Parke-Taylor formula

The Parke-Taylor formula , after Stephen Parke and Tomasz Taylor , is a formula in quantum chromodynamics . It gives the matrix element of a scattering of color-ordered gluons , of which exactly have the same helicity , in the lowest order of perturbation theory . These scattering processes are called maximum helicity violating (MHV) scattering amplitudes, since matrix elements in which all gluons have the same or only one gluon different helicity are identical zero. ${\displaystyle n}$ ${\displaystyle n-2}$

The Parke-Taylor formula is:

${\displaystyle {\tilde {\mathcal {A}}}_{n}(1^{+},2^{+},\dots ,j^{-},\dots ,k^{-},\dots ,n^{+})={\frac {\langle jk\rangle ^{4}}{\langle 12\rangle \langle 23\rangle \dots \langle n1\rangle }}}$

It denotes:

• ${\displaystyle {\tilde {\mathcal {A}}}_{n}}$ the color-ordered matrix element,
• ${\displaystyle \pm }$ the helicity of the gluons involved and
• ${\displaystyle \langle ij\rangle =e^{\mathrm {i} \phi }{\sqrt {2p_{i}\cdot p_{j}}}}$with the four-pulse of the gluons involved and any real number , as long as the pulses of the gluons themselves are real. ${\displaystyle p}$ ${\displaystyle \phi }$

The matrix element for the case that the two gluons are positive helicity can be calculated by complex conjugation of the above formula, since quantum chromodynamics is parity invariant .

Compared to a direct calculation using Feynman diagrams , the Parke and Taylor result is remarkably simple. The proof of the Parke-Taylor formula takes place via recursion in the spinor-helicity formalism and was carried out in 1988 by Frederik Berends and Walter Giele .

Individual evidence

1. ↑ Stephen Parke and Tomasz Taylor: Amplitude for -Gluon Scattering ${\displaystyle n}$ . In: Physical Review Letters . tape 56 , no. 23 , 1986, pp. 2459-2460 , doi : 10.1103 / PhysRevLett.56.2459 (English).
2. ^ Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 , pp. 550 (English).
3. ↑ Frederik Berends and Walter Giele: Recursive Calculations for Processes with Gluons ${\displaystyle n}$ . In: Nuclear Physics B . tape 306 , no. 4 , 1988, pp. 759-808 , doi : 10.1016 / 0550-3213 (88) 90442-7 (English).