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August 22, 2007


One of the nice things about travelling is that you get to hear about some of the important stuff you’ve been missing out on. A big industry was launched, several years ago by Cachazo, Svrček and Witten, who wrote down a prescription for computing Yang-Mills amplitudes, using the tree-level MHV amplitudes (suitably-continued off-shell) as vertices, and using ordinary i/p 2i/p^2 as a propagator. This proved an extremely efficient way to calculate tree amplitudes and the cut-constructible parts of higher-loop amplitudes.

But why it was correct (to the extent that it was correct) remained a mystery until a very striking paper by Paul Mansfield. He started with Yang-Mill in lightcone gauge. Pick a null vector, μ\mu, and set1 A^Aμ=0\hat{A}\equiv A\cdot\mu=0. Then Aˇ\check{A} is non-dynamical, and can be integrated out, yielding an action of the form S=4g 2trd 4x S=\frac{4}{g^2}tr\int d^4x \mathcal{L} where = ++ +++ ++ ++\mathcal{L}= \mathcal{L}^{-+} + \mathcal{L}^{++-} + \mathcal{L}^{--+} + \mathcal{L}^{--++} takes the form

(1) + =A¯(ˇ^¯)A ++ =(¯^ 1A)[A,^A¯] + =[A¯,^A](¯^ 1A¯) ++ =[A¯,^A]^ 2[A,^A¯] \begin{aligned} \mathcal{L}^{-+} &= \overline{A}(\check{\partial}\hat{\partial}-\partial\overline{\partial}) A\\ \mathcal{L}^{++-} &= - (\overline{\partial}\hat{\partial}^{-1} A) [A,\hat{\partial}\overline{A}]\\ \mathcal{L}^{--+} &= - [\overline{A},\hat{\partial}A](\overline{\partial}\hat{\partial}^{-1} \overline{A})\\ \mathcal{L}^{--++} &= - [\overline{A},\hat{\partial}A]\hat{\partial}^{-2} [A,\hat{\partial}\overline{A}] \end{aligned}

This doesn’t look much like the MHV Lagrangian: it has an ++\mathcal{L}^{++-} term, and no terms with more than two positive helicity gluons. But Mansfield shows that that there is a canonical transformation A =A(B) A¯ =A¯(B¯,B) \begin{aligned} A &= A(B)\\ \overline{A}&=\overline{A}(\overline{B},B) \end{aligned} where the latter is linear in B¯\overline{B}, but both contain all orders in BB. This transformation is cooked up so that +(A)+ ++(A) +(B) \mathcal{L}^{-+}(A)+ \mathcal{L}^{++-}(A) \equiv \mathcal{L}^{-+}(B) This transformation can be cranked out explicitly, order-by-order in BB, and, when substituted back into (1), yields the MHV Lagrangian of Cachazo et al.

Defining λ=2 1/4(p/p^ p^),λ˜=2 1/4(p¯/p^ p^) \lambda = 2^{1/4} \begin{pmatrix}-p/\sqrt{\hat{p}} \\ \sqrt{\hat{p}}\end{pmatrix},\qquad \tilde{\lambda} = 2^{1/4} \begin{pmatrix}-\overline{p}/\sqrt{\hat{p}} \\ \sqrt{\hat{p}}\end{pmatrix} (adapted to the particular choice μ=(1,0,0,1)/2\mu = (1,0,0,1)/\sqrt{2}) one finds λ αλ˜ α˙=p αα˙+aμ αα˙ \lambda_\alpha\tilde{\lambda}_{\dot{\alpha}} = p_{\alpha\dot{\alpha}} + a \mu_{\alpha\dot{\alpha}} where a=2(pˇp^pp¯)/pˇa= - 2 (\check{p}\hat{p}-p\overline{p})/\check{p} vanishes for null momenta. This is exactly the off-shell continuation that they prescribed.

Moreover, the Equivalence Theorem says that, for most purposes, you can use B,B¯B,\overline{B} external lines, instead of A,A¯A,\overline{A} external lines, in computing scattering amplitudes. The source terms trJ¯A+JA¯tr \int \overline{J}A+J\overline{A} couple to A,A¯A,\overline{A}, which are multilinear in the BB’s. But, when you apply the LSZ reduction formula, this kills the multi-BB contributions.

There are some exceptions, as shown by Ettle et al. The Equivalence theorem fails (and one gets nonzero contributions) for the tree-level ++++- anplitude and for the non-cut-constructible bits of the 1-loop amplitudes, which are exactly things that are “missed” by the “naïve” CSW prescription.

The required canonical transformation turns out to emerge very beautifully from a construction in which one lifts the Yang-Mill Lagrangian to twistor space. I’ll have to explain that some other time.

1We choose conventions where p αα˙=p μσ αα˙ μ=2(pˇ p p¯ p^)p_{\alpha\dot{\alpha}}=p_\mu \sigma^\mu_{\alpha\dot{\alpha}}=\sqrt{2}\left(\begin{smallmatrix}\check{p}&-p\\ -\overline{p}&\hat{p}\end{smallmatrix}\right) so that the Lorentz inner product AB=A^Bˇ+AˇB^AB¯A¯BA\cdot B= \hat{A}\check{B}+\check{A}\hat{B}-A\overline{B}-\overline{A}B.

Posted by distler at August 22, 2007 6:40 AM

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1 Comment & 1 Trackback


MHV is prominent in Twistor theory.

Helices are ubiquitous in ballistics, mechanics, electrical engineering [EE] an biophysiology.

Electromagnetism [EM] and bio-EM [ionic or electrochemical] appear to have solenoid like activity?

Could “strings” be of almost any length [R or 1/R] but limited to one period, repeating sometimes as a helix?

EE techniques [or ticks?] can deal with discontinuities and treat the interval [minus-infinity, plus-infinity] as one period.

Helical angles [related to periods] may be more informative than violations?

Posted by: Doug on August 25, 2007 10:58 AM | Permalink | Reply to this
Read the post Twistor Yang Mills
Weblog: Musings
Excerpt: Belatedly Boels et al.
Tracked: September 27, 2007 12:33 AM

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