Musings

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 $i/p^2$ 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, $\mu$, and set¹ $\hat{A}\equiv A\cdot\mu=0$. Then $\check{A}$ is non-dynamical, and can be integrated out, yielding an action of the form $$S=\frac{4}{g^2}tr\int d^4x \mathcal{L}$$ where $\mathcal{L}= \mathcal{L}^{-+} + \mathcal{L}^{++-} + \mathcal{L}^{--+} + \mathcal{L}^{--++}$ takes the form

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 $$\begin{aligned} A &= A(B)\\ \overline{A}&=\overline{A}(\overline{B},B) \end{aligned}$$ where the latter is linear in $\overline{B}$, but both contain all orders in $B$. This transformation is cooked up so that $$\mathcal{L}^{-+}(A)+ \mathcal{L}^{++-}(A) \equiv \mathcal{L}^{-+}(B)$$ This transformation can be cranked out explicitly, order-by-order in $B$, and, when substituted back into (1), yields the MHV Lagrangian of Cachazo et al.

Defining $$\lambda = 2^{1/4} \begin{pmatrix}-p/\sqrt{\hat{p}} \\ \sqrt{\hat{p}}\end{pmatrix},\qquad \tilde{\lambda} = 2^{1/4} \begin{pmatrix}-\overline{p}/\sqrt{\hat{p}} \\ \sqrt{\hat{p}}\end{pmatrix}$$ (adapted to the particular choice $\mu = (1,0,0,1)/\sqrt{2}$) one finds $$\lambda_\alpha\tilde{\lambda}_{\dot{\alpha}} = p_{\alpha\dot{\alpha}} + a \mu_{\alpha\dot{\alpha}}$$ where $a= - 2 (\check{p}\hat{p}-p\overline{p})/\check{p}$ 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 $B,\overline{B}$ external lines, instead of $A,\overline{A}$ external lines, in computing scattering amplitudes. The source terms $tr \int \overline{J}A+J\overline{A}$ couple to $A,\overline{A}$, which are multilinear in the $B$’s. But, when you apply the LSZ reduction formula, this kills the multi-$B$ 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.