Skip to main content
SpringerLink
Log in

Tree-level amplitudes in the nonlinear sigma model

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We study in detail the general structure and further properties of the tree-level amplitudes in the SU(N) nonlinear sigma model. We construct the flavor-ordered Feynman rules for various parameterizations of the SU(N) fields U (x), write down the Berends-Giele relations for the semi-on-shell currents and discuss their efficiency for the amplitude calculation in comparison with those of renormalizable theories. We also present an explicit form of the partial amplitudes up to ten external particles. It is well known that the standard BCFW recursive relations cannot be used for reconstruction of the the on-shell amplitudes of effective theories like the SU(N) nonlinear sigma model because of the inappropriate behavior of the deformed on-shell amplitudes at infinity. We discuss possible generalization of the BCFW approach introducing “BCFW formula with subtractions” and with help of Berends-Giele relations we prove particular scaling properties of the semi-on-shell amplitudes of the SU(N) nonlinear sigma model under specific shifts of the external momenta. These results allow us to define alternative deformation of the semi-on-shell amplitudes and derive BCFW-like recursion relations. These provide a systematic and effective tool for calculation of Goldstone bosons scattering amplitudes and it also shows the possible applicability of on-shell methods to effective field theories. We also use these BCFW-like relations for the investigation of the Adler zeroes and double soft limit of the semi-on-shell amplitudes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.A. Cronin, Phenomenological model of strong and weak interactions in chiral U(3) × U(3), Phys. Rev. 161 (1967) 1483 [INSPIRE].

    Article  ADS  Google Scholar 

  2. S. Weinberg, Dynamical approach to current algebra, Phys. Rev. Lett. 18 (1967) 188 [INSPIRE].

    Article  ADS  Google Scholar 

  3. S. Weinberg, Nonlinear realizations of chiral symmetry, Phys. Rev. 166 (1968) 1568 [INSPIRE].

    Article  ADS  Google Scholar 

  4. L.S. Brown, Field theory of chiral symmetry, Phys. Rev. 163 (1967) 1802 [INSPIRE].

    Article  ADS  Google Scholar 

  5. P. Chang and F. Gursey, Unified formulation of effective nonlinear pion-nucleon Lagrangians, Phys. Rev. 164 (1967) 1752 [INSPIRE].

    Article  ADS  Google Scholar 

  6. S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [INSPIRE].

    ADS  Google Scholar 

  7. J. Gasser and H. Leutwyler, Chiral perturbation theory to one loop, Annals Phys. 158 (1984) 142 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. J. Gasser and H. Leutwyler, Chiral perturbation theory: expansions in the mass of the strange quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].

    Article  ADS  Google Scholar 

  9. L. Susskind and G. Frye, Algebraic aspects of pionic duality diagrams, Phys. Rev. D 1 (1970) 1682 [INSPIRE].

    ADS  Google Scholar 

  10. H. Osborn, Implications of adler zeros for multipion processes, Lett. Nuovo Cim. 2S1 (1969) 717 [INSPIRE].

    Article  Google Scholar 

  11. J.R. Ellis and B. Renner, On the relationship between chiral and dual models, Nucl. Phys. B 21 (1970) 205 [INSPIRE].

    ADS  Google Scholar 

  12. M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [INSPIRE].

    Article  ADS  Google Scholar 

  13. L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359 [INSPIRE].

  14. B. Feng and M. Luo, An introduction to on-shell recursion relations, arXiv:1111.5759 [INSPIRE].

  15. J. Drummond, Hidden simplicity of gauge theory amplitudes, Class. Quant. Grav. 27 (2010) 214001 [arXiv:1010.2418] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  16. Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop N point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  17. Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].

    Article  ADS  Google Scholar 

  18. R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  20. N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  21. C. Berger, Z. Bern, L.J. Dixon, F. Febres Cordero, D. Forde et al., Precise predictions for W+ 4 jet production at the large hadron collider, Phys. Rev. Lett. 106 (2011) 092001 [arXiv:1009.2338] [INSPIRE].

    Article  ADS  Google Scholar 

  22. E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  24. L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  25. L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for scattering amplitudes, J. Phys. A 43 (2010) 485401 [arXiv:1002.2459] [INSPIRE].

    MathSciNet  Google Scholar 

  26. J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. Z. Bern, J. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  30. N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov et al., Scattering amplitudes and the positive grassmannian, arXiv:1212.5605 [INSPIRE].

  31. N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. L. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. L. Mason and D. Skinner, The complete planar S-matrix of N = 4 SYM as a Wilson loop in twistor space, JHEP 12 (2010) 018 [arXiv:1009.2225] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  34. J. Bedford, A. Brandhuber, B.J. Spence and G. Travaglini, A recursion relation for gravity amplitudes, Nucl. Phys. B 721 (2005) 98 [hep-th/0502146] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  35. F. Cachazo and P. Svrček, Tree level recursion relations in general relativity, hep-th/0502160 [INSPIRE].

  36. C. Cheung, On-shell recursion relations for generic theories, JHEP 03 (2010) 098 [arXiv:0808.0504] [INSPIRE].

    Article  ADS  Google Scholar 

  37. S.J. Parke and T. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].

    Article  ADS  Google Scholar 

  38. F.A. Berends and W. Giele, Recursive calculations for processes with N gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].

    Article  ADS  Google Scholar 

  39. S. Badger, E.N. Glover, V. Khoze and P. Svrček, Recursion relations for gauge theory amplitudes with massive particles, JHEP 07 (2005) 025 [hep-th/0504159] [INSPIRE].

    Article  ADS  Google Scholar 

  40. B. Feng, J. Wang, Y. Wang and Z. Zhang, BCFW recursion relation with nonzero boundary contribution, JHEP 01 (2010) 019 [arXiv:0911.0301] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  41. B. Feng and C.-Y. Liu, A note on the boundary contribution with bad deformation in gauge theory, JHEP 07 (2010) 093 [arXiv:1004.1282] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  42. B. Feng and Z. Zhang, Boundary contributions using fermion pair deformation, JHEP 12 (2011) 057 [arXiv:1109.1887] [INSPIRE].

    Article  ADS  Google Scholar 

  43. K. Risager, A direct proof of the CSW rules, JHEP 12 (2005) 003 [hep-th/0508206] [INSPIRE].

    Article  ADS  Google Scholar 

  44. T. Cohen, H. Elvang and M. Kiermaier, On-shell constructibility of tree amplitudes in general field theories, JHEP 04 (2011) 053 [arXiv:1010.0257] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  45. P. Benincasa and E. Conde, On the tree-level structure of scattering amplitudes of massless particles, JHEP 11 (2011) 074 [arXiv:1106.0166] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  46. B. Feng, Y. Jia, H. Lüo and M. Luo, Roots of amplitudes, arXiv:1111.1547 [INSPIRE].

  47. J. Bijnens, K. Kampf and S. Lanz, Leading logarithms in N-flavour mesonic chiral perturbation theory, Nucl. Phys. B 873 (2013) 137 [arXiv:1303.3125] [INSPIRE].

    Article  Google Scholar 

  48. R.F. Dashen and M. WEinstein, Soft pions, chiral symmetry and phenomenological Lagrangians, Phys. Rev. 183 (1969) 1261 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  49. K. Kampf, J. Novotny and J. Trnka, Recursion relations for tree-level amplitudes in the SU(N) non-linear σ-model, arXiv:1212.5224 [INSPIRE].

  50. N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  51. S. Weinberg, The quantum theory of fields. Vol. 2: modern applications, Cambridge University Press, Cambridge U.K. (1996).

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University in Prague, CZ-18000, Prague, Czech Republic

    Karol Kampf, Jirí Novotný & Jaroslav Trnka

  2. Department of Physics, Princeton University, Princeton, New Jersey, 08544, U.S.A.

    Jaroslav Trnka

Authors
  1. Karol Kampf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jirí Novotný
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jaroslav Trnka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karol Kampf.

Additional information

ArXiv ePrint: 1304.3048

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kampf, K., Novotný, J. & Trnka, J. Tree-level amplitudes in the nonlinear sigma model. J. High Energ. Phys. 2013, 32 (2013). https://doi.org/10.1007/JHEP05(2013)032

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP05(2013)032

Keywords

Search

Navigation