Matt Strassler [August 27 – September 9, 2013]
What is “Naturalness”?
[This subject is closely related to the hierarchy problem.]
What do particle physicists and string theorists mean when they refer to a particular array of particles and forces as “natural”? They don’t mean “part of nature”. Everything in the universe is part of nature, by definition.
The word “natural” has multiple meanings. The one that scientists are using in this context isn’t “having to do with nature” but rather “typical” or “or “generic” — “just what you’d have expected”, or “the usual” — as in, “naturally the baby started screaming when she bumped her head”, or “naturally it costs more to live near the city center”, or “I hadn’t worn those glasses in months, so naturally they were dusty.” And unnatural is when the baby doesn’t scream, when the city center is cheap, and when the glasses are pristine. Usually, when something unnatural happens, there’s a good reason.
In most contexts in particle physics and related subjects, surprises — big surprises, anyway — are pretty rare. That means that if you look at a physical system, it usually behaves more or less along lines that, with some experience as a scientist, you’d naturally expect. If it doesn’t, then (experience shows) there’s generally a really good reason… and if that reason isn’t obvious, the unnatural behavior of the system may be pointing you to something profound that you don’t yet know.
For our purposes here, the reason the notion of naturalness is so important is that there are two big surprises in nature that we particle physicists and our friends have to confront. The first is that the cosmological constant [often referred to as “dark `energy’ ” in public settings] is amazingly small, compared to what you’d naturally expect. The second is that the hierarchy between the strength of gravity and the strengths of the other forces is amazingly big, compared to what you’d expect.
The second one can be restated as follows: the Standard Model (combined with Einstein’s theory of gravity) — the set of equations we use to predict the behavior of all the known elementary particles and all the known forces — is a profoundly, enormously, spectacularly unnatural theory. There’s only one aspect of physics — perhaps only one aspect in all of science — that is more unnatural than the Standard Model, and that’s the cosmological constant.
The Notion of “Natural” and “Unnatural”
I think the concept of naturalness is best illuminated by a bit of story-telling.
A couple of friends of mine from college (I’ll call them Ann and Steve) got married, and now have two teenage children. Back when their kids were younger — say, 4 and 7 years old — they were pretty wild. They often played rough, got mad at each other, threw things, and generally needed at lot of supervision.
One day, Ann bought some beautiful flowers and put them in her favorite glass vase. But before she put the vase on the kitchen table, the doorbell rang. She ran to the front, carrying the vase, and as she made her way to the door, she absent-mindedly put the vase down on the small, rickety table that sits by the wall of the kids’ play room.
Half an hour later, Steve returned home with the kids, and sent them into the play room to occupy themselves while he and Ann settled in from the day and prepared dinner. They heard the usual sounds: bumps and crashes, the sounds of bouncing balls and falling blocks, yells of “no fair” and “ow! stop that!”, a moment of screaming that blissfully stopped almost as soon as it started…
It was forty-five minutes later when Ann noticed the vase with the flowers wasn’t on the kitchen table. After a moment searching the kitchen and dining room, she suddenly realized that she’d put it down and forgotten it in the most dangerous place in the house.
So she went running into the play room, hoping she wasn’t too late. And what do you think she found when she opened the door?
Guess. You get three options (Figure 1). Choose the most plausible.
- The vase was exactly where she’d left it, comfortably placed at the center of the table.
- The vase was smashed, and the flowers crushed, down on the floor.
- The vase was hanging off the table, right at the edge, within a millimeter of disaster.
Well, the answer is #3. There it was, just hanging there.
Somehow I suspect you don’t believe me. Or at least, if you do believe me, you probably are assuming there must be some complicated explanation that I’m about to give you as to how this happened. It can’t possibly be that two young kids were playing wildly in the room and somehow managed to get the vase into this extremely precarious position just by accident, can it? For the vase to end up just so — not firmly on the table, not falling off the table, but just in between — that’s … that’s not natural!
There must (mustn’t there?) be an explanation.
Maybe there was glue on the side of the table and the vase stuck to it before falling off? Maybe one of the kids was hiding behind the table and holding the vase there as a practical joke on his mom? Maybe her husband had somehow tied a string around the vase and attached it to the table, or to the ceiling, so that the vase couldn’t fall off? Maybe the table and vase are both magnetized somehow…?
Something so unnatural as that can’t just end up that way on its own… especially not in a room with two young children playing rough and throwing things around.
The Unnatural Nature of the Standard Model
Well. Now let’s turn to the Standard Model, combined with Einstein’s theory of gravity.
I want you to imagine a universe much like our own, described by a complete set of equations — a “theory”, in theoretical-physics speak — much like the Standard Model (plus gravity). To keep things simple, let’s say this universe even has all the same elementary particles and forces as our own. The only difference is that the strengths of the forces, and the strengths with which the Higgs field interacts with other known particles and with itself (which in the end determines how much mass the known particles have) are a little bit different, say by 1%, or 5%, or maybe even up to 50%. In fact, let’s imagine ALL such universes… all universes described by Standard Model-like equations in which the strengths with which all the fields and particles interact with each other are changed by up to 50%. What will the worlds described by these slightly different equations (shown in a nice big pile in Figure 2) be like?
Among those imaginary worlds, we will find three general classes, with the following properties.
- In one class, the Higgs field’s average value will be zero; in other words, the Higgs field is OFF. In these worlds, the Higgs particle will have a mass as much as ten thousand trillion (10,000,000,000,000,000) times larger than it does in our world. All the other known elementary particles will be massless (up to small caveats I’ll explain elsewhere). In particular, the electron will be massless, and there will be no atoms in these worlds.
- In a second class, the Higgs field is FULL ON. The Higgs field’s average value, and the Higgs particle’s mass, and the mass of all known particles, will be as much as ten thousand trillion (10,000,000,000,000,000) times larger than they are in our universe. In such a world, there will again be nothing like the atoms or the large objects we’re used to. For instance, nothing large like a star or planet can form without collapsing and forming a black hole.
- In a third class, the Higgs field is JUST BARELY ON. It’s average value is roughly as small as in our world — maybe a few times larger or smaller, but comparable. The masses of the known particles, while somewhat different from what they are in our world, at least won’t be wildly different. And none of the types of particles that have mass in our own world will be massless. In some of those worlds there can even be atoms and planets and other types of structure. In others, there may be exotic things we’re not used to. But at least a few basic features of such worlds will be recognizable to us.
Now: what fraction of these worlds are in class 3? Among all the Standard Model-like theories that we’re considering, what fraction will resemble ours at least a little bit?
The answer? A ridiculously, absurdly tiny fraction of them (Figure 3). If you chose a universe at random from among our set of Standard Model-like worlds, the chance that it would look vaguely like our universe would be spectacularly smaller than the chance that you would put a vase down carelessly on a table and end up putting it right on the edge of disaster, just by accident.
In other words, if (and it’s a big “if”) the Standard Model (plus gravity) describes everything that exists in our world, then among all possible worlds, we live in an extraordinarily unusual one — one that is as unnatural as a vase nudged to within an atom’s breadth of falling off the table. Classes 1 and 2 of universes are natural — generic — typical; most Standard Model-like theories would give universes in one of those classes. Class 3, of which our universe is an example, includes the possible worlds that are extremely non-generic, non-typical, unnatural. That we should live in such an unusual universe — especially since we live, quite naturally, on a rather ordinary planet orbiting a rather ordinary star in a rather ordinary galaxy — is unexpected, shocking, bizarre. And it is deserving, just like the weirdly placed vase, of an explanation. One certainly has to suspect there might be a subtle mechanism, something about the universe that we don’t yet know, that permits our universe to naturally be one that can live on the edge.
And what is the analogy to the playing children who endanger the vase, and make its balanced condition especially implausible? It is quantum mechanics itself — the very basic operating principles of our world. Quantum effects do not coexist well with accidental, unstable balance.
I’ll go on to discuss those quantum effects, and how they make the Standard Model unnatural, in a moment. But first, although I hope you liked my story, I should point out there’s one important difference between the vase on the table and the universe. If somebody bumps the table or the vase, it will probably fall off, or perhaps, if we’re lucky, slide toward the center of the table. In other words, it can easily move away from its precarious position if it is disturbed. Our universe, by contrast, is not in danger currently of smoothly shifting its properties, and becoming a universe in Class 1 or Class 2. [While it is possible that someday it could shift suddenly to become a very different universe, through a process known as tunneling or vacuum decay, this event is likely to be unimaginably far off; this is a subject for another day, but it’s not something to worry about.] The real issue for the universe is in the past: how, among the vast number of possible universes, did we end up in such an apparently unnatural one? Is there something about our universe that we don’t yet know which makes it not as unnatural as it seems? Or perhaps the fact that many (most?) natural universes don’t seem hospitable for life has something to do with it? Or maybe we humans haven’t been clever enough yet, and there some other subtle scientific explanation? Whatever the reason, either it is due to a timeless fact or due to something that happened very long ago; the universe (or at least the large region we can see with our eyes and telescopes) has been unchangingly unnatural [if the Standard Model fully describes it] for billions of years, and won’t be changing anytime in the near future.
In any case, let’s move on now, to understand the quantum physics that makes a universe described by the Standard Model (and gravity) so incredibly unusual.
Quantum Physics and (un)Naturalness
At this point, please read about quantum fluctuations of quantum fields, and the energy carried in those fluctuations, if you haven’t already done so. Along the way you’ll find out a little about another naturalness problem: the cosmological constant. After you’ve read that article, you can continue with this one.
Back to the Higgs (and Other Similar Particles)
Quantum fluctuations of fields, and their contribution to the energy density of empty space (the so-called “vacuum energy”) play a big part in our story. But our goal here requires we set the cosmological constant problem aside, and focus on the Higgs particle and on why the Standard Model is unnatural. This is not because the cosmological constant problem isn’t important, and not because we’re totally certain the two problems are completely unrelated. But since the cosmological constant has everything to do with gravity, while the problem of the Higgs particle and the naturalness of the Standard Model doesn’t have anything to do with gravity directly, it’s quite possible they’re solved in different ways. And each of the two problems is enormous on its own; if in fact we need to solve them simultaneously, then the situation just gets worse. So let’s just send the cosmological constant to a far corner to take a little nap. We do need to remember that it’s the elephant in the room that we can’t forever ignore.
Ok — about the Higgs field. There are three really important questions about the Higgs field and particle that we want to answer. [I’ll phrase all these questions assuming the Standard Model is right, or close to right, but if it isn’t, don’t worry: the ideas I’ll explore remain essentially the same, even though slightly different phrasing is required.]
- The Higgs field is “ON” — its average value, everywhere and at all times, at least since the very early universe, isn’t zero. Why is it on?
- Its average value is 246 GeV. What sets its value?
- The Higgs particle has a mass of about 125 GeV/c². What sets this mass?
I’m going to explain to you how and why these questions are related to the issue of how the energy of empty space (part of which comes from quantum fluctuations of fields) depends on the Higgs field’s average value.
The Higgs Field’s Value and the Energy of Empty Space
For any field — not just the Higgs field — how is it determined what the average value of the field is in our universe? Answer: a field’s average value must have the following property: if you change the value by a little bit, larger or smaller, then the energy in empty space must increase. In short, the field must have a value for which the energy of empty space is at a minimum — not necessarily the minimum, but a minimum. (If there is more than one minimum, than which one is selected may depend on the history of the universe, or on other more subtle considerations I won’t go into now.)
A couple of illustrative examples of how the energy of empty space in our universe, or in some imaginary universe, might depend on the Higgs field, or on some other similar field, are shown in Figure 4. In each of the two cases I’ve drawn, there happen to be two minima where the Higgs field could sit — but that’s just chance. In other cases there could be several minima, or just one. The fact that the Higgs field is ON in our world implies there’s a minimum in the universe’s vacuum energy when the Higgs field has a value of 246 GeV. While it’s not obvious from what’s I’ve said so far, we are confident, from what we know about nature and about our equations, that there is no minimum when the Higgs field is zero, and that’s why our universe’s Higgs field isn’t OFF. So in our universe, the dependence of the vacuum energy on the Higgs field probably looks more like the left-hand figure than the right-hand one, but, as we’ll see, it may not look much like either of them. If the Standard Model describes physics at energies much above and at distances much shorter than the ones we’re studying now at the Large Hadron Collider (LHC), then the form of the corresponding curve is much more peculiar — as we’ll see later.
The Higgs Particle’s Mass and the Energy of Empty Space
What about the Higgs particle’s mass? It is determined (Figure 4) by how quickly the energy of empty space changes as you vary the Higgs field’s value away from where it prefers to be. Why?
A Higgs particle is a little ripple in the Higgs field — i.e., as a Higgs particle passes by, the Higgs field has to change a little bit, becoming in turn a bit larger and smaller. Well, since we know the Higgs field’s average value sits at a minimum of the energy of empty space, any small change in that value slightly increases that overall energy a little bit. This extra bit of energy is [actually half of] what gives the Higgs particle its mass-energy (i.e., it’s E=mc² energy.) If the shape of the curve is very flat near the minimum (see Figure 4), the energy required to make a Higgs particle is rather small, because the extra energy in the rippling Higgs field (i.e., in the Higgs particle) is small. But if the shape of the curve is very sharp near the minimum, then the Higgs particle has a big mass.
Thus it is the flatness or sharpness in the curve in the plot, at the point where the Higgs field’s value is sitting — the “curvature at the minimum” — that determines the Higgs particle’s mass.
Why It Isn’t Easy to Have The Higgs Particle’s Mass Be Small
The Higgs particle’s mass is measured to be about 125-126 GeV/c², about 134 times the proton‘s mass. Now why can’t we just put that mass into our equations, and be done with this question about where it all comes from?
The problem is that the Higgs field’s value, and the Higgs particle’s mass, aren’t things you put directly into the equations that we use; instead, you extract them, by a complex calculation, from the equations we use. And here we run into some difficulty…
We get these two quantities — the average value and the mass of the field and particle — by looking at how the energy of empty space depends on the Higgs field. And that energy, as in any quantum field theory like the Standard Model, is a sum of many different things:
- energy from the fluctuations of the Higgs field itself
- energy from the fluctuations of the top quark field
- energy from the fluctuations of the W field
- energy from the fluctuations of the Z field
- energy from the fluctuations of the bottom quark field
- energy from the fluctuations of the tau lepton field
and so on for all the fields of nature that interact directly with the Higgs field… I’ve indicated these — schematically! these are not the actual energies — as blue curves in Figure 5. Each plot indicates one contribution to the energy of empty space, and how it varies as the Higgs field’s average value changes from zero to the maximum value that I dare consider, which I’ve called vmax.
[Note: Some of you may have read that these calculations of the energy of empty space give infinite results. This is true and yet irrelevant; it is a technicality, true only if you assume vmax is infinitely large — which it patently is not. I have found that many people, non-scientists and scientists alike, believe (thanks to books by non-experts and by the previous generations of experts — even Feynman himself), that these infinities are important and relevant to the discussion of naturalness. This is false. We’ll return to this widespread misunderstanding, which involves mistaking mathematical technicalities for physically important effects, at the end of this section.]
What is vmax? It’s as far as one could can push up the Higgs field’s value and still believe our calculations within the Standard Model. What I mean by vmax is that if the Higgs field’s value were larger than this (which would make the top quark’s mass larger than about vmax/c2) then the Standard Model would no longer accurately describe everything that happens in particle physics. In other words, vmax is the boundary between where the Standard Model is applicable and where it isn’t.
However, we don’t know what vmax is… and that ignorance is going to play a role in the discussion. From what we know from the LHC, vmax appears to be something like 500 GeV or larger. However, for all we know, vmax could be as much as 10,000,000,000,000,000 times larger than that. We can’t go beyond that point, because that’s the (maximum possible) scale at which gravity becomes important; if vmax were that large, top quarks would be so heavy they’d be tiny black holes! and we know that the Standard Model can’t describe that kind of phenomenon. A quantum mechanics version of gravity has to be invoked at that point… if not before!
So again, what we know is that vmax is somewhere between 500 GeV and 1,000,000,000,000,000,000 GeV or so. In Figure 5, I’ve assumed it’s quite a bit bigger than 500 GeV; we’ll look in Figure 6 at the case where vmax is close to 500 GeV.
Each one of the contributions in the upper row of Figure 5 is something we can (in principle, and to a large extent in practice) calculate, for any Higgs field value between zero and vmax, and for all quantum fluctuations with energy less than about vmax. [I’m oversimplifying somewhat here; really this energy Emax need not be quite the same as vmax, but let’s not get more complicated than necessary.] If vmax is big, then each one of these contributions is really big — and more importantly, the variation as we change the Higgs field’s value from zero to vmax is big too — something like vmax4/(hc)3 … where h is Planck’s quantum constant and c is the universal speed limit, often called “the speed of light”.
But that’s not all. To this we have to add other contributions, shown in the second row of Figure 5, which come from physical phenomena that we don’t yet know anything or much about, physics that does not directly appear in the Standard Model at all. [Technically, we absorb these effects from unknown physics into parameters that define the Standard Model’s equations, as inputs to those equations; but they are inputs, rather than something we calculate, precisely because they’re from unknown sources.] In addition to effects from quantum fluctuations of known fields with even higher energies, there may also be effects from
- the quantum mechanics of gravity,
- heavy particles we’ve not yet discovered,
- forces that are only important at distances far shorter than we can currently measure,
- other more exotic contributions from, say, strings or D-branes in string theory or some other theory like it,
some of which may depend, directly or indirectly, on the Higgs field’s value. I’ve drawn these unknown effects in red; note that these curves are pure guesswork. We don’t know anything about these effects except that they could exist (and the gravity effects definitely exist), and that some or all of them could be really big… as big as or bigger than the ones we know about in the upper row. In principle, all these unknown effects could be zero — but that wouldn’t resolve the naturalness problem, as we’ll see, so presumably they’re not all zero.
What’s crucial here is that there’s no obvious reason to expect these unknown effects in red are in any way connected with the known contributions in blue. After all, why should quantum gravity effects, or some new force that has nothing to do with the weak nuclear force, have anything to do with the energy density of quantum fluctuations of the top quark field or of the W field? These seem like conceptually separate sources of the energy density of empty space.
And here’s the puzzle. When we add up all of these contributions to the energy of empty space [Unsure how to add curves like these together? Click here for an explanation…] — each of which is big and many of which vary a lot as the Higgs field’s value changes from zero to the maximum that we can consider — we find an incredibly flat curve, the one shown in green. It’s almost perfectly flat near the vertical axis. And yet, its minimum is not quite at zero Higgs field; it’s slightly away from zero, at a Higgs field value of 246 GeV. All of those different contributions in blue and red, which curve up and down in varying degrees, have almost (but not quite) perfectly canceled each other when added together. It’s as though you piled a few mountains from Montana into a deep valley in California and ended up with a plain as flat as Kansas. How did that happen?
Well, how bad is this problem? How surprising is this cancellation? The answer is that it depends on vmax. If vmax is only 500 GeV, then there’s no real cancellation needed at all — see Figure 6. But if vmax is huge, the cancellation is incredibly precise, as in Figure 5. The larger is vmax, the more remarkable it is that all the contributions cancelled.
How remarkable? The cancellation has to be perfect to something like one part in (vmax/500 GeV)2, give or take a few. So if vmax is close to 500 GeV, that’s no big deal; but if vmax = 5000 GeV, we need a cancellation to one part in 100. If it’s 500,000 GeV, we need cancellation to one part in a million.
And if we take vmax as high as possible — if the Standard Model describes all non-gravitational particle physics — then we need cancellation of all these different effects to one part in about 1,000,000,000,000,000,000,000,000,000,000.
In the last case, the incredible delicacy of the cancellation is particularly disturbing. It means that if you could alter the W particle’s mass, or the strength of the electromagnetic force, by a tiny amount — say, one part in a million million — the cancellation would completely fail, and you’d find the theory would be in Class 1 or Class 2, with a ultra-heavy Higgs particle and either a large or absent Higgs field value (see Figure 3). This incredible sensitivity means that the properties of our world have to be, very precisely, just so — like a radio that is set exactly to the frequency of a desired radio station, finely tuned. Such extreme “fine-tuning” of the properties of a physical system has no precedent in science.
To say this another way: what’s unnatural about the Standard Model — specifically, about the Standard Model being valid up to the scale vmax, if vmax is much larger than 500 GeV or so — is the cancellation shown in Figure 5. It’s not generic or typical… and the larger is vmax, the more unnatural it is. If you take a bunch of generic curves like those in Figure 5, each of which has minima and maxima at Higgs field values that are either at zero or somewhere around vmax, and you add those curves together, you will find that the sum of those curves is a curve that also has its minima and maxima at
- a substantial fraction of vmax [Class 2 theories — see Figure 3],
- or at zero [Class 1 theories],
- but not somewhere non-zero that is much much smaller than vmax [Class 3 theories].
Moreover, if the curves are substantially curved near their minima and maxima, their sum will also typically have substantial curvature near their minima and maxima [i.e. the Higgs particle’s mass will be roughly vmax/c2, as in Class 1 and Class 2 theories], and won’t be extremely flat near any of its minimum [needed for the Higgs particle to be much lighter than vmax/c2, as occurs in Class 3 theories.] This is illustrated, for the addition of just two curves, in Figure 7, where we see the two curves have to have a very special relationship if their sum is to end up very flat.
That’s the naturalness problem. It’s not just that the green curve in Figure 5 is remarkably flat, with a minimum at a small Higgs field value. It’s that this curve is an output, a sum of many large and apparently unrelated contributions, and it’s not at all obvious how the sum of all those curves comes out to have such an unusual shape.
An Aside About Infinities, Renormalization, and Cut-offs
[You can skip this little section if you want; you won’t need to understand it to follow the rest of the article.]
Now, about those infinities that you may have read about — along with the scary-sounding word “renormalization”, in which infinities seem to be somehow swept under the rug, leading to finite predictions. These infinities, and their removal via renormalization, sometimes lead people — even scientists — to claim that particle physicists don’t know what they are doing, and that this causes them to see a naturalness problem where none exists.
Such claims are badly misguided. These technical issues (which are well understood nowadays, in any case) are completely irrelevant in the present context.
The infinities that arise in certain calculations of the Higgs particle’s mass, and of the Higgs field’s value, are a symptom of the naturalness problem, a mathematical symptom that shows up if you insist on taking vmax to infinity, which, though often convenient, is an unphysical thing to do. The infinities are not the naturalness problem, nor are they at its heart, nor are they its cause.
Among many ways to see this, one very easy way is to study the wide variety of finite quantum field theories discovered in the 1980s (a list of references can be found in an old paper of mine with Rob Leigh [now a professor at the University of Illinois].) These theories have minimal amounts of supersymmetry, as well as being finite. If you take such a theory (see Figure 8), and you ruin the supersymmetry at a scale vmax, while assuring the theory that remains at lower energies still has spin-zero fields like the Higgs field, you do not introduce any infinities. Moreover, there is no need to artificially cut the theory off at energies below vmax (as I have done in Figure 5, separating known from unknown) since in this example we know the equations to use at energies above as well as below vmax. The energy of empty space, and its dependence on the various fields, can be calculated without any ambiguity, infinities, or infinite renormalization. So — is there a naturalness problem here too? Do the spin-zero particles generically get masses as big as vmax/c2? Do the spin-zero fields have values that are either zero or roughly as big as vmax? You Bet! No infinities, no sweeping anything under a rug, no artificial-looking cutoffs — and a naturalness problem that’s just as bad as ever.
By the way, there’s an interesting loophole to this argument, using a lesson learned from string theory about quantum field theory. But though it gives examples of theories that evade the naturalness problem, neither I nor anyone else was able (so far) to use it to really solve the naturalness problem of the Standard Model in a concrete way. Perhaps the best attempt was this one.
We could also repeat this type of calculation within string theory (a technical exercise, which does not require we assume string theory really describes nature). String theory calculations have no infinities. But if vmax, the energy scale where the Standard Model fails to work, is much larger than 500 GeV, the naturalness problem is just as bad as before.
In short: getting rid of the infinities that arise in certain Higgs-related calculations does NOT by itself solve or affect the naturalness problem.
Solutions to the Naturalness Problem
On purely logical grounds, a couple of qualitatively different types of solutions to this problem come to mind. [To be continued…]