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August 31, 2007

Suboptimized

The Instiki installation on golem has performed quite solidly. It’s not the fastest Web application you’ll ever see, but it’s been pretty darned stable. Since I was giving some talks this summer, I installed a copy on my laptop, so that I could serve (and, for that matter, prepare) the slides locally.

That proved to be nothing but a headache. The damned thing would segfault at the drop of a hat. Eventually, I gave up, and relied on the wonders of the internet to deliver my slides from my office in Austin to wherever I happened to be.

Continue reading “Suboptimized” ...
Posted by distler at 11:33 PM | Permalink | Followups (9)

August 22, 2007

MHV

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^≡A⋅μ=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 2tr∫d 4xℒ S=\frac{4}{g^2}tr\int d^4x \mathcal{L} where ℒ=ℒ −++ℒ ++−+ℒ −−++ℒ −−++\mathcal{L}= \mathcal{L}^{-+} + \mathcal{L}^{++-} + \mathcal{L}^{--+} + \mathcal{L}^{--++} takes the form

(1)ℒ −+ =A¯(∂ˇ∂^−∂∂¯)A ℒ ++− =−(∂¯∂^ −1A)[A,∂^A¯] ℒ −−+ =−[A¯,∂^A](∂¯∂^ −1A¯) ℒ −−++ =−[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 ℒ ++−\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 ℒ −+(A)+ℒ ++−(A)≡ℒ −+(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 tr∫J¯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 A⋅B=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 6:40 AM | Permalink | Followups (2)