Abstract
Sum rules constraining the R-current spectral densities are derived holographically for the case of D3-branes, M2-branes and M5-branes all at finite chemical potentials. In each of the cases the sum rule relates a certain integral of the spectral density over the frequency to terms which depend both on long distance physics, hydrodynamics and short distance physics of the theory. The terms which which depend on the short distance physics result from the presence of certain chiral primaries in the OPE of two R-currents which are turned on at finite chemical potential. Since these sum rules contain information of the OPE they provide an alternate method to obtain the structure constants of the two R-currents and the chiral primary. As a consistency check we show that the 3 point function derived from the sum rule precisely matches with that obtained using Witten diagrams.
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References
D. Kharzeev and K. Tuchin, Bulk viscosity of QCD matter near the critical temperature, JHEP 09 (2008) 093 [arXiv:0705.4280] [INSPIRE].
F. Karsch, D. Kharzeev and K. Tuchin, Universal properties of bulk viscosity near the QCD phase transition, Phys. Lett. B 663 (2008) 217 [arXiv:0711.0914] [INSPIRE].
P. Romatschke and D.T. Son, Spectral sum rules for the quark-gluon plasma, Phys. Rev. D 80 (2009) 065021 [arXiv:0903.3946] [INSPIRE].
H.B. Meyer, The bulk channel in thermal gauge theories, JHEP 04 (2010) 099 [arXiv:1002.3343] [INSPIRE].
H.B. Meyer, Lattice gauge theory sum rule for the shear channel, Phys. Rev. D 82 (2010) 054504 [arXiv:1005.2686] [INSPIRE].
R.A. Ferrell and R.E. Glover, Conductivity of superconducting films: a sum rule, Phys. Rev. 109 (1958) 1398 [INSPIRE].
M. Tinkham and R.A. Ferrell, Determination of the superconducting skin depth from the energy gap and sum rule, Phys. Rev. Lett. 2 (1959) 331.
T. Springer, C. Gale, S. Jeon and S.H. Lee, A shear spectral sum rule in a non-conformal gravity dual, Phys. Rev. D 82 (2010) 106005 [arXiv:1006.4667] [INSPIRE].
T. Springer, C. Gale and S. Jeon, Bulk spectral functions in single and multi-scalar gravity duals, Phys. Rev. D 82 (2010) 126011 [arXiv:1010.2760] [INSPIRE].
R. Baier, R-charge thermodynamical spectral sum rule in N = 4 Yang-Mills theory, arXiv:0910.3862 [INSPIRE].
D.R. Gulotta, C.P. Herzog and M. Kaminski, Sum rules from an extra dimension, JHEP 01 (2011) 148 [arXiv:1010.4806] [INSPIRE].
J.R. David, S. Jain and S. Thakur, Shear sum rules at finite chemical potential, JHEP 03 (2012) 074 [arXiv:1109.4072] [INSPIRE].
K. Behrndt, M. Cvetič and W. Sabra, Nonextreme black holes of five-dimensional N = 2 AdS supergravity, Nucl. Phys. B 553 (1999) 317 [hep-th/9810227] [INSPIRE].
G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics, JHEP 09 (2002) 043 [hep-th/0205052] [INSPIRE].
P.K. Kovtun and A.O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72 (2005) 086009 [hep-th/0506184] [INSPIRE].
D.T. Son and A.O. Starinets, Hydrodynamics of R-charged black holes, JHEP 03 (2006) 052 [hep-th/0601157] [INSPIRE].
S. Jain, Holographic electrical and thermal conductivity in strongly coupled gauge theory with multiple chemical potentials, JHEP 03 (2010) 101 [arXiv:0912.2228] [INSPIRE].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE].
V.I. Arnol’d, Ordinary differential equations, Springer-Verlag, New York U.S.A. (1978).
D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT d /AdS d+1 correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [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].
D. Anselmi, D. Freedman, M.T. Grisaru and A. Johansen, Universality of the operator product expansions of SCFT in four-dimensions, Phys. Lett. B 394 (1997) 329 [hep-th/9608125] [INSPIRE].
D.Z. Freedman, K. Johnson and J.I. Latorre, Differential regularization and renormalization: a new method of calculation in quantum field theory, Nucl. Phys. B 371 (1992) 353 [INSPIRE].
D.Z. Freedman, G. Grignani, K. Johnson and N. Rius, Conformal symmetry and differential regularization of the three gluon vertex, Annals Phys. 218 (1992) 75 [hep-th/9204004] [INSPIRE].
J.I. Latorre, C. Manuel and X. Vilasis-Cardona, Systematic differential renormalization to all orders, Annals Phys. 231 (1994) 149 [hep-th/9303044] [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].
D. Marolf and S.F. Ross, Boundary conditions and new dualities: vector fields in AdS/CFT, JHEP 11 (2006) 085 [hep-th/0606113] [INSPIRE].
E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
M. Berkooz, A. Sever and A. Shomer, ’double trace’ deformations, boundary conditions and space-time singularities, JHEP 05 (2002) 034 [hep-th/0112264] [INSPIRE].
I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [arXiv:hep-th/9905104].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [arXiv:hep-th/9802150].
D.T. Son and P. Surówka, Hydrodynamics with triangle anomalies, Phys. Rev. Lett. 103 (2009) 191601 [arXiv:0906.5044].
M.J. Duff, TASI lectures on branes, black holes and Anti-de Sitter space, arXiv:hep-th/9912164.
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David, J.R., Thakur, S. Sum rules and three point functions. J. High Energ. Phys. 2012, 38 (2012). https://doi.org/10.1007/JHEP11(2012)038
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DOI: https://doi.org/10.1007/JHEP11(2012)038