Skip to main content
SpringerLink
Log in

Scattering Amplitudes

  • Chapter
  • First Online:
Book cover Dualities, Helicity Amplitudes, and Little Conformal Symmetry

Part of the book series: Springer Theses ((Springer Theses))

  • 380 Accesses

Abstract

We use the theory developed in previous chapters to examine the scattering of light by light, dyons by dyons, and how such calculations can remain perturbative even in a strongly coupled theory through an appeal to duality.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The general case at all energy scales was worked out by Karplus and Neuman [7]; details are in Appendix A.1. Additional details about the origin of the Euler-Heisenberg Lagrangian and a version for scalars is given in Appendix A.2.

  2. 2.

    This is the same result (after taking θ → 0) as equation (16) of Ref. [9], where a classical Lorentz force law analogy is used to argue for this form.

  3. 3.

    See Appendix A.4.

  4. 4.

    Taking u to be real and \(u > \Lambda ^{2}\) for simplicity.

  5. 5.

    See [11, 19] for details of the full story with non-zero θ.

  6. 6.

    For the curious: \(M_{\mu,\ell,m} = 2^{m}\left [\frac{2\ell + 1} {4\pi } \frac{(\ell-m)!(\ell+m)!} {(\ell-\mu )!(\ell+\mu )!} \right ]^{1/2}\)

  7. 7.

    In the rest frame, the choices \(k^{\mu } = \frac{m} {2} (1,\hat{\boldsymbol{k}})\) and \(q^{\mu } = \frac{m} {2} (1,-\hat{\boldsymbol{k}})\) work well, for an arbitrary unit vector \(\hat{\boldsymbol{k}}\).

  8. 8.

    Urrutia [38] worked in the extreme relativistic regime; Bazhanov et al. [39] looked at forward scattering limits and also introduced a 4-fermi vertex between monopoles to account for a particular choice of photon propagator gauge.

References

  1. D. Zwanziger, Local Lagrangian quantum field theory of electric and magnetic charges. Phys. Rev. D 3, 880 (1971); Quantum field theory of particles with both electric and magnetic charges. Phys. Rev. 176, 1489 (1968); R.A. Brandt, F. Neri, Remarks on Zwanziger’s local quantum field theory of electric and magnetic charge. Phys. Rev. D 18, 2080 (1978); R.A. Brandt, F. Neri, D. Zwanziger, Lorentz invariance of the quantum field theory of electric and magnetic charge. Phys. Rev. Lett. 40, 147 (1978); Lorentz invariance from classical particle paths in quantum field theory of electric and magnetic charge. Phys. Rev. D 19, 1153 (1979)

  2. N. Seiberg, E. Witten, Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19 (1994) [Erratum-ibid. B 430, 485 (1994)] hep-th/9407087; Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD. Nucl. Phys. B 431, 484 (1994) hep-th/9408099; R.B. Zhang, B.L. Wang, A.L. Carey, J.G. McCarthy, Seiberg-Witten monopoles in three-dimensions. Lett. Math. Phys. 39, 213 (1997). hep-th/9504005

  3. H. Bethe, F. Rohrlich, Small angle scattering of light by a Coulomb field. Phys. Rev. 86(1), 10 (1952)

  4. G. Jarlskog, L. Joensson, S. Pruenster, H.D. Schulz, H.J. Willutzki, G.G. Winter, Measurement of delbrueck scattering and observation of photon splitting at high energies. Phys. Rev. D 8, 3813 (1973)

    Article  ADS  Google Scholar 

  5. D.L. Burke et al., Positron production in multi - photon light by light scattering. Phys. Rev. Lett. 79, 1626 (1997)

    Article  ADS  Google Scholar 

  6. W. Heisenberg, H. Euler, Consequences of Dirac’s theory of positrons. Z. Phys. 98, 714 (1936). physics/0605038

  7. R. Karplus, M. Neuman, Non-linear interactions between electromagnetic fields. Phys. Rev. 80, 380 (1950); R. Karplus, M. Neuman, The scattering of light by light. Phys. Rev. 83, 776 (1951); V.B. Berestetsky, E.M. Lifshitz, L.P. Pitaevsky, Quantum Electrodynamics. Course Of Theoretical Physics, vol. 4 (Pergamon, Oxford, 1982), 652pp. V. Costantini, B. De Tollis, G. Pistoni, Nonlinear effects in quantum electrodynamics. Nuovo Cim. A 2, 733 (1971); Z. Bern, A. De Freitas, L.J. Dixon, A. Ghinculov, H.L. Wong, QCD and QED corrections to light by light scattering. J. High Energy Phys. 0111, 031 (2001). hep-ph/0109079

  8. D.A. Dicus, C. Kao, W.W. Repko, Effective Lagrangians and low-energy photon-photon scattering. Phys. Rev. D 57, 2443 (1998). hep-ph/9709415; G.V. Dunne, Heisenberg-Euler effective Lagrangians: Basics and extensions, in From fields to strings, volume 1, ed. by Shifman, M. et al., pp. 445–522. hep-th/0406216; J. Halter, An effective Lagrangian for photons. Phys. Lett. B 316 (1993) 155; F. Ravndal, Applications of effective Lagrangians in Balholm (1997), Beyond the Standard Model 5, pp. 168–177. hep-ph/9708449; R. Ruffini, S.-S. Xue, Effective Lagrangian of QED. J. Korean Phys. Soc. 49, S715 (2006). hep-th/0609081

  9. S.G. Kovalevich, P. Osland, Y.M. Shnir, E.A. Tolkachev, The effective Lagrangian of QED with a magnetic charge and dyon mass bounds. Phys. Rev. D 55, 5807 (1997). hep-ph/9702402

  10. L.C. Martin, C. Schubert, V.M. Villanueva Sandoval, On the low-energy limit of the QED N photon amplitudes. Nucl. Phys. B 668, 335 (2003). hep-th/0301022

  11. C. Csaki, Y. Shirman, J. Terning, Anomaly constraints on monopoles and dyons. Phys. Rev. D 81, 125028 (2010). hep-th/1003.0448

  12. J.H. Schwarz, A. Sen, Duality symmetric actions. Nucl. Phys. B 411, 35 (1994). hep-th/9304154

  13. G.’t Hooft, Magnetic monopoles in unified gauge theories. Nucl. Phys. B 79, 276 (1974); A.M. Polyakov, Particle spectrum in the quantum field theory. J. Exp. Theor. Phys. Lett. 20, 194 (1974) [Pisma Zh. Eksp. Teor. Fiz. 20, 430 (1974)]

  14. V.P. Gusynin, I.A. Shovkovy, Derivative expansion of the effective action for QED in (2+1)-dimensions and (3+1)-dimensions. J. Math. Phys. 40, 5406 (1999). hep-th/9804143

  15. C.G. Callan, Jr., Dyon-fermion dynamics. Phys. Rev. D 26, 2058 (1982)

    Article  ADS  Google Scholar 

  16. P.A.M. Dirac, Quantized singularities in the electromagnetic field. Proc. R. Soc. Lond. A 133, 60 (1931); The theory of magnetic poles. Phys. Rev. 74, 817 (1948)

  17. L.V. Laperashvili, H.B. Nielsen, Dirac relation and renormalization group equations for electric and magnetic fine structure constants. Mod. Phys. Lett. A 14, 2797 (1999). hep-th/9910101

  18. S.R. Coleman, The magnetic monopole fifty years later (1982). HUTP-82-A032

  19. P.C. Argyres, M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory. Nucl. Phys. B 448, 93 (1995). hep-th/9505062

  20. N. Seiberg, Electric - magnetic duality in supersymmetric nonAbelian gauge theories. Nucl. Phys. B 435, 129 (1995). hep-th/9411149; N. Seiberg, The power of duality: exact results in 4-D SUSY field theory. Int. J. Mod. Phys. A 16, 4365 (2001). Prog. Theor. Phys. Suppl. 123, 337 (1996). hep-th/9506077

  21. S. Weinberg, Photons and gravitons in perturbation theory: derivation of Maxwell’s and Einstein’s equations. Phys. Rev. 138, B988 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  22. J.J. Thomson, On momentum in the electric field. Philos. Mag. 8, 331 (1904)

    Article  MATH  Google Scholar 

  23. J.S. Schwinger, Magnetic charge and quantum field theory. Phys. Rev. 144, 1087 (1966); Sources and magnetic charge. Phys. Rev. 173, 1536 (1968); Magnetic charge and the charge quantization condition. Phys. Rev. D 12, 3105 (1975); A magnetic model of matter. Science 165, 757 (1969)

  24. V.A. Rubakov, Superheavy Magnetic Monopoles and Proton Decay J. Exp. Theor. Phys. Lett. 33, 644 (1981) [Pisma Zh. Eksp. Teor. Fiz. 33, 658 (1981)]; C.G. Callan Jr., Disappearing dyons. Phys. Rev. D 25, 2141 (1982)

  25. C. Csaki, Y. Shirman, J. Terning, Monopole-fermion scattering and the Rubakov-Callan effect. manuscript, (2016); Electroweak symmetry breaking from monopole condensation. Phys. Rev. Lett. 106, 041802 (2011) hep-ph/1003.1718

  26. C.R. Hagen, Noncovariance of the Dirac monopole. Phys. Rev. 140, B804 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  27. T.T. Wu, C.N. Yang, Dirac monopole without strings: monopole harmonics. Nucl. Phys. B 107, 365 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  28. Y. Kazama, C.N. Yang, A.S. Goldhaber, Scattering of a dirac particle with charge Ze by a fixed magnetic monopole. Phys. Rev. D 15, 2287 (1977)

    Article  ADS  Google Scholar 

  29. W.J. Marciano, I.J. Muzinich, Exact Fermion dyon scattering solutions. Phys. Rev. D 28, 973 (1983)

    Article  ADS  Google Scholar 

  30. J.S. Schwinger, K.A. Milton, W.y. Tsai, L.L. DeRaad Jr., D.C. Clark, Nonrelativistic dyon-dyon scattering. Ann. Phys. 101, 451 (1976)

  31. D. Zwanziger, Angular distributions and a selection rule in charge-pole reactions. Phys. Rev. D 6, 458 (1972)

    Article  ADS  Google Scholar 

  32. J. Kuczmarski, SpinorsExtras - mathematica implementation of massive spinor-helicity formalism. hep-ph/1406.5612

  33. R. Kleiss, W.J. Stirling, Spinor techniques for calculating p anti-p → W+- / Z0 + Jets. Nucl. Phys. B 262, 235 (1985)

    Article  ADS  Google Scholar 

  34. A. Hall, Massive quark-gluon scattering amplitudes at tree level. Phys. Rev. D 77, 025011 (2008). hep-ph/0710.1300

  35. C. Schwinn, S. Weinzierl, On-shell recursion relations for all Born QCD amplitudes. J. High Energy Phys. 0704, 072 (2007). hep-ph/0703021

  36. E. Witten, Dyons of charge e θ∕2π. Phys. Lett. B 86 (1979) 283; Monopoles and four manifolds. Math. Res. Lett. 1, 769 (1994). hep-th/9411102; On S-duality in Abelian gauge theory Selecta Math. 1, 383 (1995). hep-th/9505186; C. Vafa, E. Witten, A strong coupling test of S-duality. Nucl. Phys. B 431, 3 (1994). hep-th/9408074

  37. K. Colwell, J. Terning, S-duality and helicity amplitudes. J. High Energy Phys. 1603, 068 (2016). hep-th/1510.07627

  38. L.F. Urrutia, Zeroth order Eikonal approximation in relativistic charged particle monopole scattering. Phys. Rev. D 18, 3031 (1978)

    Article  ADS  Google Scholar 

  39. V.V. Bazhanov, V.I. Borodulin, G.P. Pronko, L.D. Solovev, Small angle electron - monopole scattering in quantum theory Of Dirac-schwinger monopole. Theor. Math. Phys. 40, 795 (1980)

    Article  Google Scholar 

  40. L.P. Gamberg, K.A. Milton, Dual quantum electrodynamics: dyon-dyon and charge monopole scattering in a high-energy approximation. Phys. Rev. D 61, 075013 (2000) hep-ph/9910526; L.P. Gamberg, K.A. Milton, Eikonal scattering of monopoles and dyons in dual QED. hep-ph/0005016

  41. W. Deans, Quantum field theory of Dirac monopoles and the charge quantization condition. Nucl. Phys. B 197, 307 (1982)

    Article  ADS  Google Scholar 

  42. A. De Rujula, Effects of virtual monopoles. Nucl. Phys. B 435, 257 (1995). hep-th/9405191

  43. A.Y. Ignatiev, G.C. Joshi, Dirac magnetic monopole and the discrete symmetries. Chaos Solitons Fractals 11, 1411 (2000). hep-ph/9710553

  44. A. Rabl, Perturbation theory for magnetic monopoles. Phys. Rev. 179, 1363 (1969)

    Article  ADS  Google Scholar 

  45. E.A. Tolkachev, L.M. Tomilchik, Y.M. Shnir, On the space reflections definition problem in a magnetic charge theory. Turk. J. Phys. 21, 546 (1997). quant-ph/9603003

Download references

Author information

Authors and Affiliations

  1. Physics, Diablo Valley College, Pleasant Hill, CA, USA

    Kitran Macey M. Colwell

Authors
  1. Kitran Macey M. Colwell
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Colwell, K.M.M. (2017). Scattering Amplitudes. In: Dualities, Helicity Amplitudes, and Little Conformal Symmetry. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-67392-9_4

Download citation

Publish with us

Policies and ethics

Search

Navigation