Abstract
We study holographic models describing an RG flow between two fixed points driven by a relevant scalar operator. We show how to introduce a spurion field to restore Weyl invariance and compute the anomalous contribution to the generating functional in even dimensional theories. We find that the coefficient of the anomalous term is proportional to the difference of the conformal anomalies of the UV and IR fixed points, as expected from anomaly matching arguments in field theory. For any even dimensions the coefficient is positive as implied by the holographic a-theorem. For flows corresponding to spontaneous breaking of conformal invariance, we also compute the two-point functions of the energy-momentum tensor and the scalar operator and identify the dilaton mode. Surprisingly we find that in the simplest models with just one scalar field there is no dilaton pole in the two-point function of the scalar operator but a stronger singularity. We discuss the possible implications.
Similar content being viewed by others
References
Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 12 (2011) 099 [ar**v:1107.3987] [INSPIRE].
Z. Komargodski, The constraints of conformal symmetry on RG flows, JHEP 07 (2012) 069 [ar**v:1112.4538] [INSPIRE].
J.L. Cardy, Is there a c theorem in four-dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].
A. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 43 (1986) 730 [Pisma Zh. Eksp. Teor. Fiz. 43 (1986) 565] [INSPIRE].
M.A. Luty, J. Polchinski and R. Rattazzi, The a-theorem and the asymptotics of 4D quantum field theory, JHEP 01 (2013) 152 [ar**v:1204.5221] [INSPIRE].
J.-F. Fortin, B. Grinstein and A. Stergiou, Limit cycles in four dimensions, JHEP 12 (2012) 112 [ar**v:1206.2921] [INSPIRE].
H. Elvang et al., On renormalization group flows and the a-theorem in 6D, JHEP 10 (2012) 011 [ar**v:1205.3994] [INSPIRE].
J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012] [INSPIRE].
E.T. Akhmedov, A remark on the AdS/CFT correspondence and the renormalization group flow, Phys. Lett. B 442 (1998) 152 [hep-th/9806217] [INSPIRE].
V. Balasubramanian and P. Kraus, Space-time and the holographic renormalization group, Phys. Rev. Lett. 83 (1999) 3605 [hep-th/9903190] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, How to go with an RG flow, JHEP 08 (2001) 041 [hep-th/0105276] [INSPIRE].
D. Freedman, S. Gubser, K. Pilch and N. Warner, Continuous distributions of D3-branes and gauged supergravity, JHEP 07 (2000) 038 [hep-th/9906194] [INSPIRE].
L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, The supergravity dual of N = 1 super Yang-Mills theory, Nucl. Phys. B 569 (2000) 451 [hep-th/9909047] [INSPIRE].
D.Z. Freedman, S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].
F. Bigazzi, RG flows toward IR isolated fixed points: some type 0 samples, JHEP 06 (2001) 068 [hep-th/0101232] [INSPIRE].
M. Berg and H. Samtleben, An exact holographic RG flow between 2D conformal fixed points, JHEP 05 (2002) 006 [hep-th/0112154] [INSPIRE].
N. Halmagyi, K. Pilch, C. Romelsberger and N. Warner, Holographic duals of a family of N =1 fixed points, JHEP 08 (2006) 083[hep-th/0506206] [INSPIRE].
K. Hotta, Y. Hyakutake, T. Kubota, T. Nishinaka and H. Tanida, The CFT-interpolating black hole in three dimensions, JHEP 01 (2009) 010 [ar**v:0811.0910] [INSPIRE].
G. Arutyunov, S. Frolov and S. Theisen, A note on gravity scalar fluctuations in holographic RG flow geometries, Phys. Lett. B 484 (2000) 295 [hep-th/0003116] [INSPIRE].
D. Martelli and A. Miemiec, CFT/CFT interpolating RG flows and the holographic c function, JHEP 04 (2002) 027 [hep-th/0112150] [INSPIRE].
E. D’Hoker and D.Z. Freedman, Supersymmetric gauge theories and the AdS/CFT correspondence, hep-th/0201253 [INSPIRE].
D. Freedman, C. Núñez, M. Schnabl and K. Skenderis, Fake supergravity and domain wall stability, Phys. Rev. D 69 (2004) 104027 [hep-th/0312055] [INSPIRE].
E. Alvarez and C. Gomez, Geometric holography, the renormalization group and the c theorem, Nucl. Phys. B 541 (1999) 441 [hep-th/9807226] [INSPIRE].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [ar**v:1011.5819] [INSPIRE].
R. Penrose and W. Rindler, Spinors and space-time. Volume 2: spinor and twistor methods in space-time geometry, Cambridge University Press, Cambridge U.K. (1986).
J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, Some calculable contributions to holographic entanglement entropy, JHEP 08 (2011) 039 [ar**v:1105.6055] [INSPIRE].
N. Boulanger, Algebraic classification of Weyl anomalies in arbitrary dimensions, Phys. Rev. Lett. 98 (2007) 261302 [ar**v:0706.0340] [INSPIRE].
W. Mueck, Correlation functions in holographic renormalization group flows, Nucl. Phys. B 620 (2002) 477 [hep-th/0105270] [INSPIRE].
I. Papadimitriou and K. Skenderis, Correlation functions in holographic RG flows, JHEP 10 (2004) 075 [hep-th/0407071] [INSPIRE].
O. DeWolfe and D.Z. Freedman, Notes on fluctuations and correlation functions in holographic renormalization group flows, hep-th/0002226 [INSPIRE].
K. Skenderis and P.K. Townsend, Gravitational stability and renormalization group flow, Phys. Lett. B 468 (1999) 46 [hep-th/9909070] [INSPIRE].
K. Skenderis and P.K. Townsend, Hidden supersymmetry of domain walls and cosmologies, Phys. Rev. Lett. 96 (2006) 191301 [hep-th/0602260] [INSPIRE].
K. Skenderis and P.K. Townsend, Hamilton-Jacobi method for curved domain walls and cosmologies, Phys. Rev. D 74 (2006) 125008 [hep-th/0609056] [INSPIRE].
A. Schwimmer and S. Theisen, Spontaneous breaking of conformal invariance and trace anomaly matching, Nucl. Phys. B 847 (2011) 590 [ar**v:1011.0696] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz, Diffeomorphisms and holographic anomalies, Class. Quant. Grav. 17 (2000) 1129 [hep-th/9910267] [INSPIRE].
M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
S.R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259 [INSPIRE].
N. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133 [INSPIRE].
M. Porrati and A. Starinets, RG fixed points in supergravity duals of 4D field theory and asymptotically AdS spaces, Phys. Lett. B 454 (1999) 77 [hep-th/9903085] [INSPIRE].
M. Berg and H. Samtleben, Holographic correlators in a flow to a fixed point, JHEP 12 (2002) 070 [hep-th/0209191] [INSPIRE].
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
S. Weinberg, The quantum theory of fields. Volume 2: modern applications, Cambridge University Press, Cambridge U.K. (1996).
D. Nickel and D.T. Son, Deconstructing holographic liquids, New J. Phys. 13 (2011) 075010 [ar**v:1009.3094] [INSPIRE].
D. Anselmi, D. Freedman, M.T. Grisaru and A. Johansen, Nonperturbative formulas for central functions of supersymmetric gauge theories, Nucl. Phys. B 526 (1998) 543 [hep-th/9708042] [INSPIRE].
D. Anselmi, J. Erlich, D. Freedman and A. Johansen, Positivity constraints on anomalies in supersymmetric gauge theories, Phys. Rev. D 57 (1998) 7570 [hep-th/9711035] [INSPIRE].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [ar**v:1006.1263] [INSPIRE].
R.A. Ferrell and D.J. Scalapino, Order-parameter correlations within the screening approximation, Phys. Rev. Lett. 29 (1972) 413 [INSPIRE].
T. Brauner, Spontaneous symmetry breaking and Nambu-Goldstone bosons in quantum many-body systems, Symmetry 2 (2010) 609 [ar**v:1001.5212] [INSPIRE].
H.B. Nielsen and S. Chadha, On how to count goldstone bosons, Nucl. Phys. B 105 (1976) 445 [INSPIRE].
M.S. Costa, Absorption by double centered D3-branes and the Coulomb branch of N = 4 SYM theory, JHEP 05 (2000) 041 [hep-th/9912073] [INSPIRE].
M.S. Costa, A test of the AdS/CFT duality on the Coulomb branch, Phys. Lett. B 482 (2000) 287 [Erratum ibid. B 489 (2000) 439] [hep-th/0003289] [INSPIRE].
I.R. Klebanov and A. Murugan, Gauge/gravity duality and warped resolved conifold, JHEP 03 (2007) 042 [hep-th/0701064] [INSPIRE].
D. Martelli and J. Sparks, Baryonic branches and resolutions of Ricci-flat Kähler cones, JHEP 04 (2008) 067 [ar**v:0709.2894] [INSPIRE].
L.A. Pando Zayas and A.A. Tseytlin, 3-branes on resolved conifold, JHEP 11 (2000) 028 [hep-th/0010088] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1207.0006
Rights and permissions
About this article
Cite this article
Hoyos, C., Kol, U., Sonnenschein, J. et al. The a-theorem and conformal symmetry breaking in holographic RG flows. J. High Energ. Phys. 2013, 63 (2013). https://doi.org/10.1007/JHEP03(2013)063
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2013)063