Inspecting the Higgs with a golden probe

Hello particle nibblers,

After recovering from a dead-diphoton-excess induced depression (see here, here, and here for summaries) I am back to tell you a little more about something that actually does exist, our old friend Monsieur Higgs boson. All of the fuss over the past few months over a potential new particle at 750 GeV has perhaps made us forget just how special and interesting the Higgs boson really is, but as more data is collected at the LHC, we will surely be reminded of this fact once again (see Fig.1).

Figure 1: Monsieur Higgs boson struggles to understand the Higgs mechanism.

Previously I discussed how one of the best and most precise ways to study the Higgs boson is just by `shining light on it’, or more specifically via its decays to pairs of photons. Today I want to expand on another fantastic and precise way to study the Higgs which I briefly mentioned previously; Higgs decays to four charged leptons (specifically electrons and muons) shown in Fig.2. This is a channel near and dear to my heart and has a long history because it was realized, way before the Higgs was actually discovered at 125 GeV, to be among the best ways to find a Higgs boson over a large range of potential masses above around 100 GeV. This led to it being dubbed the “gold plated” Higgs discovery mode, or “golden channel”, and in fact was one of the first channels (along with the diphoton channel) in which the 125 GeV Higgs boson was discovered at the LHC.

Figure 2: Higgs decays to four leptons are mediated by the various physics effects which can enter in the grey blob. Could new physics be hiding in there?
Figure 2: Higgs decays to four leptons are mediated by the various physics effects which can enter in the grey blob. Could new physics be hiding in there?

One of the characteristics that makes the golden channel so valuable as a probe of the Higgs is that it is very precisely measured by the ATLAS and CMS experiments and has a very good signal to background ratio. Furthermore, it is very well understood theoretically since most of the dominant contributions can be calculated explicitly for both the signal and background. The final feature of the golden channel that makes it valuable, and the one that I will focus on today, is that it contains a wealth of information in each event due to the large number of observables associated with the four final state leptons.

Since there are four charged leptons which are measured and each has an associated four momentum, there are in principle 16 separate numbers which can be measured in each event. However, the masses of the charged leptons are tiny in comparison to the Higgs mass so we can consider them as massless (see Footnote 1) to a very good approximation. This then reduces (using energy-momentum conservation) the number of observables to 12 which, in the lab frame, are given by the transverse momentum, rapidity, and azimuthal angle of each lepton. Now, Lorentz invariance tells us that physics doesnt care which frame of reference we pick to analyze the four lepton system. This allows us to perform a Lorentz transformation from the lab frame where the leptons are measured, but where the underlying physics can be obscured, to the much more convenient and intuitive center of mass frame of the four lepton system. Due to energy-momentum conservation, this is also the center of mass frame of the Higgs boson. In this frame the Higgs boson is at rest and the \emph{pairs} of leptons come out back to back (see Footnote 2) .

In this frame the 12 observables can be divided into 4 production and 8 decay (see Footnote 3). The 4 production variables are characterized by the transverse momentum (which has two components), the rapidity, and the azimuthal angle of the four lepton system. The differential spectra for these four variables (especially the transverse momentum and rapidity) depend very much on how the Higgs is produced and are also affected by parton distribution functions at hadron colliders like the LHC. Thus the differential spectra for these variables can not in general be computed explicitly for Higgs production at the LHC.

The 8 decay observables are characterized by the center of mass energy of the four lepton system, which in this case is equal to the Higgs mass, as well as two invariant masses associated with each pair of leptons (how one picks the pairs is arbitrary). There are also five angles (\Theta, \theta_1, \theta_2, Φ, Φ1) shown in Fig. 3 for a particular choice of lepton pairings. The angle \Theta is defined as the angle between the beam axis (labeled by p or z) and the axis defined to be in the direction of the momentum of one of the lepton pair systems (labeled by Z1 or z’). This angle also defines the ‘production plane’. The angles \theta_1, \theta_2 are the polar angles defined in the lepton pair rest frames. The angle Φ1 is the azimuthal angle between the production plane and the plane formed from the four vectors of one of the lepton pairs (in this case the muon pair). Finally Φ is defined as the azimuthal angle between the decay planes formed out of the two lepton pairs.

Figure 3: Angular center of mass observables ($latex \Theta, \theta_1, \theta_2, Φ, Φ_1$) in Higgs to four lepton decays.
Figure 3: Angular center of mass observables in Higgs to four lepton decays.

To a good approximation these decay observables are independent of how the Higgs boson is produced. Furthermore, unlike the production variables, the fully differential spectra for the decay observables can be computed explicitly and even analytically. Each of them contains information about the properties of the Higgs boson as do the correlations between them. We see an example of this in Fig. 4 where we show the one dimensional (1D) spectrum for the Φ variable under various assumptions about the CP properties of the Higgs boson.

Figure 4: Here I show various examples for the Φ differential spectrum assuming different possibilities for the CP properties of the Higgs boson.
Figure 4: Here I show various examples for the Φ differential spectrum assuming different possibilities for the CP properties of the Higgs boson.

This variable has long been known to be sensitive to the CP properties of the Higgs boson. An effect like CP violation would show up as an asymmetry in this Φ distribution which we can see in curve number 5 shown in orange. Keep in mind though that although I show a 1D spectrum for Φ, the Higgs to four lepton decay is a multidimensional differential spectrum of the 8 decay observables and all of their correlations. Thus though we can already see from a 1D projection for Φ how information about the Higgs is contained in these distributions, MUCH more information is contained in the fully differential decay width of Higgs to four lepton decays. This makes the golden channel a powerful probe of the detailed properties of the Higgs boson.

OK nibblers, hopefully I have given you a flavor of the golden channel and why it is valuable as a probe of the Higgs boson. In a future post I will discuss in more detail the various types of physics effects which can enter in the grey blob in Fig. 2. Until then, keep nibbling and don’t let dead diphotons get you down!

Footnote 1: If you are feeling uneasy about the fact that the Higgs can only “talk to” particles with mass and yet can decay to four massless (atleast approximately) leptons, keep in mind they do not interact directly. The Higgs decay to four charged leptons is mediated by intermediate particles which DO talk to the Higgs and charged leptons.

Footnote 2: More precisely, in the Higgs rest frame, the four vector formed out of the sum of the two four vectors of any pair of leptons which are chosen will be back to back with the four vector formed out of the sum of the second pair of leptons.

Footnote 3: This dividing into production and decay variables after transforming to the four lepton system center of mass frame (i.e. Higgs rest frame) is only possible in practice because all four leptons are visible and their four momentum can be reconstructed with very good precision at the LHC. This then allows for the rest frame of the Higgs boson to be reconstructed on an event by event basis. For final states with missing energy or jets which can not be reconstructed with high precision, transforming to the Higgs rest frame is in general not possible.

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Roberto Vega-Morales is currently a post-doctoral researcher in high energy theory at the University of Granada in Spain. Previously he was at the Laboratoire de Physique Thèorique in Paris France. He conducted his Ph.D studies at Northwestern University as well as Fermilab and was awarded the 2014 J.J. and Noriko Sakurai Dissertation Award in Theoretical Particle Physics. His research focuses on the phenomenology of the Higgs boson at the LHC as well as models of Supersymmetry and extended Higgs sectors. He struggled mightily with French and is happy to be speaking Spanish nowadays.

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