Abstract
We develop a new heat kernel method that is suited for a systematic study of the renormalization group flow in Hořava gravity (and in Lifshitz field theories in general). This method maintains covariance at all stages of the calculation, which is achieved by introducing a generalized Fourier transform covariant with respect to the nonrelativistic background spacetime. As a first test, we apply this method to compute the anisotropic Weyl anomaly for a (2 + 1)-dimensional scalar field theory around a z = 2 Lifshitz point and corroborate the previously found result. We then proceed to general scalar operators and evaluate their one-loop effective action. The covariant heat kernel method that we develop also directly applies to operators with spin structures in arbitrary dimensions.
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References
P. Hořava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].
P. Hořava, Membranes at quantum criticality, JHEP 03 (2009) 020 [arXiv:0812.4287] [INSPIRE].
J. Bellorín and A. Restuccia, Closure of the algebra of constraints for a non-projectable Hořava model, Phys. Rev. D 83 (2011) 044003 [arXiv:1010.5531] [INSPIRE].
V. A. Kostelecky and N. Russell, Data Tables for Lorentz and CPT Violation, arXiv:0801.0287 [INSPIRE].
P. Hořava, General covariance in gravity at a Lifshitz point, Class. Quant. Grav. 28 (2011) 114012 [arXiv:1101.1081] [INSPIRE].
A. Coates, C. Melby-Thompson and S. Mukohyama, Revisiting Lorentz violation in Hořava gravity, Phys. Rev. D 100 (2019) 064046 [arXiv:1805.10299] [INSPIRE].
S. Groot Nibbelink and M. Pospelov, Lorentz violation in supersymmetric field theories, Phys. Rev. Lett. 94 (2005) 081601 [hep-ph/0404271] [INSPIRE].
M. M. Anber and J. F. Donoghue, The emergence of a universal limiting speed, Phys. Rev. D 83 (2011) 105027 [arXiv:1102.0789] [INSPIRE].
G. Bednik, O. Pujolàs and S. Sibiryakov, Emergent Lorentz invariance from Strong Dynamics: Holographic examples, JHEP 11 (2013) 064 [arXiv:1305.0011] [INSPIRE].
T. Griffin, P. Hořava and C. M. Melby-Thompson, Conformal Lifshitz Gravity from Holography, JHEP 05 (2012) 010 [arXiv:1112.5660] [INSPIRE].
M. Baggio, J. de Boer and K. Holsheimer, Anomalous Breaking of Anisotropic Scaling Symmetry in the Quantum Lifshitz Model, JHEP 07 (2012) 099 [arXiv:1112.6416] [INSPIRE].
T. Griffin, P. Hořava and C. M. Melby-Thompson, Lifshitz Gravity for Lifshitz Holography, Phys. Rev. Lett. 110 (2013) 081602 [arXiv:1211.4872] [INSPIRE].
M. H. Christensen, J. Hartong, N. A. Obers and B. Rollier, Torsional Newton-Cartan Geometry and Lifshitz Holography, Phys. Rev. D 89 (2014) 061901 [arXiv:1311.4794] [INSPIRE].
M. H. Christensen, J. Hartong, N. A. Obers and B. Rollier, Boundary Stress-Energy Tensor and Newton-Cartan Geometry in Lifshitz Holography, JHEP 01 (2014) 057 [arXiv:1311.6471] [INSPIRE].
J. Ambjørn, A. Görlich, J. Jurkiewicz and R. Loll, Quantum Gravity via Causal Dynamical Triangulations, in Springer Handbook of Spacetime, A. Ashtekar and V. Petkov eds. (2013), DOI [arXiv:1302.2173] [INSPIRE].
P. Hořava, Spectral Dimension of the Universe in Quantum Gravity at a Lifshitz Point, Phys. Rev. Lett. 102 (2009) 161301 [arXiv:0902.3657] [INSPIRE].
A. Frenkel, P. Hořava and S. Randall, Perelman’s Ricci Flow in Topological Quantum Gravity, arXiv:2011.11914 [INSPIRE].
A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. Sibiryakov and C. F. Steinwachs, Renormalization of Hořava gravity, Phys. Rev. D 93 (2016) 064022 [arXiv:1512.02250] [INSPIRE].
A. O. Barvinsky, D. Blas, M. Herrero-Valea, S. M. Sibiryakov and C. F. Steinwachs, Hořava Gravity is Asymptotically Free in 2 + 1 Dimensions, Phys. Rev. Lett. 119 (2017) 211301 [arXiv:1706.06809] [INSPIRE].
T. Griffin, K. T. Grosvenor, C. M. Melby-Thompson and Z. Yan, Quantization of Hořava gravity in 2 + 1 dimensions, JHEP 06 (2017) 004 [arXiv:1701.08173] [INSPIRE].
A. O. Barvinsky, M. Herrero-Valea and S. M. Sibiryakov, Towards the renormalization group flow of Hořava gravity in (3 + 1) dimensions, Phys. Rev. D 100 (2019) 026012 [arXiv:1905.03798] [INSPIRE].
D. Benedetti and F. Guarnieri, One-loop renormalization in a toy model of Hořava-Lifshitz gravity, JHEP 03 (2014) 078 [arXiv:1311.6253] [INSPIRE].
D. Nesterov and S. N. Solodukhin, Gravitational effective action and entanglement entropy in UV modified theories with and without Lorentz symmetry, Nucl. Phys. B 842 (2011) 141 [arXiv:1007.1246] [INSPIRE].
A. Mamiya and A. Pinzul, Heat kernel for flat generalized Laplacians with anisotropic scaling, J. Math. Phys. 55 (2014) 063503 [arXiv:1308.2706] [INSPIRE].
G. D’Odorico, J.-W. Goossens and F. Saueressig, Covariant computation of effective actions in Hořava-Lifshitz gravity, JHEP 10 (2015) 126 [arXiv:1508.00590] [INSPIRE].
A. O. Barvinsky, D. Blas, M. Herrero-Valea, D. V. Nesterov, G. Pérez-Nadal and C. F. Steinwachs, Heat kernel methods for Lifshitz theories, JHEP 06 (2017) 063 [arXiv:1703.04747] [INSPIRE].
V. P. Gusynin, New Algorithm for Computing the Coefficients in the Heat Kernel Expansion, Phys. Lett. B 225 (1989) 233 [INSPIRE].
V. P. Gusynin, Seeley-gilkey Coefficients for the Fourth Order Operators on a Riemannian Manifold, Nucl. Phys. B 333 (1990) 296 [INSPIRE].
V. Gusynin, Asymptotics of the heat kernel for nonminimal differential operators, Ukr. Math. J. 43 (1991) 1432.
V. P. Gusynin and E. V. Gorbar, Local heat kernel asymptotics for nonminimal differential operators, Phys. Lett. B 270 (1991) 29 [INSPIRE].
V. P. Gusynin, E. V. Gorbar and V. V. Romankov, Heat kernel expansion for nonminimal differential operators and manifolds with torsion, Nucl. Phys. B 362 (1991) 449 [INSPIRE].
E. V. Gorbar, Heat kernel expansion for operators of the type of the square root of the LAplace operator, J. Math. Phys. 38 (1997) 1692 [hep-th/9602018] [INSPIRE].
V. P. Gusynin and V. V. Kornyak, Computation of the DeWitt-Seeley-Gilkey coefficient E4 for nonminimal operator in curved space, Nucl. Instrum. Meth. A 389 (1997) 365 [INSPIRE].
H. Widom, Families of pseudodifferential operators, topics in functional analysis, I. Gohberg and M. Kac, eds., Academic Press, New York, U.S.A. (1978).
H. Widom, Complete symbolic-calculus for pseudodifferential-operators, Bulletin des Sciences Mathématiques 104 (1980) 19.
D. V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].
G. Gibbons, Quantum field theory in curved space-time, pp. 639, Cambridge University Press, U.K. (1978).
J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [INSPIRE].
B. S. DeWitt, Dynamical theory of groups and fields, Gordon and Breach, New York, U.S.A. (1965).
A. O. Barvinsky and G. A. Vilkovisky, The Generalized Schwinger-Dewitt Technique in Gauge Theories and Quantum Gravity, Phys. Rept. 119 (1985) 1 [INSPIRE].
R. I. Nepomechie, Calculating heat kernels, Phys. Rev. D 31 (1985) 3291 [INSPIRE].
A. Ceresole, P. Pizzochero and P. van Nieuwenhuizen, The Curved Space Trace, Chiral and Einstein Anomalies From Path Integrals, Using Flat Space Plane Waves, Phys. Rev. D 39 (1989) 1567 [INSPIRE].
R. T. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure Math. 10 (1967) 288.
P. B. Gilkey, The Spectral geometry of a Riemannian manifold, J. Diff. Geom. 10 (1975) 601 [INSPIRE].
C. W. Misner, K. Thorne and J. Wheeler, Gravitation, W. H. Freeman, San Francisco, U.S.A. (1973).
S. Weinberg, The quantum theory of fields. Vol. 2: Modern applications. Cambridge University Press, U.K. (2013).
I. Arav, S. Chapman and Y. Oz, Lifshitz Scale Anomalies, JHEP 02 (2015) 078 [arXiv:1410.5831] [INSPIRE].
J. M. Martín-García et al., xAct: Efficient tensor computer algebra for Mathematica, www.xact.es.
P. Hořava and C. M. Melby-Thompson, General Covariance in Quantum Gravity at a Lifshitz Point, Phys. Rev. D 82 (2010) 064027 [arXiv:1007.2410] [INSPIRE].
J. Hartong and N. A. Obers, Hořava-Lifshitz gravity from dynamical Newton-Cartan geometry, JHEP 07 (2015) 155 [arXiv:1504.07461] [INSPIRE].
A. Gustavsson, Abelian M5-brane on S6, JHEP 04 (2019) 140 [arXiv:1902.04201] [INSPIRE].
K. Groh, F. Saueressig and O. Zanusso, Off-diagonal heat-kernel expansion and its application to fields with differential constraints, arXiv:1112.4856 [INSPIRE].
Y. Kluth and D. F. Litim, Heat kernel coefficients on the sphere in any dimension, Eur. Phys. J. C 80 (2020) 269 [arXiv:1910.00543] [INSPIRE].
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Grosvenor, K.T., Melby-Thompson, C. & Yan, Z. New heat kernel method in Lifshitz theories. J. High Energ. Phys. 2021, 178 (2021). https://doi.org/10.1007/JHEP04(2021)178
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DOI: https://doi.org/10.1007/JHEP04(2021)178