Journal of High Energy Physics

, 2016:172 | Cite as

Bremsstrahlung function, leading Lüscher correction at weak coupling and localization

  • Marisa Bonini
  • Luca Griguolo
  • Michelangelo Preti
  • Domenico Seminara
Open Access
Regular Article - Theoretical Physics


We discuss the near BPS expansion of the generalized cusp anomalous dimension with L units of R-charge. Integrability provides an exact solution, obtained by solving a general TBA equation in the appropriate limit: we propose here an alternative method based on supersymmetric localization. The basic idea is to relate the computation to the vacuum expectation value of certain 1/8 BPS Wilson loops with local operator insertions along the contour. These observables localize on a two-dimensional gauge theory on S 2, opening the possibility of exact calculations. As a test of our proposal, we reproduce the leading Lüscher correction at weak coupling to the generalized cusp anomalous dimension. This result is also checked against a genuine Feynman diagram approach in \( \mathcal{N}=4 \) Super Yang-Mills theory.


Wilson ’t Hooft and Polyakov loops Duality in Gauge Field Theories 1/N Expansion 


Open Access

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  1. [1]
    N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  2. [2]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].CrossRefADSMathSciNetMATHGoogle Scholar
  3. [3]
    G. Arutyunov and S. Frolov, String hypothesis for the AdS 5 × S 5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  4. [4]
    G. Arutyunov and S. Frolov, Thermodynamic Bethe Ansatz for the AdS 5 × S 5 Mirror Model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  5. [5]
    N. Gromov, V. Kazakov and P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar \( \mathcal{N}=4 \) Supersymmetric Yang-Mills Theory,Phys. Rev. Lett. 103(2009) 131601 [arXiv:0901.3753] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe Ansatz for planar AdS/CFT: A Proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum Spectral Curve for Planar \( \mathcal{N}=4 \) super-Yang-Mills Theory, Phys. Rev. Lett. 112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].CrossRefADSGoogle Scholar
  8. [8]
    J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in \( \mathcal{N}=4 \) supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  9. [9]
    N. Drukker and D.J. Gross, An Exact prediction of \( \mathcal{N}=4 \) SUSYM theory for string theory, J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [INSPIRE].CrossRefADSMathSciNetMATHGoogle Scholar
  10. [10]
    N. Drukker, 1/4 BPS circular loops, unstable world-sheet instantons and the matrix model, JHEP 09 (2006) 004 [hep-th/0605151] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, More supersymmetric Wilson loops, Phys. Rev. D 76 (2007) 107703 [arXiv:0704.2237] [INSPIRE].ADSMathSciNetGoogle Scholar
  12. [12]
    N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Supersymmetric Wilson loops on S 3, JHEP 05 (2008) 017 [arXiv:0711.3226] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  13. [13]
    V. Pestun, Localization of the four-dimensional \( \mathcal{N}=4 \) SYM to a two-sphere and 1/8 BPS Wilson loops, JHEP 12 (2012) 067 [arXiv:0906.0638] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  14. [14]
    S. Giombi and V. Pestun, Correlators of Wilson Loops and Local Operators from Multi-Matrix Models and Strings in AdS, JHEP 01 (2013) 101 [arXiv:1207.7083] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  15. [15]
    S. Giombi and V. Pestun, Correlators of local operators and 1/8 BPS Wilson loops on S 2 from 2d YM and matrix models, JHEP 10 (2010) 033 [arXiv:0906.1572] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  16. [16]
    A.M. Polyakov, Gauge Fields as Rings of Glue, Nucl. Phys. B 164 (1980) 171 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  17. [17]
    G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].CrossRefADSGoogle Scholar
  18. [18]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech. (2007) P01021 [hep-th/0610251] [INSPIRE].
  19. [19]
    Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].ADSMathSciNetGoogle Scholar
  20. [20]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A Semiclassical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  21. [21]
    D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, JHEP 08 (2012) 134 [arXiv:1203.1913] [INSPIRE].CrossRefADSGoogle Scholar
  22. [22]
    N. Drukker, Integrable Wilson loops, JHEP 10 (2013) 135 [arXiv:1203.1617] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  23. [23]
    N. Drukker and V. Forini, Generalized quark-antiquark potential at weak and strong coupling, JHEP 06 (2011) 131 [arXiv:1105.5144] [INSPIRE].CrossRefADSGoogle Scholar
  24. [24]
    N. Gromov and F. Levkovich-Maslyuk, Quantum Spectral Curve for a Cusped Wilson Line in \( \mathcal{N}=4 \) SYM,arXiv:1510.02098[INSPIRE].
  25. [25]
    D. Correa, J. Henn, J. Maldacena and A. Sever, An exact formula for the radiation of a moving quark in \( \mathcal{N}=4 \) super Yang-Mills, JHEP 06 (2012) 048 [arXiv:1202.4455] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    B. Fiol, B. Garolera and A. Lewkowycz, Exact results for static and radiative fields of a quark in \( \mathcal{N}=4 \) super Yang-Mills, JHEP 05 (2012) 093 [arXiv:1202.5292] [INSPIRE].CrossRefADSGoogle Scholar
  27. [27]
    N. Gromov and A. Sever, Analytic Solution of Bremsstrahlung TBA, JHEP 11 (2012) 075 [arXiv:1207.5489] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  28. [28]
    N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Analytic Solution of Bremsstrahlung TBA II: Turning on the Sphere Angle, JHEP 10 (2013) 036 [arXiv:1305.1944] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  29. [29]
    G. Sizov and S. Valatka, Algebraic Curve for a Cusped Wilson Line, JHEP 05 (2014) 149 [arXiv:1306.2527] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  30. [30]
    N. Drukker and J. Plefka, Superprotected n-point correlation functions of local operators in \( \mathcal{N}=4 \) super Yang-Mills,JHEP 04(2009)052 [arXiv:0901.3653] [INSPIRE].
  31. [31]
    S. Giombi, V. Pestun and R. Ricci, Notes on supersymmetric Wilson loops on a two-sphere, JHEP 07 (2010) 088 [arXiv:0905.0665] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  32. [32]
    A. Bassetto, L. Griguolo, F. Pucci and D. Seminara, Supersymmetric Wilson loops at two loops, JHEP 06 (2008) 083 [arXiv:0804.3973] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  33. [33]
    D. Young, BPS Wilson Loops on S 2 at Higher Loops, JHEP 05 (2008) 077 [arXiv:0804.4098] [INSPIRE].CrossRefADSGoogle Scholar
  34. [34]
    A. Bassetto, L. Griguolo, F. Pucci, D. Seminara, S. Thambyahpillai and D. Young, Correlators of supersymmetric Wilson-loops, protected operators and matrix models in \( \mathcal{N}=4 \) SYM, JHEP 08 (2009) 061 [arXiv:0905.1943] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  35. [35]
    A. Bassetto, L. Griguolo, F. Pucci, D. Seminara, S. Thambyahpillai and D. Young, Correlators of supersymmetric Wilson loops at weak and strong coupling, JHEP 03 (2010) 038 [arXiv:0912.5440] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  36. [36]
    M. Bonini, L. Griguolo and M. Preti, Correlators of chiral primaries and 1/8 BPS Wilson loops from perturbation theory, JHEP 09 (2014) 083 [arXiv:1405.2895] [INSPIRE].CrossRefADSGoogle Scholar
  37. [37]
    D. Correa, J. Henn, J. Maldacena and A. Sever, The cusp anomalous dimension at three loops and beyond, JHEP 05 (2012) 098 [arXiv:1203.1019] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  38. [38]
    K. Zarembo, Supersymmetric Wilson loops, Nucl. Phys. B 643 (2002) 157 [hep-th/0205160] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  39. [39]
    A.A. Migdal, Gauge Transitions in Gauge and Spin Lattice Systems, Sov. Phys. JETP 42 (1975) 743 [INSPIRE].ADSGoogle Scholar
  40. [40]
    E. Witten, On quantum gauge theories in two-dimensions, Commun. Math. Phys. 141 (1991) 153 [INSPIRE].CrossRefADSMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Marisa Bonini
    • 1
  • Luca Griguolo
    • 1
  • Michelangelo Preti
    • 1
  • Domenico Seminara
    • 2
  1. 1.Dipartimento di Fisica e Scienze della TerraUniversità di Parma and INFN Gruppo Collegato di ParmaParmaItaly
  2. 2.Dipartimento di FisicaUniversità di Firenze and INFN Sezione di FirenzeSesto FiorentinoItaly

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