Abstract
In this note we study two point functions of Coulomb branch chiral ring elements with large R-charge, in quantum field theories with \( \mathcal{N}=2 \) superconformal symmetry in four spacetime dimensions. Focusing on the case of one-dimensional Coulomb branch, we use the effective-field-theoretic methods of [1], to estimate the two-point correlation function \( {\mathcal{Y}}_n\equiv {\left|x-y\right|}^{2n{\Delta}_{\mathcal{O}}}\left\langle {\left(\mathcal{O}(x)\right)}^n{\left(\overline{\mathcal{O}}(y)\right)}^n\right\rangle \) in the limit where the operator insertion \( {\mathcal{O}}^n \) has large total R-charge \( \mathcal{J}=n{\Delta}_{\mathcal{O}} \). We show that \( {\mathcal{Y}}_n \) has a nontrivial but universal asymptotic expansion at large \( \mathcal{J} \), of the form
where \( {\tilde{\mathcal{Y}}}_n \) approaches a consstant as n → ∞, and \( {\mathbf{N}}_{\mathcal{O}} \) is an n-independent constant describing on the normalization of the operator relative to the effective Abelian gauge coupling. The exponent α is a positive number proportional to the difference between the a-anomaly coefficient of the underlying CFT and that of the effective theory of the Coulomb branch. For Lagrangian SCFT, we check our predictions for the logarithm \( {\mathrm{\mathcal{B}}}_n= \log \left({\mathcal{Y}}_n\right) \), up to and including order log \( \mathcal{J} \) against exact results from supersymmetric localization [2-5]. In the case of \( \mathcal{N}=4 \) we find precise agreement and in the case \( \mathcal{N}=2 \) we find reasonably good numerical agreement at \( \mathcal{J}\simeq 60 \) using the no-instanton approximation to the S4 partition function. We also give predictions for the growth of two-point functions in all rank-one SCFT in the classification of [6-9]. In this way, we show the large-R-charge expansion serves as a bridge from the world of unbroken superconformal symmetry, OPE data, and bootstraps, to the world of the low-energy dynamics ssof the moduli space of vacua.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Hellerman, S. Maeda and M. Watanabe, Operator Dimensions from Moduli, JHEP 10 (2017) 089 [arXiv:1706.05743] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, tt ∗ equations, localization and exact chiral rings in 4d \( \mathcal{N}=2 \) SCFTs, JHEP 02 (2015) 122 [arXiv:1409.4212] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, Exact correlation functions in SU(2) \( \mathcal{N}=2 \) superconformal QCD, Phys. Rev. Lett. 113 (2014) 251601 [arXiv:1409.4217] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, On exact correlation functions in SU(N) \( \mathcal{N}=2 \) superconformal QCD, JHEP 11 (2015) 198 [arXiv:1508.03077] [INSPIRE].
E. Gerchkovitz, J. Gomis, N. Ishtiaque, A. Karasik, Z. Komargodski and S.S. Pufu, Correlation Functions of Coulomb Branch Operators, JHEP 01 (2017) 103 [arXiv:1602.05971] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of N = 2 SCFTs I: physical constraints on relevant deformations, arXiv:1505.04814 [INSPIRE].
P.C. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of N =2 SCFTs II: Construction of special Kähler geometries and RG flows, arXiv:1601.00011 [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of N = 2 SCFTs III: enhanced Coulomb branches and central charges, arXiv:1609.04404 [INSPIRE].
P.C. Argyres, M. Lotito, Y. Lü and M. Martone, Expanding the landscape of \( \mathcal{N}=2 \) rank 1 SCFTs, JHEP 05 (2016) 088 [arXiv:1602.02764] [INSPIRE].
S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, On the CFT Operator Spectrum at Large Global Charge, JHEP 12 (2015) 071 [arXiv:1505.01537] [INSPIRE].
L. Álvarez-Gaumé, O. Loukas, D. Orlando and S. Reffert, Compensating strong coupling with large charge, JHEP 04 (2017) 059 [arXiv:1610.04495] [INSPIRE].
A. Monin, D. Pirtskhalava, R. Rattazzi and F.K. Seibold, Semiclassics, Goldstone Bosons and CFT data, JHEP 06 (2017) 011 [arXiv:1611.02912] [INSPIRE].
O. Loukas, Abelian scalar theory at large global charge, Fortsch. Phys. 65 (2017) 1700028 [arXiv:1612.08985] [INSPIRE].
S. Hellerman, N. Kobayashi, S. Maeda and M. Watanabe, A Note on Inhomogeneous Ground States at Large Global Charge, arXiv:1705.05825 [INSPIRE].
O. Loukas, D. Orlando and S. Reffert, Matrix models at large charge, JHEP 10 (2017) 085 [arXiv:1707.00710] [INSPIRE].
D. Banerjee, S. Chandrasekharan and D. Orlando, Conformal dimensions via large charge expansion, arXiv:1707.00711 [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
D. Li, D. Meltzer and D. Poland, Conformal Collider Physics from the Lightcone Bootstrap, JHEP 02 (2016) 143 [arXiv:1511.08025] [INSPIRE].
D. Li, D. Meltzer and D. Poland, Non-Abelian Binding Energies from the Lightcone Bootstrap, JHEP 02 (2016) 149 [arXiv:1510.07044] [INSPIRE].
A. Kaviraj, K. Sen and A. Sinha, Analytic bootstrap at large spin, JHEP 11 (2015) 083 [arXiv:1502.01437] [INSPIRE].
A. Kaviraj, K. Sen and A. Sinha, Universal anomalous dimensions at large spin and large twist, JHEP 07 (2015) 026 [arXiv:1504.00772] [INSPIRE].
P. Dey, K. Ghosh and A. Sinha, Simplifying large spin bootstrap in Mellin space, arXiv:1709.06110 [INSPIRE].
L.F. Alday and A. Zhiboedov, An Algebraic Approach to the Analytic Bootstrap, JHEP 04 (2017) 157 [arXiv:1510.08091] [INSPIRE].
L.F. Alday, Large Spin Perturbation Theory for Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
L.F. Alday, Solving CFTs with Weakly Broken Higher Spin Symmetry, JHEP 10 (2017) 161 [arXiv:1612.00696] [INSPIRE].
S. Hellerman and I. Swanson, Boundary Operators in Effective String Theory, JHEP 04 (2017) 085 [arXiv:1609.01736] [INSPIRE].
S. Hellerman, S. Maeda, J. Maltz and I. Swanson, Effective String Theory Simplified, JHEP 09 (2014) 183 [arXiv:1405.6197] [INSPIRE].
S. Hellerman and I. Swanson, String Theory of the Regge Intercept, Phys. Rev. Lett. 114 (2015) 111601 [arXiv:1312.0999] [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
S. Caron-Huot, Z. Komargodski, A. Sever and A. Zhiboedov, Strings from Massive Higher Spins: The Asymptotic Uniqueness of the Veneziano Amplitude, JHEP 10 (2017) 026 [arXiv:1607.04253] [INSPIRE].
A. Sever and A. Zhiboedov, On Fine Structure of Strings: The Universal Correction to the Veneziano Amplitude, arXiv:1707.05270 [INSPIRE].
D.T. Son and M. Wingate, General coordinate invariance and conformal invariance in nonrelativistic physics: Unitary Fermi gas, Annals Phys. 321 (2006) 197 [cond-mat/0509786] [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].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, Springer Briefs in Physics (2016), [arXiv:1601.05000].
D. Simmons-Duffin, The Conformal Bootstrap, Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015): Boulder, CO, U.S.A., June 1-26, 2015 (2017) 1-74, [arXiv:1602.07982] [INSPIRE].
M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.
D. Jafferis, B. Mukhametzhanov and A. Zhiboedov, Conformal Bootstrap At Large Charge, arXiv:1710.11161 [INSPIRE].
A. Zhiboedov, Conformal Bootstrap At Large Charge, in MS seminar at Kavli Institute for the Physics and Mathematics of the Universe, (October 2017), http://research.ipmu.jp/seminar/?seminar_id=1939.
D. Rodriguez-Gomez and J.G. Russo, Large-N Correlation Functions in Superconformal Field Theories, JHEP 06 (2016) 109 [arXiv:1604.07416] [INSPIRE].
M. Dedushenko, S.S. Pufu and R. Yacoby, A one-dimensional theory for Higgs branch operators, arXiv:1610.00740 [INSPIRE].
M. Baggio, V. Niarchos, K. Papadodimas and G. Vos, Large-N correlation functions in \( \mathcal{N}=2 \) superconformal QCD, JHEP 01 (2017) 101 [arXiv:1610.07612] [INSPIRE].
A. Pini, D. Rodriguez-Gomez and J.G. Russo, Large-N correlation functions \( \mathcal{N}=2 \) superconformal quivers, JHEP 08 (2017) 066 [arXiv:1701.02315] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, Aspects of Berry phase in QFT, JHEP 04 (2017) 062 [arXiv:1701.05587] [INSPIRE].
M. Billó, F. Fucito, A. Lerda, J.F. Morales, Ya. S. Stanev and C. Wen, Two-point Correlators in \( \mathcal{N}=2 \) Gauge Theories, Nucl. Phys. B 926 (2018) 427 [arXiv:1705.02909] [INSPIRE].
N.B. Agmon, S.M. Chester and S.S. Pufu, A New Duality Between \( \mathcal{N}=8 \) Superconformal Field Theories in Three Dimensions, arXiv:1708.07861 [INSPIRE].
K. Papadodimas, Topological Anti-Topological Fusion in Four-Dimensional Superconformal Field Theories, JHEP 08 (2010) 118 [arXiv:0910.4963] [INSPIRE].
D. Anselmi, J. Erlich, D.Z. Freedman and A.A. Johansen, Positivity constraints on anomalies in supersymmetric gauge theories, Phys. Rev. D 57 (1998) 7570 [hep-th/9711035] [INSPIRE].
P.C. Argyres, A.M. Awad, G.A. Braun and F.P. Esposito, Higher derivative terms in N = 2 supersymmetric effective actions, JHEP 07 (2003) 060 [hep-th/0306118] [INSPIRE].
P.C. Argyres, A.M. Awad, G.A. Braun and F.P. Esposito, Higher derivative terms in N = 2 SUSY effective actions, in Proceedings, 3rd International Symposium on Quantum theory and symmetries (QTS3): Cincinnati, U.S.A., September 10-14, 2003, (2004) 287-293, [hep-th/0402203] [INSPIRE].
P.C. Argyres, A.M. Awad, G.A. Braun and F.P. Esposito, On superspace Chern-Simons-like terms, JHEP 02 (2005) 006 [hep-th/0411081] [INSPIRE].
P.C. Argyres, A. Awad, P. Moomaw and J. Wittig, Holomorphic higher-derivative terms in supersymmetric effective actions, Proceedings, 7th International Workshop on Supersymmetries and Quantum Symmetries (SQS’07): Dubna, Russia, July 30 - August 04, 2007, (2008) 267, https://inspirehep.net/record/1391563/files/P267.pdf .
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
P.C. Argyres, M.R. Plesser, N. Seiberg and E. Witten, New \( \mathcal{N}=2 \) superconformal field theories in four-dimensions, Nucl. Phys. B 461 (1996) 71 [hep-th/9511154] [INSPIRE].
D. Xie, General Argyres-Douglas Theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
N. Bobev, H. Elvang and T.M. Olson, Dilaton effective action with \( \mathcal{N}=1 \) supersymmetry, JHEP 04 (2014) 157 [arXiv:1312.2925] [INSPIRE].
O. Aharony and E. Karzbrun, On the effective action of confining strings, JHEP 06 (2009) 012 [arXiv:0903.1927] [INSPIRE].
O. Aharony, M. Field and N. Klinghoffer, The effective string spectrum in the orthogonal gauge, JHEP 04 (2012) 048 [arXiv:1111.5757] [INSPIRE].
O. Aharony and Z. Komargodski, The Effective Theory of Long Strings, JHEP 05 (2013) 118 [arXiv:1302.6257] [INSPIRE].
S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
S.M. Kuzenko, J. Novak and G. Tartaglino-Mazzucchelli, Higher derivative couplings and massive supergravity in three dimensions, JHEP 09 (2015) 081 [arXiv:1506.09063] [INSPIRE].
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].
M. Dine and N. Seiberg, Comments on higher derivative operators in some SUSY field theories, Phys. Lett. B 409 (1997) 239 [hep-th/9705057] [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].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
E.W. Barnes, The Theory of the Double Gamma Function, Phil. Trans. Roy. Soc. Lond. A 196 (1901) 265.
I. Aniceto, J.G. Russo and R. Schiappa, Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories, JHEP 03 (2015) 172 [arXiv:1410.5834] [INSPIRE].
A.D. Shapere and Y. Tachikawa, Central charges of \( \mathcal{N}=2 \) superconformal field theories in four dimensions, JHEP 09 (2008) 109 [arXiv:0804.1957] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1710.07336
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Hellerman, S., Maeda, S. On the Large R-charge Expansion in \( \mathcal{N}=2 \) Superconformal Field Theories. J. High Energ. Phys. 2017, 135 (2017). https://doi.org/10.1007/JHEP12(2017)135
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2017)135