Carroll limit of one-loop effective action

Carroll symmetries were introduced long ago [1, 2] but they received due attention much later after the discovery of corresponding conformal symmetries [3, 4, 5]. Over the past few years considerable progress in understanding of various aspects of Carroll theories has been achieved. To give just a few examples, we mention the works [6, 7, 8] on the Carroll structure at null infinity, and on propagating Carroll fields (Carroll swiftons) [9].

Several works were dedicated to quantum Carroll theories [10, 11, 12, 13, 14, 15, 16, 17, 18, 19] mostly studying quantization, Carrollian conformal field theories, and correlation functions, but also quantum Hall effect [20], representation structure and thermodynamics [21], renormalisation group flow as an origin of Carrollian symmetries [22], and Hawking radiation [23]. It was shown, at some simple examples, that in Carroll limit the scalar partition function is divergent [24].

The purpose of present work is to perform a detailed analysis of divergent and finite terms in the Carroll (magnetic) magnetic limit of the one-loop effective action in scalar theories in various dimensions. We will consider static but otherwise quite general backgrounds.

Carroll limit is basically an ultrarelativistic limit $c\to 0$, where $c$ is the speed of light. Carroll geometry is defined through a vector $v^{\mu}$ which, roughly speaking, fixes the time direction, and a spatial metric $h_{\mu\nu}$. Together with the fields $\tau_{\mu}$ and $h^{\mu\nu}$, they satisfy the relations

Local Carroll boosts with the parameter $\Lambda_{\mu}$ act as follows

The parameter $\Lambda$ satisfies the restriction $\Lambda_{\mu}v^{\mu}=0$. It is easy to check that the volume element

is invariant under Carroll boosts.

In this work, we will be interested in scalar field theories. The so called electric and magnetic Carroll scalar theories were defined as contractions from Lorentz-invariant theories in [25] (see also an earlier work [26]) while conformal Carroll scalar theories were constructed in [27, 28]. We will consider magnetic theories only. The action for magnetic scalar on an $n+1$ dimensional manifold $\mathcal{M}$ reads

We work in the Euclidean signature. This action is invariant under local Carroll boosts if the rules (2) are supplemented by a suitable transformation rule for $\Pi$. $V$ is a potential. The field $\Pi$ generates a constraint

With this constraint, the action becomes

This action is Carroll boost invariant provide $\phi$ satisfies (5).

Consider an “electromagnetic” scalar action [29]

where $\kappa$ is a positive coupling constant having the meaning of $1/c^{2}$. In the limit $\kappa\to\infty$ the time variation of $\phi$ are suppressed so that one gets the dynamics described by (7) with the constraint(5). The purpose of present work is to see what happens in quantum theory in this limit.

Since the $\kappa\to\infty$ limit means imposing the constraint (5), one may expect that in this limit also the quantum effective action will be defined through an $n$ dimensional theory living on a hypersurface of constant $y$, where $y$ is defined through $\partial_{y}=v^{\mu}\partial_{\mu}$. However, in this singular limit the determinant of $n+1$ dimensional operator may keep the memory on the way the constant $y$ hypersurfaces are embedded in $\mathcal{M}$ and on the function $v^{\mu}$. (This happens, e.g., with Faddeev–Popov determinants in noncovariant gauges, see [30].) Moreover, even though the contribution of each $y$-dependent mode is suppresses at the $\kappa\to\infty$ limit, there are infinitely many such modes. Their collective contribution may be non-vanishing and even divergent.

For a generic scalar field theory the one-loop effective action is given by

where $L$ is an operator of Laplace type appearing in the quadratic form of classical action. We use the $\zeta$ function regularization and write the regularized determinant as

where $s$ is a complex regularization parameter and

is the spectral $\zeta$ function of $L$. The regularized effective is $W_{s}=\tfrac{1}{2}(\ln\det L)_{s}$. The physical limit corresponds to an analytic continuation to the point $s=0$. Near this point,

where $\gamma_{\mathrm{E}}$ is the Euler constant.

The heat kernel expansion [31] will be a useful tool. If $f$ is a smooth function, there is an asymptotic expansion of the smeared heat kernel at $t\to+0$,

Here $m$ is dimension of the base manifold. For $f=1$ we will use a shorthand notation $a_{k}(L)\equiv a_{k}(1,L)$. Because of the relation

the pole term in (11) is given by $a_{m}(L)$.

This work is organized as follows. We consider a scalar theory on a product manifold $\mathcal{M}=\widetilde{\mathcal{M}}\times S^{1}$ with static background fields. We define an operator $\bar{L}$ acting on quantum fluctuations and apply a local scale transformation to transform this operator to another operator $\widetilde{L}$. This latter operator is a sum of two commuting operators, one being independent of $\kappa$ and the other having an exactly known spectrum. The difference between effective actions for $\bar{L}$ and $\widetilde{L}$ is given by scale anomaly. These steps are done in Section II.1. The $\kappa$ expansion of the effective action for $\widetilde{L}$ is constructed in Section II.3 where the divergences in this expansion are analysed. Our most important observation is that all divergences in the $\kappa\to\infty$ expansion may be removed by local counterterms. A simple example in two dimensions is considered in Section II.3. The last Section III contains concluding remarks.

In this paper, we considered a Carroll magnetic limit $\kappa\to\infty$ of the one-loop effective action in an $n+1$ dimensional scalar theory. The coefficients in $\kappa$ expansion are expressed through spectral characteristics (heat kernel coefficients and $\zeta^{\prime}(0)$) of an $n$ dimensional operator $L_{n}$. The operator $L_{n}$ is not the kinetic operator $\bar{L}$ of the original theory with the term $(v^{\mu}\partial_{\mu})^{2}$ neglected, as one might have expected naively, but rather an operator obtained through a local scale transformation thereof. We observed, that the usual $1/s$ divergence of $\zeta$ regularization vanishes in the $\kappa\to\infty$ limit. Instead, several divergent terms with positive powers of $\kappa$ or with $\ln(\kappa)$ appear. All of them are local. Thus, the effective action can be made finite in Carroll limit by adding a finite number of local counterterms.

Our starting point, the action (7), is not Carroll boost invariant for finite $\kappa$. Besides, we fixed $h^{\mu\nu}$ to be an (inverse) metric over $\widetilde{\mathcal{M}}$ thus breaking the Carroll boost invariance. Somewhat surprisingly, the effective action appeared to be invariant under infinitesimal Carroll boosts, see discussion around Eq. (18). It may be, that this invariance is accidental following from a rather specific choice of the background. A more thorough study of Carroll boost invariance requires lifting the assumption made above and considering generic $h^{\mu\nu}$. Such analysis will be considerably more difficult than the one presented in this work. Technical tools like the ones which have been developed for high temperature limits on stationary (rather than static) backgrounds [37] will be useful.

A word of warning is in order. The $\kappa$ expansion of effective action has to be used with care. Since the effective metric $g$ depends on $\kappa$, the integrated quantities (like the effective action) can be of a different order in $\kappa$ than corresponding local quantities (like the stress energy tensor). Besides, the components of tensors with upper and lower indices can contain different powers of $\kappa$. An example of such situation can be found in the paper [23] where the Hawking effect on for a 2D Carroll–Schwarzschild black hole [38] was studied. As usual for massless two-dimensional theories [39], the whole information on Hawking effect is contained in the scale anomaly, which gives a contribution to the effective action suppressed by $\kappa^{-1/2}$. However, the Carroll limit of stress energy tensor contained both finite and divergent contributions. In other words, to analyse local quantities produced by taking variation derivatives of the effective action one may need to use more terms in the $\kappa$ expansion. Such terms are easily obtained by extending the summation ranges in the sums (30).

An extension of our results to more general and even $\kappa$-dependent interaction $V(\phi)$ is straightforward. It is sufficient to re-expand all terms in (30) in powers of $\kappa$. This is easy to do with the heat kernel coefficients, but may be more tedious in the case of $\zeta^{\prime}(0,L_{n})$. An expansion of the effective action for the fields of other spins can be obtained along similar lines. It seems promising to apply our methods to the $c\to 0$ limit in Conformal Field Theories [40] and to quantum theories where only external lines of Feynman diagrams are Carrollian [41].