# What are Maximally Helicity Violating (MHV) Amplitudes?

Definition of MHV amplitudes on Wikipedia:

In theoretical particle physics, 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.

But what does it mean to say they violate helicity conservation to the maximum extent possible? More details are expected. Thanks a lot!

Conservation of helicity requires that $$\sum_{i=1}^n h_i = 0$$ in any amplitude. If the above equation is not satisfied, then conservation of helicity requires that the corresponding amplitude vanish. Maximally violating helicity conservation implies that the above sum take its maximum possible value with its corresponding amplitude being non-zero.

Now, let us look at the cases which will possibly violate helicity to the maximum possible extent. This happens when $\sum_{i=1}^n h_i$ takes its maximum possible value.

The first guess is when all the $h_i$'s have the same sign, i.e. when all $h_i = +1$ or all $h_i = -1$. However, as it turns out, all such amplitudes are exactly zero, and therefore do not violate helicity conservation. The next possibility is when all have the same sign except one which has the opposite sign. It turns out that in this case also, the amplitude vanishes and helicity indeed is conserved.

Thus, the maximum violation of helicity conservation can occur only if all but two of the helicities are the same. In this case, the amplitude does not vanish, and helicity is indeed violated. Thus, amplitudes with $n-2$ gluons with $\pm$ helicity and 2 with $\mp$ helicity violate helicity conservation to the maximum possible extent.

Hence the name.

• So, for MHV amplitudes, we have $\sum_{i=1}^n h_i = \pm (n-2)$. This is the maximum violation of helicity such that the tree amplitudes do not vanish. – soliton Nov 17 '14 at 4:26
• It's $\pm(n-4)$, but the rest is true. – Prahar Nov 17 '14 at 5:03
• Neat. What is it that makes all amplitudes with all helicities the same sign (or with just one opposite sign) vanish? – QuantumDot Aug 15 '15 at 14:02