In theoretical particle physics

Particle physics

Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, maximally helicity violating amplitudes are amplitudes with n external gauge bosons, where n-2 gauge bosons have a particular helicity and the other two have the opposite helicity. These amplitudes are called MHV amplitudes, because at tree level, they violate helicity conservation to the maximum extent possible. The tree amplitudes in which all gauge bosons have the same helicity or all but one have the same helicity vanish.

MHV amplitudes may be calculated very efficiently by means of the Parke Taylor formula.

Although developed for pure gluon scattering, extensions exist for massive particles, scalars (the Higgs

Higgs boson

The Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...

) and for fermions (quarks and their interactions in QCD

Quantum chromodynamics

In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

).

The Parke–Taylor amplitudes

Work done in 1980s by Stephen Parke

Stephen Parke

Stephen Parke is a New Zealand physicist. He is currently a Senior Scientist and Head of the Theoretical Physics Department at the Fermi National Accelerator Laboratory .-Biography:...

 and Tomasz Taylor 
found that when considering the scattering of many gluons, certain classes of amplitude vanish at tree level; in particular when fewer than two gluons have negative helicity (and all the rest have positive helicity):

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:

The compactness of these amplitudes makes them extremely attractive, particularly with the impending start-up of the LHC

LHC

LHC may refer to:* Large Hadron Collider, a particle accelerator and collider located on the Franco-Swiss border near Geneva, SwitzerlandLHC also may refer to:* La hora Chanante, a Spanish comedy television show...

, for which it will be necessary to remove the dominant background of standard model

Standard Model

The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

 events.
A rigorous derivation of the Parke-Taylor amplitudes was given by Berends and Giele .

CSW rules

The MHV were given a geometrical interpretation using Witten's twistor-string theory

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).
The CSW rules can be generalised to the quantum level by forming loop diagrams out of MHV vertices.
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 construct the
amplitude at one loop. This amplitude vanishes in any supersymmetric theory, but does not in the non-supersymmetric case.

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

The MHV Lagrangian

A Lagrangian

Lagrangian

The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...

 whose perturbation theory gives rise to the CSW rules can be obtained by performing a canonical

Canonical transformation

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates  →  that preserves the form of Hamilton's equations , although it...

 change of variables on the light-cone Yang-Mills (LCYM) Lagrangian.
The LCYM Lagrangrian has the following helicity structure:

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

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

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

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