Weyl consistency conditions in non-relativistic quantum field theory

Journal of High Energy Physics, Dec 2016

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.

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Weyl consistency conditions in non-relativistic quantum field theory

Received: June Weyl consistency conditions in non-relativistic quantum eld theory Open Access 0 1 c The Authors. 0 1 0 San Diego , 9500 Gilman Drive, La Jolla, CA 92093 , U.S.A 1 Department of Physics, University of California Weyl consistency conditions have been used in unitary relativistic quantum eld theory to impose constraints on the renormalization group ow 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 nd several candidates for a C-theorem. We comment on possible candidates for a C-theorem in higher dimensions. quantum; Anomalies in Field and String Theories; Conformal and W Symmetry; Renor- 1 Introduction 2 Generalities 2.1 Dynamical exponent Listing out terms Using counter-terms Consistency conditions and vanishing anomalies Generalisation to arbitrary z value A candidate for a C-theorem in d + 1D Summary and discussion A Consistency conditions for 2 + 1d NRCFT Anomaly ambiguities B.1 @t2 sector C S-theorem: 0 + 1D conformal quantum mechanics Introduction lous dimensions of operators. However, additional information, such as dynamical (or other methods, e.g., mean eld approximation, that can give more detailed information. between these regions. diverges, jaj ! 1, exhibit non-relativistic conformal symmetry. Ultracold atom gas exseveral atomic systems, including 85Rb [9],138Cs [10, 11], 39K [12]. In the context of critical dynamics the response function exhibits dynamical scaling. the presence of ghosts,1 Horava suggested extending Einstein gravity by terms with higher using holographic methods. Wess-Zumino consistency conditions for Weyl transformations have been used in unitary relativistic quantum eld theory to impose constraints on the renormalization group Zamolodchikov's C-function [30], that famously decreases monotonically along Weyl consistency conditions can be used to show that a quantity ~a satis es is the renormalization group scale, increasing towards short distances. The equation shows that at xed points, characterized by dg =d = 0, a~ is stationary. It can be shown in perturbation theory that H is a positive de nite symmetric matrix [31]. By 1Generically, the S-matrix in models with ghosts is not unitary. However, under certain conditions on the and unitarity is a long-distance emergent phenomenon [17]. = H by explicit computations of \metric" H in perturbation theory [34{36]. However it was quantity a~ of ref. [33] that obey a C-theorem perturbatively. eld theories. The constraints imposed at xed points have been studied in ref. [23] for the critical points along the ows. As mentioned above, there are questions that can conditions on ows 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 ows. Recursive ows in the perturbative regime have been found in several examples in 4 and in 4 dimensional relativistic quantum eld theory [40{45]. Since Weyl consistency conditions in any physically meaningful sense [46, 47]; in fact, they may be removed by a not show whether generically the \metric" H has de nite sign. The question of under further investigation. The paper is organized as follows. In section 2 we set-up the computation, using a appendix C. Generalities function of elds (t; ~x), mass parameters m and coupling constants g that parametrize remains invariant under the rescaling S[ (~x; t)] = ( ~x; zt) = S[ (~x; t)] : is the matrix of canonical dimensions of the elds . In a multi- eld model the term in L is local, so that it entails powers of derivative operators, z counts the mismatch 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 eld with anisotropic scaling z in d dimensions is given by S = riz=2 riz=2 gmz=d scaling property (2.1) holds with the set of dimensionless couplings that we denote by g below. The above setup is appropriate for studies of, say, quantum criticality. However the iy in the example of eq. (2.2) the corresponding energy integral is H = riz=2 riz=2 + gmz=d The short distance divergences encountered in these models need to be regularized symmetries explicitly. Thus we consider NR eld theories in 1 + n dimensions, where the spatial dimension n = d , with d an integer. Dimensional regularization requires the introduction of a parameter with dimensions of inverse length, L 1 . Invariance under (2.1) is then broken, but can be formally recovered by also scaling 1 . For an example, consider the dimensionally regularized version of (2.2): S[ 0(~x; t); ] = riz=2 0ri1 We have written this i (...truncated)


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Sridip Pal, Benjamín Grinstein. Weyl consistency conditions in non-relativistic quantum field theory, Journal of High Energy Physics, 2016, pp. 12, Volume 2016, Issue 12, DOI: 10.1007/JHEP12(2016)012