k(x)\), \(m_r = Z(\Lambda, ...)m\), \(g_r = Z^{3/2}(\Lambda, ...)g\), \((\phi_{r2}(y, \Lambda)\phi_{r3}(y, \Lambda))_{reg} = (\phi_{r2}(y) +i\Phi_{r2}(y)) (\phi_{r3}(y)+ i\Phi_{r3}(y) )\).

We assume that the limit \(\Lambda \rightarrow \infty\) exists for the Green’s functions of the renormalized fields \(\phi_{rk}(x, \Lambda)\), namely we assume that \[( \phi_{rk}(x, \Lambda), \psi_{rk}(x, \Lambda), F_{rk}(x, \Lambda) \rightarrow ( \phi_{rk}(x), \psi_{rk}(x) , F_{rk}(x)) ~ at~ \Lambda \rightarrow \infty \, \label{ASSUMP}\tag{43}\] As a consequence of the assumption (43 ) the equation (42 ) takes the form \[m_r<0|T(\phi^{*}_{r1}(x), \phi_{r1}(y)|0> + g_r <0|T(\phi^{*}_{r1}(x), (\phi_{r2}(y)\phi_{r3}(y)))|0> = 0 \,. \label{Infinite4a}\tag{44}\] The analogous relation for the spectral densities of the KL representations for the propagators reads \[m_r\rho_{\phi_{r1}^*, \phi_{r1}}(t) + g_r\rho_{\phi_{r1}^*, \phi_{r 2}\phi_{r 3}}(t) = 0 \,. \label{Infinite7a}\tag{45}\] As a consequence of the commutation relations (23 - 25 ) we find that \[[\frac{\partial\phi^{*}_{r1}(t, \vec{x})}{\partial t}, (\phi_{2r}(t, \vec{y})\phi_{3r}(t, \vec{y}))_{reg}] = 0 \,, \label{Infinite5}\tag{46}\] For the commutation relation (46 ) the equation (31 ) takes the form \[\int_{0}^{\infty}\rho_{\phi^*_{r1}, \phi_{r2}\phi_{r3}} (t)dt = 0 \,. \label{Infinite6}\tag{47}\] As a consequence of (45 ) we find that \[\int_{0}^{\infty}\rho_{\phi^*_{r1}, \phi_{r1}} (t)dt = 0 \,. \label{Infinite66}\tag{48}\] Due to the relation (48 ) and nonegativity \(\rho_{\phi_{1r}^*, \phi_{1r}}(t) \geq 0\) of the spectral density \[\rho_{\phi^*_{r1}, \phi_{r1}} (t) = 0 \,. \label{nuwpgxky}\tag{49}\] The analogous equalities are valid for spectral densities \(\rho_{\phi^*_{r2}, \phi_{r12}} (t)\) and \(\rho_{\phi^*_{r3}, \phi_{r3}} (t)\). It means that the scalar propagators vanish and the spectrum of the model is empty that contradicts to the expectations from the perturbation theory that the spectrum contains the scalar massive state².

Another way to understand the inconsistency of the infinite wave function renormalization \(Z(\infty,...) = 0\) is the following [6]. Take the differential operator in (36 ) to be \(\frac{\delta}{\delta F_{r1}(x)}\frac{\delta}{\delta F^*_{r1}(x)}\). In the limit \(\Lambda \rightarrow \infty\) of the regularization removing one can find [6] that \[<0|T(m_r\phi_{r1}(x) + g_r\phi_{r2}(x)\phi_{r3}(x), m_r\phi^*_{r1}(y) + g_r\phi^*_{r2}(y)\phi^*_{r3}(y)|0> = 0 \,. \label{NIKOLAI1}\tag{50}\] Using the equation (44 ) one can find that \[<0|T(m_r\phi_{r1}(x), m_r\phi^*_{r1}(y))|0> = <0|T( g_r\phi_{r2}(x)\phi_{r3}(x), g_r\phi^*_{r2}(y)\phi^*_{r3}(y)|0> \,. \label{NIKOLAI2}\tag{51}\] Due to the absence of radiative corrections to the superpotential (3 ) \(m_{ij}\phi_i(x, \theta)\phi_j(x, \theta) = m_{rij}\phi_{ri}x, \theta)\phi_{rj}(x, \theta)\). As a consequence the composite renormalized operator \((\phi_{i}(x, \theta) \phi_{j}(x, \theta))_r = \phi_{ri}(x, \theta)\phi_{rj}(x, \theta)\) and the same equality \((\phi_{i}(x)\phi_{j}(x))_r = \phi_{ri}(x)\phi_{rj}(x)\) takes place for scalar fields \(\phi_{ri}(x)\) and \(\phi_{rj}(x)\). Using the KL representation for the equality (51 ) one can find that the spectral densities satisfy the equation \[m^2_r\rho_{\phi_{r1}, \phi^*_{r1}}(t) = g^2_r\rho_{(\phi_1\phi_2)_r,(\phi^*_1\phi^*_2)_r))} \, \label{NIKOLAI3}\tag{52}\] and all terms in (52 ) are well defined within perturbation theory at sufficiently small renormalization coupling constant \(g_r\). For small \(g_r\) we have \(m^2_r\rho_{\phi_{r1}, \phi^*_{r1}}(t) \sim m^2_r\delta(t - m^2_r)\) while the right hand side of the equation (52 ) is equal to \(\frac{g^2_r}{16\pi^2}(1 -\frac{4m^2_r}{t})^{1/2} + O(g^4_r)\). So we find the contradiction. The same contradiction takes place for the equation (45 ).

The limit \(Z(\infty, ...) = 0\) is described by the ultralocal Lagrangian \[L_{ultra} = \int d^2\theta [ (g_r \phi_{r1}(x,\theta)\phi_{r2}(x,\theta)\phi_{r3}(x,\theta) +\frac{m_r}{2} \phi_{r1}^2(x,\theta) + ...] + h.c. \, \label{Infinite9}\tag{53}\] without derivatives. The Lagrangian (53 ) has been considered in ref.[17]. For the Lagrangian (53 ) all scalar Green’s functions vanish. It is easy to prove this fact using the continual integral formalism for the Lagrangian (53 ). Really for the Lagrangian \(L_{utra}\) after the integration over the auxiliary fields \(F(x)\), \(F^*(x)\) we find the delta functions \[\int dF_{rk}(x)dF_{rk}^{*}(x) d...exp(iL_{ultra}) \sim \int d... \delta(m_r\phi_{r1}(x) + g_r\phi_{r2}(x)\phi_{r3}(x))... \,. \label{Infinite9a}\tag{54}\] inside of the integral that allows to calculate the continual integral for \(L_{ultra}\) exactly [17]. The spectrum for the Lagrangian \(L_{ultra}\) is an empty, i.e. it does not contain any physical states.

Note that it is possible also to consider the standard case with single scalar chiral superfield \(\phi(x, \theta)\) and with the superpotential \(W = \frac{m}{2}\phi^2(x, \theta) + \frac{g}{3} \phi^3(x, \theta)\). For such potential the equations (44 , 45 ) are also valid with the replacement \(g_r\phi_{r2}(y)\phi_{r3}(y) \rightarrow g_r\phi^2_r(y)\). In the derivation of the sum rule \(\int_{0}^{\infty}\rho_{\phi^*_{r}, \phi_{r}\phi_{r}} (t)dt = 0\) we have to use in addition the commutation relation (27 ). As in a previous case we find that the infinite renormalization \(Z(\infty, ...) = 0\) is not realized.

6 Finite renormalization \(Z(\infty, ... ) \neq 0\)↩︎

In this section we repeat the main results of [11] where we proved that finite renormalization is not realized in the WZ model with the superpotential \(W = \frac{m}{2} \phi^2(x, \theta) + \frac{g}{3}\phi^3(x, \theta)\). For finite renormalization the ultraviolet asymptotics of the scalar propagator coincides up to some factor with free massless scalar propagator. Due to the relation \(g = g_r Z^{-3/2}(\Lambda,...)\) between bare charge \(g\) and renormalized charge \(g_r\) finite renormalization corresponds to fixed point for the beta function, \(\beta(g_{r, fp}) = 0\). Moreover the ultraviolet behaviour of other Green’s functions coincides with the behaviour of the massless WZ model at fixed point \(g_r = g_{r, fp}\). It is easy to prove this fact [6] using the renormalization group. Namely the renormalization group equation in the Weinberg \(-\) t’Hooft scheme [18], [19]³ for n-point Green’s function \(G_n(p_1, ...,p_n)\) can be written in the form \[[\mu\frac{ \partial}{\partial \mu} + \beta(g_r)\frac{\partial}{\partial g_r} + m_r \gamma_m(g_r) \frac{\partial}{\partial m_r} +n\gamma(g_r)]G_n = 0 \,, \label{eqrg}\tag{55}\] where \(\beta(g_r) = 3 g_r \gamma(g_r)\) and \(\gamma_m(g_r) = 2 \gamma(g_r)\). As a consequence of the relation \(\beta(g_r) = 3g_r \gamma(g_r)\) between the \(\beta\)-function and the anomalous dimension \(\gamma(g_r)\) at fixed point \(g_{r, fp}\) the anomalous dimension \(\gamma(g_{r, fp}) = 0\) and the ultraviolet behaviour of the scalar propagator coincides up to some normalization factor with free scalar propagator. We shall use the Schwinger’s equations () with \(\frac{\delta}{\delta F_r(x)}\). As in the previous section we consider massless WZ model with three chiral scalar superfelds \(\phi_k(x, \theta)\) \((k= 1,2,3)\) and with the superpotential (). Due to the symmetry \(\phi_i(x, \theta) \rightarrow \phi_j(x, \theta )\), \(\phi_j(x, \theta) \rightarrow \phi_i(x, \theta)\) of the superpotential () \(\gamma_{i}(g_r) = \gamma_j(g_r) \equiv \gamma(g_r)\) and \(\beta(g_r) = 3 g_r \gamma (g_r)\). As a consequence of the Schwinger’s equations we find \[Z(\infty,...) \delta(x) + i Z^2(\infty, ...) <0|T(F^{*}_{r1}(x), F_{r1}(0)|0> = ig^2_r <0| T(\phi_{r2}(x)\phi_{r3}( x), \phi^{*}_{r3}(0)\phi^{*}_{r2}(0)> \,. \label{masslesseq}\tag{56}\] The analogous equations take place for the scalar fields \(\phi_{r2}\), \(\phi_{r3}\). In momentum space the equation (56 ) has the form \[Z(\infty,...) + Z^2(\infty,...) D_{F_{r1},F_{r1}^*}(p^2) = g^2_r D_{\phi_{r2}\phi_{r3}, \phi^*_{r3}\phi^*_{r2}}(p^2) \,. \label{massl1}\tag{57}\] As a consequence of the supersymmetric Ward identities [7] - [8] \(D_{F_{r1},F^*_{r1}}(p^2) = p^2 D_{\phi_{r1},\phi^*_{r1}}(p^2)\) the analog of the equation (57 ) for the spectral densities of the KL representations reads \[Z^2(\infty,...) t \rho_{\phi_{r1},\phi^*_{r1}}(t) = g^2_r\rho_{\phi_{r2}\phi_{r3}, \phi^{*}_{r3}\phi^{*}_{r2}}(t) \,. \label{KLrho}\tag{58}\] Due to assumed finiteness of \(Z^{-1}(\infty,...) = \int^{\infty}_{0} \rho_{\phi_{r1},\phi^*_{r1}}(t) dt\) we find that \(lim_{t \rightarrow \infty} (t \rho_{\phi_{r1},\phi^*_{r1}}(t)) = 0\). The spectral density \(g^{2/3}_r \rho_{\phi_{r1},\phi^*_{r1}}(t)\) has zero anomalous dimension and as a consequence of the commutation relations (27 ) \(Z(\infty,...) = (\frac{g_r}{g_{r,fp}})^{2/3}\). The spectral density \(Z(\infty,...)\rho_{\phi_{r1},\phi^*_{r1}}(t)\) can be represented in the form \[Z(\infty,...)\rho_{\phi_{r1},\phi^*_{r1}}(t) = \delta(t) + \frac{1}{t}\Delta \rho(\frac{t}{\Lambda^2}) \,, \label{ZREN}\tag{59}\]

where \(\Lambda\) satisfies the renormalization group equation \((\mu\frac{ \partial}{\partial \mu} + \beta(g_r)\frac{\partial}{\partial g_r})\Lambda =0\) and the \(\Lambda = 0\) corresponds to the fixed point \(g_{r, fp}\) with \(\beta(g_{r,fp}) = 0\). At fixed point \(Z(\infty,...)\rho_{\phi_{r1},\phi^*_{r1}}(t) = \delta(t)\) and the \(lim_{t \rightarrow \infty} \Delta \rho(\frac{t}{\Lambda^2}) = 0\). The right hand side of the equation (58 ) can be represented in the form \[g^2_r\rho_{\phi_{r2}\phi_{r3}, \phi^{*}_{r3}\phi^{*}_{r2}}(t) = Z(\infty,...)\Phi(t/\Lambda^2) \,. \label{KLrho1}\tag{60}\] As it has been mentioned before the case \(\Lambda = 0\) or the limit \(t \rightarrow \infty\) corresponds to the fixed point. According to Pohlmeyer theorem [20] if the propagator of the local scalar field is proportional to the propagator of massless free scalar field then the scalar field is free massless field. As a consequence of Pohlmeyer theorem the right hand side of the equation (58 ) at \(g_r = g_{r,fp}\) is equal to \(g^2_rZ^{-2}(\infty,...) \frac{1}{16\pi^2} \theta(t)\). It means that \[lim_{t\rightarrow \infty} (g^2_r\rho_{\phi_{r2}\phi_{r3}, \phi^{*}_{r3}\phi^{*}_{r2}}(t)) = g^2_{r}Z^{-2}(\infty,...) \frac{1}{16\pi^2} \theta(t) \neq 0 \,. \label{133}\tag{61}\] So we find the contradiction: the left hand side of the equation (58 ) vanishes in the limit \(t \rightarrow \infty\) while the right hand side of the equation (58 ) is different from zero in this limit.

Note that the proof of the absence of fixed point in the WZ model is valid also for the general case. For instance, in full analogy with the previous consideration one can prove that for the WZ model with single chiral superfield \(\phi(x, \theta)\) and with the superpotential \(W_3 = \frac{m}{2}\phi^2(x, \theta) + \frac{g}{3}\phi^3(x, \theta)\) fixed point behaviour is not possible [11].

7 Conclusions↩︎

In this paper we proved that the supersymmetric WZ model is internally contradictory. Our proof is based on the following assumptions:

1. We assumed that the WZ model is the model with positively defined metric of states - there are no ghost states in the spectrum. As a consequence of this assumption we used the KL inequality \(0 \leq Z(\infty, g_r, \mu, m_r) \leq 1\) for the wave function renormalization.

2. In the perturbation theory the Green’s functions become ultraviolet finite by the introduction of the \(\sum_{k=1}^{N} \Delta Z_k\int d^2\theta d^2\bar{\theta} \Phi^*_k(x, \theta, \bar{\theta})\Phi_k(x, \theta, \bar{\theta})\) counterterms. We assumed that this structure of the counterterms is valid irrespective of the perturbation theory.

3. We assumed that commutation relations among generalized coordinates and momenta are valid in the renormalized field theory.

Our proof of the inconsistency of the finite wave function renormalization \(Z(\infty,...) \neq 0\) was based on the use of Polhlmeyer theorem while the proof for the case of infinite wave function renormalization \(Z(\infty,...) = 0\) was based on the use of the commutation relations (23 - 26 ). So our proof excludes \(Z(\infty,...) \neq \pm \infty\). The case \(Z(\infty,...) = \pm \infty\) corresponds to the spectrum with negative norm states (so called Landau pole singularities). For \(Z(\infty, m, \mu, g) = \pm \infty\) the Schwinger’s equations (36 ) for renormalized fields are not well defined and our proof based on the use of Schwinger’s equations does not work. So our results confirm the hypothesis that the spectrum of the WZ model contains the states with negative norm, so called Landau poles. The calculations in leading log approximation confirm this hypothesis.

I am indebted to the collaborators of the INR TH department for useful comments and discussions.

References↩︎