Abstract
Motivated by the recent dilaton-based proof of the 4d a-theorem, we study the dilaton effective action for RG flows in d dimensions. When d is even, the action consists of a Wess-Zumino (WZ) term, whose Weyl-variation encodes the trace-anomaly, plus all Weyl-invariants. For d odd, the action consists of Weyl-invariants only. We present explicit results for the flat-space limit of the dilaton effective action in d-dimensions up to and including 8-derivative terms. GJMS-operators from conformal geometry motivate a form of the action that unifies the Weyl-invariants and anomaly-terms into a compact general-d structure.
A new feature in 8d is the presence of an 8-derivative Weyl-invariant that pollutes the O(p 8)-contribution from the WZ action to the dilaton scattering amplitudes; this may challenge a dilaton-based proof of an a-theorem in 8d.
We use the example of a free massive scalar for two purposes: 1) it allows us to confirm the structure of the d-dimensional dilaton effective action explicitly; we carry out this check for d = 3, 4, 5, . . . , 10; and 2) in 8d we demonstrate how the flow Δa = a UV − a IR can be extracted systematically from the O(p 8)-amplitudes despite the contamination from the 8-derivative Weyl-invariant. This computation gives a value for the a-anomaly of the 8d free conformal scalar that is shown to match the value obtained from zeta-function regularization of the log-term in the free energy.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
A. Schwimmer and S. Theisen, Spontaneous Breaking of Conformal Invariance and Trace Anomaly Matching, Nucl. Phys. B 847 (2011) 590 [arXiv:1011.0696] [INSPIRE].
Z. Komargodski, The constraints of conformal symmetry on RG flows, JHEP 07 (2012) 069 [arXiv:1112.4538] [INSPIRE].
M.A. Luty, J. Polchinski and R. Rattazzi, The a-theorem and the asymptotics of 4D quantum field theory, JHEP 01 (2013) 152 [arXiv:1204.5221] [INSPIRE].
A. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
J.L. Cardy, Is There a c Theorem in Four-Dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [INSPIRE].
N. Boulanger and J. Erdmenger, A Classification of local Weyl invariants in D = 8, Class. Quant. Grav. 21 (2004) 4305 [hep-th/0405228] [INSPIRE].
D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-theorem: N = 2 field theories on the three-sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
I.R. Klebanov, S.S. Pufu, S. Sachdev and B.R. Safdi, Entanglement entropy of 3 − D conformal gauge theories with many flavors, JHEP 05 (2012) 036 [arXiv:1112.5342] [INSPIRE].
H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, arXiv:1202.2070 [INSPIRE].
I.R. Klebanov, T. Nishioka, S.S. Pufu and B.R. Safdi, Is renormalized entanglement entropy stationary at RG fixed points?, JHEP 10 (2012) 058 [arXiv:1207.3360] [INSPIRE].
I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-theorem without supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].
H. Elvang et al., On renormalization group flows and the a-theorem in 6d, JHEP 10 (2012) 011 [arXiv:1205.3994] [INSPIRE].
T. Maxfield and S. Sethi, The conformal anomaly of M5-branes, JHEP 06 (2012) 075 [arXiv:1204.2002] [INSPIRE].
W. Nahm, Supersymmetries and their representations, Nucl. Phys. B 135 (1978) 149 [INSPIRE].
A. Bhattacharyya, L.-Y. Hung, K. Sen and A. Sinha, On c-theorems in arbitrary dimensions, Phys. Rev. D 86 (2012) 106006 [arXiv:1207.2333] [INSPIRE].
C.R. Graham, R. Jenne, L.J. Mason and G. Sparling, Conformally Invariant Powers of the Laplacian, I: Existence, J. London Math. Soc. s2-46 (1992) 557.
S.M. Paneitz, A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary), SIGMA 4 (2008) 36 [arXiv:0803.4331].
E. Fradkin and A.A. Tseytlin, One loop β-function in conformal supergravities, Nucl. Phys. B 203 (1982) 157 [INSPIRE].
E. Fradkin and A.A. Tseytlin, Asymptotic freedom in extended conformal supergravities, Phys. Lett. B 110 (1982) 117 [INSPIRE].
L.Y. Hung and R.C. Myers, unpublished.
T.P. Branson, Q-curvature and spectral invariants, Rend. Circ. Mat. Palermo (2) Suppl. 75 (2005) 11.
A. Juhl, Explicit formulas for GJMS-operators and Q-curvatures, arXiv:1108.0273 [INSPIRE].
D.E. Diaz, Polyakov formulas for GJMS operators from AdS/CFT, JHEP 07 (2008) 103 [arXiv:0803.0571] [INSPIRE].
J.M. Martın-García, xAct: efficient tensor computer algebra, http://www.xact.es.
D. Anselmi, Quantum irreversibility in arbitrary dimension, Nucl. Phys. B 567 (2000) 331 [hep-th/9905005] [INSPIRE].
G. Shore, Dimensional Regularization of Gauge Theories in Spherical Space-Time: Free Field Trace Anomalies, Annals Phys. 117 (1979) 121 [INSPIRE].
I. Drummond and G. Shore, Conformal Anomalies for Interacting Scalar Fields in Curved Space-Time, Phys. Rev. D 19 (1979) 1134 [INSPIRE].
D. Birmingham, Conformal anomaly in spherical space-times, Phys. Rev. D 36 (1987) 3037 [INSPIRE].
E.J. Copeland and D. Toms, The conformal anomaly in higher dimensions, Class. Quant. Grav. 3 (1986) 431 [INSPIRE].
A. Cappelli and G. D’Appollonio, On the trace anomaly as a measure of degrees of freedom, Phys. Lett. B 487 (2000) 87 [hep-th/0005115] [INSPIRE].
J.S. Dowker, Entanglement entropy for even spheres, arXiv:1009.3854 [INSPIRE].
J.S. Dowker, Entanglement entropy for odd spheres, arXiv:1012.1548 [INSPIRE].
H. Elvang, D.Z. Freedman and M. Kiermaier, A simple approach to counterterms in N = 8 supergravity, JHEP 11 (2010) 016 [arXiv:1003.5018] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1209.3424
Rights and permissions
About this article
Cite this article
Elvang, H., Olson, T.M. RG flows in d dimensions, the dilaton effective action, and the a-theorem. J. High Energ. Phys. 2013, 34 (2013). https://doi.org/10.1007/JHEP03(2013)034
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2013)034