Journal of High Energy Physics

, 2011:99 | Cite as

On renormalization group flows in four dimensions

  • Zohar KomargodskiEmail author
  • Adam Schwimmer


We discuss some general aspects of renormalization group flows in four dimensions. Every such flow can be reinterpreted in terms of a spontaneously broken conformal symmetry. We analyze in detail the consequences of trace anomalies for the effective action of the Nambu-Goldstone boson of broken conformal symmetry. While the c-anomaly is algebraically trivial, the a-anomaly is “non-Abelian”, and leads to a positive-definite universal contribution to the S-matrix element of 2 → 2 dilaton scattering. Unitarity of the S-matrix results in a monotonically decreasing function that interpolates between the Euler anomalies in the ultraviolet and the infrared, thereby establishing the a-theorem.


Spontaneous Symmetry Breaking Renormalization Group 


  1. [1]
    A. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 43 (1986) 730 [INSPIRE].ADSMathSciNetGoogle Scholar
  2. [2]
    J.L. Cardy, Is there a c theorem in four-dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    H. Osborn, Derivation of a four-dimensional c theorem, Phys. Lett. B 222 (1989) 97 [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    I. Jack and H. Osborn, Analogs for the c theorem for four-dimensional renormalizable field theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  5. [5]
    T. Appelquist, A.G. Cohen and M. Schmaltz, A new constraint on strongly coupled gauge theories, Phys. Rev. D 60 (1999) 045003 [hep-th/9901109] [INSPIRE].ADSGoogle Scholar
  6. [6]
    M. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hep-th/9308075] [INSPIRE].CrossRefzbMATHADSMathSciNetGoogle Scholar
  7. [7]
    D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].CrossRefADSGoogle Scholar
  8. [8]
    M. Kulaxizi and A. Parnachev, Energy flux positivity and unitarity in CFTs, Phys. Rev. Lett. 106 (2011) 011601 [arXiv:1007.0553] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  9. [9]
    N. Seiberg, Electric-magnetic duality in supersymmetric nonabelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  10. [10]
    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].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    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].ADSMathSciNetGoogle Scholar
  12. [12]
    K.A. Intriligator and B. Wecht, The exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  13. [13]
    D. Kutasov, A. Parnachev and D.A. Sahakyan, Central charges and U(1)R symmetries in N =1 super Yang-Mills, JHEP 11 (2003) 013 [hep-th/0308071] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  14. [14]
    K.A. Intriligator and B. Wecht, RG fixed points and flows in SQCD with adjoints, Nucl. Phys. B 677 (2004) 223 [hep-th/0309201] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  15. [15]
    D. Freedman, S. Gubser, K. Pilch and N. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].zbMATHMathSciNetGoogle Scholar
  16. [16]
    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].CrossRefADSMathSciNetGoogle Scholar
  17. [17]
    R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95 [INSPIRE].ADSMathSciNetGoogle Scholar
  19. [19]
    E. Witten, Global aspects of current algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  20. [20]
    E. Fradkin and G. Vilkovisky, Conformal off mass shell extension and elimination of conformal anomalies in quantum gravity, Phys. Lett. B 73 (1978) 209 [INSPIRE].ADSMathSciNetGoogle Scholar
  21. [21]
    R. Riegert, A nonlocal action for the trace anomaly, Phys. Lett. B 134 (1984) 56 [INSPIRE].ADSMathSciNetGoogle Scholar
  22. [22]
    E. Fradkin and A.A. Tseytlin, Conformal anomaly in Weyl theory and anomaly free superconformal theories, Phys. Lett. B 134 (1984) 187 [INSPIRE].ADSMathSciNetGoogle Scholar
  23. [23]
    A. Schwimmer and S. Theisen, Spontaneous breaking of conformal invariance and trace anomaly matching, Nucl. Phys. B 847 (2011) 590 [arXiv:1011.0696] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  24. [24]
    T. Pham and T.N. Truong, Evaluation of the derivative quartic terms of the meson chiral Lagrangian from forward dispersion relation, Phys. Rev. D 31 (1985) 3027 [INSPIRE].ADSGoogle Scholar
  25. [25]
    J. Distler, B. Grinstein, R.A. Porto and I.Z. Rothstein, Falsifying models of new physics via WW scattering, Phys. Rev. Lett. 98 (2007) 041601 [hep-ph/0604255] [INSPIRE].CrossRefADSGoogle Scholar
  26. [26]
    A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  27. [27]
    M. Dine, G. Festuccia and Z. Komargodski, A bound on the superpotential, JHEP 03 (2010) 011 [arXiv:0910.2527] [INSPIRE].CrossRefADSGoogle Scholar
  28. [28]
    A. Cappelli, D. Friedan and J.I. Latorre, C theorem and spectral representation, Nucl. Phys. B 352 (1991) 616 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  29. [29]
    Y. Frishman, A. Schwimmer, T. Banks and S. Yankielowicz, The axial anomaly and the bound state spectrum in confining theories, Nucl. Phys. B 177 (1981) 157 [INSPIRE].CrossRefADSGoogle Scholar
  30. [30]
    Y. Nir, Infrared treatment of higher anomalies and their consequences, Phys. Rev. D 34 (1986) 1164 [INSPIRE].ADSGoogle Scholar
  31. [31]
    I. Buchbinder, S. Kuzenko and A.A. Tseytlin, On low-energy effective actions in n = 2, N =4 superconformal theories in four-dimensions, Phys. Rev. D 62 (2000) 045001 [hep-th/9911221] [INSPIRE].ADSMathSciNetGoogle Scholar
  32. [32]
    A. Schwimmer, S. Theisen, in preparation.Google Scholar
  33. [33]
    L. Bonora, P. Pasti and M. Bregola, Weyl cocycles, Class. Quant. Grav. 3 (1986) 635 [INSPIRE].CrossRefzbMATHADSMathSciNetGoogle Scholar
  34. [34]
    S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [INSPIRE].ADSMathSciNetGoogle Scholar
  35. [35]
    D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-theorem N = 2 field theories on the three-sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].CrossRefADSGoogle Scholar
  36. [36]
    N.D. Birrell, P.C.W. Davies, Quantum fields in curved space, Cambridge University Press, Cambridge U.K. (1982).zbMATHGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  1. 1.Weizmann Institute of ScienceRehovotIsrael
  2. 2.Institute for Advanced StudyPrincetonUSA

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