Stokes stream function - Revision history

en>Crowsnest: /* Vorticity */ use consistently u for velocity

2014-11-30T22:30:25Z

Vorticity: use consistently u for velocity

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en>Crowsnest: /* Streamlines as curves of constant stream function */ replace en-dashes (rest of the article uses em dashes); 2x":"

2014-02-09T16:11:16Z

Streamlines as curves of constant stream function: replace en-dashes (rest of the article uses em dashes); 2x":"

← Older revision		Revision as of 18:11, 9 February 2014
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	~~{{Quantum field theory}}~~
	In theoretical [[particle physics]], '''maximally helicity violating amplitudes''' are amplitudes with n external gauge bosons, where n-2 gauge bosons have a particular [[helicity (particle physics)\|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 boson\|Higgs]]) and for fermions ([[quarks]] and their interactions in [[quantum chromodynamics\|QCD]]).		I'm a 47 years old and study at the college (Political Science).<br>In my spare time I'm trying to teach myself Turkish. I've been twicethere and look forward to returning sometime near future. I love to read, preferably on xbox code generator my kindle. I like to watch Game of Thrones and Arrested Development as well as documentaries about nature. I love Computer programming.<br><br>My web site ... xbox code generator password, [http://tinyurl.com/k4dcsve tinyurl.com],

	~~== The Parke–Taylor amplitudes ==~~

	~~Work done in 1980s by [[Stephen Parke]] and [[Tomasz Taylor]] <ref>[http://prola.aps.org/abstract/PRL/v56/i23/p2459_1] "Amplitude for n-Gluon Scattering", Parke~~ and ~~Taylor, Phys. Rev. Lett. 56, 2459 (1986)</ref>~~
	~~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):~~

	~~: <math> \begin{align} \mathcal{A}~~(~~1^+ \cdots n^+~~) ~~&= 0, \\~~
	~~\mathcal{A}(1^+ \cdots i^- \cdots n^+) &= 0~~. ~~\end{align}~~ <~~/math~~>

	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:

	~~: <math> \mathcal{A}(1^+\cdots i^- \cdots j^- \cdots n^+)~~
	~~= i(-g)^{n-2} \frac{\langle i \; j\rangle^4}{\langle 1\;2 \rangle \langle 2\;3 \rangle \cdots \langle (n-1)\;n\rangle \langle n \; 1 \rangle}</math>~~

	~~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.~~
	~~A rigorous derivation of the Parke-Taylor amplitudes was given by Berends and Giele~~.~~<ref>[http://arXiv.org/abs/hep-th/0] Berends and Giele, Nucl. Phys. B 306, 759 (1988)</ref>~~

	~~== CSW rules ==~~

	~~The MHV were given a geometrical interpretation using Witten~~'~~s [[twistor string theory]]~~
	~~<ref>[http://arXiv.org/abs/hep-th/0312171] "Perturbative Gauge Theory as a String Theory in Twistor Space"</ref>~~
	~~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).~~
	~~<ref>[http://arXiv.org/abs/hep-th/0403047] "MHV Vertices and Tree Amplitudes in Gauge Theory"</ref>~~

	~~The CSW rules can be generalised~~ to ~~the quantum level by forming loop diagrams out of MHV vertices.~~
	~~<ref>[http://arXiv~~.~~org/abs/hep-th/0609011] "Quantum MHV Diagrams"</ref>~~

	~~There are missing pieces in this framework, most importantly the <math>(++-)</math> 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 <math>(++++)</math>~~
	~~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]] whose perturbation theory gives rise~~ to ~~the CSW rules can be obtained by performing a [[Canonical transformation\|canonical]] change~~ of ~~variables on the [[light-cone]] Yang-Mills (LCYM) Lagrangian.<ref>[http://arXiv.org/abs/hep-th/0511264] "The Lagrangian Origins of MHV Rules"</ref>~~
	~~The LCYM Lagrangrian has the following helicity structure:~~

	~~: <math> L[A] = L^{+-}[A] + L^{-++}[A] + L^{--++}[A]. </math>~~

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

	~~: <math> L^{+-} [A] + L^{++-}[A] = L^{+-}[B]. </math>~~

	~~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:<ref>[http://arXiv.org/abs/hep-th/0605121] "Structure of the MHV-Rules Lagrangian"</ref>~~

	~~: <math> L[B] = L^{+-}[B] + L^{--+}[B] + L^{--++}[B] + L^{--+++}[B] + \cdots </math>~~

	~~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~~.
	<~~ref~~>~~[http://arXiv.org/abs/hep-th/0703286] "S-Matrix Equivalence Theorem Evasion and Dimensional Regularisation with the Canonical MHV Lagrangian"~~<~~/ref~~>

	~~An alternative approach to the MHV Lagrangian recovers the missing pieces mentioned above by using Lorentz-violating counterterms~~.~~<ref>[http://arXiv~~.~~org/abs/0704~~.~~0245] "One-Loop MHV Rules and Pure Yang-Mills"</ref>~~

	~~== BCFW recursion ==~~

	~~{{main\|BCFW recursion}}~~
	~~'''BCFW recursion'''~~, ~~also known as the Britto-Cachazo-Feng-Witten (BCFW) on-shell recursion method, is a way of calculating scattering amplitudes. <ref>~~[http://~~arxiv~~.~~org/abs/hep-th/0501052] “Direct proof of tree-level~~
	~~recursion relation in Yang-Mills theory”<~~/~~ref> Extensive use is now made of these techniques~~. ~~<ref>[http://arxiv.org/abs/1111.5759~~] ~~B. Feng and M. Luo~~, ~~“An Introduction to On-shell Recursion Relations” </ref>~~

	~~== References ==~~
	~~{{reflist}}~~

	~~[[Category:Scattering theory]]~~
	~~[[Category:Quantum chromodynamics]]~~

en>Crowsnest: /* Vorticity */ to me it seems that operator E does not add much, so I removed it

2013-11-16T21:41:02Z

Vorticity: to me it seems that operator E does not add much, so I removed it

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en>Helpful Pixie Bot: ISBNs (Build KC)

2012-05-05T23:30:01Z

ISBNs (Build KC)

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