A Quantitative Addendum to the Sensitive Creature

Allow me to be a little more quantitative about the Higgs and why it so easily can be affected by new physics. And then we’ll look at a couple of examples of why it can cause trouble for the trigger.  [This version still needs a little work; apologies if there are factor-of-two errors and things like that.  There shouldn’t be anything seriously wrong at this point.]

How does a lightweight Standard Model Higgs at 125 GeV/c² in mass or below decay? It predominantly decays (60 % of the time) to a bottom quark/anti-quark pair. It does so through the interaction H bb among the Higgs field and bottom quark fields, which has an overall strength equal to

y_b = √ 2  m_b / v ~ 0.026

This in turn implies that the lifetime of the Higgs is given roughly by

τ_H = (8 π/ 3 y_b² ) T_H ~ 10,000 T_H

(actually a tad shorter because of other decay modes, but that’s not essential at the moment) where T_H is the “heartbeat” or natural time of the Higgs

T_H = h / m_H c²

Now let’s consider the interaction of the Higgs field with a new scalar field S, which does not feel any of the three forces of the Standard Model. It feels just gravity, some effects of the Higgs, and maybe some new forces of its own. [I discussed this case at some length in this article.] When the Higgs field is zero on average, the interaction involves two Higgs fields and two S fields (i.e., H H* S S , with overall strength given by a number η), but after the Higgs field becomes non-zero on average throughout the universe ( = v) this generates an H S S interaction, with strength 2 η v. If S is lighter than m_H/2, then the Higgs particle can decay to two S particles. Just to keep formulas simple, imagine S is much lighter than the Higgs. If this were the only decay mode that the Higgs had, then the lifetime of the Higgs would be

τ ‘_H = (32 π m_H² / v² η²) T_H

Now if you take η not too small and not too large — say, η of 0.1— you find that this would make τ ‘_H ~ 1000 T_H, much shorter than in the Standard Model. In more physical terms, the effect of the interaction of the Higgs with S will completely dominate over the interaction of the Higgs with the bottom quark; decays of H to two S particles will dominate, and the lifetime, despite the other modes, will be of order τ’_H. And since S is neutral and (if it has no other interactions) stable, it is invisible to the detector. In other words, this type of Higgs decays without leaving a detectable trace 99% of the time, and to bottom quarks (and the other expected things) 100 times less often than expected.

Just one new particle has been added, and that’s enough: everything is different.

Of course, η might be much smaller, perhaps 0.005. In this case the probability of the Higgs to decay invisibly would be closer to 10%. All of the other decay modes would occur only 90% as often as expected. So how rare this decay is depends on the quantity η — which we of course don’t know yet. In short, we have to go looking for this invisible decay of the Higgs not knowing how often it happens, and therefore we want to be able to make the most precise measurement possible. We certainly would want to get down to 10% and below.

How well can ATLAS and CMS trigger on an invisible decay of the Higgs? The signal of such a Higgs is well-known to be discoverable in the context of Higgs plus two jets (for which there are contributions from all three modes, though the vector-boson-fusion process is the most important.) In such events there are two jets recoiling against the two unobserved S particles, and thus there is “missing energy” (actually missing momentum transverse to the beampipe) in the event. But let’s put it this way: the higher the rate of collisions at the LHC, and the higher the pile-up, the more difficult it becomes to trigger on missing energy, and the higher the jet transverse momenta may have to be, at the first level of the trigger. A recent study of this process at 7 TeV by Bai, Draper and Shelton requires 120 GeV of missing transverse momentum, two jets with transverse momentum greater than 30 GeV that are far apart (Delta η > 4.5) and with large invariant mass (1200 GeV). This would keep 2% of the signal events. But under current running conditions, this may be pushing the limits of the trigger. If pileup forces a requirement of 120 GeV of missing energy and two far-apart jets of 50 GeV, or of 150 and 30, then the number of signal events is reduced by 25 to 35%. At 150 and 50 the event rates drop in half. That may not be ruinous, but clearly trigger efficiency is going to be an issue.

An alternative scenario which can arise in this and a number of other models (see especially Dermisek and Gunion’s sequence of papers starting around 2004, as well as numerous later ones [including two of mine, one of them with Zurek]) is that the S and H can mix slightly, or the S can be replaced by an appropriate pseudoscalar, as can happen in certain variants of supersymmetry, with the result that the S can decay. It is quite common that its dominant decay is to a bottom quark/anti-quark pair, and its next-to-dominant decay is to a tau lepton/anti-lepton pair. Then

• The most common exotic decay of the Higgs is H –> two bottom quark/anti-quark pairs (but this may not be observable soon; one must first observe H –> one such pair, which requires specialized techniques [boosted object reconstruction], and then generalize them further)
• The next-most-common exotic decay is H –> a bottom quark/anti-quark pair plus a tau lepton/anti-lepton pair.

This last decay may be both reasonably common and relatively easy to discover, as there are several ways to reduce backgrounds. (For example, even in 2011 data, if we assume there is a 125 GeV Standard Model-like Higgs with up to a 10% branching fraction for H –> SS, then up to 1500 events of this type were produced, though many were discarded by the trigger.) Often one of the taus produces an electron or muon (or anti-electron or anti-muon) in its decay. And although the taus’ momenta cannot be fully reconstructed, as they produce neutrinos in their decays, it may be possible to reconstruct the Higgs particle with moderate accuracy using the usual tricks of assuming the neutrinos travel in the same direction as the taus. Unfortunately it may not be possible to do this in most quark+antiquark –> W Higgs events, since the anti-neutrino from a W decaying to lepton + anti-neutrino ruins the above-mentioned trick. So it may be necessary to discover this process in vector boson fusion or perhaps Higgs + one jet. But on what should one trigger? The leptons from the tau decay are too soft; I estimate only 6% of them are likely to fire the trigger even if both electron and muon trigger thresholds were at 25 GeV, and only an additional 2% on a dilepton trigger with thresholds at 12 GeV — and these are very optimistic trigger settings. Missing transverse momentum is too low. The momenta of the bottom quark jet pTs are not that high, though they may offer the best option.

Granted, maybe this is not nearly as bad as it looks; perhaps the only way to discover this process is to use precisely the few percent of the events that will easily pass triggers, even with higher thresholds. But nobody knows whether this is the case, because no theorists have done the relevant study in current conditions. So we have to decide whether an effort should be made to improve this situation even without the benefit of complete information. All in all, this looks challenging, and in need of further study.  [Note added: studies are starting to be done, as of May 2012.]

This is just one example.  There are presumably similar issues for certain other models and signatures as well, though I find that some cases that one might suspect would be challenging appear to be safe.  Clearly, though, we should not leave something this important to chance; it deserves thorough consideration by the LHC community.