# Lightweight Higgs: A Sensitive Creature’

Matt Strassler [February 6, 2012]

In a post from January 27, 2012, concerning the possibility that the Higgs particle might have exotic decays (i.e. decays of a sort not expected if the Higgs is of the “ simplest [i.e. “Standard Model”] type), I described a lightweight Higgs particle as a sensitive creature.  We might think of it as the canary in the accelerator tunnel, easily affected by new phenomena that we might otherwise overlook at the Large Hadron Collider [LHC].  It has the potential to give us our first indication of the existence of new particles and/or forces .

But what makes it so delicate?

The reason that a lightweight Higgs particle is a sensitive creature is this:  it … decays …  slowly …

Slowly??!??  In what sense can one think of a particle that typically disappears in 0.000,000,000,000,000,000,001 seconds — less than the time it takes light to cross from one side of an atom to the other — as durable?!?

Well.  In the first part of this article, I will explain in what sense the decay of a lightweight Higgs particle is typically slow.  And in the second part, I’ll explain why this long lifetime directly leads to hypersensitivity.

I want to bring back an analogy that I presented in describing why most particles decay, which I encourage you to read if you haven’t already.  There I explained that the decay of a particle is a quantum mechanical version of ordinary dissipation of waves. I used the example of a violin string (or that of a guitar or harp), which, if you pluck it, will ring for a time, but will gradually die away, as the vibrations on the string dissipate their energy into sound waves, into vibrations of the wood of the violin, and into internal heating (microscopic motion of molecules) of the string itself, and of the pegs that hold the ends of the string in place.

Just to be specific, let’s take the violin string with the second-highest pitch, the “A” string; by modern convention the string is tightened or loosened until it vibrates back and forth 440 times per second. (This is the note that orchestras use to tune their instruments, adjusting them so they will vibrate in synchrony, and assuring a pleasant collective sound.)  If you pluck the string, and don’t in any other way touch it, that string will ring for a few seconds before its tone becomes inaudible.  A few seconds, for a string vibrating hundreds of times a second, means that many thousands of back-and-forth vibrations take place before you can’t hear it anymore.

That’s a slow decay.

You can change this, if you want to. If you put a soft pillow on top of the violin, in contact with its strings, and then you pluck the A string, you will hear a completely different sound: the string will go “Plunck” and the sound instantly stops. (If you have a violin or a guitar, try it; or if you have an instrument you can strike, such as a xylophone, you can get the same effect.) The number of vibrations might be just a few before the string comes essentially to a stop. And if you really squeeze the pillow onto the violin, the string may not manage to vibrate even once.

That’s a fast decay.

Typically, vibrations (whether those of a violin string, or of a marble rolling back and forth in a bowl, or of a swing or trampoline, etc.) tend to die away in a “self-similar” fashion. As they die away, their frequency and shape don’t change; only their overall size (the distance of travel back and forth during a vibration, also called the “amplitude”) gradually decreases. [You can hear this in the violin’s sound; even though it becomes quieter, your ear continues to hear the same musical tone.]  It takes a certain amount of time for the vibrations to decrease from their initial size to half that size. Then it takes the same amount of time for the vibrations to decrease to 1/4 their original size. Then the same amount of time again to get down to 1/8th. This is known as “exponential decay”.

[A minor detail: for the violin string, this won’t be exactly right — right after the string is plucked, and at the very end when it grinds to a stop, the vibrations will do something a little different. But throughout most of the process, the decay follows this exponential law.]

What I mean by slow and fast decay is shown in Figure 3.  The difference is just whether the exponential die-off of the vibration is long compared to the back-and-forth time of the vibration (top of Figure 3) or whether it is not (bottom of Figure 3)?

A little terminology: The time it takes for vibrations to decrease to 1/2 their original size is called their “half-life”.  But physicists often use a different term,  appealing to a special number that appears in mathematics — e = 2.718…. = exp[1] —- a number which is just as special in mathematics as the number we call π. The question physicists ask is: how long does it take for vibrations to reach 1/e of their original size? That time is called the lifetime… both for waves and, appropriately adjusted, for particles. [For those who want more details: vibrations  tend to decrease over time as exp[-t/t0], where t0 is the lifetime. Meanwhile a single back-and-forth vibration lasts a time T, and behaves like cos [2 π t / T]. Altogether the vibration behaves roughly as exp[-t/t0] cos [2 π t/T], which is what is plotted in Figure 3.  This is the typical behavior of a “damped oscillator”; a long lifetime involves light damping, a short (or very short) lifetime involves strong damping (or over-damping).]

Now that’s the situation for waves.  What about particles?

A particle in particle physics (also called a “quantum”) is the smallest possible wave in a quantum field, and it vibrates, rising and falling just as any wave does. So if you had a particle (such as an electron) with a  mass m, just sitting (more or less) in front of you, and stationary (more or less), it would in some ways resemble less a little ball than a vibrating trampoline or drum-head.  And we know in quantum mechanics exactly how fast this happens — what musical tone corresponds to this particular type of particle.  The time T that it would take for a back-and-forth vibration is given by the very simple formula

T = h / m c2

where h is Planck’s constant (which is as fundamental to quantum mechanics as c, the speed of light, is to Einstein’s relativity.) The heavier the particle, the faster it vibrates and the smaller is T. (Or in fact, one may say it the other way: the faster a quantum’s vibration, the more energy it carries, and since E=mc2 for a particle at rest, the heavier it is.)  Poetically, we might say that this time scale is the particle’s natural heartbeat.

Remember the violin string, which vibrates 440 times per second, or about once ever 0.002 seconds?  Well, for the typical particles of nature, it’s a tad faster:

• 0.000,000,000,000,000,000,001 seconds for an electron.
• 0.000,000,000,000,000,000,000,000,7 seconds for a proton.
• 0.000,000,000,000,000,000,000,000,005 seconds for a lightweight Higgs particle

[Caution: just to make sure you don’t get confused, let me emphasize that this inverse relation between m and T is not the reason why a lightweight Higgs particle is sensitive.  Yes, a lighter Higgs vibrates slower, but the issue at hand is not how fast it vibrates, but how fast it decays compared to how fast it vibrates… as in Figure 3.  And see Figure 4 below.]

Now let’s turn to particle decays, and discuss what physicists call the particle’s “lifetime”.  Remember the lifetime of a wave that we discussed above?  For a particle, the lifetime is the average time that it takes for particles of this type to disintegrate. Now we do live in a quantum world, and that means that any particular particle that you might put on your shelf will decay at a random time, surprising you.  But the average over many such particles, of the same type, is something that is not random, and can be measured and in many cases calculated from the mathematical equations that describe the known particles of nature. The lifetime for a type of particle is “long”, in the current sense, if (again, speaking poetically) the particle’s heart typically beats many times before it dies. If it beats five or ten times, or just once or twice, well… I guess that’s a little sad. But for particles it is a common fate.

Most types of particles are much worse off than mayflies. For instance, a top quark typically lives a couple of hundred heartbeats; the Z particle lives half that. A muon, on the other hand, is enduring beyond Methuselah; though it decays on average after just two millionths of a second, that represents 10 million billion heartbeats. A neutron’s life is even longer. And of course a few particles are stable (or at least haven’t ever been observed to decay at all.) But all of these particles are unusual. More typical is a particle called the ρ (a type of hadron) which lives just a handful of heartbeats.  The heart of a σ (another hadron, also called the f0) hardly beats once.   For a sense of scale, consider that a human lifetime is typically two to three billion (about 2,500,000,000) heartbeats.

One of the most important and convenient things about the Standard Model Higgs particle — the simplest possible form of Higgs particle that might be part of nature, and the one that the LHC experiments are currently trying to discover or rule out for good — is that everything is known about it except its mass. More precisely, if you tell me its mass m, and thus its vibrational time T = h / m c2, I can tell you its lifetime (let’s call it t0), using the equations of the Standard Model. And I can calculate, therefore, the ratio of the lifetime divided by the back-and-forth vibrational time — t0 /T — and see whether this ratio is small (a fast decay) or large (a slow decay).  These two time scales are shown in Figure 4.

What we find from the equations is that a heavyweight Standard Model Higgs particle has a fast decay. In fact a very heavyweight Standard Model Higgs has such a fast decay that it basically doesn’t exist as a particle at all; this is part of why the Standard Model stops being sensible if the Higgs mass is heavier than about 800 GeV/c2. We’ll come back to this point later. Below 800 or so there’s clearly a particle there, just not for that long.

But below around 160 GeV/cor so, as you can see in Figure 4, the lifetime of the Standard Model Higgs particle becomes very much longer than its natural heartbeat. This is because one of the main sources of dissipation — its potential decay to W and Z particles — is no longer present.  It is a bit like taking the pillow off the violin string; when something that causes rapid dissipation is removed, the decay of the vibrations takes longer.

Now what’s special about 160 GeV/c2? Well, when a particle of mass m decays, the masses of the particles to which it decays must sum up to something less than m.  A Higgs heavier than twice the W particle’s mass can decay to two W particles, but a lighter one cannot.  The mass of the W particle is 80.4 Gev/c2. And twice 80 is 160.

Now the transition isn’t entirely sharp at 160, because the Higgs can still decay to one W particle and one W “virtual particle” (which isn’t a particle at all, but rather a more generalized disturbance in the W field.) But still, the dissipation from the decay to W’s is much reduced. And thus a Standard Model Higgs particle that is too light to decay to two W particles will ring for a much longer time than a heavier one that regularly decays to pairs of W particles (and, if above 182 GeV/c2, Z particles too.)

If the 125 GeV/c2 Higgs particle that is hinted at in current LHC data is really there, and if it is exactly (or very similar to) a Standard-Model-type Higgs, then its average lifetime is over 40,000 heartbeats… still very short by our standards, but much more like the vibrations shown at the top of Figure 3 than at the bottom of Figure 3. For the analogy of our violin A string, it is like plucking the string (which vibrates 440 times a second) and hearing the tone decrease only over a period of more than a minute! The dissipation of a lightweight Higgs particle is very slow indeed!!

It is the slowness of this decay that makes the light Standard Model Higgs a very sensitive creature.  Now let’s learn why.

The Diverse Modes of a Higgs’ Demise

Up to now, I’ve only talked about the overall lifetime of the Higgs, and the corresponding overall time for a plucked violin string to decay away. But if you’ve read about the Higgs, you know that a Standard Model Higgs particle may decay in one of a number of different ways. And you also know that a vibrating violin string can dissipate its energy in several ways simultaneously: to sound waves, to the vibrations of the violin itself, to vibrations of the pegs that hold the string in place, and to friction (internal heating, i.e. molecular motion) inside the string itself.

Clearly a very interesting question is this: for the violin string, how much of its energy is lost to sound, how much to the pegs, and how much (if relevant) to a pillow in contact with the string? And for the Higgs, the corresponding question is: how likely is the Higgs to decay to two photons, or two Z particles, or a bottom quark/anti-quark pair? In both cases, there are multiple modes of dissipation, and the question is how strong they are relative to one another. If there’s one mode that’s much stronger than the others, then that mode will determine the overall lifetime (as in the pillow which muffles the string, or the decays to W and Z particles that shorten the heavy Standard Model Higgs’ lifetime.) But even when one or two modes dominate, it is interesting to know about other modes. What fraction, small as it is, of the muffled string’s energy gets turned into sound, or into heating of the string? And how big is the admittedly small fraction of heavy Higgs particles that decay to a tau lepton/antilepton pair?

Why is the last question so interesting? Because it is something that we can predict, based on the equations of the Standard Model, and it is something we can perhaps measure — and anytime you can bring a previously untested prediction and a corresponding measurement face to face, you have the opportunity to test the validity of your equations, and possibly discover that they are not exactly correct! (Remember how particle physicists think: we would prefer to find a place where the equations are incomplete, because the current equations have many puzzles, and we won’t resolve these puzzles until we find some new and surprising phenomena in our experiments.) And similarly, if we had a detailed theory of the behavior of a violin string, we would similarly want to test it by measuring precisely all of the different sources of dissipation, and checking whether our equations predict not only the most important source accurately but also the smaller sources.

But the lightweight Higgs particle, like the unmuffled violin string, has very little dissipation — it can ring for a long time, so to speak. And that means two very important things.

1. A small new and unexpected source of dissipation, or a small change in one of the expected sources of dissipation, can have a very big effect on the Higgs’ lifetime and on the probabilities of its various decay modes. By analogy: even a small amount of unexpected dirt or moisture on the violin string, or a crack in a peg that holds it in place, or a defect in the string itself, could have a big effect on how long the string will vibrate, by introducing a new source (or changing an old source) of dissipation. (See Figure 5b.) It would also change the relative importance of different dissipation modes, perhaps dramatically.
2. A tiny new and unexpected source of dissipation would have very little effect on the Higgs’ lifetime or on the probabilities of its expected decay modes. But it still might generate a novel class of decays that would indicate something wrong with the Standard Model… perhaps something very important. By analogy: a hair lying lightly on the violin string (Figure 5c) would hardly affect the string and how its vibrations die away, but still that hair would gather a minority of the energy from the string. Small as this effect might be, if it were discovered it would indicate that the physicists who wrote the equations for the string had left something out of the theory… perhaps something very important, such as, say, the violin’s accompanying bow, or the beard of the violinist!

In summary:

• A lightweight Higgs particle is sensitive to small effects from new particles and/or forces, which can change its properties dramatically and make it look very different from a Standard Model Higgs. In this case it is easy to discover that the Standard Model is wrong, because the effects are very large. The first clue will be that either (a) the Standard Model Higgs will be missing, or (b) a Higgs candidate will be discovered that is quickly determined to behave differently from what would be expected of a Standard Model Higgs.
• A lightweight Higgs particle is even sensitive to tiny effects from new particles and/or forces, which, although they won’t change its overall properties dramatically, leaving it an apparent Standard Model Higgs at first and even second glance, can in some cases create rare exotic decay modes. In this case it is more difficult to discover that the Standard Model is wrong, because all the usual properties of the Higgs are just what you’d expect in the Standard Model, to high precision. But one must try as hard as possible, because if the new phenomena always give tiny effects, there may be no other sign of them at the LHC. The particular sensitivity of a light Higgs particle provides a special and possibly unique opportunity to discover these new particles and/or forces. And since the number of Higgs particles produced is huge — a significant fraction of a million by the end of next year — we have a good chance of discovering processes that are quite rare indeed!

I think everyone in the community would agree that measuring the production modes and decays modes of the Higgs is a crucial step in understanding it. I think we would all agree that this includes checking the production modes and decay modes predicted by the Standard Model, which involves measuring them as precisely as possible. I even think that most people in the field appreciate that we should look for exotic production modes; perhaps the Higgs is produced in the decays of new particles — such as new resonances, or supersymmetric or Kaluza-Klein partners of known particles — or even in some unexpected decay of the top quark (e.g., top quark decays to a charm quark and a Higgs particle.) I know that some experimentalists are already looking for additional evidence of the 125 GeV Higgs in places where the Standard Model wouldn’t predict you’d find it.

But I think that the importance of exotic decay modes, even in the case that the Higgs appears with moderate precision to be of Standard Model type, is underappreciated. And since the trigger system that determines which collisions should be discarded appears, according to my own naive studies, to be challenged by some of these exotic decays, I feel that raising this issue before we start taking data in 2012 is very important. We need more theoretical studies of how these processes might be discovered, and we need more experimental studies of whether in fact the current trigger strategies are deficient for some of these modes. If indeed deficiencies are found, we need to work very quickly to improve the efficiency with which the trigger accepts these events, so that we have a larger fraction of them stored permanently for later analysis.

### 9 thoughts on “Lightweight Higgs: A Sensitive Creature’”

1. Thanks Matt! I’m looking forward to reading Part 2.

Something I think I learned from this: the decay rate of a particle can be expressed as a dimensionless quantity, which you’ve called “heartbeats per lifetime”. I’ve never seen it expressed that way before; is there a more standard name for this quantity?

• What I did was convert mass to a time T, because people know about mass and can think about heartbeats as an analogy.

What we do as professionals is instead convert lifetime to an “uncertainty” in the particle’s mass, known as a *width* [denoted Γ] because when you measure a particle’s mass through a bump in a distribution, the width of that distribution is the intrinisic uncertainty in the particle’s mass. Note the term “uncertainty” here is technical and not English; we physicists may know the mass of a particular type of particle with great certainty, but when you produce a particle of this type its mass will be a little different each time, with the average being what we call the “mass” and the width of the distribution of masses being what we call the “width”. It’s a physical “uncertainty”, not an epistemelogical one, and it is intrinsic to nature, not to scientists. The shorter the lifetime, the larger this intrinsic uncertainty in the mass.

So the relevant ratio in professional particle physics is width/mass: how “intrinsically uncertain” is the particle’s mass compared to its mass.

2. Hi Matt

How does the formula “T = h / m c2” apply to photons (and gluons)?

Is the ‘T’ the particle’s proper time? If so, then your discussion would seem to indicate that photons have an infinite mass width.

This might be related to another question I have been wondering about for a while: how is it that photons come in many different varieties (with different wavelengths), whereas electrons all have the same wavelength. I’ve never heard of anyone discussing varying frequency/wavelength for particles other than photons (even the other massless particles).