Abstract

The trace anomaly and anomaly-induced action are evaluated for the two-dimensional \(2D\) vector theory with classical conformal symmetry. Implementing local conformal symmetry while preserving the gauge invariance requires either giving up locality of the classical action or, equivalently, introducing an auxiliary scalar field. The two-dimensional limit in such a theory is singular. However, in the dimensional regularization, the limit \(D \to 2\) in the one-loop divergence is smooth. As a result, we arrive at the expression for anomaly, which has a rich general structure, typical for the dimensions \(D \geq 4\). For comparison and completeness, we also evaluate anomalies for conformal scalar and fermions, also in the presence of auxiliary external scalars.

Keywords: Conformal anomaly, two dimensions, gauge field, auxiliary scalar

MSC: 81T50, 81T20, 81T40, 83C47

1 Introduction↩︎

The anomalous violation of local conformal symmetry [1], [2] was discovered in the \(4D\) (four-dimensional) semiclassical gravity. The trace (conformal or Weyl) anomaly plays a central role in many theoretical developments and important physical applications (see, e.g., [3]). On the other hand, the anomaly in \(2D\) (two-dimensional space) represents an independent area of research. One can even say that the trace anomaly in \(2D\) has a very special place in forming the content of modern high energy physics. One of the reasons is a great importance of the Weyl anomaly for string theory [4]–[6]. An important aspect is the Polyakov action, i.e., the first example of what is nowadays called anomaly-induced action. The analog of this action in \(4D\) theory [7], [8] plays an important role in the applications such as Hawking radiation [9] and Starobinsky inflationary model [10] (see earlier work [11] and also [12] for the latest theoretical developments).

For most of the applications, the anomaly-induced action is the compact and useful form of parameterizing the logarithmic form factors that emerge in quantum corrections. This parametrization works well because the leading logarithms are directly related to the UV divergences and may be reduced to the dependence on the conformal factor of the metric [13]. This feature holds independently of the dimension of spacetime.

On the other hand, there are big differences between conformal anomaly in \(2D\) and in higher \(D \geq 4\) even dimensions. One of these differences is that the \(2D\) case is much simpler. Since the \(2D\) metric has only one physical degree of freedom, in the pure metric case, the anomaly induced action does not have an “integration constant”, i.e., the ambiguous term which is an undefined conformal invariant functional. Starting from \(4D\), this term is always present, regardless of whether it has no link to the UV limit of the theory. Another distinction concerns the vector field model. The classical action of gauge fields is conformal invariant only in \(4D\), such that the \(2D\) anomaly consists of the contributions of the scalar and spinor fields, which are conformal invariant in the absence of masses. There is no contribution from vectors in \(D \neq 4\). The last critical difference concerns the existing classification of the possible terms in anomaly [14], [15]. This classification can be systematically obtained based on the theorem proved in [16] for the \(D=4\) case. Let us note that the discussion of the simpler version in [13] can be directly extended to other dimensions. According to this theorem, in the classically conformal theory, the coefficient of the pole in the one-loop divergences satisfies the conformal Noether identity. The terms that obey conformal identity may be of three distinct types. One of those is the “legitimate” conformal invariants (\(c\)-terms). There are also nonconformal structures satisfying the Noether identity (\(N\)-terms). The list of these terms includes the unique topological term and surface terms. In the even dimensions \(D \geq 4\), there are all three types of terms. On the contrary, in \(2D\) there is only the topological term because \(c\)-terms and total derivatives are algebraically impossible. For this reason, the anomaly-induced action reduces to the Polyakov action, for both boson (scalar) and fermion fields [4].

Concerning the gauge fields, there are several ways to conformally extend the vector field model to the dimension \(D \neq 4\). In particular, this can be done by giving up the gauge invariance [17]. The result of this work was used and discussed from different viewpoints in [18] and obtained in a different way in [19]. Indeed, this version of the conformal vector model is not appropriate for discussing anomalies because violating gauge invariance means extra degrees of freedom and possible loss of unitarity. On the other hand, in [19] were proposed a few alternative ways of extending the conformal vector model to \(D \neq 4\). These approaches imply either giving up the locality of the action following an earlier proposal of [20], or introducing an extra scalar field. At the classical level, two of the approaches are equivalent, but in all these models, the limit \(D \to 2\) is singular.

In what follows we shall see that, independent from the classical singularity of the \(D \to 2\), one can formulate a quantum theory of the vector field in the dimension \(D = 2 + \epsilon\), as part of the dimensional regularization scheme. The unexpected output is that the unique singularity in the one-loop divergences is the usual \(1/\epsilon\) pole. As a result, one can consistently evaluate the trace anomaly in the \(2D\) model. This anomaly depends on both metric and the extra scalar field, which is a remnant of the construction of conformal vector theory in \(D \geq 4\). Different from the purely metric case mentioned above, in this case, there are all three possible types of terms in the anomaly. For the sake of completeness, we also consider the \(2D\) one-loop trace anomaly for self-interacting scalar field and for the fermion with Yukawa coupling to external scalar.

The paper is organized as follows. In Sec. 2, we derive the one-loop divergences for scalar and fermion fields in \(2D\). Sec. 3 is devoted to the conformal vector field. We show that introducing an auxiliary scalar field to provide conformal invariance generates scalar-dependent terms in the final expression for the one-loop divergences at \(D \to 2\). The conformal anomaly and the anomaly-induced effective action are discussed in Sec. 4. Finally, in Sec. 5 we draw our conclusions.

2 One loop divergences for scalars and fermions↩︎

Using the heat-kernel technique, the one-loop divergences are related to the first order of the expansion of the evolution operator for the bilinear form of the action [21], \[\begin{align} \hat{H} &=& \mathop{\mathrm{\square}}\,+\, 2\hat{h}^\lambda\nabla_\lambda\,+\, \hat{\Pi}. \label{Hhk} \end{align}\tag{1}\]

The coefficient of the expansion that defines the logarithmic divergences for a two-dimensional theory is given by \[\begin{align} \hat{P} &=& \hat{\Pi} + \dfrac{\hat{1}}{6}R - \nabla_\lambda\hat{h}^\lambda- \hat{h}_\lambda\hat{h}^\lambda. \label{P} \end{align}\tag{2}\] The expression for the divergences at \(D \to 2\), in dimensional regularization, takes the form \[\begin{align} \Gamma^{(1)}_{div} &=& \dfrac{i}{2}\,\,sTr\,\log \,\hat{H}\,=\,-\,\dfrac{\mu^{D-2}}{\epsilon}\int d^D x\sqrt{|g|} \, \,str\,\hat{P}, \label{div} \end{align}\tag{3}\] where \(\epsilon= 2\pi(D-2)\), \(\mu\) is an artificial dimensional parameter, \(\,sTr\,\) and \(\,str\,\) are functional and usual matrix super-traces, taking into account Grassmann parities of quantum fields.

2.1 Scalar field with self-interaction↩︎

Consider a scalar field model with an arbitrary interaction term \[\begin{align} S_{sc} &=& \int d^D x\sqrt{|g|}\,\,\Big\{\dfrac{1}{2}(\nabla\varphi)^2 - \eta^2 V(\varphi)\Big\}, \label{actscal} \end{align}\tag{4}\] where we use the condensed notation \((\nabla\varphi)^2= g^{\alpha\beta} \partial_\alpha\varphi\partial_\beta\varphi\), \(V(\varphi)\) is a potential function, and \(\eta\) is an external scalar field. We consider the \(D\)-dimensional spacetime for the reason of convenience. The conformal transformations of metric and two scalars follow the rules \[\begin{align} g_{\mu\nu}=e^{2\sigma}\,\bar{g}_{\mu\nu}, \qquad \varphi=\bar{\varphi}\,e^{\frac{2-D}{2}\sigma}, \qquad \eta=e^{-\sigma}\,\bar{\eta}\,, \qquad \sigma=\sigma(x). \label{contransscal} \end{align}\tag{5}\] Using the background field method, the bilinear form of the action (4 ) corresponds to the second order of the expansion in the quantum field \(\chi\), where \[\begin{align} \varphi\rightarrow \varphi' = \varphi+\chi, \label{bfm-scal} \end{align}\tag{6}\] with \(\varphi\) being the background field. The result is \[\begin{align} \hat{H} &=& \mathop{\mathrm{\square}}\,+\,\eta^2V''(\varphi), \end{align}\] where \(V''(\varphi)\) is the second derivative of the potential. The divergences are expressed as \[\begin{align} \Gamma^{(1)}_{\rm div\,sc} &=& -\,\dfrac{\mu^{D-2}}{\epsilon} \int d^D x\sqrt{|g|}\, \Big\{\,\, \dfrac{1}{3}R + 2\eta^2V''(\varphi)\,\Big\}. \label{divsscal} \end{align}\tag{7}\] It is easy to see that the integrand of this expression includes the usual topological term \(R\) and also the \(c\)-term related to \(V''(\varphi)\). The last property holds after taking the limit \(D\to 2\) in the integrand, as it is expected in the general framework [16].

2.2 Fermions↩︎

The fermionic action of our interest has a background scalar field \(\eta(x)\), \[\begin{align} S_{fer} &=& \int d^D x \sqrt{|g|}\,\,i\, \bar{\Psi} \big(\gamma^\mu\nabla_\mu - ih\,\eta\big) \Psi \,, \label{actferm} \end{align}\tag{8}\] where \(h\) is the Yukawa coupling. The conformal transformation rule for \(\eta\) is the same as for an external scalar in (5 ), \[\begin{align} g_{\mu\nu}=e^{2\sigma}\,\bar{g}_{\mu\nu}, \qquad \Psi=e^{\frac{1-D}{2}\sigma}\,\bar{\Psi}, \qquad \eta=e^{-\sigma}\,\bar{\eta}. \label{confer} \end{align}\tag{9}\] The operator of type 1 can be derived by the doubling of the Hessian, that leads to \[\begin{align} \hat{H} &=& \mathop{\mathrm{\square}}- \dfrac{\hat{1}}{4}\,R - h^2\eta^2\,\hat{1} - 2ih\,\eta\,\gamma^\mu\nabla_\mu - ih\,\gamma^\mu(\nabla_\mu\eta), \label{Hferm} \end{align}\tag{10}\] and the divergences have the form \[\begin{align} \Gamma^{(1)}_{\rm div\,fer} &=& -\, \dfrac{\mu^{D-2}}{\epsilon} \int d^{D} x\sqrt{|g|}\, \Big\{ -\dfrac{1}{6}\,R+2h^2\,\eta^2 \Big\}. \label{divsfer} \end{align}\tag{11}\] Once again, the integrand of the divergent part of the one-loop effective action has the topological term \(R\) and the \(c\)-term \(\eta^2\). Remarkably, the invariance of this term is related to the transformation rule (9 ). As we already noted, this rule is different from the one for the scalar field \(\varphi\) in (5 ), hence \(\varphi\) and \(\eta\) should be regarded as different types of a scalar field.

3 Vector field divergences↩︎

Our starting point is the conformal model of an Abelian vector field constructed in [19]. The invariance is achieved by inserting an external scalar field \(\Phi(x)\) in the classical action \[\begin{align} S_{vec} &=& -\,\dfrac{1}{4}\int d^D x \sqrt{|g|}\, \Phi^\tau F_{\mu\nu}F^{\mu\nu}, \label{actvec} \end{align}\tag{12}\] where the constant \(\tau=\frac{D-4}{D-2}\) provides the return to the usual form of the action in the limit \(D \to 4\). On the other hand, it has a singularity in the limit \(D \to 2\). However, since the calculation for the one-loop divergences is performed in \(D\neq 2\), we can handle this calculation without problems, taking this limit only at the end of the calculation. The action (12 ) is maintaining invariance under the following local conformal transformation: \[\begin{align} g_{\mu\nu}=e^{2\sigma}\,\bar{g}_{\mu\nu}, \quad A_\mu=\bar{A}_\mu, \quad \Phi(x)=\bar{\Phi}(x)\,e^{(2-D)\sigma}. \label{convec} \end{align}\tag{13}\] In what follows, we use the parametrization \[\begin{align} \psi(x)=\Phi^\tau(x), \label{Phipsi} \end{align}\tag{14}\] such that the singularity at \(2D\) becomes implicit.

Since the action 12 is bilinear in quantum field \(A_\mu\), we can skip the use of the background field method. The Faddeev-Popov procedure requires the gauge fixing action \[\begin{align} S_{gf} &=& -\, \dfrac{1}{2}\int d^D x \sqrt{-g}\, \psi\, \big(\nabla_\mu A^\mu\big)^2\,. \label{FP} \end{align}\tag{15}\] In this case, the operator related to the bilinear form can be expressed as \[\begin{align} \hat{H} &=& \delta^\nu_\mu \mathop{\mathrm{\square}}\,-\, R^\nu_\mu \,+\, \psi^{-1} \Big[ \delta^\nu_\mu (\nabla^\lambda\psi)\nabla_\lambda - (\nabla^\nu\psi)\nabla_\mu + (\nabla_\mu \psi)\nabla^\nu \Big]\,. \label{Heq} \end{align}\tag{16}\] This is a particular example of the form (1 ), making the calculation of (2 ) a standard routine. Anyway, we shall present some intermediate formulas.

The overall expression for the divergences is given by \[\begin{align} \Gamma^{(1)}_{\rm div} &=& \dfrac{i}{2}\, \textrm{Tr log}\, \hat{H} \, -\,i\,\textrm{Tr log}\, \hat{H}_{gh}, \end{align}\] where the bilinear form of the action of gauge ghosts is \(\hat{H}_{gh}=\mathop{\mathrm{\square}}\), such that this part of the calculation is pretty much standard.

In the vector field sector, after a small algebra, we get \[\begin{align} \big[\hat{P}\big]^\nu_\mu &=& - \, R^\nu_\mu \,+ \,\dfrac{1}{6}\,\delta^\nu_\mu\,R \, + \,\dfrac{1}{2}\psi^{-2}\,\delta^\nu_\mu\,(\nabla\psi)^2 \,-\,\dfrac{1}{2}\,\psi^{-1}\,\delta^\nu_\mu\,(\mathop{\mathrm{\square}}\psi) \nonumber \\ && + \, \,\,\dfrac{(D-2)}{4}\,\psi^{-2}(\nabla_\mu\psi)(\nabla^\nu\psi). \label{Peq} \end{align}\tag{17}\] Taking the trace and summing up the ghost part, the divergences are \[\begin{align} \Gamma^{(1)}_{\rm div\,vec} \,=\, -\,\dfrac{\mu^{D-2}}{\epsilon} \int d^D x \sqrt{|g|} \,\, \bigg\{\dfrac{(D-8)}{6}R + \dfrac{(3D-2)}{4\psi^2}(\nabla\psi)^2 - \dfrac{D}{2\psi} \mathop{\mathrm{\square}}\psi \bigg\}. \label{divvec1} \end{align}\tag{18}\] In order to verify the conformal invariance of this expression in the limit \(D\to 2\), let us perform a reparametrization of the scalar field \[\begin{align} \psi &=& e^\phi \quad \Longrightarrow \quad \psi^{-1}(\nabla_\lambda\psi)\,=\,\nabla_\lambda\phi, \quad etc. \label{repaphi} \end{align}\tag{19}\] In terms of the new variable, and simplifying the integrand of (18 ) by replacing \(D \to 2\), we arrive at \[\begin{align} \Gamma^{(1)}_{\rm div\,vec} \,=\, -\, \dfrac{\mu^{D-2}}{\epsilon}\int d^2 x \sqrt{|g|}\, \Big\{- R +2(\nabla\phi)^2 - \mathop{\mathrm{\square}}\phi \Big\}. \label{divvec2} \end{align}\tag{20}\] Assuming that the scalar field \(\phi\) does not transform, i.e., \(\phi = \bar{\phi}\), the integrand of the last expression is conformal invariant. In this case, the expression (20 ) has a familiar form. Once again, we meet here more than the unique topological term \(R\), as there are also the \(c\)-term \((\nabla\phi)^2\) and the surface term \(\mathop{\mathrm{\square}}\phi\). Thus, in the presence of an extra scalar, all three types of terms emerge in the UV divergences in the vicinity of the dimension \(2D\).

4 Conformal anomaly and anomaly-induced effective action↩︎

Let us start the derivation of the conformal anomaly from the vector field case. The other two examples can be dealt with similarly. We follow the approach of [2] with small simplifications [22]. The renormalized one-loop effective action of vacuum has the form \[\begin{align} \Gamma_{R} &=& S_{\rm class} \, + \, \Gamma^{(1)} \, + \, \Delta S, \label{GaR} \end{align}\tag{21}\] where \(S_{\rm class}\) is the classical action of vacuum. According to the logic of [16], at the one-loop level \(S_{\rm class}\) may be an algebraic sum of conformal (\(c\) and \(N\)) terms without violating renormalizability. Furthermore, \(\Gamma^{(1)}=\Gamma^{(1)}_{fin}+\Gamma^{(1)}_{div}\) is the sum of divergent and finite parts of the one-loop contributions. Finally, \(\Delta S = - \Gamma^{(1)}_{div}\) are local counterterms, \[\begin{align} \Delta S &=& \frac{1}{D-2} \int_D \sqrt{|g|}\,\, \big\{ aR + \omega(\nabla\phi)^2 + b \mathop{\mathrm{\square}}\phi\big\}, \label{deS} \end{align}\tag{22}\] where we introduced the useful notations \[\begin{align} a = b = - \frac{1}{2\pi}, \quad \omega= \frac{1}{\pi}\,, \quad and \quad \int_D \,=\,\mu^{D-2} \int d^Dx\,\,. \label{intD} \end{align}\tag{23}\] According to [2] (see also [13] for detailed explanations and further discussion), the combination \(S_{\rm class}+\Gamma^{(1)}\) is conformal, so the anomaly can be obtained as \[\begin{align} \mathcal{T} &=& \big< \mathcal{T}^\mu_{\;\mu} \big> \,=\, -\,\dfrac{2}{\sqrt{|g|}}\,g_{\mu\nu} \,\dfrac{\delta\Gamma_R}{\delta g_{\mu\nu}} \,=\, - \dfrac{1}{\sqrt{|\bar{g}|}}\dfrac{\delta}{\delta\sigma} \,\Delta S\big(g_{\mu\nu} = \bar{g}_{\mu\nu} e^{2\sigma} \big)\bigg|, \label{anoma} \end{align}\tag{24}\] where the vertical bar means the limit \(D \to 2,\,\,\sigma\to 0,\,\,\bar{g}_{\mu\nu} \to g_{\mu\nu}\).

Taking into account the transformation rules (see, e.g., [23]) \[\begin{align} && \sqrt{|g|} \big(\nabla\phi\big)^2 \,=\, e^{(D-2)\sigma}\,\sqrt{|\bar{g}|} \big(\bar{\nabla}\bar{\phi}\big)^2, \nonumber \\ && \sqrt{|g|} \mathop{\mathrm{\square}}\phi \,=\, e^{(D-2)\sigma}\,\sqrt{|\bar{g}|} \big[\bar{\mathop{\mathrm{\square}}} \bar{\phi} + (D-2) \big(\bar{\nabla}^\lambda\sigma\big) \big(\bar{\nabla}_\lambda\bar{\phi} \big)\big], \nonumber \\ && \sqrt{|g|} R \,=\, e^{(D-2)\sigma}\,\sqrt{|\bar{g}|}\big[\bar{R} - 2(D-1) \bar{\mathop{\mathrm{\square}}} \sigma - (D-1)(D-2) (\bar{\nabla} \sigma)^2 \big)\big] \label{phitrans} \end{align}\tag{25}\] and using the procedure (24 ), we immediately arrive at³ \[\begin{align} \mathcal{T} &=& aR + \omega(\nabla\phi)^2 + b \mathop{\mathrm{\square}}\phi\,. \label{anomavec} \end{align}\tag{26}\]

To better understand the anomaly (26 ) and compare it to the \(4D\) case, we can derive the anomaly-induced effective action of vacuum. This implies solving the equation \[\begin{align} -\,\dfrac{2}{\sqrt{|g|}}\,g_{\mu\nu} \,\dfrac{\delta\Gamma_{\rm ind}}{\delta g_{\mu\nu}} \,=\, \mathcal{T}\,. \label{anoindac} \end{align}\tag{27}\]

It proves useful to rewrite this equation as we did earlier in (24 ), i.e., \[\begin{align} - \dfrac{1}{\sqrt{|\bar{g}|}} \dfrac{\delta}{\delta\sigma} \,\Gamma_{\rm ind}\big(g_{\mu\nu} = \bar{g}_{\mu\nu} e^{2\sigma} \big)\bigg| &=& \mathcal{T}_a + \mathcal{T}_\omega+ \mathcal{T}_b, \label{anoinsig} \end{align}\tag{28}\] where \(\mathcal{T}_{a,\,\omega,\,b}\) are the three terms in anomaly (26 ). Different from Eq. (24 ), this time vertical bar does not include the limit \(D \to 2\) because equation (28 ) should be solved in \(2D\).

The solution follows the general scheme described in [7] and formulated in a very general form in [24] as part of integrating anomaly in \(6D\). The basic element of this solution is the last formula of (25 ) in the \(2D\) limit, i.e., \[\begin{align} \sqrt{|g|} R \,=\,\sqrt{|\bar{g}|}\big[\bar{R} - 2 \bar{\mathop{\mathrm{\square}}} \sigma\big]. \label{Rtrans2} \end{align}\tag{29}\] Let us start from the \(c\)-term \(\mathcal{T}_\omega\) and guess the corresponding term in induced action, \[\begin{align} \Gamma_\omega &=& \kappa\iint\limits_{x\,y} R_x \,\,\Big(\frac{1}{\mathop{\mathrm{\square}}}\Big)_{x,y}\,(\nabla\phi)_y^2\,, \quad where \quad \int\limits_x = \int d^2x \sqrt{|g(x)|}\,. \label{Gaom} \end{align}\tag{30}\] Since \(\mathop{\mathrm{\square}}\) and \((\nabla\phi)^2\) are conformally covariant objects, using (29 ) one can easily see that (30 ) is linear in \(\sigma\) and arrive at the solution \(\kappa= \omega/2\). In a similar way, we find \[\begin{align} \Gamma_a &=& \frac{a}{4} \iint\limits_{x\,y} R_x \,\Big(\frac{1}{\mathop{\mathrm{\square}}}\Big)_{x,y}\,R_y\,. \label{Gaa} \end{align}\tag{31}\] According to (29 ) the solution for the total derivative \(N\)-term \(\mathcal{T}_b\) is the local functional \[\begin{align} \Gamma_b &=& \frac{b}{2} \int\limits_x \,R \phi. \label{Gab} \end{align}\tag{32}\] The last observation is that including the contributions of scalars (7 ) and fermions (11 ) can be reduced to the redefinition of the coefficients \(a\), \(b\) and introducing generalized \(c\)-term. For \(N_s\) copies of scalars, \(N_f\) copies of fermions and \(N_v\) copies of vectors (12 ), we get \[\begin{align} && a_t = \frac{1}{2\pi}\Big(\frac{1}{3} N_s - \frac{1}{6} N_f - N_v \Big), \tag{33} \\ && Z = \frac{1}{\pi}\,\big(V'' N_s + h^2 N_f\big) \eta^2 + \frac{\omega N_v}{2}(\nabla\phi)^2. \tag{34} \end{align}\] Using these quantities, the anomaly-induced action can be cast in the form \[\begin{align} \Gamma_{\rm ind} \,=\, S_c \,+\, \frac{a_t}{4} \iint\limits_{x\,y} R_x \,\Big(\frac{1}{\mathop{\mathrm{\square}}}\Big)_{x,y}\,R_y \,+\, \frac{1}{2} \iint\limits_{x\,y} R_x \,\Big(\frac{1}{\mathop{\mathrm{\square}}}\Big)_{x,y}Z_y \,+\, \frac{bN_v}{2} \int\limits_x \,R \phi. \quad \label{Gaind} \end{align}\tag{35}\] The remarkable special feature of this expression is the presence of an arbitrary conformally invariant functional \(S_c\). In the purely metric background this term is absent, but in the presence of external scalars \(\eta\) and \(\phi\), there is an infinite amount of terms which can constitute \(S_c\). For instance, such a term can be given by a power series of the terms \(\int (\nabla\phi)^2 \big[\mathop{\mathrm{\square}}^{-1}(\nabla\phi)^2 \big]^n\). Thus, the anomaly-induced action in \(2D\) may be qualitatively similar to the analogous constructions in the even dimensions \(D \geq 4\).

To complete the anomaly-induced approach, we reformulate the nonlocal induced action in the local form, by means of two auxiliary fields, as it is customary in \(D \geq 4\) [13], [24], [25]. One can note that the Polyakov action needs only one such scalar field and the first work on the anomaly-induced action in \(4D\) [7] also used one scalar.

The first step is to rewrite (35 ) in the symmetric, or Gaussian form, as \[\begin{align} && \Gamma_{\rm ind} \,=\, S_c \,+\, \frac{a_t}{4} \iint\limits_{x\,y} \bigg( R + \frac{1}{a_t}Z\bigg)_x \,\Big(\frac{1}{\mathop{\mathrm{\square}}}\Big)_{x,y}\, \bigg( R + \frac{1}{a_t}Z\bigg)_y \nonumber \\ && \qquad \qquad -\,\, \frac{1}{4a_t} \iint\limits_{x\,y} Z_x\,\Big(\frac{1}{\mathop{\mathrm{\square}}}\Big)_{x,y}\,Z_y \,+\, \frac{bN_v}{2} \int\limits_x \,R \phi. \quad \label{Gaindsym} \end{align}\tag{36}\] The explicit form of the local version of the action depends on the sign of the coefficient \(a_t\) in (33 ). Since our main interest is about the vector field, we arrive at the following local form of induced action: \[\begin{align} && \Gamma_{\rm ind\,loc} \,=\, S_c \,+\, \frac{bN_v}{2} \int\limits_x \,R \phi \,+\, \int_x\,\bigg\{ \frac{1}{2} \chi \mathop{\mathrm{\square}}\chi - \frac{1}{2} \psi \mathop{\mathrm{\square}}\psi \nonumber \\ && \qquad \qquad - \,\,\sqrt{- \,\frac{a_t}{2}} \chi \Big(R + \frac{1}{a_t}Z \Big) \,+ \,\sqrt{- \,\frac{1}{2a_t}} \,\psi Z \bigg\}. \quad \label{Gaindloca} \end{align}\tag{37}\] For the applications, the boundary and/or initial conditions which define the Green functions of the operator \(\mathop{\mathrm{\square}}\) in (35 ), are mapped into the corresponding conditions for the new fields \(\chi\) and \(\psi\) in the local version of the action (37 ). The terms with the field \(\psi\) are conformal, but our experience with the \(4D\) applications (see the discussion and further references in [22]) shows that it is useful not to include this part in \(S_c\), since only with these terms the induced action provides good correspondence to the UV leading logarithms.

5 Conclusions↩︎

The trace anomaly in a \(2D\) space is one of the best studied subjects in quantum field theory. However, all the existing results concern contributions of scalars (including sigma models) and fermions, required by supersymmetry. Until recently, inclusion of a vector field was not possible because this field is conformally invariant only in \(4D\), but not in \(2D\).

In this letter, we use the previously developed model [19] of conformal vector field in \(D \neq 4\) to explore the vector anomaly in \(2D\). One of the conformal models constructed in this work is especially simple, as the symmetry does not require violation of gauge invariance or introducing a nonlocal classical action. Instead, there is an auxiliary scalar field. In the original formulation of [19], the action of this model is singular in the limit \(D \to 2\). However, using dimensional regularization, we can perform calculations without taking this limit. As a result, one loop quantum corrections produce a conformal anomaly which is not degenerate in \(2D\). The anomaly depends not only on the metric but also on the auxiliary scalar field, producing interesting new structures.

The vector model in \(2D\) does not have physical degrees of freedom, different from scalars or fermions. An interesting detail is that the sign of the vector contribution to the traditional anomaly (33 ) is the same as for the spinors, opposite to the one for scalars. On the other hand, the presence of an external scalar makes the structure of the anomaly richer, similar to the one in the even dimension \(D \geq 4\). As a result, the \(2D\) anomaly possesses all possible terms of the general classification [15]. We have shown that the corresponding anomaly-induced action gains the full form with two different Green functions, or with two auxiliary scalar fields, typical for the dimension four [25] and beyond [24].

Acknowledgements↩︎

S.W.P.O. is grateful to Coordenação de Aperfeiçoamento de Pessoal de Nı́vel Superior – CAPES (Brazil) for supporting his Ph.D. project, and to Fundação de Amparo à Pesquisa e Inovação do Espı́rito Santo – FAPES (Brazil) for supporting this project. The work of I. Sh. is partially supported by Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico - CNPq under the grant 305122/2023-1.

References↩︎