Abstract
We introduce a new numerical algorithm based on semidefinite programming to efficiently compute bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and \( \mathcal{N} = 1 \) superconformal field theories. Using our algorithm, we dramatically improve previous bounds on a number of CFT quantities, particularly for theories with global symmetries. In the case of SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal technicolor. In \( \mathcal{N} = 1 \) superconformal theories, we place strong bounds on dim(Φ†Φ), where Φ is a chiral operator. These bounds asymptote to the line dim(Φ†Φ) ≤ 2 dim(Φ) near dim(Φ) ≃ 1, forbidding positive anomalous dimensions in this region. We also place novel upper and lower bounds on OPE coefficients of protected operators in the Φ × Φ OPE. Finally, we find examples of lower bounds on central charges and flavor current two-point functions that scale with the size of global symmetry representations. In the case of \( \mathcal{N} = 1 \) theories with an SU(N) flavor symmetry, our bounds on current two-point functions lie within an O(1) factor of the values realized in supersymmetric QCD in the conformal window.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B. Holdom, Techniodor, Phys. Lett. B 150 (1985) 301 [INSPIRE].
T. Akiba and T. Yanagida, Hierarchic chiral condensate, Phys. Lett. B 169 (1986) 432 [INSPIRE].
T.W. Appelquist, D. Karabali and L. Wijewardhana, Chiral hierarchies and the flavor changing neutral current problem in technicolor, Phys. Rev. Lett. 57 (1986) 957 [INSPIRE].
K. Yamawaki, M. Bando and K.-i. Matumoto, Scale invariant technicolor model and a technidilaton, Phys. Rev. Lett. 56 (1986) 1335 [INSPIRE].
T. Appelquist and L. Wijewardhana, Chiral hierarchies and chiral perturbations in technicolor, Phys. Rev. D 35 (1987) 774 [INSPIRE].
T. Appelquist and L. Wijewardhana, Chiral hierarchies from slowly running couplings in technicolor theories, Phys. Rev. D 36 (1987) 568 [INSPIRE].
M.A. Luty and T. Okui, Conformal technicolor, JHEP 09 (2006) 070 [hep-ph/0409274] [INSPIRE].
M.A. Luty, Strong conformal dynamics at the LHC and on the lattice, JHEP 04 (2009) 050 [arXiv:0806.1235] [INSPIRE].
J. Galloway, J.A. Evans, M.A. Luty and R.A. Tacchi, Minimal conformal technicolor and precision electroweak tests, JHEP 10 (2010) 086 [arXiv:1001.1361] [INSPIRE].
J.A. Evans, J. Galloway, M.A. Luty and R.A. Tacchi, Flavor in minimal conformal technicolor, JHEP 04 (2011) 003 [arXiv:1012.4808] [INSPIRE].
A. Azatov, J. Galloway and M.A. Luty, Superconformal technicolor, Phys. Rev. Lett. 108 (2012) 041802 [arXiv:1106.3346] [INSPIRE].
A. Azatov, J. Galloway and M.A. Luty, Superconformal technicolor: models and phenomenology, Phys. Rev. D 85 (2012) 015018 [arXiv:1106.4815] [INSPIRE].
H. Georgi, A.E. Nelson and A. Manohar, On the proposition that all fermions are created equal, Phys. Lett. B 126 (1983) 169 [INSPIRE].
A.E. Nelson and M.J. Strassler, Suppressing flavor anarchy, JHEP 09 (2000) 030 [hep-ph/0006251] [INSPIRE].
D. Poland and D. Simmons-Duffin, Superconformal flavor simplified, JHEP 05 (2010) 079 [arXiv:0910.4585] [INSPIRE].
N. Craig, Simple models of superconformal flavor, arXiv:1004.4218 [INSPIRE].
T. Kobayashi and H. Terao, Sfermion masses in Nelson-Strassler type of models: SUSY standard models coupled with SCFTs, Phys. Rev. D 64 (2001) 075003 [hep-ph/0103028] [INSPIRE].
A.E. Nelson and M.J. Strassler, Exact results for supersymmetric renormalization and the supersymmetric flavor problem, JHEP 07 (2002) 021 [hep-ph/0104051] [INSPIRE].
M.A. Luty and R. Sundrum, Supersymmetry breaking and composite extra dimensions, Phys. Rev. D 65 (2002) 066004 [hep-th/0105137] [INSPIRE].
M. Luty and R. Sundrum, Anomaly mediated supersymmetry breaking in four-dimensions, naturally, Phys. Rev. D 67 (2003) 045007 [hep-th/0111231] [INSPIRE].
T. Kobayashi, H. Nakano, T. Noguchi and H. Terao, Sfermion mass degeneracy, superconformal dynamics and supersymmetric grand unified theories, Phys. Rev. D 66 (2002) 095011 [hep-ph/0202023] [INSPIRE].
M. Dine, P. Fox, E. Gorbatov, Y. Shadmi, Y. Shirman, et al., Visible effects of the hidden sector, Phys. Rev. D 70 (2004) 045023 [hep-ph/0405159] [INSPIRE].
R. Sundrum, ‘Gaugomaly’ mediated SUSY breaking and conformal sequestering, Phys. Rev. D 71 (2005) 085003 [hep-th/0406012] [INSPIRE].
M. Ibe, K.-I. Izawa, Y. Nakayama, Y. Shinbara and T. Yanagida, Conformally sequestered SUSY breaking in vector-like gauge theories, Phys. Rev. D 73 (2006) 015004 [hep-ph/0506023] [INSPIRE].
M. Ibe, K.-I. Izawa, Y. Nakayama, Y. Shinbara and T. Yanagida, More on conformally sequestered SUSY breaking, Phys. Rev. D 73 (2006) 035012 [hep-ph/0509229] [INSPIRE].
M. Schmaltz and R. Sundrum, Conformal sequestering simplified, JHEP 11 (2006) 011 [hep-th/0608051] [INSPIRE].
S. Kachru, L. McAllister and R. Sundrum, Sequestering in string theory, JHEP 10 (2007) 013 [hep-th/0703105] [INSPIRE].
O. Aharony, L. Berdichevsky, M. Berkooz, Y. Hochberg and D. Robles-Llana, Inverted sparticle hierarchies from natural particle hierarchies, Phys. Rev. D 81 (2010) 085006 [arXiv:1001.0637] [INSPIRE].
T. Kobayashi, Y. Nakai and R. Takahashi, Revisiting superparticle spectra in superconformal flavor models, JHEP 09 (2010) 093 [arXiv:1006.4042] [INSPIRE].
E. Dudas, G. von Gersdorff, J. Parmentier and S. Pokorski, Flavour in supersymmetry: horizontal symmetries or wave function renormalisation, JHEP 12 (2010) 015 [arXiv:1007.5208] [INSPIRE].
T.S. Roy and M. Schmaltz, Hidden solution to the μ/Bμ problem in gauge mediation, Phys. Rev. D 77 (2008) 095008 [arXiv:0708.3593] [INSPIRE].
H. Murayama, Y. Nomura and D. Poland, More visible effects of the hidden sector, Phys. Rev. D 77 (2008) 015005 [arXiv:0709.0775] [INSPIRE].
G. Perez, T.S. Roy and M. Schmaltz, Phenomenology of SUSY with scalar sequestering, Phys. Rev. D 79 (2009) 095016 [arXiv:0811.3206] [INSPIRE].
H.D. Kim and J.-H. Kim, Higgs phenomenology of scalar sequestering, JHEP 05 (2009) 040 [arXiv:0903.0025] [INSPIRE].
N.J. Craig and D. Green, On the phenomenology of strongly coupled hidden sectors, JHEP 09 (2009) 113 [arXiv:0905.4088] [INSPIRE].
K. Hanaki and Y. Ookouchi, Light gauginos and conformal sequestering, Phys. Rev. D 83 (2011) 125010 [arXiv:1003.5663] [INSPIRE].
D. Baumann and D. Green, Desensitizing inflation from the Planck scale, JHEP 09 (2010) 057 [arXiv:1004.3801] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1133] [hep-th/9711200] [INSPIRE].
S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370 [hep-ph/9905221] [INSPIRE].
N. Arkani-Hamed, M. Porrati and L. Randall, Holography and phenomenology, JHEP 08 (2001) 017 [hep-th/0012148] [INSPIRE].
R. Rattazzi and A. Zaffaroni, Comments on the holographic picture of the Randall-Sundrum model, JHEP 04 (2001) 021 [hep-th/0012248] [INSPIRE].
A.L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, Effective conformal theory and the flat-space limit of AdS, JHEP 07 (2011) 023 [arXiv:1007.2412] [INSPIRE].
A.M. Polyakov, Conformal symmetry of critical fluctuations, JETP Lett. 12 (1970) 381 [INSPIRE].
H. Osborn and A. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
S. Ferrara, R. Gatto and A. Grillo, Positivity restrictions on anomalous dimensions, Phys. Rev. D 9 (1974) 3564 [INSPIRE].
G. Mack, All unitary ray representations of the conformal group SU(2, 2) with positive energy, Commun. Math. Phys. 55 (1977) 1 [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4d CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
V.S. Rychkov and A. Vichi, Universal constraints on conformal operator dimensions, Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE].
F. Caracciolo and V.S. Rychkov, Rigorous limits on the interaction strength in quantum field theory, Phys. Rev. D 81 (2010) 085037 [arXiv:0912.2726] [INSPIRE].
D. Poland and D. Simmons-Duffin, Bounds on 4d conformal and superconformal field theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE].
R. Rattazzi, S. Rychkov and A. Vichi, Central charge bounds in 4d conformal field theory, Phys. Rev. D 83 (2011) 046011 [arXiv:1009.2725] [INSPIRE].
R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4d conformal field theories with global symmetry, J. Phys. A 44 (2011) 035402 [arXiv:1009.5985] [INSPIRE].
A. Vichi, Improved bounds for CFT’s with global symmetries, JHEP 01 (2012) 162 [arXiv:1106.4037] [INSPIRE].
A.L. Fitzpatrick and D. Shih, Anomalous dimensions of non-chiral operators from AdS/CFT, JHEP 10 (2011) 113 [arXiv:1104.5013] [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonabelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
S. Ferrara, R. Gatto and A. Grillo, Conformal invariance on the light cone and canonical dimensions, Nucl. Phys. B 34 (1971) 349 [INSPIRE].
F. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
F. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
J.-F. Fortin, K. Intriligator and A. Stergiou, Current OPEs in superconformal theories, JHEP 09 (2011) 071 [arXiv:1107.1721] [INSPIRE].
L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev. 38 (1996) 49.
P. Parrilo, Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization, Ph.D. Thesis, California Institute of Technology, Los Angeles U.S.A. (2000) [http://resolver.caltech.edu/CaltechETD:etd-05062004-055516].
D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten Math. Ann. 32 (1888) 342.
S. Rychkov, Is conformal technicolor plausible?, talk given at Planck 2011 [http://indico.cern.ch/getFile.py/access?contribId=128&resId=0&materialId=slides&confId=112851].
G. Isidori, Y. Nir and G. Perez, Flavor physics constraints for physics beyond the standard model, Ann. Rev. Nucl. Part. Sci. 60 (2010) 355 [arXiv:1002.0900] [INSPIRE].
D. Green and D. Shih, Bounds on SCFTs from conformal perturbation theory, arXiv:1203.5129 [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [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].
K.A. Intriligator and B. Wecht, The exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
D. Poland, D. Simmons-Duffin and A. Vichi, Bounds on SQCD, to appear.
N. Beisert, C. Ahn, L.F. Alday, Z. Bajnok, J.M. Drummond, et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
D. Poland and D. Simmons-Duffin, N = 1 SQCD and the transverse field Ising model, JHEP 02 (2012) 009 [arXiv:1104.1425] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
F. Dolan and H. Osborn, Conformal partial waves: further mathematical results, arXiv:1108.6194 [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, et al., Solving the 3D Ising model with the conformal bootstrap, arXiv:1203.6064 [INSPIRE].
G. Mack, D-independent representation of conformal field theories in D dimensions via transformation to auxiliary dual resonance models. Scalar amplitudes, arXiv:0907.2407 [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B.C. van Rees, A natural language for AdS/CFT correlators, JHEP 11 (2011) 095 [arXiv:1107.1499] [INSPIRE].
http://sdpa.sourceforge.net/http://sdpa.sourceforge.net/ SDPA project, SDPA official page, http://sdpa.sourceforge.net/.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1109.5176
Rights and permissions
About this article
Cite this article
Poland, D., Simmons-Duffin, D. & Vichi, A. Carving out the space of 4D CFTs. J. High Energ. Phys. 2012, 110 (2012). https://doi.org/10.1007/JHEP05(2012)110
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2012)110