Abstract
The spinor helicity method has allowed an enormous simplification in the calculation of certain scattering amplitudes in the massless limit. By decomposing massive momenta into massless spinors, the computational difficulties of monopole and dyon scattering are greatly reduced.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The van der Waerden notation here assigns undotted indices to SU(2) L and dotted to SU(2) R .
- 2.
We will tend to use the same variable for the four-vector, the bispinor, and the spinors. Which is meant should be clear from context.
- 3.
There are occasional notational discrepancies in the literature; here, we follow thea a , \(_{\dot{a}}^{\dot{a}}\) contraction convention, which is more common. Other common notations:
$$\displaystyle\begin{array}{rcl} & k_{a} = \vert k\rangle = \vert k^{+}\rangle = u_{+}(k) = v_{-}(k),\quad k^{\dot{a}} = \vert k] = \vert k^{-}\rangle = u_{-}(k) = v_{+}(k),& {}\\ & k^{a} =\langle k\vert =\langle k^{-}\vert = \overline{u}_{-}(k) = \overline{v}_{+}(k),\quad k_{\dot{a}} = [k\vert =\langle k^{+}\vert = \overline{u}_{+}(k) = \overline{v}_{-}(k)& {}\\ \end{array}$$ - 4.
Or one can keep p μ real and switch to spacetime signature (+, −, +, −), in which case | p〉 and | p] are real and independent.
- 5.
This procedure has made calculating tree-level QCD helicity amplitudes more of a joy than a pain [4].
- 6.
The little or isotropy group is the group of transformations that leave the momentum of an on-shell particle fixed. For a massless particle in a frame where p μ = (E, 0, 0, E), we can see that the little group is SO(2) = U(1). (Technically, it’s E(2), the Euclidean group in two-dimensions, but for angular momentum concerns we only worry about the rotational subgroup.)
- 7.
We will always take all particles outgoing for symmetry.
- 8.
Unless it is a constant, as in ϕ 3 theory, or in a (+, −, +, −) spacetime signature.
- 9.
Boels [8] uses the “flat” notation p ⊥ = p ♭. When constructing spinors for massive vectors, we will assign | i〉 = (p i ⊥) a and | i′〉 = (q i ) a , etc. for notational ease.
References
A. Zee, Quantum Field Theory in a Nutshell, 2nd edn. (Princeton University, Press, Princeton, 2010), p. 486
H. Elvang, Y.t. Huang, Scattering amplitudes. hep-th/1308.1697
R. Boels, K.J. Larsen, N.A. Obers, M. Vonk, MHV, CSW and BCFW: field theory structures in string theory amplitudes. J. High Energy Phys. 0811, 015 (2008). hep-th/0808.2598
L.J. Dixon, A brief introduction to modern amplitude methods. hep-ph/1310.5353; Calculating scattering amplitudes efficiently, in Boulder (1995), QCD and Beyond, pp. 539–582. hep-ph/9601359
E. Witten, Perturbative gauge theory as a string theory in twistor space. Commun. Math. Phys. 252, 189 (2004). hep-th/0312171; R. Britto, F. Cachazo, B. Feng, New recursion relations for tree amplitudes of gluons. Nucl. Phys. B 715, 499 (2005). hep-th/0412308; R. Britto, F. Cachazo, B. Feng, E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory. Phys. Rev. Lett. 94, 181602 (2005). hep-th/0501052
S.P. Martin, A Supersymmetry primer. Adv. Ser. Direct. High Energy Phys. 21, 1 (2010) [Adv. Ser. Direct. High Energy Phys. 18, 1 (1998)] hep-ph/9709356; J. Terning, Modern Supersymmetry: Dynamics and Duality. International Series of Monographs on Physics, vol. 132 (Oxford University Press, Oxford, 2006)
S. Dittmaier, Weyl-van der Waerden formalism for helicity amplitudes of massive particles. Phys. Rev. D 59, 016007 (1998). hep-ph/9805445
R. Boels, Covariant representation theory of the Poincare algebra and some of its extensions. J. High Energy Phys. 1001, 010 (2010). hep-th/0908.0738
J. Kuczmarski, SpinorsExtras - Mathematica implementation of massive spinor-helicity formalism. hep-ph/1406.5612
S.J. Parke, T.R. Taylor, An amplitude for n Gluon scattering. Phys. Rev. Lett. 56, 2459 (1986)
F. Cachazo, P. Svrcek, E. Witten, MHV vertices and tree amplitudes in gauge theory. J. High Energy Phys. 0409, 006 (2004). hep-th/0403047; J.M. Drummond, J. Henn, G.P. Korchemsky, E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory. Nucl. Phys. B 828, 317 (2010). hep-th/0807.1095; A. Brandhuber, P. Heslop, G. Travaglini, MHV amplitudes in N=4 super Yang-Mills and Wilson loops. Nucl. Phys. B 794, 231 (2008). hep-th/0707.1153; Z. Bern, L.J. Dixon, D.A. Kosower, R. Roiban, M. Spradlin, C. Vergu, A. Volovich, The two-loop Six-Gluon MHV amplitude in maximally supersymmetric Yang-Mills theory. Phys. Rev. D 78, 045007 (2008). hep-th/0803.1465
N. Arkani-Hamed, J. Trnka, The Amplituhedron. J. High Energy Phys. 1410, 030 (2014). hep-th/1312.2007; N. Arkani-Hamed, J. Trnka, Into the amplituhedron. J. High Energy Phys. 1412, 182 (2014). hep-th/1312.7878; N. Arkani-Hamed, A. Hodges, J. Trnka, Positive amplitudes in the amplituhedron. J. High Energy Phys. 1508, 030 (2015). hep-th/1412.8478
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Colwell, K.M.M. (2017). Spinor Helicity Formalism. In: Dualities, Helicity Amplitudes, and Little Conformal Symmetry. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-67392-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-67392-9_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-67391-2
Online ISBN: 978-3-319-67392-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)