Abstract
We study the \( q\overline{q} \) potential in strongly coupled non-conformal field theories with a non-trivial renormalization group flow via holography. We focus on the properties of this potential at an inter-quark separation L large compared to the characteristic scale of the field theory. These are determined by the leading order IR physics plus a series of corrections, sensitive to the properties of the RG-flow. To determine those corrections, we propose a general method applying holographic Wilsonian renormalization to a dual string. We apply this method to examine in detail two sets of examples, 3 + 1-dimensional theories with an RG flow ending in an IR fixed point; and theories that are confining in the IR, in particular, the Witten QCD and Klebanov-Strassler models. In both cases, we find corrections with a universal dependence on the inter-quark separation. When there is an IR fixed point, that correction decays as a power ∼ 1/L4. We explain that dependence in terms of a double-trace deformation in a one-dimensional defect theory. For a confining theory, the decay is exponential ∼ e−M L, with M a scale of the order of the glueball mass. We interpret this correction using an effective flux tube description as produced by a background internal mode excitation induced by sources localized at the endpoints of the flux tube. We discuss how these results could be confronted with lattice QCD data to test whether the description of confinement via the gauge/gravity is qualitatively correct.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
S.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].
A. Adams et al., Strongly correlated quantum fluids: ultracold quantum gases, quantum chromodynamic plasmas and holographic duality, New J. Phys. 14 (2012) 115009 [arXiv:1205.5180] [INSPIRE].
O. DeWolfe, S.S. Gubser, C. Rosen and D. Teaney, Heavy ions and string theory, Prog. Part. Nucl. Phys. 75 (2014) 86 [arXiv:1304.7794] [INSPIRE].
N. Brambilla et al., QCD and strongly coupled gauge theories: challenges and perspectives, Eur. Phys. J. C 74 (2014) 2981 [arXiv:1404.3723] [INSPIRE].
M. Ammon and J. Erdmenger, Gauge/gravity duality, Cambridge University Press, Cambridge U.K. (2015).
H. Nastase, String theory methods for condensed matter physics, Cambridge University Press, Cambridge U.K. (2017).
J. Casalderrey-Solana et al., Gauge/string duality, hot QCD and heavy ion collisions, arXiv:1101.0618 [INSPIRE].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
N. Iqbal and H. Liu, Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm, Phys. Rev. D 79 (2009) 025023 [arXiv:0809.3808] [INSPIRE].
C. Eling and Y. Oz, Relativistic CFT hydrodynamics from the membrane paradigm, JHEP 02 (2010) 069 [arXiv:0906.4999] [INSPIRE].
T. Faulkner, H. Liu, J. McGreevy and D. Vegh, Emergent quantum criticality, Fermi surfaces and AdS 2, Phys. Rev. D 83 (2011) 125002 [arXiv:0907.2694] [INSPIRE].
I. Bredberg, C. Keeler, V. Lysov and A. Strominger, Wilsonian approach to fluid/gravity duality, JHEP 03 (2011) 141 [arXiv:1006.1902] [INSPIRE].
D. Nickel and D.T. Son, Deconstructing holographic liquids, New J. Phys. 13 (2011) 075010 [arXiv:1009.3094] [INSPIRE].
I. Heemskerk and J. Polchinski, Holographic and Wilsonian renormalization groups, JHEP 06 (2011) 031 [arXiv:1010.1264] [INSPIRE].
T. Faulkner, H. Liu and M. Rangamani, Integrating out geometry: holographic Wilsonian RG and the membrane paradigm, JHEP 08 (2011) 051 [arXiv:1010.4036] [INSPIRE].
C. Charmousis et al., Effective holographic theories for low-temperature condensed matter systems, JHEP 11 (2010) 151 [arXiv:1005.4690] [INSPIRE].
A. Donos and J.P. Gauntlett, Navier-Stokes equations on black hole horizons and DC thermoelectric conductivity, Phys. Rev. D 92 (2015) 121901 [arXiv:1506.01360] [INSPIRE].
J.M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].
S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [INSPIRE].
E. Kiritsis, W. Li and F. Nitti, Holographic RG flow and the quantum effective action, Fortsch. Phys. 62 (2014) 389 [arXiv:1401.0888] [INSPIRE].
I. Bakas and C. Sourdis, Dirichlet σ-models and mean curvature flow, JHEP 06 (2007) 057 [arXiv:0704.3985] [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
I.R. Klebanov and M.J. Strassler, Supergravity and a confining gauge theory: duality cascades and chi SB resolution of naked singularities, JHEP 08 (2000) 052 [hep-th/0007191] [INSPIRE].
L. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, Novel local CFT and exact results on perturbations of N = 4 superYang-Mills from AdS dynamics, JHEP 12 (1998) 022 [hep-th/9810126] [INSPIRE].
J. Distler and F. Zamora, Nonsupersymmetric conformal field theories from stable Anti-de Sitter spaces, Adv. Theor. Math. Phys. 2 (1999) 1405 [hep-th/9810206] [INSPIRE].
A. Khavaev, K. Pilch and N.P. Warner, New vacua of gauged N = 8 supergravity in five-dimensions, Phys. Lett. B 487 (2000) 14 [hep-th/9812035] [INSPIRE].
D.Z. Freedman, S.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].
K. Behrndt, Domain walls of D = 5 supergravity and fixpoints of N = 1 super-Yang-Mills, Nucl. Phys. B 573 (2000) 127 [hep-th/9907070] [INSPIRE].
A. Khavaev and N.P. Warner, A class of N = 1 supersymmetric RG flows from five-dimensional N = 8 supergravity, Phys. Lett. B 495 (2000) 215 [hep-th/0009159] [INSPIRE].
H. Lü, C.N. Pope and T.A. Tran, Five-dimensional N = 4, SU(2) × U(1) gauged supergravity from type IIB, Phys. Lett. B 475 (2000) 261 [hep-th/9909203] [INSPIRE].
M. Cvetič et al., Embedding AdS black holes in ten-dimensions and eleven-dimensions, Nucl. Phys. B 558 (1999) 96 [hep-th/9903214] [INSPIRE].
H. Nastase and D. Vaman, On the nonlinear KK reductions on spheres of supergravity theories, Nucl. Phys. B 583 (2000) 211 [hep-th/0002028] [INSPIRE].
K. Pilch and N.P. Warner, N = 2 supersymmetric RG flows and the IIB dilaton, Nucl. Phys. B 594 (2001) 209 [hep-th/0004063] [INSPIRE].
M. Cvetič et al., Consistent SO(6) reduction of type IIB supergravity on S 5, Nucl. Phys. B 586 (2000) 275 [hep-th/0003103] [INSPIRE].
K. Lee, C. Strickland-Constable and D. Waldram, Spheres, generalised parallelisability and consistent truncations, Fortsch. Phys. 65 (2017) 1700048 [arXiv:1401.3360] [INSPIRE].
F. Ciceri, B. de Wit and O. Varela, IIB supergravity and the E 6(6) covariant vector-tensor hierarchy, JHEP 04 (2015) 094 [arXiv:1412.8297] [INSPIRE].
A. Baguet, O. Hohm and H. Samtleben, Consistent type IIB reductions to maximal 5D supergravity, Phys. Rev. D 92 (2015) 065004 [arXiv:1506.01385] [INSPIRE].
K. Pilch and N.P. Warner, N = 1 supersymmetric renormalization group flows from IIB supergravity, Adv. Theor. Math. Phys. 4 (2002) 627 [hep-th/0006066] [INSPIRE].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [INSPIRE].
J. Polchinski and J. Sully, Wilson loop renormalization group flows, JHEP 10 (2011) 059 [arXiv:1104.5077] [INSPIRE].
Y. Kinar, E. Schreiber and J. Sonnenschein, \( Q\overline{Q} \) potential from strings in curved space-time: classical results, Nucl. Phys. B 566 (2000) 103 [hep-th/9811192] [INSPIRE].
K. Skenderis and P.K. Townsend, Gravitational stability and renormalization group flow, Phys. Lett. B 468 (1999) 46 [hep-th/9909070] [INSPIRE].
D.Z. Freedman, C. Núñez, M. Schnabl and K. Skenderis, Fake supergravity and domain wall stability, Phys. Rev. D 69 (2004) 104027 [hep-th/0312055] [INSPIRE].
A. Celi et al., On the fakeness of fake supergravity, Phys. Rev. D 71 (2005) 045009 [hep-th/0410126] [INSPIRE].
M. Zagermann, N = 4 fake supergravity, Phys. Rev. D 71 (2005) 125007 [hep-th/0412081] [INSPIRE].
K. Skenderis and P.K. Townsend, Hidden supersymmetry of domain walls and cosmologies, Phys. Rev. Lett. 96 (2006) 191301 [hep-th/0602260] [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. Kiritsis, F. Nitti and L. Silva Pimenta, Exotic RG flows from holography, Fortsch. Phys. 65 (2017) 1600120 [arXiv:1611.05493] [INSPIRE].
F. Nitti, L. Silva Pimenta and D.A. Steer, On multi-field flows in gravity and holography, JHEP 07 (2018) 022 [arXiv:1711.10969] [INSPIRE].
D.S. Salopek and J.R. Bond, Nonlinear evolution of long wavelength metric fluctuations in inflationary models, Phys. Rev. D 42 (1990) 3936 [INSPIRE].
A. Brandhuber et al., Wilson loops, confinement and phase transitions in large N gauge theories from supergravity, JHEP 06 (1998) 001 [hep-th/9803263] [INSPIRE].
A. Loewy and J. Sonnenschein, On the holographic duals of N = 1 gauge dynamics, JHEP 08 (2001) 007 [hep-th/0103163] [INSPIRE].
C. Núñez, M. Piai and A. Rago, Wilson loops in string duals of walking and flavored systems, Phys. Rev. D 81 (2010) 086001 [arXiv:0909.0748] [INSPIRE].
E. Caceres and R. Hernandez, Glueball masses for the deformed conifold theory, Phys. Lett. B 504 (2001) 64 [hep-th/0011204] [INSPIRE].
F. Bigazzi, A.L. Cotrone, L. Martucci and L.A. Pando Zayas, Wilson loop, Regge trajectory and hadron masses in a Yang-Mills theory from semiclassical strings, Phys. Rev. D 71 (2005) 066002 [hep-th/0409205] [INSPIRE].
Y. Kinar, E. Schreiber, J. Sonnenschein and N. Weiss, Quantum fluctuations of Wilson loops from string models, Nucl. Phys. B 583 (2000) 76 [hep-th/9911123] [INSPIRE].
O. Aharony and E. Karzbrun, On the effective action of confining strings, JHEP 06 (2009) 012 [arXiv:0903.1927] [INSPIRE].
O. Aharony and M. Field, On the effective theory of long open strings, JHEP 01 (2011) 065 [arXiv:1008.2636] [INSPIRE].
A.M. Polyakov, Fine structure of strings, Nucl. Phys. B 268 (1986) 406 [INSPIRE].
H. Kleinert, The membrane properties of condensing strings, Phys. Lett. B 174 (1986) 335 [INSPIRE].
M. Caselle, M. Panero, R. Pellegrini and D. Vadacchino, A different kind of string, JHEP 01 (2015) 105 [arXiv:1406.5127] [INSPIRE].
B.B. Brandt, Spectrum of the open QCD flux tube and its effective string description I: 3d static potential in SU(N = 2, 3), JHEP 07 (2017) 008 [arXiv:1705.03828] [INSPIRE].
B.B. Brandt, Spectrum of the open QCD flux tube and its effective string description, arXiv:1811.11779 [INSPIRE].
B. Bringoltz and M. Teper, String tensions of SU(N) gauge theories in 2 + 1 dimensions, PoS LAT2006 (2006) 041 [hep-lat/0610035] [INSPIRE].
B.B. Brandt and P. Majumdar, Spectrum of the QCD flux tube in 3d SU(2) lattice gauge theory, Phys. Lett. B 682 (2009) 253 [arXiv:0905.4195] [INSPIRE].
B.B. Brandt, Probing boundary-corrections to Nambu-Goto open string energy levels in 3d SU(2) gauge theory, JHEP 02 (2011) 040 [arXiv:1010.3625] [INSPIRE].
A. Athenodorou, B. Bringoltz and M. Teper, Closed flux tubes and their string description in D = 3 + 1 SU(N) gauge theories, JHEP 02 (2011) 030 [arXiv:1007.4720] [INSPIRE].
B.B. Brandt and M. Meineri, Effective string description of confining flux tubes, Int. J. Mod. Phys. A 31 (2016) 1643001 [arXiv:1603.06969] [INSPIRE].
O. Aharony and N. Klinghoffer, Corrections to Nambu-Goto energy levels from the effective string action, JHEP 12 (2010) 058 [arXiv:1008.2648] [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. Dubovsky, R. Flauger and V. Gorbenko, Effective string theory revisited, JHEP 09 (2012) 044 [arXiv:1203.1054] [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Evidence from lattice data for a new particle on the worldsheet of the QCD flux tube, Phys. Rev. Lett. 111 (2013) 062006 [arXiv:1301.2325] [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Flux tube spectra from approximate integrability at low energies, J. Exp. Theor. Phys. 120 (2015) 399 [arXiv:1404.0037] [INSPIRE].
S. Dubovsky and V. Gorbenko, Towards a theory of the QCD string, JHEP 02 (2016) 022 [arXiv:1511.01908] [INSPIRE].
A. Athenodorou and M. Teper, On the mass of the world-sheet ‘axion’ in SU(N) gauge theories in 3 + 1 dimensions, Phys. Lett. B 771 (2017) 408 [arXiv:1702.03717] [INSPIRE].
M. Kruczenski, L.A. Pando Zayas, J. Sonnenschein and D. Vaman, Regge trajectories for mesons in the holographic dual of large-N c QCD, JHEP 06 (2005) 046 [hep-th/0410035] [INSPIRE].
J. Sonnenschein and D. Weissman, Rotating strings confronting PDG mesons, JHEP 08 (2014) 013 [arXiv:1402.5603] [INSPIRE].
N. Brambilla, M. Groher, H.E. Martinez and A. Vairo, Effective string theory and the long-range relativistic corrections to the quark-antiquark potential, Phys. Rev. D 90 (2014) 114032 [arXiv:1407.7761] [INSPIRE].
A. Ficnar, S.S. Gubser and M. Gyulassy, Shooting string holography of jet quenching at RHIC and LHC, Phys. Lett. B 738 (2014) 464 [arXiv:1311.6160] [INSPIRE].
P.M. Chesler and K. Rajagopal, Jet quenching in strongly coupled plasma, Phys. Rev. D 90 (2014) 025033 [arXiv:1402.6756] [INSPIRE].
J. Casalderrey-Solana et al., A hybrid strong/weak coupling approach to jet quenching, JHEP 10 (2014) 019 [Erratum ibid. 09 (2015) 175] [arXiv:1405.3864] [INSPIRE].
W.A. Horowitz, Fluctuating heavy quark energy loss in a strongly coupled quark-gluon plasma, Phys. Rev. D 91 (2015) 085019 [arXiv:1501.04693] [INSPIRE].
K. Rajagopal, A.V. Sadofyev and W. van der Schee, Evolution of the jet opening angle distribution in holographic plasma, Phys. Rev. Lett. 116 (2016) 211603 [arXiv:1602.04187] [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: 1902.04279
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Casalderrey-Solana, J., Gutiez, D. & Hoyos, C. Effective long distance \( q\overline{q} \) potential in holographic RG flows. J. High Energ. Phys. 2019, 134 (2019). https://doi.org/10.1007/JHEP04(2019)134
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2019)134