Abstract
Recently, Hořava (Phys. Rev. D. 79, 084008, 2009) proposed a theory of gravity in 3+1 dimensions with anisotropic scaling using the traditional framework of quantum field theory (QFT). Such an anisotropic theory of gravity, characterized by a dynamical critical exponent z, has proven to be power-counting renormalizable at a z=3 Lifshitz Point. In the present article, we develop a mathematically precise version of power-counting theorem in Lorentz violating theories and apply this to the Hořava-Lifshitz (scalar field) models in configuration space. The analysis is performed under the light of the systematic use of the concept of extension of homogeneous distributions, a concept tailor-made to address the problem of the ultraviolet renormalization in QFT. This becomes particularly transparent in a Lifshitz-type QFT. In the specific case of the \({\phi _{4}^{4}}\)-theory, we show that is sufficient to take z=3 in order to reach the ultraviolet finiteness of the S-matrix in all orders.
References
Hořava, P.: Quantum gravity at a Lifshitz point. Phys. Rev. D. 79, 084008 (2009)
In effect, the Epstein-Glaser approach [3] avoids the infrared problem by multiplying the interaction by a test function g(x), which vanishes rapidly for \(|x| \to \infty \). In this case, the long range part of the interaction is cut-off. Infrared problem arises in the adiabatic limit g(x)→1. Performing the adiabatic limit is a delicate task, since the limit has to be taken s.t. observable quantities (cross sections) remain finite. Blanchard-Seneor [4] showed that the adiabatic limit exists (in the distributional sense) for the QED and λ: ϕ 2n : Theories.
Epstein, H., Glaser, V.: The role of locality in perturbation theory,. Ann. Inst. Henri Poincaré 19, 211 (1973)
Blanchard, P., Seneor, R.: Green’s functions for theories with massless particles (in perturbation theory). Ann. Inst. Henri Poincaré 23, 147 (1975)
Scharf, G.: Finite Quantum Eletrodynamics: the Causal Approach, 2nd edn.Springer (1995)
Stora, R.: A Note on elliptic perturbative renormalization on a compact manifold. (unpublished manuscript)
Stora, R.: A Note on elliptic perturbative renormalization on a compact manifold. (unpublished manuscript)
Stora, R.: Differential algebras in lagrangean field theory. (unpublished manuscript)
Stora, R.: Local gauge groups in quantum field theory: perturbative gauge theories. (unpublished manuscript)
Popineau, G., Stora, R.: A Pedagogical remark on the main theorem of perturbative renormalization theory. (unpublished preprint)
Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623 (2000)
Hollands, S., Wald, R.M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 231, 309 (2002)
Brunetti, R., Fredenhagen, K., Verch, R.: The generally covariant locality principle: a new paradigm for local quantum physics. Commun. Math. Phys. 237, 31 (2003)
Hollands, S.: The operator product expansion for perturbative quantum field theory in curved spacetime. Commun. Math. Phys. 273
Hollands, S.: Renormalized quantum Yang-Mills fields in curved spacetime. Rev. Math. Phys. 20, 1033 (2008)
Hollands, S., Wald, R.M.: Quantum field theory in curved spacetime the operator product expansion and dark energy. Gen. Rel. Grav. 40, 2051–2059 (2008)
Seneor, R. In: Velo, G., Wightman, A.S. (eds.) : Renormalization Theory. D. Reidel, Dordrecht (1976)
Prange, D.: Lorentz covariance in Epstein-Glaser renormalization. preprint, arXiv: hep-th/9904136 (1999)
Prange, D.: Causal perturbation theory and differential renormalization. J. Phys. A. 32, 2225 (1999)
Pinter, G.: Epstein-Glaser Renormalization: Finite Renormalizations , the S-Matrix of Φ4 Theory and the Action Principle, PhD Thesis, October 2000, DESY-THESIS-2000-047. (2000)
Gracia-Bondia, J.: Improved Epstein–Glaser renormalization in coordinate space I. Euclidean framework. Math. Phys. Anal. Geom. 6, 59 (2003)
Lazzarini, S, Gracia-Bondia, J.M.: Improved Epstein-Glaser renormalization II. Lorentz invariant framework. J. Math. Phys. 44, 3863 (2003)
Dütsch, M., Fredenhagen, K.: The master Ward identity and generalized Schwinger-Dyson equation in classical field theory. Comm. Math. Phys. 243, 275 (2003)
Dütsch, M., Fredenhagen, K.: Causal perturbation theory in terms of retarded products, and a proof of the action Ward identity. Rev. Math. Phys. 16, 1291 (2004)
Nikolov, N.M.: Cohomological analysis of the Epstein-Glaser renormalization. preprint, arXiv: 0712.2194 (2007)
Nikolov, N., Stora, R., Todorov, I.: Configuration space renormalization of massless QFT as an extension problem for associate homogeneous distributions. preprint, http://hal.archives-ouvertes.fr/hal-00703717 (2011)
Keller, K.J.: Dimensional regularization in position space and a forest formula for regularized Epstein-Glaser renormalization, PhD thesis, April 2010, DESY-THESIS-2010-012 (2010)
Falk, S., Häußling R., Scheck, F.: Renormalization in quantum field theory: an improved rigorous method. J. Phys. A. 43, 035401 (2010)
Helling, R.C.: How I Learned to Stop Worrying and Love QFT. Notes by C. Sluka and M. Flory, LMU, Summer 2011, arXiv: 1201.2714 [math-ph] (2011)
Dütsch, M., Gracia-Bondia, J.: On the assertion that PCT violation implies Lorentz non-invariance. Phys. Lett. B. 711, 428 (2012)
Bahns, D., Wrochna, M: On-shell extension of distributions. arXiv: 1210.5484 [math-ph].
Steinmann, O.: Perturbation expansions in axiomatic field theory. In: Lecture Notes in Physics, Vol. 11. Springer (1971)
Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis, 2nd edn.Springer (1990)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics: Fourier Analysis, Self-Adjointness, vol. 2. Academic Press (1975)
We recall that if u and v are functions on ℝ nd then the product u(x)v(x) is the restriction to the diagonal of the tensor product u(x)v(y) defined for (x,y)∈ℝ nd×ℝ nd. Thus, by Hörmander’s criterion on the wavefront sets, the product of distributions u and v in the same point can be defined as the pullback of the tensor product u⊗v by the diagonal map provided that there are no points (x,k 1)∈W F(u) and (x,k 2)∈W F(v) s.t. k 1+k 2=0.
Visser, M.: Lorentz symmetry breaking as a quantum field theory regulator. Phys. Rev. D. 80, 025011 (2009)
Visser, M.: Power-counting renormalizability of generalized Hořava gravity. arXiv: 0912.4757 [hep-th]
Huang, B., Huang, Q.-G.Phys. Lett. B. 683, 108 (2010)
Anselmi, D.: Weighted scale invariant quantum field theories. JHEP 0802, 051 (2008)
Anselmi, D.: Weighted power counting and Lorentz violating gauge theories I. General properties. Annals. Phys. 324, 874 (2009)
Anselmi, D.: Weighted power counting and Lorentz violating gauge theories II. Classification. Annals. Phys. 324, 1058 (2009)
Anselmi, D.: Weighted power counting, neutrino masses and Lorentz violating extensions of the Standard Model. Phys. Rev. D. 79, 025017 (2009)
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Franco, D.H.T. On the Power-counting Renormalizability of a Lifshitz-type QFT in Configuration Space. Math Phys Anal Geom 17, 139–150 (2014). https://doi.org/10.1007/s11040-014-9146-5
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DOI: https://doi.org/10.1007/s11040-014-9146-5