MHV Amplitudes

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In theoretical particle physics, maximally helicity violating amplitudes are a more efficient alternative to Feynman diagrams for calculating scattering cross sections. The technique involves decomposing amplitudes into their helicity eigenstates, and results in compact expressions (the Parke-Taylor formula) for tree amplitudes.

Although developed for pure gluon scattering, extensions exist for massive particles, scalars (the Higgs) and for fermions (quarks and their interactions in QCD).

Contents

[edit] The Parke-Taylor Formula

Work done in 1980s by Parke and Taylor [1] (and independently by Berends and Giele [2]) found that when considering the scattering of many gluons, certain classes of amplitude vanish; in particular when fewer than two gluons have negative helicity (and all the rest have positive helicity):

\mathcal{A}(1^+ \ldots n^+) = 0
\mathcal{A}(1^+ \ldots i^- \ldots n^+) = 0

The first non-vanishing case occurs when two gluons have negative helicity. Such amplitudes are known as "maximally helicity violating" and have an extremely simple form in terms of momentum bilinears, independent of the number of gluons present:

\mathcal{A}(1^+\ldots i^- \ldots j^- \ldots n^+) = i(-g)^{n-2} \frac{\langle i \; j\rangle^4}{\langle 1\; \rangle \langle 2 \; \ldots \; (n-1)\rangle \langle n \; 1 \rangle}

The compactness of these amplitudes makes them extremely attractive, particularly with the impending start-up of the LHC, for which it will be necessary to remove the dominant background of standard model events.

[edit] CSW Rules

The MHV were given a geometrical interpretation using Witten's twistor-string theory [3] which in turn inspired a technique of "sewing" MHV amplitudes together (with some off-shell continuation) to build arbitrarily complex tree diagrams. The rules for this formalism are called the CSW rules (after Cachazo, Svrcek and Witten). [4]

The CSW rules can be generalised to the quantum level by forming loop diagrams out of MHV vertices. [5]

There are missing pieces in this framework, most importantly the ( + + − ) vertex, which is clearly non-MHV in form. In pure Yang-Mills theory this vertex vanishes on-shell, but it is necessary to draw the ( + + + + ) amplitude.

The other drawback is the reliance on cut-constructibilty to compute the loop integrals. This therefore cannot recover the rational parts of amplitudes (i.e. those not containing cuts).

[edit] The MHV Lagrangian

A Lagrangian whose perturbation theory gives rise to the CSW rules can be obtained by performing a canonical change of variables on the light-cone Yang-Mills (LCYM) Lagrangian. [6] The LCYM Lagrangrian has the following helicity structure:

L[A] = L^{+-}[A] + L^{-++}[A] + L^{--++}[A].\,

The transformation involves absorbing the non-MHV three-point vertex into the kinetic term in a new field variable:

L^{+-} [A] + L^{++-}[A] \mapsto L^{+-}[B].

When this transformation is solved as a series expansion in the new field variable, it gives rise to an effective Lagrangian with a infinitie series of MHV terms: [7]

L[B] = L^{+-}[B] + L^{--+}[B] + L^{--++}[B] + L^{--+++}[B] + \cdots

The perturbation theory of this Lagrangian has been shown (up to the five-point vertex) to recover the CSW rules. Moreover, the missing amplitudes which plague the CSW approach turn out to be recovered within the MHV Lagrangian framework via evasions of the S-matrix equivalence theorem. [8]

An alternative approach to the MHV Lagrangian recovers the missing pieces mentioned above by using Lorentz-violating counterterms. [9]

[edit] References

  1. ^ [1] "Amplitude for n-Gluon Scattering"
  2. ^ Berends and Giele, Nucl. Phys. B 313, 595 (1989)
  3. ^ [2] "Perturbative Gauge Theory as a String Theory in Twistor Space"
  4. ^ [3] "MHV Vertices and Tree Amplitudes in Gauge Theory"
  5. ^ [4] "Quantum MHV Diagrams"
  6. ^ [5] "The Lagrangian Origins of MHV Rules"
  7. ^ [6] "Structure of the MHV-Rules Lagrangian"
  8. ^ [7] "S-Matrix Equivalence Theorem Evasion and Dimensional Regularisation with the Canonical MHV Lagrangian"
  9. ^ [8] "One-Loop MHV Rules and Pure Yang-Mills"