# Parity and Helicity of the Higgs Boson

I have been studying how the spin and parity of the new boson discovered at the LHC will be studied and have run into some confusion. The Standard Model Higgs is expected to be a scalar (i.e. have even parity with spin 0). My question is how is the intrinsic parity of a particle experimentally determined? Is it related to the helicity amplitudes? Why is parity even an important quantum number to determine?

Appreciating that this is a comically delayed response, as the $J^P=0^+$ nature of the higgs is already established, e.g. in the 2015 ATLAS paper (note the subtlety of the effect in Fig 4!) as well as the 2015 CMS one, the answer of principle might still be useful.

In practice, the various models are parameterized through a generic effective lagrangian, as in the standard 2013 LHC Higgs properties Handbook, and the various couplings of it are compared to experiment. Helicity amplitudes are not the best handle on the parity of the respective terms in the effective lagrangian, but the polarization of the vector decay products, by contrast, is (see below).

The reason the intrinsic parity (actually, the full CP) of the particles involved is interesting is because it sheds light on their nature (e.g. possible compositeness, multiplicities, model features, etc. For the standard higgs, the answer should be $0^+$, the value of the vacuum to which it shifts; but in multi-higgs models, other excitations of several parities may mix.) just as the intrinsic parities of hadrons half a century ago firmed up their classification and the establishment of the quark model. In the case of the Electroweak model, P is violated by the weak sector, but, discounting fermions, CP is not, so reactions such as $h \to Z Z^*\to 4l~$ yield distributions favoring a scalar versus a pseudoscalar higgs.

Let me illustrate the point of such discriminations with light mesons of half a century ago, textbook stuff (e.g. Perkins). The pseudoscalar decay $\pi^0\to \gamma \gamma$ is driven by the standard flavor anomaly term in the effective lagrangian, $\propto \pi^0 \epsilon^{\mu\nu\kappa\lambda}F_{\mu \nu}F_{\kappa \lambda}$. Note that the negative parity of the pion is matched by the negative parity of the epsilon antisymmetrizer symbol, and the lagrangian term preserves parity overall --it is EM, after all!

The decay of the σ scalar meson, so of positive parity, on the other hand, $f(500)\to \gamma \gamma$, is driven by the term $\propto f~F_{\mu \nu}F^{\mu \nu}$, also manifestly preserving parity, without much fuss.

But the respective amplitudes of the decay products are contrasting, $\propto \epsilon^{\mu\nu\kappa\lambda}{\epsilon}^{1} _{\mu} q^{1} _{\nu}\epsilon^{2}_{\kappa} q^{2}_{\lambda}$ versus $\propto (\epsilon^{1}_{\mu} q^{1}_{\nu}- \epsilon^{1}_{\nu}q^{1}_{\mu}) (\epsilon^{2~\mu}q^{2~\nu}- \epsilon^{2~\nu}q^{2~\mu})$. If the photons were virtual and decayed, e.g. to two leptons each (4 e), there would be enough vectors to access the polarization differences and discriminate between the two cases, which is how the intrinsic parity of $\pi^0$ is determined, by the polarizations of the converted photons being orthogonal to each other, essentially the two respective E fields.

This, too is virtually what is exploited in the higgs characterization measurements. The effect of contrasting $0^+$ to $0^-$ is often subtle deviations in the distributions, as commented above, but this is the basic principle behind the effort.