Weyl Consistency Conditions in Non-Relativistic Quantum Field Theory

Weyl consistency conditions have been used in unitary relativistic quantum field theory to impose constraints on the renormalization group flow of certain quantities. We classify the Weyl anomalies and their renormalization scheme ambiguities for generic non-relativistic theories in 2+1 dimensions with anisotropic scaling exponent z=2; the extension to other values of z are discussed as well. We give the consistency conditions among these anomalies. As an application we find several candidates for a C-theorem. We comment on possible candidates for a $C$-theorem in higher dimensions.


I. INTRODUCTION
Aspects of the behavior of systems at criticality are accessible through renormalization group (RG) methods. Famously, most critical exponents are determined by a few anomalous dimensions of operators. However, additional information, such as dynamical (or anisotropic) exponents and amplitude relations can be accessed via renormalization group methods near but not strictly at criticality. Far away from critical points there are often other methods, e.g., mean field approximation, that can give more detailed information. The renormalization group used away from critical points can valuably bridge the gap between these regions.
Systems of non-relativistic particles at unitarity, in which the S-wave scattering length diverges, |a| → ∞, exhibit non-relativistic conformal symmetry. Ultracold atom gas experiments have renewed interest in study of such theories. In these experiments one can freely tune the S-wave scattering length along an RG flow [1,2]: at a −1 = −∞ the system is a BCS superfluid while at a −1 = ∞ it is a BEC superfluid. The BCS-BEC crossover, at a −1 = 0, is precisely the unitarity limit, exhibiting conformal symmetry. This is a regime where universality is expected, with features independent of any microscopic details of the atomic interactions [3]. Other examples of nonrelativistic systems with accidentally large scattering cross section include few nucleon systems like the deuteron [4] and several atomic systems, including 85 Rb [5], 138 Cs [6], 39 K [7].
In the context of critical dynamics the response function exhibits dynamical scaling. This is characterized by a dynamical scaling exponent which characterizes anisotropic scaling in the time domain. There has been recent interest in anisotropic scaling in systems that are non-covariant extensions of relativistic systems. The ultraviolet divergences in quantized Einstein gravity are softened if the theory is modified by inclusion of higher derivative terms in the Lagrangian. Since time derivatives higher than order 2 lead to the presence of ghosts, 1 Horava suggested extending Einstein gravity by terms with higher spatial derivatives but only order-2 time derivatives [13]. The mismatch in the number of spatial versus time derivatives is a version of anisotropic scaling, similar to that found in the non-relativistic context. This has motivated studies of extensions of relativistic quantum field theories that exhibit anisotropic scaling at short distances. Independently, motivated by the study of Lorentz violating theories of elementary particle interactions [14], Anselmi found a critical point with exact anisotropic scaling, a so-called Lifshitz fixed point, in his studies of renormalization properties of interacting scalar field theories [15]; see Refs. [16,17] for the case 1 Generically, the S-matrix in models with ghosts is not unitary. However, under certain conditions on the spectrum of ghosts and the nature of their interactions, a unitary S-matrix is possible [8][9][10][11]. In theories of gravity Hawking and Hertog have proposed that ghosts lead to unitarity violation at short distances, and unitarity is a long-distance emergent phenomenon [12]. of gauge theories. Anomalous breaking of anisotropic scaling symmetry in the quantum Lifshitz model has been studied in Ref. [18][19][20][21][22]; see also Ref. [23] for an analysis using holographic methods.
Wess-Zumino consistency conditions for Weyl transformations have been used in unitary relativistic quantum field theory to impose constraints on the renormalization group flow of Weyl anomalies [24]. In 1+1 dimensions a combination of these anomalies gives Zamolodchikov's Cfunction [25], that famously decreases monotonically along flows towards long distances, is stationary at fixed points and equals the central charge of the 2D conformal field theory at the fixed point boundaries of the flow. Weyl consistency conditions can in fact be used to recover this result [24].
Along the same lines, in 3+1 dimensions Weyl consistency conditions can be used to show that a quantityã satisfies where µ is the renormalization group scale, increasing towards short distances. The equation shows that at fixed points, characterized by µ dg α /dµ ≡ β α = 0,ã is stationary. It can be shown in perturbation theory that H αβ is a positive definite symmetric matrix [26]. By construction the quantityã is, at fixed points, the conformal anomaly a of Cardy, associated with the Euler density conformal anomaly when the theory is placed in a curved background [27]. This is then a 4-dimensional generalization of Zamolodchikov's C function, at least in perturbation theory. Going beyond 4 dimensions, Weyl consistency conditions can be used to show that in d = 2n dimensions there is a natural quantity that satisfies (1), and that this quantity is at fixed points the anomaly associated with the d-dimensional Euler density [28]. Concerns about the viability of a C-theorem in 6-dimensions were raised by explicit computations of "metric" H αβ in perturbation theory [29][30][31]. However it was discovered in Ref. [32] that there exists a one parameter family of extensions of the the quantityã of Ref. [28] that obey a C-theorem perturbatively.
Weyl consistency conditions can also be used to constrain anomalies in non-relativistic field theories. The constraints imposed at fixed points have been studied in Ref. [18] for models with anisotropic scaling exponent z = 2 in 2-spatial dimensions; see Refs. [33,34] for studies of the Weyl anomaly at d = 4, z = 3 and d = 6. Here we investigate constraints imposed along renormalization group flows. We recover the results of [18] by approaching the critical points along the flows.
As mentioned above, there are questions that can only be accessed through the renormalization group methods applied to flows, away from fixed points. The additional information obtained from consideration of Weyl consistency conditions on flows can be used to ask a number of questions.
For example, we may ask if there is a suitable candidate for a C-theorem.
A related issue is the possibility of recursive renormalization group flows. Recursive flows in the perturbative regime have been found in several examples in 4 − ǫ and in 4 dimensional relativistic quantum field theory [35][36][37][38][39][40]. Since Weyl consistency conditions implyã does not increase along RG-flows it must be thatã remains constant along recursive flows. This can be shown directly, that is, without reference to the monotonicity of the flow; see [40]. In fact one can show that on recursive flows all physical quantities, not justã, remain constant: the recursive flow behaves exactly the same as a single fixed point. This is as it should be: the monotonicity of the flow of a implies that limit cycles do not exist in any physically meaningful sense [41,42]; in fact, they may be removed by a field and coupling constant redefinition. However, it is well known that bona-fide renormalization group limit cycles exist in some non-relativistic theories [43][44][45]. The C- The paper is organized as follows. In Sec.II we set-up the computation, using a background metric and space and time dependent coupling constants that act as sources of marginal operators. In the section we also clarify the relation between the dynamical exponent and the classical anisotropic exponent. We then use this formalism in Sec. III where we analyze the consistency conditions for the case of 2-spatial dimensions and anisotropic exponent z = 2. The Weyl consistency conditions and scheme dependent ambiguities are lengthy, so they are collected in Apps. A and B. In Sec. IV we explore the case of arbitrary z, extending some of the results of the previous section and in Sec. V we propose a candidate C-theorem for any even spatial dimension. We offer some general conclusions and review our results in Sec. VI. There is no trace anomaly equation for the case of zero spatial derivatives, that is, particle quantum mechanics; we comment on this, and present a simple but useful theorem that does apply in this case, in the final appendix, App. C.

II. GENERALITIES
We consider non-relativistic (NR) field theories with point-like interactions. Although not necessary for the computation of Weyl consistency conditions, it is convenient to keep in mind a Lagrangian description of the model. The Lagrangian density L = L(φ, m, g) is a function of fields φ(t, x), mass parameters m and coupling constants g that parametrize interaction strengths. We restrict our attention to models for which the action integral, remains invariant under the rescaling that is, Here ∆ is the matrix of canonical dimensions of the fields φ. In a multi-field model the anisotropic scaling exponent z is common to all fields. Moreover, assuming that the kinetic term in L is local, so that it entails powers of derivative operators, z counts the mismatch in the number of time derivatives and spatial derivatives. In the most common cases there is a single time derivative and z spatial derivatives so that z is an integer.
For a simple example, useful to keep in mind for orientation, the action for a single complex scalar field with anisotropic scaling z in d dimensions is given by where z is an even integer so that the Lagrangian density is local. If N = 1 + z/d the scaling property (2) holds with ∆ = d/2 (alternatively, if N ∈ Z, then z = d(N − 1) ∈ dZ). When (2) holds the coupling constant g is dimensionless. The mass parameters m have dimensions of T /L z , where T and L are time and space dimensions, respectively. One may use the mass parameter to measure time in units of z-powers of length, and this can be implemented by absorbing m into a redefinition, t = mt. In multi-field models one can arbitrarily choose one of the masses to give the conversion factor and then the independent mass ratios are dimensionless parameters of the model.
In models that satisfy the scaling property (2), these mass ratios together with the coefficients of interaction terms comprise the set of dimensionless couplings that we denote by g α below.
The above setup is appropriate for studies of, say, quantum criticality. However the calculations we present are applicable to studies of thermal systems in equilibrium since the imaginary time version of the action integral is equivalent to an energy functional in d + 1 spatial dimensions.
Taking t = −iy in the example of Eq. (3) the corresponding energy integral is The short distance divergences encountered in these models need to be regularized and renormalized. Although our results do not depend explicitly on the regulator used, it is useful to keep in mind a method like dimensional regularization that retains most symmetries explicitly. Thus we consider NR field theories in 1 + n dimensions, where the spatial dimension n = d − ǫ, with d an integer. Dimensional regularization requires the introduction of a parameter µ with dimensions of inverse length, L −1 . Invariance under (2) is then broken, but can be formally recovered by also scaling µ appropriately, µ → λ −1 µ. For an example, consider the dimensionally regularized version of (3): We have written this in terms of bare field and mass, φ 0 and m 0 , and have given the bare coupling constant explicitly in terms of the renormalized one, g 0 = µ kǫ Z g g. The coefficient k = N − 1 = z/d is dictated by dimensional analysis. It follows that In order to study the response of the system to sources that couple to the operators in the interaction terms of the Lagrangian, we generalize the coupling constants g α to functions of space and time g α (t, x). One can then obtain correlation functions of these operators by taking functional derivatives of the partition function with respect to the space-time dependent couplings, and then setting the coupling functions to constant values. Additional operators of interest are obtained by placing these systems on a curved background, with metric γ µν (t, x). One can then obtain correlations including components of the stress-energy tensor by taking functional derivatives with respect to the metric and evaluating these on a trivial, constant metric. For example, we then can define the components of the symmetric quantum stress energy tensor and finite composite operators in the following way: The square bracket notation in the last term indicates that these are finite operators, possibly differing from O α = ∂L/∂g α by a total derivative term.
Time plays a special role in theories with anisotropic scaling symmetry. Hence, it is useful to assume the background space-time, in addition to being a differential manifold M, carries an extra structure -we can foliate the space-time with a foliation of co-dimension 1. This can be thought of a topological structure on M [13], before any notion of Riemannian metric is introduced on such manifold. Now the co-ordinate transformations that preserve the foliation are of the form: We will also assume the space-time foliation is topologically given by M = R × Σ. The foliation can be given Riemannian structure with three basic objects: h ij , N i and N . This is the ADM decomposition of the metric -one can generally think as writing the metric in terms of lapse and shift functions, N (t, x) and N i (t, x), and a metric on spatial sections, h ij (t, x): Here and below the latin indices run over spatial coordinates, i, j = 1, . . . , d. We assume invariance of the theory under foliation preserving diffeomorphisms. In a non-relativistic set up, it is convenient to remove the shift N i by a foliation preserving map t → τ (t) and x i → ξ i ( x, t). The metric is then given by Once the shift functions are removed the restricted set of diffeomorphisms that do not mix space and time are allowed, t → τ (t) and x i → ξ i (x), so that N i = 0 is preserved.
In Euclidean space, the generating functional of connected Green's functions W is given by The action integral for these models is generically of the form where h = det(h ij ). We have denoted by L 0 the Lagrangian density with bare fields and couplings as arguments; these are to be expressed in terms of the renormalized fields and couplings, so as to render the functional integral finite. The term ∆S contains additional counter-terms that are solely functionals of g α and γ µν that are also required in order to render W finite. In a curved background the scaling (2) can be rephrased in terms of a transformation of the metric, Then the generalization of the formal invariance of Eq. (5) is for a suitable matrix of canonical dimensions ∆ 0 of the bare fields (appropriate to n = d − ǫ spatial dimensions).
We assume that when introducing a curved background the action integral is suitably modified so that the formal symmetry of Eq. (13) holds locally, that is, it holds when replacing λ → exp(−σ( x, t)). The modification to the action integral consists of additional terms that couple the fields φ to the background curvature.
For example, the model in Eq. (4) for z = 2 is modified to include, in addition to coupling to a background metric, additional terms Here K ij = 1 2 ∂ t h ij /N is the extrinsic curvature of the t = constant hypersurfaces in the N i = 0 gauge and K = h ij K ij (with h ij the inverse of the metric h ij ), and R is the d-dimensional Ricci scalar for the metric h ij . Under the transformation (12) with λ = exp(−σ) one has K → e 2σ (K + the action integral remains invariant. Thus, we have a one parameter family of parameters that preserves invariance of the action under anisotropic scaling. For arbitrary even z and arbitrary spatial dimension n, in the example (4) we first integrate by parts the spatial covariant derivatives: Then we replace the operator (∇ 2 ) Hence, under the Weyl rescaling h ij → e −2σ h ij , N → e −zσ N and φ → e n 2 σ φ we have following, transforming covariantly For z = 2, this construction gives This solves Eq. (14) with The extra freedom for z = 2 arises from the fact that is Weyl invariant. This special invariant quantity is available only for z = 2.
Having constructed a classically Weyl invariant curved space action, we have thatW = W − W c.t. = W + ∆S is invariant under these local transformations: We have suppressed the explicit dependence on space and time and have assumed the only dependence on the renormalization scale µ is implicitly through the couplings: using µ-independence The generating functional W is not invariant in the sense of Eq. (21). The anomalous variation of W arises purely from the counter-terms: under an infinitesimal transformation, does not vanish. Using Eqs. (6) and choosing σ to be an infinitesimal local test function, this reads Evaluating at space and time independent coupling constants and on a flat metric, so that the right hand side vanishes, we recognize this as the trace anomaly for NRQFT.
Since the Weyl group is Abelian, consistency conditions follow from requiring that These consistency conditions impose relations on the various anomaly terms on the right hand side of Eq. (22). In the following sections we classify all possible anomaly terms and derive the relations imposed by these conditions.

A. Dynamical exponent
In the theory of critical phenomena the dynamical exponent ζ characterizes how a correlation length scales with time in time dependent correlations. At the classical level (the gaussian fixed point) this just corresponds to the anisotropic exponent z introduced above. To understand the connection between these we must retain explicitly the dependence on the mass parameter(s) m in Eqs. (13) and (21). We consider for simplicity the case of a single mass parameter. In particular, we haveW By dimensional analysis and translational and rotational invariance, the correlator of fundamental fields is given by for some dimensionless function of two arguments, F (x, y). This function is further constrained by the renormalization group equation. At a fixed point, β α = 0, it takes the form where γ m and γ are the mass anomalous dimension and the field anomalous dimension, respectively. These are generally dimensionless functions of the dimensionless coupling constants, g α , here evaluated at their fixed point values, say, g α * . It follows that Here µ 0 is a reference renormalization point and f is a dimensionless function of one variable. This shows that at the fixed point the fields scale with dimension ∆ + γ and the dynamical exponent is It is important to understand that while ζ can be thought of as running along flows, the exponent z is fixed to its classical (gaussian fixed point) value.
As an example consider the following Lagrangian for a z = 2 theory in 4 + 1 dimensions: The renormalization factors in dimensional regularization in n + 1 dimensions, with n = 4 − ǫ, have the following form: FIG. 1. Self energy correction to propagator at one loop where the residues a X n are functions of the renormalized coupling constant g. Independence of the bare parameters on the scale µ requires It follows that At one loop the self-energy correction to the propagator, represented by the Feynman diagram in Fig. 1, reads where the propagator is given by The integration over p 0 and then over p gives where the ellipses stand for finite terms. We read off Form which it follows that

A. Listing out terms
We first consider 2 + 1 NRCFT with z = 2. It is convenient to catalogue the possible terms on the right hand side of Eq. (22) by the number of space and time derivatives acting on the metric, the couplings and the transformation parameter σ. Rotational invariance implies that space derivatives always appear in contracted pairs. We must, in addition, insure the correct dimensions. Table I summarizes the dimensions of the basic rotationally invariant quantities; R stands for the curvature scalar constructed from the spatial metric h ij . Since h ij is the metric of a 2 dimensional space, rotational invariants constructed from the Riemann and Ricci tensors can be expressed in terms of R only.  In order to match up the dimension of the Lagrangian, terms that only contain spatial derivatives must have exactly four derivatives. The derivatives can act on the metric or on on the dimensionless variation parameter σ. Hence we have following 2-spatial-derivatives components: where we note that in the term ∂ i N N the denominator serves to cancel off the time dimension of the numerator. To form a 4 derivative term out of above terms, we can (i) choose two terms among (35) with repetition allowed: there are 6 2 − 6 C 2 = 21 such terms; (ii) (36) can combine with any of (35) giving 6 additional terms; and (iii) we can choose one of (37) and choose another from (35), yielding an additional 2 * 6 = 12 terms. Hence we will have 21 + 12 + 6 = 39 terms with four space derivatives. Terms with derivatives of R, such as are not independent. Integrating by parts, the term ∂ i R∂ i g α can written in terms of R∇ 2 g α and R∂ i σ∂ i g α , and the term R∇ 2 N can be expressed in terms of ∂ i R ∂ i N N . The 39 four derivative terms, a R∇ 2 N can be written as ∂iR∂ i N by integration by parts, and it is for the operator ∂iR∂ i N that we use the coefficient c.
which we call the ∇ 4 sector, appear on the right hand side of (22) with dimensionless coefficients that are functions of the couplings g α , and with a factor of σ if the term does not already contain one. Table II gives our notation for the coefficients of these terms in Eq. (22). Two time derivatives are required for the sector with pure time derivatives, which we label ∂ 2 t . The terms must still have length dimension −4. The dimensions of the basic building blocks are given in Tab. III, where K ij = 1 2 ∂ t h ij /N is the extrinsic curvature of the t =constant hypersurfaces in the N i = 0 gauge and K = h ij K ij (with h ij the inverse of the metric h ij ). The combination in the first 2 × 2 block they correspond to the second entry (lowercase characters) and for those a factor of σ is implicit. The red NA labels denote terms that are either second order in infinitesimal parameter σ or terms that are not rotationally invariant.
following basic one derivative terms: The term ∂ t N is not included in the list because it is not covariant. The diffeomorphism invariant quantity is given by ∂ t N − Γ 0 00 N which vanishes identically 0. Possible anomaly terms are constructed from the 2 2 − 1 = 3 products of terms in (38); from 2 terms by combining (39) and one from (38); and we can have (40) contracted with itself. Thus in total there are 3 + 2 + 1 = 6 terms listed in Tab. IV that also gives the corresponding coefficients.
The sector with mixed derivatives has terms with one time and two spatial derivatives. For this ∂ t ∇ 2 sector we can form terms by combining one of (38) or (39) with one of (35), (36) or (37), excluding terms quadratic in σ. This gives 3 * 9 − 3 = 24 terms, as displayed with their coefficients in Tab. V. Finally, we have terms that are not constructed as products of rotationally invariant quantities. Coefficient of those terms are listed in the last row of Tab. V.

B. Using counter-terms
One can similarly list all possible terms in W c.t. . The requirements imposed by dimensional analysis and rotational invariance are as before, the only difference being that these terms are built from the metric and the couplings but not the parameter of the Weyl transformation σ.
a K∇ 2 N can be written as ∂iK∂ i N by doing integration by parts, and it is for this operator that we use the coefficient j.  In particular, in searching for an a-theorem we can use this freedom to simplify the consistency conditions. It may be possible to show then that there exist some class of subtraction schemes for which there exists a possible candidate for an a-theorem, but a general, counter-term and scheme independent statement may not be possible.

Sector
Trivial Anomalies To illustrate this, consider the variation of the K 2 and K∂ t g α terms in W c.t. : Inspecting Tabs. II, IV and V we see that the f anomaly gets contributions only from these variations, so that the change in f induced by finite changes in the counterterms is given by With a slight abuse of notation we have denoted here the arbitrary, finite, additive change to the coefficients of counterterms by the same symbol we have used for the counterterm coefficients themselves. From Eq. (41) we see that one can always choose D so as to set f arbitrarily, and it is often convenient to set f = 0. For a second example consider the R 2 anomaly, a. A similar computation gives In this case we may solve this equation so as to set a = 0 only if a = 0 at fixed points, where β α = 0. As we will see below, the Weyl consistency conditions constrain some anomalies to vanish at fixed points.
We give in App. B the complete set of ambiguities for models with z = 2 in d = 2 spatial dimensions. Terms in the effective actions whose coefficients can be varied at will are not properly anomalies, since the coefficients can be set to zero. With a slight abuse of language they are commonly referred to as trivial anomalies and we adopt this terminology here. Table VI summarizes the trivial anomalies found in each sector.

Sector Vanishing Anomalies Conditionally
Vanishing Anomalies

C. Consistency conditions and vanishing anomalies
In computing the consistency condition (24) one finds a functional that is a combination of linearly independent "operators" (combinations of σ, γ µν and g α ), each with a coefficient that is a linear combination of the coefficients in Tabs Table VII summarizes the vanishing anomalies found in each sector. The table also shows conditionally vanishing anomalies. These are vanishing anomalies but only for a specific choice of counterterms.
As explained above, some vanishing anomalies can be set to zero. For example, from Tab. VII we see that d is a vanishing anomaly, and then Eq. (B1e) informs us that one may choose D to enforce d = 0. We note, however, that by Eqs. (B1a) and (B1e) one may either choose f or d to vanish, but not both.

D. Applications
While there are many avenues for analysis in light of the relations imposed by Weyl consistency conditions on the anomalies, we concentrate on finding candidates for a C-theorem. We search for a combination of anomalies, C, a local function in the space of dimensionless coupling constants that flows monotonically, µdC/dµ ≥ 0. We try to establish this by judiciously setting some anomalies to zero by the freedom explained above and looking for a relation of the form Our first three candidates arise from the ∇ 4 sector. Consider Eq. (A3l), here reproduced: −a 5α β α + 4a + 2c + β α ∂ α n = 0 The combination 2a + c is a vanishing anomaly. One may then use (B3jj) and (B3aa) to set 2a + c = 0. Equation (B3gg) shows a 4α is a trivial anomaly and one may set a 4α = 0. Combining with Eq. (A3c) we have Similarly, Eq. (A3i) shows 2ρ 23 + c is a vanishing anomaly and using (B3y) we may set 2ρ 23 + c = 0.
We then have from Eq. (A3i) again that The difference of these equations then gives us our first candidate for a C-theorem, with C = n−h 2 : A second candidate can be found as follows. Eq. (A3s) shows χ 4 − p 4 is a vanishing anomaly.
We find one candidate for a C-theorem in the ∂ 2 t sector. Equation (A2a) shows d is a vanishing anomaly and use Eqs. (B1e) and (B1c) to set d = b α = 0. Combining (A2a) and (A2b) gives In the ∂ t ∇ 2 -sector we find the following candidates for a C-theorem: We have kept the explicit dependence on z in these equations. As we will see below the ∂ t ∇ 2sector is special in that the Weyl anomalies and the relations from consistency conditions hold for arbitrary z. Hence, the C-candidates in this sector are particularly interesting since they are candidates for any z. To derive (47) we have used that j and b are vanishing anomalies, as evident from Eqs. (A1d) and (A1f), and used B and L to set b = j = 0 in Eq. (A1f) and P 3α to set b 4α = 0 in Eq. (A1b). For (48) we used j = 0 in Eq. (A1a) and (A1n), deduce that ρ 4 is a vanishing anomaly and use P to set ρ 4 = 0 in Eq. (A1n) and P α to set ρ 1α = 0 in Eq. (A1a). For (49), we set j = ρ 4 = 0 as before and in addition we set ρ 5α = 0 using X α in (A1e), and use Eqs. (A1d), (A1e) and (A1m). In the scheme, j = ρ 1α = 0, Eq (A1o) implies that the candidates given by (49) and (48)  in NR quantum systems. Cyclic flows appear in relativistic systems too, but they differ from NR ones in that there is scaling symmetry all along the cyclic flows and, in fact, the C quantity is constant along the cyclic flow [40]. Investigating the conditions under which a theory gives positive definite metric(s) in the space of flows is beyond the scope of this work; we hope to return to this problem in the future.

IV. GENERALISATION TO ARBITRARY z VALUE
In this section, we will explore the possibility to generalize the work for arbitrary z value. It is • There is a pure ∇ 2 sector for z = 2k, k ∈ Z. It has precisely 2(k + 1) spatial derivatives. We have discussed in detail the case k = 1. Higher values of k can be similarly analyzed, but it it involves an ever increasing number of terms as z increases.
• There is a pure ∂ t sector for z = 2/k, k ∈ Z, with k + 1 time derivatives. We have analyzed the k = 1 case. Higher values of k can be similarly analyzed, but it involves an ever increasing number of terms as z decreases.
• There is a ∂ t ∇ 2 sector for arbitrary z. It has 1-time and 2-spatal derivatives regardless of z. Therefore, the classification of anomalies and counterterms is exactly as in the z = 2 case, and the consistency conditions and derived C-candidates are modified by factors of z/2 relative the z = 2 case.
In relativistic 2n-dimensional QFT the quantity that is believed to satisfy a C-theorem is associated with the Euler anomaly, that is, it is the coefficient of the Euler density E 2n in the conformal anomaly [28]. 2 It would seem natural to seek for analogous candidates in non-relativistic theories. The obvious analog involves the Euler density for the spatial sections t = constant; by dimensional analysis and scaling it should be constructed out of z+d = 2n spatial derivatives acting on the metric h ij . However, for a d-dimensional metric the Euler density E 2n with 2n − d = z > 0 vanishes. Hence, we are led to consider an anomaly of the form XE d , that is the Euler density computed on the spatial sections t = constant times some quantity X with the correct dimensions, [X] = z. This construction is only valid for even spatial dimension, d = 2n. The most natural candidate for X is K: it is the only choice if z is odd. If z is even it can be constructed out of spatial derivatives. For example, if z = dk = 2nk for some integers k and n, one may take The variation of the Euler density yields the Lovelock tensor [47], H ij , a symmetric 2-index tensor that satisfies In looking for a candidate C-theorem we consider a set of operators that close under Weyl- we are led to include terms with the Lovelock tensor and two spatial derivatives. In order to compute the consequences of the Weyl consistency conditions we assume where the ellipses denote terms that depend on derivatives of σ and are therefore independent of X. Consider therefore a subset of terms in the anomaly that appear in the consistency conditions that lead to a potential C-theorem: Correspondingly there are metric and coupling-constant dependent counter-terms with coefficients denoted by uppercase symbols: Freedom to choose finite parts of counter-terms leads to ambiguities in the anomaly coefficients as follows: In addition to the Euler density, E d , there are several independent scalars one can construct out of d derivatives of the metric in d dimensions (except for d = 2, for which the only 2-derivative invariant is the Ricci scalar and hence E d ∝ R). E d is special in that it is the only quantity that gives just the Lovelock tensor under an infinitesimal Weyl trasformation, δ σ ( In general some other d-derivative invariant 3 E constructed out of d/2 powers of the Riemann tensor We have given an example of such a term above, H ij R ij , both in the anomaly and among the contribute to the consistency conditions that lead to a potential C-theorem, they give a common contribution to all those consistency conditions and therefore effectively shift the contribution of a to the potential C-theorem -and the shift is immaterial since it is basis dependent. Consider, for example, the coefficient b of the anomaly term H ij R ij which we have retained precisely to demonstrate these points. Since  Imposing [∆ σ ′ , ∆ σ ]W = 0 we find three conditions, Here we have listed on the left the independent operators in [∆ σ ′ , ∆ σ ]W whose coefficients must vanish yielding the condition correspondingly listed on the right. We have checked that the condi- Combining these we arrive at the candidate for a C-theorem: Establishing a C-theorem requires in addition demonstrating positivity of the "metric" −2y 5αγ in Eq. (56). While we have not attempted this, it may be possible to demonstrate this in generality working on a background with positive definite Lovelock tensor and using the fact that y 5αγ gives the RG response of the contact counter-term to the obviously positive definite correlator O α O γ .
In addition, one should check that, when computed at the gaussiaan fixed point, the quantity is a measure of the number of degrees of freedom. We hope to come back to this questions in the future, by performing explicit calculations (at and away from fixed points) of these quantities -but such extensive computations are beyond the scope of this work.
The limit d = 2 is special since H ij = 2h ij . In our analysis, the term H ij R ij = 2R = 2E 2 so a and b appear in the combination a + 2b throughout. The potential C theorem reads As we have seen in Sec. IV, potential C-theorems in d = 2 for any z can be found only in the ∇ 2 ∂ t sector. Consulting Tab. V we see the only candidate for X in our present discussion is X = K. None of the potential C-theorems listed in Eqs. (47)-(50) (nor linear combinations thereof) reproduce the potential C-theorem in Eq. (57). The reason for this is that in Sec. III D we looked for C-theorems from consistency conditions that included, among others, tems with σ∇ i ∂ j σ ′ − σ ′ ∇ i ∂ j σ, wheras in this section we integrated such terms by parts. The difference then corresponds to combining the consistency conditions given in the appendix with some of their derivatives.
In fact we have found a scheme for deducing aditional C-theorem candidates in d = 2 by taking derivatives of some of our consistency conditions. The method is as follows. Take X ∈ {R, ∇ 2 N, ∂ i N ∂ i N, K}; the first three instances apply to the case z = 2 while the last is applicable for arbitrary z. Then : • Consider the consistency condition involving σ∇ 2 σ ′ X, and take a derivative to obtain an equation, say T 1 .
• Take the consistency condition involving σ∇ i σ ′ ∂ i N X. From this one may deduce a linear combination of anomalies is vanishing. Set that to 0 using the ambiguity afforded by counterterms. The remaining terms in the equation (all proportional to β α ) give an equation we denote by T 2 .
• Take the consistency condition involving σ∇ i σ ′ ∂ i g α X, contract it with β α , to get an equation, say, T 3 .
Following this scheme we obtain four new C-theorem candidates. In the following the expressions for T 1,2,3 refer to the equation numbers of the consistency conditions in the appendix: (iv) X = K. T 1 = A1f, T 2 = A1n, T 3 = A1h. Set j − ρ 4 = 0. Then We have verified that after accounting for differences in basis and notation Eq. (61) is precisely the same as the general C-theorem candidate of this section given in Eq. (57).

VI. SUMMARY AND DISCUSSION
Wess-Zumino consistency conditions for Weyl transformations impose constraints on the renormalization group flow of Weyl anomalies. As a first step in studying these constraints in nonrelativistic quantum field theories we have classified the anomalies that appear in d = 2 (spatial dimensions) at z = 2 (dynamical exponent at gaussian fixed point). There are many more anomalies than in the comparable relativistic case (3+1 dimensions): there are 39 anomalies associated with 4-spatial derivatives (Table II), 6 with 2-time derivatives (Table IV) and 32 more that contain 1-time and 2-spatial derivaties (Table V). Freedom to add finite amounts to counterterms gives in turn freedom to shift some anomalies arbitrarily. "Trivial Anomalies" are those that can thus be set to zero. We then classified all counterterms (Tables II-V), gave the shift in Weyl anomalies produced by shifts in counterterms (in App. B), and then listed the trivial anomalies (Table VI).
The consistency conditions among these 39+6+32 anomalies do not mix among the three sectors.
They are listed by sector in App. A, and from these we can read-off "Vanishing Anomalies" -those that vanish at fixed points; see Table. VII. As an application of the use of these conditions we find theorems only under conditions that do not apply to systems that exhibit cycles. We look forward to developments in this area.
Appendix C: S-theorem: 0 + 1D conformal quantum mechanics One may wonder whether the formalism that leads to the Weyl anomaly and consistency conditions can be used for the case of d = 0. One encounters an immediate obstacle when attempting this. There is no immediate generalization of the trace anomaly equation (23). The problem is that there is no extension of the action integral that gives invariance under the local version of rescaling transformations, because there is no extrinsic curvature tensor at our disposal. The naive generalisation of the Callan-Symanzik equation specialized to d = 0, H = β α O α , cannot hold. In fact, for example, the free particle is a scale invariant system with H = 0.
The inverse square potential serves as a test ground for a simple realisation of the quantum anomaly, where the classical scale symmetry is broken by quantum mechanical effects [56] leading to dimensional transmutation i.e, after renormalization the quantum system acquires an intrinsic length scale [57,58]. Studies have been made of non-self-adjointness of the Hamiltonian in the strongly attractive regime and how to obtain its self-adjoint extension, a procedure that effectively amounts to renormalisation [59,60]. The system is also shown to exhibit limit cycle behaviour in renormalization group flows [61,62]. This potential appears in different branches of physics, from nuclear physics [62,63] and molecular physics [64] to quantum cosmology [65][66][67] and the study of black holes [68]. Given this, it is of interest to understand how quantum effects break scale symmetry in non-relativist quantum mechanics. We will prove a general theorem concerning the breaking of scale symmetry.
In the quantum mechanical description of a scale invariant system, the Hamiltonian H and the generator of scale transformations D obey the following commutation relation: where z is the dynamical exponent of the theory. We will show an elementary S-Theorem, that (C1) is incompatible with H being Hermitian on a domain containing the state 4 D|E , where |E is any non-zero energy eigenstate. The S-Theorem can be used to deduce that classically scale invariant systems, e.g., the inverse square potential, cannot be quantized without loosing either unitarity or scale invariance if we insist on having bound states with finite non-zero binding energy.
To prove the theorem, we consider the eigenstates |E of the Hamiltonian H and take expecta- That the mismatch between (C3) and (C4) lies in the imaginary part hints at the fact that H can not be hermitian if we have scale invariance. We recall that hermiticity of H crucially depends on vanishing of a boundary term, which is imaginary when we consider quantities like E|H|E .
For a simple illustration of S-theorem consider the free particle with one degree of freedom, H = 1 2 p 2 and D = 1 2 (xp + px) − tH. Consider first the particle in a finite periodic box with length L. The operator algebra of the free particle holds regardless of the presence of the periodic boundaries, so the S-theorem holds and it tells us that either H is not hermitian or D|p is not a state. It is instructive to look carefully at the derivation of (C3) and (C4) in this context. An In contrast, consider the inverse square potential problem. For sufficiently strong attractive potential there are normalizable bound states |E , and the state D|E is properly normalized.
The Hamiltonian is hermitian, but this case requires reguralisation and renormalization and scale symmetry is broken.
This is in fact a statement of a more general result. A corollary of the S-theorem is that we cannot have (properly normalized) bound states with non-zero energy in a scale invariant system if we insist on the Hamiltonian being hermitian on the Hilbert space. As in the previous example, this follows from observing that if there exists a properly normalized state |E , then D|E is also a properly normalized state since the wave-function vanishes sufficiently fast at infinity. This result is consistent with representation theory: a discrete spectrum {E n } cannot form a representation of a transformation which acts by E → λ z E for continuous λ, (except if the only allowed finite energy value is E = 0). For example, it is well known that for the inverse square problem in the strongly attractive regime, continuous spectrum is an illusion since in that regime, H is no more Hermitian.
To make H hermitian, we need to renormalize the problem, breaking the scale symmetry.
The S-theorem can be generalised to to any Hermitian operator A with non zero scaling dimension α, that is, [D, A] = iαA. The operator A can not be Hermitian on a domain containing